MIGRATION OF ANTIOXIDANTS FROM POLY(LACTIC ACID), PLA , FILMS INTO FOOD SIMULANTS: A PARAMETER ESTIMATION APPROACH By Hayati Samsudin A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Packaging - Doctor of Philosophy 2015 ABSTRACT MIGRATION OF ANTIOXIDANTS FROM POLY(LACTIC ACID), PLA, FILMS INTO FOOD SIMULANTS: A PARAMETER ESTIMATION APPROACH By Hayati Samsudin Development of antioxidant active packaging has vastly increased over the year s with more focus on biodegradable polymeric films and natural antioxidants. However, few studies investigated the migration of antioxidants from films into food simulants/products by fully number of studies have simultaneously determined other migration parameters besides the diffusion coefficient ( D ), such as the convective mass transfer coefficient ( h ) and the partition coefficient ( K p,f ). Thus, this dissertation explored experimental and theoretical approaches to gain insight on the migration kinetics of antioxidants from polymer films by using parameter estimat ion approaches ( e.g., ordinary least square (OLS), sequential, bootstrap etc ). A poly(lactic acid), PLA, functional film incorporated with marigold flower extract containing carotenoid - based antioxidant ( i.e., astaxanthin) was developed and the kinetic re lease of astaxanthin from this film into 95% ethanol was investigated. The mass migrated at equilibrium ( M ) was estimated for the first time, in addition to the D . Further investigation was conducted on different migration case studies to compare the esti mation of one parameter, 1P ( i.e., the D ) versus 2P ( i.e., the D and the M ) and 3P ( i.e., the D , the M and the ratio of the mass of antioxidant migrated into the simulant to the mass of the antioxidant left in the film at equilibrium ( ) ) using general m ass transfer solutions. The 3P estimation based on the corrected Akaike information criterion (AICc) was found to better describe the migration experiment without compromising the estimation accuracy. The parameter, experimentally determined at the end of the experiment and related to K p,f by , was better estimated at ea rly times without the need to reach equilibrium; however, these migration cases did not account for the convective mass transfer coefficient. Hence, a two - step solution was developed to simultaneously assess the D , K p,f , and h from migration experiments. The first step of the solution was to identify the right local minima region for minimizing sums of squared errors (SSE ) and to provide a robust magnitude approximation for the initial guesses used in the second step of the OLS estimation. The K p,f parameter was directly used in this solution due to the ease of physical interpretation. h , the parameter that might be of im portance in a non - stirring condition, viscous food product/simulant, etc. was also estimated and related to the overall migration resistance ( Commonly, h is not estimated. Neglecting h might lead to underestimation of D , thus compromising the accuracy of the parameter estimation. Different migration case studies were used as examples and the parameters were assessed using the two - step solution. The OLS results were found comparable with the sequential estimation. Residual bootstrap was conducted to improve the residual distribution in a large population. A comparative study between the two - step solution and the general mass transfer solutions ava ilable in the literature was also performed, and model selection was performed using the AICc. A decision tree analysis consisting of the newly proposed model with the general mass transfer solutions was proposed as a tool for analyzing migration data. Fi nally, the estimation of the activation energy ( E a ) of a non - isothermal migration study was conducted using the reparameterized Arrhenius equation to identify the optimum T ref to obtain near zero correlation between the D ref and the E a , in turn, decreasing relative error of D ref . Copyright by HAYATI SAMSUDIN 2015 v This dissertation is dedicated to my beloved parents and sisters vi ACKNOWLEDGEMENTS - Abraham Lincoln. My precious appreciation to Allah S.W.T for the strength he granted me during these years. It has been a very long journey. Having lived far away from my homeland is never easy. But, I am blessed with supportive family and surrounded with beautiful friends hips. I am sincerely grateful for having Dr. Rafael Antonio Auras as my advisor. His kindness to accept me as his student despite his awareness of my health condition at the time is something that I forget. He has provided me with endless opport unity to grow as a person and has kept me grounded. He will always be my mentor , prob is Dr. James Beck for his unlimited thoughtful inputs. I would like to thank Dra. Maria Rubino for giving me a chance to be her TA and for her continuous support. She always encouraged me with her kind words of support. My sincere gratitude to Dr. Herlinda Soto - Valdez for her continuous helps and for taking a g reat care of me when I was in Hermosillo. I really appreciate her kindness and supports at all time despite being geographically apart. My deepest appreciation to Dr. Selke for her timeless support. She was the first one to welcome me to this school and I am glad to have her as part as my committee member. I am thankful to Dr. Gary Burgess for giving in answering all my doubts. Also, I want to extend my apprecia tion to Dr s . Bruce Harte and Janice vii Harte for their care and supports. To Dr. Clarke, thank you for teaching me a whole new world of packaging processes a chance that I would have never thought to have. For all my friends (not in any particular order); F l ali, Kikyung, together , and I will always cherish our friendship. A special thanks to both Mishra and Patna for world and for our timeless friendship. I also want to thank my friend, Rabiha for her continuous support. Also, I am thankful to Turk and Ploy for never fails at visiting and for being in contact with me. Special thanks to Masyitah for her friendship. My a ppreciation for the RAA research group from 2007 until now. Many have come and gone, but every chance we had, we did share lots of fun and pain together. We have learnt a lot from each other and more importantly have supported each other. I think we all ow e it to Dr. Auras Not to forget, my appreciation to the Ministry of Education Malaysia and University Sains Malaysia for the opportunity to study abroad and for their financial supports. My gratitude also goes to the School of Packaging, the College of Agricultural and Natural Resources for their financial supports and for giving me the educational opportunities on and off - campus. Enormous love and thank you to my beloved parents; Samsudin Mohd. Ali and Fatimah Jamil for their unconditional supports and love. To both my sisters; Suhaily and Nur Radzimah and my grandma, Hjh. Aisha, thank you for never giving up on me. To my late grandma, Hjh. Habibah, thank you for raising me and for your love. Hayati Samsudin viii TABLE OF CONTENTS LIST OF TABLES ................................ ................................ ................................ ..................... xiii LIST OF FIGURES ................................ ................................ ................................ .................... xv KEY TO SYMBOLS AND ABBREVIATIONS ................................ ................................ ...... xx Chapter 1 ................................ ................................ ................................ ................................ ....... 1 Background and Motivation ................................ ................................ ................................ ........ 1 1.0 Introduction ................................ ................................ ................................ ......................... 1 1.1 Research Importance and Motivation ................................ ................................ .............. 2 1.2 Objectives ................................ ................................ ................................ ............................. 6 1.3 Dissertation Overview ................................ ................................ ................................ ........ 7 REFERENCES ................................ ................................ ................................ ............................ 10 Chapter 2 ................................ ................................ ................................ ................................ ..... 15 Literature Review ................................ ................................ ................................ ....................... 15 2.0 Introduction ................................ ................................ ................................ ....................... 15 2.1 Lipid Oxidation: Brief Introduction ................................ ................................ ............... 15 2.2 Lipid Oxidation: Classification ................................ ................................ ........................ 15 2.2.1 Autoxidation ................................ ................................ ................................ ................ 15 2.2.1.1 Step 1: Initiation ................................ ................................ ................................ .... 16 2.2.1.2 Step 2: Propagation ................................ ................................ ............................... 16 2.2.1.3 Step 3: Chain branching ................................ ................................ ........................ 16 2.2.1.4 Step 4: Chain termination ................................ ................................ ..................... 17 2.2.2 Photooxidation or Photosensitized Oxidation ................................ .............................. 18 2.2.2.1 Type I: photooxidation by excitation of lipids ................................ ...................... 19 2.2.2.2 Type II: photooxidation by excitation of O 2 ................................ ........................ 19 2.2.3 Enzyme - Catalyzed Lipid Oxidation ................................ ................................ ............ 19 2.3 Factors Affecting Lipid Oxidation ................................ ................................ .................. 20 2.3.1 Chemical Structure of Fatty Acids ................................ ................................ ............... 20 2.3.2 Temperature ................................ ................................ ................................ ................. 21 2.3.3 Light ................................ ................................ ................................ ............................. 21 2.3.4 Oxygen, O 2 ................................ ................................ ................................ ................... 22 2.3.4.1 Type of O 2 ................................ ................................ ................................ ............. 22 2.3.4.2 Concentration of O 2 ................................ ................................ .............................. 22 2.3.5 Presence of Minor Compounds ................................ ................................ .................... 23 2.3.5.1 Water and Water Activity, a w ................................ ................................ ............... 23 2.3.5.2 Metals ................................ ................................ ................................ .................... 24 2.3.5.3 Free Fatty Acids ................................ ................................ ................................ .... 25 2.3.5.4 Phospholipids ................................ ................................ ................................ ........ 25 2.3.5.5 Chlorophylls ................................ ................................ ................................ .......... 26 ix 2.4 Antioxidants ................................ ................................ ................................ ....................... 27 2.4.1 Classification of Antioxidants ................................ ................................ ...................... 29 2.4.1.1 Primary Antioxidants (Chain - Breaking) ................................ ............................... 30 2.4.1.2 Secondary Antioxidants (Preventative Anti oxidants) ................................ ........... 31 2.4.1.3 An Example of an Antioxidant Reaction ................................ .............................. 33 2.5 Approaches to Extend the Shelf Life of Fatty Food Products ................................ ...... 35 2.5.1 Antioxidant Based Packaging System: Individual and Independent Antioxidant Devices ................................ ................................ ................................ .............................. 36 2.5.2 Antioxidant Functional Films ................................ ................................ ...................... 37 2.6 Migration ................................ ................................ ................................ ........................... 55 2.6.1 Thermodynamic Equilibrium ................................ ................................ ....................... 56 2.6.2 Partition Coefficient, K p,f ................................ ................................ ............................. 57 2.6.3 Diffusion Coefficient, D ................................ ................................ .............................. 58 2.6.4 Migration Models ................................ ................................ ................................ ......... 61 2.7 Parameter Estimation ................................ ................................ ................................ ....... 68 2.7.1 Parameters of Interest ................................ ................................ ................................ ... 68 2.7.2 Sensitivity Coefficient, X, and Scaled Sensitivity Coefficient, ............................... 69 2.7.3 Parameter Estimation using Ordinary Least Squares (OLS) ................................ ....... 70 2.7.3.1 Standard Errors and Correlation Coefficient of the Parameters ........................... 71 2.7.4 Sequential Estimation ................................ ................................ ................................ .. 72 2.7.5 Corrected Akaike Information Criterion (AICc) ................................ ......................... 72 2.7.6 Bootstrap ................................ ................................ ................................ ...................... 73 2.7.7 Optimal Experimental Design ................................ ................................ ...................... 74 2.7.8 Activation Energy ................................ ................................ ................................ ........ 74 REFERENCES ................................ ................................ ................................ ............................ 77 Chapter 3 ................................ ................................ ................................ ................................ ..... 92 Poly(lactic acid) membrane incorporated with marigold flower extract ( Tagetes erect a ) intended for fatty - food application ................................ ................................ ............................ 92 3.0 Introduction ................................ ................................ ................................ ....................... 92 3.1 Materials and Methods ................................ ................................ ................................ ..... 95 3.1.1 Materials ................................ ................................ ................................ ...................... 95 3.1.2 Fabrication of Antioxidant Functional Membrane ................................ ...................... 96 3.1.3 Quantification of Astaxanthin in the Fabricated Functional Membrane after Processing ................................ ................................ ................................ ................................ ............... 97 3.1 .4 Migration of Astaxanthin into A Food Simulant ................................ ......................... 97 3.1.4.1 Mathematical Models for Migration Study ................................ ........................... 98 3.1.5 Thermal Properties ................................ ................................ ................................ ....... 99 3.1.6 Number Average Molecular Weight ( M n ) and Weight Average Molecular Weight ( M w ) ................................ ................................ ................................ ................................ ..... 100 3.1.7 Scanning Electron Microscopy (SEM) ................................ ................................ ...... 101 3.1.8 Oxygen (O 2 ), Water Vapor, and Carbon Dioxide (CO 2 ) Barrier Properties .............. 101 3.1.9 Optical Properties ................................ ................................ ................................ ....... 102 3.1.10 Fourier Transform Infrared Spectrophotometer (FTIR) ................................ .......... 102 3.1.11 Oxidative Stability of Soybean Oil ................................ ................................ .......... 103 3.2 Statistical Ana lysis ................................ ................................ ................................ .......... 104 x 3.3 Results and Discussions ................................ ................................ ................................ .. 104 3.3.1 Quantification of astaxanthin in the fabricated membrane after processing .............. 104 3.3.2 Migrati on of Astaxanthin into A Food Simulant (95% ETOH) ................................ 106 3.3.3 Thermal Properties ................................ ................................ ................................ ..... 111 3.3.4 Number Average Molecular Weight ( M n ), and Weight Average Molecular Weight ( M w ) ................................ ................................ ................................ ................................ ..... 114 3.3.5 Scanning Electron Microscopy (SEM) ................................ ................................ ...... 120 3.3.6 Barrier properties ................................ ................................ ................................ ....... 122 3 .3.6.1 Oxygen (O 2 ) ................................ ................................ ................................ ........ 122 3.3.6.2 Water Vapor (WV) ................................ ................................ ............................. 122 3.3.6.3 Carbon dioxide (CO 2 ) ................................ ................................ ......................... 124 3.3.7 Optical Properties ................................ ................................ ................................ ....... 124 3.3.8 Fourier Transform Infrared Spectrophotometer (FTIR) ................................ ............ 125 3.3.9 Oxidative Stability of Soybean Oil ................................ ................................ ............ 128 3.4 Conclusion ................................ ................................ ................................ ....................... 130 REFERENCES ................................ ................................ ................................ .......................... 132 Chapter 4 ................................ ................................ ................................ ................................ ... 139 Parameter Estimation for Migration Studies of Antioxidant - Polymer Films ..................... 139 4.0 Introduction ................................ ................................ ................................ ..................... 139 4.1 Theoretical Background ................................ ................................ ................................ . 141 4.1.1 Part A: Migration ................................ ................................ ................................ ....... 141 4.1.1.1 Partition Coefficient, K p,f ................................ ................................ .................... 141 4.1.1.2 Biot number, Bi ................................ ................................ ................................ ... 142 4.1.1.3 Diffusion coefficient, D ................................ ................................ ...................... 143 4.1.2 Part B: Parameter Estimation ................................ ................................ ..................... 147 4.1.2.1 Sensitivity Coefficient and Scaled Sensitivity Coefficient ................................ . 148 4.1.2.2 Ordinary Least Squa res (OLS) Estimation ................................ ......................... 148 4.1.2.3 Corrected Akaike Information Criterion (AICc) ................................ ................ 148 4.1.2.4 Optimal Experimental Design ................................ ................................ ............. 149 4.2 Case Study ................................ ................................ ................................ ....................... 149 4.2.1 A Selected Case Study: Poly(Lactic Acid), PLA - - Tocopherol Functional Film in Contact With 100% Ethanol At 23 C ................................ ................................ ................ 150 4.2.1.1 Initial Scaled Sensitivity Coefficient, ................................ ............................ 150 4.2.1.2 Ordinary Least Square (OLS) Estimation and the Corrected Akaike Information Criterion (AICc) ................................ ................................ ................................ .............. 151 4.2.1.3 Optimal Experimental Design ................................ ................................ ............. 157 4.2.2 Other Case Studies ................................ ................................ ................................ ..... 158 4 .3 Conclusion ................................ ................................ ................................ ....................... 217 REFERENCES ................................ ................................ ................................ .......................... 218 Chapter 5 ................................ ................................ ................................ ................................ ... 224 A Two - Step Solution to Estimate Mass Transfer Parameters of Migration Experiments C ontrolled by Diffusion, Partition and Convective Mass Transfer Coefficients ................ 224 5.0 Introduction ................................ ................................ ................................ ..................... 224 5.1 Theoretical Development ................................ ................................ ................................ 227 xi 5.1.1 Assumptions and Boundary Conditions ................................ ................................ ..... 227 5.1.2.1 Step 1 ................................ ................................ ................................ .................. 228 5.1.2.2 Step 2 ................................ ................................ ................................ .................. 232 5.2 A Case Study Analysis ................................ ................................ ................................ .... 237 5.3 Assessment of the Two - Step Solutions Model ................................ .............................. 238 5.3.1 Step 1 ................................ ................................ ................................ ......................... 238 5.3.2 Scaled Sensitivity Coefficient, ................................ ................................ .............. 238 5.3.3 Step 2 ................................ ................................ ................................ ......................... 238 5.3.3.1 Ordinary Least Square (OLS) Estimation ................................ ........................... 238 5.3.3.2 Sequential Estimation ................................ ................................ ......................... 239 5.4 Kinetic Phase Diagram (KPD) ................................ ................................ ....................... 239 5.5 Bootstrap Method ................................ ................................ ................................ ........... 241 5.6 Results and Discussions ................................ ................................ ................................ .. 242 5.6.1 Step 1 ................................ ................................ ................................ ......................... 242 5.6.2 Scaled Sensitivity Coefficient, ................................ ................................ .............. 243 5.6.3 Step 2 ................................ ................................ ................................ ......................... 245 5.6.3.1 Ordinary Least Square (OLS) Estimation ................................ ........................... 245 5.7 Sequential Estimation ................................ ................................ ................................ ..... 249 5.8 Kinetic Phase Diagram (KPD) ................................ ................................ ....................... 251 5.9 Residual Bootstrap ................................ ................................ ................................ .......... 253 5.10 Conclusion ................................ ................................ ................................ ..................... 255 APPENDICES ................................ ................................ ................................ ........................... 257 APPENDIX 5A: Migration of poly(lactic acid), PLA incorporated with 2.6 wt.% - tocopherol into ethanol at 23 C. ................................ ................................ ......................... 258 APPENDIX 5B: Migration of poly(lactic acid), PLA incorporated with 1.28 wt.% catechin into 95% ethanol at 40 C. ................................ ................................ .................... 270 APPENDIX 5C: Example of MATLAB coding ................................ ................................ . 282 REFERENCES ................................ ................................ ................................ .......................... 303 Chapter 6 ................................ ................................ ................................ ................................ ... 306 Assessment of Mass Transfer Models used in Migration Experiments to Determine Diffu sion, Partition and Convective Mass Transfer Coefficients ................................ ......... 306 6.0 Introduction ................................ ................................ ................................ ..................... 306 6.1 Materials and Methods ................................ ................................ ................................ ... 308 6.1.1 Migration Case Studies ................................ ................................ .............................. 308 6.1.2 Mathematical Models ................................ ................................ ................................ . 309 6.1.2.1 Assumptions ................................ ................................ ................................ ........ 309 6.1.2.2 Model 1: A Two - development can be found in Chapter 5) ................................ ................................ ........ 309 6.1.2.2.1 Step 1 ................................ ................................ ................................ ........... 309 6.1.2.2.2 Step 2 ................................ ................................ ................................ ........... 310 K p,f and diffusion coefficient ( D ) as the governing factors (Carslaw & Jaeger, 1959; Crank, 1979) ............................ 310 D ) as t he only governing factor (Crank, 1979) ................................ ................................ ................................ ........ 311 6.1.3 Kinetic Parameter Estimation ................................ ................................ .................... 315 xii 6.1.3.1 Scaled Sensitivity Coefficient, ................................ ................................ ....... 315 6.1.3.2 Ordinary Least Square (OLS) Estimation ................................ ........................... 315 6.1.3.3 Corrected Akaike Information Criterion (AICc) ................................ ................ 316 6.2 Results and Discussions ................................ ................................ ................................ .. 317 6.2.1 Scaled Sensitivity Coefficient, ................................ ................................ .............. 317 6.2.2 Ordinary Least Square (OLS) Estimation ................................ ................................ .. 319 6.2.3 Model Selec tion Using the Corrected Akaike Information Criterion (AICc) ............ 322 6.3 Conclusion ................................ ................................ ................................ ....................... 326 APPENDICES ................................ ................................ ................................ ........................... 327 APPENDIX 6A: Migration of poly(lactic acid), PLA incorporated with 2.6 wt.% - tocopherol into ethanol at 23 C. ................................ ................................ ......................... 328 APPENDIX 6B: Migration of poly(lactic acid), PLA incorporated with 1.28 wt.% catechin into 95% ethanol at 40 C. ................................ ................................ .................... 334 REFERENCES ................................ ................................ ................................ .......................... 339 Chapter 7 ................................ ................................ ................................ ................................ ... 343 Estimation of the Activation Energy in Migration Studies ................................ ................... 343 7.0 Introduction ................................ ................................ ................................ ..................... 343 7.1 Materials and Methods ................................ ................................ ................................ ... 345 7.1.1 A Case Study ................................ ................................ ................................ .............. 345 7.1.2 Kinetic Parameter Estimation Procedure ................................ ................................ ... 345 7.1.3.1 Step 1: ................................ ................................ ................................ ................. 346 7.1.3.1.1 Scaled Sensitivity Coefficient, ................................ ................................ 346 7.1.3.1.2 Temperature Simulation ( T sim ) Approach ................................ .................... 347 7.1.3.1.3 Non - Linear Regression Estimation ................................ .............................. 347 7.1.3.2 Step 2: ................................ ................................ ................................ ................. 347 7.2 Results and Discussions ................................ ................................ ................................ .. 348 7.2.1 Initial Scaled Sensitivity Coefficient, and Reference Temperature, T ref ............... 348 7.2.1 Non - Lin ear Regression Estimation ................................ ................................ ............ 354 7.3 Additional Observations ................................ ................................ ................................ . 357 7.4 Conclusions ................................ ................................ ................................ ...................... 358 APPENDICES ................................ ................................ ................................ ........................... 360 APPENDIX 7A: Additional results of the randomly chosen T ref within the experimental temperature range. ................................ ................................ ................................ ............... 361 APPENDIX 7B: MATLAB coding for the estimation of activation energy .................... 363 REFERENCES ................................ ................................ ................................ .......................... 371 Chapter 8 ................................ ................................ ................................ ................................ ... 374 Overall Conclusion and Recommended Future Work ................................ .......................... 374 8.0 Overall Conclusion ................................ ................................ ................................ .......... 374 8.1 Recommended Future Work ................................ ................................ ......................... 379 APPENDIX ................................ ................................ ................................ ................................ 382 REFERENCES ................................ ................................ ................................ .......................... 385 xiii LIST OF TABLES Table 2 - 1 Compilation of studies reporting kinetic migration parameters for petroleum based functional films incorporated with natural antioxidants and the models used to determine these parameters. ................................ ................................ ................................ ................................ .... 40 Table 2 - 2 Compilation of studies reporting kinetic migration parameters for bio - based functional films incorpora ted with natural antioxidants and the models used to determine these parameters. ................................ ................................ ................................ ................................ ....................... 49 Table 3 - 1 Migration data of produced function al membranes. ................................ .................. 108 Table 3 - 2 Ch aracterization of the fabricated functional membranes. ................................ ........ 118 Table 4 - 1 Summarized of estimated parameters for different migration studies of antioxidant - PLA film systems. ................................ ................................ ................................ ............................... 160 Table 5 - 1 Number of terms with its corresponding percent accuracy 237 Table 5 - 2 Additional information of OLS estimation. ................................ ............................... 248 Table 5 - 3 Comparison between OLS and sequential results. ................................ ..................... 2 51 Table 5 - 4 Comparison between 95% asymptotic and 95% bootstrap confidence intervals of each parameter. ................................ ................................ ................................ ................................ .... 254 Table 5A - 1 Additional information of OLS estimation for migration of 2.6 wt.% - tocopherol into ethanol at 23 C. ................................ ................................ ................................ .................. 264 Table 5A - 2 Comparison between OLS and sequential results. ................................ ................. 265 Table 5A - 3 Comparison between 9 5% asymptotic and 95% bootstrap confidence intervals of each parameter. ................................ ................................ ................................ ................................ .... 269 Table 5B - 1 Additional information of OLS estimation for migration of 1.28 wt.% catechin into 95% ethanol at 40 C. ................................ ................................ ................................ ................. 275 Table 5B - 2 Comparison between OLS and sequential re sults. ................................ ................. 276 Table 5B - 3 Comparison between 95% asymptotic and 95% bootstrap confidence intervals of each parameter. ................................ ................................ ................................ ................................ .... 280 Table 6 - 1 Comparison of the number of terms needed to achieve a given accuracy among model 1, 2, and 3 315 xiv Table 6 - 2 OLS results for the migration study of 3 wt.% resveratrol from PLA film into ethanol at 9 C for model 1, 2, and 3. ................................ ................................ ................................ ...... 320 Table 6 - 3 AICc analysis for selecting mo del for the migration of 3 wt. % resveratrol from PLA film into ethanol at 9 C. ................................ ................................ ................................ ............. 323 Table 6A - 1 OLS results for the migration study of 2.6 wt.% - tocopherol from PLA film into ethanol at 23 C for model 1 and 2. ................................ ................................ ............................ 331 Table 6A - 2 AICc analysis for selecting model for the migration study of 2.6 wt.% - tocopherol from PLA film into ethanol at 23 C. ................................ ................................ ......................... 333 Table 6B - 1 OLS results for the migration o f 1.28 wt.% catechin from PLA film into 95% ethanol at 40 C for model 1 and 3. ................................ ................................ ................................ ......... 336 Table 6B - 2 AICc analysis for selecting model for the migration of 1.28 wt.% catechin from PLA film into 95% ethanol at 40 C. ................................ ................................ ................................ .. 338 Table 7 - 1 Correlation matrix of the estimated parameters at the average T ref =35 C. ............... 352 Table 7 - 2 Correlation matrix of the estimated parameters at the T ref =45 C. ............................ 353 Table 7 - 3 Correl ation matrix of the estimated parameters at the optimum T ref =44.94 C. ........ 355 Table 7 - 4 T ref =44.94 C. ................................ ........ 35 6 Table 7A - 1 Correlation matrix of the estimated parameters at the T ref =40 C. ........................ 361 Table 7A - 2 Correlation matrix of the estimated parameters at the T ref =50 C. ........................ 362 xv LIST OF FIGURES Figure 2 - 1 Mechanism of oxidation of linoleic acid . Figure adapted from Gordon (2001) (Gordon, 2001). ................................ ................................ ................................ ................................ ............ 18 Figure 2 - 2 Example of natural antioxidants. Figure was reproduced from Colín - Chávez et al. (2013), Gordon (2001), Mortensen & Skibsted (1997), Rice - Evans et al. (1996), and Soto - Valdez et al . (2010) (C. Colín - Chávez, H. Soto - Valdez, E. Peralta, J. Lizardi - Mendoza, & R.R. Balandrán - Quintana, 2013; Gordon, 2001; Mortensen & Skibsted, 1997; Rice - Evans, Miller, & Paganga, 1996; Soto - Valdez, Auras, & Peralta, 2010). ................................ ................................ ............... 28 Figure 2 - 3 Examples of synthetic antioxidants. Figure was reproduced from Gordon (2001) (Gordon, 2001). ................................ ................................ ................................ ............................. 29 Figure 2 - 4 Resonance of an antioxidant radical in the phenol structure. Figure was adapted and reproduced from Choe & Min (2006) (Choe & Min, 2006). ................................ ........................ 31 Figure 2 - 5 Mechanism of an antioxidant (a - tocopherol) in inhibiting lipid oxidation (linoleic acid). Figure adapted from Gordon (2001) (Gordon, 2001). ................................ ................................ .. 34 Figure 2 - 6 3D rectangular elements to represent diffusion in a plane sheet. ............................... 59 Figure 2 - 7 Migration phenonema that are controlled by the diffusion in the film. Figure was adapted from Poças et al. (2008) ( Poças et al., 2008 ) . ................................ ........................... 63 Figure 2 - 8 Migration phenomena that are controlled by the diffusion in the film with boundary layer resistance in the food/simulant. Figure was adapted from Poças et al. (2008) (Poças et al., 2008). ................................ ................................ ................................ ................................ ............ 64 Figure 2 - 9 Illustration of boundary layer resistance at the interface of film/simulant. Region 1 - 2: Overall resistance of diffusion within film; region 2 - 3: resistance at the partition between film - food/simulant; region 3 - 4: m ass transfer resistance at interface of film - food/simulant. Figure was adapted from Vitrac et al. (2007) (Vitrac et al., 2007). ................................ ................................ 65 Figure 2 - 10 Migration phenomen a that are controlled by diffusion in the film and in the food/simulant. Figure was adapted from Poças et al. (2008) (Poças et al., 2008). ....................... 67 Figure 3 - 1 Poly(lact ic acid), PLA chemica l structure. ................................ ................................ . 94 Figure 3 - 2 Astaxanthin chemical structure. ................................ ................................ ................. 95 Figure 3 - 3 The concentration of astaxanthin migrated into 95% ETOH at 30 to 40 °C. ........... 1 09 xvi Figure 3 - 4 (a) Migration of astaxanthin into 95% ETOH at 30 °C and (b) 40 °C during storage. ................................ ................................ ................................ ................................ ..................... 110 Figure 3 - 5 DSC thermogram of (a) PLA2M in contact with 95% ETOH at 40°C for 3 d; (b) PLA in contact with 95% ETOH at 40°C for 3 d; (c) PLA2M in contact with 95% ETOH at 30°C for 24 d; (d) PL A in contact with 95% ETOH at 30°C for 24 d; (e) PLA2M; and (f) PLA. ........... 114 Figure 3 - 6 Molecular weight distributions of fabricated functional membranes PLA and PLA 2M without and in contact with 95% ETOH at 30 and 40 °C for 24 and 3 d, respectively. ............. 117 Figure 3 - 7 Top SEM surface section micrograph (a) PLA; (b) PLA2M; (c) PL A in contact with 95% ETOH at 30 °C for 24 d; (d) PLA2M in contact with 95% ETOH at 30 °C for 24 d; (e) PLA in contact with 95% ETOH at 40 °C for 3 d; (f) PLA2M in contact with 95% ETOH at 40 °C for ................................ ................................ ................................ .......................... 121 Figure 3 - 8 (a) FTIR spectrum of the fabricated functional membranes and (b) focused FTIR spectrum of the fabricated functional membranes. ................................ ................................ ..... 127 Figure 3 - 9 Oxidative stability of soybean oil packaged in the glass bottles, pouches made of PLA and pouches made of PLA2M at 30 °C during 25 d. Straight green line indicated the cut off point for Codex Alimentarius. ................................ ................................ ................................ .............. 130 Figure 4 - 1 Summary of migration cases A, B, and C. Figure adapted from Poças et. al (2008) (Poças et al., 2008). ................................ ................................ ................................ ..................... 146 Figure 4 - 2 (a) Scaled sensitivity coefficients for 2P, and (b) for 3P of migration study of PLA - - tocopherol system at 23 C using forward difference approximation. Initial guesses used were: D = 0.06 10 - 9 cm 2 /s, M inf =3.95 10 - 5 g - tocopherol/g ethanol, and =0.35. ................................ 154 Figure 4 - 3 Migration of - tocopherol into 100% ethanol at 23 °C during storage for (a) 1P, (b) 2P, and (c) 3P using OLS esti mation and their corresponding residuals plot (d), (e), and (f), respectively. ................................ ................................ ................................ ................................ 155 Figure 4 - 4 (a) Final scaled sensitivity coefficients for 2P, and (b) for 3P of migration study of PLA - - tocopherol system at 23 C using forward difference approximation. Estimated values used for 2P were: D = 4.05 10 - 10 cm 2 /s, M inf =0.36 10 - 4 g - tocopherol/g eth anol. Estimated values used for 3P were: D = 0.79 10 - 10 cm 2 /s, M inf =0.37 10 - 4 g - tocopherol/g ethanol, and =0.30. Figure 4 - 5 Optimal experimental designs for 2P of migration study of PLA - - tocopherol system at 23 C 158 Figure 5 - 1 Example of multiple local minima for SSE in the non - linear estimation. (a) Surface plot with default view, (b) surface plot view set at azimuth and elevation of 28, and 28, respe ctively, and (c) contour plot for minimum region of SSE. ................................ ................. 226 xvii Figure 5 - 2 Graphical representation of the kinetics of migration . ................................ ............. 228 Figure 5 - 3 Experimental data fitting by using the simplified model of step 1. Initial approximation values obtained from this step were: D =5.88 10 - 12 cm 2 /min, K p,f =605.6 3 cm 3 PLA/cm 3 ethanol, and h =2.80 10 - 6 cm/min. ................................ ................................ ................................ ......... 243 Figure 5 - 4 Scaled sensitivity coefficient of the kinetics migration parameters using initial guesses obtained from step 1. Initial guesses were: D = 6.50 10 - 12 cm 2 /min, K p,f =605.00 cm 3 PLA/cm 3 ethanol, and h =5.90 10 - 6 cm/min. ................................ ................................ ........................... 244 Figure 5 - 5 Migration of 3 wt.% of resveratrol into ethanol at 9 C. ................................ .......... 246 Figure 5 - 6 Desorption kinetics of PLA - 3 wt.% resveratrol at 9 C in the dimensionless time space (Fourier number= . ................................ ................................ ................................ .................. 247 Figure 5 - 7 Residual plot for migration of 3 wt.% of resveratrol into ethanol at 9 C. .............. 248 Figure 5 - 8 Normalized sequential parameters as a function of time for migration of 3 wt.% of resveratrol into ethanol at 9 C. ................................ ................................ ................................ .. 250 Figure 5 - 9 (a) Sorption kinetic of 3 wt.% resveratrol into ethanol, and (b) KPD. The b lue line indicates the equilibrium state. ................................ ................................ ................................ ... 252 Figure 5 - 10 Histogram of the bootstrap residuals. ................................ ................................ ..... 254 Figure 5 - 11 Migration of 3 wt.% of resvera trol into ethanol at 9 C with added bootstrap results. ................................ ................................ ................................ ................................ ..................... 255 Figure 5A - 1 Experimental data fitting by using the simplified model of step 1. I nitial approximation values obtained from this step were: D =1.88 10 - 9 cm 2 /min, K p,f =608.11 cm 3 PLA/cm 3 ethanol, and h =4.19 10 - 4 cm/min. ................................ ................................ ............. 258 Figure 5A - 2 Scaled sensitivity coefficient of the kinetics migration parameters using initial guesses obtained from step 1. Initial guesses were: D =1.90 10 - 9 cm 2 /min, K p,f =608.11 cm 3 PLA/ cm 3 ethanol, and h =8.00 10 - 4 cm/min. ................................ ................................ ............. 259 Figure 5A - 3 Scaled sensitivity coefficient of the kinetics migration parameters ( D and K p,f ) using initial guesses obtained from step 1. Initial guesses were: D =2.00 10 - 9 cm 2 /min, K p,f =609.00 cm 3 PLA/cm 3 ethanol. ................................ ................................ ................................ ........................ 260 Figure 5A - 4 Migration of 2.6 wt.% - tocopherol into ethanol at 23 C. ................................ . 261 Figure 5A - 5 Desorption kinetics of PLA - 2.6 wt.% - tocopherol at 23 C in the dimensionless time space (Fourier number= . ................................ ................................ ............................... 262 xviii Figure 5A - 6 Residual plot for migration of 2.6 wt.% - tocopherol into ethanol at 23 C for two parameters estimation. ................................ ................................ ................................ ................ 263 Fi gure 5A - 7 Normalized sequential parameters as a function of time for migration of 2.6 wt.% of - tocopherol into ethanol at 23 C. ................................ ................................ ............................ 266 Figure 5A - 8 (a) Sorption kinetic of 2.6 wt.% - tocopherol into ethanol, and (b) KPD. The b lue line indicates the equilibrium state. ................................ ................................ ............................ 267 Figure 5A - 9 Histogram of the bootst rap residuals. ................................ ................................ ... 268 Figure 5A - 10 Migration of 2.6 wt.% - tocopherol into ethanol at 23 C with added bootstrap results. ................................ ................................ ................................ ................................ ......... 269 Figure 5B - 1 Experimental data fitting by using a simplified model of step 1. Initial approximation values obtained from this step were: D =1.39 10 - 8 cm 2 /min, K p,f =318.97 cm 3 PLA/cm 3 ethanol, and h =0.0024 cm/min. ................................ ................................ ................................ ................ 270 Figure 5B - 2 Scaled sensitivity coefficient of the kinetics migration parameters using initial guesses obtained from step 1. Initial guesses were: D =3.00 10 - 8 cm 2 /min, K p,f =318.97 cm 3 PLA/cm 3 ethanol, and h =0.0040 cm/min. ................................ ................................ ................... 271 Figure 5B - 3 Migration of 1.28 wt.% catechin into 95% ethanol at 40 C. ............................... 272 Figure 5B - 4 Desorption kinetics of PLA - 1.28 wt.% catechin at 40 C in the dimensionless time space (Fourier number = ). ................................ ................................ ................................ ......... 273 Figure 5B - 5 Residual plot for migration of 1. 28 wt.% catechin into 95% ethanol at 40 C. ... 274 Figure 5B - 6 Normalized sequential parameters as a function of time for migration of 1.28 wt.% of catechin into 95 % ethanol at 40 C. ................................ ................................ ....................... 277 Figure 5B - 7 (a) Sorption kinetic of 1.28 wt.% catechin into 95% ethanol, and (b) KPD. The b lue line indicates the equilibrium state. ................................ ................................ ............................ 278 Figure 5B - 8 Histogram of the bootstrap residuals. ................................ ................................ ... 279 Figure 5B - 9 Migration 1.28 wt.% catechin into ethanol at 40 C with added bootstrap results. ................................ ................................ ................................ ................................ ..................... 281 Figure 6 - 1 Scaled sensitivity coefficient of migration of 3 wt. % resveratrol from PLA film into ethanol at 9 C of (a) model 1 (initial guesses were: D =6.50 10 - 12 cm 2 /min, K p,f =605.00 cm 3 PLA/cm 3 ethanol, and h =5.9 10 - 6 cm/min), and (b) model 2 (initial guesses were: D =5.19 10 - 12 cm 2 /min and K p,f =430.00 cm 3 PLA/cm 3 ethanol). ................................ ................................ ... 318 xix Figure 6 - 2 Migration of 3 wt. % resveratrol from PLA film into ethanol at 9 C of (a) model 1, (b) model 2 and (c) model 3 and their corresponding residuals (d), (e), and (f), respectively. .. 321 Figure 6 - 3 Decision tree analysis for determining the kinetic mass transfer parameters ( i.e., D, K p,f , h ) of a migration study. ................................ ................................ ................................ ....... 325 Figure 6A - 1 Scaled sensitivity coefficient of migration of 2.6 wt.% - tocopherol from PLA film into ethanol at 23 C of (a) model 1 (initial guesses were: D =2.00 10 - 9 cm 2 /min, K p,f =609 cm 3 PLA/cm 3 ethanol), and (b) model 2 (initial guesses were: D =10.00 10 - 9 cm 2 /min and K p,f =500 cm 3 PLA/cm 3 ethanol). Note: the h was not estimated for model 1 due to high correlation issue with the D . ................................ ................................ ................................ ................................ ... 329 Figure 6A - 2 Migration of 2.6 wt.% - tocopherol from PLA film into ethanol at 23 C of (a) model 1 and (b) model 2 and their corresponding residuals (c), and (d), respectively. .............. 332 Figure 6B - 1 Scaled sensitivity coefficient of the migration of 1.28 wt.% catechin from PLA film into 95% ethanol at 40 C of model 1 (initial guesses were: D =3.00 10 - 8 cm 2 /min, K p,f =318.97 cm 3 PLA/cm 3 ethanol). ................................ ................................ ................................ ............... 334 Figure 6B - 2 Migration of 1.28 wt.% catechin from PLA film into 95% ethanol at 40 C of (a) model 1 and (b) model 3 and their corresponding residuals (c), and (d), respectively. .............. 337 Figure 7 - 1 Scaled sensitivity coefficient of the activation energy estimation of the migration of 1.28 wt.% catechin from PLA film into 95% ethanol ranging from 20, 30, 40, and 50 C at T ref =35 C. . Initial guesses were: D ref =1.00 10 - 9 cm 2 /min, K p,f =800 cm 3 PLA/cm 3 ethanol, h =10.00 10 - 4 cm/min, and E a =150000 J/mol). ................................ ................................ ............................... 350 Figure 7 - 2 Scaled sensitivity coefficient of the activation energy estimation at T ref =35 C of the migration of 1.28 wt.% catechin from PLA film into 95% ethanol ranging from 20, 30, 40, and 50 C. Initial guesses were: D ref =2 .00 10 - 9 cm 2 /min, K p,f =8 00 cm 3 PLA/cm 3 ethanol, and E a =15 0000 J/mol). ................................ ................................ ................................ ................................ ......... 351 Figure 7 - 3 Plot of correlation coefficient of the D ref and the E a as a function of possible T ref .. 354 Figure 7 - 4 Final scaled sensitivity coefficient of the activation energy estimation of the migration of 1.28 wt.% catechin from PLA film into 95% ethanol ranging from 20, 30, 40, and 50 C. Final estimates were: D ref =3.70 10 - 9 cm 2 /min, K p,f =436.62 cm 3 PLA/cm 3 ethanol, and E a =153 000 J/mol). ................................ ................................ ................................ ................................ ......... 357 Figure 8 - 1 Scaled sensitivity coefficient of (a) the case with Biot number < 200 (Initial guesses were: D ref =3.00 10 - 9 cm 2 /min, K p,f =608 cm 3 PLA/cm 3 ethanol, and h =8 10 - 5 cm/min), (b) the case with Biot number > 200 (Initial guesses were: D ref =3.00 10 - 9 cm 2 /min, K p,f =608 cm 3 PLA/cm 3 ethanol, and h = 1.60 10 - 3 cm/min). ................................ ................................ .......... 381 xx KEY TO SYMBOLS AND ABBREVIATIONS the ratio of the mass of migrant migrated into food/simulant to the mass of migrant left in the film at equilibrium a i chemical activity a w water activity A surface area A* antioxidant radical Bi Biot number C concentration initial concentration C the concentration of the migrant in the film at equilibrium C the concentration of migrant in the food simulant at equilibrium CO 2 carbon dioxide D diffusion coefficient diffusivity rate of the additives at E a activation energy residual F Flow rate F o Fourier number h convective mass transfer coefficient k rate constant frequency or pre - exponential factor xxi specific reaction rate at K p,f partition coefficient L film thickness M f mass of the migrant in the food or simulant M n number average molecular weight M p the mass of the migrant in the film M w weight average molecular weight M z z - average molecular weight M the mass of the migrant in the food or simulant at equilibrium i chemical potential of migrant chemical potential of migrant at a standard state O 2 oxygen 1 O 2 singlet oxygen 3 O 2 atmospheric triplet oxygen p number of parameter P concentration of food at steady state correlation coefficient the non - zero positive roots of eigenvalues R universal gas constant R rate constant R* alkyl radicals RH lipid molecule ROO* peroxy radicals xxii ROOH lipid hydroperoxides T temperature T cc cold - crystallization - temperature T d decomposition temperature T g glass transition temperature T m melting temperature reference temperature t time residual concentration volume of food/simulant volume of film sensitivity coefficient scaled sensitivity coefficient X c degree of crystallinity response variable predicted value synthetic data determinant determinant H cc enthalpies of cold crystallization H m enthalpies of melting f * heat of fusion of 100% crystalline sample AICc corrected Akaike information criterion xxiii BHA butylated hydroxyanisole BHT butylated hydroxytoluene CMC carboxymethylcellulose DSC differential scanning calorimeter EDTA ethylenediaminetetraacetic acid ETOH ethanol EVA ethylene(vinyl acetate) EVOH ethylene(vinyl alcohol) FDA Food and Drug Administration FTIR Fourier transform infrared spectrophotometer GPC gel permeation chromatography HDPE hig h density poly(ethylene) HPLC high performance liquid chromatography IP induction period KPD kinetic phase diagram LDPE low density poly(ethylene) LLDPE linear low density poly(ethylene) LOQ limit of quantification MMT montmorillonite OLS ordinary least squares PA poly(amide) PBAT poly(butylene adipate co - terephthalate) PCL poly(caprolacton) xxiv PE poly(ethylene) PEG polyethylene glycol PET poly(ethylene terep hthalate) PHBV poly(hydroxybutyrate - co - valerate) PI polydispersity index PLA poly(lactic acid) PLGA poly(lactide - co - glycolide) PP poly(propylene) PS polystyrene PTFE polytetrafluoroethylene PV peroxide value PVA poly(vinyl acetate) RH relative humidity RMSE root means square errors SEM scanning electron microscope SDS sodium dodecyl sulphate SSE sums of squared errors TBHQ tert - butylhydroquinone TGA thermogravimetric analyzer THF tetrahydrofuran UV ultraviolet UV - DAD ultraviolet - diode array detector WVP water vapor permeability 1 Chapter 1 Background and Motivation 1.0 Introduction Preservation of food is a very crucial step to maintain food quality, safety, and wholesomeness from the moment the food is produced until it is consumed. There are many technologies available and appli cable for food preservation. These include traditional preservation technologies ( e.g., the control of pH and water activity, heat treatment, and temperature control), and emerging preservation technologies ( e.g., high intensity light, irradiation, modifie d atmosphere packaging, and active packaging) ( Zeuthen & Bøgh - Sørensen, 2003 ) . Among these technologies, preservation through packaging in general is more practical and economical since the product is not modified and extra treatments are not needed. Packaging is already needed for marketing and distribution purposes; thus, the use of packaging as a preservation technique adds extra benefits to a product. One of those aforementioned examples is active packaging. Active packaging can be defined as a packaging system that provides continuous active protection to a food product/system during its shelf life by the incorporation of active substances ( e.g food additives) within the material or into the packaging system itself. Some examples of active packaging are oxygen scavengers and ethylene absorbers, which are commonly included into t he food system as a separate component. Meanwhile, antimicrobial and/or antioxidant packaging are examples of active packaging that involve the incorporation of active substances into the material itself. iously mentioned examples of incorporating the active substances directly into the packaging material, the development of antioxidant packaging system s have been extensively investigated ( Gómez - Estaca, López - de - 2 Dicastillo, Hernández - Muñoz, Cata lá, & Gavara, 2014 ; Sanches - Silva et al., 2014 ) . This type of active system is developed to control lipid oxidation in fatty food products ( Wessling , Nielsen, Leufvén, & Jägerstad, 1998 ) since lipid oxidation is among the main cause s contributing to food deterioration ( Gómez - Estaca et al., 2014 ; Sanches - Silva et al., 2014 ) . 1.1 Research Importance and Motivation The developmen t of antioxidant functional films has been seen as an effective tool in preserving the quality of food containing fats as it provides protection beyond the function of just being an inert barrier. This type of active packaging system helps to retard lipid oxidation by gradually releasing the active substances into food for an extended period of time to ensure prolonged shelf life. This technique is believed to be more efficient than that of one - time direct addition of antioxidants into food during processin g ( Balasubramanian, 2009 ) . Antioxidants are subjected to degradation and loss during processing. For an intended long shelf life product, antioxidant will be completely consumed after a short period of time, thus leaving the product unprotected from lip id oxidation. In adddition , the amount of antioxidants permitted for use in food products, singly or in combination, is limited to 0.02% by weight based on fat content of the food ( Miková, 2001 ) . This amount, sometimes, may not be sufficient, due to possible loss of antioxidants during processing and might be fully diminished before the product (food) even reaches market shelves. The addition of relativel y high concentrations of certain antioxidant s ( e.g., tocopherol and ascorbic acid) into food systems could also result in pro - oxidation reactions in lipids ( Balasubramanian, 2009 ) . Consequently, products may have a short shelf life. Even though there is other existing technology, like oxygen scaven gers, that can help to retard oxidation, there are some concerns about their application. Oxygen scavengers are 3 commonly incorporated into a packaging system in the form of sachets. Their absorbing capacity for oxygen is typically limited to 100 mL ( Smith, Hoshino, & Abe, 1995 ) , requiring multiple sachet s for use with a product designed for a long shelf life, which is not practical or acce ptable commercially ( Robertson, 2006 ) . Besides, it requires additional materials, which consumers do not perceive as good management of resources. In addition, p ossible accidental ingestion of these compounds could happen, although their ingestion does not cause adverse health impacts ( Floros, Dock, & Han, 1997 ) , which further deters consumer from buying th e products. An antioxidant functional film could be a feasible alternative technology for such applications. This technology is beneficial for both the packaging and the packaged food because ively, through a 3 - step mechanism: 1) antioxidant diffusion through the polymer bulk - phase to the polymer surface; 2) antioxidant volatilization from the packaging surface to the packaging headspace or desorption of antioxidant from the packaging surface i nto the surroundings; and 3) ( Bailey, 1995 ) . This technology prevents lipid oxidation by a controlled constant release of antioxidants from the polymer matrix to the product, thus constantly protecting product s when it is most needed, during storage. Despite the promising benefits of incorporating antioxidants into the polymeric structures over direct addition into the food products, some research has shown the effectiveness, and limitations of this technology. Oregano functional film w as found to be efficient in improving lamb steak oxidative stability ( Camo et al., 2008 ) . Ner í n et al . (2006) reported promising outcomes fr om the use of antioxidant functional films for beef products ( Nerín et al., 2006 ) . Wesslin g et al. - tocopherol (above 360 ppm) in low density poly(ethylene), 4 (LDPE) delayed the oxidation of linoleic acid at 6 °C, but not at higher temperatures (20 and 40 °C). There were also some concerns reported on the po tential changes in the mechanical properties, - tocopherol ( Wessling, Nielsen, & Leufven, 2000 ) . In antioxidant functional film applications, synthetic antioxidants , like BHT and BHA, are exploited widely. Although these antioxidants are effective and provide economic value, the safe use of these antioxidants in food products has been questioned ( Day, 2003 ; Gómez - Estaca et al., 2014 ) ; hence, there is intensive research carry out on new potential natural antioxidants ( Barbosa - Pereira et al., 2013 ; Calatayud et al., 2013 ; Chen, Lee, Zhu, & Yam, 2012 ; Contini et al., 2012 ; Hwang et al., 2012 ; Lopez de Dicastillo et al., 2011 ; Pereira de Abreu, Losada, Maroto, & Cruz, 2010 ; Sonkaew, Sane, & Suppakul, 2012 ; Zhu, Lee, & Yam, 2012 ; Zhu, Schaich, Chen, & Yam, 2013 ) . Some natural antioxidants could be more potent, efficient, and safer than synthetic ones. Natural - tocopherol, for i nstance, has higher antioxidant capability than - - tocopherol transfer protein ( HongLian et al., 2001 ) . Natural antioxidants are also generally recognized as safe (GRAS), and their use is not limited when used in accordance with good manufacturing practice (GMP) ( Rajalakshmi & Marasimhan, 1995 ) . Most of the research conducted in the area of antioxidant packaging systems focuses on non - renewable materials. Mainly, the developed antioxidant functional films are made of polyolefins such as low density poly(ethylene), (LDPE) and poly(propylene), (PP) ( Gavara, Lagarón, & Ca talá, 2004 ; Wessling et al., 1998 ) . In recent years, the re is a growing trend to use biodegradable materials to develop antioxidant functional films. This ten dency may be associated with increasing concerns about municipal solid waste and growing environmental awareness among consumers ( Endres, Siebert, & Kaneva, 2007 ) . 5 Some biodegradable materials that have gained increasing interest are poly(lactic acid) (PLA), thermo plastic starch, and poly(butylene adipate co - terephthalate) (PBAT), to name a few. PL A is a biopolymer produced from polymerization of lactic acid ( Endres et al., 2007 ) , and it can be obtained from renewable resources, like corn ( Auras, Harte, Selke, & Hernandez, 2003 ) , sugar beet and sugarcane residues ( Endres et al., 2007 ) . PLA is a transparent material, so it is good for food packaging application s. It can be formed into a variety of containers, trays, films, and other type of packaging structures. PLA is biodegradable, compostable and re cyclable, and it has been approved by the US Food Drug Administration (FDA) as suitable for food - contact packaging applications ( Auras, Harte, & Selke, 2004 ) . PLA is comparable to poly(ethylene terep hthalate), (PET), and polystyrene, (PS) in terms of its physical and mechanical properties, and it has low barrier t o gases such as carbon dioxide (CO 2 ) and oxygen (O 2 ) ( Auras, Harte, & Selke, 2003 ) . Therefore, its appli cation in food packaging, for instance, might be limited to certain types of food. For example, fatty food products packaged in PLA would e xperience lipid oxidation as a re sult of its low barrier to oxygen. For this reason, the incorporation of antioxidants into a PLA polymeric for targeted food systems. Research involving characterization of PLA with incorporated antioxidants is increasing rapidly. Both natural and synthetic antioxidants were investigated for their potential with PLA polymeric structures as antioxidant functional films. Butylated hydroxyanisole (BHA), butylated hydroxytoluene (BHT), p ropyl gallate, and te rt - butylhydroquinone (TBHQ) are among common synthetic antioxidants that have been incorporated into a PLA matrix ( Byun, Kim, & Whiteside, 2010 ; M. Jamshidian et al., 2012 ; Jamshidian, Tehrany, & Desobry, 2012 ; Ortiz - Vazque z, Shin, Soto - Valdez, & Auras, 2011 ) . C atechin, epicatechin, tocopherol, and resveratrol are examples of 6 natural antioxidants that have been incorporated into PLA ( Byun et al., 2010 ; Hwang et al., 2013 ; Hwang et al., 2012 ; - Franco et a l., 2012 ; Manzanarez - López, Soto - Valdez, Auras, & Peralta, 2011 ; Soto - Valdez, Auras, & Peralta, 2010 ) . Even though antioxidant - PLA functional films have been widely investigated, most of the information about PLA incorporated with carotenoid - based antioxidants. Therefore, functionalization of PLA with carotenoid - based antioxidants is needed to fill this gap. Ca rotenoid - based antioxidants act by a different mechanism as antioxidants than the phenolic ones; thus different outcomes on the basis of polymer - antioxidant interactions and their corresponding properties are anticipated. M ost of the research on antioxidant functional films being researched focuses on migration studies and/or characterization of the film properties. To the extent of the limited research has emphasized mathematical modeling of the kin etic release of antioxidants from functional films, by means of parameter and sequential estimations. Numerous advantages can be gained by using mathematical modeling to understand the kinetic release of antioxidant functional films such as the physical in terpretation of parameters with respect to experimental study, time and cost - saving , etc. Thus, the importance of mathematical modeling as a food safety tool and quality assurance need s to be considered. 1.2 Objectives The main objectives of this disser tation were: 1. To produce a biodegradable bilayer functional film incorporated with a carotenoid - 7 to perform an oxidative stability study and to investigate the kinetic r elease of the incorporated antioxidant into a fatty food simulant at two different temperatures. 2. To introduce a parameter estimation app roach to assess the kinetic migration parameters of antioxidant functional films. 3. To develop a new mathematical solution consisting of the three main kinetic migration parameters ( i.e ., the diffusion coefficient ( D ) , the partition coefficient ( K p,f ) , and the convective mass transfer coefficient ( h ) ) that govern most of the migration experiments, and to compare this developed solution with the general mass transfer solutions provided by Carslaw & Jaeger (1959) and Crank (1979). 4. To estimate the activation energy ( E a ) of non - isothermal migration experiments using a non - linear reparameterization approach to the Arrhenius equation. 1.3 Dissertation Overview This dissertation is organized as follows. Chapter 2 provides a literature review of three main sections; i ) lipid oxidation, its mechanism with respect to diff erent factors ( i.e., environmental conditions, presence of metal , etc. ) and approaches to prevent lipid oxidation, ii ) migration phenomena and general mathematical models used for migration studies, and iii ) parameter estimation approach ( i.e., ordinary le ast square (OLS), sequential, bootstrap, etc. ) to assess the kinetic migration parameters. Chapter 3 explores the development of PLA - functional film incorporated with a natural carotenoid - based antioxidant (astaxanthin). The developed film was subjected to various testing, which include d thermal analyses, barrier and molecular weight properties, morphological study, 8 oxidative stability and the kinetic release of astaxanthin from PLA - functional film into 95% ethanol at 30 and 40 C. Chapter 4 inv estigates the kinetic release for different migration case studies employing the general mass transfer solutions by Carslaw & Jaeger (1959) and Crank (1979) by means of the parameter estimation approach ( i.e., scaled sensitivity coefficient, , OLS estimation, opti mal experimental design). Comparison of estimating 1 parameter, 1P ( i.e., D ) versus 2P ( i.e., D and the mass of the migrant in the food or simulant at equilibrium, M ) and 3P ( i.e., D , M , and the ratio of the mass of antioxidant migrated into the simulant to the mass of the antioxidant left in the film, at equilibrium ( ) , was also done by using the corrected Akaike information criterion (AICc) and root means square errors (RMSE). Chapter 5 proposes a new two - step mathematical solution to estimate the thre e kinetic migration parameters ( i.e., D , K p,f , h ). This solution was developed based on the boundary conditions provided by Carslaw & Jaeger (1959) and Crank (1979) . Three selected migration case studies were used to demonstrate the application of this solution by means of the parameter estimation approach ( i.e., , OLS estimation, sequential estimation, kinetic phase diagram (KPD), and bootstrap method). Chapter 6 provides a comparative study between the two - step mathematical solution proposed in chapter 5 with the general mass transfer solutions by Carslaw & Jaeger (1959) and Crank (1979) using three different migration case studies. The and OLS estimation were performed. The model discrimination was evaluated using the AICc approach. Chapter 7 explores the estimation of E a of non - isoth ermal migration studies using the reparameterized Arrhenius equation. The simulation temperature ( T sim ) was introduced for visual observation of E a for plot. 9 Chapter 8 summarizes all the works in this dissertation and concludes with future work recommendations. 10 REFERENCES 11 REFERENCES Auras, R., Harte, B., & Selke, S. (2003). 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Evaluation of the effectiveness of a new active packaging film containing natural antioxidants (from barley husks) that retard lipid damage in frozen Atlantic salmon ( Salmo salar). Food Research International, 43 (5), 1277 - 1282. Rajal akshmi, D., & Marasimhan, S. (1995). Food Antioxidants: Sources and Methods of Evaluation Food Antioxidants: Technological: Toxicological and Health Perspectives (Vol. 71, pp. 65): CRC. 14 Robertson, G. L. (2006). Active and intelligent packaging. In G. L. Ro bertson (Ed.), Food Packaging: Principles and Practice (2nd ed., pp. 292 - 294). Boca Raton: CRC Press. Sanches - Silva, A., Costa, D., Albuquerque, T. G., Buonocore, G., Ramos, F., Castilho, M. C., . . . Costa, H. S. (2014). Trends in the use of natural antio xidants in active food packaging: a review. Food Additives & Contaminants: Part A (just - accepted). Smith, J. P., Hoshino, J., & Abe, Y. (1995). Active packaging in polymer films. In M. Rooney (Ed.), Active Food Packaging (pp. 143 - 172). Glasgow: Blackie Aca demic & Professional. Sonkaew, P., Sane, A., & Suppakul, P. (2012). Antioxidant activities of curcumin and ascorbyl dipalmitate nanoparticles and their activities after incorporation into cellulose - based packaging films. Journal of Agricultural and Food Ch emistry, 60 (21), 5388 - 5399. Soto - Valdez, H., Auras, R., & Peralta, E. (2010). Fabrication of Poly(lactic acid) Films with Resveratrol and the Diffusion of Resveratrol into Ethanol. Journal of Applied Polymer Science . doi: 10.1002/app.33687 Wessling, C., N ielsen, T., & Leufven, A. (2000). The influence of alpha - tocopherol concentration on the stability of linoleic acid and the properties of low - density polyethylene. Packaging Technology and Science, 13 (1), 19 - 28. Wessling, C., Nielsen, T., Leufvén, A., & J tocopherol and BHT in LDPE in contact with fatty food simulants. Food additives and contaminants, 15 (6), 709 - 715. doi: 10.1080/02652039809374701 Zeuthen, P., & Bøgh - Sørensen, L. (2003). Food preservation techniques : CRC P ress. Zhu, X., Lee, D. S., & Yam, K. L. (2012). Release property and antioxidant effectiveness of tocopherol - incorporated LDPE/PP blend films. Food Additives and Contaminants: Part A, 29 (3), 461 - 468. Zhu, X., Schaich, K. M., Chen, X., & Yam, K. L. (2013). Antioxidant Effects of Sesamol Released from Polymeric Films on Lipid Oxidation in Linoleic Acid and Oat Cereal. Packaging Technology and Science, 26 (1), 31 - 38. doi: 10.1002/pts.1964 15 Chapter 2 Literature Review 2.0 Introduction The first part of this chapter focuses on lipid oxidation and its classification, factors influencing lipid oxidation, and approaches used to inhibit lipid oxidation. The second part covers the migration process and commonly encountered migration systems. The last part o f this chapter describes the p arameter estimation technique for modeling different migration systems. 2.1 Lipid Oxidation: Brief I ntroduction Foods containing fat s, such as meats, milk, and butt er, to name a few, are subject to reduced shelf life as a re s ult of lipid degradation. Understanding the pathway of lipid oxidation is crucial to prevent this deterioration process from causing undesirable organoleptic modification of fatty - based foods ( i.e., off - flavors and aromas, detrimental effect on nutrition, and carcinogenic by - products). 2.2 Lipid Oxidation: C lassification Deterioration of lipids can be classified into th ree main types: autoxidation, photo o xidation or photosensitized oxidation, and lipoxygenase - catalyzed oxidation. 2.2.1 Autoxidation Autoxidation is the most common process contributing to oxidative deterioration ( Gordon, 2001 ) . This reaction often shows an induction period (IP). During the IP, the deterioration of lipids occurs at a very slow rate, while at the end of this period, noticeable and rapid deterioration takes 16 place. The autoxidation mechanism consists of four mechanism steps; i) initiation, ii) propagation, iii) chain branching, and iv) chain termination ( Figure 2 - 1 ) ( Gordon, 1990 ) . 2.2.1.1 Step 1: Initiation This process is called initiation because for it to occur, initiators are required. Initiators are compounds containing double bonds that are in singlet spin states . L ipid needs molecules (RH) that have similar spin states to be able to produce a lipid free radical ( Schaich, Shahidi, Zhong, & Eskin, 2013 ) . Therefore, at this early stage, an RH i s ruptured to a lipid free radical by a metal catalyst, exposure to light, by hydroperoxide decomposition or by reaction with singlet oxygen molecules or an enzyme - catalyzed reaction ( Gordon, 2001 ) . RH R* + H* 2.2.1.2 Step 2: Propagation The chain reaction is propagated by the abstraction of H at a The reaction of lipid free radicals with atmospheric triplet oxygen ( 3 O 2 ) then results in the formation of peroxy radicals (ROO*). Lipid hydroperoxides (ROOH) and alkyl radicals (R*) are formed later after peroxy radicals react with more lipid molecules ( Gordon, 2001 ; Schaich et al., 20 13 ) . This process is continuous as long as the source of H is still available or no interception of the chains occurs ( Schaich et al., 2013 ) . R* + 3 O 2 ROO* ROO* + RH ROOH + R* 2.2.1.3 Step 3: Chain branching This step involves the decomposition of hydroperoxides to generate more free radicals ROOH RO* + *OH 17 2.2.1.4 Step 4: Chain termination Free radicals are combined to form peroxide compounds. Since these compounds are unstable, they decompose to release hydrocarbons, alcohols or aldehydes ( Gordon, 2001 ; Schaich et al., 2013 ) . The by - products produced depend on the type of chain scission reactions ( composition and concentration of hydroperoxides, temperature, and/or O 2 , to name a few ( Laguerre, Lecomte, & Villeneuve, 2007 ; Schaich et al., 2013 ) . 2R* R - R R* + ROO* ROOR 18 Figure 2 - 1 Mechanism of oxidation of linoleic acid . Figure adapted from Gordon (2001) ( Gordon, 2001 ) . 2.2.2 Phot o oxidation or Photosensitized Oxidation There are two types of photo o xidation: Type I results from excitation of lipids, and Type II is associated with excitation of O 2 which may occur in the presence of light and a sensitizer 19 ( Gordon, 2001 ) . Photo o x idation has no induction period; thus the le vel of hydroperoxides formed steadily increases . This type of oxidation highly depends on the concentration of sensitizers, is independen t of O 2 concentration unless it is limited , and the rate of oxidation increases accordingly with the number of double bonds of the fatty acids ( Schaich et al., 2013 ; Terao & Matsushita, 1977 ) . 2.2.2.1 Type I: phot o oxidation by excitation of lipids This type of oxidation is characterized by hydrogen atom transfer or electron transfer between a substrate and an excited triplet sensitizer, which later forms free radical ions. This type of oxidation require a certain or specific wavelength for initiatio n ( Gordon, 2001 ) . Some of the most common photosensitizers type I are chlorophyll, hemes ( e.g., myoglobin, hemoglobin), xanthenes, anth roquinones, and food dyes ( Schaich et al., 2013 ) . 2.2.2.2 Type II: photo o xidation by excitation of O 2 In the presence of light and a sensitizer, a triplet oxygen ( 3 O 2 ) will become excited to a singlet oxygen ( 1 O 2 ). This state of oxygen ( 1 O 2 ) reacts more than 1500 times faster with a polyunsaturated fatty acid to form a hydroperoxide than triplet oxygen ( Gordon, 2001 ) . Some examples of sensitizers that fall into this category are chlorophyll, erythrosine, f lavins, and eosin ( Schaich et al., 2013 ) . 2.2.3 Enzyme - Catalyzed Lipid Oxidation Lipoxygenase is an enzyme found in both plant and animal tissues. This enzyme function s by catalyzing the incorporation of oxygen into p olyunsaturated fatty acids and/ or their esters, and acylglycerols containing the cis , cis - 1,4 - pentadiene system. This process later leads to the formation of conjugated cis , trans hydroperoxide ( Gordon, 2001 ) . 20 2.3 Factors Affecting Lipid Oxidation There are many factors that may contribute towards oxidation in fatty food products. Identifying these factors may benefit food manufacturers in protecting their products from oxidation and also aid packaging engineers in designing packaging for oxygen - sensitized products that complies with regulations as well as optimizes function and extends shelf life. The factors that affect the oxidation rate of a fatty food product are discussed in the following section. 2.3.1 Chemic al Structure of Fatty Acids Fatty food products are generally susceptible to lipid oxidation. The tendency of these products to undergo oxidation can be predicted by knowing the chemical composition of their respective fatty acids. Fatty acids that contain more conjugated double bonds are more likely to oxidize since the alkene groups reduce the bond dissociation energy of neighboring C - H bonds. Consequently, the tendency of hydrogen abstraction that leads to the formation of alkyl radicals that are readily available for interaction with oxygen increases ( Gordon, 2001 ) . In addition, the amount and the rate of formation of primary oxidation compounds at the final st age of the induction period also increases with increased degree of unsaturation ( Choe & Min, 2006 ; Martin - Polvillo, Márquez - Ruiz, & Dobarganes, 2004 ) . It is also cruc ial to high light that the isomeric form of the fatty acids influences the rate of the oxidation process. The cis isomer of fatty acids will likely oxidize faster than the trans isomer due to the unsaturated site that is ready for H abstraction ( Schaich et al., 2013 ) . 21 2.3.2 Temperature In general, an increase of 10 °C in temperature increases the rate of biological r eaction s by ( Cohen Stuart, 1912 ) . Increased temperature has been reported to accelerate lipid oxidation due to the increase in the rate of hydroperoxide decomposition ( Labuza & Dugan Jr, 1971 ; Schaich et al., 2013 ) . In addition, at temperatures greater than 150 C, the ch ances for lipid oxidation to take place are higher due to molecular degr adation and autoxidation in the presence of initi ators and air that results from t hermally - induced chain scission ( Nawar, 1969 ; Schaich et al., 2013 ) . In contrast , some literature reports that temperature has little effect on lipid oxidat ion as a result of its low activation energy (0 to 6 kcal/mole) ( Choe & Min, 2006 ; Rahmani & Csallany, 1998 ; Yang & Min, 1994 ) . 2.3.3 Light The effect of light on lipid oxidation has been reported to be more pronounced than the effect of temperature. Shorter wavelengths are reported to be more damaging than longer ones ( Sattar, DeMan, & Alexander, 1976 ) , and t he effect of light on lipid oxidation lessens as temperature increases ( Choe & Min, 2006 ; Velasco & Dobarganes, 2002 ) . Lipid oxidation of a mayonnaise product was greater at wavelengths lower than 470 nm, where a significant detrimen tal effect was observ ed with decreasing wavelengths of light (i.e., 365>405>435 nm) ( Lennersten & Lingnert, 2000 ) , due to the higher energy. Fatty acids contain chromaphores, which consist of carbonyl groups, unsaturated sites (double bonds), and peroxide ( O - O ) b onds. Among those, peroxide bonds possess bond energ i es that are in the appropriate range to interact with UV light. Moreover, UV light was reported to have more damaging effect on fatty acids than visible light because of its capability to produce 22 strong oxidizing hydroxyl radicals rather than hydroxyl ions ( Schaich et al., 2013 ) . Visible light, on the other hand, requires photosensitizers ( e.g., chlorophyll, xanthenes) to be able to absorb low energy light that later excite s the atoms from the ground state to a high energy state before forming free radicals or singlet oxygen by shifting the excitation energy to fatty acids or to oxygen, respectively ( Murray, 1979 ; Schaich et al., 2013 ) . 2.3.4 Oxygen, O 2 One of the key factors that significantly influences lipid oxidation is oxygen (O 2 ). Both the type and the concentration of O 2 may directly or indirectly affect the kinetics of lipid oxidation. 2.3.4.1 Type of O 2 There are two types of oxygen that affect lipid oxidation: i) atmospheric triplet oxygen, 3 O 2 , and ii) singlet oxygen, 1 O 2 . T he former is responsible for autoxidation by reacting with lipid radicals and the latter is a product of photosensitized oxidation of edible oils, an oxidation mechanism that takes place in the presence of light, sensitizers and atmospheric oxygen. 1 O 2 was reported to be able to directly react with lipids while 3 O 2 reacts with lipid radicals ( Choe & Min, 2006 ) . 2.3.4.2 Concentration of O 2 The kinetics of lipid oxidation not only depend on the types of O 2 , but also on the concentration of O 2 . At high temperature and in the presence of light and sensitizers, the oxidation rate in creases proportionally with increasing concentration of O 2 . High temperature increases the diffusivity of O 2 into oil; meanwhile the light and sensitizers transfer sufficient excitation energy 23 to allow the O 2 to react readily with unsaturated sites of fatt y acids, thus increasing the oxidation rate in the presence of higher O 2 concentrations ( Choe & Min, 2006 ) . Under limited O 2 concentrations, the oxidation rate increases with increasing O 2 pressure ( Schaich e t al., 2013 ) . However, in th e presence of excessively high concentration s of O 2 in comparison with the amount of fatty acids, the rate of lipid oxidation is not affected due to insufficient unsaturated sites for oxidation to occur (i.e., controlling mechanism for oxidation) . It is also worth mentioning that the rate of lipid oxidation increases with increasing surface area for food products (liqui d or solid) since the reaction between O 2 in the atmosphere with fatty acids on the pr s surface is faster than that with O 2 that diffuses into the product ( Schaich et al., 2013 ) . 2.3.5 Presence of Minor Co mpounds Lipid oxidation is a complex process that can take place by various mechanisms with different pathways. This process may end up with similar or different by - products such as hexanal, nonanal, decanal, 1 - pentene - 3 - one, 2 - pentylfuran, etc., regardle ss whether the mechanisms and pathways are similar or different . Therefore, any single or combination of different minor chemical compounds present in the food system may have an effect on the oxidation of lipids and needs to be understood to foresee the b est approach to improve the quality and the shelf life of the food system. 2.3.5.1 Water and Water A ctivity, a w Water and water activity, a w play an important role in lipid oxidation. Both can demonstrate either antioxidant or pro - oxidant effects on lipid oxidation of food. a w refers to the water that is present in the food product and is available, but it is not bound through chemical reactio ns . a w 24 ranges from 0 to 1.0 with the former being a completely dry system and the latter being a highly moist system. In the case of lipid oxidation in dry systems within a w ranges of 0 to 0.2, the rate of lipid oxidation increases with decreasing a w . A dr y system allows more accessibility towards the unsaturated chain site to chemically interact with O 2 ( Karel, 1980 ; Labuza & Dugan Jr, 1971 ; Schaich et al., 2013 ) . The rate of lipid oxidation is typically lowest in a food system that has a w 0.2 to 0.5 (monolayer region for food system) as a result of limited mobility of O 2 and catalysts. For a system that has a w between 0.5 and 0.7, lipid oxidation of a particular food system increases with increasing a w due to the increased mobility of catalysts and O 2 with reactive sit es, also due to higher catalyst and O 2 diffusivities i n the system. On the other hand, at a w > 0.7, the occurrence of lipid oxidation is at its highest and then decreases with increasing a w because of the dilution of catalysts and reactants. In addition, the non - enzymatic browning reaction that produces some antioxidants is also higher at a high a w level ( Karel, 1980 ; Labuza & Dugan Jr, 1971 ; Schaich et al., 2013 ) . 2.3.5.2 Metals Metals enhance lipid oxidation by directly/indirectly interacting with fatty acids, thus forming some compounds that increase the rate of oxidation, including peroxy radicals, 1 O 2 that are highly reactive, and hydrogen peroxides, to name a few ( Andersson, 1998 ; Choe & Min, 2006 ) . Oxidized metals are documented as more powerful than reduced metals since they can directly produce initial radicals. Some examples of metals that exist naturally in products like crude oil are copper and ferrous ions ( Schaich et al., 2013 ) . 25 2.3.5.3 Free Fatty Acids Lipids contain various free fatty acids, esters, and triacylglycerols. The nature and quantity of these groups determines the mechanism of lipid oxidation. Therefore, it is vi tal to understand the composition of lipids because of their huge impact on the kinetic s of lipid oxidation. Free fatty acids are naturally existing compounds in crude oils ( Choe & Min, 2006 ) . They can demonstrate either pro - oxidant or antioxidant effects depending on the food system and the presence of other minor components in the particular system. Free fatty acids are reported to oxidize more slowly than esters and triacylglycerols due to the involvement of acid groups in the non - radical decomposition of hydroperoxides ( Schaich et al., 2013 ) . This mechanism later initiates nucleophilic rearrangements that slow down the rate of oxidation by preventing chain branching reactions ( Schaich et al., 2013 ) . It was reported that free fatty acids increase d the O 2 diffusivity into edible oil, thus increasing the oxidation rate. The reas on was due to the chemical structure of free fatty acids that contain both hydrophilic and hydrophobic groups that help in reducing the surface tension of edible oil s ( Choe & Min, 2006 ; Mistry & Min, 1987 ) . Havi ng mentioned about some of the free fatty acids mechanisms as a pro - oxidant , to the contrary, the presence of mechanism. The carboxylic groups in free fatty acids ma y act as excellent metal complexers that help in blocking the ele ctron tr ansfer in metals, resulting in a decrease in the rate of oxidation due to the reduction of the redox potential ( Schaich et al., 20 13 ) . 2.3.5.4 Phospholipids Similar to free fatty acids, phospholipids may also act as antioxidant s or a s pro - oxidant s . The ability of phospholipids to bind water enhances the mobilization of catalysts that results in an 26 increase of oxidation rate ( Nwosu, Boyd, & Sheldon, 1997 ; Schaich et al., 2013 ) . In addition, the presence of hydrophilic and hydrophobic groups in t heir structure causes a reduction in the surface tension of their respe ctive systems and increases the O 2 diffusivity, which results in an acceleration ( Choe & Min, 2006 ; Yoon & Min, 1987 ) . The non - radical decomposition of hydroperoxides of phospholipids was reported to show an antioxidant effect since it intercepts the radical chain reactions ( Corliss & Dugan Jr, 1970 ; O'brien, 1969 ; Schaich et al., 2013 ) , in turns reducing the rate of oxidat ion. However, this mechanism does not hinder other possible pa thways for lipid oxidation. In a food system containing metals, phospholipids were reported to bind the metals from further reacting with the unsaturated sites of the system, thus slowing down the oxidation process ( Choe & Min, 2006 ; Yoon & Min, 1987 ) . 2.3.5.5 Chlorophylls Chlorophyll is a photosensitizer that accelerates the rate of oxidation by transferring the excitation energy of electrons in the form of either free radicals or 1 O 2 , in the presence of light and 3 O 2 ( Fakourelis, Lee, & Min, 1987 ; Gutiérrez - Rosales, Garrido - Fernández, Gallardo - Guerrero, Gandul - Rojas, & Minguez - Mosquera, 1992 ; Whang & Peng, 1988 ) . The by - products ( i.e., pheophytins, pheophorbides) of chlorophyll degradation can also act as sensitizers. It was reported that pheophytin is a stronger sensitizer than chlorophyll ( Endo, Usuki, & Kaneda, 1984 ; Rahmani & Csallany, 1998 ) . Although chlorophyll is most known for its pro - oxidant effect, it may act as an antioxidant under certain circumstances. Some studies have reported that chlorophyll stabilizes free radicals in the absence of light by possibly the donation of hydrogen ( Choe & Min, 2006 ; Endo, Usuki, & Kaneda, 1985 ; Gutiérrez - Rosales et al., 1992 ) . In highly unsaturated fatty 27 acid systems, the effect of chlorophyll as an antioxid ant was reported to be greater than in a system low in unsaturated fatty acids ( Endo et al., 1985 ) . 