AAA-‘11. , A--.“ bV1ESI.} RETURNING MATERIALS: P1ace in book drop to LIBRARIES remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. MODELING THERMODYNAMIC AND DIFFUSION PROPERTIES IN CONCENTRATED POLYMER SOLUTIONS By Michael John Misovich A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering 1988 ABSTRACT MODELING THERMODYNAMIC AND DIFFUSION PROPERTIES IN CONCENTRATED POLYMER SOLUTIONS By Michael John Misovich A methodology for evaluating solvent activities in concentrated polymer solutions is proposed and demonstrated. This method allows the use of any expression for the residual (enthalpic) interaction between polymer and solvent, in conjunction with a Flory-Huggins expression for the combinatorial entropy, and an empirical free volume correction. The new method is applied using several choices for the residual term, including the Analytical Solution of Groups (ASOG) group contribution equations. When adjustable parameters are determined by best fit to data, results predicted by the method generally agree with observed data from 21 isothermal binary polymer-solvent systems better than results given by the Flory-Huggins model. When parameters are determined from a single data point at low solvent concentration and extrapolated to higher concentrations, a version of the new method agrees better with observed data than the Flory-Huggins model and better than the UNIFAC-FV model which uses no binary data. Transformations of equations used by group contribution models to calculate the residual contribution to the activity coefficient are demonstrated. Using these transformations to allow more convenient analysis of the mathematical properties of the equations, bounds on the range of activity coefficients can be derived from incomplete data without knowledge of the interaction parameter values. The predicted values of activity coefficients are shown to depend on a normalization step implicit in the definition of functional group size. Three alternative models for prediction of binary diffusivities in concentrated polymer solutions are compared: a complete free volume model, a linearized form of this model, and a constant diffusivity model. A method is presented for determining when the simpler models are appropriate for calculations. The linear model is convenient to use for determining the effects of the solvent activity coefficient on the diffusivity. A new statistical technique is proposed and demonstrated for determining whether a nonlinear data fit is systematically in error with observation. Unlike many statistical techniques, the new method is valid regardless of the distribution of the observed variables. It is capable of detecting complex patterns of systematic error not found significant by other statistical methods. To my wife, Aimee, and to both our families, who can now celebrate their first Ph.D. recipient. ii ACKNOWLEDGEMENTS Special thanks go to my thesis and dissertation advisor, Dr. Eric A. Grulke, for his direction, assistance, and patience during the period of my research. Appreciation is also due to my other committee members: Dr. Donald R. Anderson, Dr. Krishnamurthy Jayaraman, Dr. Charles A. Petty, and Dr. Karel Solc, and to all the faculty members who have helpful to me during the eleven years I have spent at Michigan State University as an undergraduate and graduate student. iii TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS CHAPTER 1: INTRODUCTION VARIABLE SIZE PARAMETER METHOD FOR POLYMER SOLUTION THERMODYNAMICS RESIDUAL INTERACTIONS IN GROUP CONTRIBUTION MODELS DIFFUSION IN POLYMER SOLUTIONS ABOVE TG STATISTICS OF NONLINEAR DATA FITTING CHAPTER 2: VARIABLE SIZE PARAMETER APPROACH TO THERMODYNAMICS OF CONCENTRATED POLYMER SOLUTIONS ORIGINAL VSP SINGLE PARAMETER METHOD Generalized Correlation for Solvent Activities in Polymer Solutions Group—Contribution Models ASOG Model Modification of A806 for Polymer Solutions Comparison to Flory-Huggins Model Comparison to UNIFAC-FV Model Comparison of ASOG-VSP, UNIFAC-FV, and Flory- Huggins Models with Experimental Data Conclusions Acknowledgement Literature Cited VSP METHOD USED WITH RESIDUAL INTERACTION TERMS Prediction of Solvent Activities in Polymer Solutions Using an Empirical Free Volume Correction Introduction Generalized Thermodynamic Modeling Variable Size Parameter Revised Variable Size Parameter Approach Application with Various Residual Terms Fitting of Model Parameters iv viii xi xiii 12 13 14 14 15 16 17 l8 19 19 19 20 21 22 23 27 29 31 34 V Comparison with Solvent Activities in Polymer Solutions 40 Conclusions 50 Acknowledgement 50 Literature Cited 50 Appendix A. Example of VSP Method 52 CHAPTER 3: ANALYSIS OF RESIDUAL TERMS USED IN GROUP CONTRIBUTION MODELS 59 ANALYSIS OF RESIDUAL TERM IN SOLUTION OF GROUPS MODEL 61 Normalization and Bounding Properties Inherent in Solution of Groups Activity Coefficient Models 62 Introduction 63 Effects of Normalization on Residual Activity 66 A Normalization Independent Expression for Residual Activity Coefficients 69 Transformation of Wilson Parameters 74 Basic Properties of Infinite Dilution Normalized Residual Activity Coefficients and Bounds on Their Parameters 76 Constant Infinite Dilution Normalized Residual Activity Relationships 80 Bounding the Concentration Dependence of Normalized Residual Activity Coefficients 88 Bounding the Unknown Activity of a Second Component 93 - Normalization Dependence of Residual Activity Coefficients 97 Conclusions 101 References 102 CHAPTER 4: MODELING DIFFUSION COEFFICIENTS FOR CONCENTRATED POLYMER SOLUTIONS ABOVE TC 103 A DIFFUSION COEFFICIENT MODEL FOR POLYMER DEVOLATILIZATION 104 A Diffusion Coefficient Model for Polymer Devolatilization 106 Introduction 106 Free Volume Models for Diffusivity 107 Chemical Potential Derivative 107 Linearized Diffusivity Model 108 Comparison of Linear and Complete Models 109 Effects of Solvent WLF Parameters on the Linear Model 110 Effects of K and K2 on the Linear Model 111 Nomenclature 113 References 114 vi CHAPTER 5: STATISTICAL DETERMINATION OF SYSTEMATIC ERROR IN NONLINEAR PARAMETER ESTIMATION LINEAR AND NONLINEAR PARAMETER ESTIMATION DEFICIENCIES OF LINEAR LEAST SQUARES TEST IN DETERMINING ERROR DISTRIBUTION NONPARAMETRIC STATISTICAL TECHNIQUES A PROPOSED NONPARAMETRIC STATISTIC FOR DETERMINATION OF SYSTEMATIC ERROR AN EXAMPLE CALCULATION FOR DETERMINATION OF SYSTEMATIC ERROR CHAPTER 6: CONCLUSIONS AND RECOMMENDATIONS APPENDIX A: DATA USED IN THERMODYNAMIC MODELING APPENDIX B: RESULTS OF THERMODYNAMIC MODELING USING DATA EXTRAPOLATED FROM LOW SOLVENT CONCENTRATIONS APPENDIX C: RESULTS OF THERMODYNAMIC MODELING USING A BEST FIT OF ALL DATA APPENDIX D: PROGRAM USED TO APPLY THERMODYNAMIC MODELS USING DATA EXTRAPOLATED FROM LOW SOLVENT CONCENTRATIONS APPENDIX E: PROGRAM USED TO APPLY THERMODYNAMIC MODELS USING BEST FIT OF ALL EXPERIMENTAL DATA APPENDIX F: DERIVATION OF EQUATIONS IN "GENERALIZED CORRELATION FOR SOLVENT ACTIVITIES IN POLYMER SOLUTIONS" DERIVATION OF EQUATION 6 DERIVATION OF EQUATION 7 DERIVATION OF EQUATION 14 DERIVATION OF EQUATION 15 DERIVATION OF EQUATION 16 CONVERGENCE OF THE ITERATION PROCEDURE FOR CALCULATING 01” IN EQUATIONS 18 T0 23 116 117 120 122 129 132 142 150 165 178 209 225 242 242 244 244 245 245 246 vii APPENDIX G: DERIVATION 0F EQUATIONS IN ”PREDICTION OF SOLVENT ACTIVITIES IN POLYMER SOLUTIONS USING AN EMPIRICAL FREE VOLUME CORRECTION” DERIVATION 0F EQUATION 10 DERIVATION 0F EQUATION 148 DERIVATION OF EQUATION 17 APPENDIX H: DERIVATION 0F EQUATIONS IN ”NORMALIZATION AND BOUNDING PROPERTIES INHERENT IN SOLUTION OF GROUPS ACTIVITY COEFFICIENT MODELS" APPENDIX I: DERIVATION OF EQUATIONS IN "A DIFFUSION COEFFICIENT MODEL FOR POLYMER DEVOLATILIZATION" DERIVATION OF EQUATION 6 DERIVATION OF EQUATION 10 DERIVATION OF EQUATIONS 12, 12A, 12B, 12C LIST OF REFERENCES 252 253 255 256 260 286 286 290 291 296 CHAPTER 2 LIST OF TABLES Generalized Correlation for Solvent Activities in Polymer Solutions Table I. Entropic and Enthalpic Activity Coefficients for the Toluene-Poly(styrene) System Given by ASOG Table II. Typical Infinite Dilution Weight Fraction Activity Coefficients for Chemically Similar Systems Table III. Comparison of ASOG-VSP, Flory-Huggins, and UNIFAC-FV Models for Solvent-Polymer Systems Table IV. Comparison of Calculated and Experimental Activities for Chloroform-Poly(viny1 acetate) at 45°C. Table V. Comparison of Calculated and Experimental Activities for Benzene-Poly(ethylene oxide) at 75 1°C. Table VI. Comparison of Calculated and Experimental Activities for Benzene- Poly(isobutylene) at 25°C. Table VII. Accuracy of the ASOG-VSP, Flory- Huggins, and UNIFAC-FV Models on the Data Tested Prediction of Solvent Activities in Polymer Solutions Using an Empirical Free Volume Correction CHAPTER 3 Table 1. Parameter Values Determined by Data Fit Table 2. Comparison of Models with Experiment Table 3. Comparison of Errors Normalization and Bounding Properties Inherent in Solution of Groups Activity Coefficient Models CHAPTER 4 Table 1. Sign of dckl/dAkl as a Function of g1 and 8 Table 2. Extrema of Normalized Residual Activity Coefficient of Component 1 as a Function of Component Group Ratios A Diffusion Coefficient Model for Polymer Devolatilization Table 1. Comparison of R Values Determined from Different Values of VLF Constants. viii 15 15 18 19 19 19 19 36 46 48 80 84 110 ix Table 2. Calculated Values of K1 and K2 for Several Solvent-Polystyrene Systems. CHAPTER 5 Table 1. Acetone Viscosity Data and Prediction of Equation 5. Table 2. Calculation of Linear Correlation Coefficient, r. Table 3. Calculation of Runs Test Statistic, R. Table 4. Ranked Temperature and Error Data. Table 5. Calculation of Rank Correlation Coefficient, r . Table 6. Calcalation of Sum Square Rank Difference, Rd. APPENDIX A Table A-1. Data Used in Thermodynamic Modeling. APPENDIX B Table B-1. Results Using Thermodynamic Data Extrapolated from Low Solvent Concentrations. APPENDIX C Table C-l. Results Using Thermodynamic Data Fit to the Entire Data Set. APPENDIX D Table 0-1. File Specification for Program Execution. Table 0-2. Format of Functional Group and Compound Information File. Table D-3. Source Code for Program to Extrapolate Low Solvent Concentration Thermodynamic Data. APPENDIX E Table E-l. Format of Functional Group and Compound Information File. Table E-2. Source Code for Program to Fit All Solvent Concentration Thermodynamic Data. APPENDIX F Table F-l. Equations Used in "Generalized Correlation for Solvent Activities in Polymer Solutions". APPENDIX C Table 6-1. Equations Used in "Prediction of Solvent Activities in Polymer Solutions Using an Empirical Free Volume Correction". 111 133 133 137 138 139 140 154 166 180 211 212 213 226 227 250 258 APPENDIX H Table H-l. Equations Used in ”Normalization and Bounding Properties Inherent in Solution of Groups Activity Coefficient Models”. 282 APPENDIX I Table I-l. Equations Used in ”A Diffusion Coefficient Model for Polymer Devolatilization”. 294 LIST OF FIGURES CHAPTER 2 Generalized Correlation for Solvent Activities in Polymer Solutions Figure 1. Concentration Dependence of the interaction parameter Prediction of Solvent Activities in Polymer Solutions Using an Empirical Free Volume Correction Figure 1. Comparison of Infinite Dilution Activity Coefficients from Different Residual Terms. Figure 2. Comparison of Infinite Dilution Residual Coefficients from Different Residual Terms. Figure 3. Comparison of Size Factors from Different Residual Terms. Figure 4. Solvent Activity Coefficient as a Function of Concentration, Benzene- Poly(isobutylene) at 250 C Figure 5. Solvent Activity Coefficient as a Function of Concentration, Toluene- -Poly(methyl methacrylate) at 25° C Figure 6. Solvent Activity Coefficient as a Function of Concentration, Toluene- -Poly(styrene) at 600 C. 37 38 39 41 43 44 CHAPTER 3 Normalization and Bounding Properties Inherent in Solution of Groups Activity Coefficient Models Figure l. Constant Infinite Dilution Normalized Residual Activity Coefficient Relationships for g - 2. Figurb Constant Infinite Dilution Normalized Residual Activity Coefficient Relationships for g - 0. Figurh Bounding the Concentration Dependence of Normalized Residual Activity Coefficients for g - 2. Figurh Bounding the Concentration Dependence of Normalized Residual Activity Coefficients for - 1, g - 0. Figure 5. Bounding the Infinite Dilution Normalized Residual Activity Coefficient of the Second Component for g1 - 1, g2 - 2. xi 86 87 91 92 95 CHAPTER 4 xii Figure 6. Bounding the Infinite Dilution Normalized Residual Activity Coefficient of the Second Component for g - 1, g - 0. Figure 7. Bounding the Concengration Dependence of Residual Activity Coefficients for g - l, g2 - 2. Figure 8. Bounding the Concentration Dependence of Residual Activity Coefficients for g1 - 1, g2 - 0. A Diffusion Coefficient Model for Polymer Devolatilization CHAPTER 5 Figure 1. Comparison of the concentration dependence of diffusion coefficients calculated by the complete and linear free volume models. Toluene and Polystyrene. Figure 2. Comparison of R versus temperature for two sets of solvent VLF parameters. Figure 3. Comparison of R1 versus temperature for water-polystyrene. Figure 4. -K versus temperature for three solvents in polystyrene. Figure 5. Log D/D(0) versus log (ppm solvent) for methanol-polystyrene. Solvent free volume parameters of this study. Figure 6. Log D/D(0) versus log (ppm solvent) for methanol-polystyrene. Solvent free volume parameters of Liu (21). Figure 7. Effect of the thermodynamic term of D/D(0) for the co lete Model. Methanol- polystyrene at 155 C. Figure 1. Normal Approximation to the Runs Statistic R. Figure 2. Viscosity of Pure Acetone as a Function of Temperature. Figure 3. Error Distribution as a Function of Temperature. 96 99 100 109 111 111 112 113 114 114 126 134 135 F‘CDCD H‘ H pa. P‘H° li’ 2i 7: KWNNNKWMQD-QUUUUOO p. ”WW'U {3:13 o- 5.? LIST OF SYMBOLS groups of parameters used in free volume diffusion model group interaction parameter for group R with group 1 coefficient of quadratic term temperature-independent part of the group interaction parameter for group k with group 1 solvent activity experimental solvent activity predicted solvent activity entropic, combinatorial, or size interaction part of the solvent activity transformed group interaction parameter for group k with group 1 coefficient of linear term temperature-dependent part of the group interaction parameter for group R with group 1 transformed group interaction parameter for group k with group 1 constant term size-weighted fraction of component i in solution binary mutual diffusion coefficient preexponential constant in temperature-dependent factor of D preexponential constant in free volume-dependent factor of D self-diffusion coefficient of solvent statistical degrees of freedom difference between e and T from observation 1 base of the natural iogarithm critical energy per mole needed to overcome attractive forces size-weighted ratio of group 2 to group 1 in component i free volume coefficient in the linearized diffusivity model WLF free volume parameters of component i thermodynamic coefficient in the linearized diffusivity model a nonnegative integer molecular weight of component i occurrences of the symbol - in the runs test number of data points observed occurrences of the symbol + in the runs test number of functional groups of type k in component 1 pressure gas law constant random variable used in runs test random variable used in sum square rank difference test 1 xiii <80). Table 111 gives the details of the sets studied. For each set. the lowest concentration data point was chosen for correlation of 0.‘ by eq 18—23. This (1.0 was then used to predict the activity using the ASOG-VSP model. TheFIory-Hugginxwascalculated fromIlfand used as a constant value in the Flory-Huggins equation. The UNIFAC-FV model was applied according to Oishi and Prausnitz (1978). using their recommended values for the free volume parametersand theGmehlingetal. (1982) values for the group interaction and size parameters. Denitydata forsolventsand polymerswereobtained from Timmermans (I950). Brandup and Immergut (1975). and Marketal. (1972). lnsomemses. liquid densitydata below the normal boiling point were extrapolated to estimate liquid densities at higher temperatures. The ability ofany ofthe models to fit the data depends on the value of 0". Tables lV-Vl give typical results for three data sets: one each exhibiting negative. positive. and athermal behavior. Table VII summarises the accuracy of the three models on the given data sets. The perform- ance of the Flory—Huggins model and ASOG-VSP model was roughly equal on athermal systems. with both models accurate within 5% of the experimental activity for about 90% of the data points. In systems showing marked positive or negative deviations from athermal behavior. the ASOG-VSP model predicted activity within 5% of ex- periment for over 70% of the data. while the Flory-Hug- ginmodelwasasaccurateleasthanillfi ofthe time. The UNIFAC-W model generally performed more poorly than ASOG-VSP and Flory-Huggins. as might be expected since it utilizes no binary data in its predictions. It is possible that performance of UNIFAC-W could have been -> — -_. _.._ 1042 Ind Eng ChernflocessOes Oav.Vo1 24.140 4. 1905 Table IV. Conan“... 0' Calculated and Experimental Activities for t‘hloroferm-l‘e)y(viayl acetate) at 45 ‘43. 11" - LI, and ‘ ' —0.J92 "Hernia"! at or, I 0.093 --I\ airtiv cue-If and 1 i-rrur '1. frat-t . ‘ . ‘ FM” ‘ . _ . , “4‘. "I". A5414. \hl' Huggins IINIl‘Al :I:.\_ (1121 111111 I 47:) 4 4i I 41151 511 15:07 It II it 1:19) I 4115 I 469 4ti I not 5.9 I got 11:1 11.164 I 111'. 1411.1 :12) I ms iii 12111) I.’- :1 1115111 131% 1.455 6.5 1.48!) it 3 I 1‘17. - I I 5 021$ 1.452 145.1 1111 1 4714 1.11 I 1‘15 17 8 11.227 1.41!) 1.4411 :15 I 475 .'i .1 I 1')“: It 14 0.247 1.:1au 1443 ii; 1.47) liti I is” -I.‘17 0.276 1.382 1434 311 I464 59 1.197 ~14.) 0.295 1,3101 1 4'29 3.6 I 459 5.7 I lit-i -14.1 0.325 1.3135 1.4:" 4.0 I 45'.) 01 :1 1.111;) ~11?) (1.355 1.351 1.410 4.4 I 44) 6.7 1.1111 -12.6 0.427 1.389 1.3113 41.4 1.415 19 I 177 ~15 't 0.461 1.378 1.369 41.6 1.401) I 6 I 175 -I4 11 0.470 1:195 1:162 -2..‘1 1.39:1 4). I 1,174 -158 0.499 1.416 1.35.1 —4.3 1.382 -2.4 1.172 -I7.2 av % error 3 t t 6 14 7 Table V. Comparison of Calculated and Experimental Activities for Bensone-l’oly(ethyleae oxide) at 75.1 'C. 11,‘ I 4.48 and x I 0.210 Determined at w. ' 0.052 auIv activ coefT and % error wt fract ‘ . Flury- UNIF:AC- w“, 9‘1,“ AhOC-Vbl’ Huggins F\ 0081 3.755 3.749 (12 3.764 0 2 3.416 ":1 0 0.108 3.561 3.543 41.5 3,559 {1.0 3 247 —8 8 0.145 3.332 3.289 -|..'l 3.307 -0.7 3.1138 41.8 Av % error 0.7 0.3 8.9 Table VI. Comparison of Calculated and Experimental Activities for Beaseae-Pslytisobutylene) st 25 ’C. 11.“ - 0.41 and x I 1.09 Determined at w. - 0.043 solv activ coefl’ and % error ASOG- Florv— UNIFAC- “JI? "all VSP Huggins FV 0N3 6.409 6.274 -2.I 6.871 7.2 6.016 -6.1 0.094 5.460 5.520 1.0 6.224 13.8 5.452 -0.3 0.150 4.50s 4.5“ -2.2 5.251 14.0 4.620 0.3 0.152 4.63 4.404 -3.3 5.229 12.8 4.601 41.8 0.104 4.127 4.032 -2.3 4.752 15.1 4.199 1.8 0.245 3.404 3.370 --3.3 4.1X11 148 3.572 2.5 0.154 3.452 3.294 -4.6 3.911 13.3 3.497 13 0.” 3.070 2.940 —4.0 3. 485 13.5 3.144 2.4 0.321 2.073 2.779 -3.3 3.275 14.0 2.970 3.4 0.373 2.541 2.472 -2.7 2.801 13.4 2.642 4.0 av % error 2.9 13.2 2.3 Table V". Accuracy of the ASOGVSP. Flory-Huggins. and UNIFAC-W Models an the Data Tested model 0" < 2 3.5 < 0.‘ < 5.5 0.‘ > 8 all data % of Data Points for Which Model Was Accurate Within 5% ASOG-VSP 71 90 71 N Flory-Huggins 29 89 21 54 UNIFAC-W 0 20 25 22 % of Data Points for Which Model Was Accurate Within 10% ASOG-VSP If!) 97 83 92 Flory-Noggin 5 95 56 79 UNIFAC-W 0 59 50 46 improved if the value of the parameter used in calculating the free volume correction had been adjusted: however. no definitive guidelines for doing so are given by the authors of UNIFAC-FV. Conclusions The ASOG-VSP model was successful in predicting solvent activities in the polymer-solvent systems reviewed. Performance was equal in the Flory-Huggins model. It)- 19 perior In the UNIFAC-I'll model in athermal antenna. and sine-rim tu laitli of their: models in systems with significant enthalpic intvrm-Iinuvc ASWoVSI’ ului haul im mlvnntaigr- twer IIIt‘M' models in not requiring density (lulu for ap- plications of the model and is much simpler thiin UNI- l-‘AC-I-‘V from a cornputational standpoint. However, ANNE—VSP does require II single value of activity or an infinite dilution activity coefficient as a parameter. which UNIFAC-IV (Ii-est not. The results presented here can he extended to multi- cmnponent polymer-solvent systems. A theoretical de- rivation for systems with enthalpic interactions between polymer and solvent molecules by including the ASOG group-interaction parameters is also rmssihle. as is exten- Mun to modeling of temperature dependence of activity. We are continuing worlt on these topics and on the ap- plication of the results presented here to diffusion in polymer melts. Acknowledgment This wurlt was partially supported by the National Science Foundation (CPBWTO). During his MS. work at Michigan State University. Misovich was the recipient of fellowships from FMC Corp. and from Dow Chemical Co. Registry No. Toluene. 10888-3; poly(styrene). 900353.13; lienzene. 71-43-2; ethylbcnlcne. 10041-4; methyl ethyl lietone. 78-93-31; poly(isuhutylene), 9003-27-4; cycluhesane. 110-82-7; pentane. 109-66.0; triisopropylbenzene. 27322-34-5; carbon di- sulfide. 75-15-0; methanol, 67-56-1; polyl methyl methacrylate). 9011-14-7; poly(wnyl acetate). Mil-207; chloroform. 67-66-11; poly(cthylene oxide). 253226193. Literature Cited Abrams. O. S. Prat-n01. .1. 14. AlOfJ. 1079. 2). 116. Baum. C. E 14.; Freon-t. R. K. 3.; M A. ll. Iran. Fanny Soc. 1050. 40. 077. am. J. immenmiunaoufmwim th- Yo‘h. 1975. Chang. Y. 14.; m. D. C. J. A”. Poem- Sci. 1979. 19. 2457. Coda. 6.14 Item. .1 W. J M01519. I972. A-t. 10. 009. inn 0 i a tr." i? .11 1.3a... . . .14. any. on... 5:. urine. 124.". ii 335. iii}? g i . . . 1070. 49. 7. Fred-M A; Jones. ll; he‘ll. 4. It AlOfJ. '079. 21. 10.. . . sea 2: : :1 ii i i Kmx.;th.Wde4ME¢-OMAW: “.mw 1979. MKMD.M.MWMM.MM. I900. MKF.;GIMN.Q;M.N.K 04mm and teaming": Inter-science: New Volt. 1972. mu.i.u.s.mwmm.mm 1904. 33.1 m. 0. A. Diem. Eng. 1079. 021121. N. hear-la. .1114. list. 61v. Oren. houses Des. cm. 1002. 21. 537. has). 6.; New. .1. F. K. (M. fry. M. houses Des. Dev. 1001. 20. 204 that a.; m. .1. F. It. led. Em. main. Recess. as. an. 1004. I3. 251 see-i. r. n J. Mm. sci. tart. a4. a. isss. Ms. E. It; Ectian. C. a. 1M. Em. new. hnceasDea. Dev. 1004. 23. m . J. Cor-slams 01 has Onset: W': Eisner; Amsterdam 1950. Van. 3.11.; Vim). J aisle. Em. 5d. 1004. 30. 05). Wins. Q MJ. 14410.01”. 1004. 00.127. Received [or mm.- July 2. 1904 ‘ Revised manuscript received December 19. 1984 Accepted January 21. I985 i 20 VSP METHOD USED WITH RESIDUAL INTERACTION TERMS The manuscript which follows describes the derivation of the complete VSP method which includes an additional residual interaction term. Comparisons are made between the new method and the Flory-Huggins equation by fitting complete data sets to the adjustable parameters in each model. The new method is applied with three residual terms: one which describes no residual interaction (equivalent to the original VSP single parameter method); one which uses a term similar to the Flory- Huggins interaction term; and one which uses the ASOG-KT group contribution model to generate an interaction term from a parameter database without use of any adjustable parameters for residual interaction. Further details of the experimental data and results are given in Appendices A, C, and E. Detailed derivations for the equations proposed in the article are given in Appendix G. 21 Prediction of Solvent Activities in Polymer Solutions Using an Empirical Free Volume Correction ABSTRACT A recent correlation for solvent activities in polymer solutions is extended in scope to provide a methodology for modeling nonideal effects in polymer solutions. This new method allows the use of any expression for the residual (enthalpic) interaction between polymer and solvent in conjunction with a standard (Flory-Huggins) expression for the combinatorial entropy. An empirical free volume correction uses the infinite dilution weight fraction activity coefficient of the solvent as an adjustable parameter. The new method is applied using one residual term given by the Analytical Solution of Groups (ASOG) technique, one similar to the Flory-Huggins interaction term, and one which yields no residual interaction. The results of these three models are compared to one another and to the Flory-Huggins model for 21 isothermal binary polymer-solvent systems. When adjustable parameters are determined by best fit to the data, each of the models applying the new method results in a standard error of less than five percent for at least 16 of the systems studied. This represented a better performance than the Flory- Huggins model. 22 INTRODUCTION An understanding of the thermodynamics of polymer solutions is important in practical applications such as polymerization, devolatilization, and the incorporation of plasticizers and other additives. Diffusion phenomena in polymer melts and solutions are strongly affected by nonideal solution behavior, since chemical potential rather than concentration provides the driving force for diffusion. Proper design and engineering of many polymer processes depend greatly upon accurate modeling of thermodynamic parameters such as solvent activities. This work was an extension of previous work by the authors for correlating solvent activities in polymer solutions (Misovich et a1, 1985). In that paper, an empirical free volume correction is derived from an athermal form of the Flory-Huggins combinatorial entropy (Flory, 1953) suggested by the Analytical Solution of Groups (ASOG) group contribution model for calculation of activity coefficients in solution (Derr and Deal, 1969). The technique generally performs better than the classical Flory-Huggins equation in extrapolating solvent activity data from low solvent concentrations to higher concentrations. One deficiency of the approach is that phase separation cannot be predicted, i.e., dal/dw1 > O is always the case. In this paper, the empirical free volume correction was modified to allow the explicit inclusion of an expression for residual (enthalpic) interaction between polymer and solvent. A general scheme was given to 23 accomplish this, and three specific cases were analyzed and compared. One case used the ASOG expression for residual interaction, while a second used an interaction parameter approach similar to the Flory- Huggins equation. The third case assumed that there was no residual interaction term, and reduced to the generalized correlation previously cited (Misovich et a1, 1985). The results in this paper were based upon a best fit of the adjustable parameters in each model using a least squares evaluation of all the data, not by extrapolation from a single data point. In each of the three cases, the infinite dilution weight fraction solvent activity coefficient 01” is an adjustable binary parameter. A residual interaction parameter is a second adjustable binary parameter in the second case. The classical Flory-Huggins equation was also fit to the data for comparison. In general, regardless of which residual interaction expression was used, the new method fits the data with less error than the Flory-Huggins equation. GENERALIZED THERMODYNAMIC MODELING Nonideal interactions between molecules in solution are generally classified in one of two categories. Interactions resulting from differences in the size or shape of molecules are classified as entropic, while interactions resulting from differences in energy are classified as enthalpic. The complete expression for solvent activity a1 is typically derived by multiplying concentration (mole fraction) x1, 24 a size or entropy activity coefficient, 118, and a enthalpy or group interaction activity coefficient, 716, or by adding their logarithms as shown in eq 1a. It is also common to lump the concentration with one of the activity coefficients (usually the entropic coefficient) to give eq lb. 1n a - 1n x + In S + 1n C (Is) 1 1 71 71 S G In a1 - 1n a1 + In 71 (1b) A statistical approach allows entropic interactions to be handled combinatorially, as is done by the athermal Flory-Huggins equation (Flory, 1953), giving for the entropic contribution to activity, a1S S S In a1 - 1n (x111 ) - 1 - ¢1 + 1n ¢1 (2) where x1 is the mole fraction, 118 is the entropic activity coefficient, and ¢1 is the volume or segment fraction of component 1 (solvent). Staverman (1950) has also given an expression for combinatorial entropy which includes surface area variables as well as volume variables. The modeling of enthalpic interactions generally involves the use of some type of binary interaction parameters. For similarly sized molecules, the entropic term is often considered small and the activity coefficient model consists wholly of the enthalpic term. In cases where both effects must be considered, the enthalpic or group interaction contribution to the activity coefficient, 116, is taken as the residual remaining after the combinatorial entropic term is removed from the 25 total activity coefficient. In the Flory-Huggins equation, this term is given by G 2 11 - X¢2 (3) where x is the adjustable interaction parameter. Several models for solution thermodynamics incorporate both types of effects. Analytical Solution of Groups, or ASOG, (Derr and Deal, 1969) uses a Flory-Huggins combinatorial entropy along with a residual enthalpy similar to Wilson (1964). Universal Quasi-Chemical, or UNIQUAC (Abrams and Prausnitz, 1975) and UNIFAC (Fredenslund et a1, 1975) are similar, but use a Staverman combinatorial entropy, and use surface area fraction rather than mole fraction as the independent variable. ASOG and UNIFAC also differ from UNIQUAC in that a group-contribution concept is used to analyze a solution in terms of interactions between functional groups rather than molecules. In both models, a database of functional group interaction parameters has been built. This allows prediction of residual interactions without use of binary data for the molecular components. All necessary binary data for functional groups is available from the database. Group-contribution models can be particularly useful in describing polymer solutions. Although polymer molecules are distributed in molecular weight, they are identical in their functional group composition regardless of their size. 26 Predictions of classical Flory-Huggins theory and the group-contribution models show deficiencies when compared to actual data for concentrated polymer solutions. The interaction parameter in the Flory-Ruggins equation, x, does not correlate directly to the enthalpic interaction between molecules. This is evidenced by the fact that significantly nonzero values of x are required for accurate fit of data for systems with little enthalpic interaction, like polystyrene-toluene. The presently accepted interpretation of x is that of a free energy interaction parameter incorporating both entropic and enthalpic effects. When functional group interaction parameters (which are derived from small molecules in ASOG and UNIFAC databases) are used to predict solvent activities in polymer solutions, the results are significantly poorer than those found for solutions of small molecules. Again, this seems to be due to the existence of a noncombinatorial entropy effect. Free volume differences contribute to such nonideal interactions. Chemically similar polymers and solvents still differ in their free volume, as evidenced by the difference in densities between polystyrene and toluene. To account for such effects, Flory (1970) proposes an equation of state approach for analysis of polymer solution properties in terms of pure component properties. This is adapted to the UNIFAC model by Oishi and Prausnitz (1978); the resulting UNIFAC-FV model is more accurate than UNIFAC in fitting activity data from polymer solutions. Other equations of state for polymer solutions have also been proposed (Lacombe and Sanchez, 1976; Liu and Prausnitz, 1979; 27 Scholte, 1982). Derr and Deal (1973) note that the ASOG model is not accurate when applied to polymer solutions. By choosing an ”effective" size parameter for the polymer molecule, they are able to improve predictions. A technique for choosing size parameters, referred to as Variable Size Parameter (VSP), results in a correlation for solvent activities in polymer solutions which shows good accuracy (Misovich et a1, 1985). However, it is deficient in that residual interactions are not properly modeled. That drawback was eliminated in this paper. VARIABLE SIZE PARAMETER The following discussion reviews the development of the VSP technique. An expression similar to the combinatorial entropy given by eq 2 is used in the ASOG model, shown in eq 4, with the volume fraction ¢1 replaced by the size ratio R defined in eq 5. 