2.4 Antioxidants Lipid oxidation contributes to undesirable changes in fatty food products, which in turn th the use of antioxidants. Antioxidants are naturally occurring compounds that can be found in plants ( Wang & Lin, 2000 ) . Some of the examples of natural antioxidants are astaxanthin (algae, shrimp), lycopene (tomato, autumnberry), catechin (green tea), resveratrol (grapes), and others ( Figure 2 - 2 ). Antioxidants can also be produced synthetically. Synthetic a ntioxidants include gallates, butylated hydroxytoluene (BHT), butylated hydroxyanisole (BHA), and others (Figure 2 - 3 ). In some cases, natural antioxidants can also be produced synthetically and considered as antioxidants of natural origin (but not as a nat ural antioxidant), like tocopherol (Figure 2 - 2 ). 28 Figure 2 - 2 Example of natural antioxidants . Figure was reproduced from Colín - Chávez et al. (2013), Gordon (2001), Mortensen & Skibsted (1997), Rice - Evans et al. (1996), and Soto - Valdez et al. (2010) ( C. Colín - Chávez, H. Soto - Valdez, E. Peralta, J. Lizardi - Mendoza, & R.R. Balandrán - Quintana , 2013 ; Gordon, 2001 ; Mortensen & Skibsted, 1997 ; Rice - Evans, Miller, & Paganga, 1996 ; Soto - Valdez, Auras, & Peralta, 2010 ) . 29 Figure 2 - 3 Examples of synthetic antioxidants. Figure was reproduced from Gordon (2001) ( Gordon, 2001 ) . 2.4.1 Classification of Antioxidants In general, antioxidants can be classified into two groups based on their mechanisms, which are primary antioxidants (chain - breaking), and secondary antioxidants (preventive antioxidants) ( Gordon, 2001 ; Laguerre et al., 2007 ; McClements & Decker, 2000 ) . Primary antioxidants work by scavenging free radicals ( Gordon, 2001 ) . These compounds are mainly phen olic substances and act as electron donors ( Kochhar & Rossell, 1990 ) . They are also consumed du ring the induction 30 period. Secondary antioxidants function by binding metal ions, scavenging oxygen, transforming hydroperoxides into non - radical species, absorbing UV radiation or deactivating singlet oxygen. These compounds are normally effective in the presence of a second minor compound ( Gordon, 2001 ) . 2.4.1.1 Primary Antioxidants (Chain - Breaking) The mechanism of chain - breaking antioxidants is based on hydrogen atom donation ( Laguerre et al., 2007 ; Schaich et al., 2013 ) . Once lipid oxidation has started, this process forms non - stable by - products, namely, peroxyl radicals. Since peroxyl radicals are non - stable, they are highly reactive and continuously react with more lipid molecules to fo rm more hydroperoxides and alkyl radicals. Then, the hydrope roxides decompose to generate more free radicals and alkyl radicals continuously react with atmospheric triplet oxygen ( 3 O 2 ) to produce other peroxyl radicals ( Choe & Min, 2006 ; Gordon, 2001 ; Schaich et al., 2013 ) . Based on the nature of this con tinuous r eaction, chain - breaking antioxidants function by donating their hydrogen atoms to those aforementioned non - stable by - products, thus transforming these by - products into more stable radicals or non - radical products ( McClements & Decker, 2000 ) . The efficacy of chain - breaking antioxidants relies on their higher affinity toward free radicals than that of lipids and also their ability to produce less reactive antioxidant radicals than those of lipids and peroxyl radicals ( McClements & Decker, 2000 ) . In addition, chain - breaking antioxidants should pose lower reduction potentials in comparison to free radicals to be able to don ate their hydrogen atoms ( Buettner, 199 3 ; Choe & Min, 2006 ) . Free radicals of polyunsaturated fatty acids such as alkoxy, peroxy, and alkyl radicals were reported to have standard 1 - electron reduction potential of 1600, 1000 , and 600 mV, respectively, w hile antioxidants generally have standard reduction potentials 31 - tocopherol (500 mV) and ascorbic acid (282 mV)) ( Buettner, 1993 ; Choe & Min, 2006 ) . Most of the chain - breaking antioxidants are phenoli c compounds. The capability of these compounds to donate their hydrogen atoms is measured by their bond dissociation energy. The lower the bond dissociation energy of these compounds, the greater is their ability to donate hydrogen atoms ( Laguerre et al., 2007 ) . Moreover, the presence of phenolic groups in t he chain - breaking antioxidant molecules not only helps to stabilize the free radicals, but also helps to decelerate the oxidation process by the resonance stabilization of antioxidant radicals ( Choe & Min, 2006 ; Laguerre et al., 2007 ; Schaich et al., 2013 ) ( Figure 2 - 4 ). Therefore, in general, the higher the number of phenolic groups, the greater antioxidant activity is anticipated. However, the location of this ph enolic group in the antioxidant structure may also determine the antioxidant activity ( Schaich et al., 2013 ) . Figure 2 - 4 Resonance of an antioxidant radical in the phenol structure. Figure was adapted and reproduced from Choe & Min (2006) ( Choe & Min, 2006 ) . 2.4.1.2 Secondary Antioxidants (Preventative Antioxidants) Preventative antioxidants work even before the oxidation starts as a preventative measure. This type of antioxidant typically inhibits lipid oxidation by either chelating the metals or scavenging the 1 O 2 ( Schaich et al., 2013 ) . 32 Chelation of metals may involve metal chelators or metal complexers. Commonly used metal chelators are ethylenediaminetetraacetic acid (EDTA) and phytate. M etal complexers include citric acid, ascorbic acid, and diamines . Both metal chelators and complexers can perform efficiently as preventative antioxidants by having a structure that inhibits electron transfers by surrounding the metals present in the food system ( Schaich et al., 2013 ) . The reason behind this mechanism is to be able to hinder the metals from reacting with lipids, thus inhibiting the formation of alkyl radicals and some other r eactive species like 1 O 2 and hydroperoxides ( Andersson, 1998 ; Choe & Min, 2006 ) . In ad dition, they must have sufficiently high conc entration to retard lipid oxidation ( Schaich et al., 2013 ) . Some metal chelators a nd metal complexers may alter the redox poten tial or may increase the diffusivity of the metals present in the food system, which in turn, enhances pro - oxidant effects of these metals ( Decker, 1998 ; McClements & Decker, 2000 ) . There are t wo quenching mechanisms: physical quenching and chemical quenching. Physical quenching involves transformation of 1 O 2 into 3 O 2 by transferring the energy or the charge without antioxidant oxidation. Chemical quenching occurs when antioxidants chemically re act with 1 O 2 , thus forming oxidized products ( Min & Boff, 2002 ) . Singlet oxygen quenchers are generally carotenoid - based antioxidants. These compounds contain conjugated double bonds in their structure. In 1 O 2 oxidation, the type of double bond is not as important as the number of double bonds in the lipid structure ( Min & Boff, 2002 ) . Since 1 O 2 is electrophilic in nature, it is always in need of filling the vacancy of its molecular orbital, which targets electron - riched conjugated double bonds ( Adam, 1975 ; Min & Boff, 2002 ; Stahl & Sies, 2005 ) . Therefore, in the presence of singlet oxygen quenchers ( i.e., carotenoid - based antioxidants), its tendency to react with the extended conjugated double bonds of carotenoid - based antioxidants is higher than that of the unsaturated sites of the lipid ( Schaich et al., 2013 ; Stahl & Sies, 2005 ) . 33 2.4.1. 3 An Example of a n Antioxidant Reaction The alkyl peroxy radical (ROO*) is readily reduced to the related anion, and converted to a hydroperoxide by an electron donor. ROO* +e ROO - H+ ROOH The presence of an antioxidant (AH) acts to interrupt the propagation step, and form an antioxidant radical (A*). This radical has a low reactivity, thus inhibiting further oxidation steps ( Yanishlieva - Maslarova, Pokorny, Yanishlieva, & Gordon, 2001 ) . ROO* + AH ROOH + A* RO* + AH ROH + A* Figure 2 - 5 shows visual comparison of an antioxidant mechanism to inhibit lipid oxidation. 34 Figure 2 - 5 Mechanism of an antioxidant (a - tocopherol) in inhibiting lipid oxidation (linoleic acid). Figure adapted from Gordon (2001) ( Gordon, 2001 ) . 35 2.5 Approaches to Extend the Shelf Life of Fatty Food Products Lipid oxidation is the most critical issue when dealing wi th fatty food products. This type of food deterioration is mainly caused by the high content of unsaturated fatty acids in the fatty food products. As has been described in the previous sections, the chances for oxidation to occur are very high because of the nature of the product itself and the environmental conditions. Therefore, it is crucial to find the right approach to protect this type of food system in order to be able to provide acceptable organoleptic properties with maximum quality and safety ass urance for the product. Many approaches are available to help in inhibiting lipid oxidation. However, it is quite challenging to find a single approach that can resolve this issue due to the wide variety of fatty food products and the complexity of the f ood systems themselves. Nevertheless, the efforts to control such a critical issue have been extensively investigated and some are commercially available. Among the typical approaches used to inhibit lipid oxidation of fatty food products are direct addit ion of antioxidants into the systems, controlled environment s (exclusion of O 2 , dark and chilled storage), and antioxidant packaging systems. The common practice in the food industry involves direct addition of antioxidants into the food system. Neverthel ess, some concerns about this approach are the loss of antioxidants during processing due to the thermal conditions required in most food practices, the limits on use of antioxidants to comply with regulations (whenever applicable), the pro - oxidant issue t hat may happen due to the presence of minor components in the food, and others. Even though a certain amount of added antioxidant might still be left in the food system, the question is how much antioxidant is sufficient, and can it maintain the quality of the product during distribution and storage before reaching consumers? 36 packaging system. Not only does the product from the beginning need a pac kage to carry and contain it , but also the package may beneficially enhance the nutritional value of the product while - based packaging systems can be divided into two categories: i) individual and independent antioxidan t devices, ii) antioxidant incorporated polymeric films by means of blending, functional layers via multilayer structures or chemically/ enzymatically - modified structures, and coatings. 2.5.1 Antioxidant Based Packaging System: Individual and Independent Antioxidant Devices For this category, the antioxidant is kept separately in another extra packaging material like a sachet/pouch . This individual and independent device will then be incorporated into the primary package of the product. The most common a pplication for this category is O 2 scavenger, a type of device that is capable of removing O 2 from the package headspace. To a certain extent this device can also be tailored to be able to reduce the amount of O 2 significantly. However, the obstacles for s uch packaging systems to be implemented rely on the type of the intended product, the practicality of usage, possible interaction with other reactants or compounds in the food product and the solid products, and it seems impractical to use this device for an extended shelf life due to its limited absorbing capacity. It was reported that the capacity of this device to absorb O 2 is limited to 100 mL ( Smith, Hoshino, & Abe, 1995 ) . In addition, Lee (2014) estimated that iron - based O 2 scavenger could absorb approximately 300 mL of O 2 in the presence of 0.43 g of water ( D. S. Lee, 2014 ) . Moreover, additional packaging material is also required apart from the primary and secondary packaging that are normally used for packaging food products, th us g enerating more waste. The presence of other reactants such as 37 CO 2 is also known to reduce the efficiency of iron - based scavengers in absorb ing O 2 ( D. S. Lee, 2014 ) . Extra efforts need to be taken to educat e consumers not to mistake these device s for p art of the food product, as that might lead to a food safety hazard. Many issues have been associated with metal - based individual and independent devices such as the d anger of microwave arcing in certain applications, detection by metal detectors, etc. ( Cruz, Camilloto, & dos Santos Pires, 2012 ) . Therefore, some initiatives have been made to use organic substrates like ascorbic acid and catechol. These new alternative substrates may contain a small amount of metal to control thei r self - oxidation mechanisms ( Cruz et al., 2012 ; D. S. Lee, 2014 ) . 2.5.2 Antioxidant Functional Films In recent years, focus on modification of polymer films has intensified to gain positive perception from the public and to fulfill the needs of the industry. Commonly used compounds for individual and independent antio xidant devices ( i.e., metal - based scavengers, organic substrates) and antioxidants ( i.e., natural - based compounds) have been incorporated into polymer films to create a functional film with single or multiple layers. The incorporation of antioxidants into polymer films is beneficial not only to retard lipid oxidation of intended food systems but also to stabilize the polymer during processing. Direct incorporation of antioxidant compounds into polymer films has a long history worldwide. Incorporation of i ron compounds into polymer films has resulted in products such as Oxyguard® of Toyo Seikan (Japan), Shelfplus O 2 ® of Ciba Specialty Chemicals (Switzerland), Oxycap® of Standa Industrie (France), and ActiTUF® of M&G (Italy) ( D. S. Lee, 2011 , 2014 ) . Chemically modified unsaturated hydrocarbons were also introduced as part of polymeric films to 38 scavenge O 2 in package - product systems ( Cruz et al., 2012 ; Ferrari et al., 2009 ) . Immobilized - yeast was incorporated into the liner of beer bottle caps to consume O 2 and to release C O 2 from/into the headspace, respectively, without changing the organoleptic properties of the product ( Cruz et al., 2012 ; Edens, Farin, Ligtvoet, & Van Der Plaat, 1992 ) . Nanocrystalline titania was also added into polymeric films for scavenging O 2 to inhibit oxidation by activating it via UV light ( Azeredo, 2009 ; D. S. Lee, 2014 ; Mills, Doyle, Peiro, & Durrant, 2006 ; Xiao - e, Green, Haque, Mills, & Durrant, 2004 ) . BHT and - tocopherol are among the antioxidants that have been widely investigated as part of the packaging system ( Bailey, 1995 ; Byun, Kim, & Whiteside, 2010 ; Galindo - Arcega, 2004 ; Granda - Restrepo et al., 2009 ; Hwang et al., 2013 ; Jurina, Azizah, Siah, & Ngadiman, 2011 ; Manzanarez - López, Soto - Valdez, Auras, & Peralta, 2011 ; Ortiz - Vazquez, Shin, Soto - Valdez, & Auras, 2011 ; Wessling, Nielsen, & Giacin, 2001 ; Wessling, Nielsen, & Leufven, 2000 ; Wessling, Nielsen, Leufvén, & Jägerstad, 1998 ; Yanidis, 1989 ) . Currently, the tendency of using natural antioxidants ( e.g., - tocopherol, catechin, plant extract) is increasing due to the abundance and availability of these compounds, positive perception by consumers, non - limited use by regulation, and others. However, the majority of the developed antioxidant functional films are made of petroleum - based polymers such as low den sity poly(ethylene) (LDPE), and poly(propylene) (PP) ( Gavara, Lagarón, & Catalá, 2004 ; Wessling et al., 1998 ) . Table 2 - 1 shows examples of these system s . Nevertheless, the growth in the development of bio - based functional films with natural antioxidants added has increased (Table 2 - 2). The driving factors behind this evolution are mainly the rising environmental awareness and continuous oil price volatility ( Jiang & Zhang, 2013 ; Siracusa, Rocculi, Romani, & Rosa, 2008 ) . Fabricated bio - based functional films include polymers 39 that are extracted directly fr om biomass, particularly polysaccharides, synthesized from bio - derived monomers, and/or produced from natural or genetically modified organisms ( Siracusa et al., 2008 ; Tuil, Fowler, Lawther, & Weber, 2000 ) , with the majority being made of polyester that is synthesized from bio - derived monomers ( i.e., poly(lactic acid), PLA). 40 Table 2 - 1 Compilation of studies reporting kinetic migration parameters for petroleum based functional films incorporated with natural antioxidants and the models used to determine these parameters . Petroleum - based functional films Nat ural antioxidant compounds Parameters estimated Mathematical equations applied for estimating the kinetic migration parameters References PE and PA laminate with PS tray - Oregano extract - Rosemary extract NE ** ( Camo, Beltrán, & Roncalés, 2008 ) EVOH LDPE EVA EVA and LDPE PP - Quercetin - Tocopherol D and K p,f ( Chen, Lee, Zhu, & Yam, 2012 ) PET - DFC Amosorb 4020 (containing cobalt salt) NE ( Galdi, Nicolais, Di Maio, & Incarnato, 2008 ) 41 Table 2 - Petroleum - based functional films Natural antioxidant compounds Parameters estimated Mathematical equations applied for estimating the kinetic migration parameters References LDPE: EVOH: HDPE+ 7% Titanium dioxide - - Tocopherol D and K p,f ( Granda - Restrepo et al., 2009 ) LDP E LDPE adsorbed on Syloblock LDPE adsorbed on SBA - 15 EVA - - Tocopherol D ( H eirlings et al., 2004 ) PVA - Chitosan - Mint extract - Pomegranate extract NE ( Kana tt, Rao, Chawla, & Sharma, 2012 ) 42 Table 2 - Petroleum - based functional films Natural antioxidant compounds Parameters estimated Mathematical equations applied for estimating the kinetic migration parameters References Ziegler - Natta LLDPE Metallocene LLDPE - - Tocopherol - - Cyclodextrin + - tocopherol - Quercetin - - Cyclodextrin + quercetin NE ( Koontz et al., 2010 ) HDPE + Surlyn/EVA® - - Tocopherol N E ( Y. S. Lee, Shin, Han, Lee, & Giacin, 2004 ) 43 Table 2 - Petroleum - based functional films Natural antioxidant compounds Parameters estimated Mathematical equations applied for estimating the ki netic migration parameters References EVOH - Quercetin - Catechin D and K p,f ( ) EVOH - Green tea extract D and K p,f ( Lopez de Dicastillo et al., 2011 ) 44 Table 2 - Petroleum - based functional films Natural antioxidant compounds Parameters estimated Mathematical equations applied for estimating the kinetic migration parameters References EVOH - Green tea extract - Ascorbic acid - Ferulic acid - Quercetin D and K p,f ( López - de - Dicastillo, Gómez - Estaca, Catalá, Gavara, & Hernández - Muñoz, 2012 ) PP photografted HEMA - Caffeic acid NE ( Arr ua, Strumia, & Nazareno, 2010 ) Grafted PP - Poly(acrylic acid) (metal chelator) NE ( Tian, Decker, & Goddard, 2012 ) 45 Table 2 - Petroleum - based functional films Natural antioxidant compounds Parameters estimated Mathematical equations applied for estimating the kinetic migration parameters References HDPE - Carvacrol D ( Peltzer, Wagner, & Jiménez, 2009 ) LDPE - Barley husks derived antioxidant NE ( Pereira de Abreu, Losada, Maroto, & Cruz, 2010 ) PET - Green tea extract - Green coffee extract - Grapefruit extract NE ( Colon & Nerin, 2012 ) 46 Table 2 - Petroleum - based functional films Natural antioxidant compounds Parameters estimated Mathematical equations applied for estimating the kinetic migration parameters References PP - Rosemary extract NE ( Nerín, Tovar, & Salafranca, 2008 ) EVOH - Cocoa extract D and K p,f ( Calatayud et al., 2013 ) LDPE - TOCOBIOL - TOCOBIOL® GL - NUTRABIOL® T90 - TOCOBIOL® PV - NUTRABIOL® T50 PV NE ( Barbosa - Pereira et al., 2012 ) 47 Table 2 - Petroleum - based functional films Natural antioxi dant compounds Parameters estimated Mathematical equations applied for estimating the kinetic migration parameters References LDPE - - tocopherol - - cyclodextrin + - tocopherol D and K p,f ( Siró et al., 2006 ) LDPE/PP blend - Tocopherol D ( Zhu, Lee, & Yam, 2012 ) LLDPE:HDPE + A*:HDPE HDPE:HDPE+A* :EVA - Sesamol NE ( Zhu, Schaich, Chen, & Yam, 2013 ) 48 Table 2 - *PE - Poly(ethylene); PA - Poly(amide); PS - Poly(styrene); EVOH - Ethylene(vinyl alcohol); LDPE - Low densi ty poly(ethylene); HDPE - High density poly(ethylene); EVA - Ethylene(vinyl acetate); PP - Poly(propylene); PET - Poly(ethylene terephthalate); PVA - Poly(vinyl acetate); LLDPE - Linear low density poly(ethylene).**A* - Antioxidant.***NE - Not estimated. Pet roleum - based functional films Natural antioxidant compounds Parameters estimated Mathematical equations applied for estimating the kinetic migration parameters References PP - Catechin - Green tea extract (gallic acid, quercetin, caffeine) D ( Dic ) Recycled PET - Citrus extract NE ( Contini et al., 2012 ) 49 Table 2 - 2 Compilation of studies reporting kinetic migration parameters for bio - based functional films incorporated with natural antioxidants and the models used to determine these parameters . Bio - based functional films Natural antioxidant compounds Parameters estimated Mathematical equations applied for estimating the kinetic migration parameters References Chitosan - - Tocopherol NE ( Martins, Cerqueira, & Vicente, 2012 ) Chitosan - MMT - Rosemary essential oil NE ( Abdollahi, Rezaei, & Farzi, 2012 ) Grafted chitosan - Gallic acid NE ( Schreiber, Bozell, Hayes, & Zivanovic, 2013 ) Sodium caseinate Calcium caseinate - Carvacrol NE ( Arrieta, Peltzer, Garrigós, & Jiménez, 2013 ) 50 Table 2 - Bio - based functional films Natural antioxidant compounds Parameters estimated Mathematical equations applied for estimating the kinetic migration parameters References CMC - MMT - Murta leaves extract NE ( Quilaqueo - Gutiérrez, Echeverría, Ihl, Bifani, & Mauri, 2012 ) Chitosan - Green tea extract NE ( Siripatrawan & Noipha, 2012 ) Cellulose - Curcumin nanoparticles - Ascorbyl dipalmitate nanoparticles NE ( Sonkaew, Sane, & Suppakul, 2012 ) 51 Table 2 - Bio - based functional films Natural antioxidant compounds Parameters estimated Mathematical equations applied for estimating the kinetic migration parameters References PLA - Resveratrol D and K p,f ( Soto - Valdez et al., 2010 ; Soto - Valdez , Peralta, & Auras, 2008 ) PLA - - Tocopherol NE; D , K p,f and ( Goncalves et al., 2012 ; Manzanarez - López et al., 2011 ) 52 Table 2 - Bio - based functional films Natural antioxidant compounds Parameters estimated Mathematical equations applied for estimating the kinetic migration parameters References PLA - - Tocopherol - Ascorbyl palmitate D and K p,f ( Jamshidian, Tehrany, & Desobry, 2012 ) 53 Table 2 - Bio - based functional films Natural antioxidant compounds Parameters estimated Mathematical equations applied for estimating the kinetic migration parameters References PLA - Catechin - Epicatechin D , K p,f , and M ( - Franco et al., 2012 ) PLA/Starch blend - - Tocopherol - Resveratrol D , K p,f , and M ( Hwang et al., 2013 ) 54 Table 2 - Bio - based functional films Natural antioxidant compounds Parameters estimated Mathematical equations applied for estimating the kinetic migration parameters References PLGA - - Tocopherol NE ( Van Aardt e t al., 2007 ) PLA PCL PHBV - - Carotene NE ( López - Rubi o & Lagaron, 2010 ) *MMT - Montmorillonite; CMC - Carboxymethylcellulose; PLA - Poly(lactic acid); PLGA - Poly(lactide - co - glycolide); PCL - Poly(caprolacton); PBHV - Poly(hydroxybutyrate - co - valerate). **NE - Not estimated. 55 Even though the fabrication of antioxidant functional film systems is continuously growing with more trends toward the use of bio - based polymers with natural antioxidants, issues related to migration of the incorporated antioxidants into food systems are s till not fully addressed. It is very important to understand the migration phenomena in order to ensure optimized release of the incorporated antioxidants into the intended food system so that an extended shelf life may be achieved. 2.6 Migration Migration is a phenomenon involving the transfer of substances originating from the packaging material into a packaged product. These substances could be monomers, solvents, additives ( e.g., antioxidants), etc., and are known as migrants. Migration could b e a desirable or undesirable event that occurs in a packaged product. Residual solvents ( e.g., toluene and hexane), for instance, when they migrate into food product will result in unwanted odor and taste. However, migration is desired when it is intention ally designed in order to protect a polymer and packaged product, as in the case of antioxidant packaging systems. The initial concentration of migrants in the packaging systems and the partition coefficient between the package and the packaged food determ ine the extent of migration ( Selke, Culter, & Hernandez, 2004 ) . The better the understanding of the migration phenomena, the more efficient the prediction of the shelf life of a pro duct and the better the assessment of specific migration limits in accordance with regulation can be achieved ( Poças, Oliveira, Oliveira, & Hogg, 2008 ) . Migration phenomena can be described through established mathematical models with the following assumpt ions: i) initial concentration of the migrants is uniformly distributed in the film, ii) migration happens on the side of the fil m that is in contact with food/ simulant, iii) the food/simulant is well mixed and has a large surface mass transfer coefficient , h (Biot no. >100), 56 iv) Fickian diffusion controls the migration in the film, v) migration depends only on temperature and the diffusion coefficient, D , and the partition coefficient, K p,f , are constants, vi) the film interface and the food are always at equilibrium, and vii) no interaction exists between the film and the food/simulant and the edge effect is negligible ( Chung, Papadakis, & Yam, 2001 , 2002 ; Crank, 1979 ; Poças et a l., 2008 ) . 2.6.1 Thermodynamic Equilibrium The chemical potential is the driving force that causes a molecule to diffuse within a film or to transfer between a film and a surrounding phase, and it is described as follows: = (Eq. 2 - 1) = chemical potential of migrant I = chemical potential of migrant I at a standard state R = universal gas constant T = temperatu re, K = chemical activity Substances like migrants naturally tend to move from a high chemical potential to a low which means the migrants tend to reach thermodynamic equilibrium. In addition, the general mass balance equation can be used to describe the migration model of a product - package system, assuming no chemical reaction or evaporation process is involved ( Poças et al., 2008 ) . M p (0)= M p (t) + M f (t) or (Eq. 2 - 2) 57 M p (0)= M p f (Eq. 2 - 3) M p (0) = Initial mass of the migrant in the film M p (t) = Mass of the migrant in the film at time, t M f (t) = Mass of the migrant in the food or simulant at time, t M p M f = Mass of the migrant in the food or simulant 2.6 .2 Partition C oefficient, K p,f The partition coefficient, K p,f , can be described as the equilibrium concentration and distribution of migrants in a film and in a food/simulant ( De Meulenaer, 2009 ; Franz & Störmer, 2008 ; Selke et al., 2004 ) . (Eq. 2 - 4) C = the concentration of the migrant in the film at equilibrium. C = the concentration of migrant in the food simulant at equilibrium. The value of K p,f is always a good indication of the behavior of the migrant in correlation with the type of food/simulant used. For example, w hen the migrant is hydrophobic and the food/simulant nature is non - polar like oil, a K p,f < 1 is anticipated du s nature, meaning that most or all of the migrant will migrate from the film into the food/simulant. However, if the food/simulant is polar in nature, a K p,f >1 is expected, in which most of the migrant stays in the film inste ad of migrating into the food/simulant ( Poças et al., 2008 ) . The extent of K p,f will also be influenced by factors like temperature and the nature of the film. 58 2.6.3 Diffusion C oefficient, D The diffusion coefficient, D is used to describe the migration of a migrant from a film into a food/simulant. D is a function of temperature, and it may increase when the concentration of the migrant is relatively high in the film ( Baner, Franz, & Piringer, 1994 ) flow of diffusio n by the equation below (for a one - dimensional diffusion process) ( Selke et al., 2004 ) : (Eq. 2.5) F = flow rate D = d iffusion coefficient c = migrant concentration in the film x = the distance (in the direction of the diffusion) process. The following equation is used when diffusion occ urs in one dimension ( Selke et al., 2004 ) : (Eq. 2 - 6) Both equations can be visualized further by considering a 3D rectangular element with the parallel sides to the axes coordinates represented with length 2 dx, 2 dy, 2dz. The center of this 3D element is set at P (x,y,z) with the concentration of diffusing migrant C (Figure 2 - 6). 59 2dz 4dy dz (F x - ) 4dy dz(F + ) Figure 2 - 6 3D rectangular elements to represent diffusion in a plane sheet. 4dy dz (F x - ) (Eq. 2 - 7) 4dy dz (F + ) (Eq. 2 - 8) - 8 dx dy dz ( ) (Eq. 2 - 9) From the other two faces of axes: - 8 dx dy dz ( ) and - 8 dx dy dz ( ) The rate that the concentration of diffusing migrant increases: 8 dx dy dz ( ) (Eq. 2 - 10) =0 (Eq. 2 - 11) If the diffusion coefficient is constant, F x , F y , F z , substituting Eq. 2 - 5 into Eq. 2 - 11, =0 obtained: (Eq. 2 - 12) D A A C C D B B 2dx 2dy 60 If the diffusion is one - dimensional where the gradient concentration is only in the direction of the x - axis, then this is simplified to : (Eq. 2 - 13) Moreover, the diffusion process is affected by: i) approach that can be directly/indirectly linked to the molecular wei ght and molecular weight distribution, solubility parameters (polarity, dispersion forces, and hydrogen bonding), crystallinity, orientation and density; ii) the nature of the migrant such as its hydrophilicity or hydrophobicity and molecular weight; iii) the nature of the food/simulant that is in contact with the film in terms of its aggressiveness, solvency (polar vs. non - polar); iv) film - migrant - food/simulant interaction effects such as plasticization; and v) experimental/ environmental temperature that temperature and/or melting temperature ( De Meulenaer, 2009 ; Limm & Hollifield, 1996 ; Poças et al., 2008 ) . Generally, the behavior of the film based on relaxation rate and its correlation to Fickian diffusion can be classified into three ma in cases ( De Meulenaer, 2009 ; Schlotter & Furlan, 1992 ) : Case 1: Fickian diffusion takes place in a situation where the rate of diffusion is lower than the relaxation rate of the film. This type of diffusion commonly occurs for polymers like LDPE and HDPE with their glass transition temperatures well below room temperature so both exhibit a rubbery nature in their amorphous regions. Case 2: Whe n the rate of diffusion is faster than the relaxation rate of the film, then so called apparent Fickian diffusion occurs. In this case, the mass sorption is proportional to time. This case may occur due to the aggressive nature of food/simulant to penetrat e the film at a constant velocity, thus resulted in rapid diffusivity of the migrant. This situation could happen in the case of a glassy 61 polymer such as PLA, due to solvent - induced phenomena that are time - dependent, which influence the diffusion rate of t he migrant. Case 3: Non - Fickian or anomalous diffusion takes place if the rate of diffusion and the relaxation rate of the film are similar. 2.6.4 Migration M odels In general, there are three migration models that are used to describe migration of a migr ant from a film into a liquid food/simulant: Model A : Film in contact with a finite volume of food/simulant and negligible external mass transfer coefficient (Figure 2 - 7). This model is normally in conjunction with K p,f >1 which occurs when most of the migrant stays in the film instead of migrating into the food/simulant. Normally, the solution for this model (Eq. 2 - 14) is applied in the case of migration at a relatively low temperature. The final solution for this model is as follows: (Eq. 2 - 14) where M f,t is the mass of migrant release at time t , and M f, is the mass of migrant at t = , = V F / K p,f V P , q n are the non - zero positive roots of tan q n = q n , and L is the thickness of the sample , D is the diffusion coefficient . Model B : Film in contact with an infinite volume of food/simulant and negligible external mass transfer coefficient (Figure 2 - 7). This model is often used in the case where as a r esult of a larger volume of food/simulant (20 - 50 times) than the volume of the film ( Ham dani, Feigenbaum, & Vergnaud, 62 1997 ) , t hat denotes most of the migrant, if not all, migrates into the food/simulant. The final solution for this model is as follows: (Eq. 2 - 15) Both model A and B are often known as Piringer models. These models are controlled by diffusion in the film. For short migration times, the U.S Food and Drug Administration (FDA) model is applied ( Poças et al., 2008 ) . The final solution for short migration times is shown in Figure 2 - 7. Model C : Film in contact with an infinite volume of food/simulant and non - negligible external mass transfer coefficient (Figure 2 - 8). This model involves a convection process, in which resistance exists at the interface between the film and the food/simulant. In this case, a dimensionless Biot number (Bi) is calculated by taking into consideration the thickness of the film, the convective mass transfer coefficient, and diffusion that takes place in the film ( Vitrac, Mougharbel, & Feigenbaum, 2007 ) . Commonly, in the case of Bi < 2 00 which suggests high resistance at the film - food/simulant interface (Figure 2 - 8 and Figure 2 - 9 ), the following solution is used: (Eq. 2 - 16) In a situation wher e the food/ liquid is strongly stirred (Bi>2 00) ( Mascheroni, Guillard, Nalin, Mora, & Piergiovanni, 2010 ) , Eq. 2 - 15 is then used as a solution. 63 Figure 2 - 7 Migration pheno men a that are controlled by the diffusion in the film. Figure was adapted from Poças et al. (2008) ( Poças et al., 2008 ) . Initial conditions Boundary condition Balance equation Solutions = the non - zero positive roots of tan If because and/ or a simplified solution: FDA model: For short migration time: If because and/ or a simplified solution: L 0 x film food C p C f 64 L 0 x film food C f C p Figure 2 - 8 Migration phenomena that are controlled by the diffusion in the film with boundary layer resistance in the food/simulant. Figure was adapted from Poças et al. (2008) ( Poças et al., 2008 ) . Initial conditions Boundary condition Balance equation Solutions Biot number (Bi)= tan = the non - zero positive roots or eigenvalues If is very high (>2 00): = convective mass transfer coefficient L = thickness of film 65 L 0 x film food C p diffusion C f partition convection - diffusion 1 2 3 4 t>0 Figure 2 - 9 Illustration of boundary layer resistance at the interface of film/simulant. Region 1 - 2: Overall resistance of diffusion within film; region 2 - 3: resistance at the partition between film - food/simulant; region 3 - 4: mass transfer resistance at interface of film - food/simulant. Figure was adapted from Vitrac et al. (2007) ( Vitrac et al., 2007 ) . Meanwhile, in the case of migration phenomena in volving solid or semi solid food, a numerical solution using the finite difference method can be applied (Figure 2 - 10). The solution to this case can be approached based on discretization of time and/or discretization of space ( Piringer & Beu, 2000 ; Poças et al., 2008 ) . The accuracy of the solution is dependent on the degree of implicitness: fully explicit, fully implicit or Crank - Nicholson. The fully explicit solution uses forward finite differences, and it is not a stable solution. Often, this approach results in algorithm oscillations that grow exponentially as a function of time. The fully implicit solution applies 66 backwa rd finite differences, and it is quite an accurate solution. Even though it is not as accurate as Crank - Nicholson, it does not produce the algorithm oscillations. The Crank - Nicholson method takes an average of both fully explicit and fully implicit and pro vides highly accurate and stable solutions ( Piringer & Beu, 2000 ) . By having all migration models portrayed, it is very important to further estimate the parameters of interest, and thus be able to predict their behavior as well as to assist the experimental design of a variety of migration study. 67 L 0 x film food C p C f Figure 2 - 10 Migration phenomena that are controlled by diffusion in the film and in the food/simulant. Figure was adapted from Poças et al. (2008) ( Poças et al., 2008 ) . Initial conditions Boundary condition Balance equation Solutions Numerical solution using finite differences method: Discretization of time: Discretization of space: = degree of implicitness: Fully implicit, Fully explicit, Crank - Nic olson, 68 2.7 Parameter Estimation The use of the modeling approach aids data simula tion and prediction; thus time and cost needed fo r performing an experiment can b e reduced to a certain extent. The m odeling approach can be classified into two parts, which are the forward problem and the inverse problem. The forward problem is a direct approach using an explicit or differential soluti on where the parameters are known. The observational data are not required for a for ward problem operation. T he inverse problem requires data to be able to determ ine parameters or functions of the model ( Dolan & Mishra, 2013 ) . Experimental studies are often designed to collect observational data (dependent variable) with unknown functions or parameters. This approach is an inverse problem and also is known as para meter estimation. Parameter estimation helps to estimate the parameters /constants of interest involved in mathematical models and to a certain extent it may provide some physical meaning for parameters relevant to the experiment. Beck and Arnold (1977) de that provides tools for the efficient use of data in the estimation of constants appearing in ( Beck & Arnold, 1977 ) . 2.7.1 Parameters of Interest Generally, in a migration study, one is very interested in determining the parameter D , that is the diffusion coefficient. This parameter helps to explain the kinetic release of a migrant from/to film to/from food/simulant, respectively. When the experimen t is set up for a range of temperatures, with the parameter values obtained, the activation energy may be assessed. Recently, some researchers have started the quest of estimating the mass transfer coefficient, h in the case of a resistance boundary layer ( Mascheroni et al., 2010 ; Vitrac et al., 2007 ) . Other parameters that are worth estimati ng are the and the , that may provide extra 69 information for the migration study. Although some researchers have estimated the former, none has taken a step further to either report or explain it ( C. Colín - Chávez et al., 2013 ; Citlali Colín - Chávez, Herlinda Soto - Val dez, Elizabeth Peralta, Jaime Lizardi - Mendoza, & René Renato Balandrán - Quintana, 2013 ; Hwang et al., 2013 ; - Franco et al., 2012 ; Manzanarez - López et al., 2011 ; Ortiz - Vazquez et al., 2011 ) . 2.7.2 Sensitivity Coefficient, X , and Scaled Sensitivity Coefficient, Depending on the particular research, parameter estimation may involve one particular parameter or more. As stated earlier, in a migration study, estimation of D is a nec essity. However, by taking a more upfront approach, estimating some other parameters ( i.e., and ) that are importance in the research could result in more fruitful findings. When there are more parameters involve d , it is vital to investigate w h ether those parameters can be estimated simultaneously, easily and accurately; thus, more interpretable result s can be anticipated. By taking the first derivative of the dependent variable (response variable) with respect to the parameter of interest, a sensitivity coefficient can be obtained (Eq. 2 - 17). The sensitivity coefficient is an important tool to determine the correlation among parameters estimated, the ease of accurately estimating each of the parameters involved, and in turn, to find the parame ter that results in the smallest relative error ( Dolan & Mishra, 2013 ) . The sensitivity coefficient also provides insight about the magnitude of change with respect to response as a result of perturbations in the parameters ( B eck & Arnold, 1977 ; Dolan & Mishra, 2013 ) . In MATLAB®, a sensitivity coefficient matrix is constructed and known as the Jacobian matrix (Eq. 2 - 18). (Eq. 2 - 17) 70 (Eq. 2 - 18) However, for the purpose of comparing the parameters involved on the same scale, a scaled sensitivity c oefficient figure is often plotted. By using the finite forward difference method (Eq. 2 - 19), the scaled sensitivity coefficient can be numerically approximated (Eq. 2 - 20) ( Beck & Arnold, 1977 ) . The scaled sensitivity coefficient is obtained by multiplying the sensitivity coefficient with its respective parameter (Eq. 2 - 21) ( Dolan & Mishra, 2013 ) . (Eq. 2 - 19) (Eq. 2 - 20) (Eq. 2 - 21) 2.7.3 Parameter Estimation using Ordinary Least Squares ( OLS ) Parameter estimatio n can be performed by OLS using the non - linear regression (nlinfit) command in MATLAB®. Statistical assumptions need to be analyzed before data fitting. Statistical assumptions that need to be taken into account include (but are not limited to) ( Beck & Arnold, 1977 ) : 1. Errors are additive in the measurement 2. Errors in the measurement contain zero mean 3. The measurement errors have constant variance 71 4. The measurement errors are uncorrelated 5. Errors are normally, independently, identically distributed 6. Statistical parameters describing errors are known 7. Independent variables are errorless 8. Th e nature of the parameters (constant vs. random vector parameter; prior information vs. unknown statistics of the parameter) Regardless of the outcome, the results of how the model fit s and meets the aforementioned statistical assumptions should be reported, and based on that, further changes can be considered whenever necessary and applicable ( Do lan & Mishra, 2013 ) . 2.7.3.1 Standard Errors and Correlation Coefficient o f t he Parameters Standard errors of the parameters can be obtained by using OLS through a variance - covariance matrix (Eq. 2 - 22). In this context, the sensitivity coefficient or J acobian matrix, X , is also directly correlated with the determination of standard errors of the parameters. The standard error of an individual parameter, can be further divided by the parameter vector itself to obtain its relative standard error ( Mishra, Dolan, & Yang, 2008 ) . (Eq. 2 - 22) The correlation coefficient is an important tool to determine the correlation among the estimated parameters (Eq. 2 - 23). The val ue obtained is absolute, ranges from 0 to 1, and the closer the value is t o 1, the more highly correlated the parameters are. As a result, the parameters may be difficult to estimate accurately. The correlation coefficient matrix is also shown as follows (Eq. 2 - 24). (Eq. 2 - 23) 72 (Eq. 2 - 24) 2.7.4 Sequential Estimation In addition to the OLS method, sequential estimation can be used to estimate parameters of interest. This metho d was developed based on the Gauss minimization method by using the matrix inversion lemma ( Beck & Arnold, 1977 ) . This method requires iteration steps in the case of a non - linear model like in most migration models. It is more powerful than OLS in the sense that it may provide the duration required for an experimental study , and it is still able to estimate the parameters accurately. This method also updates the parameter whenever new responses are added. However, prior information is always required ( Beck & Arnold, 1977 ; Dolan & Mishra, 2013 ) . It is always good practice to compare the outcomes between OLS and sequential est imation. 2.7.5 Corrected Akaike Information Criterion (AICc) Mathematical models used for migration phenomen a contain parameters such as D and the h. Some researcher s, as discussed earlier, have started to focus on other relevant parameter s for migration such as the . Therefore, the idea of investigating more parameters is very crucial to correlate the physical meanings behind these parameters with respect to the migration experiment. However, the more parameters are estimated, the more uncertainty w ill be introduced, in turn, decreasing the accuracy of the estimation. Therefore, it is important to be able to select the right model containing a sufficient number of parameters to justify the estimation process. Often, the sums of square d errors (SSE) or the root mean square error (RMSE) is used to get an indication of a better model by selecting the model with the lowest corresponding SSE or 73 RMSE. This approach may be useful for comparing different models with a similar number of parameters. However, f or comparing models (nested or non - nested) with different number s of parameters, the use of SSE or RMSE will introduce bias since generally the more parameters estimated, the better the SSE or RMSE will be. Thus, in such cases, the corrected Akaike informa tion criterion (AICc) can be used. The AICc is the second order of the AIC. This approach penalizes additional parameters, thus elim inating the bias introduced by having more parameters. The AICc is recommended for case s involving small sample size s ( n ). However, it is applicable for all cases since with larger n , the correction term ( ) becomes trivial; thus the AICc is reduced to the AIC expression ( Motulsky & Christopoulos, 2004a) . (Eq. 2 - 25) where n =number of data; p =number of parameter s ; K=p+1 2.7.6 Bootstrap The b ootstrap method is a resampling approach to draw relevant information that represents the population. Bootstrap is beneficial when the error distribution is unknown ( Mishra, D olan, & Yang, 2011 ) . This method can provide accurate statistical inferences in cases when the number of data points is insufficient or the data are ill - posed ( Fox, 2015 ) . There are three main type s of bootstraps; i) parametric, ii) residuals, and iii) data. The parametric bootstrap is the strict one since it relies on how better is the model for some parameters. For this type of bootstrap, the model is first estimated and the simulation is done from the estimated model. The residuals bootstrap does not rely on the model and does not assume the residuals distribution ( Fox, 2015 ) . This method in particular is beneficial for a small data set and when the magnitude response of the parameter is 74 large at a certain data interval. Once the model is estimated, the residuals of the estimates are then simulated. The resampled residuals are then added to the fitted values, thus produci ng synthetic data. T he data bootstrap ignores the model and the data is then resampled from the data r ange, thus m aking this type of bootstrap the safest choice ( Anonymous , 2013 ) and produces widened confidence and prediction bandwidths. 