1 ln'ylS-l-R1+lnR1 (4) R1 - S1 / (Slx1 + 82x2) (5) where Si is the size parameter of component i, and xi is the mole fraction of component i. The size parameter is intended to correlate with the molar volume of a component, and is calculated by counting the number of atoms other than hydrogen in the molecule, with a few exceptional cases such as H O. 2 28 At infinite dilution of component 1 (pure polymer limit), and taking 81 << S2 because of the size disparity of the molecules, eqs 4 and 5 yield a mole fraction activity coefficient Mole fraction concentration variables are seldom used for polymer solutions because the difference in component molecular weights makes them impractical. Weight fraction w1 is typically used, and weight fraction activity coefficients 0 are defined by 1 a1 - 01w1 (7) If the ratio of polymer size parameter to solvent size parameter, 82/81, is assumed equal to the ratio of molecular weights, eq 6 can be rewritten in terms of weight fraction activity coefficient at infinite dilution. 01 - e (8) Experimental values of 01” range from 1.5 for chloroform in poly(vinyl acetate) (Ju, 1981) to over 100 for water in polystyrene (Gunduz and Dincer, 1980). Much of the discrepancy can be attributed to residual interactions which are not accounted for in eq 4. However, data for toluene in polystyrene yield 01” values between 3.7 and 5.5 (Covitz and King, 1972; Newman and Prausnitz, 1972), yet little residual interaction is expected for this system. The discrepancy in this case can be explained only in terms of the noncombinatorial entropy. The data for 29 other chemically similar systems show a similar pattern. Originally (Misovich et a1, 1985), an empirical correction was proposed for the size ratio R1 in eq 5. V1 R - .. <9) w1 + (e/O1 )w2 This results in a correct value of weight fraction activity coefficient at infinite dilution when used in eqs 4 and 7. Reasonably accurate results are obtained for the variation of activity coefficient with ‘concentration for most systems for which data are available. However, the approach lacks theoretical correctness for systems with residual interactions since a term like the one given by eq 3 is not employed in addition to eq 4. Also, the parameter 01” describes the complete activity coefficient containing residual effects as well as combinatorial and noncombinatorial entropy effects. Hence, including 01” in the size ratio R incorrectly places residual effects in an 1 entropic factor. REVISED VARIABLE SIZE PARAMETER APPROACH A more correct treatment of the size ratio given by eq 9 was made by canceling the effect of residual interactions from 01”. This was accomplished by placing the infinite dilution value of the residual activity coefficient, 716m, in the numerator of the ratio e/Olm. 30 w1 R - (10) Go a w1 + (e11 /01 )w2 Eq 1 can then be used in an appropriate manner to calculate solvent activity. The first term on the right side of eq 1 will account for size and free volume interactions between molecules according to eqs 4, 5, and 10. The second term on the right side of eq 1b will account for residual interactions. Any functional expression may be used to generate the term 116, e.g., the Flory-Huggins interaction parameter term (eq 2) could be used. The factor 716” in eq 10 has the value given by the expression for 116 with w taken as zero, i.e., 1 G - fw / [w + (e/0 ”w 1 1 01 _ 1 2 1.. 1 2 (16) w1 + (e/O1 )w2 This expression contained a single adjustable parameter, 01”, which was selected to minimize the residual error in 1n 01 compared to experiment. A numerical minimization technique was necessary. The residual interaction given by eq 14 also allowed an expression to be * written for 01. In this expression, 116” was used in place of exp(x ), which gave G00 60 Geo an (61 /0 M? (cur /0 )w a exp [ 1 1 2 [1 + 1 1 2 1n 01 ]] w Goo on Gun an a - 1 + (e11 /01 )w2 w1 + (e11 /01 )w2 (17) 1 w + (e1 GQ/fl on)w l 1 1 2 Both 11c” and 01co were taken as adjustable parameters. They were chosen in the same way as described for eq 16. When the ASOG model given by eqs 15 was used for residual interaction, constants from Kojima and Tochigi (1979) were used. (This version of ASOG is called ASOG-KT.) Only 01co was taken as an adjustable parameter, because 116co is given by eq 15e as a function of the ASOG-KT constants only; hence, 1160 is itself a constant for a given polymer-solvent system and temperature. 34 FITTING OF MODEL PARAMETERS Experimental data for 116 points in 21 sets of isothermal polymer- solvent activities were used to test the VSP approach with each of the three residual expressions. The classical Flory-Huggins model, eqs 1-3, was also applied for comparison. For each data set and each equation, the best fit of adjustable parameters was made to minimize the sum of squares residual of 1n a1, i.e., to minimize the relative error in al. An example of the technique is given as Appendix A. The parameters adjusted were 01co (VSP with eq 13 and VSP with eqs 15), 01co and 11cm (VSP with eqs 14), and x (Flory-Huggins). Table 1 contains all values of the adjustable parameters which were derived from experimental data. In addition, the value of 116° given by eq 15e from the ASOG-KT parameter database is given for comparison. Table 1 shows a remarkable consistency in 01do values in the VSP results using different residual expressions. This indicates the physical significance of the parameter, as distinguished from a mere data fit. As long as there is a reasonable model for the enthalpic term, the VSP method yields similar values for 01”. In Figure 1, the values of 01° given using eqs 13 have been arbitrarily taken as x-coordinates, and the values given using eqs 14 and 15 are plotted as y-coordinates. The plot shows little scatter from the line x - y. Values of 01m given using eq 13 exceeded those given using the other equations when 11Gdo was greater than unity (positive enthalpic deviations from Raoult's Law); the opposite was true when 116” was less than unity. 35 Comparison of the value of 116‘ between eqs 14 and 15 showed that some data sets agreed well, while others had no apparent correlation, as Figure 2 indicates. In particular, there were three data sets where eq 14 predicted a best fit value of 116” of unity or less while eq 15 predicted a value substantially larger than unity. The apparent disagreement was due to the nature of the calculation of 116° in the VSP model with eqs 14 and 15. In eq 14, 716” was an adjustable parameter, while in eq 15, it was not adjustable but was given as a function of the ASOG-KT constants. Noting this distinction, the results from eq 15 would generally have been considered preferable as they had a more fundamental basis in a solution model than the parameter fitting results from eq 14. The general agreement between parameters derived from numerical fit and those estimated from the ASOG-KT database was encouraging in many cases. In Figure 3, the size factors defined as (e/fllm) in eq 13, and as (eylco/Ola) in eqs 14 and 15 were compared. Again, there is a sizable amount of scatter in the plot. Of the total of 63 data fits (21 sets with three models), only in eight cases was a size factor greater than unity predicted by any model. In no case did all three models predict a size factor greater than unity for a given data set. These results are consistent with the observation of Derr and Deal (1973) that the "effective size factor" must be less than the actual size ratio of the molecules; in our models, size factors less than unity indicated they were less than the actual weight ratio. 36 Table 1. Parameter Values Determined by Data Fit. a Go Go Solvent- wt no. 0 11 x 1 Polymer frac of VSP wifh eqs VSP with Flory- VSP with Temp, 0C range pts l3 14 15 eq 14 Huggins eq 15e (not data fit) toluene-poly(styrene)a 25 0.111-0.918 11 4.95 4.56 4.94 60 0.102-0.261 3 4.85 4.63 4.84 80 0.246-0.67l 3 5.17 4.72 5.15 .73 0.34 1.01 .59 0.29 1.01 .58 0.32 1.00 F‘P‘P‘ methyl ethyl ketone-poly(styrene)a 25 0.091-0.298 4 8.93 8.23 7.77 1.65 0.71 1.88 benzene-poly(isobutylene)b 10 0.225-0.454 3 10.66 7.96 25 0.044-0.373 11 8.79 8.18 0‘ .70 .92 0.84 .35 1.73 0.92 H .82 .71 \J P‘P‘ cyclohexane-poly(isobutylene)c 25 0.128-0.569 8 4.97 4.90 4.94 1.25 0.39 1.06 n-pentane-poly(isobutylene)d 25 0.029-0.584 9 8.76 8.33 8.76 1.64 0.68 1.00 triisopropylbenzene-poly(styrene)e 165 0.030-0.086 3 12.34 12.25 12.05 1.22 1.00 1.07 175 0.020-0.066 3 10.61 9.84 10.51 2.66 0.92 1.06 carbon disulfide-poly(styrene)e 115 0.014-0.041 3 3.73 3.73 3.70 1.00 0.41 3.61 140 0.008-0.029 4 3.48 3.48 3.48 1.00 0.34 4.15 methanol-poly(methyl methacrylate)e 120 0.003-0.009 3 16.65 16.33 16.23 2.71 130 0.003-0.008 3 12.73 10.79 12.56 0.19 .28 .01 .97 .84 F‘P‘ has: toluene-poly(methyl methacrylate)e 130 0.017-0.112 3 9.68 9.68 9.69 .00 0.79 0.97 160 0.006-0.037 5 10.95 10.95 11.07 1.00 0.95 0.90 H toluene-poly(vinyl acetate)f 35 0.084-0.195 4 9.71 8.41 8.26 2.06 0.78 1.40 40 0.051-0.171 7 9.26 8.35 8.26 2.06 0.77 1.38 47 0.052-0.107 3 8.87 7.63 8.18 3.09 0.76 1.35 chloroform-poly(vinyl acetate)f 35 0.163-0.464 7 1.49 1.49 1.62 1.00 -0.41 0.41 45 0.093-0.499 16 1.44 1.40 1.48 0.67 -0.46 0.45 References: aBawn ea a1 (1950). bEichinger and Flgry (1968a). gEichinger and Flory (1968b). Eichinger and Flory (1968c). Liu (1980). Ju (1981). 11,- (Eq.140r15) 37 20 D Eq. 14 X Eq.15 IS - O 10 - D 8‘" D X 5 . 0 u v u 0 S 10 IS 20 a,'(Eq.13) Figure 1. Comparison of Infinite Dilution Activity Coefficients from Different Residual Terms. Squares, eq 14; crosses, eq 15. n“'(fil‘5) 38 S D 4. 0 3d 0 D 2- D E C) 1—1 [In [DU 0 0 D 0 I I I I o 2 3 4 rf- (Eq.14) Figure 2. Comparison of Infinite Dilution Residual Coefficients from Different Residual Terms. Afip 5A.. Sp he .1. v 2 \UUV‘.‘ g~9w 6. el" 39 a X {R t 3‘ 0 Q T x 0’ 3. ,P. CJqud X 8115 5 2‘ U S 3 :3 D 1 D “ dp x 6 o D . I 0 1 2 3 Size Factor 15: (Eq.13) Figure 3. Comparison of Size Factors from Different Residual Terms. Squares, eq 14; crosses, eq 15. . 40 Disagreement among the models on the parameter values did not appear to be random. Rather, certain systems seemed prone to good agreement or poor agreement on certain parameters, as can be seen from examination of Table 1. The systems benzene-poly(isobutylene), methyl ethyl ketone- poly(styrene). and toluene-poly(vinyl acetate) had large relative deviations among 01co values; the first two also had small relative deviations among 116“ values. The opposite was true for the system carbon disulfide-poly(styrene). Finally, the systems toluene- poly(methyl methacrylate) and cyclohexane-poly(isobutylene) showed small relative deviations in both parameter values. The other systems showed either intermediate levels of deviation among parameters or showed different trends at different temperatures. COMPARISON WITH SOLVENT ACTIVITIES IN POLYMER SOLUTIONS Some specific results which illustrate the accuracy and flexibility of the method are given in Figures 4 through 6. Solvent weight fraction activity coefficient 01 was plotted versus solvent weight fraction for a given polymer-solvent system at a given temperature. Experimental points were shown along with lines or curves representing the best fit results of certain models. In Figure 4, data for benzene-poly(isobutylene) at 25°C is shown, along with the VSP model using eq 13 and the Flory-Huggins model. (Both of these models contain one adjustable parameter.) The VSP predictions were more accurate in this case, particularly at the extremes of 41 8 VSP(Eq.13) 7‘ D —-—»Hoq+mggns 6 -4 5. 4. 3. 2 v I I 0 I .2 3 4 w! Figure 4. Solvent Activity Coefficient as a Function of Concentration, Benzene-Poly(isobutylene) at 25°C. Curves, equations; squares, experiment. 42 concentration that were used. Some investigators prefer to express activity results as a variation of the interaction parameter x with concentration, often referred to as "reduced residual chemical potential." The curve in Figure 4 labeled ”Flory-Huggins" would represent a constant x value. The experimental data would show x decreasing with concentration because the slope of the data is more steeply negative than the "Flory-Huggins" curve. The curve representing VSP with eq 13 also correctly showed this decrease. Figure 5 compares VSP using eq 15 with Flory-Huggins for the system toluene-poly(methyl methacrylate) at 160°C. Neither model performed well on this data set, although VSP with eq 15 did correctly model the fact that x decreases with concentration, although not the magnitude of decrease. In Figure 6, data for the system toluene-poly(styrene) at 60°C showed a very slight increase in x with concentration, and this was correctly modeled by VSP with eq 14, since it predicted a less steeply negative slope than the Flory-Huggins model. The examples in Figures 4 through 6 show that the VSP method is capable of modeling systems in which x either decreases or increases with solvent concentration. For each data set and equation, a standard error was defined by pred _ 1n a exptl 2 2 (1n a ) s _[ 1 1 11/2 (n-d) (18) where the sum was over all n points in the data set, and where d was the number of adjustable parameters (degrees of freedom) in the model used. 43 13 . VSP(Eq.15) 12 .. —— — Flory-Huggins 11- 6'10- 38-, . . -n.‘ —-.- Figure 5. Solvent Activity Coefficient as a Function of Concentration, Toluene—Poly(methy1 methacrylate) at 160°C. Lines, equations; squares, experiment. 44 33 VSP(Eq.14) 3.7 q \ — —- Flory-Huggins 3.5 f 413* 3.1 1 29‘ 17 u 1 I . .08 .12 .16 .20 .24 .28 Figure 6. Solvent Activity Coefficient as a Function of Concentration, Toluene-Poly(styrene) at 60°C. Lines, equations; squares, experiment. 45 For the VSP model with eqs 14, d - 2; for all the other models, d - 1. Hence, in cases where the VSP model with eqs 14 produced the same standard error as the other models, it must have resulted in a smaller deviation from experiment on the average. The standard error defined by eq 18 in effect penalizes eq 14 because it has more adjustable parameters. The standard error results are given as Table 2, and were generally quite good for all the models. Even the Flory-Huggins model, when fit to the data, had a standard error of less than five percent in 14 of 21 data sets. The VSP models were somewhat more accurate, with standard errors less than five percent for 16 of 21 data sets using eqs 13 and 15 for the residual term, and 17 of 21 data sets using eq 14. Previous work (Misovich et a1, 1985) has shown that the VSP model using eqs 13 is superior to the Flory-Huggins model when data from low concentration is extrapolated to higher concentrations. The same results were found here in a best fit of all data, for all the models using the VSP method regardless of the residual expression used. Because of the generally good performance of all the models, it was not clear that any given model was significantly better or poorer than the others for a particular system in many cases, outside of the general trend noted in the previous paragraph. (VSP, eq 14) > (VSP, eq 13) - (VSP, eq 15) > (Flory-Huggins) 46 Table 2. Comparison of Models with Experiment. Solvent- wt no. Std % error Polymeg frac of VSP with eqs Flory- Temp, C range pts 13 14 15 Huggins toluene-poly(styrene) 25 0.111-0.918 11 2.1 1.1 2.0 1.6 60 0.102-0.261 3 1.3 0.4 1.3 0.8 80 0.246-0.67l 3 1.0 0.1 1.0 0.6 methyl ethyl ketone-poly(styrene) 25 0.091-0.298 4 1.9 0.6 1.6 2.4 benzene-poly(isobutylene) 10 0.225—0.454 3 2.1 0.8 2.3 25 0.044-0.373 11 2.6 1.2 4.4 5.2 cyclohexane-poly(isobutylene) 25 0.128-0.569 8 2.4 2.6 2.4 2.4 n-pentane-poly(isobutylene) 25 0.029-0.584 9 2.2 0.9 2.2 2.2 triisopropylbenzene-poly(styrene) 165 0.030-0.086 3 2.9 4.1 2.9 5.4 175 0.020-0.066 3 15.4 21.4 15.4 15.1 carbon disulfide-poly(styrene) 115 0.014-0.041 3 0.5 0.6 0.7 0.7 140 0.008-0.029 4 13.8 16.9 13.8 13.9 methanol-poly(methyl methacrylate) 120 0.003-0.009 3 1.4 1.4 1.0 1 130 0.003-0.008 3 6.9 0.6 6.2 6. toluene-poly(methyl methacrylate) 130 0.017-0.112 3 20.2 28.6 20.2 24.1 160 0.006-0.037 5 12.4 14.3 11.9 14.3 toluene-poly(vinyl acetate) 35 0.084-0.195 4 2.8 0.3 0.5 40 0.051-0.l7l 7 2.9 1.4 1.4 47 0.052-0.107 3 4.6 2.4 2 7 tor-'0 \me chloroform-poly(vinyl acetate) 35 0.163-0.464 7 3.9 4.3 45 0.093-0.499 16 2.7 2.6 WU NL‘ O‘U‘ 47 It was also difficult to define an average error over all the systems tested because the presence of large errors in a few data sets tended to obscure the behavior in the majority of data sets in which standard errors were relatively small. The average error, defined by the arithmetic mean over all 21 data sets, was greatly affected by this. At the same time, the average defined by the geometric mean over all data sets was affected most strongly by the presence of very small errors in a few data sets. Both these averages, as well as the median standard error for the 21 data sets, are given for each model as part of Table 3. VSP with eqs 14 and 15 performed best according to these measurements; VSP with eqs 13 and Flory-Huggins performed worst, but still showed small standard errors on many sets. Also included in Table 3 are the number of times each model had the lowest (or highest) standard error for a single data set. (The numbers total more than 21 because of ties.) VSP with eqs 14 and 15 again outperformed the other two models. Finally, for each data set in which a given model had the lowest (or highest) standard error, an average amount by which the error in the other models exceeded that of the best model (or the error in the worst model exceeded that of the other models) was calculated. Both absolute (differences in standard errors) and relative (ratios of standard errors) amounts are listed in Table 3. As was the case with arithmetic and geometric means above, the absolute amounts gave greater weight to data sets in which all models had large standard errors, while the relative amounts gave greater weight to data sets in which standard errors were small. On an absolute 48 Table 3. Comparison of Errors. Model VSP VSP VSP Flory- eq 13 eq 14 eq 15 Huggins Average standard error: arithmetic mean 5.0 5.1 4.7 5.2 geometric mean 3.2 1.8 2.8 2.9 median 2.7 1.4 2.4 2.6 Number of data sets where standard error was lowest 5 12 8 5 highest 9 6 5 8 In sets with lowest standard error, average amount by which error was larger in other models absolute 0.3 0.8 0.5 0.2 relative 1.2 2.7 1.2 1.1 In sets with highest standard error, average amount by which error was smaller in other models absolute 0.5 0.9 0.1 0.4 relative 2.0 1.2 1.5 1.5 49 basis, VSP with eqs 15 was the only model which outperformed the other models by a wider margin when having the lowest error than the other models outperformed it when it had the highest error. On a relative basis, the same was true of only VSP with eqs 14. By most measurements of average performance, the VSP model using eqs 14 or 15 produced a lower standard error than the VSP model using eq 13 or the Flory-Huggins model. This was attributed to the fact that the Flory-Huggins equation does not correctly model nonideal solution interactions due to free volume differences, while the VSP model using eq 13 does not include a term for nonideal residual interactions. However, due to their simplicity, they were more convenient to use than the more accurate models. Table 2 indicates that their performance was generally in the same order of magnitude of standard error as the more complicated, more accurate VSP models using eqs 14 or 15. There are, however, certain situations in which behavior in the infinite dilution limit of zero solvent is important, e.g., thermodynamic modeling for polymer devolatilization. In such cases, as Table 1 shows, the choice of model may produce a large difference in the predicted value of infinite dilution parameters. This can be true even when all models perform relatively equally over a larger concentration range as shown by the standard errors in Table 2. For modeling behavior near the pure polymer limit, the VSP models using eqs 14 or 15 would be preferable to the other models tested. 50 CONCLUSIONS The VSP method using various residual terms allowed accurate prediction of solvent activities in most of the polymer-solvent systems reviewed. Choosing terms which modeled nonideal residual interactions in solution gave the best results. When all the points in a given experimental data set were fit to determine adjustable parameters, the VSP method generally performed better than the Flory-Huggins model. Use of the VSP method with residual interaction given by the ASOG-KT equations produced accurate results with only one adjustable parameter representing the infinite dilution solvent activity coefficient on a weight fraction basis. Even better results were sometimes obtained by using a residual term containing an additional adjustable parameter. ACKNOWLEDGEMENT During his Ph.D. work at Michigan State University, Mr. Misovich was the recipient of a fellowship from the R.L. Gerstacker Foundation. LITERATURE CITED Abrams, D.S.; Prausnitz, J.M. Alghfi_l, 1975, 21, 116. Bawn, C.E.H.; Freeman, R.K.; Kamaliddin, A.R. Trans, Faraday Soc, 1950, 46, 677. Covitz, F.H.; King, J.W. J, 20119, Sci, 1972, A-l, 19, 689. Derr, E.L.; Deal, C.H. Inst, Chem, Eng, Symp, Ser, 1969, No, 32, 40. 51 Derr, E.L.; Deal, C.H. Aay_,___C_ham,,__§_ar_L 1973, Na, 129, 11. Eichinger, B.E.; Flory, P.J. Irana, Earaday Sag, 1968a, gr, 2053. Eichinger, B.E.; Flory, P.J. Iragar_Earaaay_§aar 1968b, $5. 2061. Eichinger, B.E.; Flory, P.J. Iraaa, Faraday Soc, 1968c, 63, 2066. Flory, P.J. ”Principles of Polymer Chemistry"; Cornell University Press: Ithaca, N.Y., 1953. Flory, P.J. Diaausa, Faraday 809, 1970, $2, 7. Fredenslund, A.; Jones, R.L.; Prausnitz, J.M. arggarl, 1975, 21, 1086. Gunduz, S.; Dincer, S. Polymer 1980, 21, 1. Ju, S. Ph.D. Thesis, The Pennsylvania State University, University Park, Pennsylvania, 1981. Kojima, K.; Tochigi, M. "Prediction of Vapor-Liquid Equilibria Using ASOG"; Elsevier: Amsterdam, 1979. Lacombe, R.H.; Sanchez, 1.0. 4, Phys, ghaa, 1976, 80, 2568. Liu, D.D.; Prausnitz, J.M. J, Anal, Polya, $91, 1979, 23, 725. Liu, H. Ph.D. Thesis, The Pennsylvania State University, University Park, Pennsylvania, 1980. Misovich, M.J.; Grulke, E.A.; Blanks, R.F. Ina, Eng, Chem, Eragesa Dea, Dev, 1985, 23, 1036. Newman, R.D.; Prausnitz, J.M. J, zhya, Chaa, 1972, 19, 1492. Oishi, T.; Prausnitz, J.M. Ina, Eng, Chan, Prgaeaa Qea, Dav, 1978, 11, 333. Scholte, W. Ina, Eng, Chaa, Prgceaa Des, Dev, 1982, 21, 289. Staverman, A.J. Reg, Tray, tha, Pays-baa 1950, Q2, 163. Wilson, G.M. J, Am, Chem, Sag, 1964, 86, 127. 52 APPENDIX A. EXAMPLE OF VSP METHOD. The following experimental data are given for toluene(l)- poly(styrene)(2) at 80°c. w1 a1 0 246 0.706 0 458 0.914 0.671 0.984 To fit the Flory-Huggins parameter x in eqs 1-3, weight fraction data must be converted to volume fraction data. Density data can be used for this transformation. w /p i i "1/91 + w2/82 Densities: p - 0.8075 p2 - 1.068 W1 81 ‘1 0.246 0.706 0.301 0.458 0.914 0.528 0.671 0.984 0.730 The least squares condition results in the following equation which can be directly solved for x. Subscripts 11 and 21 refer to components 1 and 2, data point i. -2<¢21n (1-2) X 1 21 11 11 21 1 21 x - 0.319 53 Applying eqs 1-3 gives these results. pred '1 a1 81 0.246 0.706 0.708 0.458 0.914 0.909 0.671 0 984 0.979 To apply VSP using eq 13, it is necessary to minimize the error between the activity calculated using eqs 1, 2, 4, 10, and 13, and the measured activity. 01do is an adjustable parameter, but the least squares condition cannot be solved directly for it. The simplest way to proceed is to assume a value for 01w, generate R1 values from eq 10 (using 116” - l as given by eq 13), and calculate the sum of squares residual given pred 2 by adding [1n(a1/a1 )] for each data point. A good initial choice for 01co comes from the Flory-Huggins model Q 01 - (92/91) eXP (1 + x) (A-3) using the known density values and x. The two tables below illustrate m and the best fit value of 0 Q. the results using this initial 01 1 0 ° - 4.948 e/o ° - 0.549 1 1 red r d 2 w, a1 a, 61" [haul/a," e )1 0.246 0.706 0.373 0.698 1.36x10:: 0.458 0.914 0.606 0.899 2.87::10_4 0.671 0.984 0.788 0.974 1.04x10_4 sum of squared residuals 5.26x10 no no 01 - 5.166 e/O1 - 0.526 54 pred pred 2 w1 a1 R1 a1 [ln(a1/a1 )] 0.246 0.706 0.383 0.710 2.45x10:2 0.458 0.914 0.616 0.905 1.09x10_5 0.671 0.984 0.795 0.976 6.9lx10_4 sum of squared residuals 2.03x10 To apply VSP using eq 14, two adjustable parameters must be fit to the data, 01” and ylcm. As in the previous case, the simplest way to proceed is to assume values for these parameters, generate R1 values from eq 10, and calculate the sum of squares residual given by adding pred 2 [ln(a1/a1 )] for each data point. Initial choices for the parameters can be made using the results from the previous case (or eqs A-2 and A-3) for 0 illustrates the results using the best fit values. a and setting 11G0° equal to unity. The table below The initial values are identical to the best fit results from the previous case. 0 01 - 4.719 w1 R1 0.246 0.264 0.458 0.481 0.671 0.691 w1 a1 0.246 0.706 0.458 0.914 0.671 0.984 To apply VSP using eq 15, only 0 is given a priori from the ASOG equations. a - 1.580 S 1 0.551 0.809 0.941 a a pred 1 0.706 0.914 0.983 Go e11 G 11 1.281 1.131 1.044 O /01 - 0.526 a pred 1 0.706 0.914 0.983 sum of squared residuals 1 pred)]2 [ln(a1/a1 1.69x10:§ 2.64x10_7 5.53x10_7 8.34x10 a must be fit to the data because 116 The necessary parameters for use of these equations for the example are given by Kojima and Tochigi (1979). Molecular components toluene and poly(styrene) are defined in 55 terms of functional groups CH2 and ArCH as follows. v - number of functional groups k occuring in molecule or repeat k1 unit 1 MW - molecular weight of molecule or repeat unit 1 V ki CH2 ArCH MW toluene 1.0 6.0 92.0 PS 1.8 6 0 104.0 ASOG-KT gives functional group interaction parameters Ak1 used in eqs 15 as the sum of a temperature-independent and a temperature-dependent term given by eq A-4. Values of these constants are listed for the groups in this example. Ak1 ' ex? (akl + bk1 / T) (A-4) “R1 bkl CH2 ArCH CH2 ArCH CH2 0 -0.7457 0 146.0 ArCH 0.7297 0 -176.8 0 In the example, the temperature is 80°C or 353.16 K, giving interaction parameter values of Akl CH2 ArCH CH2 1.000 0.717 ArCH 1.257 1.000 * which are used in eqs 15. Consider the calculation of 1n Pk in eq 15c. 56 Since x1 equals one in this calculation, eq 15d gives this result for group mole fractions. x1 - 1.0 / (1.0 + 6.0) - 0.143 X2 - 1 - 0.143 - 0.857 Applying these group mole fractions in eq 15b gives 1n r1* - - 1n (0.143 1 + 0.857-0.717) + 1 0 143 1 0.857-1.257 - 0.143 1 + 0 857 0.717 - 0.143-1.257 + 0.857-1 ln r1* - 0.037 1n r2* - - 1n (0.143 1.257 + 0.857-1) + 1 0.143-0.717 0.857-1 - 0.143 1 + 0.857-0.717 - 0.143 1.257 + 0.857 1 1n r2* - 0.005 The same procedure is used to calculate ln Pk at any concentration. The only additional step needed is the conversion of component or repeat unit weight fraction to mole fraction. X I 1 0.246/92 / (0.246/92 + (l-0.246)/104) - 0.269 1 - 0.269 - 0.731 N I X1 - (0.269-1.0 + 0.731-1.8) / (0.269~7.0 + 0.731-7.8) - 0.209 N I l - 0.209 - 0.791 1n F In P 1n P ln F2 57 - 1n (0.209-1 + 0.791-0.717) + 1 0.209-1 0.209-l + 0.791-0.717 0.040 0.791-1.257 0.209-1.257 + 0.791-1 - 1n (0.209-1.257 + 0.791-1) + 1 0.209-0.717 0.209-l + 0.791-0.717 0.004 0.791-l 0.209-1.257 + 0.791-l The activity coefficient 116 for this concentration is given by eq 15a. 71° - exp ( 1.0 (0.040-0.037) + 6.0 (0.004-0.005) ) - 1.003 Results for all data points as well as pure components 1 and 2 are given in the table. w l 1 0 0.246 0.458 0.671 The adjustable parameter 0 1.000 0 0.269 0.489 0.697 x1 0.143 0.231 0.209 0.190 0.172 x2 0.857 0.769 0.791 0.810 0.828 1n P1 0.049 0.037 0.040 0.043 0.045 In P2 0.002 0.005 0.004 0.003 0.003 c 71 1.005 1.003 1.001 1.000 G can now be fit to the data. pure toluene pure polymer A good initial choice for this parameter is the result from VSP using eq 13 or from eqs A-2 and A-3. The tables below give results for the initial value and best fit value. O 01 - 5.166 w1 R1 0.246 0.382 0.458 0.615 0.671 0.794 WI 81 0.246 0.706 0.458 0.914 0.671 0.984 G 01 - 5.152 w1 R1 0.246 0.381 0.458 0.615 0.671 0.794 W1 81 0.246 0.706 0.458 0.914 0.671 0.984 Go Go 0 11 - 1.005 e 11 01 - 0.529 a S G a pred 1 11 1 0.708 1.003 0.710 0.904 1.001 0.905 0.976 1.000 0.976 pred pred 2 81 [1n(81/81 ) ] 0 710 3.38x10:g 0.905 9.52x10_5 0.976 6.40x10-4 sum of squared residuals 1.93x10 C00 Co) a: 11 - 1.005 e 11 / 01 - 0.530 a S G a pred 1 11 1 0.708 1 003 0.709 0.904 1.001 0.905 0.976 1.000 0.976 pred pred 2 al [ln(a1/a1 )] 0 709 2.34x10:z 0.905 1.03x10_S 0.976 6.58x10_h sum of squared residuals 1.92x10 CHAPTER 3 ANALYSIS OF RESIDUAL TERMS USED IN GROUP CONTRIBUTION MODELS One of the important advances in modeling of solution behavior has been the isolation of residual (enthalpic or energetic) effects and combinatorial (entropic) effects. The recent approach to both types of interaction has become fairly standardized. In the case of combinatorial effects, some form of combinatorial entropy (such as Flory, 1953 or Staverman, 1950) is used. For residual effects, a local composition model similar to Wilson (1964) is applied. The synthesis of both types of interaction in a single model is typified by UNIQUAC (Abrams and Prausnitz, 1975). The use of distinct combinatorial and residual terms is commonplace in group contribution models; in fact, the original development of the ASOG model (Derr and Deal, 1969) predates UNIQUAC by several years. The unique feature of group contribution models such as ASOG and UNIFAC (Fredenslund, Jones, and Prausnitz, 1975) is the treatment of summed functional group interactions rather than individual molecular interactions. This makes data reduction possible in terms of functional groups, so that binary molecular data is not required once a functional group interaction database has been tabulated. 59 60 The concept of deriving molecular solution properties, e.g., activity coefficients, by summing properly weighted and normalized functional group properties is the basis of the residual interaction terms in ASOG and UNIFAC. The summations used make sense from an intuitive standpoint, and the residual interaction given by a Wilson-like equation has a theoretical basis in local composition and like-unlike pair interaction. However, a careful study of the mathematical properties inherent in the residual terms of group contribution models shows an implicit dependence of the model predictions on the choice of unit used to describe functional group size. This dependence arises from the fact that the summation of functional group activity coefficients is done in a linear fashion, but the Wilson-like equation used to derive these coefficients is nonlinear in all its parameters and variables. In this chapter, this idea is developed and studied in depth for the simplest possible non-trivial case of a binary solution containing at most two distinct functional groups. One consequence of the detailed study of such systems is that the group contribution model equations for residual interaction can be transformed to make their behavior more explicit in some fashion. Doing so allows the additional constraint of molecular composition (in terms of the different ratios of functional groups present in different molecules) to modify the rather weak constraint on activity coefficients given by a Wilson-like equation. A framework is thus given for determining bounds on activity coefficients without sufficient knowledge to actually fit all the interaction parameters for functional groups in solution. 