2.7.7 Optimal Experimental Design By maximizing the determinant (Eq. 2 - 25), the optimal time to perform an experimental study can be determined. Optimal experimental design helps in finding the optimal point at which the parameters can be estimated and have lower errors. The C matrix is needed in order to achieve a des irable maximum determinant (Eq. 2 - 26). This approach is beneficial in terms of optimizing not only the resources used, but also the time spent for a given experiment ( Beck & Arnold, 1977 ; Dolan & Mishra, 2013 ) . (Eq. 2 - 26) (Eq. 2 - 27) 2.7.8 Activation Energy All of the kinetic reactions are temperature dependent. A well - recognized mathematical expression used to describe the dependency of kinetic reactions on temperat ure is the Arrhenius equation (Eq. 2 - 28). (Eq. 2 - 28) where k = rate constant; =frequency or pre - exponential factor; E a =activation energy; R=gas constant; T=temperature . 75 The Arrhenius equation is used to obtain information about the activation energy, E a . The E a is the rate of migration changes with temperature . The mathematical expression of the Arrhenius equation is known to complicate the estimation process due to the high correlation that is commonly found between the and the . As a result, the linearize d form of this equation is commonly applied to obtain the estimation of the . For such approach, the error structure of the is not known since it is not attained experimentally. Since this equation is a non - linear model, it s error structure is more complex than the error structure of a linear model, which the latter can be obtained from observational data ( Schwaab & Pinto, 2007 ; Watts, 1994 ) . Thus, to avoid the correlation issue between parameters and the risk of introducing more error to the estimation process, a reparameterization approach is recommended ( Agarwal & Brisk, 1985a , 1985b ; Schwaab, Lemos, & Pinto, 2008 ; Schwaab & Pinto, 2007 ) . This approach was first introduced by Box, 1960 and later Himmelblau, 1970 . The reparameterized form of the Arrhenius equation is as follows; (Eq. 2 - 29) where =specific reaction rate at ; =reference temperature. Numerous advantages can be gaine d from using t he reparameteriz ed Arrhenius equation. Among them are; i) the ease of the simultaneous estimation of parameters following the optimum T ref resulted in a lower correlation, thus minimized errors of the parameters ( Schwaab et al., 2008 ; Schwaab & Pinto, 2007 ) ; ii) the need for heavy computational work to achieve the minimization of the objective function is eliminated ( Espie & Macchietto, 1988 ) and iii) the improvement of the elliptical confidence region can be obtained ( Schwaab et al., 2008 ; Schwaab & Pinto, 2007 ; Watts, 1994 ) . 76 T he reparameterized form of the Arrhenius equation has been used in various applications such as microbial inactivation, starch gelatinization, in situ vib rational spectroscopy etc. ( Dolan, Valdramidis, & Mishra, 2013 ; Furusjö, Svensson, & Danielsson, 2003 ; Sulaiman, Dolan, & Mishra, 2013 ) . 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Packaging Technology and Science, 26 (1), 31 - 38. doi: 10.1002/pts.1964 92 Chapter 3 Poly(lactic acid) membrane incorporated with marigold flower extract ( Tagetes erecta ) intended for fatty - food application 3.0 Introduction A functional membrane used in food packaging can be described as a membrane or packaging system that provides a continuous active protection to a food product during its shelf life by the incorporation of active substances (food additives) within the membrane. Some examples of active functional membranes are oxygen scavenger, ethylene absorber, and antimicrobial and antioxidant packaging ( Rooney, 1995 ) . Among those, the information and research related to antioxidant packaging are considerably limited ( Camo, Beltrán, & Roncalés, 2008 ) - tocopherol, ascorbic a cid, butylated hyroxyanisole (BHA), and butylated hydroxytoluene (BHT), to name a few. Antioxidant packaging is the active system developed to control, for example, lipid oxidation in fatty food products ( Wessl ing, Nielsen, Leufvén, & Jägerstad, 1998 ) . Even though there is other existing technology like oxygen scavenger, that can help to retard oxidation, there are some concerns regarding their application. Oxygen scavenger is commonly incorporated into a pac kaging system in the form of a sachet. Its capacity to absorb oxygen is limited to 100 mL ( Smith, Hoshino, & Abe, 1995 ) , requiring multi ple sachets for use with a product that is designed for a long shelf life, which is not practical and commercially acceptable ( Robertson, 2006 ) . It also requires additional materials, which consumers do not perceive as good management of resources, besides the accidental ingestion of these compounds could happen. Although their ingestion does not cause adverse health impacts ( Floros, D ock, & Han, 1997 ) , it can further deter consumer from buying these products. 93 The addition of antioxidants into food during processing seems to provide another solution for lipid oxidation. However, generally, the amount of antioxidants permitted for us e in food products, singly or in combination, is limited to 0.02% by weight based on fat content of the food ( Miková, 2001 ) . This antioxidant amount may not be sufficient to protect the food due to possible loss of antioxidants during processing and may gradually diminish before the product (food) even reaches market shelves. The addition of relatively high concentration of certain antioxidants ( e.g., tocopherol and ascorbi c acid) into food systems could also result in pro - oxidation reactions in lipids, subjecting products to a short shelf life ( Balasubramanian, 2009 ) . Antioxidant functional membrane systems are a feasible alternative technology for such applications. This technology is beneficial for both the packaging and the packaged food because the incorporated antioxidants stabilize the polymer during processing an oxidation. However, most of the research conducted in this area focuses on non - renewable materials mainly made of polyolefins ( Gavara, Lagarón, & Catalá, 2004 ; Wessling et al., 1998 ) . Current consumer and environmental trends are increasing pressure to develop antioxidant functional membranes from bio - based polymers such as poly(lactic acid), PLA, poly (hydroxybutyrate - co - valerate), PHBV, soy - protein, starch - based polymers, to name a few, mainly due to driving factors such as price volatility of petrole um resources, replacement of non - renewable resources that may lead to the reduction of environmental burden and contribution towards a sustainable packaging industry with positive perceptions from consumers ( Robertson, 2006 ) . Among those bio - based materials, PLA (Figure 3 - 1), a bioplastic produced from the ring - opening polymerization of lactide ( Endres, Siebert, & Kaneva, 2007 ) has increasingly gained interest. PLA can be obtained from renewable resources, like corn ( R. Auras, B. Harte, & S. Selke, 94 2004a ) , sugar beet and biomass residues. It is a transparent polymer, which is good for food packaging applications. It can be formed into a variety of containers, trays, membranes, and other types of packaging structures. PLA is biodegradable, compostable and recyclable ( Auras et al., 2004a ) , and it has been approved by the Food Drug Administration (FDA) for intended use in fabricated materials for food - contact applications ( Conn et al., 1995 ; Mutsuga, Kawamura, & Tanamoto, 2008 ; Ortiz - Vazquez, Shin, S oto - Valdez, & Auras, 2011 ; Soto - Valdez, 2011 ) . PLA is comparable to poly(ethylene terepthalate), PET, and polystyrene, PS, in terms of its physical and mechanical properties. However, its lower barrier property to gases such as carbon dioxide (CO 2 ) and oxygen (O 2 ) than PET imposes limited fu nction for food packaging application. For example, fatty food products packaged in PLA material would experience lipid oxidation as a result of its low oxygen barrier property. For this reason, the incorporation of antioxidants from natural sources like a and a preservation tool for targeted food systems. Figure 3 - 1 Poly(lactic acid), PLA chemical structure. Astaxanthin is a keto - carotenoi d that can be found naturally ranging from crustacean products and by - products, plants, and yeast (Figure 3 - 2). Astaxanthin is commonly fed to crustacean, salmonids, and farmed fish to enhance their color appearance as well as to provide healthy growth and reproduction systems ( Higuera - Ciapara, Félix - Valenzuela, & Goycoolea, 2006 ) . Astaxanthin has been reported to have more antioxidant capacity in comparison to lutein, - carotene, and lycopene ( Naguib, 2000 ) . Quenching of singlet O 2 and inhibition of lipid oxidation 95 are the most noticeable antioxidant activities of astaxanthin. In this study, marigold flower extract was used directly as the astaxanthin source. The utilization of this extract was preferred due to its accessibility (abundance in nature) and economical value (cheaper than the commercial there is scarce information about PLA membranes incorporated with carotenoid - based antioxidants preferably from natural sources. Thus, the objectives of this study were: 1) to fabricate a bilayer bio - based membrane made of PLA and incorporated with marigo ld flower extract, 2) to study the migration of astaxanthin from the fabricated membrane towards a fatty food simulant, 3) to determine the thermal, molecular, surface, and barrier properties of the produced functional membrane, and 4) to assess the effect iveness of the fabricated membrane in the oxidative stability of soybean oil during storage under accelerated condition. Figure 3 - 2 Astaxanthin chemical structure. 3.1 Materials and Methods 3.1.1 Materials PLA 4043D (94% L - lactic acid content) was bought from PromaPlast (Monterrey, México) with a weight average molecular weight ( M w ) of 120,980 ± 2,425 Da and number average molecular weight ( M n ) of 81,375 ± 955 Da. Marigold flower extract (Florafil 93 TM ) with an oil - 96 based consistency of fatty acid esters containing 0.7 wt.% astaxanthin in a water emulsion was donated by Vepinsa (Industrias Vepinsa, S.A. de C.V., Los Mochis, México). The amount of astaxanthin in the marigold flower extract was quantified by using a Hi gh Performance Liquid Chromatography (HPLC) equipment equipped with a UV - DAD detector at 474 nm . Astaxanthin standard was obtained from Alexis Biochemicals with a purity of 97 wt.% (San Diego, CA, USA). Additive - free soybean oil was provided by Industriali sulfate (SDS) and potato starch were purchased from J.T. Baker (New Jersey, USA). Sodium thiosulfate solution (0.1 N) wa s provided by Golden Bell (México City, México), and potassium iodide A.C.S reagent was obtained from Fermont (Monterrey, México). Ethanol 99.9% (HPLC grade), methanol (HPLC grade), and tert - methyl butyl ether (HPLC grade), water (HPLC grade), and formic a cid (A.C.S reagent grade) were obtained from Sigma - Aldrich (MO, USA). 3.1.2 Fabrication of Antioxidant Functional Membrane PLA resins were dried at 40 °C for 8 h in a vacuum oven and was extruded to produce bilayer membranes without (PLA), and with 2 nomi nal wt.% (PLA2M) marigold flower extract in the inner layer by using a pilot plant size blow - extrusion machine (Beutelespacher, México) at the Centro de Investigación en Alimentación y Desarrollo, A.C. campus, Hermosillo, México. The L/D ratio of both extr uders is 26:1 with a screw diameter of 2 cm. Marigold flower extract in a form of liquid was mixed directly with the PLA resin on the weight basis and this mixture was then introduced into the hopper of the extruder 1 for membrane processing. The temperatu res of extruder 1 for zone 1, 2, 3 and 4 were 140, 150, 150, and 150 °C, respectively, with an extruder velocity of 23 rpm. Meanwhile the temperatures set for extruder 2 were the same in all zones but 97 3, of which was 155 °C, with an extruder velocity of 35 rpm. The total thickness of the produced 3.1.3 Quantification of Astaxanthin in the Fabricated Functional Membrane after Processing Pieces of the membranes weighing approximately 0.5 g (0.25×0.25 cm) were ex tracted with methanol with constant stirring in the darkness at 40 °C for a designated period of time (24, 48, 72 h). A fourth extraction after 96 h was performed to ensure a complete extraction. The ormance Liquid Chromatography (HPLC) equipment equipped with a UV - DAD detector at 474 nm. The antioxidants quantification was performed by using an YMC Carotenoid 3 µm 2.0x250 mm column (Waters Corporation, Milford, Massachusetts, US) with a gradient mode mobile phase of methanol (81%): tert - butyl methyl ether (15%): acidified water (4%), and a standard flow of 0.3 mL/min. An external calibration curve was prepared by using astaxanthin standard (97%) diluted with methanol in the range of 0.01 to 1 h a R 2 0.9905. Limit of quantification (LOQ) was determined and was found to be 3.1.4 Migration of Astaxanthin into A Food Simulant Migration of astaxanthin was studied in an accordance with ASTM 4754 - 98 with some modification ( ASTM, 2003a ) . Th e produced membranes were used to make a 4 x 4 cm pouch with the active layer outside. The pouch was then inserted into a 40 mL screw cap amber glass vial containing 30 mL of 95% ethanol (ETOH) (volume - area ratio of 0.9375 mL/cm 2 ) as fatty food simulant, a nd then was kept at 30 ± 2 °C, and 40 ± 0.5 °C for a designated period of time until the equilibrium was reached. The quantification of astaxanthin in the simulants was periodically 98 determined as previously described. 3.1.4.1 Mathematical Models f or Migration Study The migration process involves the transfer of substances ( e.g., antioxidant, monomer, etc.) originated from the packaging membrane into a packaged product, and the diffusion coefficient is second law. The following assumptions were taken into account to describe the migration process of this study: 1) the antioxidant (astaxanthin) was distributed well and uniformly in the PLA antioxidant membrane layer; 2) the migration only occurs from one side of membrane layer that was in contact with the simulant; 3) the simulant was well - mixed; 4) the migration process was only affected by the temperature where the partition coefficient ( K p,f ) and diffusion coefficient ( D ) were constants during experimental period; and 5) the interaction between the PLA membrane and the simulant was negligible. The analytical solution for a finite volume of food simulant and negligible mass transfer coefficient for the membrane thickness of 2L can be described as follow: (Eq.3 - 1) where M t is the concentration of migrant migrating into the food simulant at time t ; M is the concentration of the migrant migrating into food simulant at equilibrium; is the non - zero positive roots of tan where ; V f is the volume of food; V p is the volume of membrane; K p,f is the partition coefficient where where C is the concentration of the migrant in the membrane at equilibrium ; and C is the concentration of migrant in the food simulant at equilibrium. D is the diffusion coefficient of the migrant; is the thickness of the 99 membrane containing the antioxidant ( Baner, 2000 ; Crank, 1979 ; De Meulenaer, 2009 ; Hamdani, Feigenbaum, & Vergnaud, 1997 ) . In a case where the food simulant is considered to have a larger volume than the polymer resulting in negligible mass transfer coefficient for the polymer thickness of 2L, and (when V f >> V p and/or K p,f <1), the following simplified solution can be applied to determine the diffusion D ): (Eq. 3 - 2) MATLAB R2011b (MathWorks, Na tick, MA, USA) was utilized to solve for equations 1 & 2 for the two parameters, D and M , by using the non - linear regression function ( Colín - Chávez, Soto - Valdez, Peralta, Lizardi - Mendoza, & Balandrán - Quintana, 2012 ; Dhoot, Auras, Rubino, Dolan, & Soto - Valdez, 2009 ; - Franco et al., 2012 ) . 3.1.5 Thermal Properties The glass transition temperature ( T g ), cold - crystallization - temperature ( T cc ), melting temperature ( T m ), enthalpies of cold crystallization ( H cc ) and melting ( H m ), and the degree of crystallinity ( X c ) of the produced functional membranes were characterized in accordance with ASTM 3418 - 03 ( ASTM, 2003b ) by using a differential scanning calorimeter (DSC) Q100 (New Castle, DE). Samples of about 6 8 mg were heated up from - 10 to 200 °C at 10 °C/min with a N 2 flow of 50 mL/min with heat - cool - heat cycle. The data obtained was then analyzed by using the Thermal Analysis Universal 2000 version 4.5A software. The degree of crystallinity ( X c ) of PLA and PLA2M was calculated as follow: % Crystallinity, (Eq. 3 - 3) 100 where f * = heat of fusion of 100% crystalline sample (PLA= 93.7 J/g); x is the amount of antioxidant in the membrane. The decomposition temperature ( T d ) of the fabricated samples (approximately 6 - 10 mg) was determined by using a thermogravimetric analyzer (TGA Q50, TA Instruments, New Castle, DE). Temperature was set to start at 0 °C and heated up to 700 °C with a rate of 10 °C/min in a presence of air of a 60 mL/min flow rate to simulate oxidation condition. The analysis was carried out in accordance with ASTM E 1131 - 08 ( ASTM, 2008 ) . 3.1.6 Number Average Molecular Weight ( M n ) and Weight Average Molecular Weight ( M w ) Fabricated samples of about 20 mg were weighted and transferred into a 10 mL volumetric flask. These sa mples were then dissolved with tetrahydrofuran (THF). The solvent containing the dissolved sample was then filtered through polytetrafluoroethylene (PTFE) filter membrane, and the filtrate was then transferred into a 2 mL glass vial with a PTFE septum. The n, M n , M w and the polydispersity index ( PI ) were determined by using a Water Gel Permeation Chromatography (GPC) (Waters 1515, Waters, Milford MA, USA) equipped with a Refractive Index detector (Waters 2414) with a flow rate of 1 mL/min. The columns used were HR Styragel® HR4, HR3, HR2 (300 mm x 7.8 mm (I.D)) with a temperature of 35 °C. The universal calibration curve (R 2 =0.9983) was prepared by using a polystyrene standard with a range of molecular weights of 1.20 x 10 3 to 3.64 x 10 6 Da. The Mark - Houwink - Sakurada equation ( was employed with mL/g to find the correlation between eluted volume of intrinsic viscosity of polymer and the absolute M w with THF as a particular solvent. 101 3.1.7 Scanning Electron Microscopy ( SEM ) P LA and PLA2M were investigated for their corresponding surface changes before and after being in contact with 95% ETOH at 30 and 40 °C for 24 and 3 d, respectively by using a Scanning Electron Microscope (SEM), model JEOL JSM 6610LV (JEOL Inc., MA, USA) wi th an accelerating voltage of 10keV. Samples were dried prior to analysis, and were coated with gold sputter coater by an EMSCOPE SC500 Sputter Coater (Kent, Britain). 3.1.8 Oxygen ( O 2 ), Water Vapor, and Carbon Dioxide ( CO 2 ) Barrier Properties The sample for each barrier test was prepared with a masking aluminum to obtain the exposure area of 3.14 cm 2 . O 2 permeability was determined using an Illinois Oxygen Permeation Analyzer 8001 (Illinois Instruments Inc., Johnsburg, IL) in accordance with ASTM D3985 - 0 5 ( ASTM, 2005a ) . The testing temperature was 23 °C, 0% relative humidity (RH), and 21% permeant concentration with a carrier gas of 2% H 2 : 98% N 2 . Water vapor permeability (WVP) of the sample was performed in accordance with ASTM F1249 - 06 by using a MoCON Permatran W3/33 (MOCON Inc., Minneapolis, MN) ( ASTM, 2006 ) . The temperature and RH applied during this test were 38.7 °C and 100%, respectively, with N 2 as a carrier gas. Carbon dioxide (CO 2 ) permeabi lity was done in accordance to ASTM F2476 - 05 by using a MOCON Permatran 4/41 Module C (MOCON Inc., Minneapolis, MN) ( ASTM, 2005b ) . The testing conditions were 23 °C, 0% RH, and a permeant concentration of 100%. N 2 was used as a c arrier gas. All tests were run continuously until steady state was achieved with less than 5% variation for at least the last 10 data points. Three replicates were run per sample. The gases and water vapor permeabilities were calculated as follow: Gases an d water permeability= (Eq. 3 - 4) 102 where TR = Transmission rate of each gas or vapor (kg); l = sample thickness (m); A = area of exposure (m 2 ); t =time (s); p =partial pressure (Pa). 3.1.9 Optical Properties The light transmission of the PLA and PLA2M was performed by using a Perkin - Elmer Lambda 25 UV - Vis spectrophotometer (Waltham, MA, USA) with an integrating reflectance spectroscopy accessory (RSA - E - 20, Labsphere, North Sulton, NH, USA). The measurements we re carried out at a wavelength range of 200 to 800 nm in transmittance (%) mode with a rate of 240 nm/min. Samples were measured in triplicate. Color measurements of PLA and PLA2M was done by a LabScan XE (HunterLab, Reston, VA, USA) and the L* , a* , b* val ues were analyzed by Easymatch QC version 3.8. Samples were samples using Eq. 5: (Eq. 3 - 5) where L = L PLA2M - L PLA , a = a PLA2M - a PLA , and b = b PLA2M - b PLA . 3.1.10 Fourier Transform Infrared Spectrophotometer ( FTIR ) Fabricated membranes were scanned to assess their chemical structural properties by using a Fourier transform infrared spectrophotometer (Shimadzu IR Pres tige - 21, Shimadzu Scientific Instruments, Columbia, MD) with an attachment of attenuated total reflectance (ATR) set from 4000 to 650 cm - 1 wavenumber with 40 scans at 4 cm - 1 . The main and additional functional group absorption bands were identified and obs erved for any optical changes. 103 3.1.11 Oxidative Stability of Soybean Oil Sample preparation: Deodorized soybean oil was added into 40 mL glass bottles (control) and pouches made of those fabricated membranes (6 x 4 cm) in the quantity of 15 mL for each package. Pouches formed resulting in a total contact area of 48 cm 2 , in turns, volume - area rati o of 0.3125 mL/cm 2 . These pouches were then kept at 30 ± 2 °C with exposure to fluorescence light (900 - 1000 lux (Luxometer SP 840020, Neurtek Instruments, Guizpucoa, Spain)) for 25 days of storage. The samples were analyzed for peroxide values (PV) at day 0, 3, 5, 7, 10, 15, and 25. While performing the test, the pouches were left sat still and the measurement was performed in triplicate. Peroxide values (PV): Determination of peroxide values was conducted in accordance with ciety (AOCS) official methods Cd - 8b - 90: Peroxide value acetic acid: isooctane method - Reapproved 2009 ( AOCS, 2009 ) . Approximately, 2.50 ± 0.01 g of each sample was weighed and kept under refrigerated condition until analyzed. The sample was first dissolved with 50 mL acetic acid: isooctane (3:2 (v/v)). A 0.5 mL of potassium iodide solution was added into the sample - acetic acid: isooctane solution and was continuously shaken for exactly 1 min, then 30 mL of distilled water was immediately added. This solution was then titrated gradually with 0.01 M sodium thiosulfate with constant and vigorous agitation until the color of the solution faded to light yellow. This solution was then added with 0.5 mL of 10% sodium dodecyl sulphate SDS (w/v) followed by 0.5 mL of starch indicator solution before further titrat ion. When close to the end point, drops of the thiosulfate solution were added until the dark brown color just disappeared. The volume of thiosulfate solution used for each titration was then recorded for PV calculation. A blank determination of the reage nts was also conducted in each 104 analyzed day. PV for each sample was expressed in milliequivalents peroxide/1000 g test portion unit and was calculated as follows: (Eq. 3 - 6) where S is the volume of tit rant of test portion (mL); B is the volume of titrant of blank (mL); and M is the molarity of sodium thiosulfate solution. 3.2 Statistical Analysis The data was analyzed by using one - way analysis of variance (ANOVA), and post - hoc pairwise comparisons were conducted by using Tukey - Kramer test with 95% level of confidence SAS (Version 9.1, SAS Institute Inc., Cary, NC, US). 3.3 Results and Discussions 3.3.1 Qu antification of astaxanthin in the fabricated membrane after processing PLA2M originally contained 1.49 wt.% extract. From the extraction, it was found that losi ng 82% due to processing. Antioxidant loss during processing was expected due to the nature of astaxanthin which is susceptible towards degradation with a decomposition temperature of approximately 200 °C as reported by Guo, Jones, & Ulrich (2010) ( Guo, Jones, & Ulrich, 2010 ) , and measured by TGA (Figure not shown). Even so, ast axanthin in oil - based extract was reported to have a considerable stability at room temperature ( Rao, Sarada, & Ravishankar, 2007 ) . Therefore, it can be expected that astaxanthin membrane are stable during typical packaging membrane storage conditions. Other reported studies have also shown that it is not uncommon to 105 lose antioxidant during processing. Butylated hydroxytoluene (BHT) incorporated into low density poly(ethylene), LDPE experienced losses from as low as 1.5% up to 58% ( Galindo - Arcega, 2004 ; Soto - Cantú et a l., 2008 ; Wessling et al., 1998 ) . Lopez - de - Dicastillo et al. (2010) reported that 1 and 5 wt.% quercetin added into ethylene vinyl alcohol, EVOH, polymer experienced a loss of 19.9, and 24.4%, respectively, as well as 0.5, and 2 wt.% catechin added to that of EVOH had lost 32.9, and 33.2%, respectively ( 2010 ) - tocopherol, resveratrol, catechin, and epicatechin added to PLA has been reported in the range of 15 to 30% during processing. These losses were function of the extrusion process, processing temper ature, residence time of PLA in the extruder, and the concentration of the antioxidants used ( - Franco et al., 2012 ; Manzanarez - López, Soto - Valdez, Auras, & Peralta, 2011 ; Soto - Valdez, Auras, & Peralta, 2010 ) . Colín - Chávez et al. (2012) used marigold flower extract as an antioxidant in LDPE, and co - extruded LDPE/HDPE membranes processed at 130, and 150 °C (temperature at which the molten polymer h ad entered the die), respectively, and they reported approximately 63 - 79% losses of astaxanthin with significant losses for the coextruded LDPE/HDPE than that of monolayer LDPE due to the higher processing temperature ( Colín - Chávez et al., 2012 ) . Anderson & Sunderland (2002) reported that astaxanthin have a tendency to degrade in the presence of moisture during processing of extruded fish feed ( Anderson & Sunderland, 2002 ) . The amount of water content in PLA was below 0.02%, of which it could be speculated that during polymer processing the surrounding moisture level might as well contribute tow ards the loss of astaxanthin. Antioxidants are utilized to protect polymer degradation as a result of oxidation during its processing ( Al - Malaika, Goodwin, Issenhuth, & Burdick, 1999 ; Byun, Kim, & Whiteside, 106 2010 ; Wessling et al., 1998 ) . However, in this case, the loss of astaxanthin during processing could not be accounted for its mechanism to protect PLA from oxidation. 3.3.2 Migration of Astaxanthin into A Food Simulant (95% ETOH ) The migration of astaxanthin into 95% ETOH at both 3 0 and 40 °C followed the second - 3). The partition coefficient, K p,f was calculated based on the ratio of the concentration of astaxanthin left in the PLA and the concentration of the astaxanthin migrated into 95% ETOH at equilibrium. The K p,f were found to be 61.75 and 18.7 at 30 and 40 °C, respectively (Table 3 - 1). A reduction of K p,f with temperature was expected since the solubility of the antioxidant in the food simulant increases as the temperature increases ( Brandsch, Mercea, R üter, Tosa, & Piringer, 2002 ) . Astaxanthin was released gradually into 95% ETOH before finally reached equilibrium at 8 d at 30 °C. Meanwhile, the release rate of astaxanthin at 40 °C reached equilibrium at 3 d (Figure 3 - 3). The diffusion coefficients were found to be 12.7 ± 4.1 × 10 - 11 cm 2 /s (Figure 3 - 4a), and 22.8 ± 4.7 × 10 - 11 cm 2 /s at 30 and 40 °C (Figure 3 - increment of temperature of 10 °C enhances the rate of diffusion by 2 to 3 fold. Likewise, as the rate of the diffusion increased, the intermolecular interaction between ethanol polymeric chains was also enhanced ( Lassalle & Ferreira, 2007 ; Peltonen, Koistinen, Karjalainen, Häkkinen, & Hirvonen, 2002 ) . The sorption of ethanol by the polymer was conjectured to behave as a plasticizer by increasing the segmental mobility of the polymer chains ( Mascheroni, Guillard, Nalin, Mora, & Piergiovanni, 2010 ) , thus creating void that eventually diffuse the migrating compound as in this case astaxanthin into ethanol. Similar behavior was also reported by other studies with PLA - based functional - 107 tocopherol, resveratrol, catechin, epicatechin, and BHT into ethanol - based simulants ( - Franco et al., 2012 ; Manzanarez - López et al., 2011 ; Ortiz - Vazquez et al., 2011 ; Soto - Valdez et al., 2010 ) . However, it was found that the rate of astaxanthin released into 95% ETOH as a function of time was higher than the previously mentioned antioxidants at 30 °C (12.7 × 10 - 11 (astaxanthin) vs. 5.29 × 10 - 11 ( - tocopherol), 8.95 × 10 - 11 ( BHT), 22.6 - 41.7 × 10 - 11 ( 1 - 3% resveratrol), 13.1×10 - 11 (catechin), and 13.7 × 10 - 11 (epicatechin ) cm 2 /s ). Meanwhile, at 40 °C, similar trend was also observed with the exception for the release of BHT that was higher than that of astaxanthin (22.8 × 10 - 11 (astaxanthin) vs. 38 × 10 - 11 ( - tocopherol), 190.4 × 10 - 11 ( BHT), 85.1 - 82.6 × 10 - 11 ( 1 - 3% resveratrol), 47.9 ×10 - 11 (catechin), and 51.2 × 10 - 11 (epicatechin ) cm 2 /s ). In general, the diffusivity of BHT is expected to be fast due to its non - bulky structure with only one hydroxyl group as can be observed in the case of migration at 40 °C. - Franco et al. (2012) speculated that the presence of more number of hydroxyl group in the antioxidant structure may be the determinant factor that had caused the lower release rate of antioxidant into ethanol since more interaction could occur be tween PLA polymeric chains and the incorporated antioxidants. Since astaxanthin chemical structure contains two hydroxyl groups (Figure 3 - 2), it is expected to have higher release rate that that of resveratrol, catechin and epicatechin in which they consis ts of three and five - tocopherol and BHT consist of one hydroxyl group, - tocopherol was 4 to 5 magnitude lower than astaxanthin, which may be associated with its longer alkane chain with a m ethylated phenolic group. On the other hand, the release of astaxanthin from monolayer LDPE and bilayer LDPE/HDPE was reported lower than the release of astaxanthin obtained in this study. Colín - Chávez et al. (2013) reported the D value of 7 ×10 - 11 cm 2 /s of astaxanthin from monolayer LDPE membrane into ethanol at 30 °C. Meanwhile, a D value of 6 ×10 - 11 cm 2 /s was reported for bilayer 108 LDPE/HDPE at 30 °C in the same study ( Co lín - Chávez, Soto - Valdez, Peralta, Lizardi - Mendoza, & Balandrán - Quintana, 2013 ) . The large release in PLA and ethanol can be attributed to the modification of the membrane due the presence of ethanol, which disrupt the microstructure of PLA, and its further discussed in the next sections. In addition, it is worth mentioning that no degradation products of astaxanthin ( i.e., 9 - cis and 13 - cis ) was detected during the migration test by HPLC. Table 3 - 1 Migration data of prod uced function al membranes. Temperature (°C) *** 30 40 K p,f * 61.75 ± 7.36 a 18.7 ± 7.17 b * 1.33 ± 0.15 a 4.92 ± 2.36 b D × 10 - 11 (cm 2 /s)** 12.7 ± 4.1 a 22.8 ± 4.7 b D, Relative error 0.33 0.21 95% CI ×10 - 9 0.04, 0.21 0.13, 0.32 M inf/Predicted × 10 - 8 (g Astaxanthin/ g ETOH) ** 9.89 ± 0.64 a 10.12 ± 0.63 b M inf/Predicted, Relative error 0.06 0.06 95%CI × 10 - 8 8.61, 11.18 8.86, 11.38 M inf/Experimental × 10 - 8 (g Astaxanthin/ g ETOH)** 18.86 ± 2.35 52.16 ± 5.58 b RMSE × 10 - 8 (g Astaxanthin/ g ETOH) 2.19 1.43 Correlation coefficient, 0.53 0.76 *The values are reported as mean ± standard deviation. ** The values are reported as mean ± standard error. *** Values in the same row with same alphabetic symbol are not statistically significantly different (p>0.05). 109 M inf/Experimental = M total - M membrane extracted,inf ; 95% CI is reported as asymptotic; RMSE= Root mean square error. **** All measurements were performed in triplicate. Figure 3 - 3 The concentration of astaxanthin migrated into 95% ETOH at 30 to 40 °C. 110 Figure 3 - 4 (a) Migration of astaxanthin into 95% ETOH at 30 °C and (b) 40 °C during storage. 111 3.3.3 Thermal Properties Table 3 - 2 shows the T g , T m , and X c between PLA and PLA2M. The addition of approximately 2 wt.% of marigold flower extract did not affect the T g , T m , and X c . S imilar observation was also reported by other studies ( Manzanarez - López et al., 2011 ; Ort iz - Vazquez et al., 2011 ; Soto - Valdez et al., 2010 ) . However, it is worth mentioning that a decrease of 1 °C in T g was observed for all these systems. On the other hand, a slight and significant decrease in the T g was reported for PLA - - tocopherol, ascorbyl palmitate, BHT, or tert - butyl - hydroquinone (TBHQ) by other studies ( Goncalves et al., 2012 ; Jamshidian, Tehrany, Imran, et al., 2012 ) . These antioxidants were found responsible for a plasticization effect, thus resulted in the reduced T g . A reduction of 3 - 5 °C was observed for PLA - based membrane - tocopherol when the antioxidants were introduced at a vari ous combination of high concentration (1 - 4 wt.%) ( Hwang et al., 2012 ) - tocopherol, BHT, and TBHQ was reported to show no impact in the T m of PLA - based membrane ( Goncalves et al., 2012 ; Jamshidian, Tehrany, Imran, et al., 2012 ; Manzanarez - López et al., 2011 ; Ortiz - Vazquez et al., 2011 ) , similar to this stud y. However, the incorporation of antioxidants ( i.e., - tocopherol, BHT, TBHQ, and resveratrol) at a higher concentration (> 3 wt.%) was reported to induce lower T m , which may possibly be due the greater plasticization effect ( Goncalves et al., 2012 ; Hwang et al., 2012 ) , as the antioxidants cause a h indrance in the formation of crystallites that resulted in less energy required for melting the crystallites. In addition, a study on the effect of different plasticizers ( i.e., glycerol, citrate ester, polyethylene glycol (PEG), PEG - monolaurate, and oligo meric lactic acid) on PLA found that the presence of those plastizers in the polymeric system did reduced the T m of this polymer by 10 - 15 °C; however, the T m reduction was not greatly affected by the plasticizer 112 concentration as it was observed in the T g ( Martin & Averous, 2001 ) . No effect of marigold flower extract was also observed on the X c of PLA2M. The difference effect of antioxidants on the X c of PLA - based membrane was also explained in other studies ( Goncalves et al., 2012 ; Hwang et al., 2012 ; Jamshidian, Tehrany, Cleymand, et al., 2012 ; Sawalha, Schroën, & Boom, 2010 ; Soto - Valdez et al., 2010 ) . Table 3 - 2 shows also detailed information about T g , T m , and %X c of the PLA - based functional membranes before and after contact with 9 5% ETOH at 30 °C (24 d), and 40 °C (3 d). The T g of PLA that was originally 57.8 °C reduced to 50.5 °C, and increased to 73.6 °C after being in contact with 95% ETOH at 30 and 40 °C, respectively. As for PLA2M, similar behavior was also demonstrated at bot h respective temperatures under similar condition. In terms of the T m , it is worth noticing that two melting peaks appeared (Figure 3 - 5). This behavior was explained by Sato et al. (2012) from the differences type of crystalline formation due to solvent in duced crystallization. This type of behavior was classified due to the degree of cloudiness produced by the solvent with 7 - 20% swelling. This type of behavior was observed and associated with PLA membrane that had been in contact with ethanol, methanol, 1 - propanol, 2 - propanol, butanol, and 3 - methyl - 1 - butanol di - n - butylphthalate isopropyl ether ( Sato, Gondo, Wada, Kanehashi, & Nagai, 2012 ) . Even though two T m was seen in this study, only the distinctive peak data was reported, and only slight changes were observed for both PLA and PLA2M t hat were in contact with 95% ETOH at 30, and 40 °C compared to that of counterpart membranes that were not in contact with 95%ETOH. For X c , a significant increment was demonstrated comparing before and after both membranes had been in contact with 95% ETOH at both temperatures (p<0.0001); however, PLA2M exhibited significant increase in X c than that of PLA (p<0.0001), confirming the solvent induced crystallization with ETOH. This effect can be clearly observed in Figure 3 - 5, of which the 113 cold crystallizatio n diminished completely for both PLA and PLA2M after had been in contact with 95% ETOH. Similar phenomena was also observed by Chen et al. (2013), and was reported to be associated with the reduction of locally ordered structured as a result of hydrolytic degradation ( Chen et al., 2013 ) . It can also be seen that the p resence of antioxidant did further induce the increment of X c with the exposure to 95% ETOH at both temperatures. This circumstance could possibly be attributed to the presence of more - OH groups in the PLA2M membrane; thus introducing more intermolecular forces that consequently causing a greater increase in the X c . This result has important regulatory implications for testing migration of fatty foods in PLA membranes using ETOH solution as simulant. Thus, a new simulant should be designed for testing migr ation in fatty foods for PLA membranes. Both PLA and PLA2M exhibited no significant difference in their respective T d (p>0.05). Some antioxidants like - tocopherol and resveratrol showed significant enhancement in the thermal stability of PLA - based membrane, in which these antioxidants extended the temperature at which the mass loss of the sample can be observed ( Hwang e t al., 2012 ) . However, an opposite behavior was observed for PLA - based membrane incorporated with BHT where the T d of this membrane was reduced in comparison to the control PLA - based membrane ( Ortiz - Vazquez et al., 2011 ) . This phenomenon was reported to be as a result of the degradation product during processing ( Ortiz - Vazquez et al., 2011 ) . 114 Figure 3 - 5 DSC thermogram of (a) PLA2M in contact with 95% ETOH at 40°C for 3 d; (b) PLA in contact with 95% ETOH at 40°C for 3 d; (c) PLA2M in contact with 95% ETOH at 30°C for 24 d; (d) PLA in contact with 95% ETOH at 30°C for 24 d; (e) PLA2M; and (f) PLA. 3.3.4 Number Average Molecular Weight ( M n ), and Weight Average Molecular Weight ( M w ) In comparison with PLA resin, significant reduction in the M n and M w , was observed for both PLA and PLA2M of which could be attributed to the ther mal, and possible hydrolytic degradation of PLA during processing (p<0.0001). Despite the fact that the PLA resin was truly dried prior to processing (moisture content: <0.02 wt.%), the presence of heat and residual moisture from the surrounding could rein troduce moisture during processing causing the chain scission of the polymer (Table 3 - 2). PLA2M experienced a slight decrease in its M n and M w than 115 that of PLA. The incorporation of the marigold flower extract seemed to induce the membrane degradation. It could be speculated that the presence of the extract containing astaxanthin enhanced occurred at the OH chain ends. This phenomenon is a non - radical process that degrades the chain into a lactide molecule, an oligomeric ring, or acetaldehyde with carbon monoxide accordingly to the site in the backbone at which the reactions take place ( Lim, Auras, & Rubino, 2008 ; McNeill & Leiper, 1985 ) . Even though the occurrence of this phenomenon was reported at an excessive temperature (>270 °C) ( McNeill & Leiper, 1985 ) , and the maximum processing temperature used in this study was only at 155 °C, the presence of additional OH groups from astaxanthin could be conjectured to cause the chain degradation that finally resulted in the decreased of M w and M n . On the contrary, O rtiz - Vazquez et al. (2011 ) reported no significant impact on M w and M n of adding the pure antioxidant BHT in a PLA - based membrane ( Ortiz - Vazquez et al., 2011 ) . Meanwhile, Hwang et al. (2011) found that the M w and M n of PLA - based membrane incorporated - tocopherol and resveratrol increased gradually with increasing concen tration of these antioxidants ( e.g. , estimated relative change of 25%, and 11% of M w , and M n, , respectively by - tocopherol). This behavior was speculated to be due to the interaction that could have occurred between PLA, and ant ioxidant chains in the amorphous regions, thus increased the chain entanglements ( Hwang et al., 2012 ) . Physical crosslinking between PLA, and antioxidant chains was also conjectured as one of the reasons that resulted in the higher M w and M n ( Hwang et al., 2012 ) . The PI changed for PLA and PLA2M as it did for the studies by Ortiz - Vazquez et al. (2011 ) and Hwang et al. (2011) ( Hwang et al., 2012 ; Ortiz - Vazquez et al., 2011 ) . 116 Figure 3 - 6 shows molecular weight distribution o f the fabricated functional membranes before and after had been in contact with 95% ETOH. It was observed that M w of both PLA and PLA2M had reduced approximately by 67% and 69%, respectively (in comparison with their respective M w after processing) after had been in contact with 95% ETOH at 30 and 40 °C during the experimental period (p<0.0001). The significant shift toward lower molecular weight for those membranes that had been in contact with 95% ETOH suggested the possibility o f the chain scission phenomena. Meanwhile, M n for both produced functional membranes indicated a decrease of approximately 69 - 73% under similar condition (p<0.0001). There was neither pronounced effect of the temperature seen, nor the contact time (Table 3 - 2). In addition, it was observed that both PLA, and PLA2M had greater reduction in M w compared to those of PLA - based membranes incorporated with antioxidants ( i.e., BHT, resveratrol, - tocopherol, catechin, and epicatechin) after had been in contact with 95% ETOH at similarly reported temperatures (30 and 40 °C) ( - Franco et al., 2012 ; Manzanarez - López et al., 2011 ; Ortiz - Vazquez et al., 2011 ; Soto - Valdez , Peralta, & Auras, 20 08 ) . In this study, it was found that the M w of both membranes decrease by 3 orders of magnitude from its original M w (after processing). In the other afore mentioned studies, the decreased observed in the M w of their membranes was not larger than 1 order of magnitude, regardless of the temperatures and contact times with ethanol. It could be speculated that the polymer processing method bilayer blown extrus ion - and the temperature conditions used in this study may be responsible for inducing these significant changes in the morphologies of the PLA surfaces. It can also be concluded that 95% ETOH is not a good food simulant for studying fatty foods migration in PLA membranes. 117 Figure 3 - 6 Molecular weight distributions of fabricated functional membranes PLA and PLA2M without and in contact with 95% ETOH at 30 and 40 °C for 24 and 3 d, respectively. 118 Table 3 - 2 Ch aracterization of the fabricated functional membranes. Properties After processing After in contact with 95%ETOH 30 °C (24 d) After in contact with 95% ETOH 40 °C (8 d) Thermal PLA PLA 2M PLA PLA 2M PLA PLA 2M T g ,°C 56.8 ± 0.1 a1 56.9 ± 0.3 a1 50.5 ± 1.9 a2 51.7 ± 3.8 a2 73.6 ± 1.7 a3 75.5 ± 0.4 a3 T m ,°C 151.05 ± 0.1 a1 150.5 ± 0.6 a1 151.3 ± 0.1 a1,2 151.5 ± 0.5 a1,2 152.1 ± 0.0 a2 151.8 ± 0.4 a2 X c ,% 0.52 ± 0.3 a1 1.3 ± 0.4 a1 25.5 ± 0.7 a2 38.2 ± 1.5 b2 23.7 ± 0.8 a2 35.8 ± 1.02 b2 T d ,°C 355.9 ± 0.4 a 354.7 ± 0.7 a ND ND ND ND Molecular weight M w , kDa 113.0 ± 0.5 a1 110.0 ± 1.6 b1 36.1 ± 0.1 a2 34.5 ± 0. 1 a2 36.3 ± 0. 9 a2 34.1 ± 1.1 a2 M n , kDa 73.8 ± 1.0 a1 71.4 ± 1.7 a1 22.3 ± 0.5 a2 21.0 ± 0. 7 a2 22.1 ± 2.2 a2 19.1 ± 2.7 a2 M z , kDa 143.0 ± 0. 3 a1 141.0 ± 1.2 b1 46.8 ± 0.1 a2 45.5 ± 0. 1 b2 47.1 ± 0. 3 a3 45.7 ± 0.5 a2 PI 1.53 ± 0.02 a1 1.55 ± 0.02 a1 1.62 ± 0.03 a1 1.64 ± 0.05 a2,3 1.65 ± 0.13 a1 1.81 ± 0.21 b3 Barrier ***, kg.m/m 2 .s.Pa Water vapor × 10 - 15 26.4 ± 6.8 a 20.7 ± 6.8 b ND ND ND ND O 2 × 10 - 17 55.6 ± 53.6 a 45.9 ± 7.1 a ND ND ND ND CO 2 x 10 - 17 4.7 ± 0.8 > 4.7 ± 0.8 ND ND ND ND * Values are reported as mean ± standard deviation. ** Values with same alphabetic, and numerical symbol are not statistically significantly different (p>0.05). 119 Table 3 - Alphabetic symbol indicates comparison between samples within the corresponding properties. Numerical symbol indicates co mparison between the corresponding properties within samples. *** Water was measured at 37.8 °C, 100% RH, and O 2 , CO 2 were measured at 23 °C, 0% RH. The CO 2 sensitivity limit for MoCON Permatran 4/41 Module C is 1.14 × 10 - 5 kg/ m 2 .s based on standard maski ng area size (PLA= 4.39 × 10 - 8 kg/ m 2 .s with smaller masking area size). **** All measurements were performed in triplicate. Note: PLA resin M w = 121.8 ± 1.0 kDa; M n = 81.2 ± 1.1kDa ; M z = 152.6 ± 0.7 kDa; PI = 1.5 ± 0.01 120 3.3.5 Scanning Electron Microscopy ( SEM ) The SEM micrograph indicates initial homogenous surfaces for both PLA and PLA2M (Figure 3 - 7a and b). Both PLA and PLA2M showed a modified fracture - like morphology on their surface after being in contact with 95% ETOH at 30 °C (24 d) (Figu re 3 - 7c and d). However, at 40 °C (3 d), more noticeable effect of 95% ETOH was observed on the surface of both membranes in terms of roughness and fracture - like effects compared to that of 30 °C (Figure 3 - 7e and f). These morphological changes were antici pated due to the strong chemical interaction between simulant and membrane, especially at 40 °C. Similar looking surfaces classified as rib - like structures were reported by Chen et al. (2013). These structures were found to be correlated with hydrolytic de - form lamellar structure. Chen et al. (2013) found more distinguishable pattern than those shown in this study since samples were subjected to an extreme condition of 60 °C, 40 h in sodium hydroxide (NaOH) solution ( Chen et al., 2013 ) . In addition, shish kebab like structure was also reported to be associated with the effect of ethanol - richer solvent mixtures on PLA surface ( Gao et al., 2012 ) . This type of structure is more of a resemblance of the structure found in this study and it could be ju stified that 95% ETOH greatly induced the morphological properties of the membranes by means of thermally assisted solvent degradation. This finding was also in conjunction with the data obtained in DSC (Figure 3 - 5), and molecular weight analysis (Figure 3 - 6), in which the absences of amorphous regions, and the reduction of molecular weight, respectively, were observed. 