61 Derivation of such bounds can also assist in the design of experiments to take the necessary data for fitting interaction parameters. ANALYSIS OF RESIDUAL TERM IN SOLUTION OF GROUPS MODEL The manuscript which follows contains the analysis of bounding and normalization properties inherent in typical solution of groups model residual expressions. Transformations of the model which allow more convenient analysis are developed and some typical results are shown for a binary solution containing at most two distinct functional groups. Extension of the technique to multicomponent, multifunctional group solutions should be possible, but is not described here. Details of the derivation of new equations which are presented in this manuscript are given in Appendix H. 62 Normalization and Bounding Properties Inherent in Solution of Groups Activity Coefficient Models ABSTRACT Recent thermodynamic models for activity coefficients such as UNIFAC and ASOG use a form of Wilson's equation to calculate the residual contribution to the activity coefficient. These equations can be transformed to allow more convenient analysis of their mathematical properties. Two important results have been obtained from such an analysis. Bounds on the range of activity coefficients can be derived without knowledge of the interaction parameter values. The predicted values of activity coefficients are shown to depend on a normalization step implicit in the definition of functional group size. 63 INTRODUCTION The equation proposed by Wilson [1] for modeling nonideal liquid solutions is a popular and useful tool in the design of chemical processes. Comparisons of the Wilson equation to other activity coefficient correlations such as the Margules and Van Laar equations have shown the Wilson equation to have superior predictive ability for binary systems and particularly for multicomponent systems [2]. Furthermore, the Wilson equation embodies the concept of local composition as distinct from overall solution composition, thus modeling the molecular segregation which occurs in nonideal solutions. The success of the original Wilson equation has led to its adoption as a basis or component of more sophisticated solution models. Among these are the Nonrandom, Two-liquid (NRTL) equation [3], the Analytical Solution of Groups (ASOG) model [4], the Universal Quasi-chemical (UNIQUAC) model [5], and the UNIQUAC Functional Group Activity Coefficient (UNIFAC) model [6]. These models utilize the form of the Wilson equation because of its theoretical basis and good predictive ability, but allow prediction of anomalous behavior such as phase separation which the original Wilson equation is incapable of modeling. Of these models, ASOG and UNIFAC include the concept of functional group contribution. This concept allows a solution to be treated as if it were composed not of interacting molecules, but rather of interacting functional groups, and considers the interaction of a molecule to be the 64 sum of its functional group interactions. By correlating available equilibrium data, a database of functional group interaction parameters can be derived and used to make predictions about substances for which no equilibrium data are available, but which contain only functional groups with parameters in the database. Progress has been made toward constructing such databases for both UNIFAC [6,7] and ASOG [8-11]. Comparison of these two models shows both to have approximately equal predictive ability and accuracy, and to be superior to other models applying the group contribution concept [12]. Both UNIFAC and ASOG consider the activity of a component in solution to be composed of two parts: a size interaction (entropic or combinatorial) and a group interaction (enthalpic or residual). In both models, the group or residual interaction term is given by a form of the multicomponent Wilson equation. The ASOG model uses group mole fraction as the independent variable for residual interaction while the UNIFAC model uses group surface area fraction. The unit of surface area in the UNIFAC model was originally chosen as the surface area of a single methylene (CH2) group in an infinitely large polymethylene molecule. Skjold-Jorgensen, Rasmussen, and Fredenslund [13] showed that the predictions made by UNIFAC are quite sensitive to the selection of surface area unit size, and indicated that the database could more accurately model solution behavior if the interaction parameters were derived again based on a different normalization of the surface area and segment size parameters. 65 The ASOG model, since it employs mole fractions rather than surface area fractions, contains a natural normalization of its independent variable in the entity of a single functional group of any type. However, this may be somewhat misleading since functional groups themselves vary in mass and size: for example, is it consistent to assign the same importance to the interaction of a large carboxylic acid (COOH) group as a small methylene (CH2) group? It is exactly this problem which UNIFAC addresses by using functional group segment size and surface area parameters. Recent revisions of A806, such as ASOG-KT [9], have also attempted to address this problem in a somewhat systematic way by assigning to each functional group a weighting factor equal to the number of non-hydrogen atoms it contains, and including some special cases as well. In doing so, ASOG makes explicit the normalization of functional group size. There are several methodological ideas which are useful in describing and analyzing normalization effects in the calculation of residual contributions to the activity coefficient within the solution of groups framework. The ASOG model is used throughout to illustrate these proposals and comments; however, they are applicable in the most part to UNIFAC and other similar models. The standard equations for residual activity coefficient in the ASOG model are reduced to simpler forms applicable to binary systems containing at most two distinct functional groups. This simple case can be representative of many binary solutions, and was chosen to enable discussion and graphical representation of the effects of changes in system parameters. The 66 approach taken can be extended to multicomponent solutions containing multiple distinct functional groups. In addition to facilitating the discussion of size normalization, the approach also allows the behavior of the ASOG model to be analyzed for cases in which insufficient data are available to specify complete sets of interaction parameters for the functional groups. In such cases, conclusions about residual activity coefficients can be derived as bounds rather than single values. These bounds can be made on the concentration dependence of activity for either component of a binary system based on a single measurement. EFFECTS OF NORMALIZATION ON RESIDUAL ACTIVITY Consider a binary solution whose molecules contain two distinct functional groups, e.g., ethanol and methanol contain the functional groups CH3 (or CH2) and OH. Denote the component mole fractions by x1 and x2. In order to apply the ASOG model, it is necessary to define group mole fractions X1 and X2 according to x+ X xk _ “k1 1 “k2 2 (1) (“11+“21)x1 + (“12+“22)x2 where nkj is proportional to some measure of the number of functional groups of type k found in molecule j. Derr and Deal [4,8] consider this measure to be the number of functional groups, whereas others [9-11] consider it to be the number of functional groups multiplied by an 67 appropriate weighting factor accounting for relative group sizes. The use of a size-weighting factor to model group size in ASOG makes the method equivalent to UNIFAC in its definition of the functional group concentration variables denoted here by X and X . As studied by 1 2 Skjold-Jorgensen et al [13], there is an implicit normalization step in the definition of group size. UNIFAC applies this normalization by choosing the methylene (CH2) group to have unit volume and unit surface area. ASOG does essentially the same thing in a less precise manner by considering the number of non-hydrogen atoms in a group to be its size measurement, with a few explicit exceptions such as water and multiple- substituted carbon atoms (>CH- or >C<). ASOG gives the residual part of component i activity coefficient for a binary system containing two distinct functional groups by the following equations. G - i i 1n 71 - n1i(1n F1 - 1n P1 ) + n21(1n P2 - ln F2 ) (2) X A X A 1n P - -1n(x Ak + x Ak ) + 1 - 1 1k - 2 2k (3) k 1 1 2 2 X A +X A X A +X A l 11 2 12 l 21 2 22 i 1n Pk - 1n Pk (xi - l) (4) In these equations, 11G is the residual (or group interaction, hence the letter C) part of component 1 activity coefficient, and Pk is the functional group activity coefficient for group type k. I‘k1 is the functional group activity coefficient for group type k, evaluated at the functional group composition of pure component i, and Ak1 are group 68 interaction parameters, with Akk - 1. Eq 3 is the Wilson equation, applied to functional groups in solution rather than the actual molecular components. Eq 2 gives the logarithm of component i activity coefficient as the sum of its functional group activity coefficients, 1n P1 - 1n P11 and In F2 - 1n F 1 relative to a pure component basis. 2 9 The functional group activity coefficients in pure component i are subtracted from the functional group activity coefficients in solution; if this were not done, activity coefficients would not approach unity in the pure component limit for molecules containing more than one distinct functional group type. The effect of group size normalization is to change the absolute values of the factors n11 and n21 in eq 2, although not their ratio. (ASOG would give a different ratio than UNIFAC, since each measures a different type of size, but once a method is selected, the unit of size will not affect the ratio.) If eq 3, the Wilson equation for group activity coefficients, were linear in the group interaction parameters Akl’ the magnitudes of n11 and 1121 would not affect overall predictions of the equation set. The Wilson equations are obviously nonlinear in the group interaction parameters (as well as the composition variables), therefore an activity coefficient result in eq 2 cannot be associated, independent of normalization, with any single set of group interaction parameters A12 and A21 in eqs 3 and 4. Since the technique used by both ASOG and UNIFAC is to construct a database of group interaction parameters based upon reduction of 69 experimental activity data, it is apparent that such a database must depend on the normalization of group size in a nonlinear way. This is the underlying cause behind the discovery by Skjold-Jorgensen et a1 [13] that varying the group size normalization within UNIFAC results in changes in the group interaction parameter database. Some normalizations produce a database which gives more accurate prediction of concentration and temperature dependence of activity coefficients than other normalizations. The relative merit of different normalization schemes will not be discussed here; the relevant issue in this paper is means of analyzing such effects. A NORMALIZATION INDEPENDENT EXPRESSION FOR RESIDUAL ACTIVITY COEFFICIENTS It is possible to derive an expression related to residual activity coefficient which contains no implicit or explicit dependence on the unit of functional group size. The complexity of this expression can be minimized by introduction of a conveniently weighted composition variable, c1, for the molecular species in a binary solution, defined as follows. (n +n )x c1 _ 11 21 1 (5) (“11+“21)x1 + (“12+“22)x2 Such composition variables represent size-weighted fractions in that ci equals the total size (as measured by number of functional groups) of all molecules of component i in solution divided by the total size of 70 all molecules in solution. Although c1 depends explicitly on the nkj values, it does not depend on the unit of functional group size. Since eq 5 contains one occurrence on an nkJ in each term of the numerator and denominator, size effects will cancel in the overall expression. Following through the calculations for a binary solution containing two distinct functional groups, but using composition variables c1 and c2 rather than the actual mole fractions x1 and x2, simplified results for group mole fraction can be found. Define group ratios 81 ' n21 / “11 (6) giving the size-weighted ratio of group 2 to group 1 in each component molecule, then c c 1 + 2 1 + g1 l + g2 X1 - (7) defines the group mole fraction, X1 in eq 1, in terms of component size-weighted fractions c1 and c2. Group ratios are particularly useful in polymer solutions, because polymer molecules are typically distributed in their molecular weight, hence in their absolute size. This fact can make eqs 1-4 difficult to apply to a solvent molecule in polymer solution since there is no single x2. Characterization in terms of group ratios is sizeaindependent, thus all polymer molecules of a given type have the same group ratios regardless of their molecular weights. Eq 5 can also be rewritten for 71 polymer solutions by using weight fraction w1 rather than mole fraction x on the right side and adding average molecular weight factors to the i equation. The definition of group ratios also allows eq 2 to be rewritten as In G 1 - 1n P i) + (1n F - 1n F i) (8) 1 81 2 2 - (ln P1 n 11 The left side of eq 8 is the normalized residual activity coefficient of component i. (It is termed "normalized" because it contains the term n11, inversely proportional to the unit chosen for functional group size, in its denominator.) The right side contains no explicit dependence on the nkj’ since the group ratio g1 has been substituted. The implicit dependence of F1, P11, F2, and [‘21 on nkj can be removed by substituting eq 7 into eqs 3 and 4, and using the property that both component size-weighted fractions and group mole fractions sum to unity. The resulting lengthy equation is 72 1n 11° (1 + g1)(l + A1231) - ln 01, (1 + sj)(1 + A1281) + (8J - 8,)(A12 - 1)cj '(1 + 8])(A21 + 81) + g1 1n . (1 + 81)“21 + 81) + (8J - 81)(1 - A21)cJ + (1 + 8,)(8J - 81)Cj A12 (1 + gj)(1 + A1281) + (8J - 81)(A12 - 1)c ( J A21 - 9 (1 + gj)(A21 + 81) + (gJ - 31)(1 - A21)cJ ) ( ) The result for component 1, (1n 11G)/n11, is given by setting i - 1 and j - 2, while the result for component 2, (1n 726)/n12, is given by setting 1 - 2 and j - 1. In eq 9, the normalized residual activity coefficient (1n in)/n11 depends upon three distinct sets of variables. The first of these, group ratios g1 and g2, describe the functional group composition of the molecular components. These two ratios replace the four functional group variables nkj in the original form of the ASOG model. The second set of variables, A and A are the Wilson 12 21’ parameters for the functional groups. The third variables are size- weighted fractions c or c2, which describe molecular component 1 composition in the solution. None of these three sets of variables depends on the functional group size unit. The Wilson parameters are constants for given functional groups, while g1 and g2 are ratios of two nkj values which depend on the size unit in the same linear way. The discussion following eq 5 showed 73 that c1 and c2 are independent of the functional group size unit for similar reasons. Hence, the right side of eq 9 will describe the same function of composition for a given set of Wilson parameters regardless of the size unit chosen for normalization. All of the normalization dependence of this equation is given explicitly by the denominator of the left side. This result applies to any solution of functional groups methods which treat component activities as the sum of functional group activities given by the Wilson equation. The only distinction will be in the definition of size-weighted fraction in eq 5. For example, in UNIFAC, the size-weighted fraction will actually represent a molecular surface area fraction, whereas in ASOG-KT, it will essentially represent a molecular fraction of atoms other than hydrogen (as mentioned above, there are a few special cases in ASOG-KT which do not follow the general rule for determining group and molecule size). The result given by eq 9 can also be extended to multicomponent solutions containing several distinct functional groups. This is done by defining additional group ratios so that the right side of the equation contains only group ratios, Wilson parameters, and component size-weighted composition variables. Such a generalized result will not be attempted in this paper. Instead, the dependence of eq 9 upon its existing parameters and variables will be interpreted. In the remainder of this paper, the variables 1 and j in eq 9 will 74 arbitrarily be taken as l and 2. Hence, the results given will apply to the activity coefficient of component 1. Equivalent equations for component 2 can be obtained by interchanging g1 with g2, and c1 with c2 on the right side, giving (1n 126)/n12 on the left side. TRANSFORMATION OF WILSON PARAMETERS The expression for normalized residual activity coefficient (1n 116)/n11 given by eq 9 is rather complicated; however, by appropriate transformations of the Wilson parameters, simpler forms of the expression can be written. Begin by defining transformed parameters (8 - 8 )(A - 1) 312 _ (12+ 1 12 (10) . 82)(1 + A1281) (82 - 81)(1 - A21) B21 - (11) (1 + 82)(A21 + 81) Each parameter Bij is a function of the group ratios and only one of the Wilson parameters, so that B1.1 can be regarded as the transformation of Aij' When eq 9 is written in terms of these parameters, it simplifies to 1n 716 - - 1n (1 + Blzcz) - g1 1n (1 + B21c2) n 11 c (8 -8 ) + (1+8 )8 (8 -8 ) - (1+8 )8 B 2 2 l 2 12 2 1 2 l 21 + ( - ) (12) 1+g2 l + Blzc2 1 + B21c2 75 At infinite dilution of component 1 in component 2, c2 approaches unity, and eq 12 can be further simplified to 1n 116 Q ) - - 1n (1 + B ( ln (1 + B n 12) ' g1 21) 11 1 (8 '8 ) + (1+8 )8 (8 -8 ) - (1+8 )8 B + ( 2 1 2 12 _ 2 1 2 1 21) (13) 1+g2 1 + 812 1 + 321 A further transformation of parameters B and 821 provides additional 12 simplification. (1 + g )(1 + A 8 ) c - 1 + B - 1 12 2 (14) (1 + 82)(1 + A1281) (1 + 8 )(A + 8 ) c - 1 + B - 1 21 2 (15) (1 + 82)(A21 + 81) Application of these parameters to eq 13 gives C In 11 w 1+g1 1 g2 ( - - 1n C12 - g1 1n C21 - (-—- + -——) + (1+g1) (16) n11 1+82 C12 c21 Eq 16 is the simplest possible form of the infinite dilution normalized residual activity coefficient in a binary solution. The only parameters required for calculation of this quantity are the group ratios g1 and g2, which measure the functional group composition of the molecular components, and C12 and C21, transformations of the Wilson parameters A12 and A21. This equation is simple enough so that its properties can be thoroughly investigated. 76 BASIC PROPERTIES OF INFINITE DILUTION NORMALIZED RESIDUAL ACTIVITY COEFFICIENTS AND BOUNDS ON THEIR PARAMETERS The quantity calculated by eq 16 classifies solution behavior into positive or negative deviation (from Raoult's Law) or athermality, depending upon its sign. By inspection of eq 16, the condition C12 ' C21 ' 1 (17) is seen to be sufficient for prediction of athermal behavior, since it forces the expression to zero. Three distinct types of athermal behavior can be described, dependent upon the group ratios and Wilson parameters. The first type is true athermality due to identical functional group composition of components, occuring when g1 equals g2. An example of this would be the binary system methanol-ethylene glycol, in which the ratio of hydroxyl to hydrocarbon groups is unity in both molecules. (ASOG-KT counts -CH -, -CH3, and -OH all as a having a size of one; this would not be true in UNIFAC.) Eqs 14 and 15 are seen to reduce to eq 17 when g1 and g2 are equal. A second type is true athermality due to non-interaction of functional groups, occuring when A12 and A21 both equal unity, the standard value of Wilson parameters in an ideal solution. Again, eqs 14 and 15 reduce to satisfy the condition given by eq 17 when this is the case. th« eq i. 77 The final type of athermality is accidental athermality, occuring when the value given by eq 16 is zero, but the sufficient condition given by eq 17 is not met. Examples of this behavior will be given later. It is possible for the group ratios g1 and g2 to take on any nonnegative values, including zero and infinity. A group ratio will equal zero (or infinity) when the molecular component it describes contains only a single functional group, while the other molecular component contains both functional groups, e.g., water and ethanol. If each of the two molecular components in a binary solution contain a single different functional group, one group ratio will equal zero while the other becomes infinite, e.g., hexane and water. This case represents the most nonideal extreme of functional group composition, with increasing ideality occuring in order for the following cases: one group ratio zero (or infinite), the other finite and nonzero; both group ratios finite and nonzero (e.g., l-hexanol and ethanol); group ratios equal. For each case where at least one of the group ratios becomes zero or infinite, special forms of eq 16 are possible. When g1 is zero, eq 16 reduces to G In 11 a 1 1 g2 ) - - ln C12 - (-—- + -——) + l (18) “11 1+82 C12 C21 when g2 is zero, it reduces to 78 In 1 G l 1 Q ) - - ln €12 - g1 1n C21 + (1+g1)(l - -——) (l9) n11 C12 ( and when g1 is zero and g2 is infinite, eq 16 becomes In 110 1 Q n ) - - 1n C12 - E—- + 1 (20) 11 21 ( There is no need to consider the situation when only one group ratio is infinite and the other is nonzero. By relabeling the groups, this becomes a situation where one group ratio is zero. Interaction parameters A12 and A21 are physically interpreted as resulting from energy differences between like-like and like—unlike pairs in solution. Quantitatively, this is given by [14] as A v (A - ) A __18xp[_ 11 11 11 v1 RT ] (21) where v is the molar volume of component i and A is the interaction 1 11 energy of an i-j pair (Aij - 111). When applied to functional groups rather than molecules, the preexponential factor vj/v1 is of magnitude unity. The argument of the exponential factor varies from negative values when like-like interactions are favored to positive values when like-unlike interactions are favored. The magnitude of this argument depends on the exact strength of secondary bonds, and can probably be bounded by the maximum bond energy of a hydrogen bond, about 50 kJ/mole [15]. Division by R gives a bound of approximately 6000/T, where T is 79 in K. Interestingly, this is similar to the maximum values assigned to the argument in several versions of the UNIFAC parameter tables (3000/T in [6] and lOOOO/T in [7]). At normal temperatures, then, the interaction parameters Ak1 can probably be bounded by the values exp(-20) and exp(+20). This essentially allows them to take on any positive values. However, the transformed parameters, Ckl’ are generally more restricted in their domain. If eqs 14 and 15 are differentiated with respect to A12 and A21, respectively, several properties follow for unequal values of the group ratios. First, C12 and C21 are either monotone increasing or monotone decreasing functions of A and A . Second, if C 12 21 12 increasing function of A12, 021 will be a decreasing function of A21 a vice versa. Third, the direction of variation is given in all cases by is an nd either g2-g1 or gl-gz, as Table 1 indicates. Taking limits on eqs 14 and 15 for these cases results in a general set of bounds on C and 12 C 21° 1+g1 l+g1 g2 --— < C12 , 021 < -— when g2 > g1 (22) 1+g2 1+g2 g1 1+g1 g2 1+g - < C12 , C21 < when g2 < g1 (23) 1+g2 g1 1+g2 Equality of g1 and g2 results in athermal behavior as discussed above, with the condition of eq 17 holding. 80 Table 1. Sign of deI/dAk1 as a Function of g1 and g2. 1(32 g1'32 31f32 8 dCIZ/dA12 + g chl/dAZl - + CONSTANT INFINITE DILUTION NORMALIZED RESIDUAL ACTIVITY RELATIONSHIPS In the presence of a large amount of experimental data, optimal values of the parameters C12 and C21 can be derived. The interaction parameter databases of UNIFAC and ASOG are derived in such a way from many sets of multicomponent, multifunctional group experimental data. For the simplified binary component-binary group case presented here, two experimental data points (e.g., two infinite dilution activity coefficients) suffice to determine the interaction parameters 012 and 021. There are cases, however, where only one data point can be determined at infinite dilution. An example would be the case of a concentrated polymer solution for which only an infinite dilution solvent activity coefficient was available. An analogous situation arises in the multifunctional group case even when several experimental values are available, because the number of interaction parameters required for a system containing n distinct functional groups is n(n+l)/2. Given a constant value of the infinite dilution normalized residual activity coefficient, as might be derived from experimental data using suitable choices for the non-residual part of component activity and the unit size normalization, a relationship between the parameters C and 12 81 621 can be defined. Such a relationship would be given by eq 16 (or eq 18, 19, or 20 for special cases of group ratios). In addition, eq 22 or 23 would place bounds on C and C . It is possible to solve eqs 18-20 12 21 for an explicit function of one parameter in terms of the other at constant [(1n 11G)/n11]m; such a solution is not possible for the general case of eq 16 where the relationship remains implicit. In that general case, a numerical solution for the (C ) relation can be 12'021 made. Bounds on the maximum and minimum possible values of [(1n 116)/n11]co for a given set of group ratios are also possible. Eq 16 can be differentiated with respect to each interaction parameter C or C 12 21 with the other held constant. This allows necessary and sufficient conditions for [(1n 11G)/n11]co to be an increasing function of each interaction parameter to be written. 1+g1 —— > 612 (24) l+g2 l+g g -——l —3 > c (25) 21 1+g2 g1 Combining these results with those given in Table l, the maximum value of [(1n ylc)/n11]co (as a function of 012 and C21) will always occur when 82 1+g c - —-—l 12 (26) 1+g2 c - 1+g1 g3 21 (27) l+g2 g1 and the minimum value of [(ln 116)/n11]co will always occur when 1+g1 g2 C - 12 (28) l+g2 g1 1+g1 c - 21 (29) 1+g2 These maxima and minima are given by G In 11 a l+g2 g1 ) - (1+g1) 1n + g1 1n —- (30) 11 max l+g1 g2 C 1n 11 a l+g g1 l > - (1+g1> 1n + 1n -— + (82‘81>(“ - 1) (31) n11 min l+g1 g2 g2 ( n ( If eq 30 is differentiated with respect to either group ratio with the other held constant, it can be shown that the expression takes on a minimum value when g1 and g2 are equal. Since the normalized residual activity coefficient equals zero under that condition, eq 30 necessarily predicts that the maximum normalized residual activity coefficient for distinct values of g1 and g2 must be positive. Similar arguments using eq 31 show that the minimum normalized residual activity coefficient for distinct values of g1 and g2 must be negative. 83 Maximum and minimum values of [(1n 116)/n11]do are listed in Table 2 for various finite values of g1 and g2. Some trends are apparent. Wider ranges of normalized residual activity coefficients result when g1 and g2 differ considerably from each other. The physical interpretation of this result is that greater nonideality is expected when the functional group similarity between two components decreases. When g2 or both of the group ratios are zero or infinite, the range of normalized residual activity coefficients is unbounded both positively and negatively. This represents an even more nonideal case of functional group dissimilarity. In all cases of finite group ratios, the minimum value of [(1n 116)/n11]co is larger in magnitude than the maximum. This has no physical significance; in fact, most nonideal systems exhibit positive deviations from Raoult's Law. It appears to be an artifact of Wilson's equation, indicating a mathematical tendency to predict negative values of the residual activity. Large values of g1 lead to wider ranges of [(1n ylc)/n11]do since the component activity coefficient is resulting from a sum of a larger number of functional group activity coefficients. The values in Table 2 may not be indicative of the magnitude of actual residual activity coefficients 1n 116 because of the normalization effect of dividing by the measurement n11 of functional groups of type 1 in component 1. .a;.~.32r31110000000 Ft 84 Table 2. Extrema of Normalized Residual Activity Coefficient of Component 1 as a Function of Component Group Ratios. Group Ratios [(1n 116)/n 11] g1 g2 minimum max mum l 2 -0 382 0.118 1 5 -2.612 0.588 1 10 -6.993 1.107 2 4 -0.661 0.146 5 10 -1.556 0.171 2 1 -0.523 0.170 5 1 -4.982 1.456 1 0.5 -0.382 0.118 1 0.2 -2.612 0.588 1 0.1 -6.993 1.107 0 l - a 0.693 0 2 - o 1.099 0 5 - m 1.792 0 10 - a 2.398 0 0.5 - o 0.405 0 0.2 - o 0.182 O 0.1 - m 0.095 For a given set of group ratios, eq 16 can be solved numerically for the G a: set of parameters (C12, C21) that result in a given [(ln 71 )/n11] . Figures 1 and 2 show this representation as a set of constant [(ln 116)/n11]co curves in (C12, C21) space for two sets of group ratios. Figure 1 illustrates constant [(ln 710)/n11]m curves for a case when both group ratios are finite and nonzero. The case g1 - 1, g2 - 2 (e.g., methanol-ethanol) is relatively close to ideality. Parameters C12 and 021 are restricted to the domain between 2/3 and 4/3, and normalized residual activity coefficients between -0.382 and 0.118 can be predicted. Because g2 > g1, a decrease in [(1n 1lc)/n11]co is seen as the curves are crossed in a clockwise direction from the 021 axis to the 612 axis. Clockwise rotation in this quadrant means an increase in C12 85 at constant C21 or a decrease in C21 at constant C12. Figure 2 represents a case where g2 equals zero, meaning that component 2 contains only a single functional group (e.g., water). Parameters C12 and 021 are restricted to the domain between zero and two by eqs 22 and 23, but the range of [(1n 116)/n11]co that can be predicted is unbounded. In this case, g2 < g1, resulting in an increase in activity coefficient with clockwise rotation about the axes. The set of curves collapses into the 012 axis as C12 approaches zero; no asymptotic relationship between 012 and C21 seems to exist for this case. The presence of a curve for [(1n 116)/n11]no - 0 in Figures 1 and 2 illustrates the situation termed accidental athermality. Although eq 16 predicts [(1n 116)/n11]co - 0 for all points on this curve, only the point (1,1) represents true athermality due to either identical functional group composition of molecular components or non-interaction of functional groups. All other points on this curve result from cancelling effects of positive deviations from ideality by one functional group and negative deviations by the other. Figures 1 and 2 illustrate the wide range of behavior which can be predicted by eq 16. This is true both in terms of the possible values of the normalized residual activity coefficient which can be predicted I as well as the (C12, 621) relationship which can generate a single activity value. In the absence of additional experimental data beyond a single point, it is not possible to determine which (612, 621) point on Cu 86 1.33 1.25 - 1.17 -* 1.08 -< 1.00 -‘ .917 J .833 - .750 -' .667 I I I I .667 .750 .833 .917 1.00 1.08 —-— V 1.17 1.25 1.33 Figure l. Constant Infinite Dilution Normalized Residual Activity Coefficient Relationships for g1 - l, g2 - 2. cl! 87 2.00 Int? :34 "n In I." .= 1 1154 n" 1.50 - 1.25 - . , (in I" )80 m. 100‘ OJS« . 1 r" 0.50 - (.151) =1 . 1 rs ' 0.25 - _, (Jf—) =2 -2 0 " I I v v I I 0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 Ct: Figure 2. Constant Infinite Dilution Normalized Residual Activity Coefficient Relationships for g1 - l, g2 - 0. 2.00 88 a constant [(1n 116)/n11]co curve to use in predicting the variation of activity coefficient with concentration. However, the restriction of interaction parameter values to a single curve can still provide useful information regarding bounds on the activity coefficient, as shall be seen. BOUNDING THE CONCENTRATION DEPENDENCE OF NORMALIZED RESIDUAL ACTIVITY COEFFICIENTS Since the results of the previous section indicated that a wide range of (C12, 021) points defined a given value of [(ln 116)/n11]w, it is enlightening to consider the concentration dependence of normalized residual activity as a function of the interaction parameters for a fixed value of [(1n 116)/n11]o. This can be investigated by taking the numerical results from eq 16 discussed above, applying eqs 14-15 to transform from interaction parameters (C12, C21) to interaction parameters (B12, B21). Eq 12 can then be used to give the concentration dependence of [(1n 116)/n11]. Consideration of this point is useful in two regards. First, the concentration dependence of normalized residual activity for a given set of group ratios and infinite dilution value can be bounded over the entire concentration range. This helps in estimation of the concentration dependence when only a single infinite dilution property is known. 89 Second, this point leads directly into consideration of the normalization of n11. For a given infinite dilution residual activity [(1n 116)]m, the value of the normalized infinite dilution residual will depend upon the normalization of n11. If the concentration dependence of [(1n 110)/n11] changes markedly with activity [(1n 1lc)/n11]co changes in [(1n ylc)/n11]Q, then normalization of n11 will have a noticeable change on the concentration dependence of residual activity coefficients. This will be true even when a large body of experimental data is used to find the optimal values of interaction parameters, as in the databases of UNIFAC and ASOG. The interaction parameters databases must then be considered size-dependent or normalization-dependent as shown in [13]. The technique described above was applied to produce Figures 3 and 4. These plots illustrate bounds on the normalized residual activity coefficient as a function of concentration for various fixed values of the infinite dilution normalized residual activity coefficient. For positive fixed values of [(1n ylc)/n11]m, the bounds could always be derived assuming C12 and C21 values at the endpoints of a specific curve in Figures 1 and 2. Such endpoints can be found from numerical solution of eq 16, or by algebraic solution of eq 18, 19, or 20 in special cases. This is not necessarily the case for negative values of [(ln ylc)/n11]°, which are not shown in Figures 3 and 4. Figure 3 is based upon the same group ratios as Figure 1, representing a solution not far from ideality. In this case, the bounding curves shown 90 are quite tight. Knowledge of the infinite dilution value allows estimation of the activity at any concentration with little uncertainty. Also, infinite dilution values near zero and near the maximum possible (0.118 for this case) result in the narrowest bounds on concentration dependence. The fact that there is some uncertainty for the case of zero infinite dilution value provides another example of accidental athermality, as discussed above. Figure 4, corresponding to the same group ratios as in Figure 2, illustrates a case which is more nonideal than shown in Figure 3. As a consequence of g2 equaling zero, a lower bounding curve could not be derived for this case, and only upper bounding curves are shown. In general, molecular components which are more dissimilar in their functional group composition result in more uncertainty in the concentration dependence of residual activity. That is why Figure 3 illustrates narrow bounding curves while Figure 4 illustrates a situation which is unbounded in one direction. However, functional groups which are more dissimilar in terms of their secondary interactions result in less uncertainty in concentration dependence. The uppermost set of bounding curves in Figure 3 represent the greatest deviation from ideality by functional groups, yet show less uncertainty than the bounding curves for an infinite dilution value of 0.05. Combining these results, it seems that the tightest bounds occur for systems in which the functional groups themselves interact strongly, but the molecules are not too different in their functional group makeup, 91 .10 (1.5.) =o.1o . . .05 ~ FAR?) =o.os -.05 q d Figure 3. Bounding the Concentration Dependence of Normalized Residual Activity Coefficients for g1 - l, g2 - 2. Labels are infinite dilution values; curves are upper and lower bounds. 92 2 1.5- (7.” r ) =2 :é 1- in I“ .:‘ 0.5- '"nn (lnnr,‘)'.o 0 v T x v 0 .2 .4 6 8 1 Figure 4. Bounding the Concentration Dependence of Normalized Residual Activity Coefficients for g1 - l, g2 - 0. Labels are infinite dilution values; curves are upper bounds. 93 e.g., methanol-ethanol. The relevance of this type of analysis is that it allows bounds on the concentration dependence of normalized residual activity to be accurately made by some analytical means. Such bounds are important in themselves, as they allow estimation of concentration dependence from a single data point without recourse to a group interaction database. They are also useful in providing bounds on the effect of normalization of n11 upon residual activity, as discussed later. BOUNDING THE UNKNOWN ACTIVITY OF A SECOND COMPONENT An approach similar to that of the previous section can be used to provide bounds for estimation of the activity of the second molecular component. Again, only a single value of the activity of the first component at infinite dilution is needed. The procedure for this calculation is similar to that for bounding the concentration dependence shown previously. An additional step is required because eqs 10-31 are specific to component 1 activity calculation. The first step consists of finding the (612, C21) endpoints of the constant infinite dilution residual activity curve for component 1 as described in the previous section. Since the transformed parameters Cij are component-specific, it is necessary to invert eqs 14 and 15 to generate A1.1 interaction parameters. The inverted equations are 94 (1 + g )C - (l + g ) A12 _ 2 12 1 (32) 82(1 + 31) - 81(1 + 22)012 81(1 + 82)C21 ' 82(1 + 81) A21 - (33) At this point, Cij values specific to component 2 can be generated by 1 replacing c2 in addition to interchanging g1 with g2, then gives the concentration interchanging g1 with g2 in eq 14. Eq 12, with c dependence of component 2 activity, namely, (1n 12G)/n12. Bounding curves like Figures 3 and 4 can be generated for component 2. The only qualitative difference between these curves and those for component 1 will be that the infinite dilution value for component 2 will not be a single point, i.e., the upper and lower bounding curves for component 2 will not merge at c2 - 0. Since a single infinite dilution value for component 1 can be used to generate bounds for the concentration dependence of component 2 activity, it can be used in particular to bound the infinite dilution activity coefficient of component 2. This provides another graphical relationship, shown in Figures 5 and 6. Values of [(1n 126)/n12]co are plotted versus values of [(1n 11G)/n11]no ranging from zero to the maximum allowable. The bounding curves show the allowable range of component 2 activity at infinite dilution corresponding to a known component 1 activity at infinite dilution. Figure 5 corresponds to the fairly ideal case used for Figures 1 and 3; both upper and lower bounds are available. In Figure 6, corresponding to the less ideal case of 95 .20 .15-1 .)‘ l' ' F’.101 (in .05 - Figure 5. Bounding the Infinite Dilution Normalized Residual Activity Coefficient of the Second Component for g1 - l, g2 - 2. Curves are upper and lower bounds. a» .c. .c. x 96 ). in 1'" n.. '5 l ( (#1). Figure 6. Bounding the Infinite Dilution Normalized Residual Activity Coefficient of the Second Component for g1 - l, g2 - 0. bound. Curve is upper 97 Figures 2 and 4, only an upper bounding curve is possible. This approach provides a more powerful tool than the Gibbs-Duhem relationship between activity coefficients of different components. Since the Gibbs-Duhem equation relates differential changes in the activity coefficients, it cannot be used to derive the activity of one component from that of a second component. The added power of this technique results from the assumption of a particular activity coefficient relationship given by the solution of groups model. However, the accuracy of the estimates depends on the validity of the solution of groups model, whereas the Gibbs-Duhem relationship is always thermodynamically correct. Such a bounding approach is most useful for systems in which limited data are available, where interaction parameters themselves cannot be fit. In such cases, the bounding result can be used to help design an experiment to take additional data. NORMALIZATION DEPENDENCE OF RESIDUAL ACTIVITY COEFFICIENTS Previously, a technique for bounding the concentration dependence of [(1n 116)/n11] was developed. In the reduction of experimental data to interaction parameter databases, a given normalization for n11 is assumed, and interaction parameters are chosen to best fit [(1n 11G)/n11] as a function of concentration. (In.UNIFAC and ASOG, data points from various concentrations are used, not merely from 98 infinite dilution.) Using the bounding technique of the previous section, the effect of varying normalization of n11 can be quantitatively illustrated. This will be done within the framework of fitting concentration dependence curves to an infinite dilution residual activity coefficient. Taking (1n 116)no as a fixed value, but allowing n to vary, values of 11 [(1n 11G)/n11]co corresponding to different normalizations of n are 11 produced. Each of these infinite dilution normalized residual activity coefficients has associated bounds as shown previously. If the bounds upon [(1n 116)/nll] given by these curves are multiplied by n a set 11’ of bounds for (In 11G) is produced for each normalization of n11 which is considered. Figures 7 and 8 illustrate the results of this procedure for the same group ratios shown previously, with a sample value of (In 716)do chosen for each. Bounds derived from n11 values of l and 4 are compared. It is evident from this plot that increases in n11, which are equivalent to decreases in the size of the unit of normalization, result in a wider possible variation in the concentration dependence of residual activity. This shows in Figure 7 as increases in the upper bound and decreases in the lower bound. In Figure 8, only an upper bound can be derived, and it increases with increasing n11. It is not necessarily true that wider bounds on the concentration dependence of residual activity result in a more inaccurate fit of 99 in 1‘,‘ Figure 7. Bounding the Concentration Dependence of Residual Activity Coefficients for g1 - l, g2 - 2 (ln ylc)° - 0.1 Solid curves are upper and lower bounds for n11 bounds for n11 - 4. - 1; dashed curves are upper and lower 100 2 \ \ \ \\ -—- n,,=4 \ \ fL=1 \ \ \ LS- \ \ \ \ \ \ \ \ e, \\ “ \ c 1 . \ "‘ \ \ \ \ \ \ \ \ \ \ .5- \ \ \ \ \ \\ \ \ \\\ \ \ \ \\\‘ 0 l 1 1 I ~~— 0 .2 .4 .6 8 C1 0 Figure 8. Bounding the Concentration Dependence of Residual Activity Coefficients for g1 - 1, g2 - 0 (ln 110)co - 2. Both curves are upper bounds, labels are n11 values. 101 experimental data. As shown in [13], the average accuracy of UNIFAC was increased when the unit of surface area was decreased in size. When attempting to use the results given here to make predictions for systems for which no interaction parameters are available, narrower bounds are preferable, which seems to imply that larger functional group size units would work best. The results given here do not imply that normalization unit can be varied indiscriminately in applying the residual activity equations within solution of functional groups models. Such a procedure would produce chaotic and meaningless results. What is illustrated here is the effect of changes of normalization unit upon some aspects of residual activity coefficient prediction, specifically, the bounds upon concentration dependence given a fixed infinite dilution value. Such an approach may prove useful in determining a proper value for the normalization unit in solution of functional groups models. CONCLUSIONS The residual activity coefficient given by solution of groups models using forms analogous to Wilson's equation can be conveniently analyzed by the transformations presented here. Transformation of interaction parameters allows simple expressions for component activity coefficient to be written. The transformed parameters also are restricted to a narrow range of values in many cases. In the case of a binary solution with two functional groups, the concentration dependence of both 102 residual activity coefficients can be bounded using only a single infinite dilution activity value. Group contribution models measure functional groups present in a component molecule in various ways. Regardless of the measurement used, the size of the unit chosen for normalization has an effect on the predicted concentration dependence of activity coefficients given by such a model. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] Wilson, C.H., J, Am, Chem, Soc, 1964 86 127. Holmes, H.J., Van Winkle, M., Ind, Eng, Chem, 1 1970 62 21. Renon, H, Prausnitz, J.H., Algh§_;+ 1968 14 135. Derr. ELL. Deal. C.H.. WEI... 1969 12 40. Abrams, D.S., Prausnitz, J.M., A1§h§4;, 1975 21 116. Fredenslund, A., Jones, R.L., Prausnitz, J.M., AIQQE J, 1975 21 1086. Gmehling, J., Rasmussen, P., Fredenslund, A., Ind En em Process Deg, Dev, 1982 21 118. Derr, E.L., Deal, C.H., Adv, Chem, Seg, 1973 124 ll. Kojima, R., Tochigi, M., Prediction of Vapor-Liquid Equilibria using ASOG, Elsevier, Amsterdam, 1979. Rizzi, A., Huber, J.F.K., Ind, Eng, Chem, Process De§1 Dev, 1984 g; 251. Vera, J.H., Vidal, J., Chem, Enggg, §ci, 1984 32 651. Rizzi, A.. Huber. J.F.K.. MW 1981 22 251. Skjold-Jorgensen, S., Rasmussen, P., Fredenslund, A., Chem,_finggg, £31, 1982 11 99. Orye, R.V., Prausnitz, J.M., Ing,_fing*_§hgm‘ 1965 51 18. CHAPTER 4 MODELING DIFFUSION COEFFICIENTS FOR CONCENTRATED POLYMER SOLUTIONS ABOVE TG Diffusion phenomena in polymer solutions have been difficult to study and interpret, due to the variety of effects observed. Differences in behavior occur dependent upon the state of the system, e.g., glassy, melt, dilute solution. In most cases, the behavior is non-Fickian, since the diffusion coefficient varies with composition and, under some conditions, relaxation occurs on the same time scale as diffusion. At temperatures sufficiently above T8, relaxation occurs more quickly than diffusion and may be ignored. In concentrated polymer solutions or melts, the mobility of polymer molecules can be neglected in comparison to solvent molecules. The remaining problems in determining binary mutual diffusivities are to model the self-diffusion coefficient (some authors refer to this as the tracer diffusion coefficient) of solvent in the system and to model the nonideal thermodynamic effects which cause the chemical potential gradient to differ from the concentration gradient. Both these effects must be considered as functions of temperature and of solvent concentration. Typically, an increase of solvent concentration results in an increase in solution free volume which tends to increase the diffusivity, while it simultaneously results 103 104 in a decrease in solvent activity coefficient which tends to decrease the diffusivity. The major quantitative analysis of this phenomenon has been made by two sets of investigators over the last 25 years. Fujita (1961,1968) originally proposed a model for the dependence of diffusion coefficients upon free volume. Vrentas and Duda (1977) extended the model and relaxed many of its original assumptions. The complexity of their model and its use of different independent variables for the free volume term and for the chemical potential term somewhat obscured its interpretation. As an example of this, Fujita was unable to show that water, unlike organic solvents, seemed to show very little increase in diffusivity with increasing concentration in polymer. Vrentas and Duda were able to show the correct concentration dependence with their model. They apparently attributed this behavior to free volume effects. In this chapter, a reprint describing the prediction of diffusion coefficients in polymer solution is presented. The model given here shows clearly that it is thermodynamic (chemical potential) effects which cause the seemingly anomalous diffusion behavior of water. A DIFFUSION COEFFICIENT MODEL FOR POLYMER DEVOLATILIZATION The following reprint article develops a model for the prediction of binary mutual diffusivities in concentrated polymer solutions and melts. A general form of the model is based upon the work of Vrentas and Duda, but applies a version of the new thermodynamic results given in Chapter 105 2. A linearized version of this model is also described. In certain cases, e.g., polymer devolatilization, the linearized model or a constant diffusivity model is shown to be accurate for describing diffusion phenomena. Details of the derivation of new equations proposed in the article are given in Appendix 1. Reprinted with permission from Polymer Engineering and Science, 21, 303 (1987). Copyright (c) 1987. 106 A Diffusion Coefficient Model for Polymer Devolatilization' MICHAEL J. MISOVICII and ERIC A. CRUI.KE Michigan State University East Lansing. Michigan 48824-1226 and ROBERT F. BLANKS Amoco Chemical Corporation Napcruiilc. Illinois 60566 Polymer devolatilizers are in widespread use in the poly- mer industry for removing solvents and monomers from polymer melts prior to product fabrication. Design equa- tions for describing the solvent flux usually include both the diffusion coefficient of the solvent in the polymer melt and the equilibrium ctmeentratlon of the solvent at the polymer-vapor interface. Several models make the as- sumption that the solvent diffusivity is constant over the ranges of solvent concentrations and temperatures in the devolatiltzer. This is a critical assumption that may be difficult to check without obtaining diffusivity data at the operating temperatures and concentrations of the process equipment. There are three models that can be used for diffusion coefficients in devolatilizer design: the free vol- ume model developed by Duda. Vrentas. and coworkers: a new linear model proposed in this study: and a constant diffusivity model. The linear model is obtained by combin. ing a new correlation for solvent activity coefficients in molten polymers with free volume theory and lineartzing the resulting equation. The error between using the com- plete free volume theory and using the linear model. or alternatively. using a constant diffusion coefficient. is alculatcd for several solvent-polymer systems. The linear model is convenient to use for determining the effects of the solvent activity coefficient on the diffusion coefficient. A method is presented for determining whether the corn- plete model. the linear model. or the constant diffusivity model is appropriate for a given devotatiliaer design. INTRODUCTION Diffusion processes play an important role in the manufacture and processing of com- mereial polymers. Processing steps such as polymerization. devolatilization. plasticization. and addition of additives require a knowledge of diffusion within polymer solutions and melts. Accurate modeling of diffusion coefficients of solutes in polymer systems above their glass transition temperatures is necessary for proper design of these . Molten polymer devolatilization is often done ’ .W.M.~ansisua—uwmussnaa little-hm” momma-Illumination. 1.7. “It,“d in either rotating equipment. such as a vented extruder or a thin f itm evaporator. or in equip- ment which foams the polymer. Models for pre- dicting the solvent flux in this equipment (I) of ten need diffusion coefficients of the solvent through the polymer at operating conditions. Some models. such as that of Newman and Simon (2) for foam devolatilization. are imple- mented with constant solvent diffusion coef f i- cients even though the calculations are per- formed ovcr a temperature range in which the temperature dependence of the diffusivity ls significant. Devolatilization is frequently car- ried out with less than 5 weight percent solvent in the polymer. Over the temperature and con- centration ranges in most commercial equip- 107 M. J. Nisan-it'll. h' A. (hulk-241ml R. F. iiimilts‘ ment. there can be a conventration-dependence of the diffusion coefficient. While it is simple from a computational point of vicw to assume a consmiii value for the diffusion coefficient. there can he significant errors in doing so. Duda ct til. (ill have predicted that the diffusion coefficient can vary signifi- cantly with temperature and concentration for the system. toluene-polystyrene. Their model shows good agreement with diffusivity data in some athermal polymer solutions above the glass transition temperature. Unfortunately. none of their comparisons are in the tempera- ture and concentration ranges of actual devol- atilization processes. There seem to be discrepo aneles between diffusivities determined from data taken in commerical devolatilizers (2) and diffusivities estimated by model extrapolation using parameters found at lower temperatures (3). in this work. we use the free volume diffusion model and employ an improved correlation for the thermodynamic factor (4] to analyze the diffusion coefficient predictions at conditions typical of devolatilization for polystyrene. At temperatures well above T,. solvent diffusion coefficients can be modeled by an equation lin- ear in solvent weight fraction. For small solvent concentrations. the diffusion coefficient can be taken as a constant. FREE VOLUME MODELS FOR DIFFUSIVITY Free volume diffusion models for transport of solvent in polymers are based on previous de- scriptlons of transport properties in liquid sys- tems. Cohen and Tumbull (5. 6) derived an expression for self diffusion coefficients as a function of free volume. Fujita (7. 8] used their work for describing solvent-polymer diffusion. Fujita's model is qualitatively correct but does not give quantitative agreement with available data. Several assumptions of the Fujita model were relaxed by Vrentas and Duda (9. 10) to derive a free volume model showing good agreement with data. Modifications and improvements have been made to this model in a series of papers since 1977. The most recent version gives excellent agreement with data for the sys- tems. toluene-polystyrene and cthylbcnzene- polystyrene. over the temperature range of I 10 to 178'C and concentration ranges up to 70 weight percent solvent (3). The binary mutual diffusion coefficient is given by (3): 0217»: (flag) T.’ RT do. (l, 0.0. D. on the right hand side of Eq l computes the effect of free volume changes on the diffusion coefficient: and the second group. the chemical potential derivative. computes the effect of thermodynamic changes. The self diffusion coeflicient of solvent. 1).. is given by: “11”“ {,I: T fl"); 01.1 l). =1)...cxp - v," (2] where thc nvcrugc holc lrcc volume. 17“,, is given by: 9. K —'—" - —'—! mm. + T — m T 7 (3) K + J 1011K" + 1' - T“) T The preexponential factor describing the en- ergy needed to overcome neighboring attractive forces. 1).... is given by: (41 Equations 1 through 4 define the binary mutual diffusion coefficient as a function of thermo- dynamic parameters. free-volume parameters. and an activation energy for diffusion. using solvent weight fraction as a basis. The free- volume parameters can be obtained from WLF equation data (3]. I)". 8 Du CXH’EIIITI CHEMICAL POTENTIAL DERIVATIVE in their solution for liq l. Duda and coworkers (3) used the Flory-Huggins theory and obtained the following equation for the thermodynamic factor: 2.2.91: 2:: - _ 2 _ RT (80),, u «in 2w.) (5) For systems that are athermal (the enthalpy change on mixing is zero). the interaction pa- rameter. x. can be taken as a constant. The athermal assumption is good for a system such as toluene-polystyrene. However. for a number of solvent-polymer pairs. enthalpic interactions occur and x is expected to vary with solvent concentration. in these cases. the variation of x with concentration should be included in the model equations. This could be done by writing the chemical potential in terms of a concentra- hon-dependent 1. taking the derivative with re- spect to mass concentration. and substituting the result for Eq 5. There is now no generally accepted model for describing the concentration dependence of x. Misovich and coworkers (4) have recently de- veloped a correlation for solvent activity coeffi- cients in concentrated polymer solutions which fits data for systems with enthalpic interactions at least as well as the Flory-liuggins equation. The correlation gives an improved result for the concentration dependence of the chemical p0. tenttal and can be used to determine the value of the derivative In Eq l. The result is: rummauoscema w, 1'7. CH. 11.“. 4 108 A ”illusion Cinjuii‘ient Ali-ileljur Polymer fh'iuluiihuunui i291»; («11.) ’ r! In a. m9 a... - ' tI-ITIT. . (GI ( I 97' "'7 ( “I + " H’- ' iz.‘ ‘ it.‘ is the weight fraction activity coefficient of solvent in polymer at infinite dilution of solvent and can be determined by a variety of methods. For values of ll.‘ between 2 and 20. the new correlation predicts the concentration depend. ence of activity coefficients in binary solvent- polymer systems. Since il.‘ in this correlation is a true constant at a given temperature. this equation can he used without revision for poly- mer solutions that are athermal and for some solutions with enthalpic interactions. An additional advantage of Bq 6 is that the weight fraction is used as the independent con- centration variable. whereas the Flory-Huggins equation uses volume fraction. Applying Eq 5 requires equilibrium and density data for the solvent and polymer at the temperature of in- terest. while Eq 6 only requires equilibrium data. Blanks. et at. (I It show that the assump- tion of a constant density ratio between solvent and polymer is not a good one for devolatiliza- tion problems. The chemical potential derivative could be obtained by differentiating expressions for the chemical potential. There are methods for ob— tainin the chemical potential based on equa- tion-o -state approaches (l2). lattice fluid the- ory (l3). and UNlFACoFV (l4). UNIFAC-W is based upon statistical mechanics and contains separate entropic (combinatorial) and enthalpic (residual) terms. One of its advantages is that many polymerosolveot systems can be de- scribed by the database built for UNIFAC (15. l6). Van den Berg (l7) has recently proposed a method for generating UNIFAC-W activity coef- ficients using a UNIFAC program. The disad- vantage of using any of these methods to get the chemical potential derivative Is that their differentials are complicated expressions which are difficult to analyze except by numerical means. Eqs 1 to4and6can becombinedtogetan equation for the diffusion coefficient: 9 2 (7| th Q... and 9,..[7 are dependent on temperature. Vrulv is also dependent on concentration. Even though Eq 7 includes concentration and tem. ““[jww' + o.-......] sumac-mum M. an. an. 12,... perature dependence. it is not a convenient form to use for modeling and design. In the next section. we will show how to inudify liq 7 in get a form that is easy to apply to (levolatiIi/er design. LINEARIZED DIFFUSIVITY MODEL Polymer devolatilization often takes place at solvent concentrations of less than 5 weight percent and temperatures well above 1‘”. lie- cause the solvent diffusivity is required at low solvent weight fractions. we choose to linearize Eq 7 with weight fraction at the point. to. - 0.The value of the diffusion coefficient at zero weight fraction of solvent is easy to determine and the differential of l) with w. is easy to evaluate. Linearized models have been proposed for describing the concentration dependence of the solute diffusivity both for polymer diffusion in dilute solutions ( l 8) and for solvent diffusion in concentrated solutions (i9). The free volume terms in Eq 3 vary with temperature. These terms are grouped as shown below and inserted in Eq 7: K A. - —;— (Kn + T - m (81 and Kn A: - ‘7' (K2: + T - 7}.) (9) giving e a —. m D 3 Do a. e wt + '—.'. w: I . . (10) 4 V.'w. + Vg'uag E I” - - - Aiw. + Aawa RT for the diffusion coefficient. Equation 10 as- sumes that the solvent and the polymer are in thermodynamic equilibrium at the vaporopoly- mer interface. This would seem to be met for most polymer-solvent systems. Even for those systems in which anomalous polymer behavior is claimed (such as the T. 1 transition in polysty- rene) (20). the equilibrium requirement should be met if the temperature is greater than 1.2 1“,. An implicit condition on the application of the thermodynamic model is that the solvent molecular weight should be much smaller than the polymer molecular weight (4|. Free volume parameters based on the WLF equation are usually assumed valid up to lOO‘C above T,. Some commercial devolatilization conditions may exceed this temperature. There is no generally accepted method for estimating the polymer frecwolumc parameters for tem- peratures greater than T, + lOO‘C. it is not clear that the WLF equation is a good model for ex- trapolating solvent-free volume parameters. M. .I. Misovich, E. A. Griilke. and R. P. “tanks The linearized model is: 7!) (mini a Din..-" + '—— (m, - 0) (i ii «hit. ,w'," wliii'li lit-emiies; Dita.) = ()(0NI + (K. - Kmtnl (l2) where '_o _ f, e K. a Aitvl “A: i “25!) C a ll.“ l l i we 0(0) 1).. exp{- (RT+ A: )] il2ci The term. K .. is the free volume factor. and the term. K3. is the thermodynamic factor. The exponential term in £q 12c includes a term describing the attractive forces between neigh- boring molecules and a term describing the ratio of critical molar jumping units for the solvent and polymer. COMPARISON OF LINEAR AND COMPLETE MODELS The three levels of model complexity for de- scribing the effects of solvent concentration on diffusivity in polymer devolatillzers: the com- plete model (Eq l0). the linear model proposed here (Eq l2). and the constant diffusivity model. provide a good range of choices for the design engineer. An advantage of the linearized model is that. at a given temperature. the dif- ference between two constants describes the concentration dependence of D. The errors as- sociated with using the simpler models depend on both the temperature and concentration ranges over which devolatilization is taking place. As shown in Fig. i. there can be a signif- 4. U 1 I U . b "c .m.— g h. o.-- 3 . 3 ,. soft: 3 so". —— ' i- I A . 1 8 8 s ‘ bout-Iota... Fig. l. Conn-arisen (j the cancenmuton dependence of «illusion mjfh'icnts calculated by the complete and tin- qsullrn- iotunie models. Toluene and (ratgstyrene. icant effect of concentration. but its magnitude depends on the temperature. Equation 12 should only he used for modeling after its accuracy has been evaluated. We have compared these models for the system. piilysty- relic-toluene. at a temperature Just alxwe T” (l lO°Ci and a temperature typical of commer- cial iievolatilizers (240‘Ci. The ratio between the diffusion coefficient at the specified weight fraction and that at zero solvent weight fraction is used to determine the difference between the two models. Figure l shows the error associated with a linear model at the two temperatures. At l iO°C. the calculations show that. below lOO ppm sol- vent. the diffusion coefficient can be considered constant. There is less than 2.4 percent error in the value of the diffusivity by this assump- tion. Up to IOOO ppm solvent. the linear model diffusivity is within 2 percent of the complete model diffusivity. The accuracy of the linear model decreases rapidly at greater solvent weight fractions and. in this case. underpre- diets the diffusivity. It is not clear whether the condition of thermodynamic equilibrium at the interface is met for this system at l l0°C. Anom- alous transitions in the polymer melt might make the polymer relaxation time the same or- der of magnitude as the solvent diffusion time. Figure l shows calculations for the same sys. tem at 240'C. a temperature in the range of typical devolatilization temperatures for poly. styrene. At the higher temperature. diffusivity can be considered constant at solvent weight fractions less than lOOO ppm. in this case. the linear model value is within l percent of the value for the complete model up to 100.000 ppm or to weight percent solvent. The diffusivity of toluene in polystyrene at 5 weight percent (a typical concentration of solvent in polymer at the start of a devolatilization process) would be 2.2! times the value at zero weight fraction solvent. suggesting that a model using a con- stant diffusion coefficient could be in error. Figure I does not show the temperature de. pendence of the diffusivity. which can be sig- nificant. We have calculated the infinite dilu- tion solvent diffusion coefficients at two tem- peratures based on Eq 12c and using the con- stants su ed by Duda. et at. (3) for toluene] polystyrene. At t IO°C. the diffusion coefficient is 6.l x lO‘" cm’ls: and at 240‘C. the diffusion coefficient is 5.5 x 10“ cm’ls. in changing the temperature from near 1‘, to l.4 T,. the difoo slon coefficient has increased by about 5 orders of magnitude. Since the free volume parameters of Eqs 1 to 3 are temperature-dependent. the scalingof diffusivity with temperature does not follow a simple Arrhenius equation. There are practical problems associated with determining the free volume parameters and thermodynamic parameters for Eqs 10 and 12. The concentration dependence of the solution free volume parameters is taken to be linear (Bq mmmmmw. manna: A Uiuusam (.‘ijn‘u-m Mullet/m Polymer Uetulatilizutuui 3|. The polymer and solvent free volume param- eters are determined by fitting viscosity data with the WM: equation. For polymers. the Vis- cosity can usually be determined over the tem- perature range of interest. For solvents. the WM" parameters are usually determined below the normal solvent boiling point (at atmospheric pressure}. The thermodynamic terms in Eqs 7 and 12 describe the coneentrationodcpcndenec of the activity coefficients. However. 11.‘ changes with temperature. as does the interac- tion parameter. Typical errors associated with these estimation techniques are discussed in the next section. EFFECTS OF SOLVENT WLF PARAMETERS ON THE LINEAR MODEL The WI.F equation (3) may not describe the free volume changes of the solvent well. partic~ ularly if it is extrapolated to temperature well above T... + lOO‘C. 79i°/Kii 13 K3|+T-T,| ( i l’l '. 3 I" A. + Furthermore. the fits of some solvent viscos- lty data by the WLF equation seem to show systematic deviations rather than random er- ror. Such deviations suggest that this model may not correctly predict the changes in solvent free volume with temperature. If the WLF model is used. it is preferable to determine its param- eter values as close to the devolatilization tem- perature as possible. The comparisons below show typical differences in the linear diffusion model parameters caused by differences in the WLF parameters. Table 1 lists solvent free volume constants and the values of K. for acetone and methyl acetate obtained from two different WLF fits of viscosity data. Liu (21) apparently combined two data sets (22) and (23). while only one set was used in this work (22). The two data sets covered similar temperature ranges. The WLF parameters appear to be sensitive to small changes in viscosity data. Polytmethyl methac- rylate) is the polymer considered and has a Tn of about 303 K. Both sets of WLF parameters generate viscosity models which average 1 per- cent relative error with the data. The K. values. which describe the concentra- tion-dependence of the diffusion coefficient. are compared in the lower portion of Table 1. We computed K. values at 378 K(1.25 T“) and 453 ML!) 1‘...) since this might be the range of tem- peratures used in devolatilizing such polymer solutions. The K. values calculated for acetone in PMMA are similar for both sets of WLF pa- rameters. However. the K. values for methyl- acetate differ by factors of 3 to 4. Since both sets of WLF parameters describe the viscosity data about the same. it is not clear which set of K. values is the better description of the free mmmmmmx 8.7. H. ".hl labial. munvmwmmtim Vabos st W1! Gustavus. Con-torus 79“] “M h ‘ ‘1. “is".s M. Acetone -3 23 $08 -53 3 (2i) -3 ii 468 -599 ttvssiudy Methyl—W -3 64 662 -38 5 (21) -2.26 lbs ~16!) this study System I. 