121 Figure 3 - 7 Top SEM surface section micrograph (a) PLA; (b) PLA2M; (c) PLA in contact with 95% ETOH at 30 °C for 24 d; (d) PLA2M in contact with 95% ETOH at 30 °C for 24 d; (e) PLA in contact with 95% ETOH at 40 °C for 3 d; (f) PLA2M in contact with 95% ETOH at 40 °C for 12 2 3.3.6 Barrier properties 3.3.6.1 Oxygen (O 2 ) In general, antioxidant may introduce a plasticization effect into membrane, thus changing membrane crystallinity, increasing the amount of free volume, hence resulted in the increase of gas permeability. However, in this study the incorporation of marigol d flower extract did not affect the X c of the membrane, and this finding further confirming the non - significant different obtained in the O 2 permeability between PLA and PLA2M (55.6 ± 53.6 × 10 - 17 vs. 45.9 ± 7.1 × 10 - 17 kg - m/m 2 - Pa - s, respectively) (p=0.5784) (Table 3 - 2). Similarly, Ortiz - Vazquez et al. (2011 ) also reported that the addition of 1.5 wt.% BHT did not significantly affect the O 2 permeability of the PLA - based membrane at the same testing conditions (23 °C, 0 %RH) ( Ortiz - Vazquez et a l., 2011 ) . Meanwhile, the O 2 permeability of PLA - based membranes added with 4 and 10 wt.% TBHQ were significantly lower than that of PLA - based membrane at 20 °C ( Goncalves et al., 2012 ) . However, of the same PLA - based membranes but with 2 wt.% - tocopherol, and 4 wt.% B HT, no significant reduction were observed in their O 2 permeability. Gonçalves et al., (2012) reported that due to TBHQ lower molar volume (166.1 cm 3 /mol) compared to - tocopherol (453.4 cm 3 /mol) and BHT (244.3 cm 3 /mol), increase of free spaces are anticipated for the orientation of polymer chains, thus reducing the gas permeability of the membranes. This decreased was also attributed to the fast diffusivity of gas ( Goncalves et al., 2012 ) . Since astaxanthin has a molar volume of 55 7.0 cm 3 /mol, the finding in this study matched the aforementioned explanation. 3.3. 6.2 Water V apor (WV) WV permeability for both PLA and PLA2M were 26.4 ( ± 6.79) × 10 - 15 , and 20.7 ( ± 6.80) × 10 - 15 kg.m/m 2 .s.Pa , respectively at 37.8 °C and 100% RH (Table 3 - 2). These values were 123 comparable and of the same order of magnitude to those reported for PLA - based membrane with and without antioxidants. WVP of PLA - based membrane was widely reported and ranged from approximately 1.4 to 15 × 10 - 1 5 kg.m/m 2 .s.Pa at 37 - 38 °C, 90 - 100% RH ( Auras et al., 2004a ; R. Auras, B. Harte, & S. E. M. Selke, 2004b ; J amshidian, Tehrany, Cleymand, et al., 2012 ; Jamshidian, Tehrany, Imran, et al., 2012 ; Ortiz - Vazquez et al., 2011 ) . Meanwhile, PLA - based membrane with BHT were reported to have WVP bet ween 1.2 - 2.5 × 10 - 15 kg.m/m 2 .s.Pa, and PLA - based membranes with other antioxidants i.e., - tocopherol, ascorbyl palmitate, butylated hydroxyl - anisole (BHA), propyl gallate, TBHQ had WVP ranged from approximately 1.2 - 3.0 × 10 - 15 kg.m/m 2 .s.Pa under the same condition ( Jamshidian, Tehrany, Cleymand, et al., 2012 ; Jamshidian, Tehrany, Imran, et al., 2012 ; Ortiz - Vazquez et al., 2011 ) . Other study performed at 23 °C, 45% RH, reported a value of 19.9 × 10 - 15 kg.m/m 2 .s.Pa for PLA, and a range of value from 12 - 20 × 1 0 - 15 kg.m/m 2 .s.Pa for PLA - - tocopherol, BHT, and TBHQ at three different concentrations (2, 4, 10 wt.%) ( Goncalves et al., 2012 ) . The incorporation of marigold flower extract was found to significantly reduce WVP of PLA, which could be attributed to the hydrophobic nature of this extract (p=0.03). On the contrary, no significant differences were observed for WVP of PLA - based membrane s with - tocopherol, BHT, BHA, propyl gallate (PG), TBHQ, and ascorbyl palmitate ( Goncalv es et al., 2012 ; Jamshidian, Tehrany, Cleymand, et al., 2012 ; Jamshidian, Tehrany, Imran, et al., 2012 ; Ortiz - Vazquez et al., 2011 ) except when the antioxidants were added at a concentration >2 wt.%. However, the addition of BHT or TBHQ at a nominal concentratio n >2 wt.% did not show any significant decrease in WVP of the PLA - based membranes ( Goncalves et al., 2012 ) . The reason behind this difference could be due to the chemical structure of - tocopherol that contains a long hydrophobic side chain, thus possess greater hydrophobi city than that of BHT, and TBHQ. 124 Moreover, it can be observed that the addition of 2 wt.% marigold flower extract containing astaxanthin resulted in approximately 21 % decrease in WVP of PLA, while the addition of 2 wt.% - tocopherol did not change WVP of the same membrane. The significant decreased of WVP of PLA - based membrane in this study (p=0.03) could be as well due to the presence of longer hydrophobic side chain possess by astaxanthin. 3.3.6.3 Carbon dioxide (CO 2 ) CO 2 permeability of PLA was found to be 4.7 ( ± 0.77) × 10 - 17 kg.m/m 2 .s.Pa at 23 °C, 0% RH (Table 3 - 2). This value was closed to the value reported by Auras et al. (2004b) (1.99 ( ± 0.06) × 10 - 17 kg.m/m 2 .s.Pa at 25 °C, 0% RH) ( Auras et al., 2004a ) . Gonçalves et al. (2012) also did report a value within this range ( Goncalves et al., 2012 ) . Meanwhile, the permeability value of PLA2M can only be assumed to be higher than the PLA since it was not possible to obtain the value due to the machine limitation. Gonçalves et al. (2012) also did report a higher permeability value for those PLA membranes added wi th 4 wt.% - tocopherol and BHT, even though the opposite effect was demonstrated for PLA membranes added with 4 wt.% TBHQ. They speculated that the - tocopherol and BHT was due to this antioxi dant higher molar volume compared to that of TBHQ ( Goncalves et al., 2012 ) . Therefore, it was reasonable that PLA2M showed higher permeability in comparison to that of PLA as had been described before. 3.3.7 Optical P roperties PLA and PLA2M were visually transparency wit h the latter pose a pale orange - like color. PLA and PLA2M did show slightly lower light transmission from 250 to 480 nm. PLA2M 125 indicated absorption behavior within the aforementioned region due to the chromophore compounds (conjugated double bonds) in the astaxanthin structure (Figure not shown). The CIELAB color parameters for PLA were a = - 1.0 ± 0.0, b = 0.9 ± 0.0 and L =92.1 ± 0.2 and for PLA2M were a = - 1.1 ± 0.1, b = 4.6 ± 0.8 and L =91.2 ± 0.2. The E value between the PLA and PLA2M membranes was 3.8 ± 0.8 indicating that the samples were not much different in color, mainly attributed to the yellow (+ b ) difference. 3.3.8 Fourier Transform Infrared Spectrophotometer ( FTIR ) The IR spectrum of PLA, PLA2M, and both membranes in contact with 95% ETOH at 40 °C is shown in Figure 3 - 8. The IR spectrum of PLA indicated the free OH stretching around 3300 - 3600 cm - 1 that reflects the OH side chain end in the PLA chemical structure. The CH stretch, and asymmetrical CH stretch bands can also be observed around 2877 and 2997 cm - 1 , respectively ( Auras et al., 2004b ) . Meanwhile, at around 1749 cm - 1 , the C=O carbonyl st retching appeared as a sharp and large band and the band was reported to be due to the A 1 , and E 1 active modes ( Gonçalves, Coutinho, & Marrucho, 2010 ) . The CH 3 bending was also observed at around 1458 cm - 1 . The symmetric and asymmetric CH deformatio n bending regions were also seen as a twin peaks at 1387, and 1366 cm - 1 , respectively. The C - O stretch was observed at around 1189 cm - 1 . The - C - O - C aliphatic esters, and C - O - C asymmetric stretching mode corresponded to the wavenumber at 1132, and 1076 cm - 1 , respectively. The OH bending mode can also be slightly observed around 1040 cm - 1 ( Auras et al., 2004a ) . Additionally, the CH 3 rocking mode can be observed at 955 cm - 1 , and this mode characterizes the helical backbone vibrations. The bands for amorphous, and crystalline regions of PLA were found at 870, and 757 cm - 1 , re spectively ( Gonça lves et al., 2010 ) . 126 The incorporation of 2 wt.% marigold flower extract was found to affect the PLA - based membrane, and the most profound changes can be observed for carbonyl band stretching, - C - O - C aliphatic esters stretching, and - C - O - C asymmetric st retching at 1744, 1132, and 1076 cm - 1 , respectively ( Colín - Chávez et al., 2012 ) . The - C - O stretching of the ester group band was also found at 1210 cm - 1 . All these changes were anticipated due to the presence of carbonyl group, and more polyunsaturated alkenes in the structure of astaxanthin as well as the fatty acids and triglycerid es presence in the marigold flower extract (Figure 3 - 2). In the case of PLA, and PLA2M that were in contact with 95% ETOH at 30 and 40 °C , more changes were observed especially for carbonyl band stretching, and - C - O stretching at 1749, and 1189 cm - 1 , resp ectively. The OH stretching band was observed to be broader for PLA2M that was in contact with 95% ETOH, and it could be attributed to the rapid penetration of the ethanol into the membrane at such a high temperature, hence modifying the membrane chains. A new and strong formation of peaks was also seen at 1968, and 2025 cm - 1 PLA stored in 95% ETOH at 40 °C (indicated by arrows in Figure 3 - 8(b)) . These peaks could be associated with C=C asymmetric stretching, and could possibly due to depolymerization of t he membrane chains after an exposure at 40 °C with 95% ETOH for 3 days. 127 Figure 3 - 8 (a) FTIR spectrum of the fabricated functional membranes and (b) focused FTIR spectrum of the fabricated functional membranes. 128 3.3.9 Oxidative Stability of Soybean Oil The most influential deteriorative reaction that could shorten the shelf life of oil in general is lipid degradation. Soybean oil that consists of significant amount of polyunsaturated fatty acids (PUFA) is highly susceptible to oxidation. In this study, t he effectiveness of the fabricated membranes in the oxidative stability of soybean oil was investigated by the determination of peroxide value (PV). PV is one of the most commonly technique used to determine the degree of lipid degradation by measuring the formation of hydroperoxides, the primary product of lipid oxidation ( Yildiz, Wehling, & Cuppett, 2003 ) . Soybean oil packaged in all pouches made of PLA and PLA2M underwent oxidation at more or less similar rate during the first 10 days as can be seen from Figure 3 - 9. This result was not surprising since these pouches were transparent in nature causing them susceptible towards oxidation due to light penetration, thus accelerating oxidation mechanism, in turns resulting in the formation of h ydroperoxides. Light was also reported to have greater effect on singlet O 2 oxidation than the temperature. However, in the case of PLA2M pouches, it was expected that they could reduce the oxidation due to its antioxidant reaction with the oil by quenchin g of singlet O 2 . Then, starting at day 15, the PV value of soybean oil in the PLA2M pouches was significantly lowered than that of PV of soybean oil in the glass bottle and PLA pouches (32.7 ± 2.58 vs 43.5 ± 2.89 and 41.5 ± 2.51 Meq/1000g test portion, res pectively) (p<0.0001). It seemed that astaxanthin had migrated slowly from the membrane of the pouches to soybean oil before it accumulated sufficiently to slow down the oxidation. Based on the results obtained, it can be concluded that the shelf life of soybean oil packaged in PLA2M pouches was 5 days or less, and in PLA and glass bottle was 3 days or less as compliance with Codex Alimentarius, in which the maximum acceptable level of PV for refined 129 vegetable oils is 10 milliequivalent/kg ( Codex - Alimentarius, 1999 ) . In an agreement, similar behavio r was also observed for soybean oil packaged in the pouches made of mono - and bilayer LDPE, and LDPE/HDPE membranes containing marigold flower extract ( Colín - Chávez et al., 2012 ) - tocopherol was reported to maintain the acceptable limit of PV of soybean oil for 60 days at 30 °C ( Manzanarez - López et al., 2011 ) . This difference could be associated with the concentration of antioxidant incorporated (2.5 wt.% - tocopherol vs 1.49 wt.% marigold extract (of which contained only 0.7 % astaxanthin)), the nature of soybean oil (riched - additive vs additive - free oil), the compatibility of the antioxidant utilized with the membrane, and the nature of the contact surface of membrane (discs vs pouch). 130 Figure 3 - 9 Oxi dative stability of soybean oil packaged in the glass bottles, pouches made of PLA and pouches made of PLA2M at 30 °C during 25 d. Straight green line indicated the cut off point for Codex Alimentarius. 3.4 Conclusion Bilayer antioxidant functional membrane made of PLA incorporated with Marigold flower extract was successfully fabricated. Migration study of astaxanthin towards 95% ETOH at 30 and 10 - 11 cm 2 /s, respectively. No effect of marigold flower extract was seen in the thermal properties, and O 2 permeability of the PLA - based membrane. The incorporation of marigold flower extract was found to affect the molecular weight, IR spectra, and significantly r educed the WVP of PLA - 131 based membrane. However, it was also found that the gas permeability of this membrane was significantly increased possibly due to the presence of the components of the marigold flower extract introducing more free spaces for gas diffu sivity to happen. In addition, PLA2M did not show any effect in prolonging the freshness of soybean oil within the Codex Alimentarius guideline (measured at 30 ± 2 °C, and 900 - 1000 lux of exposure) due to the slow release of astaxanthin. However, PLA2M did retard the oxidation of soybean oil from 15 to 25 d. Future work should be oriented to control the release of astaxanthin from PLA - based membrane by tailoring the membrane structure either by blending with another polymer or by incorporating another anti oxidant that can facilitate the migration process optimizing the migration process, and extending the shelf life of the intended product. Ethanol at 95% is not a good food simulant for studying fatty foods release in PLA membranes. An alternative fatty foo d simulant for PLA membranes should be investigated in future studies. 132 REFERENCES 133 REFERENCES Al - Malaika, S., Goodwin, C., Issenhuth, S., & Burdick, D. (1999). The antioxidant role of [alpha] - tocopherol in polymers II. Melt stabilising effect in polypropylene. Polymer Degradation and Stability, 64 (1), 145 - 156. Anderson, J. S., & Sunderland, R. (2002). Effect of extruder moisture and dryer p rocessing temperature on vitamin C and E and astaxanthin stability. Aquaculture, 207 (1 - 2), 137 - 149. AOCS. (2009). 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Journal of the American Oil Chemists' Society, 8 0 (2), 103 - 107. 139 Chapter 4 Parameter Estimation for Migration Studies of Antioxidant - Polymer Films 4.0 Introduction Migration in packaging is referred as a diffusivity process of chemical substances from polymeric films into products. These chemical substances include but are not limited to surfactants, contaminants, polymer hydrolysis by - products such as monomers, oli gomers and others, and regulated chemical additives such as antioxidants, antimicrobials, and heat stabilizers. Migration could be a desirable or an undesirable process. It is desirable when the incorporated additives are intended for a gradual release ove r time for protecting the product from any adverse additives migrate into the product due to the incompatibility of polymer film - additives, e.g., surfactan ts, unwanted organoleptic changes and/or unforeseen food safety hazards may occur, depending on the reactivity of the additives; thus this migration process is undesirable. Both types of migration have been widely investigated for multiple reasons, particu larly shelf life extension and safety concerns of intended products. However, in recent years the development of antioxidant - functional film systems for food products has tremendously increased due to optimized packaging technology with dual functions: pro tection of the film and the food product during proces sing and storage, and also nutritional enhancement of intended products. Developments in the area of antioxidant functional films have covered a wide range of polymers: petroleum - based and/or bio - based and various types of antioxidants: natural and/or synthetic (Barbosa - Pereira et al., 2013; Byun, Kim, & Whiteside, 2010; Calatayud et al., 2013; Chen, Lee, Zhu, & Yam, 2012; C. Colín - Chávez, H. Soto - Valdez, E. Peralta, J. Lizardi - Mendoza, & R.R. Balandrán - Quintana, 2013; Citlali Colín - Chávez, Herlinda Soto - Valdez, Elizabeth Peralta, 140 Jaime Lizardi - Mendoza, & René Renato Balandrán - Quintana, 2013; Goncalves et al., 2012; Graciano - Verdugo et al., 2010; Granda - Restrepo et al., 2009; Hwang et al., 2013; Iñiguez - Franco & Soto - Valdez 2011; M. Jamshidian, E. A. Tehrany, et al., 2012; M. Jamshidian, E.A. Tehrany, & S. Desobry, 2012; Majid Jamshidian, Elmira Arab Tehrany, & Stéphane Desobry, 2012; Lee, Shin, Han, Lee, & Giacin, 2004; Lopez de Dicastillo et al., 2011; López - de - Dicastillo, Gómez - Estaca, Catalá, Gavara, & Hernández - Muñoz, 2012; Manzanarez - López, Soto - Valdez, Auras, & Peralta, 2011; Nerín et al., 2006; Ortiz - Vazquez, Shin, Soto - Valdez, & Auras, 2011; Pereira de Abreu, Losada, Maroto, & Cruz, 2010; Sonkaew, Sane, & Suppakul, 2012; Soto - Valdez, Auras, & Peralta, 2010; Soto - Valdez , Peralta, & Auras, 2008; Wessling, Nielsen, & Giacin, 2001; Zhu, Lee, & Yam, 2012; Zhu, Schaich, Chen, & Yam, 2013) . Most of these studies investigated the diffusion coefficient, D , of migrants from different polymeric films with some extended focus on the antioxidant capacity and shelf life of various products. Nevertheless, these studies were designed on a trial and error basis with limite d fruitful findings. Even though this approach may have seemed feasible, a considerable amount of money, time, and effort could have been wasted on resources without meeting sufficient expectations. Therefore, a parameter estimation approach can be used to improve estimation of the parameters of interest, and to assist experimental design by providing insights into the physical process of the experiment. The e stimation of parameters involved in migration processes was often explor ed with little considerati on of statistical assumptions and the accuracy of the estimation was scarcely reported and discussed with the exception of few works. In addition, to the best of the migration studies often focus on the D values. Only a few works invest igated parameters like mass transfer coefficient, h (Reynier, Dole, & Feigenbaum, 2002a, 2002b; Vitrac & Hayert, 2006; Vitrac, Mougharbel, & Feigenbaum, 2007) , other parameters are commonly neglected. 141 This study explored the parameter estimation aspects of three migration models addressing the following conditions : i) film in contact with a finite volume of food/simul ant and negligible external mass transfer coefficient, ii) film in contact with an infinite volume of food/simulant and negligible external mass transfer coefficient, and iii) film in contact with infinite volume of food/simulant and non - negligible externa l mass transfer coefficient. The parameters considered for estimation were D , M , and . 4.1 Theoretical Background 4.1.1 Part A: Migration 4.1.1.1 Partition C oefficient, K p,f In a migration study, the partition coefficient is often determined when the concentration of migrant from the film that has migrated into the food/simulant reaches equilibrium. The partition coefficient, K p,f can be defined as the ratio of the migrant con centration left in the film to the migrant concentration in the food simulant/system (Poças, Oliveira, Oliveira, & Hogg, 2008) . The value of K p,f film - food system. The value of K p,f can also be used to describe the anti cipated physical process of the film - food system in terms of the chemical affinity of the migrant towards the film and the food system. Other factors such as temperature, film area of exposure to the food system, and the degree of solvency of the food syst em will also influence the value of K p,f (Tehrany & Desobry, 2004) . A value of K p,f equal to 1 means that the migrant concentrations are similar in both the film and the food system at equilibrium. Meanwhile, K p,f >1, and K p,f <1 describe a higher affinity of the migrant towards the film, and a higher affinity of the migrant towards the food system, respectively. The former is commonly preferred when food safety is of concern, while the later is 142 preferred in the case of a functional film where the additives incorporated into the film are expected to provide controlled release into a food system for shelf life extension. From K p,f , one can determine , which can also be estimated from the non - zero positive roots of q n (tan q n = - q n ). is also a ratio of the mass of migrant migrated into food/simulant to the mass of migrant left in the film, at equilibrium. (Eq. 4 - 1) C = the concentration of the migrant in the film at equilibrium (g of migrant/cm 3 film). C = the concentration of migrant in the food/simulant at equilibrium (g of migrant/cm 3 food/simulant). (Eq. 4 - 2) volume of food/simulant (cm 3 ) volume of film (cm 3 ) = mass of the migrant in food/simulant at equilibrium ( g or g) mass of the migrant left in film at equilibrium ( g or g) 4.1.1.2 Biot number, Bi Besides K p,f , another important parameter that may affect migration is the convective mass transfer coefficient, h . This parameter describes the interfacial mass transfer coeffi cient with or without the presence of fluid flow (bulk convection). However, the importance of the parameter depends on the system, specifically in the case where resistance exists at the surface boundary between film and food/simulant. This parameter is e mbedded inside the Biot number together with the diffusion coefficient, D as the overall migration resistance series. Bi is a dimensionless number 143 representing a ratio between the convective mass transfer coefficient, h and the diffusion coefficient, D , ta king into consideration the film thickness, L (Eq. 4 - 3) (Bird, Stewart, & Lightfoot, 2007; Galotto, Torres, Guarda, Moraga, & Romero, 2011; Vitrac et al., 2007) . (Eq. 4 - 3) In addition, determination of h is crucial for a system of film - food/simulant that imposes the film is thicker (Vitrac et al., 2007) and in the case of hydrophilic polymer - oil like simulant system, thus increases the resistance of interfacial mass transfer. 4.1.1.3 Diffusion coefficient, D The diffusion coefficient, D is a parameter used to describe the migration process that takes ion process that occurs from higher to lower migrant concentration with respect to spatial discretization (Eq. 4 - in an unsteady state, where the concentration o f the migrant changes as a function of time (Eq. 4 - 5) (Crank, 1979) . The rate of diffusion is usually affected by the temp erature, the partition coefficient, chemical affinity between film - food/simulant, and the effect of stirring, to name a few. (Eq. 4 - 4) (Eq. 4 - 5) F = flow rate D = diffusion coefficient c = migrant concentration in the film 144 x = the distance (in the direction of the diffusion) t=time The determination of K p,f , , h , and D are among the most important parameters that describe migration. This phenomenon can be further described through established mathematical models with the following assumptions: i) initial concentration of the migrants is uniformly distributed in the film, ii) migration happens on the side of the film th at is in contact with the food/ simulant, iii) the foo d/simulant is well mixed and has large surface mass transfer coefficient, h ( Biot no. >100), iv) Fickian diffusion controls the migration in the film, v) migration depends only on temperature, and the diffusion coefficient, D and the partition coefficient, K p,f, are constants, vi) the film interface and the food are always at equilibrium, and vii) no interaction occurs between the film and the food/simulant and the edge effect is negligible (Chung, Papadakis, & Yam, 2001, 2002; Crank, 1979; Poças et al., 2008) . Model A: Film in contact with finite volume of food/simulant and negligible external mass transfer coefficient This model is commonly used for a migration study at relatively low temperature or in the case where the migrant has higher affinity towards the film than that of the food/simulant, which results in <1 because K p,f >1 (Figure 4 - 1). The final solution in this model i s as follows: ( Eq. 4 - 6) Model B: Film in contact with an infinite volume of food/simulant and negligible external mass transfer coefficient This model is applicable when m ost of the migrant in the film migrates into the food/simulant, resulting in K p,f <1, and thus 1 (Figure 4 - 1). The final solution in this model is as follows: 145 (Eq.4 - 7) It is worth mentioning that in the case where the ratio of V f /V p >> 20, the same outcome will result from both Eq. 4 - 6 and Eq.4 - 7 (Hamdani, Feigen baum, & Vergnaud, 1997) . Model C: Film in contact with an infinite volume of food/simulant and non - negligible external mass transfer coefficient This type of model is commonly used when there is interfacial resistance at the boundary layer between the film and the food/simulant, like in the case of a film in contact with an oil - like food/simulant and/or when a thicker film is used ( Bi <200) (Figure 4 - 1) (Vitrac et al., 2007) . The f inal solution for this model is as follows: (Eq. 4 - 8) In the case that the food/simulant is stir red vigorously and/or a thin layer of film is used, thus resulting in Bi >200, Eq. 4 - 8 is reduced/simplified to Eq. 4 - 7 (Mascheroni, Guillard, Nalin, Mora, & Piergiovanni, 2010) . 146 Migration controlled by diffusion in the film Migration with boundary layer resistance nd Law of Diffusion Initial conditions Boundary conditions Boundary conditions = the non - zero positive roots of tan = the non - zero positive roots of Biot # ( Bi )= tan If because and/ or a simplified solution: If is very high (>100), a simplified solution: Figure 4 - 1 S ummary of migration models A, B, and C. Figure adapted from Poças et. al (2008) (Poças et al., 2008) . L 0 x film food C C f 0 x film food C f C p L 147 4.1.2 Part B: Parameter E stimation Many migration models are available to determine the diffusion coefficient through mainly determ inistic mathematical modeling. Deterministic models often have an input of variables as a single and c onstant value that results in the same output value of variables due to its zero error assumption (Poças et al., 2008) . Some studies employ a curve - fitting or optimization approach; however this approach considers mostly minimi zing the sum of squared errors rather than evaluating the importance of the parameters (Dolan & Mishra, 2013) . In reality, most experimental studies are designed by collecting observational data (dependent varia ble) with unknown functions or parameters. This approach is an inverse problem and is also known as parameter estimation. Parameter estimation helps to estimate the parameters/constants of interest using mathematical models, and to a certain extent it may provide some physical meaning for parameters that are relevant to the experiment. Beck and Arnold (1977) estimation of constants appearing in mathematica l models and for aiding in modeling of (Beck & Arnold, 1977) . Parameter estimation takes into consideration the imp ortance of parameters and how these parameters affect each other, and in turn the whole experimental design. Experimentally, in most cases, more than one parameter is estimated through mathematical models. Lack of information on how these parameters affec t each other often results in higher variability in observational data, thus causing higher statistical error and/or overestimation or underestimation of parameters of interest. Therefore, the first step to consider in parameter estimation is to investigat e the correlation among parameters through the sensitivity coefficients and scaled sensitivity coefficients. 148 4.1.2.1 Sensitivity Coefficient and Scaled Sensitivity Coefficient Sensitivity coefficient and scaled sensitivity coefficient were described in Cha pter 2 section 2.7.2. 4.1.2.2 Ordinary Least Squares ( OLS ) Estimation Parameter estimation can be performed by an OLS using the non - linear regression (nlinfit) command in MATLAB®. Statistical assumptions need to be analyzed before data fitting. Statistica l assumptions that need to be taken into account include (but are not limited to); i) the errors are additive in the measurement, ii) the errors in the measurement contain zero mean, iii) the measurement errors have constant variance, iv) the measurement e rrors are uncorrelated, v) the errors are normal, independent, and identically distributed, vi) the statistical parameters describing the errors are known, vii) the i ndependent variables are errorless, and viii) the nature of the parameters (constant vs. random vector parameter; prior information vs. unknown statistics of parameter) (Beck & Arnold, 1977; Dolan & Mishra, 2013) . 4.1.2.3 Corrected A kaike Information Criterion (AICc) The corrected Akaike Information Criterion (AICc) was used to compare models with 1P, 2P and/or 3P. This approach is commonl y used for the non - nested model comparison. Commonly, the higher the number of the parameters the more likely the goodness of fit seems to improve and vice versa. Th us, AICc eliminates the bias that may be caused by different numbers of parameter s among models. The smaller the value of AICc is, the more likely the model is correct ( Motulsky & Christopoulos, 2004) . 149 (Eq. 4 - 9 ) where n =number of data; p =number of parameter; K=p+1 4.1.2.4 Optimal Experimental Design Optimal experimental design helps in finding the optimal point at which the parameters can be estimated and have lower errors. The C matrix is needed in order to achieve a desirable maximum determinant (Eq. 4 - 15). By maximizing the determinant (Eq. 4 - 16), the optimal time to perform an experimental study can be determined. This approach is beneficial in terms of optimizing not only the resources used, but also the time spent for a given experiment (Beck & Arnold, 1977; Dolan & Mishra, 2013) . (Eq. 4 - 10 ) (Eq. 4 - 11 ) In addition, providing that all standard statistical assumptions are met, the OLS, maximum likelihood (ML), Gauss - Markov, and Maximum A Posteriori (MAP) method produce similar estimators and variance. Among all the criter ia suggested for , Beck & Arnold (1977) recommended the maximization of the determinant due to its implications of minimizing the hypervolume of the confidence region (Beck & Arnold, 1977) . 4.2 Case Study A selected case study is presented to demonstrate the difference between estimation of 1, 2, and 3 parameters. Data analyses were performed using a non - linear regressi on fitting function in MATLAB® R2011b (MathWorks, Natick, MA, USA). 150 Statistical analysis was performed by using a SPSS Statistics (version 22, 2013, IBM Corporation©, Armonk, NY, USA). Mean comparison of more than two parameters was done using an analysis of variance (ANOVA) and post - - test was performed for comparing means between two parameters. 4.2.1 A Selected Case Study: Poly (Lactic Acid), PLA - - Tocopherol Functional Film in Contact With 100% Ethanol At 23 C A poly(lactic acid), PLA film incorporated with 2.58 0.18 wt.% - tocopherol was prepared in the form of round discs with a total area/disc of 6.28 cm 2 . Six discs were separated by beads o n a stainless steel wire and placed in a vial containing 30 mL 100% ethanol (volume - area ratio= 0.8 mL/ cm 2 ). The experiment was run at 23 C and sampling was performed frequently during an interval of 8 days (Manzanarez - López et al., 2011) . The K p,f was measured at the end of the experiment and was reported to be 7 96.62 45.4. The value was also calculated and was found to be 0.366 (Manzanarez - López et al., 2011) , which me e t s the boundary conditions and the requirements of model A presented earlier. Thus, the analytical solution of Eq. 4 - 6 was used to fit the model. For this selected case study, three parameters ( D , M and ) were estimated for 1, 2, and 3P models . 4.2.1.1 Initial Scaled Sensitivity C oefficient, The initial scaled sensitivity coefficient, involvi ng two and three parameters was analyzed and plotted based on initial parameter guesses by using an approximation of the forward difference method. Figure 4 - 2a and 4 - 2b show the plots for 2 and 3 parameters, respectively. From Figure 4 - 2a, it can be seen that D and M were not corre lated, thus providing an initial 151 indication that they can be estimated easily and accurately. Since the absolute magnitude of change for M was larger, it was expected that the estimation for this particular parameter could be performed accurately with low er relative error than that of D . Meanwhile, Figure 4 - 2b indicates that based on the magnitude of the plots, M can be accurately estimated with the lowest relative error, followed by and D . Even though it may have appeared on the plot that there could be some correlation between , and D , the ratio between these two parameters was not constant. It is also worth noticing that a closer look into provides an initial indication that this parameter would be best estimate d at an early stage of the experimen t due to its sensitivity towards perturbation as well as to avoid correlation with D (Figure 4 - 2b). In addition, the plot can also be used to give an approximation of the duration actually needed to sufficiently estimate all parameters of interest. This approximation can later be compared with the optimal experimental design. 4.2.1.2 Ordinary Least Square ( OLS ) Estimation and the Corrected Akaike Information Criterion (AICc) D values of 1.06 0.04, 4.5 0.23, and 0.79 0.08 10 - 10 cm 2 /s were obtained from OLS estimation of 1, 2, and 3P, respectively. No significant different (p>0.05) was observed between the M values of 2P vs. 3P estimation (0.36 0.003 vs. 0.37 0.004 10 - 4 g - tocopherol/ g ethanol). Meanwhile, an estimated val ue was found to be 0.30 0.01 (Table 4 - 1). Root mean square errors (RMSE) of 1, 2, and 3P were 1.51 10 - 6 , 1.52 10 - 6 and 1.32 10 - 6 g - tocopherol/ g ethanol, respectively. In general, the RMSE represents the accuracy of estimation by taking into consideration how much deviation occurs between predicted and observed data. A smaller RMSE value is often anticipated with an increasing number of parameters estimated as more factors are 152 being weighted into the model fitting, thus increasing the accuracy of estimation as can be seen from the results obtained. However, a limit on the number of parameters estimated should be considered carefully s ince over - parameterized models introduce more uncertainty; thus, the expected accuracy could be compromised. Therefore, to avoid the bias introduced by having different number of parameters in a model, AICc can provide better and more fair indication than that of RMSE. In this case, it was found that 3P estimation is more likely to be the correct one over 1P and 2P, since its AICc value was the lowest among the others ( - 2269). This result gave an indication that the estimation of those aforementioned three parameters should be considered for this particular case study. From an overall perspective, it can be concluded that 3P estimation did give a lower, better RMSE and lower AICc over 1P and 2P estimation. Therefore, by estimating three parameters, the accu racy of the estimation can be improved and additional insight on the kinetics behind the migration experiment can be obtained. For 2P estimation, M had the lowest relative error (0.009) compared to D (0.052) as had been anticipated based on the initial plot. The asymptotic 95% confidence interval, for which indicates the reliability of the estimate, was found to be tighter for M than for D (0.35 - 0.36 10 - 4 vs. 4.04 - 4.97 10 - 10 , respectively). A s imilar expectation was also met for 3P estimation, whic h M had the lowest relative error (0.012), followed by (0.039), and D (0.097). The correlation coefficient between the parameters estimated for both models (2P and 3P) was also in an agreement with the initial plot (Figure 4 - 2a,b), which demonstrated that they were all not correlated (Table 4 - 1). Figure 4 - 3a,b,c shows the migration plots of this case study for all 3 estimations. Residual plots were also plotted for 1P, 2P, and 3P estimation for visual interpretation of how the assumption of no rmal, i dentically, independent distribution of data was met (Figure 4 - 3d,e,f). A signature residual was found visually apparent for 1P estimation in comparison to that 153 of the residual distribution observed for 2P and 3P. For this particular case study, it was obs erved that the addition of extra parameters increased the accuracy of the estimates based on the results obtained from OLS estimation. A final plot was also constructed based on the estimated values obtained from OLS estimation to demonstrate the final outcomes (Figure 4 - 4a,b). 154 Figure 4 - 2 (a) Scaled sensitivity coefficients for 2P, and (b) for 3P of migration study of PLA - - tocopherol system at 23 C using forward difference approximation. Initial guesses used were: D = 0.06 10 - 9 cm 2 /s, M inf =3.95 10 - 5 g - tocopherol/g ethanol, and =0.35. 155 Figure 4 - 3 M igration of - tocopherol into 100% ethanol at 23 °C during storage for (a) 1P, (b) 2P, and (c) 3P using OLS estimation and their corresponding residuals plot (d), (e), and (f), respectively. 156 Figure 4 - 4 (a) Final scaled sensitivity coefficients for 2P, and (b) for 3P of migration study of PLA - - tocopherol system at 23 C using forward difference approximation. Estimated values used for 2P were: D = 4.05 10 - 10 cm 2 /s, M inf =0.36 10 - 4 g - tocopherol/g ethanol. Estimated values used for 3P were: D = 0.79 10 - 10 cm 2 /s, M inf =0.37 10 - 4 g - tocopherol/g ethanol, and =0.30. 157 4.2.1.3 Optimal Experimental Design A plot of the optimal experimental design for 2P was constructed to demonstrate the (Figure 4 - 5) . It was observed that the required time to collect enough data to obtain the best estimates of D and M was approximately 2 d and 8 d, respectively. However, it was also apparent that the delta ( ) was maximized at approximately 8 d, which suggested that to estimate both parameters simultaneously, at least 8 d of experimental duration was needed. Therefore, it can be concluded that both the and the optimal experimental design were in agreement, and were informative and beneficial for designing experimental plans . 158 Figure 4 - 5 Optimal experimental designs for 2P of migration study of PLA - - tocopherol system at 23 C. 4.2.2 Other Case Studies Other case studies are summarized in Table 4 - 1. All data were extracted from published and unpublished works from our research group and reanalyzed to showcase different scenarios of migration of antioxidants from PLA - based films into food simulants based on the parameter estimation approach (Table 4 - 1). Overall, the introduction of more than one parameter did show an improvement in models by resulting in a significant decrease in RMSE. However, in some cases ( i.e., PLA - catechin at 20 C, PLA - epicatechin at 20 C, and PLA - rutin at 40 C), a detail observation is needed due to the high correlation coefficient values obtained (>0.93), which could affect the accuracy of estimating those parameters ( i.e., D and M . It was also observed t hat there 159 was not much difference between the AICc values when comparing 1P versus 2P estimations. This indicates that the additional parameter of M might not really contribute much to the overall physical interpretation of the experiment s . In addition , the optimal experimental design was found useful to predict the sufficient time needed for an experiment to be able to accurately estimate parameters of interest. For example, in the case of PLA - 3 wt. % resveratrol at 23 C, the experiment was performed beyon d necessary since it was found the time needed for this experiment is actually around 17 d instead of 43 d. In contrast, PLA - 3 wt. % resveratrol at 9 C that was performed for 278 d , longer experimental time (417 d) is needed in order to achieve better estimation of all parameters. Thus, optimal experimental design should be implemented to efficiently use the resources ( i.e., cost and labor) with compromi sing the accuracy of the parameters estimated. 