'K (.01) K. (this study) Acetone-PUMA 378 $6 61 45.3 20 3| Methyl W 378 as it? 453 10 66 volume term. These calculations merely illusv irate the sensitivity of K. to the values chosen for the solvent's WLF parameters. Vrentas et at. (24] (Eq. 5] suggest that defi- nition of Kin/‘7 Pcrmits a bound to be placed on this parameter. which results in a lower bound for the group. 7V.'/K... For acetone. both val- ues of yV.‘/K.. are above the lower bopnd of 450. For methylacetate. the value for 7V.'/K.. determined in this study is below the lower bound (380). Presumably. different WLF con- stants could be obtained by forcing this group to equal the lower bound and varying the other constants to fit the viscosity data with similar precision. The bounding of this group depends on the assumption that the WLF equation cor- rectly describes free volume changes of the sol. vent. We calculated the K. parameters for toluene. methanol. and water with polystyrene over the temperature range. 1.02 T, to 1.42 T,. The WLF parameters for the polymer were taken from Liu (21). Figure 2 compares the results for tol- uene and methanol. For both solvents. the dif- ferences between the K. parameters are large near 1‘, and become smaller at high tempera- tures. Figure 3 compares K. values based on WLF parameters from water viscosltica below the normal boiling point (50 to 100°C) (20] with those based on WLF parameters for water vis- cosities taken between 110 and 160‘C (23). The viscosity data between 50 and lOO‘C lead to negative values of K.. Negative values of K. imply that the polymer expands with tempera- ture more than the solvent. which is not ex- pected. Thesereaultssuggestthatitwouldbepref- erabie to determine the solvent WLF parame- ters as close to the devolatilization temperature range as possible. For many cases. this would mean determining solvent viscosities at high pressures. An alternative approach might be to determine solvent free volume parameters from viscosity data of polymer melts containing sol- vent concentrations in the range of interest. A capillary rhcometer might be used to talte such data. ‘ N? 111 M .l .‘ltauui‘lt‘fl. 9:. A. Grulke. and R. i". liluuks (a) 20° , Toluene ‘ s '4 .r \ \ m " \ 1 \ \ P \\ 4 \ \‘ ‘ ~ 0 A A - — — l” ‘50 ZIO 160 Tomp..°C .0 V V (b) .0 _. “ethanol 4 a. \ \ 3. I \ d \ \\ \ \ \ ‘\‘ 0 A r_—_— ‘3. ‘“ 3‘0 1.. To”. '6 Fly. 2. Comparison 1mm temperature/or two sets 1 solvent WLF parameters. in) toluene-polystyrene. (bl ‘ “‘ -..:. Solid line-Lia (2". dashed line- —_AL A U U this work.r EFFECTS O!" K. AND K. ON THE LINEAR MODEL Equation i2 provides a convenient method for determining the effects of the thermody- namic and free volume terms on the concentra- tion dependence of the diffusion coefficient. The difference between K. and K, gives the slope of diffusivity versus solvent weight frac- tion curve (as long as the linear model is valid). Table 2 compares values of K. and K; for an athermal system (toluene/polystyrene). a sys- tem with moderate enthalpic interactions (methanol/polystyrene). and a system with strong enthalpic interactions (water/polysty- renel. The WLF parameters for the solvents were determined by fitting viscosity data taken below the normal boiling point. The thermody- namic data were obtained by condo: and Din. eer (26). who measured weight fraction activity coefficients for 42 solvents in polystyrene as a function of temperature. Although their data seems internally consistent. the activity coef f i- cients they report are factorsof LE: to 2.0 higher than coefficients reported by other researchers .0 fl 1 .0 ‘0 x. 30 0 . l A "0 ‘CO 3'0 3‘0 Tampa 'C Fig. .‘l. Comparison for K. versus temperature-for “Inter- puiystyrene. Solid line-WU" parameters jrom water vis- eosiites bdrm-en 50 and lOO‘C (221 Dashed iine-WLF parameters from water Munsttics between 110 and lGO'C (27L Tablaz. WVahnsotK.ar-dl.tor$ovoral$olvant- Polystyrene Systems. System Toluene/PS moans my}? Temporal-m K.‘ K.‘ .' Ka' Kc‘ Kn‘ i62°C ' as so as so 33 no l72‘C x 5.4 20 25 27 l” 220°C is 4.0 t7 12 14 57 m l3 3.6 ‘2 7.5 13 56 mum ‘u-nn 'l.astsrna-d-v-.rtaaas~.hba- net-raven ‘ma—fi‘uwuwwm (27. 28). We use their values beause they seem to be the only values available for our solvents. For toluene-polystyrene. the difference be- tween K. and K, is always positive. and. while the linear model applies. the diffusivity will increase as the weight fraction of solvent in- creases. The difference between K. and K. for the methanol-polystyrene system is much less. The solvent diffusivity for this system should show very little concentration dependence. It should be noted that the ASOG-VSP model has successfully represented the dependence of the activity coefficient on solvent concentration for methanol in poly(methyl methacrylate) (4). it is not ltnown whether this model adequately de- scribes the solubllity of methanol in polysty- rene. We consider the calculations for the water-polystyrene system to be speculative. since the ASOG-VSP model has not been used on data with such large infinite dilution weight fraction activity coefficients. The negative dif~ ference between the free volume and thermo- dynamic terms suggests that there may be a range of water weight fractions for which the diffusion coef f icient decreases with increasing water concentration. Performing measure- rammed-communion “Luthflad 112 A Hillitsinu Can'flh'n'ni Male! for Polymer Uc'tuluiili/uiiuu inents on systems in which the concentration dependence of the solvent diffusivity was near zero. or negative. would constitute an interest- ing test of the free volume theory. Figure 4 shows the slope of the diffusivitv versus weight fraction eutve lor the three sul- vents from lit) to 2ii0"(.‘. it.“ values were ex- trapolated using a model linear in temperature. Over this temperature range. the diffusivity of toluene should always increase as its weight fraction increases (until the linear model is no longer valid). 0n the other hand. methanol shows very little concentration dependence of the diffusivity above IGO‘C. The model predicts that water should have the unusual property of a decreasing diffusivity through polystyrene as its weight fraction is increased. Again. this re- sult should be considered Spt‘t’ulallvc since the ASOG-VSP model has not been verified for sys- tems with such large enthalpic interactions. Using the linear model to analyze the effects of thermodynamic and free volume terms is valid as long as the linear model provides a good approximation to the complete model. The error associated with the linear model depends on the solvent-polymer system and the temperature. For the toluene-polystyrene system. the free volume term dominates the concentration-de- pendence of the diffusivity. Since diffusivity in the methanol-polystyrene system is much less dependent on solvent concentration. the linear model should approximate the complete model over larger concentration ranges than for the toluene-polystyrene system. Figures 5 and 6 show this effect for two different sets of solvent free volume parame- ters. For most temperatures. the linear model will describe the complete model up to 10.000 ppm. The improved range of fit to the complete model is due to the lower concentration depend- enoeof this system. The linear model will either predict a positive or negative (rarely zero) con- ccntratlon dependence to the diffusivity and will not predict maxima or minima in D versus w. curves. Comparisons of Figs. 5b and 5d with Figs. 6b and 6d illustrate the sensitivity of the diffusion coefficient to the solvent-free volume parameters. For both Figs. 6b and 6d. K. - K. is slightly above zero and the complete model should'go through a maximum value. Figures 4 to 6 show how the thermodynamic and free volume term affect the linear model and over what concentration ranges the linear model is valid. Figure 7 illustrates the effect of the thermodynamic term on the diffusivity of the complete model for methanol-polystyrene at l55’C. For a 25 percent change in the value of il.'. the diffusivity can change from monotoni- cally decreasing to going through a small max. imum. While there is good agreement between mea- sured solvent diffusivities and the free volume model in the papers of Duda and Vrentas. sol. vent dif f usivities measured in actual devolati- edema (MEN m “a WY. I’ll. VOL 27. Na. d V V (a) 10° __ Toluene 4 ~ 4 n: O u. .i d \ ‘~~_-_.— o A ‘ — — no no 3‘0 no temperature.°c .0 v v (b) _ “OON d x“ .- m b ‘ " \ L V \\ a ” \‘ ——’——” -20 L ‘ H0 160 2‘0 200 Tmeretaoe.’c ” Y 1 470 A no no tempera-103C Fig. 4. K.-K. versus temperature for three solvents in polystyrene-Jar ‘ - ‘rc‘, ‘, ouNMMlDlNWy. rene. kl outer-polystyrene. Lines identgfled per figures 2 and .‘l. A "O m llzers do not agree well with predicted values. For example. in the foaming devolatilizer work of Newman and Simon (2). the estimated value for the diffusivity of styrene in polystyrene is l x l0" cm'ls. This value was assumed constant for fitting data between 200 and 250‘C. Al log OIDlOl log 0/0t0) la. 0’0“) be 0! Dial M. .I. .‘fisui'it it, if A Giullu', uinl it. I". [thinks 1 T 1 (a) . i I ,4 _ not I . b 1 b 0 - . 0 t 1 to. ppm “00” " (b) \ \ \ . l - 33 C .5 ‘ ‘1 -u o i a a 4 s to. ppm IleOH J j Y j ' (c) \ . \ \ -3 800°C -‘J 4‘ A A A 0 I I 3 0 I to. ”a 800“ a j Y 1 f (a) on 1 cm 1 -u au'c -.17 -a L J A _L 0 ' I I 0 0 lo. "m Icon 240’C. the free volume model predicts that the diffusivity should be 4.2 x l0" emf/s. There is obviously a significant error associated with assuming a constant diffusivity over this tem- perature range. There are also discrepancies between diffu- sivities measured in commermal equipment and those measured in research equipment. in an cxtruder devoiatilizer. liiesenberger and Kessi- dis (29) report a diffusivity of styrene in poly- styrene of l.5 x l0” cm’/s at l77‘C. Dnda ei at. (3) measure a value of 3 x l0" cm’ls at 178°C for ethyibenzene (which should be simi- lar to styrene). The linear diffusion coefficient model pro- posed in this work has the potential to be a convenient tool for designing and controlling the operation of commercial devolatilizers. The designer can determine by calculation whether to use the complete difquion model. the linear diffusion model. or a constant diffusivity for his equipment conditions. The concentration-de- pendence of the solvent diffusivity is sensitive to extrapolations with the solvent-free volume parameters. Because of this sensitivity. it is preferable either to use solvent viscosities ob- tained at devolatilization temperatures or to de- vise another method for obtaining them. Fi- nally. the effects of thermodynamics on the concentrationdependenceof solvent diffusivity may be the same order of magnitude as the free- volume effects for some solvent-polymer sys- tems. NOMENCLATURE a. - activity of the solvent. A.. A. - groups of parameters def lned by Eqs 8 and 9. D - binary mutual diffusion coefficient. 0. - self -dlf fusion coefficient of solvent. 0. - defined by Eq 4. Do. - defined by Eq 3. e - base of the natural logarithm. E - critical energy per mole needed to overcome attractive forces. K. - free volume coefficient in the line- arized model. Sq 12a. K. - thermodynamic coefficient in the linearized model. Eq 12b. K... Kn - free-volume parameters of solvent. K... K.. - f ree-volume parametersof the poly- mer. pressure. ideal gas constant. temperature. glass transition temperature of component i. partial specific volume of compo- nent i. q IIII S I Pig. 5. Log UlilOi versus log ippm soiuentljor nail-anoi- poiysigrema Solvent free tar-fume parameter; c] this study. lot i lO‘C. (bi l55‘C. itl m. idl215'C. Solid line- 13¢ :2. dashed line-flu to. ram mews m m WY. tfll. W. 27. No. 4 111+ /\ Ihlfusnm Fur-(intent Motle'flnr l'ultpm'i f)~'i\iliillfi/.ttflitli .. 1 v v V (a) ,. 110‘ 2 .0 y C o \ O o 2 .8 b o A _L 0 t 2 to. ppm Moo“ " v t t Y (b) b 1 155°C / .. / o A 5 0 \ O o 2 i- . n.‘ F A A J A 0 t 2 3 o g M ”on not)“ of t 1 § —oe- aoo‘c O \ O O 2 c.“ I o... #1 A A a ti I 3 3 4 II to. MW V 1 V V (d) 1 / / 0 a O 8 us'c \ O r + o 2 c“ p 1 0 l 3 3 o I '09 Dean noon ”until (W M0 906m. rem. in". H- 17. “a ‘ .05 v v v v '95 h —30 ——2. --- as ‘-“'"’30 I” 01' 0(0) A j A 0 t 2 3 to. pa- IoON Fig 7. LfUu-t of the thermodynamic term of D/finjor the complete Model. Methanol-polystyrene at l55'C. .L (If = Specific critical hole free volume of . component i required for ajump. V,-,, = average hole free volume per gram of mixture. to, - weight fraction of component i. x. = mole fraction of component i. Greek Letters 7 =- overlap factor for free volume. a. - chemical potential of solvent. £ a ratio of critical molar volume of solvent jumping unit to critical molar volume of jumping unit of polymer. mass concentration of component i. Pi '- a. - volume fraction of component i. x - Flory-Huggins interaction parameter. 0. - solvent weight fraction activity coeffi- cient. o.‘ - solvent weight fraction activity coeffi- cient at infinite dilution of solvent. Subscripts . - solvent. , - polymer. REFERENCES l. J. A. Biesenberger and D. it. Solution. 'Prtnelples of Polymerization Engineem‘.’ p 573. Wileybintersci- ence. New York “983). 2. R. S. Newman and R. H. M. Simon. 'A Mathematical Model of Devolatilization Promoted by mbble Foam-o tha.‘ presented at 77th Annual m Meeting. San Francisco. California "0843. 3. J.t..DudI.J. S. Vrentaa.S.T.Ju.andl-l.1'.uu.NCh£ J.. as. 279 0982i c. H. J. Misovich. R. r. Hanks. and E. A. Grulke. 'A Generalized Correlation for Solvent Activities In Poly- mer Solutions.‘ ind. Eng. Chem. Process Des. Dev.. 24. l036tl985l. ‘ 5. M. It. Cohen and D. Tami-ill. J. Chem. Phys. 31. It“ "959). 6. D. Turntaull and M. ti. Cohen. J. Chem. Phys. 34. too Fig. 6. (119 DIIXOI m by w sdmtljor methanol- poiysiyreue. Solvent free intone parameters 9] Liu (2". h! l lO'C. (hi lSS'C. id WC. WZCS‘C. Solid line—Ea '2. dashed line—F4 IO. 3" 9. '0. l2. l3. l4. l5. l6. l7. l0. ll. Yamakawa. 'Mudern Theory of Polymer Solutions.‘ 115 M. J. Misum‘t'it. (I. A. Grulke, and R. F. iiimtlts “96”. . H. Fujtia. Foliu'hr. Unehpoiym.-f'tv.\¢‘ti.. 3. l “96” . ll. Fulila. 'Organlr Vapors Above the Class Transition Tetn;rraiure.' in J Crank and C S l'arlt. eds, Uifln stun in Polmm-rs. Academic Press. New York “90“). J. S. Vrentas and J I. lhula. J. l'otym. $11.. l‘oigm. Phys. Ed" l5. 403(I977). J. S. Vrentas and .l. l. Outta. op, t‘ll.. l5. 4l7 “977) R. F. lllanlis. J A. Meyer. and E. A. Crnihe. i‘uiym. ling. 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Duda. li.-C. Ling. and A.-C. lion. .1. i’oiym. Sci. Polym. Phys. 84.. 23. 889 “985). it. N. liaward. J. Macronni. Sci. Revs. Macromoi. Chem.. C4. l9l -242 “970). S. Gunduz and S. Dincer. Polymer. 2|. l04l "980). M. Calm and H. C. Ruppreeht. I’iotgmer. 19. 506(l978l. R. D. Newman and J. M. Prausnitz. J. Phys. Chem" 76. l492 “972). J. A. Uiesenberger and C. Kassidis. Poiym. Eng. Sci.. 22. 832 “982). ram EM!” no ”a rem. i”). Vd. 27. No. 4 CHAPTER 5 STATISTICAL DETERMINATION OF SYSTEMATIC ERROR IN NONLINEAR PARAMETER ESTIMATION Statistical parameter estimation involves the determination of the value or values of some unknown quantity based upon data which may be inconsistent or contain error. Normally, the quantity or quantities to be estimated are in some ways characteristic of the sample from which the data were taken. The determination of an equation or model for some physical phenomenon normally involves the selection of an expression on theoretical (or empirical) grounds followed by a parameter estimation step to determine the unknown parameters of the model. A set of estimated parameters is usually considered good if the predicted values of the dependent variable generated from the model do not deviate substantially from the observed values. A global criterion, such as the sum of the squared deviations of the predicted values from the observed values, is typically applied for this purpose. Use of a global criterion of this type may mask conditions which cause the model to be inadequate in other ways. One such problem is the existence of systematic error within the model, causing overprediction and underprediction of the dependent variable to be correlated to the independent variable rather than random. 116 117 Systematic error may indicate an underlying lack of agreement between the model and the physics of the problem. It is a particularly crucial type of error when results from a parameter estimation must be extrapolated outside the domain of the independent variable over which parameters were found. For this reason, testing for systematic error in a parameter estimation may be important in certain situations, even when the global fit of the model seems acceptable. In order to do statistical testing, it is necessary to have a hypothesis, usually in the assumption of a particular random distribution of the variable or variables being studied. The standard approach is to evaluate the distribution of the test statistic under this random (or null) hypothesis. If the value of the test statistic calculated from the observed variables is unlikely to have occurred with random variables chosen under the null hypothesis, the null hypothesis can be rejected. A good test statistic will be able to discriminate between values taken under the null hypothesis and those taken under some alternative hypothesis; the ability to discriminate in this manner defines the power of the statistical test. LINEAR AND NONLINEAR PARAMETER ESTIMATION When the expression is linear in the unknown parameters, linear parameter estimation techniques based upon least squares can be applied 118 (Mendenhall, 1968; Graybill, 1961). Least squares means that values of the unknown parameters are chosen such that the sum of the squared deviation between each observed variable and predicted variable is minimized. It is possible for the expression to be linear in the unknown parameters even though it is nonlinear in the independent variables. For example, if y is the dependent variable, and x is the independent variable, the equation 2 y - ax + bx + c (1) is linear in the parameters a, b, and c even though it is nonlinear in the independent variable x (since it contains a term in x2). If x and y are variables which can be measured, linear least squares can be used to determine the best values of a, b, and c from measured data. Least squares provides an optimal solution to the parameter estimation problem when the distribution of error is normal. If x1 and y1 are the observed or measured values, the error e is defined by 1 red -yp - ‘1 1 Y1 (2) which becomes, in the case of eq 1. 2 e - ax 1 1 + bx1 + c - yi (3) When the error distribution is not normal, the equations of least squares do not necessarily provide an optimal solution to the parameter estimation problem. Such a situation may occur when the data contains 119 outlier values due to measurement error or some other problem, or when the equation used to model the physical situation is systematically in error . In cases where the expression is nonlinear in the unknown parameters, least squares analysis still provides a solution to the parameter estimation problem. However, the classical equations used for the linear case are not applicable, and often the sum of squared error must be minimized by a numerical method. Two examples of nonlinear parameter estimation are given by equations used in Chapters 2 and 4. exp { (e/o ”)w / [w + (em ”)w 1 ) 0 _ 1 2 1 1 2 (a) 1 o w1 + (e/O1 )w2 £3 "7K1 1n "1 - 1n A1 + 1 1 (5) K21 + T - '1‘81 In eq 4, the dependent variable is 01, and w1 is the independent variable, with w - 1 - w . The parameter to be estimated is 0 In Q 2 1 1 ' eq 5, the dependent variable is al, and T is the independent variable. Three parameters to be estimated are: Al, the grouping £Vl*/K11, and the grouping (K21 - Tgl)' In deriving the results in Chapters 2 and 4, eq 2 was applied numerically to data used with eqs 4 and 5, allowing "best fit" values to be determined for the necessary parameters. 120 DEFICIENCIES OF LINEAR LEAST SQUARES TEST IN DETERMINING ERROR DISTRIBUTION One means of estimating the accuracy of a parameter fit is the calculation of a confidence interval for each parameter, or a confidence ellipsoid in parameter space. The Student's t statistic (Mendenhall, 1968) provides a means of generating a confidence interval for a linear least squares parameter, but is not applicable to the nonlinear case. One technique recommended for overcoming this problem is to linearize the equation using one term of a Taylor expansion. This technique is not useful if the domain and range of the measured variables fall partially outside the region in which the linearization is accurate. Confidence intervals estimated in this way may also be inaccurate if the parameters are not independent. The amount of information available to determine the ”goodness of fit" in parameter estimation may be limited in nonlinear cases. Taking this to be true, other, possibly simpler statistical procedures can be employed. As an example, consider the use of eq 5, with parameters estimated from data taken with the independent variable T in the range: 0 5,T S 100. Eq 5 is then to be used for prediction with values of T in the range: 140 5 T S 240. Extrapolation beyond the domain of the independent variable in the observed data is not always avoidable. A predominant consideration for the modeler is whether the distribution of error in the equation is a random function of the independent variable 121 T, or whether the equation has a tendency to systematically give inaccurate predictions. Distribution of error can be analyzed statistically by using linear regression to examine the correlation between the variables 6, the error, and T, the independent variable. The correlation coefficient for these variables is defined by eq 6. n n n n U2 T c - 2 T 2 e i i i-l ii-l i [1153‘ 'L. 1 ]m ['23: -[,_1 ] T” If the variables 6 and T are not correlated, meaning that e is not a function of T, the correlation coefficient, r, will be zero or near zero. When that is true, the null hypothesis that e is not related to T can be accepted, and systematic error in the equation used for fitting data, eq 5, is assumed not to exist over the domain of T values observed. Two problems arise in such an analysis. First, the test for correlation between two variables assumes both are normally distributed. If this is not true, eq 6 does not give an accurate test of correlation. Even if error is normally distributed, most physical data are taken at uniform intervals, resulting in a nonnormal distribution. 122 Second, nonlinear functions like eq 5, because of their curvature, have a tendency to exhibit an unusual pattern of systematic error in cases where systematic error is present. A typical example is overprediction of the dependent variable near either end of the domain of the independent variable, and underprediction in the middle of the domain of the independent variable. Since correlation examines the linear relationship between two variables, the effects will cancel and no correlation between c and T will be observed. Yet, systematic error is present despite the lack of correlation, and extrapolation in this case would be greatly in error. NONPARAMETRIC STATISTICAL TECHNIQUES It is possible to devise statistical tests which do not assume a particular distribution (e.g., normal) for the random variables. Such techniques are termed nonparametric or distribution-free. Many nonparametric techniques are similar to standard parametric techniques, but with the actual data values replaced by their rank statistics, i.e., their position among the data values when the data is ordered. Since the distribution of ranks is known (from 1 to N, where N is the number of data items), it is possible to determine the distribution of various statistics which are functions of ranks (Kendall and Stuart, 1961). Nonparametric tests are generally less powerful than parametric tests because less information is used. Information is lost when the actual 123 data values are replaced by their ranks. However, parametric tests are valid only when the distributions of the random variables being studied are the same as assumed by the test. Generally, this means normal distributions. If the distribution is not normal, parametric tests may be invalid, and even if valid, may become less powerful than nonparametric tests. Another advantage of nonparametric statistics arises in the calculation of the distribution of a test statistic. Without knowledge of the distribution of a test statistic, inferences regarding statistical hypotheses cannot be made. The distribution provides a basis for deciding that a particular value of the test statistic observed in the data would be unlikely to occur by chance. The known discrete distribution of rank statistics can make it possible to derive the distribution of statistics in the nonparametric case which would be difficult to derive for a continuous normal distribution, or which might have to be estimated, or derived under impractical assumptions. Two nonparametric statistical procedures relevant to this discussion are the runs test for randomness (Wald and Wolfowitz, 1940) and the rank correlation coefficient (Spearman, 1904). Both of these can be applied to the problem posed in the previous section: to determine whether the relationship between error, e, and an independent variable, T, indicates systematic error within the equation being used. 124 The runs test for randomness consists of ordering the individual errors, :1, in order of the corresponding T values. Each error is then i assigned a symbol, +, if it is positive and a symbol, -, if it is negative, producing a ordered sequence of the symbols, + and -. Each subsequence of successive symbols of the same type is termed a run. The underlying principle of the runs test is that a small number of runs indicates that similar e values occur for values of T near one another, while a large number of runs indicates that 6 values have little relationship to T values. Hence, the former situation describes a pattern of correlation between 6 and T, or systematic deviation in the predictions of the model when compared to observation. For the case where there are n occurences of the symbol + and m occurences of the symbol -, the probability of an even number of runs, 2k, or an odd number of runs, 2k+l, is given by eqs 7 and 8 for a random (null) distribution of c. A table of the distribution of the number of runs, R, as a function of m and n can be compiled using these equations. 2 n-l m-l k-l k-l P(R - 2k) - (7) n+m n n-l m-l + n-l m-l k k-l k-l k P(R - 2k+l) - (3) [“2“] For large values of m and n, a normal approximation, 2, to the distibution 125 of the random variable R can be used. This approximation is given by eqs 9, 10, and 11. The distribution of 2 will be approximately N(0,1) (normal with zero mean and unit variance) so that a table of the normal distribution can be used to determine the probability that Z 5 2. Figure 1 illustrates the approximation for typical values of m and n. The additional term 0.5 arises in the numerator of eq 11 because a continuous distribution of z is being used to approximate a discrete distribution of R. The best approximation to the probability of a given discrete value of R is given by the probability that the normal 2 calculated by eq 11 lies between Z(R - 0.5) and Z(R + 0.5). 2mn - E(R) - “R +1 (9) m-i-n 2mn(2mn- m - n) Var(R) - 2 (10) (m + n) (m + n - l) R + 0.5 - ”R z - (11) [Var11/2 To apply the runs test for randomness, the number of runs R is counted, and the probability of observing R runs or fewer is calculated from eqs 7 and 8, or eqs 9, 10, and 11. If this probability is less than some small number, a, the hypothesis that the error distribution is random with respect to T is rejected with probability l-a. The strength of the runs test lies in its flexibility, ease of 126 Figure 1. Normal Approximation to the Runs Statistic R. normal distribution; points denote distribution of R.) 25 (Line denotes 127 application, and lack of assumptions about the underlying distributions of e and T. However, the test is not very powerful, because it uses only a small amount of the available information: namely, whether each individual e value is positive or negative. The magnitude of deviation from zero is ignored. The rank correlation coefficient proposed by Spearman (1904), rs, is analogous to the correlation coefficient used in linear regression. The difference is that the ranks of the data are correlated (as integers from 1 to n), rather than the actual data values. The Spearman rank correlation for the problem posed here would consist of replacing each error value, 61, by its rank when the 61 values were ordered, and i by its rank when the Ti values were ordered. Once this is done, the rank correlation coefficient is replacing each independent variable T computed by eqs 12 and 13, which are a simplified case of eq 6 when the variables being correlated each contain an arrangement of the integers from 1 to n. di - rank(ei) - rank(Ti) (12) n 62d12 i-l r - 1 - -——--——- (13) s n(n2 - 1) To apply the rank correlation, the values of c and T are ranked, rank differences for each data point are calculated by eq 12, and the rank correlation coefficient is calculated by eq 13. Critical values of rs 128 are available in statistics references (Mendenhall and Scheaffer, 1973; Bradley, 1967) as a function of these parameters: n, the number of data points, and a, the probability that a value as large or larger than rs would be observed in correlating two random distributions of ranks. If the absolute value of rs exceeds the critical value for a particular a, the hypothesis that the error distribution is random with respect to t is rejected with probability 1-2a. (Since either positive or negative correlation indicates systematic error, the test described is two-sided, rejecting the randomness hypothesis if rs is either too large or too small.) Like the runs test, the rank correlation is flexible, easily applied, and makes no assumptions about the underlying distributions of e and T. The rank correlation is generally more powerful than the runs test, although not quite as powerful as the ordinary correlation of linear regression, eq 6, when the underlying distributions are normal. This is because the information about the actual deviations of the 6 values from one another is not used; only the relative ranks are. The rank correlation satisfies one of the objections to the correlation coefficient from ordinary linear regression: the possibly erroneous assumption of normal distribution of the variables being correlated. However, the second problem discussed earlier still exists. If the equation used to model the data is nonlinear, predicted values may systematically overshoot and undershoot the actual observations over 129 ranges of the independent variable. Like the correlation coefficient from linear regression of the actual observations, the rank correlation coefficient will tend to cancel these effects, producing a ”false negative" conclusion of no correlation. For this reason, it may also be a poor statistical test of the accuracy of extrapolation. A PROPOSED NONPARAMETRIC STATISTIC FOR DETERMINATION OF SYSTEMATIC ERROR The strengths of both the runs test and rank correlation test lie in their nonparametric, distribution-free nature. This allows application to any data, regardless of the form assumed for its underlying distribution or even the knowledge of its distribution. Furthermore, the distribution of the runs statistic, R, and the rank correlation statistic, rs, and their critical values are relatively easy to calculate, because the distributions are discrete and involve only functions of positive integers. Besides their general character as nonparametric procedures, the runs test and rank correlation have strengths and weaknesses that complement one another. The runs test lacks power because it reduces each data value to a simple binary value, indidated above by the symbols, + and -. Yet it is flexible in that it measures the deviation from randomness in gradations from complete monotonicity (e.g., all + symbols precede all - symbols; there are two runs), through randomness, to complete periodicity (the sequence of + and - symbols alternate; there are n 130 runs). The rank correlation retains a considerable amount of information contained in the original data within the ranks. However, it detects only a monotonic deviation from randomness. A new statistical procedure, referred to as the Sum Square Rank Difference (SSRD), combines the strong points of runs test and rank correlation. The procedure begins by ordering the data values of the independent variable, T, in either increasing or decreasing order, just as the runs test did. The ranks of the e values corresponding to each 1 T1 value are used in calculating the statistic, Rd, by eq 14. n-l 2 Rd -i§1[Rank(ei+1) - Rank(ei)] (14) If the error values are similar at neighboring values of the independent variable, the difference in ranks will be small and the statistic, Rd’ will be relatively small. If the value of Rd calculated from data is so small as to be unlikely to have occurred by chance, this would indicate that systematic error is present within the equation when fit to this data. If the distribution of error is random, Rd will tend to take on larger values, and the null hypothesis of no association between error and the independent variable could be accepted. In order to determine critical values of Rd, its distribution must be derived. For small values of n, this can be done by exhaustive listing of all possible orderings of the ranks (integers from 1 to n) and 131 calculation of Rd for each case. For larger values of n, numerical approximation of the distribution can be made by Monte Carlo techniques. If the distribution of Rd obeys the Central Limit Theorem (assumed here without proof), a normal approximation to the distribution of Rd (analogous to eq 11 for the runs test) can be used. This is given by eqs 15 to 17. R - E(R ) Z - d I/Z (15) [Var(Rd)] n(n - l)(n + 1) E(Rd) - (16) 6 n(n - 2)(n + l)(5n2 - 2n - 9) Var(Rd) - (17) 180 The expected value (mean) and variance formulas were derived from exhaustive listing for the cases n - 2 up to n - 8. Since the largest value that a single term in the summation of eq 14 can have is (n - 1)2, and since there are (n - 1) terms, R is bounded above by (n - 1)3. d Therefore, the distribution mean, E(Rd), can be represented as a function of n which is no larger than a polynomial of degree three. The result of fitting a cubic polynomial with unknown coefficients to the mean derived from exhaustive enumeration of all cases from n - 2 to n - S was eq 16. Similar arguments apply to the variance: since it results from the difference of the square of the expected value of R d and the expected value of Rdz, degree six or less. Eq 17 resulted from fitting a sixth degree it can be represented as a polynomial of 132 polynomial to calculated distribution variances for n - 2 to n - 8. Since the SSRD statistic, R , is nonparametric, it is valid even when the underlying distributions of the variables are nonnormal. The use of ranks retains more of the information contained in the actual observations than the binary value (+ or -) used by the runs test. At the same time, the comparison of neighboring values allows systematic patterns of similarity to be detected when the rank correlation would find no overall linear correlation. For these reasons, the SSRD statistic appears to be a useful procedure for determining whether a nonlinear parameter fit exhibits systematic error. AN EXAMPLE CALCULATION FOR DETERMINATION OF SYSTEMATIC ERROR The data in Table 1 represent a typical example of data fitting using eq 5. The dependent variable, n1, is solvent viscosity as a function of the independent variable, T1, which is temperature. The predicted value of the dependent variable is labeled nipred, and the relative error in prediction is labeled :1. (Since eq 5 actually predicts the logarithm of "1, the :1 values shown are derived from subtracting logarithms, which makes them the logarithms of relative errors in the dependent variable.) Figure 2 shows the observed data and predictions, and a visual inspection seems to indicate the fit is good. The relative error is plotted versus temperature in Figure 3, and the plot does not show a regular linear pattern of systematic error; there appears to be Table l . 