160 Table 4 - 1 Summarized of estimated parameters for different migration studies of antioxidant - PLA film systems. Case studies PLA - - tocopherol (L= 5.46 × 10 - 3 cm) (Manzanarez - López et al., 2011) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 (cm 2 /s) M × 10 - 4 g AOx/g ETOH 23 1P Estimated value ± Standard error (95% Asymptotic CI) 1.0582 ± 0.036 a (0.987 - 1.129) RMSE ×10 - 6 ( g AOx/g ETOH) 1.5107 Relative error 0.034 NA AICc - 2249 161 Table 4 - Case studies PLA - - tocopherol (L= 5.46 × 10 - 3 cm) (Manzanarez - López et al., 2011) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 (cm 2 /s) M × 10 - 4 g AOx/g ETOH 23 2P Estimated value ± Standard error (95% Asymptotic CI) 4.5025 ± 0.23 b 0.3576 ± 0.0033 1 (4.037 - 4.968) (0.3511 - 0.3641) RMSE ×10 - 6 ( g AOx/g ETOH) 1.5153 Relative error 0.052 0.009 0.7507 AICc - 2247 162 Table 4 - 1 Case studies PLA - - tocopherol (L= 5.46 × 10 - 3 cm) (Manzanarez - López et al., 2011) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 (cm 2 /s) M × 10 - 4 g AOx/g ETOH 23 3P Estimated value ± Standard error (95% Asymptotic CI) 0.7894 ± 0.076 a (0.638 - 0.941) 0.3664 ± 0.0044 1 (0.3577 - 0.3751) 0.3034 ± 0.012 (0.28 - 0.3268) RMSE ×10 - 6 ( g AOx/g ETOH) 1.322 Relative error 0.0963 0.012 0.0387 0.8627; 0.8132; 0.5560 AICc - 2269 163 Table 4 - Case studies PLA - - tocopherol (L= 5.46 × 10 - 3 cm) (Manzanarez - López et al., 2011) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 (cm 2 /s) M × 10 - 4 g AOx/g ETOH 33 1P Estimated value ± Standard error (95% Asymptotic CI) 0.4855 ± 0.036 a (0.4185 - 0.5525) RMSE ×10 - 6 ( g AOx/g ETOH) 9.050 Relative error 0.0691 NA AICc - 1576 164 Table 4 - Case studies PLA - - tocopherol (L= 5.46 × 10 - 3 cm) (Manzanarez - López et al., 2011) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 (cm 2 /s) M × 10 - 4 g AOx/g ETOH 33 2P Estimated value ± Standard error (95% Asymptotic CI) 0.4857 ± 0.0055 a 0.9251 ± 0.027 (0.3753 - 0.5962) (0.8701 - 0.98) RMSE ×10 - 6 ( g AOx/g ETOH) 9.118 Relative error 0.1139 0.0298 0.7912 AICc - 1574 165 Table 4 - Case studies PLA - - tocopherol (L= 5.46 × 10 - 3 cm) (Manzanarez - López et al., 2011) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 (cm 2 /s) M × 10 - 4 g AOx/g ETOH 43 1P Estimated value ± Standard error (95% Asymptotic CI) 3.8023 ± 0.2298 a (3.343 - 4.262) RMSE ×10 - 6 ( g AOx/g ETOH) 9.4088 Relative error 0.0604 NA AICc - 1386 166 Table 4 - Case studies PLA - - tocopherol (L= 5.46 × 10 - 3 cm) (Manzanarez - López et al., 2011) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 (cm 2 /s) M × 10 - 4 g AOx/g ETOH 43 2P Estimated value ± Standard error (95% Asymptotic CI) 3.8038 ± 0.288 a 1.3179 ± 0.0205 (3.2256 - 4.3820) (1.2769 - 1.3588) RMSE ×10 - 6 ( g AOx/g ETOH) 9.4896 Relative error 0.0759 0.0155 0.5966 AICc - 1383 167 Table 4 - Case studies PLA - BHT (L= 5.08 × 10 - 3 cm) (Ortiz - Vazquez et al., 2011) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 23 1P Estimated value ± Standard error (95% Asymptotic CI) 0.2224 ± 0.0103 a (0.2017 - 0.2430) RMSE ×10 - 6 ( g AOx/g ETOH) 9.3148 Relative error 0.0464 NA AICc - 1480 168 Table 4 - Case studies PLA - BHT (L= 5.08 × 10 - 3 cm) (Ortiz - Vazquez e t al., 2011) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 23 2P Estimated value ± Standard error (95% Asymptotic CI) 0.223 ± 0.013 a 1.622 ± 0.018 (0.1966 - 0.2485) (1.5840 - 1.6591) RMSE ×10 - 6 ( g AOx/g ETOH) 9.3896 Relative error 0.0583 0.0116 0.596 AICc - 1477 169 Table 4 - Case studies PLA - BHT (L= 5.08 × 10 - 3 cm) (Ortiz - Vazquez et al., 2011) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 33 1P Estimated value ± Standard error (95% Asymptotic CI) 0.6780 ± 0.0483 a (0.5820 - 0.7741) RMSE ×10 - 6 ( g AOx/g ETOH) 19.3447 Relative error 0.0712 NA AICc - 1798 170 Table 4 - Case studies PLA - BHT (L= 5.08 × 10 - 3 cm) (Ortiz - Vazquez et al., 2011) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 33 2P Estimated value ± Standard error (95% Asymptotic CI) 0.678 ± 0.06 a 2.085 ± 0.034 (0.5586 - 0.7975) (2.0174 - 2.1527) RMSE ×10 - 6 ( g AOx/g ETOH) 19.464 Relative error 0.0866 0.0163 Correlation 0.588 AICc - 1796 171 Table 4 - Case studies PLA - BHT (L= 5.08 × 10 - 3 cm) (Ortiz - Vazquez e t al., 2011) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 43 1P Estimated value ± Standard error (95% Asymptotic CI) 19.0376 ± 0.5073 a (18.032 - 20.044) RMSE ×10 - 6 ( g AOx/g ETOH) 8.2746 Relative error 0.0267 NA AICc - 2431 172 Table 4 - Case studies PLA - BHT (L= 5.08 × 10 - 3 cm) (Ortiz - Vazquez et al., 2011) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 43 2P Estimated value ± Standard error (95% Asymptotic CI) 19.04 ± 0.617 a 1.9635 ± 0.011 (17.812 - 20.26) (1.9414 - 1.9857) RMSE ×10 - 6 ( g AOx/g ETOH) 8.3151 Relative error 0.0324 0.0057 0.5639 AICc - 2429 173 Table 4 - Case studies PLA - catechin (L= 6.49 × 10 - 3 cm) - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 20 1P Estimated value ± Standard error (95% Asymptotic CI) 0.4329 ± 0.016 a (0.4006 - 0.4652) RMSE×10 - 6 ( g AOx/g ETOH) 0.2095 Relative error 0.0371 NA AICc - 1473 174 Table 4 - Case studies PLA - catechin (L= 6.49 × 10 - 3 cm) - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 20 2P Estimated value ± Standard error (95% Asymptotic CI) 0.407 ± 0.051 a 0.049 ± 0.0016 1 (0.304 - 0.511) (0.046 - 0.052) RMSE ×10 - 6 ( g AOx/g ETOH) 0.211 Relative error 0.126 0.033 0.9572 AICc - 1471 175 Table 4 - Case studies PLA - catechin (L= 6.49 × 10 - 3 cm) - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 20 3P Estimated value ± Standard error (95% Asymptotic CI) 0.4794 ± 0.11 a 0.0476 ± 0.0021 1 0.3869 ± 0.0288 (0.2638 - 0.6949) (0.0434 - 0.0518) (0.329 - 0.4448) RMSE ×10 - 6 ( g AOx/g ETOH) 0.2095 Relative error 0.2233 0.0434 0.0743 0.9623; 0.8763;0.7661 AICc - 1471 176 Table 4 - Case studies PLA - catechin (L= 6.49 × 10 - 3 cm) - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 30 1P Estimated value ± Standard error (95% Asymptotic CI) 0.9734 ± 0.1016 a (0.7683 - 1.1785) RMSE ×10 - 6 ( g AOx/g ETOH) 1.9495 Relative error 0.104 NA AICc - 1101 177 Table 4 - Case studies PLA - catechin (L= 6.49 × 10 - 3 cm) - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 30 2P Estimated value ± Standard error (95% Asymptotic CI) 0.9744 ± 0.131 a 0.2234 ± 0.005 (0.7105 - 1.238) (0.2126 - 0.2342) RMSE ×10 - 6 ( g AOx/g ETOH) 1.974 Relative error 0.134 0.0239 0.6153 AICc - 1099 178 Table 4 - Case studies PLA - catechin (L= 6.49 × 10 - 3 cm) - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 40 1P Estimated value ± Standard error (95% Asymptotic CI) 6.8511 ± 0.276 a (6.289 - 7.413) RMSE ×10 - 6 ( g AOx/g ETOH) 0.9691 Relative error 0.0403 NA AICc - 911 179 Table 4 - Case studies PLA - catechin (L= 6.49 × 10 - 3 cm) - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 40 2P Estimated value ± Standard error (95% Asymptotic CI) 6.853 ± 0.3895 a 0.2438 ± 0.0032 (6.0581 - 7.647) (0.2373 - 0.2503) RMSE ×10 - 6 ( g AOx/g ETOH) 0.9846 Relative error 0.0568 0.0131 0.6939 AICc - 908 180 Table 4 - Case studies PLA - catechin (L= 6.49 × 10 - 3 cm) - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 50 1P Estimated value ± Standard error (95% Asymptotic CI) 3.7609 ± 0.1525 a (3.449 - 4.073) RMSE×10 - 6 ( g AOx/g ETOH) 1.2862 Relative error 0.0406 NA AICc - 810 181 Table 4 - Case studies PLA - catechin (L= 6.49 × 10 - 3 cm) guez - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 50 2P Estimated value ± Standard error (95% Asymptotic CI) 3.769 ± 0.396 a 0.271± 0.009 (2.9582 - 4.5806) (0.252 - 0.2895) RMSE ×10 - 6 ( g AOx/g ETOH) 1.3089 Relative error 0.1051 0.0337 0.9194 AICc - 808 182 Table 4 - Case studies PLA - catechin (L= 6.49 × 10 - 3 cm) - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH *40 1P Estimated value ± Standard error (95% Asymptotic CI) 0.8840 ± 0.04 a (0.799 - 0.9693) RMSE ×10 - 6 ( g AOx/g ETOH) 0.6394 Relative error 0.0478 NA AICc - 1195 183 Table 4 - Case studies PLA - catechin (L= 6.49 × 10 - 3 cm) - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH *40 2P Estimated value ± Standard error (95% Asymptotic CI) 0.8831 ± 0.0744 a 0.1136 ± 0.0029 (0.7328 - 1.033) (0.1077 - 0.1196) RMSE ×10 - 6 ( g AOx/g ETOH) 0.065 Relative error 0.0842 0.0258 0.8187 AICc - 1192 184 Table 4 - Case studies PLA - epicatechin (L= 6.19 × 10 - 3 cm) guez - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 20 1P Estimated value ± Standard error (95% Asymptotic CI) 0.3492 ± 0.013 a (0.3229 - 0.3755) RMSE ×10 - 6 ( g AOx/g ETOH) 0.1952 Relative error 0.0374 NA AICc - 1387 185 Table 4 - Case studies PLA - epicatechin (L= 6.19 × 10 - 3 cm) - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 20 2P Estimated value ± Standard error (95% Asymptotic CI) 0.3226 ± 0.048 a 0.04795 ± 0.0018 1 (0.2246 - 0.4206) (0.0443 - 0.05161) RMSE ×10 - 6 ( g AOx/g ETOH) 0.197 Relative error 0.1507 0.0378 0.97 AICc - 1385 186 Table 4 - Case studies PLA - epicatechin (L= 6.19 × 10 - 3 cm) - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 20 3P Estimated value ± Standard error (95% Asymptotic CI) 0.525 ± 0.091 b (0.3415 - 0.709) 0.044 ± 0.0013 1 (0.042 - 0.047) 0.3767 ± 0.026 (0.3251 - 0.4284) RMSE ×10 - 6 ( g AOx/g ETOH) 0.1886 Relative error 0.1734 0.0296 0.068 0.932; 0.8318; 0.6599 AICc - 1388 187 Table 4 - Case studies PLA - epicatechin (L= 6.19 × 10 - 3 cm) guez - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 30 1P Estimated value ± Standard error (95% Asymptotic CI) 1.168 ± 0.122 a (0.9195 - 1.4161) RMSE×10 - 6 ( g AOx/g ETOH) 1.5292 Relative error 0.105 NA AICc - 961 188 Table 4 - Case studies PLA - epicatechin (L= 6.19 × 10 - 3 cm) - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 30 2P Estimated value ± Standard error (95% Asymptotic CI) 1.168 ± 0.143 a 0.1824 ± 0.004 (0.8764 - 1.460) (0.1748 - 0.1899) RMSE ×10 - 6 ( g AOx/g ETOH) 1.5515 Relative error 0.123 0.0204 0.5021 AICc - 958 189 Table 4 - Case studies PLA - epicatechin (L= 6.19 × 10 - 3 cm) - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 40 1P Estimated value ± Standard error (95% Asymptotic CI) 5.349 ± 0.214 a (4.912 - 5.785) RMSE ×10 - 6 ( g AOx/g ETOH) 1.003 Relative error 0.040 NA AICc - 908 190 Table 4 - 1 Case studies PLA - epicatechin (L= 6.19 × 10 - 3 cm) - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 40 2P Estimated value ± Standard error (95% Asymptotic CI) 5.348 ± 0.3023 a 0.2538 ± 0.003 (4.732 - 5.965) (0.2471 - 0.2605) RMSE ×10 - 6 ( g AOx/g ETOH) 1.019 Relative error 0.057 0.013 0.6943 AICc - 906 191 Table 4 - Case studies PLA - epicatechin (L= 6.19 × 10 - 3 cm) iguez - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH *40 1P Estimated value ± Standard error (95% Asymptotic CI) 0.7746 ± 0.032 a (0.7104 - 0.8388) RMSE ×10 - 6 ( g AOx/g ETOH) 5.6021 Relative error 0.041 NA AICc - 1206 192 Table 4 - Case studies PLA - epicatechin (L= 6.19 × 10 - 3 cm) - Franco et al., 2012) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH *40 2P Estimated value ± Standard error (95% Asymptotic CI) 0.7751 ± 0.0547 a 0.1113 ± 0.0023 (0.6644 - 0.8857) (0.1067 - 0.1159) RMSE ×10 - 6 ( g AOx/g ETOH) 0.5672 Relative error 0.0706 0.0203 0.8085 AICc - 1204 193 Table 4 - Case studies PLA - Resveratrol 1% (L= 5.08 × 10 - 3 cm) (Soto - Valdez et al., 2010) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 23 1P Estimated value ± Standard error (95% Asymptotic CI) 0.2389 ± 0.0033 a (0.232 - 0.245) RMSE×10 - 6 ( g AOx/g ETOH) 0.666 Relative error 0.0140 Correlation NA AICc - 1817 194 Table 4 - Case studies PLA - Resveratrol 1% (L= 5.08 × 10 - 3 cm) (So to - Valdez et al., 2010) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 23 2P Estimated value ± Standard error (95% Asymptotic CI) 0.2385 ± 0.0048 a 0.3562 ± 0.0015 (0.2289 - 0.2480) (0.3531 - 0.3593) RMSE ×10 - 6 ( g AOx/g ETOH) 0.671 Relative error 0.02 0.0043 0.7113 AICc - 1815 195 Table 4 - Case studies PLA - Resveratrol 1% (L= 5.08 × 10 - 3 cm) (Soto - Valdez et al., 2010) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 33 1P Estimated value ± Standard error (95% Asymptotic CI) 2.8025 ± 0.184 a (2.4359 - 3.1692) RMSE ×10 - 6 ( g AOx/g ETOH) 3.5020 Relative error 0.0655 NA AICc - 1705 196 Table 4 - Case studies PLA - Resveratrol 1% (L= 5.08 × 10 - 3 cm) (Soto - Valdez et al., 2010) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 33 2P Estimated value ± Standard error (95% Asymptotic CI) 2.8036 ± 0.21 a 0.4708 ± 0.0056 (2.384 - 3.223) (0.4596 - 0.4819) RMSE ×10 - 6 ( g AOx/g ETOH) 3.5284 Relative error 0.075 0.012 0.4735 AICc - 1703 197 Table 4 - Case studies PLA - Resveratrol 1% (L= 5.08 × 10 - 3 cm) (Soto - Valdez et al., 2010) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 43 1P Estimated value ± Standard error (95% Asymptotic CI) 9.5287 ± 0.4481 a (8.627 - 10.430) RMSE ×10 - 6 ( g AOx/g ETOH) 1.956 Relative error 0.047 NA AICc - 1259 198 Table 4 - Case studies PLA - Resveratrol 1% (L= 5.08 × 10 - 3 cm) (Soto - Valdez et al., 2010) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 43 2P Estimated value ± Standard error (95% Asymptotic CI) 9.56 ± 0.56 a 0.421 ± 0.004 (8.44 - 10.68) (0.4128 - 0.4285) RMSE ×10 - 6 ( g AOx/g ETOH) 1.977 Relative error 0.0581 0.0093 Correlation 0.5733 AICc - 1256 199 Table 4 - Case studies PLA - Resveratrol 3% (L= 5.08 × 10 - 3 cm) (So to - Valdez et al., 2010) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 23 1P Estimated value ± Standard error (95% Asymptotic CI) 0.3988 ± 0.017 a (0.3649 - 0.4327) RMSE ×10 - 6 ( g AOx/g ETOH) 4.6825 Relative error 0.0423 NA AICc - 1666 200 Table 4 - Case studies PLA - Resveratrol 3% (L= 5.08 × 10 - 3 cm) (Soto - Valdez et al., 2010) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 23 2P Estimated value ± Standard error (95% Asymptotic CI) 0.3988 ± 0.021 a 0.9756 ± 0.0083 (0.3576 - 0.44) (0.959 - 0.9921) RMSE ×10 - 6 ( g AOx/g ETOH) 4.7179 Relative error 0.0517 0.0085 0.559 AICc - 1664 201 Table 4 - Case studies PLA - Resveratrol 3% (L= 5.08 × 10 - 3 cm) (Soto - Valdez et al., 2010) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 33 1P Estimated value ± Standard error (95% Asymptotic CI) 4.8085 ± 0.2960 a (4.2170 - 5.3999) RMSE ×10 - 6 ( g AOx/g ETOH) 8.0134 Relative error 0.06 NA AICc - 1499 202 Table 4 - Case studies PLA - Resveratrol 3% (L= 5.08 × 10 - 3 cm) (Soto - Valdez et al., 2010) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 33 2P Estimated value ± Standard error (95% Asymptotic CI) 4.8084 ± 0.3321 a 1.2735 ± 0.013 (4.1446 - 5.4723) (1.2484 - 1.2986) RMSE ×10 - 6 ( g AOx/g ETOH) 8.0778 Relative error 0.0691 0.0099 0.4393 AICc - 1497 203 Table 4 - Case studies PLA - Resveratrol 3% (L= 5.08 × 10 - 3 cm) (Soto - Valdez et al., 2010) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 43 1P Estimated value ± Standard error (95% Asymptotic CI) 8.9965 ± 0.470 a (8.0508 - 9.9421) RMSE×10 - 6 ( g AOx/g ETOH) 7.2709 Relative error 0.052 NA AICc - 1133 204 Table 4 - Case studies PLA - Resveratrol 3% (L= 5.08 × 10 - 3 cm) (Soto - Valdez et al., 2010) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 43 2P Estimated value ± Standard error (95% Asymptotic CI) 8.9968 ± 0.5684 a 1.2279 ± 0.015 (7.8527 - 10.14) (1.1973 - 1.2585) RMSE ×10 - 6 ( g AOx/g ETOH) 7.3495 Relative error 0.0632 0.0124 0.5487 AICc - 1130 205 Table 4 - Case studies PLA - Resveratrol 3% (L= 5.08 × 10 - 3 cm) (Soto - Valdez et al., 2010) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH **43 1P Estimated value ± Standard error (95% Asymptotic CI) 8.9959 ± 0.4700 a (8.0503 - 9.9415) RMSE ×10 - 6 ( g AOx/g ETOH) 5.7834 Relative error 0.052 NA AICc - 1155 206 Table 4 - Case studies PLA - Resveratrol 3% (L= 5.08 × 10 - 3 cm) (Soto - Valdez et al., 2010) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH **43 2P Estimated value ± Standard error (95% Asymptotic CI) 8.9968 ± 0.5684 a 0.9691 ± 0.012 (7.8527 - 10.14) (0.945 - 0.993) RMSE ×10 - 6 ( g AOx/g ETOH) 5.8 Relative error 0.0632 0.0124 0.5487 AICc - 1153 207 Table 4 - Case studies ***PLA - Rutin (L= 4.83 × 10 - 3 cm) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 40 1P Estimated value ± Standard error (95% Asymptotic CI) 0.2127 ± 0.022 a (0.1687 - 0.2567) RMSE ×10 - 6 ( g AOx/g ETOH) 0.5640 Relative error 0.1032 NA AICc - 1637 208 Table 4 - 1 Case studies ***PLA - Rutin (L= 4.83 × 10 - 3 cm) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 40 2P Estimated value ± Standard error (95% Asymptotic CI) 0.1003 ± 0.036 a 0.052 ± 0.0068 1 (0.028 - 0.1726) (0.0383 - 0.0657) RMSE ×10 - 6 ( g AOx/g ETOH) 0.5533 Relative error 0.3593 0.1313 0.9699 AICc - 1638 209 Table 4 - Case studies ***PLA - Rutin (L= 4.83 × 10 - 3 cm) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 40 3P Estimated value ± Standard error (95% Asymptotic CI) 0.7695 ± 0.278 b (0.2123 - 1.3267) 0.0322 ± 0.0021 1 (0.028 - 0.036) 0.8086 ± 0.4324 (0.0583 - 1.6756) RMSE ×10 - 6 ( g AOx/g ETOH) 0.5514 Relative error 0.3611 0.0653 0.5348 0.8236; 0.8778; 0.5839 AICc - 1637 210 Table 4 - Case studies ***PLA - Quercetin ( L= 4.83 × 10 - 3 cm) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 40 1P Estimated value ± Standard error (95% Asymptotic CI) 0.6248 ± 0.028 a (0.5670 - 0.6825) RMSE ×10 - 6 ( g AOx/g ETOH) 0.4262 Relative error 0.0463 NA AICc - 2021 211 Table 4 - Case studies ***PLA - Quercetin ( L= 4.83 × 10 - 3 cm) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 4 (cm 2 /s) g AOx/g 95%ETOH 40 2P Estimated value ± Standard error (95% Asymptotic CI) 0.6362 ± 0.039 a 0.061 ± 0.0008 (0.5584 - 0.7140) (0.059 - 0.0623) RMSE ×10 - 6 ( g AOx/g ETOH) 0.4288 Relative error 0.0613 0.0135 Correlation 0.6402 AICc - 2019 212 Table 4 - Case studies PLA - Astaxanthin (L= 5.02 × 10 - 3 cm) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 8 (cm 2 /s) g AOx/g 95%ETOH 30 1P Estimated value ± Standard error (95% Asymptotic CI) 1.269 ± 0.347 a (0.5694 - 1.9683) RMSE ×10 - 6 ( g AOx/g ETOH) 0.022 Relative error 0.2735 NA AICc - 1585 213 Table 4 - Case studies PLA - Astaxanthin (L= 5.02 × 10 - 3 cm) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 8 (cm 2 /s) g AOx/g 95%ETOH 30 2P Estimated value ± Standard error (95% Asymptotic CI) 1.269 ± 0.4145 a 9.8946 ± 0.637 (0.4332 - 2.1051) (8.6086 - 11.18) RMSE ×10 - 6 ( g AOx/g ETOH) 0.022 Relative error 0.3266 0.0644 0.5311 AICc - 1583 214 Table 4 - Case studies PLA - Astaxanthin (L= 5.02 × 10 - 3 cm) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 8 (cm 2 /s) g AOx/g 95%ETOH 40 1P Estimated value ± Standard error (95% Asymptotic CI) 2.2838 ± 0.299 a (1.6811 - 2.8864) RMSE ×10 - 6 ( g AOx/g ETOH) 0.014 Relative error 0.1309 NA AICc - 1623 215 * Simulant used was 50% ETOH. ** Simulant used was 100% water. Table 4 - Case studies PLA - Astaxanthin (L= 5.02 × 10 - 3 cm) Temperature (°C) Number of estimation (P) Criteria estimated D × 10 - 10 M × 10 - 8 (cm 2 /s) g AOx/g 95%ETOH 40 2P Estimated value ± Standard error (95% Asymptotic CI) 2.2816 ± 0.4677 a 10.12 ± 0.63 (1.3385 - 3.2247) (8.861 - 11.39) RMSE ×10 - 6 ( g AOx/g ETOH) 0.014 Relative error 0.205 0.062 0.7632 AICc - 1621 216 Table 4 - *** Residuals for these studies did show a significant signature. The data needs to be log transformed to meet the standard s tatistical assumptions. Remarks: Values within a case study of a particular temperature in the same row with same alphabetic or numeric symbol are not statistically significantly different (p>0.05). RMSE=Root mean square error. 217 4.3 Conclusion Parameter estimation was implemented to analyze selected p ublished and unpublished data for migration of antioxidant(s) from PLA - based polymeric films into fatty food simulants, and in limited cases into an aqueous food simulant based on 1P, 2P, and 3P est imation s . Significant improvement in terms of meeting standard statistical assumptions was observed in the case where additional parameters were being considered for model fitting ( i.e., 3P estimation) based on the residual scatter plot, RMSE and the AICc observations. Initial , and optimal experimental design were found helpful in predicting and designing the right time for collecting sufficient data to obtain better P estimates with lower errors. Model s A and B presented in this chapter were successfully integrated into fitting the data. 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(2001). Antioxidant ability of BHT tocopherol impregnated LDPE film in packaging of oatmeal. Journal of the Science of Food and Agriculture, 81 (2), 194 - 201. Zhu, X., Lee, D. S., & Yam, K. L. (2012). Release property and antioxidant effectiveness of tocopherol - incorporated LDPE/PP blend films. Food Additives and Contaminants: Part A, 29 (3), 461 - 468. Zhu, X., Schaich, K. M., Chen, X., & Yam, K. L. (2013). An tioxidant Effects of Sesamol Released from Polymeric Films on Lipid Oxidation in Linoleic Acid and Oat Cereal. Packaging Technology and Science, 26 (1), 31 - 38. doi: 10.1002/pts.1964 224 Chapter 5 A Two - Step Solution to Estimate Mass Transfer Parameters of Migration Experiments Controlled by Diffusion, Partition and Convective Mass Transfer Coefficients 5.0 Introduction Migration in food packaging applications involves mass transfer phenomenon of additives from polymeric membranes into products and/or sim ulants. These additives could be antimicrobials, antioxidants, or any chemical substances that may serve multifunctional purposes to a food - package system. Normally, these additives are intended for prolonging the shelf life of a food product by means of inhibiting microbial growth, retarding lipid oxidation, etc. Often times, these additives add extra value such as enhan cing the flavor or nutritional aspect s of the product or protecting the polymer from degradation during processing - Franco et al., 2012; Ortiz - Vazquez, Shin, Soto - Valdez, & Auras, 2011; Samsudin, Valdez, & Auras, 2014) , or they are introduced to improve the polymeric membrane s ageing properties. Despite all the positive outcomes, the se additives or the by - products of the interaction between polymer and ad ditives can migrate into the food, thus reaching a safety threshold limit, which may adversely affect migration parameters determining this mass transfer phe nomenon, which may have safety concern and implications for shelf life determination. Since migration study is complex and time consuming, an increasing number of researche r s have focused on investigating the kinetics of migration by means of mathematical modeling. In depth physical understanding of the particular factor/parameters that govern the kinetics of migration can be obtained by solving the mathematical models describing these experiments. There are various types of mathematical models available , including deterministic , stochastic, 225 mechanistic, dynamic, etc. In the area of mass transfer, most of the mathematical m odels available are based on (Crank, 1979) (Carslaw & Jaeger, 1959) of the governing equations for the diffusion of chemical compounds through thin membranes. These solutions are using a deterministic approach and are expressed analytically . Even though several numerical approximation techniques have been developed and are available to estimate the kinetics of migration, the u se of analytical solutions is favored due to their simplicity and relationship to the physical phenomena driving the migration . Most of these mass transfer analytical solutions describe migration by solving for t wo kinetic migration parameters, the diffusi on ( D ) and the partition coefficients ( K p,f ) (Dhoot, Auras, Rubino, Dolan, & Soto - Valdez, 2009; G - Franco et al., 2012; Manzanarez - López, Soto - Valdez, Auras, & Peralta, 2011; Mascheroni, Guillard, Nalin, Mora, & Piergiovanni, 2010; Ortiz - Vazquez et al., 2011; Reinas, Oliveira, Pereira, Machado, & Poças, 2012; Samsudin et al., 2014) . To the solutions describing t he migration process consid er the convective mass transfer coefficient ( h ) between the films and the food / simulant (Galotto, Torres, G uarda, Moraga, & Romero, 2011; Gandek, Hatton, & Reid, 1989; Mascheroni et al., 2010; Pocas, Oliveira, Brandsch, & Hogg, 2012 ; Vitrac, Mougharbel, & Feigenbaum, 200 7 ) . The existence of one analytical solution capable of assessing the three kinetic migration parameter s ( D , K p,f and h ) can provide a method to unde rstand the physical process driving these experiments. Since there are many combinations of these parameters that can satisfy the solution of this equation, it is difficult to be sure that the solution used has the right global minimum that minimize s the s ums of squared errors (SSE). Often times, a complex non - linear mathematical equation results in many 226 local minima (Figure 5 - 1 ); thus finding the right global minimum region that provides the true solution can be a challenge. Therefore, the aims of this wo rk are: i ) to propose a two - step solution to estimate the D , K p,f and h kinetic migration parameters, ii ) to assess the proposed two - step solution by using experimental migration data, iii ) to introduce the kinetic phase diagram (KPD) as previously present ed by Vitrac and Hayert (2006) based on the forward approximation, and iv ) to use a bootstrapping technique on the residuals to find the true data distribution. Figure 5 - 1 Example of mult iple local minima for SSE in non - linear estimation. (a) Surface plot with default view, (b) surface plot view set at azimuth and elevation of 28, and 28, respectively, and (c) contour plot for minimum region of SSE. 227 5.1 Theoretical Development A two - step solution consists of step 1: a simplified model, and step 2: an ordinary least square estimation , presented in the next section. Step 1 is used to determine the initial guess for step 2. 5.1.1 Assumptions and Boundary Conditions Let us assume that we are conducting a migration ex periment where a piece of film added with an additive or chemical compound is in contact with a food or a simulant on two sides (Figure 5 - 2). To find the concentration of the additive as a function of time in the polymeric membrane and/or in the food simul ant, we need to make a number of assumptions to satisfy the boundary conditions used to derive the analytical solution. These assumptions include, but are not limited to: i) the initial concentration of additive is uniformly distributed inside of the film ii) th e food/food simulant is assumed well - mixed iii) the overall mass transfer is balanced iv) the system of film - liquid food simulant is closed v) no interaction occurs between the film and food/food simulant vi) the concentration profile is symmetrical across the film vii) the c oncentration of the additive far away from the film in the simulant is homogenous viii) the diffusion coefficient ( D ) is constant through out the experiment and it does not depend on the concentration of the additive ix) the partition coefficient, K p,f is constan t. 228 Figure 5 - 2 Gra phical representation of the kinetics of migration . Initial conditions; (Eq. 5 - 1) (Eq. 5 - 2) Boundary conditions; (Eq. 5 - 3) (Eq. 5 - 4) Now, the main differential equation describing the mass transfer process is provided by: (Eq. 5 - 5) 5.1.2.1 Step 1 I n this step, a simplified model is proposed based on the homogeneous constant - coefficient linear differential equation by using the mass balance. The motivation of proposing step 1 is to be able to find the right region of SSE global minimum for obtaining robust order magnitude Polymeric film F ood product/ simulant Polymeric film F ood product/ simulant 229 approximation for simultaneously estimate of all kinetic migration parameters ( i.e., K p,f , D , h ) before estimating the parameters by using the ordinary least square (OLS) estimation method. By using step 1, the accuracy of the estima tion is obtained since deviation from the true solution is avoided as the right region of SSE global minimum can be identified. Consequently, the initial guesses for the kinetic migration parameters can be obtained. Finding an appropriate initial guess for h in the mass transfer area is particularly challenging since this property imposes experimental difficulty due to the need to measure the additive concentration at the initial time of the experiment , resulting in two main issues: i ) high possibility of introducing significant experimental error, and ii ) the values obtained experimentally do not represent the t rue value at the interfacial boundary layer (since the concentration of additives is obtained via mathematical expression s assuming the spatial coordinate). Meanwhile, approximate values for D and K p,f can be obtained from published/known data for a simila r polymer - additive - food simulant system. Although the quest of initial guesses for D and K p,f are not quite as challenging as for h , the importance of having robust initial guesses when estimating these parameters simultaneously cannot be neglected to avo id over and/or under fitting estimation. The mass balance for the additive between the membrane an d the food and/or simulant can in itially be expressed as a linear relation between the different concentrations as shown: (Eq. 5 - 6) Although this is a large simplification since we are assuming that the mass coming out of the membrane i s an average between C 1 and C 2. ,w e can use this initial approach to calculate initial guesses to be used in step 2. Vitrac and Hayert (2006), for example, used one linear and quadratic solution to determine the concentration profile of additives inside a membrane to avoid this initial 230 simplification. However, an initia l parabolic profile does not produce guesses that are much different from this initial assumption. The interface property is expressed as follows : (Eq. 5 - 7) Solving for C 4 ( C f at time t ): (Eq. 5 - 8) (Eq. 5 - 9) (Eq. 5 - 10) (Eq. 5 - 11) Assuming a solution of the type for Eq. 5 - 11 (Eq. 5 - 12) (Eq. 5 - 13) By substituting Eq. 5 - 12 and 5 - 13 into Eq. 5 - 11, the following equation is obtained; (Eq. 5 - 14) (Eq. 5 - 15) (Eq. 5 - 16) Thus, Eq. 5 - 16 becomes Eq.5 - 17; (Eq. 5 - 17) Then; 231 ; ; Thus, at , resulting in By substituting into Eq. 5 - 17, Eq. 5 - 18 is obtained; (Eq. 5 - 18) The right side of Eq. 5 - 18 is equivalent to concentration of additives in the food or food simulant at e quilibrium similar to what can be obtained by the exact analytical solution. Note for ; at satisfied after a short time ( ) since has decreased and has increased resulting in Eq.5 - 7. Therefore, at , . As a result, a simplified model (Eq. 5 - 19) is obtained; where Eq. 5 - 19) By fitting Eq. 5 - 19 to the experimental data, a combination of P and R with the lowest sums of squared errors (SSE)= can be obtained. The best range of R can be selected accordingly to resulting in the best fit of P (Eq. 5 - 20) for a given range of R. ( Eq. 5 - 20 ) The combination of values of P and R that giv e the lowest SSE is used to first obtain the initial guess of K p,f by using . T hen by substituting K p,f inside of , the initial guesses of D and h can be obtained as follows ; 232 (Eq. 5 - 21) where and are numerical values. The expression above (Eq. 5 - 21) is extracted from R since and are two resistance in the series with only the total resistance determined by the rate of R . T hus D and h cannot be separated or obtained individually to establish the order of magnitude approximations as they both are physically influencing each other. The approximation of the initial guesses of both D and h based on Eq. 5 - 21 are and , respectively. The initial guesses obtained for K p,f , D and h are then used as the starting point of parameter estimation via the ordinary least square s estimation using the analytical solution in step 2. 5.1.2.2 Step 2 By using the Laplace transform, the analytical solution that satisfies the boundary conditions previously described in Eqs. 5 - 1 to 5 - 4 is derived. Laplace transforms: Partial differential equation: (Eq. 5 - 22) By rearranging Eq. 5 - 22, the following equation is obtained: , which has the solution , from which ; (Eq. 5 - 23) Boundary conditions; 233 (Eq. 5 - 24) (Eq. 5 - 25) Laplace transforms (side of food/food simulant): From Eq. 5 - 25; Eq. 5 - 26) Substituting this and Eq. 5 - 23 back into Eq. 5 - 25, the following equation is obtained: (Eq. 5 - 27) (Eq. 5 - 28) (Eq. 5 - 29) By using the residue theorem (complex variable theory), the inverse Laplace transform can be performed (to change to time domain): (Eq. 5 - 30) where the poles of in Eq. 5 - 29 are the roots of ; s is an infinite complex number corresponding to infinite number of roots of eigenvalues Let where and 234 By anticipating purely imaginary roots, set with where Residue at s =0; (Eq. 5 - 31) By dividing Eq. 5 - 31 with s, the following is obtained; (Eq. 5 - 32) Since the last limit is indeterminate (0/0), rule is employed: (Eq. 5 - 33) Residue (s=0)= (Eq. 5 - 34) Residue at s = s n ; (Eq. 5 - 35) Since s n is the root of the denominator rule is employed: (Eq. 5 - 36) Eq. 5 - 37) 235 Substitute Eq. 5 - 37 into Eq. 5 - 35, where s (= s n )= ; Residue at s=s n = (Eq.5 - 38) Substitute Eq. 5 - 34 and into Eq. 5 - 38; (Eq.5 - 39) Substitute Eq. 5 - 39 into Eq. 5 - 30; (Eq. 5 - 4 0) The mass that remains in the polymer at time t is: (Eq. 5 - 41) (Eq. 5 - 42) Now, if we divide numerator and denominator of Eq. 5 - 42 by , where and is the non - zero roots of ; 236 (Eq. 5 - 43) Since , the following equation is obtained; (Eq. 5 - 44) As , thus Eq. 5 - 44 becomes; (Eq. 5 - 45) Since , the following analytical solution is obtained; (Eq. 5 - 46) where ; and In order to ensure the analytical solution of Eq. 5 - 46 converge, the number of terms needed for a given accuracy (~98%) can be calculated as follows: Let z = where The solution converges faster when z decreases, thus in the worst case scenario is when z = 1 (slowest convergence). 237 % Accuracy= where sum of finite series= and sum of infinite series = . The infinite series was estimated with 100,000 terms, which is a good estimation. Table 5 - 1 Number of terms with its corresponding percent accuracy . Number of terms % Accuracy 1 99.99 2 100.00 3 100.00 . . . . 20 100.00 5.2 A Case Study Analysis A case study was selected to assess the proposed two - step solutions model. A data set of poly(lactic acid) (PLA) polymer incorporated with 3 wt.% resveratrol in contact with 100% ethanol and kept at 9 C (Soto - Valdez , Peralta, & Auras, 2008) was chosen to demonstrate the kinetics migration of the parameter estimation. The data was analyzed with different approaches by using MATLAB ® R2011b (MathWorks, Natick, MA, USA). 238 Statistical analysis was performed using an independent t - test (SPSS Statistics, version 22, 2013, IBM Corporation©, Armonk, NY, USA) to compare means between two parameters. 5.3 Assessment of the Two - Step Solutions Model 5.3.1 Step 1 Step 1 was performed by employing Eq. 5 - 19 using a code deve loped for MATLAB (see Appendix 5C). The combination of P and R that resulted in the lowest SSE was then used to determine the initial guesses of the kinetics migration parameters ( i.e., K p,f , D , and h ). 5.3.2 Scaled Sensitivity Coefficient, Sensitivity coefficient and scaled sensitivity coefficient are as described in Chapter 2 section 2.7.2. 5.3.3 Step 2 5.3.3.1 Ordinary Least Square (OLS) Estimation For step 2, Eq. 5 - 46 was employed. The parameters were estimated by minimizing the SSE using the non - l inear regression fitting function (nlinfit) of MATLAB. Additional information such as residuals, correlation coefficient matrix and asymptotic confidence interval (CI), to name a few , were also obtained. The r elative error of each parameter was calculated by dividing the standard error of the d . 239 5.3.3.2 Sequential Estimation Sequential estimation is a powerful and generic method for parameter estimation. It was developed based on the Gauss minimization method by usin g the matrix inversion lemma (Beck & Arnold, 1977) . This method updates parameter s each time new responses are added and is pa rticularly relevant for a time - dependent experime nt. This method does rely on prior information such as good initial guesses, the covariance matrix of the parameter, etc. (Beck & Arnold, 1977; Dolan & Mishra, 2013) . The s equential estimation method can be used to validate the OLS estimation. It can also offer additional insight ( i.e., time needed to collect sufficient data) in compari son to that of OLS estimation. The s equential estimation method was performed on the data set and the results were compared with the OLS results. 5.4 Kinetic Phase Diagram (KPD) The idea behind the kinetic phase diagram (KPD) is to extrapolate the migration phenomenon at equilibrium ( i.e., concentration of additives at time= t ). It is an algebraic differential equation that substitute s for the commonly used partial differential equation to descr ibe physical relationship s among parameters, kinetic parameters and measurement. This concept helps to avoid many difficulties encounter ed in solving partial differential equation s (Vitrac & Hayert, 2006) . A d imensionless analytical expression of KPD is obtained by taking the first derivative of the concentration changes with respect to time and as a function of the residual concentratio n in the polymer phase (Eq. 5 - 47 ). The dimensionless analytical expression was developed by Vitrac and Hayert (2006) by using the parabolic solution. (Eq. 5 - 47 ) Residual concentration, (Eq. 5 - 48 ) 240 (Vitrac & Hayert, 2006) : (Eq. 5 - 49 ) where (dimensionless concentration) ; (dimensionless position); (dimensionless time or Fourier, F o number) The p arabolic profile is defined based on its boundary conditions: ) (Eq. 5 - 50 ) Mass balance approximation is as follows : = (Eq. 5 - 51 ) where (the initial concentration in food/food simulant); (characteristic of reservoir volume of food/food simulant); (dimensionless length/thickness) . By combining Eq. 5 - 50 and E q. 5 - 51 , a more practical form of the boundary conditions is obtained; ( Eq. 5 - 52 ) (Eq.5 - 53 ) where (dimensionless flux); (concentration at the interface based on the assumption of local thermodynamic equilibrium); ( concentration in the food/food simulant at equilibrium). (Eq. 5 - 54 ) By taking the dimensionless flux of and Eq. 5 - 48, Eq. 5 - 54 becomes; (Eq. 5 - 55 ) 241 The KPD expression (Eq. 5 - 47) is implied from Eq. 5 - 52 and by substituting with its calculated va lue using Eq. 5 - 56 . The mass balance for the polymer phase is also defined from as it changes with time; = (Eq. 5 - 56 ) Thus, Eq. 5 - 58 becomes; (Eq. 5 - 57 ) More details on the KPD profile solution can be found in Vitrac and Hayert (2006). The data set of the selected case study was analyzed using the KPD approach to acquire additional understanding of experiment kinetics, in turns, the kinetic migration parameters. 5.5 Bootstrap Method Bo otstrap is a rando m resampling method based on statistical inference. This method is very powerful and useful since it better approximates the true distribution of the population in the case of an insufficient data set. This method counteracts any experime ntal errors that could have caused any disproportionate skewness of the data/and or residual distribution. There are many types of bootstrapping methods and residual bootstrapping is one of them. This method in particular is beneficial for a small data set and for when the magnitude of response of a parameter is large at a short time interval. where (residual); (response variable); (predicted value). Synthetic data ) is created by adding a random and resampled er ror term : 242 This synthetic data is then generated for n =1000 and is run through the model for estimated bootstrap parameters. 5.6 Results and Discussions The case study chosen to illustrate the two - step solution was PLA film incorpora ted with 3% wt. resveratrol tested at 9 C in ethanol. The thickness of the film was 2 mil (5.08 10 - 5 m) and eight round discs of film attached to a stainless steel wire and separated by glass beads were used for the migration study (Soto - Valdez et al., 2008) . Additional case studies were also analyzed and the results can be found in Appendix 5 A and 5 B. 5.6.1 Step 1 The newly proposed simplified model provided a good fit to the experimental data. Figure 5 - 3 shows experimental data sets with fitted values computed using Eq. 5 - 19. The combination of P and R values that gave the lowest SSE ( 4.08 from Eq.5 - 19 was found to be 0.0013 and 9.00 , respectively. These values were then used to compute the initial guesses for K p,f , D , and h , which were found to be 605.63 cm 3 PLA/cm 3 ethanol, 5.88 cm 2 /min, and 2.80 , respectively. It is worth mentioning that the K p,f value that is always experimentally determined at the end of the experiment by extracting the total amount of additive remaining in the film can be estimated using early data and at the beginning of the migration process. 243 Figure 5 - 3 Exp erimental data fitting by using the simplified model of step 1. Initial approximation values obtained from this step were: D =5.88 10 - 12 cm 2 /min, K p,f =605.63 cm 3 PLA/cm 3 ethanol, and h =2.80 10 - 6 cm/min. 5.6.2 Scaled Sensitivity Coefficient, Figure 5 - 4 shows the scaled sensitivity coefficient, of all the kinetic migration parameters. Based on this figure, it was found that K p,f had the highest absolute magnitude of response; thus this parameter can be estimated easier with highest accuracy, followed by D and h . There were no high correlations ( ) found among the parameters (Table 5 - 2 ). The highest correlation coefficient ob served among the parameters was between D and K p,f , which was 0.95 . 244 From initial observation, it can be anticipated that K p,f will hav e the lowest relative error followed by D and h . Consequently, all these parameters were then estimated simultaneously by using the OLS estimation method. Figure 5 - 4 Sc aled sensitivity coefficient of the kinetics migration parameters using initial guesses obtained from step 1. Initial guesses were: D = 6.50 10 - 12 cm 2 /min, K p,f = 605.00 cm 3 PLA/cm 3 ethanol, and h = 5.90 10 - 6 cm/min. 245 5.6.3 Step 2 5.6.3.1 Ordinary Least Square (OLS) Estimation All three kinetic migration parameters were successfully estimated by the proposed analytical solution (Eq. 5 - 46) (Figure 5 - 5). The estimated K p,f , D and h values were 548.87 21.76 cm 3 PLA/cm 3 ethanol, 5.42 0.62 10 - 12 cm 2 /min, and 2.56 0.38 10 - 6 cm/min , respecti vely. The reported experimental K p,f value of this case study was 506.10 cm 3 PLA/cm 3 ethanol, which was not to o far off from the estimated K p,f value, thus indicating an advantage from using the newly proposed model. Additionally, h was estimated for th e first time without having an experimental set up. The importance of estimating h may have not be great for this particular case study since it seems not to be controlling the mass transfer process; however, it can be neglected since it could lead to over estimation or underest imation of a migration scenario . The feasibility of estimating h helps in reducing the experimental error due to the difficulty in measuring this parameter at the initial time. Vitrac et al. (2007) reported h of a range of 1.2 10 - 4 to 1.2 10 - 3 cm/min for alcohol homologous from a low density poly(ethylene), (LDPE) film into ethanol at 40 C, which was higher than that estimated in this study. The low affinity of the migrant ( i.e., alcohol homologous) to ethanol besides the higher e xperimental temperature used could be responsible for the higher h value than that observed in this study. The existence of a slight interfacial resistance at the food simulant - PLA polymer interface was observed as shown by the kine tic desorption plot (Fig ure 5 - 6). On the other hand, the estimated D value found in this study was significantly lower by o ne magnitude order than that reported by Soto - Valdez et al. (2010) (5.42 cm 2 /min vs. 2.04 cm 2 /min. This discrepancy could be due to the diffe rent model selection. The accuracy of model selection can be assessed using the corrected Akaike information criterion (AICc) as a guideline. Further discussion can be found in Chapter 6. 246 Relative errors of K p, , D and h were found to be 3.97, 11.38, and 14 .97 %, respectively, thus confirming the indication observed from the scaled sensitivity coefficient plot (Figure 5 - 3). Residual plot (Figure 5 - 7) appears to be normally scattered with constant variance and additive errors, although a slight signature can be seen at time < 1 10 5 min . This issue could be due to experimental error as can be observed in Figure 5 - 5. Additional information such as correlation coefficients and confidence interval can be found in Table 5 - 2 . Figure 5 - 5 Migration of 3 wt.% of resveratrol into ethanol at 9 C. 247 Figure 5 - 6 Desorption kinetics of PLA - 3 wt.% resveratrol at 9 C in the dimensionless time space (Fourier number= ). 248 Figure 5 - 7 Re sidual plot for migration of 3 wt.% of resveratrol into ethanol at 9 C. Table 5 - 2 A dditional information of OLS estimation . Parameters 95% Confidence interval Correlation coefficients, D (cm 2 /min) Kp,f (cm 3 PLA/cm 3 ethanol) h (cm/min) D (cm 2 /min) 4.19 6.65 10 - 12 Symmetric Kp,f (cm 3 PLA/cm 3 ethanol) 505.56 592.17 0.95 h (cm/min) 1.80 - 3.33 10 - 6 0.72 0.83 249 5.7 Sequential Estimation Sequential estimation result s for each parameter were normalized by the estimates of dividing the i th parameter by its final value to counteract the magnitude order differences among parameters. Figure 5 - 8 demonstrates the normalized sequential result for each parameter. All the estimated parameters were found to reach a constant state at around 2.0 10 5 min (~139 days) ; thus implying that the measurement does not need to be continued for much longer time afterward since it did not affect the pa rameters. Additionally, the parameters are expected to reach a constant state before an experiment ends. In the case that the constant state is not reached for a parameter, among the reasons that could have been responsible are insufficient data, number of parameters (too little or too many) , and an imperfect physical model (Beck & Arnold, 1977) . This particular information is an added insight that could not be gained from the OLS. Besides, the addition al effect of new responses can be observed through the sequential estimation method since it continuously updates the parameters (Beck & Arnold, 1977) . Table 5 - 2 shows the comparison between the OLS and the sequential estimation results. The results from both methods were in an agreement with each other. 25 0 Figure 5 - 8 N ormalized sequential parameters as a function of time for migration of 3 wt.% of resveratrol into ethanol at 9 C. 251 Table 5 - 3 Comparison between OLS and sequential results. Note: RMSE (Root mean square errors) unit= cm 3 ethanol/cm 3 PLA; Data is presented as mean ± standard error ; similar superscript s represent no statistical differe nce at p>0.05 within the same row. 5.8 Kinetic Phase Diagram (KPD) An approximation space, known as kinetic phase diagram (KPD) was used to assess information of the kinetics of migration (Vitrac & Hayert, 2006) . This approach allows an easy approximation for sorption and desorption kineti cs of migration. In a case where equilibrium state is not reached, this approach may be used to extrapolate the equilibrium state values by a linear approximation theory. Figure 5 - 9 (a) shows the sorption (migration) kinetic of 3 wt.% resveratrol Parameters OLS Sequential Estimates Standard error Relative errors (%) RMSE 10 - 5 Estimates Standard error Relative errors (%) RMSE 10 - 5 D 10 - 12 (cm 2 /min) 5.42 0.62 a 11.38 7.91 5.42 0.62 a 11.47 7.91 Kp,f (cm 3 PLA/cm 3 ethanol) 548.87 21.76 a 3.97 548.87 21.95 a 3.99 h 10 - 6 (cm/min) 2.56 0.38 a 14.97 2.56 0.39 a 15.07 252 into etha nol, and its corresponding KPD. The noise due to experimental errors was filtered using a non - deterministic filtering method (weighting kernels) based on the local polynomial approximants (Ducruet et al., 2007; Vitrac & Hayert, 2006) . As a result, the noise seen at time < 57 days becomes smoother as indicated by the filtered data (square dark cyan symbol) (Figur e 5 - 9 (a)). Figure 5 - 9(b) shows the derivative of concentration as a function of time plotted vs concentration ( C f ). A linear extrapolation that can be used to predict the theoretical equilibrium state ( i.e., the concentration of additive at equilibrium) b y making the first de rivative of the concentration as a function of time equ al to zero is indicated by the blue line intersection in Figure 5 - 9 (a and b). Also, the initial slope of the KPD (Figure 5 - 9 (b)) provides an initial estimation of the rate of the compound leaving the polymer film. Additional interpretation of the KPD method can be obtained elsewhere (Ducruet et al., 2007; Vitrac & Hayert, 2006) . Figure 5 - 9 (a) Sorpti on kinetic of 3 wt.% resveratrol into ethanol, and (b) KPD. The b lue line indicates the equilibrium state. 253 5.9 Residual Bootstrap Residual bootstrap was performed and the results as expected did improve in terms of confidence interval (Table 5 - 3) and resi dual distribution (Figure 5 - 10) in comparison to the OLS results. However, a slight shift was observed in the lower bound of the bootstrap confidence interval and the upper bound shifted to lower level s than that of the asymptotic confidence interval for D and h . Si milar outcomes were reported for the kinetic degradation of anthocyanins in grape pomace, indicating the flexibility of bootstrap from constrained need of being symmetric (Mishra, Dolan, & Yang, 2011) . Clear improvement of the bootstrap method can be visualized in Figure 5 - 11. The bootstrap bands for both confidence and prediction are tighter than the asymptotic ones. Therefore, accuracy of estimation can be improved since the bootstrap method did statistical inference based on the size of the population ( i.e., for the 84 experimental data points and n =1000, we obtained 84,000 new resa mpling residual points), thus apparent results can be anticipated. Specifically, the information from bootstrap provides higher accuracy of a migration limit since it represents the population size with a tighter bandwidth of confidence and prediction inte rvals. Therefore, the information generated by this method could be used improve the uncertainty on migration limit that are of concern in food safety and provide tighter shelf life determination. Additionally, the bootstrap method does not rely on the dis tribution assumption, thus providing the actual representation of residuals, even when the sample size is insufficient and the data is ill - posed. 254 Figure 5 - 10 Hist ogram of the bootstrap residuals. Table 5 - 4 C omparison between 95% asymptotic and 95% bootstrap confidence intervals of each parameter. Parameters 95% Asymptotic confidence interval 95% Bootstrap confidence interval D 10 - 12 (cm 2 /min) 4.19 - 6.65 4.46 - 6.55 K p,f (cm 3 PLA/cm 3 ethanol) 505.56 - 592.17 513.02 - 584.87 h 10 - 6 (cm/min) 1.80 - 3.33 1.97 - 3.31 255 Figure 5 - 11 M igration of 3 wt.% of resveratrol into ethanol at 9 C with added bootstrap results. 5.10 Conclusion A two - step solution model was developed based on obtain an analytical solution that provides a reasonable approximation for initial guesses needed for sim ultaneous parameter estimation allowing the estimation of thre e important kinetic migration parameters ( D , K p,f , and h ) . This model was used to determine the D , K p,f , and h of the migration of PLA added with 3%wt resveratrol. The new two - step solution was also successfully used to analyze a few selected case studies ( i.e., PLA incorporated with 2.6 wt.% - tocopherol into ethanol at 23 C and PLA incorporated with 1.28 wt.% catechin into 95% ethanol at 40 C) that are shown in the Appendix 256 5 A an d 5 B. The KPD approach was used to give additional insight on the sorption and desorption kinetics of the migration phenomenon. The residual bootstrap method provides insight information regarding the parameter estimation since they are estimated from a la rger sample size population. This technique is particularly important since most of the data sets analyzed had a small sample size and large magnitude response was concentrated at the small region of time space. Further data validation is needed to ensure the robustness of the two - step solution model. 257 APPENDICES 258 A PPENDIX 5 A : Migration of poly(lactic acid), PLA incorporated with 2.6 wt.% - tocopherol into ethanol at 23 C. Step 1 Figure 5A - 1 shows the fitting of the step 1 solution Eq. 5 - 19 to PLA incorporated with 2.6 wt.% - tocopherol into ethanol at 23 C. Figure 5A - 1 Experimental data fitting by using the simplified model of step 1. Initial approximation values obtained from this step were: D =1.88 10 - 9 cm 2 /min, K p,f =608.11 cm 3 PLA/cm 3 ethanol, and h =4.19 10 - 4 cm/min. 259 Scaled Sensitivity Coefficient, Figure 5A - 2 d emonstrates the for PLA incorporated with 2.6 wt.% - tocopherol into ethanol at 23 C for three parameters. Since D and h are highly correlated ( ), only D and K p,f were estimated. h was not estimated due to it having the smallest magnitude of response. Figure 5A - 3 shows the for for D and K p,f . Figure 5A - 2 Scaled sensitivity coefficient of the kinetics migration parameters using initial guesses obtained from step 1. Initi al guesses were: D =1.90 10 - 9 cm 2 /min, K p,f =608.11 cm 3 PLA/cm 3 ethanol, and h =8.00 10 - 4 cm/min. 260 Figure 5A - 3 Scaled sensitivity coefficient of the kinetics migration parameters ( D and K p,f ) using initial guesses obtained from step 1. Initial guesses were: D =2.00 10 - 9 cm 2 /min, K p,f =609.00 cm 3 PLA/cm 3 ethanol. 261 Step 2 Figure 5A - 4 shows the migration of 2.6 wt.