133 Acetone Viscosity Data and Predictions of Equation 5. The values of the parameters are: A1 - -3.603 evl /x11 - -642.7 T1, °c "1' c? qipred, cP -92 50 2.1480 2.1100 -80.00 1.4870 1.5030 -59 60 0.9320 0.9557 -42.50 0.6950 0.7026 -30.00 0.5750 0.5792 -20.90 0.5100 0.5102 -13.00 0.4700 0.4608 -10 00 0.4500 0.4441 0.00 0.3990 0.3954 7.86 0.3638 0.3633 11.72 0.3495 0.3492 15.00 0.3370 0.3379 15.24 0.3376 0.3371 19.02 0.3258 0.3250 23.01 0.3131 0.3130 25.00 0.3160 0.3073 27.22 0.3007 0.3012 30.00 0.2950 0.2938 32.43 0.2863 0.2877 36.00 0.2772 0.2790 40.04 0.2675 0.2698 41.00 0.2800 0.2677 44.12 0.2584 0.2611 47.62 0.2503 0.2540 52.20 0.2405 0.2453 53.86 0.2377 0.2423 References: Weast, 1979; Washburn, 1929. (K21 ‘1 .0179 .0107 .0251 .0109 .0072 .0004 .0198 .0131 .0090 .0014 .0010 .0027 .0015 .0025 .0004 .0279 .0016 .0039 .0048 .0066 .0087 .0448 .0105 .0148 .0198 .0191 - T 81) 240.3 Table 2. Calculation of Linear Correlation Coefficient, r. n - 26 2 T1 - 172.84 2 £1 - -0.0003 2 T e - 0.3069 2 T 2 - 41205.9 2 e 2 - 0.0059309 1 1 1 1 26 - 0 3069 - 172.84 . (-0.0003) r- (26 - 41205.9 - (172.84)2)1/2 (26 - 0 0059309 - (-0.0003)2)1/2 r - 0.0005 Viscosity. cP 134 1L5 in l 1% nun---- -100 -50 0 50 100 Temperature, C Figure 2. Viscosity of Pure Acetone as a Function of Temperature. Relative error 135 .04 .029 J----------------- D -.02 « 0 odin-0------COCOC-nnn-n-0-- -100 -50 Temperature, C Figure 3. Error Distribution as a Function of Temperature. 100 136 considerable scatter. The T1 values given in Table 1 would probably not have come from a normal distribution. Such a contention could be demonstrated by a statistical procedure, such as the Kolmogorov-Smirnov goodness of fit test (Kolmogorov, 1941; Smirnov, 1948), which allows an empirical distribution (like the T values) to be compared to a hypothesized i distribution function (the normal distribution). That type of demonstration will not be pursued here; mere observation of the values will be used as evidence against an underlying normal distribution. Applying the linear correlation coefficient, eq 6, using the summations of the data given in Table 2, results in a correlation coefficient of r - 0.0005, indicating nearly perfect lagk of correlation between 6 and T. The chance of observing a correlation coefficient with magnitude at least this large in a chance arrangement of 26 normally distributed pairs of values would be 99.8 percent! The correlation coefficient gives us no reason to suspect the error in eq 5 is systematic. This example shows that the linear regression correlation coefficient can be a poor statistical test for systematic error in a nonlinear parameter fit. The details of the calculation of the runs test are given in Table 3. Replacement of the data with the + and - symbols gives 10 runs, with 14 positive data values and 12 negative data values. Since m and n are 137 both larger than 10, use of the limiting normal distribution is valid. The mean and variance of this distribution are calculated according to eqs 9 and 10, then eq 11 is applied. The resulting normal variable, 2 g -l.379, would occur by chance in only 8.5 percent of randomly distributed pairs of values. The runs test allows the rejection of the null hypothesis (no correlation between 6 and T) at the 90 percent confidence level, although not at the 95 percent level. This rejection would be evidence for the presence of systematic error. - + + + + + ----- + - - - - + - + + + - + + + + l 2 3 4 S 6 7 8 9 10 R - 10 runs n - 14 (+) m - 12 (-) 2 - 12 - 14 "R - E(R) - + 1 - 13.923 12 + 14 2 - 12 - 14 - (2 - 12 - 14 - 12 - l4) Var(R) - - 6.163 (12 + 14)2(12 + 14 - 1) P(R S. 10) ' P(2 _<. 2) (10.5 - 13 923) z - - -l.379 (6.163)1/2 p(Z g -1.379) - 0.085 The data in Table 4 are used for calculation of both the rank correlation coefficient and the SSRD statistic. These data were produced by replacing the observed values in Table 1 by their ranks within the 26 data points. The third and fourth columns contain quantities used in the statistic calculations. 138 Table 4. Ranked Temperature and Error Data. Rank(Ti) Rank(ci) diz, eq 12 [Rank(ei+1)-Rank(ei)]2, eq 14 l 4 9 289 2 21 361 25 3 26 529 16 4 22 324 16 5 18 169 25 6 13 49 100 7 3 16 4 8 5 9 1 9 6 9 16 10 10 0 1 11 11 0 16 12 15 9 36 13 9 16 1 14 8 36 16 15 12 9 100 16 2 196 144 17 14 9 49 18 7 121 81 19 16 9 1 20 17 9 4 21 19 4 324 22 1 . 441 361 23 20 9 9 24 23 1 4 25 25 0 1 26 24 4 26 2 25 2 2 d1 - 2348 Z [Rank(ei+1)-Rank(ei)] - 1640 i-l i-l 139 The data in Table 4 were used to determine the rank correlation coefficient in Table 5, by use of eq 13. The value, rs - 0.1973, is not significant at the 80 percent level, i.e., a value of this magnitude would arise more than 20 percent of the time from a randomly chosen sample. The hypothesis that the variation of c with T in the data is random could not be rejected. The rank correlation seems to show considerably more relationship between a and T than the linear correlation coefficient based on the original data. Even though the data in Table 4 contain less information than the data in Table 1 from which they were derived, the fact that the T1 observations are not normally distributed makes the linear correlation coefficient an inappropriate statistical test for this data. The rank correlation coefficient assumes no form for the distribution of the original data: hence, it is appropriate and in fact detects some correlation, although not at a statistically significant level. Table 5. Calculation of Rank Correlation Coefficient, rs. 6 - 2348 rs - 1 - 2 - 0.1973 26(26 - 1) p(lrsl 2 0 259) - 0.20 p(|rs| 2 0.329) - 0.10 Calculation of the SSRD statistic using eqs 15 to 17 is shown in Table 6. The required summation in eq 14 which defines R is already given in d Table 4. When the mean and variance are calculated and substituted into the normal approximation formula, the resulting normal variable, 140 Z 5 -2.305, would occur only 1.06 percent of the time by chance if the pairs were random. The null hypothesis can be rejected at virtually the 99 percent confidence level when the SSRD statistic is used. The test gives strong evidence to what may not be apparent to a casual viewer of the data: that a systematic, nonlinear pattern of overprediction and underprediction is present. Table 6. Calculation of Sum Square Rank Difference, R d. 25 2 Rd ' 2 [Rank(€i+1)-Rank(ei)] - 1640 1-1 26 ' (25 ' 1) ' (26 + 1) E(Rd) ' - 2925 6 1/2 26 ' <26 ~ 2><26 + 1)o~a:o~01c>uao:a~o~ .489 .488 .485 .480 .478 .475 .471 .464 .459 .451 .441 .415 .400 .393 .382 F‘P‘h‘h‘h‘P‘F‘F‘P‘P‘P‘h‘h‘h‘h‘ a>c>rahaa\c\u:uaoxuapoaic~uwu: 41> 0‘ Benzene-Polyethylene oxide at 70 C By correlating activity at finite conc Infinite dilution wt frac activity coefficient was Flory-Huggins chi parameter was Wt Frac Solvent 0.067 0.099 0.139 0.193 0.261 0.388 Exptl .231 .810 .474 .093 .690 .153 NNWWWL‘ Avg pct error 0.3345 kl—‘GONMNQO‘UQWUQ‘O@ 0.061 F‘P‘F‘F‘P‘P‘F‘P‘P‘h‘h‘P‘P‘P‘F‘ .207 .204 .200 .195 .195 .192 .190 .187 .186 .183 .181 .177 .175 .174 .172 ~14. ~14. ~15. ~12. ~17. ~14. ~13. ~14. ~14. ~13. ~12. ~15. ~14. ~15. ~17. 14.7 Activity Coefficients and Percent Error ASOGVSP hbhauauauac~ .253 .936 .591 .192 .779 .207 2.7 wwwwwo U'leJ-‘WUI Flory-Huggins UNIFAC-EV .291 .982 .642 .244 .826 .239 NMUU’W“ 4.1 PU-l-‘J-‘l-‘i—l coconut-5 NNNWWU .524 .317 .085 .809 .510 .070 ~16. ~12. ~11. ~9. ~6. ~3. 10.1 .4937 woooouaswi-‘chnwmuuw .0642 WNNNON 175 Table B-1 (cont'd.). Benzene-Polyethylene oxide at 70 C (second run) By correlating activity at finite conc 0.050 Infinite dilution wt frac activity coefficient was 4. Flory-Huggins chi parameter was 0.2403 Wt Frac Activity Coefficients and Percent Error Solvent Exptl ASOGVSP Flory-Huggins UNIFAC-EV 0.089 3.743 3.765 0 6 3.785 1 1 3.379 ~9. 0.142 3.344 3.368 0.7 3.392 1 5 3.066 ~8. 0.201 2.996 3.001 0.2 3.026 1.0 2.771 ~7. 0.265 2.702 2.663 ~1 4 2.686 -0 6 2.494 ~7. Avg pct error 0.7 1.0 8.3 Benzene-Polyethylene oxide at 75.1 C By correlating activity at finite conc 0.052 Infinite dilution wt frac activity coefficient was 4 Flory-Huggins chi parameter was 0.2102 Wt Frac Activity Coefficients and Percent Error Solvent Exptl ASOGVSP Flory-Huggins UNIFAC-EV 0.081 3.755 3.749 ~0.2 3.764 0.2 3.416 ~9. 0.108 3.561 3.543 ~0.5 3.559 ~0.0 3.247 ~8. 0.145 3.332 3.289 ~1.3 3.307 ~0.7 3.038 ~8. Avg pct error 0.7 0.3 8.9 Benzene-Polyethylene oxide at 88.1 C By correlating activity at finite conc 0.026 Infinite dilution wt frac activity coefficient was 4 Flory-Huggins chi parameter was 0.2147 Wt Frac Activity Coefficients and Percent Error Solvent Exptl ASOGVSP Flory-Huggins UNIFAC-FV 0.050 3.986 4.021 0.9 4.032 1.2 3.563 ~10. 0.066 3.850 3.879 0.7 3.893 1.1 3.454 ~10. 0.090 3.667 3.688 0.6 3.704 1.0 3.305 ~9. Avg pct error 0.7 1.1 10.3 6088 \IUIWN .4790 mono .5008 \OWO i 3 ' i E t g E E e 5 E 3 § :5 .' Table B~1 (cont'd.). 176 Benzene-Polyethylene oxide at 102 C By correlating activity at finite conc 0.020 Infinite dilution wt frac activity coefficient was 4 Flory-Huggins chi parameter was 0.2209 Wt Frac Activity Coefficients and Percent Error Solvent Exptl ASOGVSP Flory-Huggins UNIFAC-FV 0.021 4.280 4.292 0.3 4.298 0.4 3.662 ~14. 0.025 4.272 4.262 ~0.2 4.269 ~0.0 3.641 ~14. 0.029 4.225 4.222 ~0.0 4.230 0.1 3.613 ~14. 0.036 4.148 4.150 0.0 4.160 0.3 3.562 ~14. 0.044 4.070 4.084 0.3 4.094 0.6 3.515 ~13. 0.047 4.042 4.051 0.2 4.063 0.5 3.492 ~13. 0.058 3.950 3.959 0.2 3.972 0.6 3.426 ~13. 0.077 3.713 3.802 2.4 3.818 2.8 3.314 ~10. 0.091 3.520 3.685 4.7 3.703 5.2 3.230 ~8. 0.118 3.169 3.491 10.2 3.510 10.8 3.087 ~2. Avg pct error 1.9 2.1 12.0 Benzene-Polyethylene oxide at 125.4 C By correlating activity at finite conc 0.010 Infinite dilution wt frac activity coefficient was 4 Flory-Huggins chi parameter was 0.1810 Wt Frac Solvent 0.017 0.023 0.032 Exptl 4.141 4.080 3.993 Avg pct error Activity Coefficients and Percent Error ASOGVSP 4.187 4.131 4.054 1. Flory-Huggins UNIFAC-EV 1.1 4.190 1.2 3.611 ~12. 1.2 4.136 1.4 3.571 ~12. 1.5 4.059 1.7 3.515 ~12. 3 1.4 12.4 .5132 GNN‘NO‘O‘HU‘Q? .3520 OU'OG 177 Table B-1 (cont'd.). Benzene-Polyethylene oxide at 125.7 C By correlating activity at finite conc 0.011 Infinite dilution wt frac activity coefficient was 4 Flory-Huggins chi parameter was 0.1583 Wt Frac Activity Coefficients and Percent Error Solvent Exptl ASOGVSP Flory-Huggins UNIFAC-FV 0.017 4.073 4.101 0.7 4.103 0.7 3.613 ~11. 0.024 3.945 4.038 2.4 4.041 2.4 3.566 ~9. 0.033 3.874 3.969 2.4 3.973 2.5 3.513 ~9. Avg pct error 1.8 1.9 10.1 Benzene-Polyethylene oxide at 150.4 C By correlating activity at finite conc 0.007 Infinite dilution wt frac activity coefficient was 4 Flory-Huggins chi parameter was 0.1876 Wt Frac Activity Coefficients and Percent Error Solvent Exptl ASOGVSP Flory-Huggins UNIFAC-FV 0.011 4.202 4.280 1.9 4.282 1.9 3.569 ~15. 0.016 4.061 4.233 4.2 4.236 4.3 3.538 -12. 0.022 3.956 4.176 5.6 5.7 3.501 ~11. 4.180 Avg pct error 3.9 4.0 13.1 .2540 wow» .3883 APPENDIX C. Results of Thermodynamic Modeling Using a Best Fit of All Data The results given in this appendix were produced as output by a computer program using the data in Appendix A. This output was used as results in Chapter 2 of the dissertation. The program itself and instructions for its execution are given in Appendix E. Experimental data for a given polymer-solvent system and given temperature were fit to the VSP model assuming no residual interaction, the Flory-Huggins model, the VSP model assuming a Flory-Huggins type residual interaction term, and the VSP model assuming an interaction term given by Analytical Solution of Groups (ASOG). For each data set, a heading is given, followed by the values of adjustable parameters determined by a least squares best fit criterion. (The ASOG-VSP enthalpic coefficient is determined a priori from the ASOG interaction parameter tables, not from fitting to the data, but is included in this section for comparison.) The next section contains a comparison of experimental and predicted weight fraction solvent activity coefficients for each concentration in the data set. At the bottom of each column, the root mean square error for the data set is given. The following section gives the results of nonparametric statistical tests of the randomness of the error in each model 178 179 prediction compared to experiment, as discussed in Chapter 5 of the dissertation. In cases where the Flory-Huggins or FH-VSP models predict phase separation, the concentration at which it is predicted to occur is given. (The VSP model using ASOG residual term is also capable of predicting phase separation, but such prediction was not included in this table.) No ASOG interaction parameters are available for the ether oxygen (-O~) functional group with the aromatic hydrocarbon (ArCH) functional group. For this reason, calculations with the VSP model using ASOG residual term could not be made for benzene-polyethylene oxide. There are values given in the table for this system, but the ASOG-VSP results are invalid, as the computer program generating the table set the interaction parameters to zero for these functional groups. The results for the other three models are valid for benzene-polyethylene oxide, since those models do not use the ASOG parameter tables. 180 Table C-l. Results Using Thermodynamic Data Fit to the Entire Data Set. Toluene-Polystyrene at 25 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 4.9495 Flory-Huggins chi parameter: 0.3394 FH-VSP inf diln parameters: wt frac act coeff 4.5644 enth coeff 1.7256 ASOG-VSP inf diln parameters: wt frac act coeff 4.9391 enth coeff 1.0064 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG—VSP 0.111 3.631 3.769 3.7 3.739 3.0 3.673 1.2 3.767 3.7 0.156 3.353 3.411 1.7 3.403 1.5 3.375 0.7 3.410 1.7 0.191 3.199 3.168 ~1.0 3.171 ~0.9 3.164 ~1.1 3.168 ~1.0 0.236 2.983 2.893 ~3.1 2.905 ~2.6 2.917 ~2.2 2.894 -3.0 0.273 2.711 2.694 ~0.6 2.710 ~0.0 2.732 0.8 2.695 ~0.6 0.304 2.602 2.543 ~2.3 2.562 ~1.6 2.589 ~0.5 2.545 ~2.2 0.380 2.279 2.226 ~2.3 2.245 ~1.5 2.278 ~0.0 2.228 ~2.3 0.476 1.929 1.908 ~1.1 1.924 ~0.3 1.954 1.3 1.909 ~1.0 0.599 1.618 1.597 ~1.3 1.607 ~0.7 1.628 0.6 1.598 ~1.2 0.744 1.340 1.325 ~1.1 1.329 ~0.8 1.338 ~0.1 1.325 ~1.1 0.918 1.089 1.088 ~0.1 1.088 ~0.1 1.089 0.0 1.088 ~0.1 Standard pct err 2.1 1.6 1.1 2 0 Analysis of model error randomness Sum sqr rank difference test: mean - 220.00 sd - 61.55 Test statistic 158 187 210 158 Normal (2) ~1.007 ~0.536 ~0.162 ~1.007 Reject level 0.843114 0.704061 0.564541 0.843114 Sum abs rank difference test: mean - 40.00 sd - 6.66 Test statistic 34 35 40 34 Normal (2) -0.900 ~0.750 0.000 ~0.900 Reject level 0.816062 0.773481 0.500000 0.816062 Phase separation behavior prediction FH-VSP model: wt frac - 0.919 181 Table C-l (cont'd.). Toluene-Polystyrene at 60 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 4.8456 Flory-Huggins chi parameter: 0.2938 FH-VSP inf diln parameters: wt frac act coeff 4.6321 ASOG-VSP inf diln parameters: wt frac act coeff 4.8393 enth coeff 1.5868 enth coeff 1.0052 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.102 3.755 3.792 1.0 3.777 0.6 3.750 ~0.1 3.791 1.0 0.179 3.218 3.213 ~0.2 3.220 0.1 3.227 0.3 3.213 ~0.1 0.261 2.778 2.734 ~1.6 2.750 ~1.0 2.773 ~0.2 2.735 ~l.5 Standard pct err 1.3 0.8 0.4 1.3 Analysis of model error randomness Sum sqr rank difference test: mean - 4.00 sd - 1.41 Test statistic 2 2 5 2 Normal (2) ~l.4l4 ~1.414 0.707 ~1.4l4 Reject level 0.921358 0.921358 0.760243 0.921358 Sum abs rank difference test: mean - 2.67 sd - 0.47 Test statistic 2 2 3 2 Normal (2) ~l.4l4 ~1.4l4 0.707 ~1.414 Reject level 0.921358 0.921358 0.760243 0.921358 182 Table C~1 (cont'd.). Toluene-Polystyrene at 80 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 5.1655 Flory-Huggins chi parameter: 0.3195 FH-VSP inf diln parameters: wt frac act coeff 4.7192 enth coeff 1.5798 ASOG-VSP inf diln parameters: wt frac act coeff 5.1523 enth coeff 1.0045 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.246 2.870 2.884 0.5 2.880 0.3 2.870 ~0.0 2.884 0.5 0.458 1.996 1.975 ~1.0 1.984 ~0.6 1.997 0.1 1.976 ~l.0 0.671 1.466 1.454 ~0.8 1.459 ~0.5 1.465 ~0.1 1.455 ~0.8 Standard pct err 1.0 0.6 0.1 1.0 Analysis of model error randomness Sum sqr rank difference test: mean - 4.00 sd - 1.41 Test statistic 5 5 5 5 Normal (Z) 0.707 0.707 0.707 0.707 Reject level 0.760243 0.760243 0.760243 0.760243 Sum abs rank difference test: mean - 2.67 sd - 0.47 Test statistic 3 3 3 3 Normal (2) 0.707 0.707 0.707 0.707 Reject level 0.760243 0.760243 0.760243 0.760243 183 Table C-l (cont'd.). Methyl ethyl ketone-Polystyrene at 25 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 8.9345 Flory-Huggins chi parameter: 0.7101 FH-VSP inf diln parameters: wt frac act coeff 8.2319 enth coeff 1.6469 ASOG-VSP inf diln parameters: wt frac act coeff 7.7699 enth coeff 1.8783 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.091 5.681 5.774 1.6 5.515 ~3.0 5.674 ~0.1 5.575 ~l.9 0.215 3.758 3.730 ~0.8 3.817 1.6 3.777 0.5 3.805 1.2 0.279 3.161 3.116 ~1.4 3.230 2.2 3.169 0 2 3.208 1.5 0.298 3.040 2.968 ~2.4 3.082 1.4 3.019 -0 7 3.059 0.6 Standard pct err 1.9 2.4 0.6 1.6 Analysis of model error randomness Sum sqr rank difference test: mean - 10.00 sd - 3.74 Test statistic 3 9 9 9 Normal (2) ~1.87l ~0.267 ~0.267 ~0.267 Reject level 0.969310 0.605367 0.605367 0.605367 Sum abs rank difference test: mean - 5.00 sd - 1.00 Test statistic 3 5 5 5 Normal (2) ~2.000 0.000 0.000 0.000 Reject level 0.977241 0.500000 0.500000 0.500000 Phase separation behavior prediction Flory-Huggins model: wt frac - 0.636 184 Table C-l (cont'd.). Benzene-Polyisobutylene at 25 C Results of least squares fit: VSP inf diln wt frac activity coefficient: Flory-Huggins chi parameter: 0.9213 FH-VSP inf diln parameters: wt frac act coeff 8.1759 ASOG-VSP inf diln parameters: wt frac act coeff 7.3466 8.7866 enth coeff 1.7336 enth coeff 1.7073 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.044 6.842 7.039 2.8 6.291 ~8.4 6.809 -0.5 6.397 ~6.7 0.063 6.409 6.435 0.4 5.942 ~7.6 6.305 ~1.6 6.023 ~6.2 0.095 5.468 5.637 3.1 5.437 ~0.6 5.612 2.6 5.486 0.3 0.150 4.608 4.575 ~0.7 4.665 1.2 4.636 0.6 4.678 1.5 0.152 4.636 4.552 ~1.8 4.647 0.2 4.615 -0.5 4.659 0.5 0.184 4.127 4.083 ~1.1 4.262 3.2 4.164 0.9 4.262 3.2 0.245 3.484 3.401 ~2.4 3.647 4.6 3.489 0.1 3.634 4.2 0.254 3.452 3.323 ~3.8 3.572 3.4 3.410 ~1.2 3.558 3.0 0.297 3.070 2.966 ~3.5 3.217 4.7 3.045 ~0.8 3.199 4.1 0.321 2.873 2.795 ~2.7 3.040 5.7 2.869 ~0.1 3.021 5.0 0.373 2.541 2.483 ~2.3 2.704 6.2 2.545 0.2 2.685 5.5 Standard pct err 2.6 5.2 1.2 4.4 Analysis of model error randomness Sum sqr rank difference test: mean - 220.00 sd - 61.55 Test statistic 43 22 202 38 Normal (2) ~2.876 ~3.217 ~0.292 ~2.957 Reject level 0.997978 0.999349 .615025 .998441 Sum abs rank difference test: mean - 40.00 sd 6.66 Test statistic 19 14 38 18 Normal (2) ~3.152 -3.902 ~0.300 ~3.302 Reject level 0.999184 0.999952 .617967 .999517 Phase separation behavior prediction Flory-Huggins model: wt frac - 0.531 FH-VSP model: wt frac - 0.852 185 Table C~l (cont'd.). Benzene-Polyisobutylene at 10 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 10.6574 Flory-Huggins chi parameter: 0.8446 FH-VSP inf diln parameters: wt frac act coeff 7.9552 enth coeff 1.9196 ASOG-VSP inf diln parameters: wt frac act coeff 6.7035 enth coeff 1.8245 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.225 3.721 3.772 1.3 3.646 ~2.0 3.727 0.1 3.661 ~1.6 0.357 2.691 2.632 ~2.2 2.714 0.9 2.674 ~0.6 2.711 0.8 0.454 2.159 2.130 ~1.4 2.224 3.0 2.170 0.5 2.218 2.7 Standard pct err 2.1 2.6 0.8 2.3 Analysis of model error randomness Sum sqr rank difference test: mean - 4.00 sd - 1.41 Test statistic 5 2 5 2 Normal (2) 0.707 ~1.414 0.707 ~1.4l4 Reject level 0.760243 0.921358 0.760243 0.921358 Sum abs rank difference test: mean - 2.67 sd - 0.47 Test statistic 3 2 3 2 Normal (2) 0.707 ~1.4l4 0.707 ~1.414 Reject level 0.760243 0.921358 0.760243 0.921358 Phase separation behavior prediction Flory-Huggins model: wt frac - 0.584 FH-VSP model: wt frac - 0.683 186 Table C~1 (cont'd.). Cyclohexane~Polyisobuty1ene at 25 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 4.9719 Flory-Huggins chi parameter: 0.3891 FH-VSP inf diln parameters: wt frac act coeff 4.8958 enth coeff 1.2541 ASOG-VSP inf diln parameters: wt frac act coeff 4.9417 enth coeff 1.0595 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.128 3.609 3.636 0.8 3.597 ~0.3 3.624 0.4 3.632 0.6 0.165 3.350 3.352 0.1 3.336 ~0.4 3.347 ~0.1 3.350 0.0 0.188 3.242 3.193 ~1.5 3.187 ~1.7 3.192 -1.6 3.192 ~l.5 0.235 2.953 2.905 ~1.7 2.914 ~1.3 2.908 ~1.5 2.906 ~1.6 0.281 2.523 2.660 5.3 2.678 5.9 2.666 5.5 2.662 5.4 0.303 2.589 2.553 ~1.4 2.573 ~0.6 2.560 ~1.1 2.556 ~1.3 0.400 2.187 2.154 ~l.5 2.176 ~0.5 2.160 ~1.2 2.156 ~1.4 0.569 1.688 1.667 ~1.3 1.681 ~0.4 1.671 ~1.0 1.668 ~1.2 Standard pct err 2.4 2.4 2.6 2 4 Analysis of model error randomness Sum sqr rank difference test: mean - 84.00 sd - 26.61 Test statistic 88 87 84 88 Normal (2) 0.150 0.113 0.000 0.150 Reject level 0.559757 0.544895 0.500000 0.559757 Sum abs rank difference test: mean - 21.00 sd - 3.87 Test statistic 20 21 20 20 Normal (2) -0.258 0.000 -0.258 -0.258 Reject level 0.601875 0.500000 0.601875 0.601875 187 Table C-l (cont'd.). N-pentane-Polyisobutylene at 25 C Results of least squares fit: VSP inf diln wt frac activity coefficient: Flory-Huggins chi parameter: 0.6795 FH-VSP inf diln parameters: wt frac act coeff 8.3268 wt frac act coeff 8.7630 8.7630 enth coeff 1.6386 enth coeff 1.0000 ASOG-VSP inf diln parameters: Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.029 7.423 7.556 1.8 7.087 ~4.6 7.336 ~1.2 7.556 1.8 0.073 6.057 6.161 1.7 6.030 ~0.5 6.120 1.0 6.161 1.7 0.133 4.849 4.863 0.3 4.933 1.7 4.917 1.4 4.863 0.3 0.185 4.153 4.069 ~2.1 4.200 1.1 4.145 ~0.2 4.069 ~2.1 0.212 3.820 3.744 -2.0 3.886 1.7 3.822 0.1 3.744 ~2.0 0.267 3.303 3.208 ~2.9 3.349 1.4 3.280 ~0.7 3.208 ~2.9 0.322 2.872 2.786 -3.1 2.910 1.3 2.846 ~0.9 2.786 ~3.1 0.328 2.808 2.750 ~2.1 2.872 2.3 2.809 0.0 2.750 ~2.1 0.584 1.694 1.680 ~0.8 1.718 1.4 1.696 0.1 1.680 ~0.8 Standard pct err 2.2 2.2 0.9 2.2 Analysis of model error randomness Sum sqr rank difference test: mean - 120.00 sd - 36.37 Test statistic 35 117 102 35 Normal (2) ~2.337 -0.082 ~0.495 ~2.337 Reject level 0.990267 0.532877 0.689643 0.990267 Sum abs rank difference test: mean - 26.67 sd - 4.75 Test statistic 15 27 24 15 Normal (2) ~2.457 0.070 ~0.561 ~2.457 Reject level 0.992975 0.527987 0.712757 0.992975 Phase separation behavior prediction Flory-Huggins model: wt frac - 0.654 188 Table C~1 (cont'd.). Triisopropylbenzene-Polystyrene at 165 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 12.3352 Flory-Huggins chi parameter: 1.0003 FH-VSP inf diln parameters: wt frac act coeff 12.2476 enth coeff 1.2180 ASOG-VSP inf diln parameters: wt frac act coeff 12.0454 enth coeff 1.0724 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.030 9.936 9.875 ~0.6 9.377 ~5.8 9.853 ~0.8 9.793 ~1.4 0.066 7.625 7.864 3.1 7.995 4.7 7.873 3.2 7.893 3.5 0.086 7.191 7.004 ~2.6 7.324 1.8 7.018 ~2.4 7.062 ~1.8 Standard pct err 2.9 5.4 4.1 2.9 Analysis of model error randomness Sum sqr rank difference test: mean - 4.00 sd - 1.41 Test statistic 5 5 5 5 Normal (Z) 0.707 0.707 0.707 0.707 Reject level 0.760243 0.760243 0.760243 0.760243 Sum abs rank difference test: mean - 2.67 sd - 0.47 Test statistic 3 3 3 3 Normal (2) 0.707 0.707 0.707 0.707 Reject level 0.760243 0.760243 0.760243 0.760243 Phase separation behavior prediction Flory-Huggins model: wt frac - 0.406 189 Table C~1 (cont'd.). Triisopropylbenzene~Polystyrene at 175 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 10.6137 Flory-Huggins chi parameter: 0.9154 FH-VSP inf diln parameters: wt frac act coeff 9.8435 enth coeff 2.6559 ASOG-VSP inf diln parameters: wt frac act coeff 10.5095 enth coeff 1.0559 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory—Huggins FH-VSP ASOG-VSP 0.020 9.971 9.296 ~7.0 9.043 ~9.8 9.014 ~10.1 9.261 ~7.4 0.038 7.039 8.366 17.3 8.375 17.4 8.373 17.3 8.369 17.3 0.066 8.041 7.178 ~11.4 7.445 ~7.7 7.474 ~7.3 7.215 ~10.8 Standard pct err 15.4 15.1 21.4 15.4 Analysis of model error randomness Sum sqr rank difference test: mean - 4.00 sd - 1.41 Test statistic 5 5 5 5 Normal (2) 0.707 0.707 0.707 0.707 Reject level 0.760243 0.760243 0.760243 0.760243 Sum abs rank difference test: mean - 2.67 sd - 0.47 Test statistic 3 3 3 3 Normal (Z) 0.707 0.707 0.707 0.707 Reject level 0.760243 0.760243 0.760243 0.760243 Phase separation behavior prediction Flory-Huggins model: wt frac - 0.452 FH-VSP model: wt frac - 0.435 190 Table C~1 (cont'd.). Carbon disulfide-Polystyrene at 115 C Results of least squares fit: VSP inf diln wt frac activity coefficient: Flory-Huggins chi parameter: 0.4079 FH-VSP inf diln parameters: wt frac act coeff 3.7286 ASOG-VSP inf diln parameters: wt frac act coeff 3.7048 3.7286 enth coeff 1.0000 enth coeff 3.6140 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.014 3.655 3.637 ~0.5 3.629 ~0.7 3.637 ~0.5 3.626 ~0.8 0.024 3.566 3.574 0.2 3.573 0.2 3.574 0.2 3.572 0.2 0.041 3.465 3.476 0.3 3.484 0.6 3.476 0.3 3.487 0.6 Standard pct err 0.5 0.7 0.6 0.7 Analysis of model error randomness Sum sqr rank difference test: mean - 4.00 sd - 1.41 Test statistic 2 2 2 2 Normal (2) ~1.4l4 ~1.414 ~1.414 ~1.4l4 Reject level 0.921358 0.921358 0.921358 0.921358 Sum abs rank difference test: mean - 2.67 sd - 0.47 Test statistic 2 2 2 2 Normal (Z) ~1.414 ~1.414 ~1.4l4 ~1.414 Reject level 0.921358 0.921358 0.921358 0.921358 191 Table C~1 (cont'd.). Carbon disulfide-Polystyrene at 140 C Results of least squares fit: VSP inf diln wt frac activity coefficient: Flory-Huggins chi parameter: 0.3394 FH-VSP inf diln parameters: wt frac act coeff 3.4823 enth coeff 1.0000 ASOG-VSP inf diln parameters: wt frac act coeff 3.4761 enth coeff 4.1537 3.4823 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.008 3.885 3.438 ~12.2 3.433 ~12.4 3.438 ~12.2 3.432 ~12.4 0.011 3.831 3.423 ~11.3 3.418 ~11.4 3.423 ~11.3 3.417 ~11.4 0.018 2.995 3.384 12.2 3.383 12.2 3.384 12.2 3.379 12.1 0.029 2.948 3.329 12.2 3.332 12.2 3.329 12.2 3.324 12.0 Standard pct err 13.8 13.9 16.9 13.8 Analysis of model error randomness Sum sqr rank difference test: mean - 10.00 sd - 3.74 Test statistic 6 3 6 6 Normal (Z) ~1.069 ~1.871 ~1.069 ~1.069 Reject level .857484 .969310 .857484 .857484 Sum abs rank difference test: mean - 5.00 sd - 1.00 Test statistic 4 3 4 4 Normal (Z) ~1.000 ~2.000 ~1.000 ~1.000 Reject level .841351 .977241 .841351 .841351 192 Table C~1 (cont'd.). Methanol-Polymethyl methacrylate at 120 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 16.6476 Flory-Huggins chi parameter: 1.2827 FH-VSP inf diln parameters: wt frac act coeff 16.3347 enth coeff 2.7097 ASOG-VSP inf diln parameters: wt frac act coeff 16.2298 enth coeff 2.9717 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.003 16.070 16.148 0.5 15.924 ~0.9 15.989 ~0.5 15.934 ~0.9 0.006 15.397 15.555 1.0 15.572 1.1 15.568 1.1 15.571 1.1 0.009 15.303 15.063 ~1.6 15.271 ~0.2 15.211 ~0.6 15.262 ~0.3 Standard pct err 1.4 1.0 1.4 1.0 Analysis of model error randomness Sum sqr rank difference test: mean - 4.00 sd - 1.41 Test statistic 5 5 5 5 Normal (Z) 0.707 0.707 0.707 0.707 Reject level 0.760243 0.760243 0.760243 0.760243 Sum abs rank difference test: mean - 2.67 sd - 0.47 Test statistic 3 3 3 3 Normal (2) 0.707 0.707 0.707 0.707 Reject level 0.760243 0.760243 0.760243 0.760243 Phase separation behavior prediction Flory-Huggins model: wt frac - 0.279 FH-VSP model: wt frac - 0.312 193 Table C~1 (cont'd.). Methanol-Polymethyl methacrylate at 130 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 12.7268 Flory-Huggins chi parameter: 1.0138 FH-VSP inf diln parameters: wt frac act coeff 10.7857 enth coeff 0.1860 ASOG-VSP inf diln parameters: wt frac act coeff 12.5597 enth coeff 2.8431 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.003 11.601 12.442 7.0 12.366 6.4 11.581 ~0.2 12.365 6.4 0.006 12.217 12.135 ~0.7 12.152 ~0.5 12.274 0.5 12.152 ~0.5 0.008 12.724 11.902 ~6.7 11.987 ~6.0 12.686 ~0.3 11.988 ~6.0 Standard pct err 6.9 6.2 0.6 6.2 Analysis of model error randomness Sum sqr rank difference test: mean - 4.00 sd - 1.41 Test statistic 2 2 5 2 Normal (2) ~1.4l4 ~1.414 0.707 ~1.414 Reject level 0.921358 0.921358 0.760243 0.921358 Sum abs rank difference test: mean - 2.67 sd - 0.47 Test statistic 2 2 3 2 Normal (Z) ~1.4l4 ~1.4l4 0.707 ~1.414 Reject level 0.921358 0.921358 0.760243 0.921358 Phase separation behavior prediction Flory-Huggins model: wt frac - 0.367 194 Table C~1 (cont'd.). Toluene-Polymethyl methacrylate at 130 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 9.6790 Flory-Huggins chi parameter: 0.7953 FH-VSP inf diln parameters: wt frac act coeff 9.6790 enth coeff 1.0000 ASOG-VSP inf diln parameters: wt frac act coeff 9.6920 enth coeff 0.9658 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.017 10.638 8.772 ~19.3 8.224 ~25.7 8.772 ~19.3 8.779 ~19.2 0.060 5.823 6.980 18.1 6.921 17.3 6.980 18.1 6.985 18.2 0.112 4.955 5.517 10.7 5.706 14.1 5.517 10.7 5.526 10.9 Standard pct err 20.2 24.1 28.6 20.2 Analysis of model error randomness Sum sqr rank difference test: mean - 4.00 sd - 1.41 Test statistic 5 5 5 5 Normal (2) 0.707 0.707 0.707 0.707 Reject level 0.760243 0.760243 0.760243 0.760243 Sum abs rank difference test: mean - 2.67 sd - 0.47 Test statistic 3 3 3 3 Normal (Z) 0.707 0.707 0.707 0.707 Reject level 0.760243 0.760243 0.760243 0.760243 Phase separation behavior prediction Flory-Huggins model: wt frac - 0.536 195 Table C~1 (cont'd.). Toluene-Polymethyl methacrylate at 160 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 10.9509 Flory-Huggins chi parameter: 0.9547 FH-VSP inf diln parameters: wt frac act coeff 10.9509 ASOG-VSP inf diln parameters: wt frac act coeff 11.0656 enth coeff 1.0000 enth coeff 0.9014 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.006 12.699 10.514 -18.9 10.174 ~22.2 10.514 ~18.9 10.590 ~18.2 0.014 9.936 9.951 0.2 9.791 ~1.5 9.951 0.2 9.982 0.5 0.023 9.345 9.413 0.7 9.409 0.7 9.413 0.7 9.408 0.7 0.025 8.462 9.261 9.0 9.299 9.4 9.261 9.0 9.247 8.9 0.037 7.563 8.624 13.1 8.822 15.4 8.624 13.1 8.576 12.6 Standard pct err 12.4 14.3 14.3 11.9 Analysis of model error randomness Sum sqr rank difference test: mean - 20.00 sd - 7.28 Test statistic 4 4 4 4 Normal (2) ~2.198 ~2.198 ~2.198 ~2.198 Reject level 0.986006 0.986006 0.986006 0.986006 Sum abs rank difference test: mean - 8.00 sd - 1.61 Test statistic 4 4 4 4 Normal (Z) ~2.481 ~2.481 ~2.481 ~2.481 Reject level 0.993433 0.993433 0.993433 0.993433 Phase separation behavior prediction Flory-Huggins model: wt frac - 0.426 196 Table C~1 (cont'd.). Toluene-Polyvinyl acetate at 35 C Results of least squares fit: VSP inf diln wt frac activity coefficient: Flory-Huggins chi parameter: 0.7772 FH-VSP inf diln parameters: wt frac act coeff 8.4101 enth coeff 2.0621 ASOG-VSP inf diln parameters: wt frac act coeff 8.2575 enth coeff 1.4046 9.7100 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory—Huggins FH-VSP ASOG-VSP 0.084 6.081 6.240 2.6 6.038 ~0.7 6.067 ~0.2 6.039 ~0.7 0.117 5.377 5.408 0.6 5.391 0.3 5.396 0.4 5.391 0.3 0.161 4.665 4.565 ~2.2 4.676 0.2 4.663 ~0.0 4.677 0.3 0.195 4.207 4.068 ~3.4 4.224 0.4 4.202 ~0.1 4.225 0.4 Standard pct err 2.8 0.5 0.3 0.5 Analysis of model error randomness Sum sqr rank difference test: mean - 10.00 sd - 3.74 Test statistic 3 9 11 9 Normal (Z) ~1.87l ~0.267 0.267 ~0.267 Reject level 0.969310 0.605367 0.605367 0.605367 Sum abs rank difference test: mean - 5.00 sd - 1.00 Test statistic 3 5 5 5 Normal (Z) ~2.000 0.000 0.000 0.000 Reject level 0.977241 0.500000 0.500000 0.500000 Phase separation behavior prediction Flory-Huggins model: wt frac - 0.564 FH-VSP model: wt frac - 0.598 197 Table C~1 (cont'd.). Toluene-Polyvinyl acetate at 40 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 9.2644 Flory-Huggins chi parameter: 0.7733 FH-VSP inf diln parameters: wt frac act coeff 8.3495 enth coeff 2.0578 ASOG-VSP inf diln parameters: wt frac act coeff 8.2587 enth coeff 1.3804 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.051 6.831 7.057 3.3 6.777 ~0.8 6.812 ~0.3 6.782 ~0.7 0.076 6.201 6.287 1.4 6.197 ~0.1 6.211 0.2 6.199 ~0.0 0.089 5.936 5.943 0.1 5.925 ~0.2 5.930 ~0.1 5.926 ~0.2 0.094 5.732 5.826 1.6 5.830 1.7 5.833 1.7 5.831 1.7 0.128 5.289 5.081 ~4.0 5.202 ~1.7 5.189 -l.9 5.202 ~1.7 0.139 5.055 4.871 ~3.7 5.017 ~0.8 5.000 ~1.1 5.016 ~0.8 0.171 4.444 4.346 ~2.2 4.536 2.1 4.512 1.5 4.535 2.0 Standard pct err 2.9 1.4 1.4 1.4 Analysis of model error randomness Sum sqr rank difference test: mean - 56.00 sd — 18.58 Test statistic 36 59 67 60 Normal (Z) ~1.076 0.161 0.592 0.215 Reject level 0.859100 0.564134 0.723042 0.585218 Sum abs rank difference test: mean - 16.00 sd - 3.06 Test statistic 12 17 17 16 Normal (2) ~1.309 0.327 0.327 0.000 Reject level 0.904794 0.628285 0.628285 0.500000 Phase separation behavior prediction Flory-Huggins model: wt frac - 0.567 FH-VSP model: wt frac - 0.602 198 Table C~1 (cont'd.). Toluene-Polyvinyl acetate at 47.5 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 8.8654 Flory-Huggins chi parameter: 0.7609 FH-VSP inf diln parameters: wt frac act coeff 7.6321 enth coeff 3.0918 ASOG-VSP inf diln parameters: wt frac act coeff 8.1821 enth coeff 1.3458 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.052 6.506 6.815 4.6 6.704 3.0 6.587 1.2 6.704 3.0 0.071 6.370 6.257 ~1.8 6.262 ~1.7 6.248 ~1.9 6.258 ~1.8 0.107 5.613 5.380 ~4.2 5.522 ~1.6 5.653 0.7 5.516 ~1.7 Standard pct err 4.6 2.7 2.4 2.7 Analysis of model error randomness Sum sqr rank difference test: mean - 4.00 sd - 1.41 Test statistic 2 5 5 5 Normal (Z) ~1.414 0.707 0.707 0.707 Reject level 0.921358 0.760243 0.760243 0.760243 Sum abs rank difference test: mean - 2.67 sd - 0.47 Test statistic 2 3 3 3 Normal (2) ~1.4l4 0.707 0.707 0.707 Reject level 0.921358 0.760243 0.760243 0.760243 Phase separation behavior prediction Flory-Huggins model: wt frac - 0.577 FH-VSP model: wt frac - 0.467 199 Table C~1 (cont'd.). Chloroform-Polyvinyl acetate at 35 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 1.4938 Flory-Huggins chi parameter: ~0.4l68 FH-VSP inf diln parameters: wt frac act coeff 1.4938 ASOG-VSP inf diln parameters: wt frac act coeff 1.6218 enth coeff 1.0000 enth coeff 0.4051 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.163 1.587 1.464 ~8.1 1.450 ~9.l 1.464 -8.1 1.501 -5.6 0.231 1.421 1.447 1.8 1.443 1.6 1.447 1.8 1.462 2.8 0.276 1.407 1.435 1.9 1.436 2.1 1.435 1.9 1.438 2.2 0.327 1.376 1.419 3.1 1.426 3.6 1.419 3.1 1.413 2.7 0.381 1.364 1.401 2.6 1.412 3.4 1.401 2.6 1.388 1.7 0.416 1.368 1.388 1.4 1.400 2.3 1.388 1.4 1.372 0.2 0.464 1.373 1.368 ~0.3 1.383 0.7 1.368 ~0.3 1.349 ~1.7 Standard pct err 3.9 4.5 4.3 3.1 Analysis of model error randomness Sum sqr rank difference test: mean - 56.00 sd - 18.58 Test statistic 25 25 25 47 Normal (Z) ~1.668 ~1.668 ~1.668 ~0.484 Reject level 0.952360 0.952360 0.952360 0.685906 Sum abs rank difference test: mean - 16.00 sd - 3.06 Test statistic 11 11 11 13 Normal (2) ~1.637 ~1.637 ~1.637 ~0.982 Reject level 0.949147 0.949147 0.949147 0.836951 Table C~1 (cont'd.). 200 Chloroform-Polyvinyl acetate at 45 C Results of least squares fit: VSP inf diln wt frac activity coefficient: Flory-Huggins chi parameter: FH-VSP inf diln parameters: ASOG-VSP inf diln parameters: Wt Frac Solvent Exptl 0.093 1.478 0.121 1.408 0.139 1.405 0.164 1.416 0.198 1.366 0.206 1.452 0.227 1.400 0.247 1.380 0.276 1.382 0.295 1.380 0.325 1.365 0.355 1.351 0.427 1.389 0.461 1.378 0.478 1.395 0.499 1.416 Standard pct err ~0.4 604 1.4417 wt frac act coeff 1.4048 wt frac act coeff 1.4801 Activity Coefficients and Percent Error VSP F‘F‘P‘P‘P‘P‘F‘P‘F‘P‘P‘P‘P‘F‘P‘F‘ .432 .428 .425 .421 .416 .414 .410 .406 .400 .396 .388 .380 .358 .347 .341 .332 Analysis of model error Sum sqr rank difference Test statistic Normal (Z) Reject level 0 Sum abs rank difference Test statistic Normal (Z) Reject level Flory-Huggins FH-VSP ~3.2 1 1.4 1 1.5 l 0.4 1 3.6 1 ~2.7 1 0.8 1 1.9 l 1.3 1 1.1 1 1.7 1 2.2 l ~2.2 1 ~2.3 1 ~3.9 1 ~6.l l 2.7 randomness test: mean 496 ~1.137 .872136 test: mean 70 ~1.222 .889111 .407 .409 .410 .411 .411 .411 .410 .409 .406 .404 .400 .395 .378 .367 .362 .354 0. -4. uac-a:aan>uwa>\ar-4«anus-c-c>m> 0‘ 680.00 481 ~1.229 .890503 85.00 63 ~1.792 963423 P‘P‘P‘P‘P‘P‘P‘P‘F'P‘P‘P‘P‘F‘P‘P‘ sd sd .412 .413 .413 .413 .412 .411 .410 .408 .405 .402 .397 .392 .374 .363 .357 .349 - l I UJNDF‘F‘NDCDNDUSC>C’C>¢‘ GD\JH‘F‘C>UIO\O\C>\JWDUDNDO\930\ 0‘ 61.90 475 ~1.266 .897291 12.28 63 ~1.792 .963423 enth coeff 0.6656 enth coeff 0.4490 ASOG-VSP 1.448 ~2.1 1.439 2.1 1.433 2.0 1.425 0.6 1.415 3.5 1.412 ~2.8 1.406 0.4 1.400 1.4 1.391 0.6 1.385 0.4 1.376 0.8 1.367 1.2 1.343 ~3.3 1.332 ~3.4 1.325 ~5.l 1.318 ~7.2 3.0 393 ~1.773 0.961857 61 ~1.955 0.974690 201 Table C~1 (cont'd.). Benzene-Polyethylene oxide at 70 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 4.9056 Flory-Huggins chi parameter: 0.2870 FH-VSP inf diln parameters: wt frac act coeff 4.9056 ASOG—VSP inf diln parameters: wt frac act coeff 4.6698 enth coeff 1.0000 enth coeff 1.1931 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.062 4.312 4.204 ~2.5 4.177 ~3.2 4.204 ~2.5 4.118 ~4.6 0.067 4.231 4.149 -2.0 4.125 ~2.5 4.149 ~2.0 4.073 ~3.8 0.099 3.810 3.850 1.1 3.840 0.8 3.850 1.1 3.823 0.4 0.139 3.474 3.523 1.4 3.526 1.5 3.523 1.4 3.541 1.9 0.193 3.093 3.144 1.6 3.156 2.0 3.144 1.6 3.198 3.4 0.261 2.690 2.747 2.1 2.764 2.7 2.747 2.1 2.824 4.9 0.388 2.153 2.192 1.8 2.208 2.5 2.192 1.8 2.269 5.3 Standard pct err 2.0 2.5 2.2 4.1 Analysis of model error randomness Sum sqr rank difference test: mean - 56.00 sd - 18.58 Test statistic 9 9 9 6 Normal (2) ~2.529 ~2.529 ~2.529 ~2.691 Reject level 0.994274 0.994274 0.994274 0.996426 Sum abs rank difference test: mean - 16.00 sd - 3.06 Test statistic 7 7 7 6 Normal (2) ~2.946 ~2.946 ~2.946 ~3.273 Reject level 0.998385 0.998385 0.998385 0.999466 202 Table C~1 (cont'd.). Benzene-Polyethylene oxide at 70 C (second run) Results of least squares fit: VSP inf diln wt frac activity coefficient: 4.5994 Flory-Huggins chi parameter: 0.2298 FH-VSP inf diln parameters: wt frac act coeff 4.5553 ASOG-VSP inf diln parameters: wt frac act coeff 4.4055 enth coeff 1.2775 enth coeff 1.1931 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.050 4.103 4.096 ~0.2 4.080 ~0.6 4.078 ~0.6 4.007 ~2.4 0.089 3.743 3.759 0.4 3.754 0.3 3.753 0.3 3.727 ~0.4 0.142 3.344 3.364 0.6 3.368 0.7 3.369 0.8 3.385 1.2 0.201 2.996 2.998 0.1 3.008 0.4 3.009 0.5 3.052 1.9 0.265 2.702 2.661 ~1.5 2.673 ~1.1 2.675 ~1.0 2.733 1.1 Standard pct err 0.8 0.8 0.9 1.7 Analysis of model error randomness Sum sqr rank difference test: mean - 20.00 sd - 7.28 Test statistic 13 15 15 10 Normal (2) ~0.962 ~0.687 ~0.687 ~1.374 Reject level 0.831861 0.753889 0.753889 0.915226 Sum abs rank difference test: mean - 8.00 sd - 1.61 Test statistic 7 7 7 6 Normal (2) ~0.620 ~0.620 ~0.620 ~1.240 Reject level 0.732418 0.732418 0.732418 0.892587 203 Table C~1 (cont'd.). Benzene-Polyethylene oxide at 75.1 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 4.5056 Flory-Huggins chi parameter: 0.2106 FH-VSP inf diln parameters: wt frac act coeff 4.4221 ASOG-VSP inf diln parameters: wt frac act coeff 4.3698 enth coeff 1.4920 enth coeff 1.1635 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.053 3.984 4.004 0.5 3.996 0.3 3.977 ~0.2 3.959 ~0.6 0.081 3.755 3.766 0.3 3.765 0.3 3.761 0.2 3.756 0.0 0.108 3.561 3.557 ~0.1 3.560 ~0.0 3.567 0.2 3.574 0.4 0.145 3.332 3.300 ~0.9 3.308 ~0.7 3.326 ~0.2 3.344 0.4 Standard pct err 0.6 0.5 0.2 0.5 Analysis of model error randomness Sum sqr rank difference test: mean - 10.00 sd - 3.74 Test statistic 3 3 11 6 Normal (2) -1.871 ~1.871 0.267 ~1.069 Reject level 0.969310 0.969310 0.605367 0.857484 Sum abs rank difference test: mean - 5.00 sd - 1.00 Test statistic 3 3 5 4 Normal (Z) ~2.000 ~2.000 0.000 ~1.000 Reject level 0.977241 0.977241 0.500000 0.841351 204 Table C~1 (cont'd.). Benzene-Polyethylene oxide at 88.1 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 4.4720 Flory-Huggins chi parameter: 0.2049 FH-VSP inf diln parameters: wt frac act coeff 4.4720 enth coeff 1.0000 ASOG-VSP inf diln parameters: wt frac act coeff 4.3992 enth coeff 1.0940 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH~VSP ASOG-VSP 0.027 4.234 4.209 ~0.6 4.202 ~0.7 4.209 ~0.6 4.175 ~1.4 0.050 3.986 3.999 0.3 3.998 0.3 3.999 0.3 3.994 0.2 0.067 3.850 3.859 0.2 3.861 0.3 3.859 0.2 3.870 0.5 0.091 3.667 3.670 0.1 3.676 0.2 3.670 0.1 3.701 0.9 Standard pct err 0.4 0.5 0.5 1.0 Analysis of model error randomness Sum sqr rank difference test: mean - 10.00 sd - 3.74 Test statistic ll 11 ll 3 Normal (2) 0.267 0.267 0.267 ~1.87l Reject level 0.605367 0.605367 0.605367 0.969310 Sum abs rank difference test: mean - 5.00 sd - 1.00 Test statistic 5 5 5 3 Normal (Z) 0.000 0.000 0.000 ~2.000 Reject level 0.500000 0.500000 0.500000 0.977241 205 Table C~1 (cont'd.). Benzene-Polyethylene oxide at 102 C Results of least squares fit: VSP inf diln wt frac activity coefficient: Flory-Huggins chi parameter: 0.2020 FH-VSP inf diln parameters: wt frac act coeff 4.4430 enth coeff 1.0000 ASOG-VSP inf diln parameters: wt frac act coeff 4.3920 enth coeff 1.0280 4.4430 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.021 4.303 4.241 ~1.5 4.233 ~1.6 4.241 ~1.5 4.214 ~2.1 0.022 4.280 4.230 ~1.2 4.222 ~1.4 4.230 ~1.2 4.205 ~1.8 0.025 4.272 4.201 ~1.7 4.194 ~1.8 4.201 ~1.7 4.179 ~2.2 0.029 4.225 4.163 ~1.5 4.157 ~1.6 4.163 ~1.5 4.145 ~1.9 0.037 4.148 4.093 ~1.3 4.089 ~1.4 4.093 ~1.3 4.083 ~1.6 0.044 4.070 4.029 ~l.0 4.026 ~1.1 4.029 ~1.0 4.026 ~1.1 0.048 4.042 3.998 ~1.1 3.996 ~1.1 3.998 ~1.1 3.998 ~1.1 0.058 3.950 3.908 ~1.1 3.909 ~1.0 3.908 ~1.1 3.917 ~0.8 0.077 3.713 3.757 1.2 3.760 1.3 3.757 1.2 3.779 1.8 0.092 3.520 3.643 3.4 3.649 3.6 3.643 3.4 3.675 4.3 0.118 3.169 3.455 8.6 3.463 8.9 3.455 8.6 3.500 9.9 Standard pct err 3.2 3.3 3.4 3.8 Analysis of model error randomness Sum sqr rank difference test: mean - 220.00 sd - 61.55 Test statistic 52 47 52 33 Normal (Z) ~2.729 ~2.811 ~2.729 ~3.038 Reject level 0.996821 0.997522 0.996821 0.998805 Sum abs rank difference test: mean - 40.00 sd - 6.66 Test statistic 20 19 20 17 Normal (Z) ~3.002 ~3.152 ~3.002 ~3.452 Reject level 0.998652 0.999184 0.998652 0.999720 206 Table C~1 (cont'd.). Benzene-Polyethylene oxide at 125.4 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 4.3079 Flory-Huggins chi parameter: 0.1700 FH-VSP inf diln parameters: wt frac act coeff 4.3079 ASOG-VSP inf diln parameters: wt frac act coeff 4.2956 enth coeff 1.0000 enth coeff 0.9330 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.011 4.249 4.207 ~1.0 4.206 -l.0 4.207 ~l.0 4.202 ~1.1 0.018 4.141 4.147 0.1 4.146 0.1 4.147 0.1 4.145 0.1 0.024 4.080 4.093 0.3 4.093 0.3 4.093 0.3 4.094 0.3 0.033 3.993 4.017 0.6 4.018 0.6 4.017 0.6 4.023 0.8 Standard pct err 0.7 0.7 0.8 0.8 Analysis of model error randomness Sum sqr rank difference test: mean — 10.00 sd - 3.74 Test statistic 3 3 3 3 Normal (2) ~1.87l ~1.87l ~1.871 ~1.87l Reject level 0.969310 0.969310 0.969310 0.969310 Sum abs rank difference test: mean - 5.00 sd - 1.00 Test statistic 3 3 3 3 Normal (2) ~2.000 ~2.000 ~2.000 ~2.000 Reject level 0.977241 0.977241 0.977241 0.977241 207 Table C~1 (cont'd.). Benzene-Polyethylene oxide at 125.7 C Results of least squares fit: VSP inf diln wt frac activity coefficient: Flory-Huggins chi parameter: 0.1434 FH-VSP inf diln parameters: wt frac act coeff 4.1940 ASOG-VSP inf diln parameters: wt frac act coeff 4.1816 4.1940 enth coeff 1.0000 enth coeff 0.9319 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.011 4.155 4.098 ~1.4 4.097 ~1.4 4.098 ~1.4 4.092 ~1.5 0.017 4 073 4.046 ~0.7 4.046 ~0.7 4.046 ~0.7 4.044 ~0.7 0.025 3.945 3.985 1.0 3.986 1.0 3.985 1.0 3.987 1.1 0.033 3.874 3.918 1.1 3.919 1.2 3.918 1.1 3.924 1.3 Standard pct err 1.2 1.3 1.5 1.4 Analysis of model error randomness Sum sqr rank difference test: mean - 10.00 sd - 3.74 Test statistic 3 3 3 3 Normal (2) ~1.871 ~1.871 ~1.87l ~1.871 Reject level 0.969310 0.969310 0.969310 0.969310 Sum abs rank difference test: mean - 5.00 sd - 1.00 Test statistic 3 3 3 3 Normal (2) ~2.000 ~2.000 ~2.000 ~2.000 Reject level 0.977241 0.977241 0.977241 0.977241 208 Table C~1 (cont'd.). Benzene-Polyethylene oxide at 150.4 C Results of least squares fit: VSP inf diln wt frac activity coefficient: 4.2609 Flory-Huggins chi parameter: 0.1576 FH-VSP inf diln parameters: wt frac act coeff 4.2609 ASOG-VSP inf diln parameters: wt frac act coeff 4.2603 enth coeff 1.0000 enth coeff 0.8493 Wt Frac Activity Coefficients and Percent Error Solvent Exptl VSP Flory-Huggins FH-VSP ASOG-VSP 0.008 4.311 4.189 ~2.9 4.188 ~2.9 4.189 ~2.9 4.189 ~2.9 0.011 4.202 4.160 ~l.0 4.159 ~l.0 4.160 ~1.0 4.160 ~l.0 0.016 4.061 4.116 1.4 4.116 1.4 4.116 1.4 4.116 1.4 0.023 3.956 4.063 2.7 4.064 2.7 4.063 2.7 4.063 2.7 Standard pct err 2.5 2.5 3.0 2.5 Analysis of model error randomness Sum sqr rank difference test: mean - 10.00 sd - 3.74 Test statistic 3 3 3 3 Normal (Z) ~1.87l ~1.871 ~1.871 ~1.871 Reject level 0.969310 0.969310 0.969310 0.969310 Sum abs rank difference test: mean - 5.00 sd - 1.00 Test statistic 3 3 3 3 Normal (2) ~2.000 ~2.000 ~2.000 ~2.000 Reject level 0.977241 0.977241 0.977241 0.977241 APPENDIX D. Program Used to Apply Thermodynamic Models Using Data Extrapolated from Low Solvent Concentrations. The program listed below was used to generate the results in Appendix B from the original data in Appendix A. These results were presented in Chapter 2 of the dissertation. Input in the form of polymer-solvent activity data at a given temperature is processed to fit adjustable parameters if necessary and then the predictions of the VSP, Flory- Huggins, and UNIFAC-EV models are compared to experimental results. Refer to Appendices A and B for a more detailed description of the input data format and the output produced by the program. The source code given below was written in IBM Pascal Version 2 for the IBM Personal Computer XT. Since Pascal, unlike many version of Fortran and Basic, has a fairly standardized language description, this code should run with few revisions under any Pascal compiler. One possible source of incompatibility is the use of string types, which are an IBM Pascal extension not part of standard Pascal. Most Pascal compilers support this or a similar extension (a type equivalent to array of char). The use of file names in the program statement may not work in other versions of Pascal, or may not work in the same way. In IBM Pascal, the user is prompted for file names at the time execution 209 210 begins. To run this program, it should be compiled and linked. When execution begins, the user is prompted to name the four files: infile, outfile, display, and auxfile. Infile is the input to the program as described in Appendix A. Outfile is the program output as shown is Appendix B. Display is a file which receives prompt lines when input is expected from infile. Auxfile receives auxiliary output containing intermediate values of calculation, useful mostly for debugging purposes, but not well labeled or documented. Allowing file specification gives the program flexibility to accept data either from an already created file or directly from user keyboard input, and to produce output either to the monitor screen, or to the printer, or to an external disk file for later review and use. To accept input data from a file, give the file name (including the drive designator and extension, e.g., A:MYFILE.DAT). To accept input data from the keyboard, type USER. (USER is the IBM DOS filename for keyboard input.) To produce output to a file, give the file name; to produce it at the monitor, type USER; to produce it at the line printer, type PRN: (the IBM DOS device designation for the printer). If you have chosen to enter input from the keyboard, it is helpful to specify USER for the Display file. This will result in messages appearing on the monitor every time the program requires input. It is probably not a good idea to specify USER for the Outfile file in this 211 case, as the program output will intermix with the prompt messages at certain points of program execution. If you have chosen to enter input from an external file, specify NUL for the Display file so that prompt messages are not displayed at the monitor. Since the Auxfile output is not generally useful, specify NUL for this file also. File specification is summarized here as Table D-l. Table D-l. File Specification for Program Execution. To use an external data file: To enter data from keyboard: INFILE: INFILE: user OUTFILE: OUTFILE: DISPLAY: nul DISPLAY: user AUXFILE: nul AUXFILE: nul To send output to the monitor: OUTFILE: user To send output to the printer: OUTFILE: prn: To save output on an external file (can be printed or sent to monitor at a later time using the PRINT, TYPE, or COPY commands in DOS.) OUTFILE: There are some points in program execution where terminal input may be necessary even if Infile is taking input from an external file. This will occur if a new compound name (not previously used during any 212 execution of this program) is specified on line 2 or 3 of a data set. In this case, prompt messages will appear on the monitor for input to be entered from the keyboard (regardless of your choices for Infile and Display). The input will consist of the functional group description of the compound, its molecular weight, and, if any new functional groups are specified, UNIFAC interaction parameters must also be supplied as input. The functional group and compound information is stored on a file named ASOGVSP.TAB. The format of this file is given as Table D-2. Table D-2. Format of Functional Group and Compound Information File. Line 1: N, the number of functional groups (limit of 20). Lines 2 to N+l: Each line contains this information for one functional group. The UNIFAC surface area parameter, q , as a real value, followed by the UNIFAC segment volume parameter, r , as a real value, followed by a group name (maximum 6 characters). Lines N+2 to 2N+1: Each line contains the UNIFAC interaction parameters, a j’ for group i with each of the N groups j, in order, as real values. Line 2N+2: M, the number of compounds (limit of 50). Lines 2N+3 to 4N+2: Each two lines contain this information for one compound. The first contains K, the number of different groups in the compound as an integer, followed by the compound molecular weight as a real value, followed by the compound name (maximum 20 characters). The second contains 2K integers, which represent K pairs of group information. Each pair is the number of that particular group found in that compound followed by the position of the group in rows 2 to N+l of this file. Position is given as an integer between 1 and N (not between 2 and N+l). Dr. Eric A. Grulke of Michigan State University has a disk copy of this program and necessary files. Source code for the program is found on 213 file ASOGVSP.PAS, and the executable version of the program is found on file ASOGVSP.EXE. This file can be executed by typing its name at the DOS prompt, i.e., A:ASOGVSP (assuming the floppy drive is device A:). Table D-3. Source Code for Program to Extrapolate Low Solvent Concentration Thermodynamic Data. program asogvsp(infile,outfile,disp1ay,auxfile,input,output); type setptr - Adataset; dataset - record concenzreal; activityzreal; nextzsetptr end; modeltype - (asogvsp,flory,unifacfv,asog); nametype - string(20); const e - 2.7182818; const compoundtablesize - 50; grouptablesize - 20; solutiontablesize - 10; var wl,omegalexp,omegalinf,conc,act,ml,m2,m2r,chi,densityratiozreal; tempomegal,rhol,rh02,lastwl,moment0error:real; omegal,pctdiff,1astpctdiff,momentlerror:array[modeltype] of real; rpoly,qpoly,rsolv,qsolv:rea1; rk,qk:array[l..grouptablesize] of real; a:array[l..grouptablesize,l..grouptablesize] of real; tkzreal; count,i,j,solvindex,polyindex:integer; numgroupszo..grouptablesize; numcompounds:0..compoundtablesize; endofdata,found:boolean; concunit,actunit,rhounit,ch:char; heading:string(80); compoundname:array[l..compoundtablesize] of nametype; compoundmw:array[l..compoundtablesize] of real; compoundgroups:array[l..compoundtablesize] of integer; groupsinit:array[l..compoundtablesize] of integer; mw:array[1..compoundtablesize] of real; numgroup,group:array[l..compoundtablesize,l..solutiontablesize] of integer; groupname:array[1..grouptablesize] of string(7); ptr,firstset,lastset,nextset:setptr; modelzmodeltype; infile,outfile,display,auxfile,data:text; Table D~3 (cont'd. ) . procedure getactunits; begin writeln(display,' writeln(display,' writeln(display,' end; procedure getconcunits; begin writeln(display,' writeln(display,' writeln(display,' end; procedure getmolecwts; begin ‘9 for for for for for for 214 activity'); wt frac activity coef'); mol frac activity coef'); weight fraction solvent'); mass ratio solvent/polymer'); mole fraction solvent'); write(display,'Enter MW of polymer, MW of solvent '); readln(infile,m2,m1) end; function convertconc(conc:real;concunit:char):real; begin case concunit of 'w': convertconcz-conc; 'x': convertconc:-conc/(conc+(m2/m1)*(1.0-conc)); 'm': convertconc:-1.0-1.0/(1.0+conc) end end; function convertact(act:real;actunit:char):real; begin case actunit of 'w': convertact:-act; 'a': convertact:-act/w1; ’x': convertact:-act*conc/wl; end end; function convertrho(rho:real;rhounit:char;mw:real):real; begin case rhounit of 'd': convertrhoz-rho; 'v': convertrho:-l.0/rho; 'm': convertrhoz-mw/rho; end end; procedure getcompound(solvorpoly:nametype;var r,q:rea1; var name:nametype; izinteger; getnewzboolean; var indexzinteger); 215 Table D~3 (cont'd.). procedure getsizeparams; var name:string(7); k:integer; procedure getgroupparams; var i,same:integer; begin same:-0; if k > 1 then begin writeln(output,'Enter group interaction parameter a for ' groupname[k],' with the following groups'); for i:-l to k-l do if same - 0 then begin write(output,groupname[i],' '); readln(input,a[k,i]); if a[k,i] - 0 then same:-i end; readln(input); if same - 0 then begin writeln(‘Now enter group interaction parameter a for each of ', 'the following groups with ',groupname[k]); for iz-l to k-l do write(output,groupname[i]); begin write(output,groupname[i]); readln(input,a[i,k]) end end else for i:-l to k-l do a[i,k]:-a[i,same]; end; a[k,k]:-0; end; begin writeln(‘For ',compoundname[index]); endofdataz-false; r:-0; Cit-0; repeat writeln(output,'Enter number of groups followed by group name'); write(output,'or: 0 ~ end of groups for component '); read(input,numgroup[index,1+1]); 216 Table D-3 (cont'd.). if numgroup[index,i+1] > 0 then begin i:-i+l; readln(input,name); k:-0; if numgroups > 0 then repeat k:-k+1; until (name - groupname[k]) or (k - numgroups); if (numgroups - 0) or (name <> groupname[k]) then begin numgroups:-numgroups+l; groupname[numgroups]:-name; write(output,:Enter Rk, Qk, for group ',name); readln(input,rk[numgroups],qk[numgroups]); kz-numgroups; getgroupparams; end; group[index,i]:-k; r:-r+numgroup[index,i]*rk[k]; q:-q+numgroup[index,i]*qk[k]; writeln(‘Groups entered so far:'); for kz-l to i do write(numgroup[index,k],group[index,k]); writeln; end else begin readln(input); endofdata:-true; end; until endofdata; end; begin readln(infile,name); index:-0; getnewz-false; if numcompounds > 0 then repeat index:-index+l; until (compoundname[index] - name) or (index - numcompounds) else getnew:-true; if (numcompounds > 0) and (compoundname[index] <> name) then getnewz-true; 217 Table D-3 (cont'd.). if getnew then begin numcompounds:-numcompounds+l; compoundname[numcompounds]:-name; index:-numcompounds; i:-0; getsizeparams; groupsinit[index]:-i; write('Enter molecular weight of the ',solvorpoly); readln(mw[index]); end else begin r:-0; q=-0: for i:-1 to groupsinit[index] do begin r:-r+numgroup[index,i]*rk[group[index,i]]; q:-q+numgroup[index,i]*qk[group[index,i]]; end; end; r:-r/mw[index]; q:-q/mw[index]; end; function findact(model:modeltype):real; const 2 - 10.0; b - 1.28; c1 - 1.1; var y,phi1,phi2,thetal,tempomegalinfzreal; vlred,vmred,fv:rea1; arfrac:array[l..2,l..10] of real; grpindex:array[l..10] of integer; factor:array[l..2] of real; term,term2,residual,solvtotal,solntotalzreal; i,j,k,1,solvgroups,solngroups:integer; begin case model of asogvsp: begin y:-wl+(e/omegalinf)*(l.0-w1); findact:-exp((y~wl)/y)/y end; 218 Table D~3 (cont'd.). flory: begin phil:-densityratio*wl/(densityratio*wl+(1.0-wl)); phi2:-l.0-phil; findact:-exp(ln(phi1)+chi*phi2*ph12+ph12)/wl end; unifacfv: begin thetal:-qsolv*w1/(qsolv*w1+qpoly*(1.0-w1)); phil:-rsolv*wl/(rsolv*wl+rpoly*(1.0-wl)); ph12z-l.0~phil; findact:-ln(phil)+phi2; findact:-result(findact)+(z/2.0)*ml*qsolv*ln(thetal/phil); findact:-result(findact)~(z/2.0)*m1*qsolv*(1.0-phil/thetal); findact:-exp(resu1t(findact))/wl; write(auxfile,w1:8:3); write(auxfile,resu1t(findact):8:3); solvtotal:-0; solvgroups:-groupsinit[solvindex]; for i:-1 to solvgroups do begin grpindex[i]:-group[solvindex,i]; arfrac[1,i]:-qk[grpindex[i]]/m1*numgroup[solvindex,i]*wl; arfrac[2,i]:-arfrac[l,i]; solvtotal:-solvtota1+arfrac[l,i]; end; solngroupsz-solvgroups; solntotal:-solvtotal; for i:-l to groupsinit[polyindex] do begin Ji-O; repeat j:-j+l until (group[polyindex,i] - grpindex[j]) or (j - solngroups); if group[polyindex,i] - grpindex[j] then begin arfrac[2,j]:-arfrac[2,j]+qk[grpindex[j]]/m2r*numgroup [polyindex,i]*(l.0~w1); end 219 Table D~3 (cont'd.). else begin solngroups:-solngroups+1; grpindex[solngroups]:-group[polyindex,i]; arfrac[2,solngroups]:-qk[grpindex[solngroups]]/m2r* numgroup[polyindex,i]*(1.0~w1); arfrac[l,solngroups]:-0.0; j:-solngroups; end; solntotal:-solntotal+qk[grpindex[j]]*numgroup[polyindex,i]/ m2r*(l.0~w1); end; for i:-l to solngroups do begin arfrac[1,i]:-arfrac[1,i]/solvtotal; arfrac[2,i]:-arfrac[2,i]/solntotal; end; residualz-O; for j:-l to solvgroups do begin for iz-l to 2 do begin term:-0; for k:-1 to solngroups do begin term:-term+arfrac[i,k] *exp(-a[grpindex[k],grpindex[j]]/tk); end; factor[i]:-l.0-1n(term); term2z-0; for kz-l to solngroups do begin termz-O; for l:-l to solngroups do term:-term+arfrac[i,l] *exp(~a[grpindex[l],grpindex[k]]/tk); term2:-term2+arfrac[i,k] *exp(~a[grpindex[j],grpindex[k]]/tk) /term; end; factor[i]:-factor[i]~term2; end; residual:-residual+qk[grpindex[j]]*(factor[2]~factor[1]) *numgroup[solvindex,j]; end; 220 Table D~3 (cont'd.). residual:-exp(residual); write(auxfile,residual:8:3); findact:-result(findact)*residual; v1red:-1.0/(rhol*15.l7*b*rsolv); vmred:-(w1/rhol+(l.0-w1)/rh02)/(15.17*b*(rsolv*w1+rpoly* (1.0-v71)»: fv:-3.0*c1*ln((exp(1n(vlred)/3.0)~l)/(exp(ln(vmred)/3.0)~1)); fv:-fv-cl*((v1red/vmred-l.0)/(l.0-exp(-1n(v1red)/3.0))); fv:-exp(fv); write(auxfile,fv:8:3); findact:-resu1t(findact)*fv; writeln(auxfile); end; asog: begin tempomegalinf:-omegalinf; omegalinfz-e; findact:-findact(asogvsp); omegalinfz-tempomegalinf end end end; function findinfact(wl,omegal:real):real; var y,newy,lnomegal:real; convergentzboolean; begin convergentz-false; lnomegal:-ln(omega1); y:-exp(l.0-1nomegal); while not convergent do begin newy:-exp(l.0-wl/y-lnomegal); convergent:-abs(newy~y) < (l.0e-5*newy); yz-newy; end; findinfact:-e*(1.0-wl)/(y~wl); writeln(outfile,'By correlating activity at finite conc ',w1:8:3); end; begin reset(infile); assign(data,'asogvsp.tab'); reset(data); rewrite(outfile); rewrite(display); rewrite(auxfile); readln(data,numgroups); for i:-1 to numgroups do readln(data,rk[i],qk[i],groupname[i]); 221 Table D~3 (cont'd.). for iz—l to numgroups do begin for j:-l to numgroups do read(data,a[i,j]); readln(data) end; readln(data,numcompounds); for i:-1 to numcompounds do begin readln(data,groupsinit[i],mw[i],compoundname[i]); for jz-l to groupsinit[i] do read(data,numgroup[i,j],group[i,j]); readln(data); end; repeat writeln(display,'Enter a heading for this data set'); readln(infile,heading); writeln(outfile,heading); writeln(outfile); writeln(auxfile,heading); write(display,'What is the solvent? '); getcompound('solvent molecule ',rsolv,qsolv,solvindex); ml:-mw[solvindex]; write(display,'What is the polymer? '); getcompound('polymer repeat unit ',rpoly,qpoly,polyindex); m2r:-mw[polyindex]; write(display,'Enter the temperature in K '); readln(infile,tk); writeln(display, 'Enter inf diln wt frac act coef, or 0 if unknown'); readln(infile,omegalinf); if omegalinf - 0.0 then begin writeln(display, 'Enter known activity or act coef, followed by:'); getactunits; read(infile,act,actunit); while actunit - ' ' do read(infile,actunit); readln(infile); writeln(display,'Enter concentration, followed by unitsz'); getconcunits; read(infile,conc,concunit); while concunit - ' ' do read(infile,concunit); readln(infile); if concunit - 'x' then getmolecwts; wl:-convertconc(conc,concunit); tempomegal:-convertact(act,actunit); omegalinf:-findinfact(wl,tempomegal); end 222 Table D-3 (cont'd.). else begin writeln(display,'Enter units of activity or act coef data'); getactunits; repeat read(infile,actunit); until actunit <> ' '; readln(infile); writeln(display,'Enter units of concentration data'); getconcunits; repeat read(infile,concunit); until concunit <> ' '; readln(infile); end; writeln(display,'Enter polymer density or sp vol followed by'); writeln(display,' d for density'); writeln(display,' v for specific volume'); writeln(display,' m for molar volume'); read(infile,rhoZ,rhounit); while rhounit - ' ' do read(infile,rhounit); readln(infile); if (concunit <> 'x') and (rhounit - 'm') then getmolecwts; rho2:-convertrho(rh02,rhounit,m2); write(display,'Enter solvent '); case rhounit of 'd': write(display,'density '); 'v': write(display,'specific volume '); 'm': write(display,'molar volume ') end; readln(infile,rhol); rhol:-convertrho(rhol,rhounit,m1); densityratio:-rh02/rhol; chi:-~ln(e/omegalinf*densityratio); firstsetz-nil; repeat writeln(display,'Enter conc followed by activity or act coef'); writeln(display,'or an out of range concentration to stop'); readln(infile,conc,act); wl:-convertconc(conc,concunit); omegalexp:-convertact(act,actunit); endofdata:-(wl>l.0) or (wl<0.0); if not endofdata then begin nextsetz-firstset; lastsetz-firstset; foundz-false; while not found do begin if nextset - nil then foundz-true 223 Table D-3 (cont'd.). else begin found:-nextset“.concen > wl; if not found then begin lastsetz-nextset; nextset:-nextset‘.next end end end; new(ptr); ptr“.concen:-wl; ptr“.activity:-omega1exp; ptr“.next:-nextset; if lastset - nextset then firstsetz-ptr else lastset“.next:-ptr; end until endofdata; write(outfile, 'Infinite dilution wt frac activity coefficient was '); writeln(outfile,omegalinf:10:4); writeln(outfile,'Flory-Huggins chi parameter was ',chi:8:4); writeln(outfile); writeln(outfile, ' Wt Frac Activity Coefficients and Percent Error'); writeln(outfile, ' Solvent Exptl ASOGVSP Flory-Huggins UNIFAC-FV ASOG'); writeln(outfile); ptrz-firstset; momentOerror:-0.0; for model :- asogvsp to asog do momentlerror[mode1]:-0.0; while ptr <> nil do begin wl:-ptr‘.concen; omegalexpz-ptr‘.activity; ptr:-ptr“.next; momentOerror:-moment0error+1.0; write(outfile,wl:8:3,0megalexp:8:3); for model :- asogvsp to asog do begin omegal[model]:-findact(model); pctdiff[mode1]:-(omegal[model]/omega1exp~1.0)*100.0; momentlerror[model]:nmomentlerror[model] +abs(pctdiff[model]); write(outfile,omegal[model]:8:3,pctdiff[model]:6:l) end; 224 Table D-3 (cont'd.). writeln(outfile); end; writeln(outfile); write(outfile,'Avg pct error '); for model :- asogvsp to asog do write(outfile,momentlerror[model]/ momentOerrorzl4zl); writeln(outfile); page(outfile); until eof(infile); rewrite(data); writeln(data,numgroups); for i:-1 to numgroups do writeln(data,rk[i]:7:4,qk[i]26:3 .groupnamelil): for iz-l to numgroups do begin for j:-1 to numgroups do write(data,a[i,j]:10:2); writeln(data) end; writeln(data,numcompounds); for i:-l to numcompounds do begin writeln(data,groupsinit[i],mw[i]:10:2,compoundname[i]); for j:-l to groupsinit[i] do write(data,numgroup[i,j],group[i,j]); writeln(data); end end. APPENDIX E. Program Used to Apply Thermodynamic Models Using Best Fit of All Experimental Data. The program listed below was used to generate the results in Appendix C from the original data in Appendix A. These results were presented in Chapter 2 of the dissertation. Input in the form of polymer-solvent activity data at a given temperature is processed to fit adjustable parameters and then the predictions of the VSP method using no residual interaction, a Flory-Huggins type residual interaction, and an ASOG-KT residual interaction are compared to experimental results. Refer to Appendices A and C for a more detailed description of the input data format and the output produced by the program. Directions for program compilation, linking, and execution are identical to those given in Appendix D. Much of the program presented in this appendix is identical to that presented in Appendix D. Only the data analysis itself is substantially different. There are some points in program execution where terminal input may be necessary even if Infile is taking input from an external file. This will occur if a new compound name (not previously used during any execution of this program) is specified on line 2 or 3 of a data set. In this case, prompt messages will appear on the monitor for input to be 225 226 entered from the keyboard (regardless of your choices for Infile and Display). The input will consist of the functional group description of the compound, its molecular weight, and, if any new functional groups are specified, ASOG-KT interaction parameters must also be supplied as input. The functional group and compound information is stored on a file named VSP.TAB. The format of this file is given as Table E-l. Table E-l. Format of Functional Group and Compound Information File. Line 1: N, the number of compounds (limit of 50), followed by M, the number of functional groups (limit of 20). N sets of lines follow, each set containing for a compound: Line 1: J, the number of functional groups in the compound as an integer, followed by the molecular weight of the compound or repeat unit as a real value, followed by the compound name (maximum 20 characters). Lines 2 to J+1: For each functional group in the compound, the number of times that group occurs in the compound as a real value (ASOG-KT rules for counting groups allow fractional weighting) followed by the position in which the group appears in the list of M functional groups later in the file. After all N sets have been completed, there are M lines, each containing a group name (maximum 7 characters). Following these, there are two sets of M lines containing the ASOG-KT interaction parameters. Each line in the first set contains the ASOG-KT temperature-independent interaction parameters, ai , for group i with each of the M groups j, in order, as real values. Ealh line in the second set contains the ASOG-KT temperature-dependent interaction parameters, b , for group i with each of the M groups j, in order, as real values. 