% - tocopherol into ethanol at 23 C with the predicted values, 95% confid ence and prediction intervals. Meanwhile, Figure 5A - 5 shows the desorption kinetic of the aforementioned migration in the dimensionless time. Ordinary Least Square (OLS) Estimation Figure 5A - 4 Migration of 2.6 wt.% - tocopherol into ethanol at 23 C. 262 Figure 5A - 5 Desorption kinetics of PLA - 2.6 wt.% - tocopherol at 23 C in the dimensionless time space (Fourier number= ). 263 Figure 5A - 6 demonstrates the residuals plot for the two estimated parameters ( i.e., D and K p,f ). Signature of the residuals can be observed, in particular at the beginning of the experimental duration. Figure 5A - 6 Residual plot for migration of 2.6 wt.% - tocopherol into ethanol at 23 C for two parameters estimation. 264 OLS results can be found in Table 5A - 1 and the comparison between the two estimation approaches ( i.e., OLS and sequential) can be observed in Table 5A - 2. Figure 5A - 7 shows the normalized sequential parameters, which indicated that the parameters had reached their constants state toward the end of experimental duration. Table 5A - 1 Additional information of OLS estimation for migration of 2.6 wt.% - tocopherol into ethanol at 23 C. *NE: Not estimated Parameters 95% Confidence interval Correlation coefficients, Estimates D (cm 2 /min) Kp,f (cm 3 PLA/cm 3 ethanol) h (cm/min) D 10 - 10 (cm 2 /min) 64.41 0.075 62.92 - 65.91 Symmetric Kp,f (cm 3 PLA/cm 3 ethanol) 631.54 7.08 617.46 - 645.63 0.72 h (cm/min) NE 265 Table 5A - 2 Comparison between OLS and sequential results. Note: RMSE (Root mean square errors) unit= cm 3 ethanol/cm 3 PLA Parameters O LS Sequential Estimates Standard error Relative errors (%) RMSE 10 - 5 Estimates Standard error Relative errors (%) RMSE 10 - 5 D 10 - 10 (cm 2 /min) 64.4130 0.753 1.17 12.29 64.4323 0.734 1.14 13.03 Kp,f (cm 3 PLA/cm 3 ethanol) 631.54 7.08 1.12 631.83 7.06 1.12 266 Figure 5A - 7 Normalized sequential parameters as a function of time for migration of 2.6 wt.% of - tocopherol into ethanol at 23 C. 267 Kinetic Phase Diagram (KPD) Figure 5A - 8 (a) shows the KPD for migration of 2.6 wt.% of - tocopherol into ethanol at 23 C fitted with the experimental data and Figure 5A - 8(b) shows the linear slope that can be used to extrapolate the equilibrium state of - tocopherol. Figure 5A - 8 (a) Sorption kinetic of 2.6 wt.% - tocopherol into ethanol, and (b) KPD. The b lue line indicates the equilibrium state. 268 Residual Bootstrap Figure 5A - 9 Histogram of the bootstrap residuals. The results from residual bootstrapping can be found in Figure 5A - 9 and Table 5A - 3. Results indicated the improvement of the residual distribution as can be observed in Figure 5A - 10. 269 Table 5A - 3 Comparison between 95% asymptotic and 95% bootstrap confidence intervals of each parameter. Parameters 95% Asymptotic confidence interval 95% Bootstrap confidence interval D 10 - 10 (cm 2 /min) 62.91 - 65.91 63.49 - 66.13 Kp,f (cm 3 PLA/cm 3 ethanol) 617.46 - 645.63 618.26 - 641.84 Figure 5A - 10 Migration of 2.6 wt.% - tocopherol into ethanol at 23 C with added bootstrap results. 270 APPENDIX 5 B: Migration of poly(lactic acid), PLA incorporated with 1.28 wt.% catechin into 95% ethanol at 40 C. Step 1 Figure 5B - 1 shows the experimental data fitting of migration of 1.28 wt.% catechin into 95% ethanol at 40 C from PLA with Eq. 5 - 19. Figure 5B - 1 Experimental data fitting by using a simplified model of step 1. Initial approximation values obtained from this step were: D =1.39 10 - 8 cm 2 /min, K p,f =318.97 cm 3 PLA/cm 3 ethanol, and h =0.0024 cm/min. 271 Scaled Sensitivity Coefficient, Figure 5B - 2 indicates that all parameters can be estimated simultaneously for the migration of 1.28 wt.% catechin into 95% ethanol at 40 C from PLA. Figure 5B - 2 Scaled sensitivity coefficient of the kinetics migration parameters using initial guesses obtained from step 1. Initial guesses were: D =3.00 10 - 8 cm 2 /min, K p,f =318.97 cm 3 PLA/cm 3 ethanol, and h =0.0040 cm/min. 272 Step 2 Ordinary Least Square (OLS) Estim ation Figure 5B - 3 and Figure 5B - 4 show the migration of 1.28 wt.% catechin into 95% ethanol at 40 C and desorption kinetics of the similar migration study in the dimensionless time space, respectively. Figure 5B - 3 Migration of 1.28 wt.% catechin into 95% ethanol at 40 C. 273 Figure 5B - 4 Desorption kinetics of PLA - 1.28 wt.% catechin at 40 C in the dimensionless time space (Fourier number= ). 274 Residual plot of migration of 1.28 wt.% catechin into 95% ethanol at 40 C showed better normal residual distribution in comparison with the other two case studies (Figure 5B - 5). Comparison between OLS and sequential estimations can be found in Table 5B - 2. Addition al OLS results are shown in Table 5B - 1. Figure 5B - 5 Residual plot for migration of 1.28 wt.% catechin into 95% ethanol at 40 C. 275 Table 5B - 1 Additional information of OLS estimation for migration of 1.28 wt.% catechin into 95% ethanol at 40 C. Parameters 95% Confidence interval Correlation coefficients, Estimates D (cm 2 /min) Kp,f (cm 3 PLA/cm 3 95% ethanol) h (cm/min) D 10 - 10 (cm 2 /min) 225.86 13.43 198.64 - 253.08 Symmetric Kp,f (cm 3 PLA/cm 3 95% ethanol) 324.04 4.15 315.59 - 332.49 0.64 h 10 - 4 (cm/min) 44.00 4.26 36.00 - 53.00 0.60 0.56 276 Table 5B - 2 Comparison between OLS and sequential results. Note: RMSE (Root mean square errors) unit= cm 3 95% ethanol/cm 3 PLA. Parameters OLS Sequential Estimates Standard error Relative errors (%) RMSE 10 - 5 Estimates Standard error Relative errors (%) RMSE 10 - 5 D 10 - 10 (cm 2 /min) 225.86 13.40 5.92 9.6412 225.84 13.34 5.91 9.6413 Kp,f (cm 3 PLA/cm 3 95% ethanol) 324.04 4.15 1.28 324.04 4.14 1.28 h 10 - 4 (cm/min) 44.00 4.26 9.62 44.00 4.25 9.60 277 Sequential plot of estimated parameters of migration of 1.28 wt.% of catechin into 95% ethanol at 40 C reached a constant about 8 hrs (500 min) (Figure 5B - 6). Figure 5B - 6 Normalized sequential parameters as a function of time for migration of 1.28 wt.% of catechin into 95% ethanol at 40 C. 278 Kinetic Phase Diagram (KPD) Figure 5B - 7 (a) and (b) indicates the sorption kinetic of 1.28 wt.% catechin into 95% ethanol and its KPD space that corresponding to the theoretical sorption equilibrium. Figure 5B - 7 (a) Sorption kinetic of 1.28 wt. % catechin into 95% ethanol, and (b) KPD. The b lue line indicates the equilibrium state. 279 Residual Bootstrap The bootstrap residual histogram of migration of 1.28 wt.% catechin into 95% ethanol had a normal Gaussian distribution (Figure 5B - 8). Th e 95% bootstrap confidence interval had a tighter and asymmetrical band compared to 95% asymptotic confidence interval (Table 5B - 3), which indicated the flexibility of bootstrap under no assumption of the shape of the residual distribution. Figure 5B - 8 Histogram of the bootstrap residuals. 280 Table 5B - 3 Comparison between 95% asymptotic and 95% bootstrap confidence intervals of each parameter. Parameters 95% Asymptotic confidence inte rval 95% Bootstrap confidence interval D 10 - 10 (cm 2 /min) 198.64 - 253.08 203.68 - 251.37 Kp,f (cm 3 PLA/cm 3 ethanol) h 10 - 4 (cm/min) 315.59 - 332.49 36.00 - 53.00 318.09 - 333.00 39.00 - 54.00 281 Figure 5B - 9 Migration 1.28 wt.% catechin into ethanol at 40 C with added bootstrap results. 282 APPENDIX 5C: Example of MATLAB coding %% CONVECTIVE MASS TRANSFER STUDY %% HouseKeeping & Data close all; clear all; clc format long data =xlsread('PLA_RESV3_9C_REPS');% To read raw data from excel global L global alpha %% %% File Directory switch localname case '13 - 138 - 76.client.wireless.msu.edu' local = '/Users/HAYATISAMSUDIN/Documents/MATLAB_EXP/MATLAB_CIADEXP/MODELS COMPARIS ON'; papertype = 'A4'; paperposition = [0.3397 10.1726 20.3046 9.3322]; %cm otherwise local = pwd; papertype = 'usletter'; 283 paperposition = [0.3397 10.1726 20.3046 9.3322]; %update the values to usle tter format warning('Please set the case for your computer') end datafile = 'PLA_RESV3_9C_REPS.xls'; outputfolder = fullfile(local,'Figures_PLARESV3_9C'); if ~exist(outputfolder,'dir'), mkdir(outputfolder), end [~,outputfile] = fileparts(datafile); %% Extracted Info L=0.00254;%L= half of the thickness, cm A=3.1416; %A=area, cm2 Co=0.028115; % Co=Initial concentration, g/cm3 Vf=1.227 % Vf=volume of food,cm3 t=data(:,1); yobs=data(:,2); yobs2=data(:,2)./Co; %% Step 1: Simplified Model sum1=0; sum2=0; R=linspace(0.00000850,0.000009000,100); N=length(data); for j=1:length(R) for i=1:N if i == 1 284 fit(i) = 0 else z=1 - exp( - R(j)*t(i)); sum1=sum1+z.*yobs2(i); sum2=sum2+z.^2 P(j)=sum1/sum2 fit=P(j)*z; end end fit reverse=fit'; SSE(j)=sumsqr(yobs2 - fitreverse) end comparison=[P' R' SSE'] Index=0:1:N; [M,I]=min(SSE);%I is index refers to min value of SSE showminSSE=[P(I) R(I)] % to display P and R corresponding to I=index of min SSE figure [hAx,hLine1,hLine2]=plotyy( SSE,P,SSE,R) title('Combination of P and R with SSE','FontSize',16,'FontWeight','bold') xlabel('SSE','FontSize',16,'FontWeight','bold') ylabel(hAx(1),'P','FontSize',16,'FontWeight','bold') % left y - axis ylabel(hAx(2),'R','FontSize',16,'FontWeight','bold') % right y - axis 285 %% for i=1:N if i == 1 fit(i) = 0 else z=1 - exp( - R(I)*t(i)); final_fit(i)=P(I)*z; end end display(final_fit) predict_obs=final_fit'; Table=[t yobs final_fit' ] figure hold on set(gca, 'fontsize',14,'fontweight', 'bold'); pl(1)=plot(t, yobs2, 'o','LineWidth',1.01) pl(2)=plot(t, final_fit,'r','LineWidth',1.01) xlabel('Time (min)','fontsize',16,'fontweight','bold'); ylabel('Exp data, final fit','fontsize',16,'fontweight','bold'); pll=legend (pl,'exp data','final fit' ); set(pll,'box','off','location','Best'); set(gca,'box','on','xticklabelmode','auto','yticklabelmode','auto') %% Printing 286 print_pdf(600,get(gcf,'filename'),outputfolder,'nocheck') print_png(300,get(gcf,'filename'),outputfolder,[],0,0,0) %% Initial Guesses Approximations m0=Vf/(A*L); m0=P(I)*(Vf/(A*L)); Kpf=(1 - m0)/P(I); %P(I)= value contain the lowest SSE from before m1= (2*(Kpf+(Vf/(A*L)))) %term 1 of R m2=m1/R(I) D=1; h=(2*Kpf)/((m2/(Vf/A)) - (L/D)) h1=1 D=L/((m2/(Vf/A)) - (2*Kpf/h1)) % R= %(lowest SSE fr om before);To obtain the D and h as initial guesses D=6.5e - 12; % Initial guess for D (cm2/min)6.5E - 12 h=5.9e - 6; %% Step 2: OLS %Solving for Eigenvalues format long global qn global Bi alpha=Vf/(Kpf*A*L); % qn; 287 Bi=(h*L)/D; Lo=1; Up=8 0; z=@(qn) (Bi - Kpf*alpha*qn^2)*sin(qn)+alpha*Bi*qn*cos(qn); for i=Lo: Up; ev(i)=fzero(z,i); %Newton - Raphson end; EV=unique(ev); EV' ev=EV(2:end); qn=ev' %% Initial parameter guesses beta0(1)=6.5 ; % Initial guess for D (cm2/min) x 10^ - 12 beta0(2)=6.05 ; %Kpf x 10^2 beta0( 3)=5.9 ;%h x 10^ - 6 beta=beta0; % Set beta to the initial guesses %% X' = scaled sensitivity coefficients using forward - difference % This is a forward problem with known approximate parameters Xp=SSC_convecdiff(beta,t,@myfunconvecdiff); %% printing print_pdf(600,get(gcf,'filename'),outputfolder,'nocheck') print_png(300,get(gcf,'filename'),outputfolder,[],0,0,0) 288 %% ypredict = myfunconvecdiff(beta,t) % To check and compare with experimental values and to check for matrices dimension figure plot(t,yobs2,'o') hold on plot(t,ypredict,' - ') %% Nlinfit regression Nt=length(t); [beta,resids,J,COVB, MSE] = nlinfit(t,yobs2,@myfunconvecdiff,beta0); ci=nlparci(beta,resids,J,0.05) %asymptotic 95% confidence interval [ypredict, delta] = nlpredci( 'myfunconvecdiff',t,beta,resids,J,0.05,'on','curve'); %CI for mean [ypredict, deltaobs] = nlpredci('myfunconvecdiff',t,beta,resids,J,0.05,'on','observation'); beta; % parameters estimated [R, sigma]=corrcov(COVB); R sigma % parameter standard errors relati ve_error=sigma./beta'% >0.6 the likeliness of CI contains 0 is high (estimate is useless since it is not statistically diff than 0) MSE RMSE=MSE^(1/2) %% Model Discrimination n = length(ypredict); 289 p = length(beta_n); SS= MSE*(n - p) K=p +1; AICC= n*(log(SS/n ))+ 2*K+(((2*K)*(K+1))/(n - K - 1)) %% figure grid on hold on plot(t,yobs2,'or') plot(t,ypredict,' - b') xlabel('Time (min)','fontsize',16,'fontweight','bold'); ylabel('yobserved, ypredicted','fontsize',16,'fontweight','bold'); %% Plotting the diffusion of anti oxidant into simulant over time and its corresponding residual plot figure asyCImax1 = ypredict + delta;% Asymtotic confidence interval (CI) asyCImin1 = ypredict - delta;% Asymtotic confidence interval (CI) predCImax1 = ypredict + deltaobs;% Prediction in terval (PI) predCImin1 = ypredict - deltaobs;% Prediction interval (PI) hold on set(gca, 'fontsize',14,'fontweight','bold'); h1(1) = plot(t,yobs2,'ob'); %Plot y observed over time h1(2) = plot(t,ypredict,'r', 'LineWidth',1.2); % Plot y predicted over time 290 h1(3) = plot(t,asyCImax1,' - .g','LineWidth',1.05); % Plot CI as dotted line h1(4) = plot(t,predCImax1,' - c','LineWidth',1.05'); % Plot upper PI as solid line plot(t, asyCImin1,' - .g','LineWidth',1.05);% Plot lower CI as dashed line plot(t, predCImin1,' - c','Li neWidth',1.05); % Plot lower PI as dashed line xlabel('Time (min)', 'fontsize', 16, 'fontweight','bold'); ylabel('C_{f}(t)/C','fontsize', 16, 'fontweight','bold'); plf=legend (h1,'exp data','best fitted line','95% confidence interval','95% prediction inte rval'); set(plf,'box','off','location','Southeast'); set(gca,'box','on','xticklabelmode','auto','yticklabelmode','auto') %% printing print_pdf(600,get(gcf,'filename'),outputfolder,'nocheck') print_png(300,get(gcf,'filename'),outputfolder,[],0,0,0) %% Resid ual figure hold on set(gca, 'fontsize',14,'fontweight','bold'); h1(1) = plot(t,resids,'*');% Plot residual over time xlabel('Time (min)','fontsize', 16, 'fontweight','bold'); ylabel('Residuals','fontsize', 16, 'fontweight','bold'); YLine = [0 0]; XLine = [ 0 7000]; plot (XLine, YLine,' -- r') set(gca,'box','on','xticklabelmode','auto','yticklabelmode','auto') 291 %% printing print_pdf(600,get(gcf,'filename'),outputfolder,'nocheck') print_png(300,get(gcf,'filename'),outputfolder,[],0,0,0) %% Biot no. Biot=( beta(3)*L)/beta(1) %% Residual Bootstrap %% simultaneous confidence bands for regression line CBu=ypredict+delta; CBl=ypredict - delta; %% simultaneous prediction bands for regression line PBu=ypredict+deltaobs; PBl=ypredict - deltaobs; %% bootstrap CI for b eta nboot=1000; mm=2;%use x1, y1; x2, y2;...method of bootstrapping %for data bootstraping m=1, use myfunconvecdiff_boot..otherwise %myfunconvecdiff %mm=2;%use x1, Ypred1+e1; x2, Ypred2+e2;...residual bootstrapping nlinfitcheck=statset('nlinfit'); nlinfit check.FunValCheck='off'; %options = statset('FunValCheck','on'); for j=1:nboot r=round(1 + (n - 1).*rand(n,1));%index of random integers from 1 to n 292 for i=1:n if mm==1 tt(i)=t(r(i));% tt(i) each time for bootstrapped datum yboot(i)=yobs2(r(i));%yboot(i) is the value for each bootstraped datum end if mm==2 tt=t; yboot(i)=ypredict(i)+resids(r(i)); if i==n yboot=yboot' end end end % yboot' %[betab(j,:),rr(j,:),J2,COVB2,mse2]= nlinfit(tt,yboot,'myfunconvecdiff',beta0,options);%betab are the paramters from the bootstraps [betab(j,:),rr(j,:),J2,COVB2,mse2]= nlinfit(tt,yboot,'myfunconvecdiff',beta0); ypredb(j,:)=myfunconvecdiff(betab (j,:),t); %qq=2; clear yboot end r2=rr(1,:)'; for j=2:nboot 293 r2=[r2;rr(j,:)']; end bsort=sort(betab,1); ysort=sort(ypredb,1); %sorts along columns K=round(0.05*nboot); if K==0; K=1; end; U=round(0.975*nboot); cib(1,1)=bsort(K,1); cib(1,2)=bsor t(U,1);%bootstrap 95% CI for first betaeter cib(2,1)=bsort(K,2); cib(2,2)=bsort(U,2);%bootstrap 95% CI for second betaeter cib(3,1)=bsort(K,3); cib(3,2)=bsort(U,3); for i=1:n ybci(i,1)=ysort(K,i); ybci(i,2)=ysort(U,i);%ybci is a n - by - 2 matrix with boot strap CI for y at each time end %% Compute bootstrap prediction bands D=RMSE*tinv(.975,n - p); CIwb(:,1)=ybci(:,1) - ypredict; CIwb(:,2)=ypredict - ybci(:,2) %upper (column 1) and lower (column 2) bootstrap CIwidths PIwb(:,1)=sqrt(CIwb(:,1).^2+D^2); PIwb(:,2)=s qrt(CIwb(:,2).^2+D^2)%upper and lower widths of PI PIb(:,1)=ypredict+PIwb(:,1); PIb(:,2)=ypredict - PIwb(:,2) %PI values %% %% Residual histogram for bootstrap residuals [n1, xout] = hist(r2,6); figure 294 hold on set(gca, 'fontsize',14,'fontweight','bold'); bar(xout, n1) % plots the histogram xlabel('Observed data - Predicted data','fontsize',16,'fontweight','bold') ylabel('Frequency','fontsize',16,'fontweight','bold') set(gca,'box','on','xticklabelmode','auto','yticklabelmode','auto') %% printing print_pdf(6 00,get(gcf,'filename'),outputfolder,'nocheck') print_png(300,get(gcf,'filename'),outputfolder,[],0,0,0) %% %Monte - Carlo %%plot Cobs, Cpred line, confidence band for regression line figure hold on set(gca, 'fontsize',14,'fontweight','bold'); L4 = ['Time (m in)']; xlabel(L4,'fontsize',16,'fontweight','bold'); % ylabel('log{ \ itS}_a','fontsize',16,'fontweight','bold'); ylabel('C_{f}(t)/C_','fontsize',16,'fontweight','bold'); h1(1)=plot(t,yobs2,'ob'); h1(2) = plot(t,ypredict,'r','LineWidth',2); h1(3) = plot(t,CB u,' - .g','LineWidth',2); plot(t,CBl,' - .g','LineWidth',2); 295 %% Plot prediction band for regression line h1(4) = plot(t,PBu,'c','LineWidth',2); plot(t,PBl,'c','LineWidth',2); %% Plot bootstrap bands h1(5) = plot(t,ybci(:,1),' -- k','LineWidth',2); plot( t,ybci(:,2),' -- k','LineWidth',2); h1(6) = plot(t,PIb(:,1),' - m','LineWidth',2); plot(t,PIb(:,2),' - m','LineWidth',2); pf=legend(h1,'exp data','best fitted line','95% confidence interval','95% prediction interval','95% confidence interval bootstrap','95% pred iction interval bootstrap') set(pf,'box','off','location','Best'); set(gca,'box','on','xticklabelmode','auto','yticklabelmode','auto') %% printing print_pdf(600,get(gcf,'filename'),outputfolder,'nocheck') print_png(300,get(gcf,'filename'),outputfolder,[],0 ,0,0) meanres=mean(resids); %% Residual scatter plot figure hold on set(gca, 'fontsize',14,'fontweight','bold'); plot(t, resids, 'square', 'Markerfacecolor', 'b') plot([0,max(t)],[0,0], 'R') ylabel('Observed Data - Predicted Data','fontsize',16,'fontweight ','bold') 296 xlabel('time (min)','fontsize',16,'fontweight','bold') %% Residual histogram [n1, xout] = hist(resids,6); figure hold on set(gca, 'fontsize',14,'fontweight','bold'); bar(xout, n1) % plots the histogram xlabel('Observed Data - Predicted Data','fo ntsize',16,'fontweight','bold') ylabel('Frequency','fontsize',16,'fontweight','bold') beta = beta; for i=1:length(beta) d=0.001; betain = beta; betain(i) = beta(i)+beta(i)*d; yhat{i} = myfunconvecdiff(betain, t); ysens{i}=(yhat{i} - ypredict)/d; %scaled sens coefficient X{i} = (yhat{i} - ypredict)/(beta(i)*d); end figure hold on set(gca, 'fontsize',14,'fontweight','bold'); L4 = ['Time(min)']; 297 xlabel(L4,'fontsize',16, 'fontweight','bold'); ylabel('Scaled Sensitivity Coefficient','fontsize',16,'fontweight','bold'); YLine =[0 0]; XLine = [0 max(t)]; plot (XLine, YLine,'k'); h2(1) = plot(t,ysens { 1 } ,'o - b','LineWidth',1); h2 (2) = plot(t,ysens { 2 } ,'s - r','LineWidth',1); h2(3) = plot(t,ysens{3},'^ - m','LineWidth',1); pf=legend(h2,'X''_{D}','X''_{Kpf}','X''_{h}') set(pf,'box','off','location','Best'); set(gca,'box','on','xticklabelmod e','auto','yticklabelmode','auto') %% printing print_pdf(600,get(gcf,'filename'),outputfolder,'nocheck') print_png(300,get(gcf,'filename'),outputfolder,[],0,0,0) %% Sequential Estimation %% n=length(data); p=3; %b1=D, b2=Kpf, b3=h b_old=[6.5 6.05 5.9 ]';%must use initial guess for b, because of model structure sigma=7.9851 e - 5; sig=sigma*ones(n,1);%close to RMSE value tol=5e - 3; %stopping criterion % start small and increased over time ratio = 1; %ratio compares new b to old b d=0.001; %delta for compu ting sensitivity coefficients count=1; %counts how many iterations 298 while ratio==1 %run loop while parameter change is greater than tol Pz=1.0e+5; b= b_old; ypred=myfunconvecdiff(b,t);%PUT YOUR FUNCTION IN THIS LINE e=yobs2 - ypred;%this replaces the line f or eq. 5.9.8.d in linear sequential for i=1:length(b)%loop to compute sensitivity coefficient for each parameter bin=b; bin(i)=b(i)*(1+d); yhat{i}=myfunconvecdiff(bin,t);%PUT YOUR FUNCTION IN THIS LINE XX{i}=(yhat{i} - ypred)/(b(i)*d); %sens itivity coefficient if i==1 X=XX{i}; else X=[X XX{:,i}]; end end P=diag((b_old.^2)*10);% squared guesses B=b_old'; % B is a row for ii=1:n; %SEQUENTIAL A=P*X(ii,:)'; %eq.(5.9.8a) %SEQUENTIAL Delta=sig(ii)^2+X(ii,:)*A; %eq (5.9.8b) %SEQUENTIAL K=A/Delta; %eq.(5.9.8c) %SEQUENTIAL 299 b=b+K*(e(ii) - X(ii,:)*(b - b_old));%eq.(7.8.22e) %SEQUENTIAL P=P - K*A'; %eq.(5.9.8f)%SEQUENTIAL B=[B;b']; if ii==1 PP=[P(1,1) P(1,2) P(1,3) P(2,2) P(2,3) P(3,3)];% Matrix 3 by 3 else PP=[PP; P(1,1) P(1,2) P(1,3) P(2,2) P(2,3) P(3,3)]; end end b_new=b;%last b is the new b ratioall=abs((b_new - b_old)./b_old); for i=1:p if ratioall(i) Tagetes erecta) intended for fatty - food application. Food Control, 46 , 55 - 66. doi: 10.1016/j.foodcont.2 014.04.045 Soto - Valdez , H., Peralta, E., & Auras, R. (2008). Poly(lactic acid) films added with resveratrol as active packaging with potential application in the food industry. Paper presented at the 16th IAPRI World Conference on Packaging Bangkok, Thail and. Vitrac, O., & Hayert, M. (2006). Identification of diffusion transport properties from desorption/sorption kinetics: an analysis based on a new approximation of fick equation during solid - liquid contact. Industrial & engineering chemistry research, 45 (23), 7941 - 7956. 306 Chapter 6 Assessment of Mass Transfer Models used in Migration Experiments to Determine Diffusion, Partition and Convective Mass Transfer Coefficients 6.0 Introduction Migration studies in food packaging have been extensively invest igated mainly due to safety concern and for compliance with food contact regulation s . A s ignificant amount of money and time have been investe d to perform experimental studies with different polymer - additive - food simulant combination s . Therefore, a conside rable amount of effort has been given to understand and assess the kinetics of migration by solving different mathematical models with different boundary conditions to simulate migration experiments and real case scenarios. This modeling approach allows re searcher s to identify important factor(s) governing the sorption and/or desorption kinetics of migration. a model is not verifiable directly by an experiment. For a ll models are both true and false. The - R.Levins, American Scientist 54:421 - 31,1966 (as cited in (Motulsky & Christopoulos, 2004c) ). There are many models available to determine migration parameters which have diverse complexity including molecular simulation, short - contact simulation, analytical solution, and numerical approximation, to name a few - Franco et al., 2012; Pocas, Oliveira, Brandsch, & Hogg, 2012; Poças, Oliveira, Oliveira, & Hogg, 2008; Reynier, Dole, & Feigenbaum, 2002a, 2002b; Samsudin, Valdez, & Auras, 2014; Soto - Valde z, Auras, & Peralta, 2010; Vitrac & Hayert, 2006; Vitrac, Mougharbel, & Feigenbaum, 2007) . Off all the listed models, analytical solutions have been widely used due to their simplicity 307 and direct physical i nterpretat ion of the migration kine tics. Analytical solutions applied to migration studies are commonly focused on two main kinetic parameters, which are the diffusion ( D ) and the partition ( K p,f ) coefficients. Another parameter commonly known as the convective mass transfer coefficient ( h ) is often considered negligible. As evidence, only a limited number of works ha ve investigated or tried to estimate h (Galotto, Torres, Guarda, Moraga, & Romero, 2011; Gandek, Hatton, & Reid, 1989; Mascheroni, Guillard, Nalin, Mora, & Piergiovanni, 2010; Pocas, Oliveira, Brandsch, & Hogg, 2012 ; Vitrac, Mougharbel, & Feige nbaum, 2007 ) . h is a key factor for understanding the kinetic s of migration at the interface of the polymer - food/food simulan t. In the case that convection is not present (continuous stirring or a well - mixed food medium), the h becomes larger and approaches infinity; thus its effect is negligible. However, in real case scenarios, most of the liquid food in contact with the packa ging system is left still at the market shelves, the storage temperature is fairly low to extend shelf life, the diffusion process is slow and/or the food has considerably high viscosity. For such aforementioned conditions, if the h effect is not considered, not only its effect as a governing kinetic factor is overlooked, the D estimation will also be underestimated since these two parameters are kinetically correlated in the series of migration resistance as indicated by the Biot nu mber definition (i.e., the ratio of the diffusion resistance in the liquid to the internal diffusion resistance in the polymer or ). Therefore, all the kinetic migration parameters ( i.e., D , h , and K p,f ) should be taken into consideration to fully qu antify the mass transfer process, and to understand how their individual effect s and possible interaction s with each other could influence the migration kinetics. A two - step solution to simultaneously determine these three parameters ( D , h , and K p,f ) was p roposed in Chapter 5. However, can we assure that this model is sufficient for all migration studies? or are previous solutions and available mathematical expressions used for many years to deter mine one 308 of these parameters sufficient? The aim of this stu dy was to compare the proposed two - step solutions presented in chapter 5 with two commonly used mathematical expressions using several selected case studies. 6.1 Materials and Methods 6.1.1 Migration Case Studies Several migration case studies were chosen to evaluate the different model approaches, which were i) poly(lactic acid), PLA film incorporated with 3 wt.% resveratrol in contact with ethanol at 9 C (Soto - Valdez , Peralta, & Auras, 2008) , ii) PLA film incorporated with 2.6 wt.% - tocopherol in contact with ethanol at 23 C (Manzanarez - López, Soto - Valdez, Auras, & Peralta, 2011) - presented in Appendix 6A and iii) PLA film incorporated with 1.28 wt.% catechin in contact with 95% ethanol at 40 C - Franco, Soto - Valdez, P - Meza, 2012) - presented in Appendix 6B. The data was analyzed using MATLAB ® R2011b (MathWorks, Natick, MA, USA) and s tatistical analysis was performed using an independent t - test (SPSS Statistics, version 22, 2013, IBM Corporation©, Armonk, NY, USA) for means comparisons. 309 6.1.2 Mathematical Models 6.1.2.1 Assumptions Analytical solutions were derived ba sed on the defined boundary conditions, and assumptions were made and hold true to satisfy those boundary conditions. Those assumptions include, but are not limited to: i) the initial concentration of the migrants is uniformly distributed in the films ii) the mig ration happens on the side of the film that is in contact with the food/ simulant iii) the food/simulant is well mixed iv) the film interface and the food is always at an equilibrium v) no interaction exists between the films and the food/simulant, and the edge effect is negligible (Chung, Papadaki s, & Yam, 2001, 2002; Crank, 1979; Poças, Oliveira, Oliveira, & Hogg, 2008) vi) the overall mass transfer is balanced vii) the D , K p,f and h do not change with the concentration viii) the mass transfer parameters are temperature dependent. 6.1.2.2 Model 1: A Two - Step Solution (A detail ed be found in Chapter 5) 6.1.2.2.1 Step 1 (Eq. 6 - 1) 310 where = concentration of antioxidant in food simulant; ; 6.1.2.2.2 Step 2 (Eq. 6 - 2) where ; and where = concentration of antioxidant in the food simulant; = initial concentration of antioxidant in the polymer; =the ratio of the mass of antioxidant migrated into food simulant to the mass of antioxidant l eft in the polymer, at equilibrium; =volume of food simulant; A =area; L =Biot number; = the non - zero roots. The number of terms needed for the analytical solution of model 1 to converge to a given accuracy (~98%) can be calculated as shown in Chapter 5. K p,f and diffusion coefficient ( D ) as the governing factors (Carsla w & Jaeger, 1959; Crank, 1979) Model 2 is a special case of model 1 for large Biot number. Biot number could become large as or or or any combination thereof. In the case that the food is well mixed ( i.e., continuous stirring), convection is fast and local equilibrium at the interface is obtained (boundary condition at ; ). Model 2 can be derived from step 2 of model 1 as follows: 311 (Eq. 6 - 2) where and (Eq. 6 - 3) As , Eq. 6 - 2 becomes; (Eq. 6 - 4) where and the eigenvalues satisfy The number of terms needed for a given accuracy (~98%) for the analytical solution of model 2 to c onverge can be calculated as follows: where The solution converges faster when z decreases, thus the worst - case scenario is when z=1 (slowest convergence). % Accuracy= where sum of finite series= and sum of infinite series= . The infinite series was estimated with 100,000 terms, which is a good estimation. n with diffusion coefficient ( D ) as the only governing factor (Crank, 1979) Model 3 is a special case for model 1 and mod el 2. It can be solved simply by the separation of variable s technique. As and , the boundary conditions at the interface ( 312 and ) are satisfied with a partial differential equation (PDE) of . Thus, model 3 can be derived as follows; (Eq. 6 - 5) Substitute Eq. 6 - 5 into the PDE ; (Eq. 6 - 6) Divide Eq. 6 - 6 by ; (Eq. 6 - 7) Separate Eq. 6 - 7 into two equations; (Eq. 6 - 8) (Eq. 6 - 9) where are constants. At : , (Eq. 6 - 10) At : (Eq. 6 - 11) Since cannot be 0, otherwise a trivial solution will be obtained, = ; thus Substitute Eq. 6 - 10 and Eq. 6 - 11 into Eq. 6 - 5; (Eq. 6 - 12) where the are arbitrary constants. Eq. 6 - 12 satisfies the boundary conditions and the PDE. The initial condition (at requires that; 313 = (Eq. 6 - 13) Multiplying by the m th cosine function and integrating, (Eq. 6 - 14) where , thus / (Eq. 6 - 15) The mass that remains in the polymer at time t , is (Eq. 6 - 16) (Eq. 6 - 17) (Eq. 6 - 18) (Eq. 6 - 19) (Eq. 6 - 20) the mass of antioxidant in the food simulant at time ; = =the initial mass of antioxidant in the polymer; the mass of antioxi dant remaining in the polymer at time (Eq. 6 - 21) where concentration of antioxidant in the food simulant 314 Thus, Eq. 6 - 21 becomes; (Eq. 6 - 22) The number of terms needed for a given accuracy (~98%) for the analytical solution of model 3 to converge can be calculated as follows: The term of is expanded to where The solution converges faster when z decreases, thus the worst case scenario is when z=1 (slowest convergence). % Accuracy= where sum of finite series= and sum of infinite series= . The infinite series was estimated with 100,000 terms, which is a good estimation. 315 Table 6 - 1 Comparison of the number of terms needed to achieve a given accuracy among model 1, 2, and 3. Number of terms % Accuracy Model 1 Model 2 Model 3 1 99.99 40.28 81.06 2 100.00 60.84 90.06 3 100.00 71.64 93.31 . . . . . . . . 70 100.00 98.67 99.71 6.1.3 Kinetic Parameter Estimation 6.1.3.1 Scaled Sensitivity Coefficient, Sensitivity coefficient and scaled sensitivity coefficient are as described in Chapter 2 section 2.7.2. 6.1.3.2 Ordinary Least Square (OLS) Estimation The parameters were estimated using the non - linear regression fitting function (nlinfit) of MATLAB® R2011b (MathWorks, Natick, MA, USA) by minimizing the sums of squared errors (SSE). Additional information such as residuals and relative error were also obtained. Relative error of each parameter was calculated by di viding the standard error of parameter with the 316 6.1.3.3 Corrected Akaike Information Criterion (AICc) The model selection was based on the corrected Akaike Information Criterion (AICc). AICc is useful to compare models that are differ ent from each other. Commonly, the higher the number of the parameters the more likely the goodness of fit seems to improve and vice versa. Thus, AICc eliminates the biasness that may be caused by different numbers of parameter among models. The smaller th e value of AICc is, the more likely the model is correct (Motulsky & Christopoulos, 2004a) . (Eq. 6 - 22) where n =number of data; p =number of parameter; K=p+1 were also calculated as follows (Motulsky & Christopoulos, 2004a; Wagenmakers & Farrell, 2004) ; (Eq. 6 - 23) where m =model 1... M (Eq. 6 - 24) where =the absolute difference of AICc between model s . weights or probability provide the information of how much more likely the model with the lower AICc is to be correct. Meanwhile, the evidence ratio informs the likelihood of favoring one model over the others. 317 6.2 Results and Discussions A c ase study of migration of PLA - 3 wt.% resveratrol into ethanol at 9 C was used to demonstrate and discuss the use of model 1, 2, and 3. For model 1 (Eq. 6 - 2), model 2 (Eq. 6 - 4), and model 3 (Eq. 6 - 22); three ( i.e., D, K p,f , h ), two ( i.e., D, K p,f ), and one ( i.e., D ) parameter(s), respectively, were estimated for these models, and the solutions were fitted to the experimental data. 6.2 .1 Scaled Sensitivity Coefficient, For parameters to be estimated simultaneously, the correlation among them is expected to be as low as possible since each individual parameter has its unique physical meaning in the migration experiment. Thus, a high cor relation among parameters means the key factor that governs a migration process cannot be identified separately. The s caled sensitivity coefficient, , of model 1 showed that the three kinetic migration parameters can be simultaneously estimated, so di d the two parameters of model 2 (Figure 6 - 1). Since model 1 was only estimating one parameter, its plot is not shown. It can also be observed that in both models, the parameter K p,f had the largest absolute magnitude of r esponse, which is expected to gi ve the low est relative error compared to the other estimated parameters. 318 Figure 6 - 1 Scaled sensitivity coefficient of migration of 3 wt. % resveratrol from PLA film into ethanol at 9 C of (a) model 1 (initial guesses were: D =6.50 10 - 12 cm 2 /min, K p,f =605.00 cm 3 PLA/cm 3 ethanol, and h =5.9 10 - 6 cm/min), and (b) model 2 (initial guesses were: D =5.19 10 - 12 cm 2 /min and K p,f =430.00 cm 3 PLA/cm 3 ethanol). 319 6.2.2 Ordinary Least Square (OLS) Estimation The OLS results of all three models can be observed in Table 6 - 1. The D values between model 1 and 2 were significantly different from each other (p<0.05) . This difference could be attributed to the different count s of estimated parameter in each model. When only one parameter is estimated in a model, only this parameter change s its value within the range to find the estimate that minimized the sums of squared error (SSE). In the case that more parameters are estimated, the combin ed value of those parameters changes until they end up with the lowest SSE (Motulsky & Christopoulos, 2004b) . Similar behavior was also observed in the case of estimated K p,f for both model 1 and 2. The published results of this case study reported a D value of 20.9 ×10 - 12 cm 2 /min (Soto - Valdez , Peralta, & Auras, 2008) , which was higher than that of the estimated D value of model 1, 2, and 3. The difference could be due to the different structure of the models. Meanwhile, the estimated K p,f of model 2 (491.24 cm 3 PLA/cm 3 ethanol) was found to have a closer value to the reported experimental K p,f (506.10 cm 3 PLA/cm 3 ethanol) than that of the estimated K p,f of model 1 (548.87 cm 3 PLA/cm 3 ethanol). No comparison can be made for the estimated h value since it is the first time this parameter is being estimated for this particular migration case study. However, higher o rder of magnitude for h has been reported by Vitrac, Mougharbel, & Feigenbaum, 2007 and c ould be due to the testing temperature and the type of polymer and food simulant. h depends on the polymer/simulant selection and the flow environment between the film and the simulants. Meanwhile, the estimated K p,f for model 1 and 2 was found to have the lowest relative error compared to the other estimated parameters as anticipated from the plot (Figure 6 - 1). For model 1, the following parameters were found to be easily and accurately estimated in decreasing ord er K p,f , D and h . The same order was observed for model 2, except the h since it was not estimated. 320 Model 1 had the lowest root mean square errors (RMSE) followed by model 3 and model 2. In general, the more parameters incorporated into the model, the lowe r the RMSE would be. However, model 3 with only one parameter resulted in lower RMSE than model 2 with two parameters and this could possibly be due to th e ease of fitting less number of parameter to the data set. The experimental data for the migration o f 3 wt.% resveratrol from PLA film into ethanol at 9 C was fitted using model s 1, 2, and 3 (Figure 6 - 2). The predicted fitting for model 3 (Figure 6 - 2(c)) was slightly off around 3.7 to 4.0 10 5 min compared to those of model 1 and 2 (Figure 6 - 2 (a),(b)) . Table 6 - 2 OLS results for the migration study of 3 wt.% resveratrol from PLA film into ethanol at 9 C for model 1, 2, and 3. Parameter & Additional Info Model 1 2 3 D ×10 - 12 (cm 2 /min) 5.42 ± 0.62 a 7.50 ± 1.61 b 0.61 ± 0.015 c Relative error (%) 11.38 21.45 2.43 K p,f (cm 3 PLA/cm 3 ethanol) 548.87 ± 21.76 a 491.24 ± 33.00 b N/A Relative error (%) 3.97 6.72 h ×10 - 6 (cm/min) 2.56 ± 0.38 N/A N/A Relative error (%) 14.97 RMSE×10 - 5 (cm 3 ethanol/cm 3 PLA) 7.91 12.09 9.07 Note: Data represents as mean ± standard error ; a,b superscripts represent statistical difference at p<0.05 within the same row. 321 Figure 6 - 2 Migration of 3 wt. % resveratrol from PLA film into ethanol at 9 C of (a) model 1, (b) model 2 and (c) model 3 and their corresponding residuals (d), (e), and (f), respectively. 322 The residual distribution of the three models can be observed in Figure 6 - 2(d ), (e), and (f). Model 1 had a better residuals distribution than model 2 and 3. Model 3 showed a significant signature trend. In general, the plot of residuals is expected to be normally, independently distributed with additive errors, constant variance a nd zero mean, to name a few, to meet the standard statistical assumptions. However, these conditions seemed to be violated to certain extent particularly for model 3 , which shows a clear residual signature; thus further data transformation may be needed fo r improvement of fitting or the model is not a good fit for this experimental data . 6.2.3 Model Selection Using the Corrected Akaike Information Criterion (AICc) Since three models were presented in this study, question are raised: which model should be used to represent this particular migration case study?, and what is the criterion needed to provide us enough information for selecting the model? As mentioned earlier, the standard selection criteria can be based on the RMSE and AICc. Often, RMSE introd uces bias in the model selection since experimental data is better fitted as an incre asing number of parameters is used in a model. The AICc approach can be employe d to assist in mode l selection eliminating bias since it finds a balance between the goodnes s of fit with the incorporated number of parameters. The AICc i s the second order of AIC, which it is applicable when the number of observations ( n ) is smaller and the number of estimated parameters ( p ) is larger. As n becomes larger, the correction term ( ) becomes trivial; thus AICc converges to AIC. T herefore the use of AICc is safe and provides better accuracy when n is small like in migration studies since it penalizes the addition of parameter s more than the AIC ( Motulsky & Christopoul os, 2004a) . Table 6 - 2 shows the results based on the AICc approach for selecting the appropriate model for the migration of 3 wt. % resveratrol from PLA film into ethanol at 9 C. It can be observed that 323 model 1 resulted in the lowest AICc in comparison with model 2 and 3. The probability that model 1 is a better model than model 2 and 3 was found to be 0.9999 (~99.99%). The evidence ratio of selecting model 1 over model 2 was overwhelming ( 1.59×10 15 ). In addition, model 1 is 2.20 10 4 times more likely to be the right model over model 3, which is enough evidence for choosing model 1 over model 3. Despite having known which model is likely to be correct, AICc cannot be used to solely reject or accep t a particular model. It only gives indication s based on how can a certain model best fit a data set and the likelihood of selecting the right model (Motulsky & Christopoulos, 2004a) . It is up to researchers to interpret the physical me aning of a constructed experiment in correlation with the parameter of interests within a mathematical model. Therefore, in this particular migration case study, it was found that model 1 is more likely the correct model with D , K p,f , and h as the driving factors of the migration. D is the controlling factor inside of the film, while K p,f indicates that the partition effect between the film and food simulant phase is important as not all of the antioxidant ( i.e., resveratrol) migrated into the food simula nt ( i.e., ethanol). The effect of h seemed to be taken place at an early time as indicated in the plot, and it is small. Table 6 - 3 AICc analysis for selecting the model for the migration of 3 wt. % resveratrol from PLA film into ethanol at 9 C. Model AICc Probability Evidence Ratio 1 - 1581 0.9999 2.20×10 4 2 - 15 11 6.30 ×10 - 1 6 1.59 ×10 1 5 3 - 1561 4.54×10 - 5 324 Overall, three different models consist ing of different combination s or one particular parameter(s) that govern the kinetics of a migration study have been presented. The comparison among the results based on the OLS estimation was made and model selection was discussed. How can all this information be combined as a well - constructed guideline for researchers to decide which model they should select that best represents their data set obtained from migration experiments? To address this ques tion, Figure 6 - 4 presents a decision tree analysis as a guideline with extended flexibility to estimate kinetic migration parameters. The dec ision tree analysis started with model 1 since it includes all the important kinetic migration parameters. This mo del also provides the magnitude approximation of initial guesses in the first step to be used in the second step of the non - linear regression estimation. This model allows the estimation of h , which is difficult to determine experimentally, hence, the calc ulation of the Biot number. By using the Biot number as the key question for the next step, researchers will know if model 1 can appropriately support their experimental data. Otherwise, they may proceed to the next key question for selecting either model 2 or 3. The advantage of assessing model 1 as the initial step is to avoid the assumption that h is negligible as presented in the boundary conditions of model 2 and 3. This is particularly important since h and D are kinetically related in the overall mig ration resistance series per the Biot number definition. For instance, in the absence of convection ( i.e., non - stirring food simulant/food), low temperature, and/or viscous food simulant/food etc., the h may no longer be assumed negligible. If researchers assume such condition s , not only are they neglecting the possibly important effect of this kinetic parameter, but they also end up underestimating the D value. Thus, the estimated D value without considering h when it is necessary, may be far off from the true D value. For such occurrence with respect to 325 food safety and shelf life estimation, negative consequences are of concern. Additionally, the presented AICc approach can be used to further assist the likelihood of selecting the right model. Figure 6 - 3 Decision tree analysis for determining the kinetic mass transfer parameters ( i.e., D, K p,f , h ) of a migration study. 326 6.3 Conclusion Comparative studies between the two - step solution (model 1) with solutions (model 2 and 3) were performed on three different migration case studies. The estimation of the scaled sensitivity coefficient gave insight into how well each individual parameter in each model can be predicted and the co rrelation among them. The OLS estimation results were successfully compared among models in each selected migration case study to calculate the D, K p,f , h coefficients. The AICc approach was employed based on its values, probability and likelihood comparis ons to assist in the model selection. This approach is not purely statistical sinc e it does not consider hypotheses, but it informs researchers on the data fitting to the model and the likelihood of selecting the correct model. The results obtained from th e OLS estimation and AICc were found to be in an agreement with each other. It is worth mentioning that although the AICc can provide information on the likelihood that one model is more likely to be correct than the others, it does not take into considera tion the residual distribution. This is because the AICc approach is developed based on the maximum likelihood, information and the entropy of decision on the AIC c approach to choose a model without investigating if all the statistical assumptions have been met could lead to another issue like over fitting, etc . Therefore, it is u p to the researcher to weigh all the information obtained from both the OLS results an d the AICc before choosing a particular model for a data set. In addition, a decision tree analysis was introduced to offer a constructed guideline to select the appropriate model with its respective kinetic migration parameters. Further experimental stud ies should be conducted to design experiments where h is a driving factor during the migration study. 327 APPENDICES 328 APPENDIX 6A: Migration of poly(lactic acid), PLA incorporated with 2.6 wt.% - tocopherol into ethanol at 23 C. Scaled Sensitivity Coefficient, For both model 1 and 2 of the migration of 2.6 wt.% - tocopherol from PLA film into ethanol at 23 C , D and K p,f were not highly correlated to each other (Figure 6A - 1), which allows the simultaneous estimation of them. However, for model 1, the h parameter was introduced as a constant parameter since it was highly correlated with the D ( ) . Besides, the ma gnitude response of change of this parameter towards pertubation was relatively small. 329 Figure 6A - 1 Scaled sensitivity coefficient of migration of 2.6 wt.% - tocopherol from PLA film into ethanol at 23 C of (a) model 1 (initial guesses were: D =2.00 10 - 9 cm 2 /min, K p,f =609 cm 3 PLA/cm 3 ethanol), and (b) model 2 (initial guesses were: D =10.00 10 - 9 cm 2 /min and K p,f =500 cm 3 PLA/cm 3 ethanol). Note: the h was not estimated for model 1 due to high correlation issue with the D . 330 Ordinary Least Square (OLS) Estimation The estimated D values for both model 1 and 2 were not significantly different (p>0.05) from each other (Table 6A - 1). However, these estimated D values were different than the published result of 1.90 ×10 - 9 cm 2 /min (Manzanarez - López, Soto - Valdez, Auras, & Peralta, 2011) . Similarly, the reported K p,f ( 796.62 cm 3 PLA/cm 3 ethanol) (Manzanarez - López, Soto - Valdez, Auras, & Peralta, 2011) was slightly higher th an the estimated K p,f of model 1 and 2. Closer estimation values of D and K p,f were anticipated between model 2 and the published result of this migration case study. Lowest relative error of the K p,f in both model 1 and 2 were observed than that of the D as expected based on the plot (Table 6A - 1). The plots of the migration 2.6 wt.% - tocopherol from PLA film into ethanol at 23 C (Figure 6A - 2(a)) demonstrated that model 2 fits the experimental data better with a normally scattered residual distribut ion (Figure 6A - 2(c)) than that of model 1 since the residual are normally, independently distributed ( NID ) . Motulsky & Christopoulos (2004b) discussed that due to the symmetricallity of the asymptotic confidence interval, different way s of expressing a mod el can result in different output. In addition, the AICc of model 2 turned out to be lower than that of model 1 (Table 6A - 2), with overwhelming evidence ratio supporting model 2 as more likely to be the correct model. These outcomes are in agreement with e ach other. 331 Table 6A - 1 OLS results for the migration study of 2.6 wt.% - tocopherol from PLA film into ethanol at 23 C for model 1 and 2. Parameter & Additional Info Model 1 2 D ×10 - 9 (cm 2 /min) 6.44 ± 0.08 a 6.74 ± 0. 44 a Relative error (%) 1.17 6.59 K p,f (cm 3 PLA/cm 3 ethanol) 631.54 ± 7.08 a 482.71 ± 5.57 b Relative error (%) 1.12 1.15 RMSE×10 - 4 (cm 3 ethanol/cm 3 PLA) 1.23 0.84 Note: Data represents as mean ± standard error ; a,b superscripts represent statistical difference at p<0.05 within the same row. 332 Figure 6A - 2 Migration of 2.6 wt.% - tocopherol from PLA film into ethanol at 23 C of (a) model 1 and (b) model 2 and their corresponding residuals (c), and (d), respectively. 333 Model Selection Using the Corrected Akaike Information Criterion (AICc) Table 6A - 2 AICc analysis for selecting the model for the migration study of 2.6 wt.% - tocopherol from PLA film into ethanol at 23 C. Model AICc Probability Evidence Ratio 1 - 1508 7.68 ×10 - 15 1.30 ×10 14 2 - 1573 1.00 334 APPENDIX 6B: Migration of poly( lactic acid), PLA incorporated with 1.28 wt.% catechin into 95% ethanol at 40 C. Scaled Sensitivity Coefficient, The plot of the migration of 1.28 wt.% catechin from PLA film into 95% ethanol at 40 C indicated no high correlation among the three parameters. Since model 1 was only compared with model 3, the plot for model 3 is not shown (only one parameter estimated). Model 2 was not considered for the comparison per the decision tree guideline ( Figure 6B - 1 Scaled sensitivity coefficient of the migration of 1.28 wt.% catechin from PLA film into 95% ethanol at 40 C of model 1 (initial guesses were: D =3.00 10 - 8 cm 2 /min, K p,f =318.97 cm 3 PLA/cm 3 ethanol). 335 Ordinary Least Square (OLS) Estimation s The only parameter that can be compared between model s 1 and 3 was the D and the result was significantly different (p<0.05). Interestingly, the estimated D value of model 1 was comparable to the reported D value (2.87 ×10 - 8 cm 2 /min) of this case study - Meza, 2012) . As anticipated, with more parameters estimated in model 1, the resulting RMSE value was lower than that of model 3 (Table 6B - 1 ). The model fitting of the experimental data for model s 1 and 3 can be observed in Figure 6B - 2. Although model 1 seemed to better fit the experi mental data, the residual plot distribution showed the signature in the residuals for both models; therefore, further data transformation may be needed to improve the fitting of both models. The AICc approach re sults indicated that model 1 had a higher lik elihood of being the right model than model 3. This result supported the outcomes from the OLS estimation. 336 Table 6B - 1 OLS results for the migration of 1.28 wt.% catechin from PLA film into 95% ethanol at 40 C for model 1 and 3. Parameter & Additional Info Model 1 3 D ×10 - 8 (cm 2 /min) 2.26 ± 0.13 a 10.13 ± 0.74 b Relative error (%) 5.92 7.29 K p,f (cm 3 PLA/cm 3 ethanol) 324.04 ± 4.15 Relative error (%) 1.28 h ×10 - 3 (cm/min) 4.40 ± 0.43 Relative error (%) 9.62 RMSE×10 - 4 (cm 3 ethanol/cm 3 PLA) 0.96 3.54 Note: Data represents as mean ± standard error ; a,b superscripts represent statistical difference at p<0.05 within the same row. 337 Figure 6B - 2 Migration of 1.28 wt.% catechin from PLA film into 95% ethanol at 40 C of (a) model 1 and (b) model 3 and their corresponding residuals (c), and (d), respectively. 338 Model Selection Using the Corrected Akaike Information Criterion (AICc) Table 6B - 2 AICc analysis for selecting model for the migration of 1.28 wt.% catechin from PLA film into 95% ethanol at 40 C. Model AICc Probability Evidence Ratio 1 - 659 1.002 3.49×10 19 3 - 569 .86×10 - 20 339 REFERENCES 340 REFERENCES Carslaw, H. S., & Jaeger, J. C. (1959). Conduction of heat in solids. 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Psychonomic bulletin & review, 11 (1), 192 - 196. 343 Chapter 7 Estimation of the Activation Energy i n Migration Studies 7.0 Introduction Migration studies are usually conducted to gain in sight on the migration limit and the kinetic behaviors of additives or possible contaminant(s) from a base polymer into a food/f ood simulant. Kinetic behavior in a migration study provide s useful information such as the chemical affinity between a polymer and a food/food simulant (partition coefficient, K p,f ), the rate of the transfer mechanism (diffusion coefficient, D ), and the resistance at the interface (convective mass transfer coefficient, h ) as shown in Chapters 4 to 6. Unlike the K p,f that can be me asured experimentally at the end of an experiment , the D and h can only be approximated by means of mathematical expression s . Therefore, assessment of these parameters holds its own significance for describing a migration phenomenon. Since migration is a t ransfer process involving the law of diffusion, the mass transfer behavior is temperature dependent and can be described using the Arrhenius equation: (Eq. 7 - 1) where D = diffusion coefficient; =pre - exponential factor; E a =activation energy; R=gas constant; T=temperature. The activation energy ( E a ) term embedded in the Arrhenius equation is commo nly estimated and can be defined as how the kinetic rate changes with temperatures . Often, the E a is estimated using its linearize d form by taking the natural logarithm ( ln ) of the equation to eliminate the exponential term (Eq. 7 - 2). A p lot of ln ( D ) as a function of reciprocal temperature (1/T ( K)) can be constructed and the E a value can be obtained from the slope R. 344 (Eq. 7 - 2) This practice is commonly considered to simplify the difficulty of estimating the E a and to avoid the numerical complication of having a high correlation between the two important factors of the A rrhenius equation, which are D o and E a . Several authors investigated the fitting of linear versus non - linear approximation s of the Arrhenius equation and the outcomes were inconclusive (Brauner & Shacham, 1997; Chen & Aris, 1992; Klicka & Kubácek, 1997; Sundberg, 1998) . Despite t hat, several other authors recommended the reparameterization of the Arrhenius equation (Eq. 7 - 3) to reduce the correlation tendency between the D o and E a by introducing the reference temperature ( T ref ) that corresponds to the reference D ( D ref ) (Agarwal & Brisk, 1985a, 1985b; Ahmed, Dolan, & Mishra, 2012; K. D. Dolan, Valdramidis, & Mishra, 2013; Pritchard & Bacon, 1975, 1978; Schwaab, Lemos, & Pinto, 2008; Schwaab & Pinto, 2007; Sulaiman, Dolan, & Mishra, 2013) . This concept was introduced by Box, 1960 . (Eq. 7 - 3) where = the diffusivity rate of the additives at . The concept of reparameterizing by finding an optimum T ref is crucial to obtain near zero correlation between the rate constant of the reference diffusion, D ref and the E a since by minimizing the correlation between the two parameters, their respective relative errors will also be minimized. Therefore, the non - lin ear approximation of the Arrhenius equation is recommended and by us ing it as a secondary model in an analytical kinetic migration expression, simultaneous estimation of the parameters can be performed. o published work in the food packaging area about the reparameterization of the Arrhenius equation. So, the objectives of this work were; i) to find the optimum T ref that resulted in near zero correlation between the D ref and the E a by the 345 insertion of the Arrhenius equation as a secondary model into an analytical kinetic migration equation and ii) to estimate all uncorrelated kinetic migration parameters simultaneously. Additionally, the temperature simulation ( T sim ) approach was introduced for the first time in the foo d packaging area to evaluate non - isothermal migration studies based on the work of K. Dolan, 2015 . 7.1 Materials and Methods 7.1.1 A Case Study A case study was selected for this work to demonstrate the estimation of E a : the migration of 1.28 wt.% catec hin from poly(lactic acid), PLA, film into 95% ethanol at 20, 30, 40, and 50 C - Franco et al., 2012) . All computational data analyses were coded (Appendix 7B) and performed using MATLAB® R2011b (MathWorks, Natick, MA, USA). 7.1.2 Kinetic Parameter Estimation Procedure The analytical kinetic migration equation describing the movement of additives from polymer films can be represented with model 1 as presented in chapter 5; (Eq. 7 - 4) where ; = and where = concentration of antioxidant in the food simulant; = initial concentration of antioxidant in the polymer; = the ratio of the mass of antioxidant migrated into food simulant to 346 the mass of antioxidant left in the polymer, at equilibrium; = volume of food simulant; A = area; L = = Biot number; = the non - zero roots of eigenvalues . The secondary model (Eq. 7 - 2 (below)) was inserted into Eq. 7 - 4, resulted in Eq. 7 - 5; (Eq. 7 - 5) where ; 7.1.3.1 Step 1: 7.1.3.1.1 Scaled Sensitivity Coefficient, Data set of a case study at each temperature was set up for the non - isothermal estimation. The scaled sensitivity coefficient, , was plotted for the fitting parameters ( D ref , K p,f , h and E a ). Since this is a forward problem, initial guesses were mostly acquired from published data with some modification. The f orward problem provides an explicit soluti on and the parameters are given; thus it can be run without any data. By using a forward difference approximation, the scaled sensitivity coefficient, , was computed by taking the first derivatives of the observational data with respect to the parameter and multiplying by the parameter itself. The plot of as a function of time was plotted to investigate the correlation among the parameters. The magnitude of the chang e of the response of each parameter to perturbation was observed. 347 7.1.3.1.2 Temperature Simulation ( T sim ) Approach Since temperature is not constant and E a depends on how D is changing with temperature, the T sim approach was proposed to plot the of E a (K. Dolan, 2015) by inserting an anonymous (Eq. 7 - 6 ) where = the lowest temperature ( ; = the highest temperature ( ; =maximum time duration; = linearly spaced time. 7.1.3.1.3 Non - Linear Regression Estimation The non - linear regression fitting (nlinfit) function in MATLAB® R2011b (MathWorks, Natick, MA, USA) was used to estimate the parameters. The correlation coe fficient matrix obtained was used to confirm any correlation among the parameters as observed in the plot. Since there is no available information about the T ref for migration studies, different values were fixed to find a better estimation. 7.1.3.2 St ep 2: The best guess of the T ref was then used as the initial guess to find the correlation between the D ref and the E a . The plot of the correlation between the D ref and the E a as a function of the possible range of the T ref was constructed to find the opt imum T ref . The optimum T ref value was then used to estimate the parameters ( D ref , K p,f , h and E a ) for final estimation. 348 7.2 Results and Discussions 7.2.1 Initial Scaled Sensitivity Coefficient, and Reference Temperature, T ref Since there is no available information on the T ref of this particular case study, it is important to start off with the best - considered T ref value to initiate the forward problem estimation process. It has been widely cited that the use of a T ref that app roaches large values can cause high correlation between the D ref and the E a and correspondingly large standard errors. This is because the reparameterized Arrhenius equation converges to its traditional form (Eq.7 - 1). Most of the researchers recommend the average temperature within the experimental temperature range as the starting point, rather than arbitrarily choosing a particular temperature (Ahmed, Dolan, & Mishra, 2012; K. D. Dolan, 2003; Schwaab & Pinto, 2007) . Meanwhile, Datta (1993) and Ahmed, Dolan, & Mishra (20 12) suggested the T ref to be closer to the upper range of the experiment temperature range. For this case study, the initial T ref chosen was the average within the migration testing temperature range ( i.e., 20 to 50 C), which was 35 C. The plot was then constructed to find any possible correlation among all the parameters of interest. Initial observation from the plot indicated that the h parameter had the smallest magnitude change of response among the rest of the parameters, which is e xpected to contribute to the largest relative error (Figure 7 - 1). Therefore, this parameter was kept as constant value since it presence had negligible effect on the overall kinetic s of migration and to a certain extent it may introduce more difficulty in estimating the other parameters (as discussed in chapter 5). Figure 7 - 2 shows the plot consisting only of the three parameters ( i.e., D ref , K p,f , E a ) (Figure 7 - 2) which has a better improvement in terms of the change in their magnitude of response tha n Figure 7 - 1. Both plots were constructed by implementing the T sim approach as explained in section 7.1.3.1.2. The K p,f was identified as the easiest parameter to be estimated; hence, the lowest 349 relative error s followed by the E a and D ref . The correlation coefficients among the estimated parameters using the initial T ref can be found in Table 7 - 1. The correlation coefficient, , between D ref and E a was fairly high ( since near zero correlation is desired (Schwaab, Lemos , & Pinto, 2008; Schwaab & Pinto, 2007) , thus the iterative search for T ref was conti nued until showed a possible lower co rrelation. Additional results for the corr elation coefficients between D ref and E a for different and randomly chosen temperatures are shown in Appendix 7A. 350 Figure 7 - 1 Scaled sensitivity coefficient of the activation energy estimation of the migration of 1.28 wt.% catechin from PLA film into 95% etha nol ranging from 20, 30, 40, and 50 C at T ref =35 C . Initial guesses were: D ref =1.00 10 - 9 cm 2 /min, K p,f =800 cm 3 PLA/cm 3 ethanol, h =10.00 10 - 4 cm/min, and E a =150000 J/mol). 351 Figure 7 - 2 Scaled sensitivity coefficient of the activation energy estimation at T ref =35 C of the migration of 1.28 wt.% catechin from PLA film into 95% ethanol ranging from 20, 30, 40, and 50 C. Initial guesses were: D ref =2.00 10 - 9 cm 2 /min, K p,f =800 cm 3 PLA/cm 3 ethanol, and E a =150000 J/mol). 352 Table 7 - 1 Correlation matrix of the estimated parameters at the average T ref =35 C. After the iterative search was performed, it was found that at T ref 45 C, the correlation between D ref and E a was 0.01 with relative errors of 8.34 and 4.21 %, respectively (Table 7 - 2). Thus, the estimated values obtained at this T ref were then used as the initial guesses to construct the plot of correlation coefficient between the D ref and the E a as a function of the experimental temperature range (Figure 7 - 3). From Figure 7 - 3, the optim um T ref that resulted in near zero correlation between the D ref and the E a was identified to be 44.94 C ( 10 - 4 ). Parameters Correlation coefficients, Relative Error (%) D ref K p,f E a D ref Symmetric 11.07 Kp,f 0.11 4.93 E a - 0.63 - 0.29 7.33 353 Table 7 - 2 Correlation matrix of the estimated parameters at the T ref =45 C. Parameters Correlation coefficients, Relative Error (%) D ref K p,f E a D ref Symmetric 8.34 K p,f 0.33 4.16 E a 0.01 - 0.34 4.21 354 Figure 7 - 3 Plot of correlation coefficient of the D ref and the E a as a function of possible T ref 7.2.1 Non - Linear Regression Estimation The non - linear regression estimation was performed by using the optimum T ref and the results can be found in Table 7 - 3 and Table 7 - 4. The lowest correlation between D ref and E a was found at the identified optimum T ref . As has been widely discussed in several publications (Ahmed, Dolan, & Mishra, 2012; K. D. Dolan, 2003; Schwaab, Lemo s, & Pinto, 2008; Schwaab & Pinto, 2007; Sulaiman, Dolan, & Mishra, 2013) , the estimation of E a using the optimum T ref not only is crucial to reduce the correlation issue between the E a and the D ref , but also is critical to minimize the relative error of D ref . The refore, the linear estimation of activation energy as shown in Eq. 7 - 2 should be avoided whenever possible to avoid the risk of over fitting, which means fitting noises 355 over the actual data. In addition, the re - parameterization approach (Eq. 7 - 3) helps to minimize the correlation issue as discussed earlier. The estimated D ref and K p,f were found to be 3.70 10 - 9 cm 2 /min, and 436.62 cm 3 PLA/cm 3 ethanol, respectively at the optimum T ref =44.94 C . Meanwhile the estimated E a in this study was found to be 153.00 kJ/mol, which was si gnificantly higher than that reported by I - Franco et al. (2012) of 110.43 kJ/mol. This huge difference could be due to the linearization of the Arrhenius equation. Figure 7 - 4 shows the final constructed plot. Significant improvement can be observed in all parameters by taking into consideration their magnitude change of response. The K p,f had the lowes t relative error followed by E a and D ref (Table 7 - 3) as compared using the initial observation from Figure 7 - 2. Table 7 - 3 Correlation matrix of the estimated parameters at the optimum T ref = 44.94 C. *3.15 10 - 4 Parameters Correlation coefficients, Relative Error (%) D ref K p,f E a D ref Symmetric 8.34 K p,f 0.33 4.16 E a 0.00* - 0.34 4.21 356 Table 7 - 4 The parameter estimates at the optimum T ref = 44.94 C. Note: RMSE (Root mean square errors) unit= cm 3 ethanol/cm 3 PLA. Parameters Estimates RMSE D ref 10 - 9 (cm 2 /min) 3.70 0.31 5.06 10 - 4 K p,f (cm 3 PLA/cm 3 ethanol) 436.62 18.16 E a 10 5 (J/mol) 1.53 0.06 357 Figure 7 - 4 Final scaled sensitivity coefficient of the activation energy estimation of the migration of 1.28 wt.% catechin from PLA film into 95% ethanol ranging from 20, 30, 40, and 50 C. Final estimates were: D ref =3.70 10 - 9 cm 2 /min, K p,f =436.62 cm 3 PLA/cm 3 ethanol, and E a =153000 J/mol). 7.3 Additional Observations While performing the estimation of the E a , several difficulties were experienced. Since it is the first time the estimation of the E a using the reparameterization approach was done for migration studies, findin g adequate initial guesses was pretty challenging due to the nature of this non - isot hermal experiment. Although there are available published data on these parameters for 358 other type of experiments such as microbial inactivation, thermal iso merization of compounds, gelatinization of starch, etc. , locating appropriate ones that satisfy the overall experimental temperature ranges was difficult. This issue was resolved by using step 1 of model 1 of the two - step solution for mass transfer additiv es from polymer films developed in Chapter 5. In addition, the fact that the Arrhen ius equation was employed as a secondary model increases the complexity of the estimation procedure. Since in this study more than one kinetic constant was estimated, the pr ocess of eliminating the correlation issue among parameters cannot be assured. Hence, simultaneous estimation of all parameters may not be possible. Moreover, the T ref may also change accordingly with different mathematical expression. More details on this issue were discussed by Schwaab & Pinto (2008). Two key fact ors to solve this issue are appropriate initial guesses and the selection of initial T ref , while continuously monitoring the correlation coefficient among the parameters. It was also observed tha t there was a scaling issue since the initial guesses for D ref , K p,f and E a were all of different order s of magnitude deficient, rank = 3, tol = 1.768837e - ) indicating that the matrix had an issue with rank deficiency, which means the generated matrix did not having linearly independent rows and columns. The approach to solve this issue is by normalizing the initial guesses of the estimated parameters, so all the parameters are of the same order of magn itude. 7.4 Conclusion s The estimation of E a for a migration case study was performed for the first time using the reparameterization approach of the Arrhenius equation to reduce the correlation between D ref and E a , minimizing the relative errors . The op timum T ref at 44.94 C was successfully obtained by plotting the correlation coefficient of D ref and E a within the possible reference temperature ranges 359 (40 to 60 C) . Subs equently, all the parameters ( i.e., D ref , K p,f and E a ) were estimated simultaneously with the correlation coefficients ranging between almost 0.00 ( 10 - 4 at 44.94 C) and 0.34 . The optimum T ref resulted with the lowest relative errors of parameters was anticipated. The T sim approach allowed the plot ting of E a , thus the dependency criterion among parameters can be visualized. Further evaluation of different case studies should be performed. 360 APPENDICES 361 APPENDIX 7A: Additional results of the randomly chosen T ref within the experimental temperature range. Table 7A - 1 Correlation matrix of the estimated parameters at the T ref =40 C. Parameters Correlation coefficients, Relative Error (%) D ref K p,f E a D ref Symmetric 9.18 K p,f 0.45 4.16 E a - 0.42 - 0.34 4.21 362 Table 7A - 2 Correlation matrix of the estimated parameters at the T ref = 5 0 C. Parameters Correlation coefficients, Relative Error (%) D ref K p,f E a D ref Symmetric 9.17 K p,f - 0.16 4.16 E a 0.42 - 0.34 4.21 363 APPENDIX 7B: MATLAB coding for the estimation of activation energy %% MIGRATION STUDIES: ESTIMATION OF ACTIVATION ENERGY (REPARAMETERIZATION APPROACH) %% Notes % 1. Start Tref with average temperature within the experimental temperature % ranges %2. Use step 1 to find approximation initial guesses %3. Tsim approach was used to plot Ea. %Formula: T_sim=@(t) T_low +(T_high - T_low)/t_max t ; y=c + mX %% close all; clear; clc f ormat long global L % This variable is being shared with all the file that is declared global global alpha global qn global D global T global h %% data =xlsread('PLACate_95etoh_EA.xls');% To read raw data from excel %% switch localname case '13 - 138 - 76 .client.wireless.msu.edu' local = '/Users/HAYATISAMSUDIN/Documents/MATLAB_EXP/MATLAB_CIADEXP/ACTIVATIO N ENERGY/PLA_CATE_EA_TRIAL'; papertype = 'A4'; 364 paperposition = [0.3397 10.1726 20.3046 9.3322]; %cm otherwise local = pwd; papertype = 'usletter'; paperposition = [0.3397 10.1726 20.3046 9.3322]; %update the values to usletter format warning('Please set the case for your computer') end datafile = 'PLACate_95etoh_EA.xls'; outputfolder = fullfile(local,'Figures_PLACate_EA'); if ~exist(outputfolder,'dir'), mkdir(outputfolder), end [~,outputfile] = fileparts(datafile); %% Additional Info L=0.003616667;%L= half of the thickness, cm A=3.1416; %A=area, cm2 Co=0.011591; % Co=Initial concentr ation, g/cm3 Vf=1.227 % Vf=volume of food,cm3 %% Data Extraction t=data(:,1);% time variable (min) yobs=data(:,2); %Concentration at time t, g/cm3 yobs2=data(:,2)./Co;% Concentration at time t/Concentration at initial time T=data(:,3); % Temp in Kelvin t1 =linspace(0,max(t),1000)'; sum1=0; sum2=0; %% Step 1 R=linspace(0.100,0.000560,200); %R=linspace(0.00090,0.000400,100); %% N=length(data); for j=1:length(R) for i=1:N 365 if i == 1 fit(i) = 0 else z=1 - exp( - R(j)*t(i)); sum1=sum1+z.* yobs2(i); sum2=sum2+z.^2 P(j)=sum1/sum2 fit=P(j)*z; end end fitreverse=fit'; SSE(j)=sumsqr(yobs2 - fitreverse) end comparison=[P' R' SSE']; Index=0:1:N; [M,I]=min(SSE);%I is index refers to min value of SSE showminSSE=[P(I) R(I)] % to display P and R corresponding to I=index of min SSE %% for i=1:N if i == 1 fit(i) = 0 else z=1 - exp( - R(I)*t(i)); final_fit(i)=P(I)*z; end end display(final_fit) predict_obs=final_fit'; Table=[t yobs2 final_fit' ] figure 366 hold on set( gca, 'fontsize',14,'fontweight','bold'); pl(1)=plot(t, yobs2, 'o','LineWidth',1.2) pl(2)=plot(t, final_fit,'r','LineWidth',1.2) xlabel('Time (min)','fontsize',16,'fontweight','bold'); ylabel('exp data, final fit','fontsize',16,'fontweight','bold'); pll=leg end (pl,'exp data','final fit'); set(pll,'box','off','location','Best'); set(gcf,'Color',[1 1 1]); %% m0=Vf/(A*L); m0=P(I)*(Vf/(A*L)); Kpf=(1 - m0)/P(I); %P(I)= value contain the lowest SSE from before m1= (2*(Kpf+(Vf/(A*L)))) %term 1 of R m2=m1/R(I) D=1 ; %m3=(Vf/A)*((L/D)+(2*Kpf/h)) h=(2*Kpf)/((m2/(Vf/A)) - (L/D)) h1=1 D=L/((m2/(Vf/A)) - (2*Kpf/h1)) %% %% Kpf=800; h=10e - 4; D=2E - 9; %% global Bi alpha=Vf/(Kpf*A*L); % qn; Bi=(h*L)/D; 367 Lo=1; Up=60; z=@(qn) (Bi - Kpf*alpha*qn^2)*sin(qn)+alpha*Bi*qn*cos(qn); for i=Lo: Up; ev(i)=fzero(z,i); %Newton - Raphson end; EV=unique(ev); EV'; ev=EV(2:end); qn=ev' %% Initial parameter guesses b1=2 ;% Initial guess for Dr...fix close to publish data 2 x e - 9 b2=8 ; % Initial guess for Kpf x 10^2 b3=1.5 ;% Ea (J/mol)... fix as close to publish data 110 kJ/mol x 10^2 beta0(1)=b1; beta0(2)=b2; beta0(3)=b3; %% X' = scaled sensitivity coefficients using forward - difference % This is a forward problem with known approximate parameters Xp=SSC_EA3P( beta0,t1,@myfunconvecdiff_EASim3P); %title('Scaled Sensitivity Coefficients using initial guesses')%% %printing print_pdf(600,get(gcf,'filename'),outputfolder,'nocheck') print_png(300,get(gcf,'filename'),outputfolder,[],0,0,0) %% %printing print_pdf(600,ge t(gcf,'filename'),outputfolder,'nocheck') print_png(300,get(gcf,'filename'),outputfolder,[],0,0,0) %% ypredict = myfunconvecdiff_EASim3P(beta0,t1); 368 %% [beta,resids,J,COVB, MSE] = nlinfit(t,yobs2,@myfunconvecdiff_EA3P,beta0); ci=nlparci(beta,resids,J,0.05 ) %asymptotic confidence interval [ypredict, delta] = nlpredci('myfunconvecdiff_EA3P',t,beta,resids,J,0.05,'on','curve'); %CI for mean [ypredict, deltaobs] = nlpredci('myfunconvecdiff_EA3P',t,beta,resids,J,0.05,'on','observation'); [R, sigma]=corrcov(COVB) ; R; sigma; % parameter standard errors relative_error_n=sigma./beta';% >0.6 the likeliness of CI contains 0 is high (estimate is useless since it is not statiscially diff than 0) beta; % parameters estimated MSE; RMSE=MSE^(1/2); standard_residuals=resids/ RMSE; [R, sigma]=corrcov(COVB); n = length(ypredict); p = length(beta); SS= MSE*(n - p) Step 2: %% clc clear format long; %% switch localname case '13 - 138 - 76.client.wireless.msu.edu' local = '/Users/HAYATISAMSUDIN/Documents/MATLAB_EXP/MATLAB_CIADEXP/ACTIVATIO N ENERGY/PLA_CATE_EA_TRIAL'; papertype = 'A4'; 369 paperposition = [0.3397 10.1726 20.3046 9.3322]; %cm otherwise local = pwd; papertype = 'uslett er'; paperposition = [0.3397 10.1726 20.3046 9.3322]; %update the values to usletter format warning('Please set the case for your computer') end datafile = 'PLACate_95etoh_EA.xls'; outputfolder = fullfile(local,'Figures_PLACATE95ETOH _CorrEa_Tref'); if ~exist(outputfolder,'dir'), mkdir(outputfolder), end [~,outputfile] = fileparts(datafile); %% TrV=40:1:6 0; TrV=TrV+273.15; for i = 1:length(TrV) Tr = TrV(i) corrEA(i) = EaTry(Tr) end TrV=TrV - 273.15; figure hold on set( gca, 'fontsize',14,'fontweight','bold'); h = plot(TrV,corrEA,' - ob', 'linewidth',1.05); % xlabel('Reference Temperature \ it{T_r}, (^oC_','FontSize',16,'fontweight','bold'); xlabel('Reference Temperature, \ it{T_{ref}} \ rm \ bf(^oC)','FontSize',16,'fontweig ht','bold'); % ylabel('Correlation Coefficient of \ itd_r and \ itz','FontSize',16,'fontweight','bold'); ylabel('Correlation Coefficient of \ it{D_{ref}} \ rm \ bf and \ itEa','FontSize',16,'fontweight','bold'); % ylabel('Correlation Coefficient of \ it{k} and \ i tE','FontSize',16,'fontweight','bold'); plot([min(TrV),max(TrV)],[0,0], 'R','linewidth',1.05) set(gca,'box','on','xticklabelmode','auto','yticklabelmode','auto'); grid on 370 %% %printing print_pdf(600,get(gcf,'filename'),outputfolder,'nocheck') print_png(300, get(gcf,'filename'),outputfolder,[],0,0,0) %% 371 REFERENCES 372 REFERENCES Agarwal, A. K., & Brisk, M. L. (1985a). Sequential experimental design for precise parameter estimation. 1. Use of reparameterization. Industrial & Engineering Chemistry Process Design and Development, 24 (1), 203 - 207. Agarwal, A. K., & Brisk, M. L. (1985b). Sequential experimental design for precise parameter estimation. 2. Design criteria. Industrial & Engineering Chemistry Process Design and Development, 24 (1), 207 - 210. Ahmed, J., Dolan, K. D., & Mishra, D. K. (2012). Chemical reacti on kinetics pertaining to foods. Handbook of food process design , 113. Box, G. E. (1960). Fitting empirical data. Annals of the New York Academy of Sciences, 86 (3), 792 - 816. Brauner, N., & Shacham, M. (1997). Statistical analysis of linear and nonlinear correlation of the Arrhenius equation constants. Chemical Engineering and Processing: Process Intensification, 36 (3), 243 - 249. Chen, N. H., & Aris, R. (1992). Determination of Arrhenius constants by linear and nonlinear fitting. AIChE journal, 38 (4), 626 - 628. Datta, A. K. (1993). Error estimates for approximate kinetic parameters used in food literature. Journal of Food Engineering, 18(2), 181 - 199. Dolan, K. (2015, July 9th, 2015). [Simulation Temperature Approach for Non - Isothermal Migration Studies]. Dolan, K. D. (2003). Estimation of kinetic parameters for nonisothermal food processes. Journal of Food Science, 68 (3), 728 - 741. Dolan, K. D., & Mishra, D. K. (2013). Parameter estimation in food science. The Annual Review of Food Sci ence and Technology, 4 , 401 - 422. 373 Dolan, K. D., Valdramidis, V. P., & Mishra, D. K. (2013). Parameter estimation for dynamic microbial inactivation: which model, which precision? Food Control, 29 (2), 401 - 408. - Franco, F., Soto - Valdez, H., Peralta, - Meza, N. (2012). Antioxidant Activity and Diffusion of Catechin and Epicatechin from Antioxidant Active Films Made of Poly (l - lactic acid). Journal of Agricultural and Food Chemistry, 60 (26), 6515 - 6523. Klicka, R., & Kubácek, L. (1997). Statistical properties of linearization of the Arrhenius equation via the logarithmic transformation. Chemometrics and intelligent laboratory systems, 39 (1), 69 - 75. Pritchard, D. J., & Bacon, D. W. (1975). Statistical ass essment of chemical kinetic models. Chemical Engineering Science, 30 (5), 567 - 574. Pritchard, D. J., & Bacon, D. W. (1978). Prospects for reducing correlations among parameter estimates in kinetic models. Chemical Engineering Science, 33 (11), 1539 - 1543. S chwaab, M., Lemos, L. P., & Pinto, J. C. (2008). Optimum reference temperature for reparameterization of the Arrhenius equation. Part 2: Problems involving multiple reparameterizations. Chemical Engineering Science, 63 (11), 2895 - 2906. Schwaab, M., & Pinto , J. C. (2007). Optimum reference temperature for reparameterization of the Arrhenius equation. Part 1: Problems involving one kinetic constant. Chemical Engineering Science, 62 (10), 2750 - 2764. Sulaiman, R., Dolan, K. D., & Mishra, D. K. (2013). Simultane ous and sequential estimation of kinetic parameters in a starch viscosity model. Journal of Food Engineering, 114 (3), 313 - 322. Sundberg, R. (1998). Statistical aspects on fitting the Arrhenius equation. Chemometrics and intelligent laboratory systems, 41 (2), 249 - 252. 374 Chapter 8 Overall Conclusion and Recommended Future Work 8.0 Overall Conclusion Migration of additives from a polymer film into food product s has been continuously studied since the early 1960s due to its importance in food safety , quality assurance and shelf life of foods. Earlier on a large amount of research was conducted to determine if the polymer films were safe to be in contact with food product s . Lately, a large amount of research was directed into finding natural occurrin g additives to be incorporated into polymer films and in investigating the effectiveness of these additives to prolong the shelf life of the intended product while monitoring its threshold limit (Barbosa - Pereira et al., 2013; Chen, Lee, Zhu, & Yam, 2012; Colín - Chávez - Franco et al., 2012; Lopez de Dicastillo et al., 2011; Lopez - de - Dicastillo, Alonso, Catala, Gavara, & Hern 2009; Pereira de Abreu, Losada, Maroto, & Cruz, 2010; Sanches - Silva et al., 2014; Sonkaew, Sane, & Suppakul, 2012; Zhu, Schaich, Chen, & Yam, 2013) . Meanwhile, a small number of s tudies have been performed to assess the kinetic s of migration using mathematical modeling or centered on determining the migration parameters . Most of the work published in this area used previously developed mathematical modeling and solutions to identif y the rate at which the diffusion of additives takes place known as the diffusion coefficient, D . To a certain extent, the chemical affinity between a polymer and a food/food simulant known as the partition coefficient, K p,f is determined experimentally at the end of the experimental duration (Baner, 2000; Baner, 375 . The measurement of these parameters often takes time and is costly. Therefore, this dissertation aimed at understand ing the kinetics of migration of additives , e special ly antioxidants , from polymer films . M igration of these additives from a biodegradable polymer, poly(lactic acid), PLA into regulated food simulants by using parameter estimation approach was conducted an d the method to calculate these parameters is presented . Para meter estimation is defined as provides tools for the efficient use of data in the estimation of constants appearing in mathematical models and for aiding in modeling of phenom (Beck & Arnold, 1977) . The first initiative taken to perform the parameter estimation approach was to produce a bilayer P LA film incorporated with marigold flower extract via a blown extrusion process (Chapter 3). This produced film was subjected to three different parts; i ) migration study of astaxanthin (the dominant antioxidant presence in the Marigold flower extract) into 95% ethanol at 30 and 40 C, ii ) characterization of the produced film by thermal, barrier, physical and morphological analyses, and iii ) the oxidative stability assessment of the antioxidant toward a real fatty food product ( i.e., soybean oil). This study found that the addition of the marigold flower extract did not affect the polymer properties except for molecular weight, water vapor permeability and polymer infrared (IR) spectra. The migration of astaxanthin to 95% ethanol was est imated using the general Crank mathema tical solution and followed astaxanthin into soybean oil was too slow to be a ble to retain the freshness of the product within the limit of Codex Alimentarius, whi ch may be attributed to the lower chemical affinity between the astaxanthin and soybean oil. In this work, parameter estimation was done using ordinary least square (OLS) estimation . Instead of estimating only D , the concentration of the migrant migrating 376 into the food simulant at equilibrium, M was also estimated to understand the migration kinetic s in the food simulant. From chapter 3, it was observed that the assessme nt of an additional parameter did provide additional infor mation on the kinetic s of a migration phenomenon. Therefore, effort was allocated to further underst and the mass transfer process in polymer films from the parameter estimation point of view, presen ted in chapter 4. The impact of estimating one (1P), two (2P) and three (3P) parameters by analyzing the scaled sensitivity coefficient, before performing the OLS estimation and the optimal experimental design was examined. By assessing a different numb er of parameters inside a mathematical expression, the physical meaning behind a migration experiment was better understood. Also, by conducting this assessment the issue of under parameterized or over parameterized migration experiments was explored. Asse ssment of only one parameter estimate could cause an under parameterized issue when there is a possibility of obtaining additional insight from other parameters. Meanwhile, assessment of more parameters could have induced the complexity of the estimation p rocess by introducing more uncertainty and reduction of estimation accuracy. Therefore, the plot was introdu ced and constructed to foresee the perturbatio n. The desirable parameter s to be estimated would be the ones with larger magnitude change of response that are uncorrelated with the other parameters. Optimal experimental design was employed to identify the time needed for a migration experiment to be ab le to accurately estimate the parameter s of interest . The AICc approach in addition to RMSE value was used as a tool for model selection. Several selected migration case studies (based on PLA) were chosen to demonstrate the parameter estimation approach in troduced chapter 4. Interesting results included ; i) the assessment of the third migration parameter known as (ratio of the mass of migrant 377 migrated into food/simulant to the mass of migrant left in the polymer, at equilibrium) was observed to be better estimate d at the initial experimental time instead of at equilibrium, which goes against the general assumption that it should be determine d at the end of the migration experiment, ii) both the and the optimal experimental design plots indicated that m o st of the migration studies were performed beyond the necessary time. Therefore, with adequate initial guesses for t he polymer - additive system, one can predict the optimal time needed to conduct a migration experiment while being able to accurately estimat e the parameters via maximization of the determinant. The overall highlight from this chapter was that the kinetic migration parameter that is directly related to K p,f should be investigated at an early time in the experiment. Additionally, another kine tic parameter known as the convective mass transfer coefficient, h , is presented at the early time of experiment, which holds large importance in migration experiments when the simulant is not stirred or viscous simulants. Consequently, these highlights le d to the development of a two - step solution in chapter 5 to estimate h in addition to the other two kinetic migration parameters ( i.e., D and K p,f ). In general mathematical solutions based on Fic diffusion was used to e stimate D , M , and to understand the kinetics of migration (Crank, 1979) . Based on our finding from chapters 3 and 4, we identified that D and did have significant impact on the overall kinetics of migration. Hence, in chapter 5, three driving factors that govern the sorption and/or desorption kinetics of migration from polymer films were estimated; D , K p,f and h. These three parameters are beneficial for providing in - depth insight on the physical meaning behind a migration phenomenon. Therefore, a two - step solution based on the boundary conditions - step solution was used to find the combination of D , K p,f and h that minimized the sums of squared errors (SSE). Unlike the linear 37 8 equation that contains a unique local minimum, a non - linear equation contains many local minima, thus the use of step 1 served to find the right local minima region to start the estimation of the migration parameters and can provide the true solution of the parameters. From the initial guesses from step 1, OLS estimation could be used in step 2. The OLS estimation was performed by us ing the proposed analytical solution containing the D , K p,f and h . A migration case study of PLA incorporated with 3 wt.% resveratrol in contact with ethanol at 9 C was used to demonstrate the use of this two - step solution, which was later named as model 1 in chapter 6. Additional parameter estimation approaches such as the sequential, bootstrap and the kinetic phase diagram were also performed to acquire a better knowledge about the kinetics of migration. Chapter 6 is presented as the continuation of chap ter 5, where the two - step solution (model general mathematical solutions containing D and K p,f ( model 2) and containing only D (model 3) by using three different migration case studies. The OLS estimation and the model select ion were based on the corrected Akaike information criterion (AICc) approaches. Consequently, a decision tree analysis containing all three models was proposed as a tool for selecting the appropriate model to analyze migration studies of additives from pol ymer films. After an extensive focus on the kinetics of migration at one particular temperature (isothermal), a quest for the kinetics of migration involving several temperatures (non - isothermal) was pursued in chapter 7. The Arrhenius equation was employe d as the secondary model to model 1 presented in chapter 5. The reparameterization approach was applied to find the optimum reference temperature, T ref corresponding to reference D , D ref , to solve the corre lation issue associated with D ref and the activation energy, E a The simulation temperature, T sim approach was introduced in this chapter for the purpose of 379 constructing E a of the plot. The reparameterization approach improved the estim ation of E a as indicated by the observed relative errors of parameters . Hence, it was proven that with the T ref (optimum) that gave the n ear zero correlation between D ref and E a , the lowest relative error of D ref was achieved. This finding highlights the importance of having the optimum T ref instead of arbitrarily choosing any temperature within the experimental range to obtain better accuracy of the parameters estimation. All things considered, the work performed in this dissertation is just the beginnin g of introducing a prediction technique to determine the kinetic migration parameters and to reduce the ir uncertainty . Additional and more comprehensive and extensive works should be theoretically and experimentally executed to further gain understanding of the kinetics of migration of additives from polymer films. 8.1 Recommended Future Work All the migration case studies selected for demonstrating the use of different models were performed in the presence of continuous stirring (presence of turbulent fl ow), thus the effect of h on the overall migration kinetics may have not been observed much. Figure 8 - 1 demonstrated two different cases (a) small h , thus Biot number < 200 ( Bi =146), and (b) larger h , thus Biot number >200 ( Bi = 2920 ). Future work should fo cus on conducting and getting experimental data for a migration experiment that could be reflected by Figure 8 - 1(a), which means the resistance at the interface between the polymer and the food simulant is larger. Several scenarios such as the absence of c onvection, the use of viscous food simulants such as oil, Miglyol®, emulsion etc. and the effect of various polymer thickness should be designed and investigated in a migration experiment to assess the kinetics parameters using the two - step solution (model 1) proposed in 380 chapt er 5. Comparative modeling should also be performed with other available mathematical models provided by other authors (Carslaw & Jaeger, 1959; Crank, 1979; Gandek, Hatton, & Reid, 1989; Piringer & Beu, 2000; Vitrac & Hayert, 2006; Vitrac, Mougharbel, & Feigenbaum, 2007) . While model 1 can provide an exact solution to a migration study and direc t physical interpretation on the kinetics of migration, it does not assess additional cases when the polymer film may degrade or the diffusion coefficient changes with the concentration. So, parameter estimation through numerical approximation considering all three kinetic migration parameters should also be explored to obtain migration kinetic parameters and to complement the analytical solution approach when D , K p,f and h change as a function of time . 381 Figure 8 - 1 Scaled sensitivity coefficient of (a) the case with Biot number < 200 (Initial guesses were: D ref =3.00 10 - 9 cm 2 /min, K p,f =608 cm 3 PLA/cm 3 ethanol, and h =8 10 - 5 cm/min), (b) the case with Biot number > 200 (Initial guesses were: D ref =3.00 10 - 9 cm 2 /min, K p,f =608 cm 3 PLA/cm 3 ethanol, and h =1 .60 10 - 3 cm/min). 382 APPENDIX 383 APPENDIX : List of Conference Presentations and Publications Generated from this Dissertation Peer Reviewed Journal Article Samsudin, H., Soto - Valdez, H. & Auras, R. (2014). Poly(lactic acid) membrane incorporated with marigold flower extract ( Tagetes erecta ) intended for fatty - food application. Food Control, Volume 46, pages 55 - 66. Book Chapter Samsudin, H. & Auras, R. Food Packaging Interaction in Introduction to Food Packaging. Submitted to John Wiley & Sons and Institute of Food Technologists (IFT) Press, February 2015. (Submitted and under review) Grant Application Samsudin, H., Auras, R., Soto - Valdez, H. T he NineSigma RFP (2013) - Best Anti - Oxidants to Improve Oxidation Stability of Synthetic Polymers (RFP# 69117) Conferences 1. Hayati Samsudin, Rafael Auras, Gary Burgess, and Herlinda Soto - Valdez. A Decision Tree Analysis for Determining Mass T ransfer Parameters for Migration Studies. 3 rd International Meeting of Material/Bioproduct Interaction (MATBIM) 2015, Spain 384 2. Javiera Rubilar, Rafael Auras, Hayati Samsudin, and Franco Pedreschi. Release of Citral, Carvacrol and Eugenol from Poly(lactic aci d) Nanocomposite Films. 3 rd International Meeting of Material/Bioproduct Interaction (MATBIM) 2015, Spain 3. Hayati Samsudin, Rafael Auras, Kirk Dolan, Dharmendra Mishra, and Herlinda Soto - Valdez. Assessing The Kinetics of A Migration Study by Estimating A Two or Three - Parameter Models. Inverse Problems Symposium (IPS) 2015, East Lansing (Poster) 4. Hayati Samsudin, Rafael Auras, Kirk Dolan, Dharmendra Mishra, and Herlinda Soto - Valdez. Application of Parameter Est imation to Predict Migration of Antioxidant Films. The Shelf Life International Meeting (SLIM) 2014, New Jersey (Poster). 3 rd place in the poster competition sponsored by the Elsevier©. Seminar: Hayati Samsudin and Herlinda Soto - Valdez. Migración de Astaxantina de una Película de Ácido Poliláctico (Migration of Astaxanthin from a Poly(lactic acid) Film), Center for Food and Research Development (CIAD), September, 2011, Sonora, Mexico. 385 REFERENCES 386 REFERENCES Baner, A. L. (2000). Partition coefficient. In A. L. Baner & O. - G. Piringer (Eds.), Plastic Packaging Materials for Food, Barrier Function,Mass Transport, Quality Assurance and Legislation (pp. 79 - 123). Weinheim: Wiley - VCH. Baner, A. 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