3 Dr. Eric A. Grulke of Michigan State University has a disk copy of this program and necessary files. Source code for the program is stored in file VSP.PAS, and the executable version of the program is stored in 227 file VSP.EXE. The program is executed by giving the name of the file at the DOS prompt, i.e., A:VSP (assuming the floppy drive is device A:). Table D~2. Source Code for Program to Fit All Solvent Concentration Thermodynamic Data. program asogvsp(infile,outfile,display,auxfi1e,input,output); type setptr - Adataset; dataset - record concen:rea1; activityzreal; nextzsetptr end; modeltype - (vsp,flory,fhvsp,asogvsp); nametype - string(20); realarray - array[l..20] of real; intarray - array[l..20] of integer; const e - 2.7182818; grouptablesize - 20; compoundtablesize - 50; blank - ' '; var wl,omegalexp,omegalinf,conc,act,m1,m2,m2r,chi,densityratiozreal; omegal,tempomega1,lng1,rhol,rh02,tk:real; sumsqrerror:array[modeltype] of real; olinf,olinf2,glinf,fholinf,fhglinf,res:real; mean,sd:real; z,tstat:array[mode1type] of real; wlray,olray,lnglray:realarray; relerr:array[modeltype] of realarray; count,i,j,solvindex,polyindex:integer; nptszinteger; stat:array[modeltype] of integer; rank:array[modeltype] of intarray; numcompounds:0..compoundtablesize; numgroupszo..grouptablesize; endofdata,found:boolean; concunit,actunit,rhounit,ch:char; heading:string(80); compoundname:array[l..compoundtablesize] of nametype; compoundgroups:array[1..compoundtablesize] of integer; grouptable:array[l..compoundtablesize,l..grouptablesize] of integer; groupcount:array[l..compoundtablesize,l..grouptablesize] of real; mw:array[1..compoundtablesize] of real; groupname:array[1..grouptablesize] of nametype; groupm,groupn,groupa:array[l..grouptablesize,l..grouptablesize] of real; 228 Table E~2 (cont'd.). ptr,firstset,1astset,nextset:setptr; modelzmodeltype; infile,outfile,display,auxfile,data:text; procedure getactunits; begin writeln(display,' a for activity'); writeln(display,' w for wt frac activity coef'); writeln(display,' x for mol frac activity coef'); end; procedure getconcunits; begin writeln(display,' w for weight fraction solvent'); writeln(display,' m for mass ratio solvent/polymer'); writeln(display,' x for mole fraction solvent'); end; procedure getmolecwts; begin write(display,'Enter MW of polymer, MW of solvent '); readln(infile,m2,ml) end; function convertconc(conc:real;concunit:char):real; begin case concunit of 'w': convertconcz-conc; 'x': convertconc:-conc/(conc+(m2/ml)*(l.0-conc)); 'm': convertconc:-1.0~l.0/(1.0+conc) end end; function convertact(act:rea1;actunit:char):real; begin case actunit of 'w': convertact:-act; 'a': convertactz-act/wl; 'x': convertact:-act*conc/wl; end end; function convertrho(rho:real;rhounit:char;mw:real):real; begin case rhounit of 'd': convertrhoz-rho; 'v': convertrho:-l.0/rho; 'm': convertrhoz-mw/rho; end end; procedure getcompound(solvorpoly:nametype;var indexzinteger); var name:nametype; izinteger; getnewzboolean; 229 Table E~2 (cont'd.). procedure getgroups; var i,gindex:integer; gnameznametype; getnewzboolean; procedure getgroupparams; var i:integer; begin if numgroups > 1 then writeln('Enter interaction parameters m and n'); for i:-1 to numgroups-1 do begin write('(',gname,groupname[i],') ’); readln(groupm[numgroups,i],groupn[numgroups,i]); end; for i:-l to numgroups-1 do begin write('(',groupname[i],gname,') '); readln(groupm[i,numgroups],groupn[i,numgroups]); end; groupm[numgroups,numgroups]:-0.0; groupn[numgroups,numgroups]:-0.0; end; begin iz-O; repeat writeln('Enter name of the next group in ',name); write('or return to stop entering groups '); readln(gname); if gname <> blank then begin i:-i+1; gindex:-0; getnew:-false; if numgroups > 0 then repeat gindex:-gindex+1; until (groupname[gindex] - gname) or (gindex - numgroups) else getnew:-true; if (numgroups > 0) and (groupname[gindex] <> gname) then getnewz-true; 230 Table E~2 (cont'd.). if getnew then begin numgroups:-numgroups+1; groupname[numgroups]:-gname; gindexz-numgroups; getgroupparams; end; grouptable[index,i]:-gindex; write('How many are in ', name); readln(groupcount[index,i1); end; until gname - blank; compoundgroups[index]:-i; end; begin readln(infile,name); index:-0; getnewz-false; if numcompounds > 0 then repeat index:-index+l; until (compoundname[index] - name) or (index - numcompounds) else getnew:-true; if (numcompounds > 0) and (compoundname[index] <> name) then getnewz-true; if getnew then begin numcompounds:-numcompounds+1; compoundname[numcompounds]:-name; indexz-numcompounds; writeln('For ',name); write('Enter molecular weight of the ',solvorpoly); readln(mw[index]); getgroups; end; end; procedure generatea; var i,jzinteger; begin for iz-l to numgroups do for j:-1 to numgroups do groupa[i,j]:- exp(groupm[i.j1+groupnli.J]/tk); end; 231 Table E~2 (cont'd.). function enthpart(w1:real):real; type grouptype - array[l..grouptablesize] of real; var kzinteger; xl,x2,total:real; group:array[l..grouptablesize] of integer; num:array[l..grouptablesize] of real; bigx,bigxstar:grouptype; function gamma(x:grouptype; k:integer):real; var l,m:integer; sum,den:real; begin sum:-0.0; for 1:-1 to numgroups do sum:-sum+x[l]*groupa[k,l]; gamma:-~ln(sum)+l.0; sum:-0.0; for 1:-1 to numgroups do begin den:-0.0; for m:-1 to numgroups do den:-den+x[m]*groupa[l,m]; sum:-sum+x[1]*groupa[l,k]/den; end; gammaz-result(gamma)~sum; end; begin x1:-wl/m1/(w1/m1+(l.0-w1)/m2r); x2:-1.0-xl; total:-0.0; for k:-l to numgroups do bigxstar[k]:-0.0; for k:-1 to compoundgroups[solvindex] do begin group[k]:-grouptab1e[solvindex,k]; num[k]:-groupcount[solvindex,k]; bigxstar[group[k]]:-num[k]; total:-total+bigxstar[group[k]]; end; for kz-l to numgroups do bigxstar[k]:-bigxstar[k]/total; total:-0.0; for kz-l to numgroups do bigx[k]:-0.0; for kz-l to compoundgroups[solvindex] do bigx[group[k]]:-x1*num[k]; for k:-1 to compoundgroups[polyindex] do bigx[grouptable[polyindex,k]]:-bigx[grouptab1e[polyindex,k]1+ x2*groupcount[polyindex,k]; for kz-l to numgroups do total:-total+bigx[k]; for kz-l to numgroups do bigx[k]:-bigx[k]/total; 232 Table E~2 (cont'd.). enthpart:-0.0; for k:-1 to compoundgroups[solvindex] do enthpart:-resu1t(enthpart)+num[k]*(gamma(bigx,group[k]) ~gamma(bigxstar,group[k])); writeln(auxfile,'enthpart of ',w1,': ',result(enthpart)); end; function findact(mode1:modeltype):real; var y,phil,phi2:rea1; begin case model of vsp: begin y:-w1+(e/olinf)*(1.0~w1); findact:-exp((y-w1)/y)/y end; flory: begin phil:-densityratio*wl/(densityratio*wl+(1.0-wl)); phiZ:-l.0-phi1; findact:-exp(ln(phi1)+chi*phi2*ph12+phi2)/wl end; fhvsp: begin y:-w1+(e*fhglinf/fholinf)*(1~w1); findact:-exp((y-w1)/y*(y+(y-wl)*ln(fhglinf))/y)/y; end; asogvsp: begin y:-wl+(e*glinf/olinf2)*(l.0-w1); findact:-exp((y-wl)/y+1ngl)/y end; end end; procedure fitparams(var chi,olinf,olinf2,glinf,fholinf,fhglinf,res:real; wlray,olray:rea1array; nptszinteger); const delta - 0.0001; var i:integer; resl,res2,res3,change,size:real; deriv,deriv2:rea1; procedure findchi; var i:integer; phi1,ph12,num,den:real; begin numz-O; denz-O; 233 Table E~2 (cont'd.). for i:-1 to npts do begin phiZ:-(l~wlray[i])/(densityratio*wlray[i]+l~wlray[i]); phil:-1~phi2; num:-num+sqr(phi2)*(ln(w1ray[i]*olray[i]/phil)~phi2); den:-den+sqr(sqr(phi2)); end; chi:-num/den; end; procedure findolinf(var res:real; olinf,g1inf:real); var i:integer; gl,wtavg,size:real; begin resz-O; for i:-1 to npts do begin size:-e*glinf/olinf; wtavg:-wlray[i]+size*(l~wlray[i]); if glinf <> 1.0 then g1:-1nglray[i] else g1:-0.0; res:-res+sqr(ln(olray[i])+1n(wtavg)~size*(1~w1ray[i])/wtavg~gl); end; end; procedure findolgl(var res,ol,g1:real; sizezreal); var i:integer; r2,num,den,chi:real; rlzrealarray; begin numz-O; denz-O; for i:-1 to npts do begin r1[i]:-wlray[i]/(wlray[i]+size*(1~wlray[i])); r2:-l-rl[i]; num:-num+sqr(r2)*(ln(w1ray[i]*olray[i]/rl[i])~r2); den:-den+sqr(sqr(r2)); end; chiz-num/den; gl:-exp(chi); ol:-e*gl/size; resz-O; 234 Table E~2 (cont'd.). for i:-1 to npts do begin r2:-l~rl[i]; res:-res+(ln(wlray[i]*olray[i]/rl[i])~r2~chi*sqr(r2)) *(1-1/rl[i]+2*chi*r2)*size/ol*(l-wlray[i])/wlray[i] *sqr(r1[1l); end; count:-count+l; end; begin countz-O; findchi; olinf:-densityratio*exp(l+chi); olinf2z-olinf; writeln(auxfile,'vsp'); repeat findolinf(resZ,olinf+delta,1.0); findolinf(resl,olinf—delta,1.0); findolinf(re33,olinf,1.0); deriv:-(resZ-resl)/(2*delta); deriv2:-(res2+resl~2*res3)/sqr(de1ta); if deriv2 > 0 then changez-deriv/deriv2 else change:-~deriv/deriv2; olinfz-olinf-change; writeln(auxfile,resZ,resl,res3,olinf,deriv,deriv2); until abs(change) < 0.0001*abs(olinf); glinf:-exp(enthpart(0.0)); for iz-l to npts do lnglray[i]:-enthpart(wlray[i]); writeln(auxfile,'asogvsp, glinf - ',glinf); repeat findolinf(res2,olinf2+delta,glinf); findolinf(resl,olinf2-delta,glinf); findolinf(res3,olinf2,glinf); deriv:-(re32~resl)/(2*de1ta); deriv2:-(resZ+resl~2*res3)/sqr(delta); if deriv2 > 0 then changez-deriv/deriv2 else change:-~deriv/deriv2; olinf2:-olinf2~change; writeln(auxfile,resZ,resl,res3,olinf2,deriv,deriv2); until abs(change) < 0.0001*abs(olinf2); sizez-l; repeat findolgl(resZ,fholinf,fhglinf,size+delta); findolgl(resl,fholinf,fhglinf,size~delta); findolgl(res,fholinf,fhglinf,size); deriv:-(re52-resl)/(2*delta); changez-res/deriv; 235 Table E~2 (cont'd.). if change < size then size:-size~change else sizez-size/Z; if fhglinf > fholinf then size:-e/(4*olinf); writeln(auxfile,size,change,deriv,res); until abs(change) < 0.00001*size; writeln(auxfile,'Total calls to fhvsp: ',countz4); end; procedure sortrank(error:realarray; var rank:intarray; nzinteger); var i,j,tempzinteger; trankzintarray; begin for iz-l to n do trank[i]:-i; for i:-l to n-l do for j:-i+l to n do if error[trank[i]] > error[trank[j]] then begin temp:-trank[i]; trank[i]:-trank[j]; trank[j]:-temp; end; for i:-1 to n do rankltrank[i]]:-i; end; procedure stattest(test:integer; rank:intarray; nzinteger; var statzinteger; var mean,sd,z,tstat:real); var i:integer; tzreal; begin statz-O; case test of l: begin for i:-l to n do stat:-stat+i*rank[i]; mean:-n*(n+l)*(n+l)/4; sd:-n*(n+1)/12*sqrt(float(n~1)); end; 2: begin for i:-1 to n-l do stat:-stat+sqr(rank[i+l]-rank[i]); mean:-n*(n~l)*(n+l)/6; sd:-sqrt(n*(n~2)*(n+1)*(5*n*n~2*n-9)/180); end; 3: begin for i:-l to n-l do stat:-stat+abs(rank[i+l]~rank[i]); mean:-(n-l)*(n+l)/3; sd:-sqrt((n-2)*(4*n*n~3*n~7)/90); end; end; 236 Table E~2 (cont'd.). z:-(stat-mean)/sd; t:-1/(l+0.33267*abs(z)); tstat:-exp(~z*z/2)/sqrt(2*3.14159)* t*(0.4361836+t*(~0.1201676+t*0.9372980)); if test > 1 then tstatz-l-tstat else tstat:-l~2*tstat; end; begin (* main program *) reset(infile); ' assign(data,'vsp.tab'); reset(data); rewrite(outfile); rewrite(display); rewrite(auxfile); readln(data,numcompounds,numgroups); for i:-l to numcompounds do begin readln(data,compoundgroups[i],mw[i],compoundname[i]); for j:-l to compoundgroups[i] do readln(data,groupcount[i,j],grouptab1e[i,j]); end; for i:-1 to numgroups do readln(data,groupname[i]); for iz-l to numgroups do begin for j:-1 to numgroups do read(data,groupm[i,j]); readln(data); end; for iz-l to numgroups do begin for j:-1 to numgroups do read(data,groupn[i,j]); readln(data); end; repeat writeln(display,'Enter a heading for this data set'); readln(infile,heading); writeln(outfile,heading); writeln(outfile); writeln(auxfile,heading); write(display,'What is the solvent? '); getcompound('solvent molecule ',solvindex); ml:-mw[solvindex]; write(display,'What is the polymer? '); getcompound('polymer repeat unit ',polyindex); m2r:-mw[polyindex]; write(display,'Enter temperature in Kelvin '); readln(infile,tk); writeln(display, 'Enter inf diln wt frac act coef, or 0 if unknown'); readln(infile,omegalinf); 237 Table E~2 (cont'd.). if omegalinf - 0.0 then begin writeln(display, 'Enter known activity or act coef, followed by:'); getactunits; read(infile,act,actunit); while actunit - ' ' do read(infile,actunit); readln(infile); writeln(display,'Enter concentration, followed by units:'); getconcunits; read(infile,conc,concunit); while concunit - ' ' do read(infile,concunit); readln(infile); if concunit - 'x' then getmolecwts; wl:-convertconc(conc,concunit); tempomegal:-convertact(act,actunit); new(firstset); firstset‘.concen:-wl; firstset“.activity:-tempomegal; firstset“.next:-nil; end else begin writeln(display,'Enter units of activity or act coef data'); getactunits; repeat read(infile,actunit); until actunit <> ' '; readln(infile); writeln(display,'Enter units of concentration data'); getconcunits; repeat read(infile,concunit); until concunit <> ' '; readln(infile); firstsetz-nil; end; writeln(display,'Enter polymer density or sp vol followed by'); writeln(display,' d for density'); writeln(display,' v for specific volume'); writeln(display,' m for molar volume'); read(infile,rhoz,rhounit); while rhounit - ' ' do read(infile,rhounit); readln(infile); if (concunit <> 'x') and (rhounit - 'm') then getmolecwts; rhoZ:-convertrho(rh02,rhounit,m2); write(display,'Enter solvent '); case rhounit of 'd': write(display,'density '); 'v': write(display,'specific volume '); 'm': write(display,'molar volume ') end; 238 Table E~2 (cont'd.). readln(infile,rhol); rhol:-convertrho(rhol,rhounit,m1); densityratioz-rhOZ/rhol; repeat writeln(display,'Enter conc followed by activity or act coef'); writeln(display,'or an out of range concentration to stop'); readln(infile,conc,act); wl:-convertconc(conc,concunit); omegalexp:-convertact(act,actunit); endofdata:-(wl>l.0) or (wl<0.0); if not endofdata then begin nextset:-firstset; 1astset:-firstset; foundz-false; while not found do begin if nextset - nil then foundz-tru else ‘ begin found:-nextset‘.concen > wl; if not found then begin 1astsetz-nextset; nextset:-nextset“.next end end end; new(ptr); ptr“.concen:-w1; ptr“.activityz-omegalexp; ptr‘.next:-nextset; if lastset - nextset then firstset:-ptr else lastset‘.next:-ptr; end until endofdata; ptr:-firstset; npts:-0; while ptr <> nil do begin npts:-npts+1; wlray[npts]:-ptr“.concen; olray[npts]:-ptr“.activity; ptr:-ptr“.next; end; 239 Table E~2 (cont'd.). generatea; fitparams(chi,olinf,olinf2,glinf,fholinf,fhglinf,res,wlray,olray, nPCS): writeln(outfile,'Results of least squares fit:'); writeln(outfile); write(outfile,'VSP inf diln wt frac activity coefficient: '); writeln(outfile,olinf:10:4); writeln(outfile,'Flory-Huggins chi parameter: ',chi:8:4); write(outfile,’FH-VSP inf diln parameters: '); write(outfile,’ wt frac act coeff',fholinf:8:4); write(outfile,’ enth coeff',fhglinf:8:4); writeln(outfile); write(outfile,’ASOG-VSP inf diln parameters: '); write(outfile,’ wt frac act coeff',olinf2:8:4); write(outfile,’ enth coeff',g1inf:8:4); writeln(outfile); writeln(outfile); writeln(outfile, ' Wt Frac Activity Coefficients and Percent Error'); writeln(outfile, ' Solvent Exptl VSP Flory-Huggins FH-VSP', ' ASOG-VSP'); writeln(outfile); for model :- vsp to asogvsp do sumsqrerror[mode1]:-0.0; for i:-l to npts do begin wl:-w1ray[i]; omegalexp:-olray[i]; lngl:-lnglray[i]; write(outfile,wl:8:3,0mega1exp:8:3); for model :- vsp to asogvsp do begin omega1:-findact(model); relerr[model,i]:-ln(omegal/omegalexp)*100; write(outfile,omegal:8:3,relerr[model,i]:6:l); sumsqrerror[mode1]:-sumsqrerror[model]+sqr(relerr[model,i]); end; if sumsqrerror[model] < 0.0 then sumsqrerror[model]:-0.0; writeln(outfile); end; writeln(outfile); write(outfile,'Standard pct err'); for model :- vsp to flory do write(outfile, sqrt(sumsqrerror[modell/(npts-l)):l4:l); if npts > 2 then write(outfile,sqrt(sumsqrerror[fhvsp]/(npts~2)):1421); write(outfile,sqrt(sumsqrerror[asogvsp]/(npts~l)):14:l); writeln(outfile); writeln(outfile); writeln(outfile); 240 Table E~2 (cont'd.). for model :- vsp to asogvsp do begin sortrank(relerr[model],rank[mode1],npts); end; writeln(outfile,'Analysis of model error randomness'); writeln(outfile); write(outfile,'Sum sqr rank difference test: '); for model :- vsp to asogvsp do stattest(2,rank[mode1],npts,stat[model],mean, sd,z[mode1],tstat[model]); writeln(outfile,'mean - ',mean:7:2,' sd - ',sd:6:2); writeln(outfile); write(outfile,'Test statistic '); for model :- vsp to asogvsp do write(outfile,stat[model]:14); writeln(outfile); write(outfile,'Normal (Z) '); for model :- vsp to asogvsp do write(outfile,z[model]:14:3); writeln(outfile); write(outfile,'Reject level '); for model :- vsp to asogvsp do write(outfile,tstat[model]:14:6); writeln(outfile); writeln(outfile); write(outfile,'Sum abs rank difference test: '); for model :- vsp to asogvsp do stattest(3,rank[model],npts,stat[model],mean, sd,z[model],tstatlmodel]); writeln(outfile,'mean - ',mean:7:2,' sd - ',sd:6:2); writeln(outfile); write(outfile,'Test statistic '); for model :- vsp to asogvsp do write(outfile,stat[mode1]:14); writeln(outfile); write(outfile,'Norma1 (Z) '); for model :- vsp to asogvsp do write(outfile,z[mode1]:14:3); writeln(outfile); write(outfile,'Reject level '); for model :- vsp to asogvsp do write(outfile,tstat[model]:l4z6); writeln(outfile); writeln(outfile); writeln(outfile); if (chi >- 0.5) or (fhglinf >- sqrt(e)) then begin writeln(outfile,'Phase separation behavior prediction'); writeln(outfile); if chi >- 0.5 then writeln(outfile,'Flory-Huggins model: ', 'wt frac - ',1/(l+densityratio*(2*chi~1)):5 3); if fhglinf >- sqrt(e) then writeln(outfile,'FH-VSP model: ', 'wt frac - ',1/(l+fholinf/(e*fhg1inf)* (2*ln(fhglinf)-l)):5:3); end; 241 Table E~2 (cont'd.). page(outfi1e); until eof(infile); rewrite(data); writeln(data,numcompounds,numgroups); for iz-l to numcompounds do begin writeln(data,compoundgroups[i]:3,mw[i]:7:2,compoundname[i]); for j:-l to compoundgroups[i] do writeln(data,groupcount[i,j]:5:2,grouptable[i,j]:3); end; for iz-l to numgroups do writeln(data,groupname[i]); for iz-l to numgroups do begin for jz-l to numgroups do write(data,groupm[i,j]:12:4); writeln(data); end; for i:-1 to numgroups do begin for j:-1 to numgroups do write(data,groupn[i,j]:12:4); writeln(data); end; end. APPENDIX F. Derivation of Equations in "Generalized Correlation for Solvent Activities in Polymer Solutions" This appendix contains a more detailed derivation of the equations presented in the reprint article "Generalized Correlation for Solvent Activities in Polymer Solutions". This article was included as part of Chapter 2 of the dissertation. The major source of the equations which were used to derive the results was the ASOG model (Derr and Deal, 1969). In this appendix, equation numbers refer to the reprint article itself, beginning on page 13 of the dissertation. New equations not included in the article are numbered with a preceding letter F, e.g., F-l, F-2, etc. Eqs 1 to 4 are taken directly from the ASOG model. Only eqs 1 and 2 are used to derive further results; eqs 3 and 4 were presented for the sake of completeness. Eq 5, actually an inequality, merely states the assumption that the solvent molecule is much smaller than the polymer molecule. DERIVATION 0F EQUATION 6 Eq 6 is the first equation which was derived. The superscript w in the 242 243 equations shown refers to the limiting value of a variable as the mole fraction (or weight fraction) of solvent (component 1) approaches zero. Beginning with eq 2 and letting x1 approach zero (x2 will approach one) R1 - s1 / (51x1 + 82x2) (2) Q R1 - s1 / (51(0) + 32(1)) - sl / 52 (F-l) then substituting this value into eq 1 to give the mole fraction activity coefficient at infinite dilution 1n 115 - 1 - R1 + 1n R1 (1) In 715” - 1 - R1” + 1n R1” - 1 - (51/32) + 1n (51/52) (F-2) and finally using eq 5 to eliminate one of the terms in eq F-2 gives eq 6. 51 << 32 (5) San 1n 11 - 1 + 1n (51/82) (F-3) 118m - exp [1 + 1n (81/32)] - exp(l) - Sl/S2 - e-Sl/S2 (6) In the article, the superscript S was suppressed since only the size component of activity was being considered. It should be pointed out that eq 1, taken from the ASOG model, leaves out a factor which appears in the athermal Flory-Huggins equation. For our purposes, this factor would equal (1 ~ 81/82), and would differ negligibly from one after the assumption of eq 5 is made. Eq 5 restricts the application of the results to binary solutions of low molecular weight solvents in high molecular weight polymers. 244 DERIVATION 0F EQUATION 7 Eq 7 follows from the definition of mole fraction and weight fraction concentration variables and activity coefficients at infinite dilution, where x approaches zero and x 1 approaches one. 2 a1 - 11x1 - Olwl (F-4) 01 - ylxl/w1 (F-5) w1 - Mlx1 / (Mlx1 + M2x2) (F'6) 01 - 11 (Mlx1 + szz) / M1 (F-7) 0‘” wMO+M1 M ”M M (7) 1 - 71 ( 1( ) 2( )) / 1 71 2 / 1 Eq 8 follows from substitution of eq 6 into eq 7, and eq 9 follows from substitution of eq 10 into eq 8. Eq 10 expresses the fact that the size ratio equals the ratio of molecular weights for chemically similar polymer-solvent pairs. This is the assumption which is removed by the variable size parameter concept. Instead, the effective size ratio, shown in eq 11, is given by rearrangement of eq 8, treating the size ratio as an unknown and the activity coefficient 01do as a known value. Eqs 12 and 13 are transformations from weight fraction composition variables to mole fraction, analogous to eq F~6 above. DERIVATION OF EQUATION 14 Eq 14 for the size ratio R1 was derived in several steps, as shown below. Eq 2 was divided through by S then eq 11 was substituted for 1’ S2/Sl’ and eqs 12 and 13 were substituted for x1 and x2. The final 245 expression was simplified by multiplying through by Ml/MZ. R1 - s1 / (81x1 + 82x2) (2) R1 - 1 / (x1 + (82/51) x2) (F-8) 82/31 - (e/nl°> (Hz/M1) (11) R1 - 1 / (x1 + (e/Olm) (Mz/Ml) x2) (F-9) x1 - (M2/M1) w1 / (("2/M1) w1 + w2> (12) x2 - w2 / ((MZ/Ml) w1 + “2) (13) R1 - ((“2/M1) w, + "2) / (("2/"1) w1 + (e/ol“) (Hz/M1) "2) R1 - (w1 + (Ml/M2) “2) / (w1 + (e/nl”) “2) (14) In the article, w2 was replaced by 1~w to reinforce the fact that the l expressions derived were functions of a single concentration variable. The variable w2 has been retained in this appendix for clarity in all equations. DERIVATION OF EQUATION 15 Eq 15 is easily derived from substitution of eq 12 into eq F-S, cancelling w1 and multiplying through by Ml/Mz. 01 - ylxl/w1 (F-5) x1 - (Hz/M1) w1 / ((“2/M1) w1 + w2> <12) 01 - 71 ("z/M1) w1 / w1 (042/141) w1 + "2) (F41) 0, - 11 / (w1 + (Ml/M2) ”2) (15) DERIVATION 0F EQUATION 16 246 To derive eq 16, begin by substituting eq 14 into eq 1 (superscript S suppressed in eq 1), noting the simplification for 1~R1 given as eq F-12. 1n 11 - 1 - R1 + 1n R1 (1) R1 - (wl + (Ml/M2) w2) / (w1 + (e/ol°) w2) (14) 1 - R1 - (e/nlco - Ml/Mz) w2 / (wl + (e/nl”) w2) (F~12) 1n 11 - (e/le - Ml/MZ) w2 / (wl + (e/nl”) w2) + 1n (w1 + (Ml/M2) wz) - 1n (w1 + (e/nl”) wz) (F-l3) Take the exponential of eq F-l3 and substitute it into eq 15 to produce eq F~15. Then use the assumption given by eq 17 to eliminate the molecular weight term, giving eq 16. The assumption is that the molecular weight of solvent is much smaller than that of polymer, and that 01co is not too large (the solution is not too nonideal). 11 - exp I (e/nl"o - Ml/M2> w2 / (w1 + (e/nl°) "2) 1 (pl + (Ml/M2) "2) / (w1 + e/nlco >> Ml/Mz (17) 01 - exp [ (e/nl”) w2 / (w1 + (e/01°) w2> 1 / (x1 + (e/nl°) “2) (16) CONVERGENCE OF THE ITERATION PROCEDURE FOR CALCULATING 01” IN EQUATIONS 18 TO 23 Eqs 18 to 23, defining an iteration procedure for calculating 0100 from 257 a finite concentration activity value, all follow from straightforward substitution involving eq 16 and eqs 18 to 23 themselves. The convergence properties of the iteration scheme were not examined in the article, except for the comment that the ”procedure converges quickly". Since the method consists of successive substitutions, a necessary condition for convergence in eq 20 is given by the magnitude of the derivative of the right hand side, i.e., |F'(Y)| < l. exptl exptl Y - exp (1 - w1 /Y - In 01 ) - E(y) (20) F'(Y) _ exp (1 _ wlexptl/Y _ 1n “lexpt1) . (w 1exptl/Y2 ) _ Y . (w 1exptl/Y2 )_ w lexptl/Y (F-16) Since Y is calculated as the result of the exponential function, it is necessarily positive. The experimental weight fraction of solvent, w exptl exptl l 1 iteration scheme would be unnecessary since the experimental activity , is also a positive quantity. (If w were zero, the value would already be at infinite dilution of solvent.) Consider the first two approximations of Y, Yo (given by eq 21), and Y1 (given by eq 22 with n - 1, shown here as eq F-l7). Yo - exp (1 - ln Olethl) (21) _ - exptl _ exptl _ Y1 exp (1 w1 /Yo In 01 ) (F 17) Yl - exp (1 - ln Olethl) - exp(- w ethl/Y0) _ - exptl _ Y0 exp( w1 /Yo) (F 18) Because the argument of the exponential function in eq F-18 is negative, and Y0 is positive, Y1 < YO can be concluded. Identical logic applies to the general case of Yn’ leading to this result. 248 Yo > Y1 > Y2 > . . ..> Yn_1 > Y“ (F-l9) Assume that the expression for F'(Y) given by eq F-l6 does not meet the necessary condition for convergence. IF'(Y)I - wlethl/Y 2 1 (p-20) This expression will take on its smallest value when its denominator is largest, i.e., when Y is largest. According to eq F-l9, this will occur when n equals zero on the initial iteration. For eq F-20 to hold for any value of Y, it must hold for Y as shown in eq F-Zl. The expression 0 for Yo from eq 21 can then be substituted to give eq F-22, and since the denominator is positive, the direction of inequality remains the same in eq F-23. wleXPtl/Yo 2 1 (F-21) Yo - exp (1 - ln olethl) (21) wleth1 / exp (1 - 1n olethl) 2 1 (p-22) wlethl 2 exp (1 - ln Olethl) (F-23) Working through the exponential function gives eq F-24, which can then be rearranged to eq F-25 using the definition of activity coefficient. wlexptl exptl exptl _ exptl w1 01 a1 2 exptl 2 e / 01 (F-24) e (F-25) Since the activity of a component cannot exceed unity when based on a pure component standard state, eq F-25 is a contradiction. Thus the original assumption of eq F-2O must be false, and the necessary 249 condition for convergence holds for the iteration scheme of successive substitutions described by eqs 18 to 23. Eq 24 is the Flory-Huggins equation for solvent activity, and eqs 25 and 26 are definitions of the volume fraction in a binary system with no volume of mixing. When eq 24 is rearranged for x, and eqs F-h, 16, 25 and 26 are substituted, the results are In a1 - 1n ¢1 + ¢2 + x¢22 (24) x - (1n a1 - 1n p1 - p2) / p22 (F-26> al - Olwl (F-h) 01 - exp 1 (e/01”> w2 / (w1 + (e/nl”) W2) 1 / (w1 + (e/ol”> w2> (16) In a1 - 1n ¢1 + ¢2 + x¢22 (24) ¢1 - (pZ/pl) "1 / ( (pz/pl) “1 + w2 ) (25) ¢2 - w2 / < (”z/P1) w1 + “2 > (26) In a1 - 1n ¢1 + ¢2 + x¢22 (24) x - [ (e/fllno + 1n w1 ) w2 / (w1 + (e/Olm) w2) - ln (“1 + (e/Olm) w2) - 1n (”z/P1) "1 + 1n ( (pz/pl) "1 + “2 ) - w2 / ( (pz/p1> "1 + w2 ) 1 - ( (”z/P1) w1 + w2 )2 / w22 (F-27) This result as shown in in eq F-29. through the error in eq denominator shown here. is simplified by grouping together all the logarithm terms eq F-28 and by dividing out individual w2 factors as shown When the factor ( (pz/pl) wl/w2 + l )2 is multiplied other three terms, eq 27 results. There is a typographical 27 of the article which shows a division by w2 in the of the logarithm term; multiplication by w is correctly 2 250 x - I (e/ol”> wz / (w1 + (e/01”> w2> wz / ( (”z/P1) "1 + "2 ) + 1n ( (wl + (pl/p2) w2) / (w1 + “2) > 1 ° ( (pZ/pl) wl + wz )2 / w22 x - [ (e/01°> wz / (w1 + (e/01”> w2> 1 / ( (pZ/pl) wl/w2 + 1 ) + 1n < (w1 + (pl/p2) w2> / (w1 + (e/01”> w2> ) 1 ° ( (pz/pl) wl/wz + 1 )2 x - ( (Pp/Pl) wl/w2 + 1 )2 - (e/nl”) w2 / (w1 + (e/nl”> "2) ( (pz/pl) "l/wz + 1 ) + < (pz/pl) wl/w2 + 1 )2 - 1n ( (w, + (pl/p2) w2> / (w1 + (e/01”> wz) > (F-28) (F-29) (27) Table F-l. Equations Used in "Generalized Correlation for Solvent Activities in Polymer Solutions". In 118 - l - R + In R 1 1 R1 - s1 / (81x1 + 82x2) 1 G 2 1 r 2 1 r * n 71 ' ”k1 “ k ' "k1 “ k k k x A In Pk - - 1n 2 XlAk1 + l - 2 ——l—lE- 1 1 z x A m In m 51 << 52 Sea 71 - exp [1 + 1n (81/82)] - exp(l) - 81/82 - e-Sl/S2 01 - 71 (M1(0) + M2(1)) / M1 - 11 M2 / M1 (1) (2) (3) (4) (5) (6) (7) (3) 251 Table F-l (cont'd.). (D 01 - e (9) 52/51 - 142/111 (10) 52/31 - (6/01 ) (HZ/M1) (11) x1 - (Hz/M1) W1 / ((MZ/Ml) wl + “2) (12) x2 - w2 / ((M2/Ml) w1 + w2> (13) R1 - (w1 + (Ml/M2) w2> / (w1 + (e/nl”) "2) (14) 01 - ‘11 / (W1 + (Ml/M2) 1:72) (15) 01 - exp [ (e/ol°) "2 / (w1 + (e/nl°) w2> 1 / (w1 + (e/01°) w2> (16) e/ol” >> 141/M2 (17) _ exptl m Y w1 + (e/O1 ) w2 (18) olethl - exp [ (Y - wleth1> / Y 1 / Y (19) Y - exp (1 - wlethl/Y - ln Olethl) - P(y) (20) YO - exp (1 - ln Olethl) (21) Yn - exp (1 - wl‘eXPtl/Yn.1 - 1n olethl) (22) film _ e (1 _ wlexptl) / (Y _ wlexptl) (23) ln a1 - ln ¢1 + ¢2 + x¢22 (24) ¢1 - (pz/pl) "1 / ( (pz/pl) “1 + w2 ) (25) p2 - w2 / ( (pz/pl) w1 + w2 ) (26) x - ( (Pz/Pl) wl/w2 + 1 )2 - (e/nl”) w2 / (w1 + (e/01°> w2) - < (pz/pl) wl/w2 + 1 ) + < (pz/pl) wl/w2 + 1 >2 - 1n < (w1 + (pl/p2) w2> / (w1 + (e/olm) w2> > (27) APPENDIX C. Derivation of Equations in "Prediction of Solvent Activities in Polymer Solutions Using an Empirical Free Volume Correction" This appendix contains a more detailed derivation of the equations presented in the manuscript article ”Prediction of Solvent Activities in Polymer Solutions Using an Empirical Free Volume Correction". This article was included as part of Chapter 2 of the dissertation. In this appendix, equation numbers refer to the manuscript article itself, beginning on page 21 of the dissertation. New equations not included in the article are numbered with a preceding letter G, e.g., G-l, G-2, etc. Eqs la and 1b are defining equations for solvent mole fraction activity coefficient size component and group interaction component and for the size component of solvent activity. Eqs 2 and 3 illustrate these definitions using the terms which make up the Flory-Huggins equation. Eqs 4 to 9 were presented as background from "Generalized Correlation for Solvent Activities in Polymer Solutions": these equations were either taken from the ASOG model (Derr and Deal, 1969) or were derived in Appendix F of this dissertation. 252 253 DERIVATION OF EQUATION 10 The first new equation in this article is eq 10, an extension of eq 9, which was derived in Appendix F. The key step in the derivation of eq 9 was assuming the activity coefficient 0 m was known and rearranging for l the unknown size ratio 82/81. The same approach is used to derive eq 10, but instead of considering only the size interaction component of solvent activity, als, the complete solvent activity consisting of both size and group interaction components is used as shown below. Eq 1a is written as a product rather than a sum of logarithms in eq G-l, then combined with the definition of weight fraction activity coefficient to give eq G-2. The conversion of weight fraction to mole fraction is made using eq G-3 to give eq G-4, which is evaluated at infinite dilution (as solvent concentration x1 approaches zero) to give eq G-S for the infinite dilution weight fraction activity coefficient 0 a 1 . lna-lnx+1n S+1 G (1) 1 1 11 n 11 a s G a1 - x111 11 (G-l) 0 - a /w - x S G (G 2 1 1 1 171 71 /"1 ' ) w1 - Mlx1 / (Mlx1 + M2x2) (5'3) 8 G 01 - 11 71 (Mlx1 + M2x2) / M1 (G-4) on San Goo Sco Geo 01 - 71 11 (111(0) + M2(1)) / M1 - 11 71 M2 / M1 (G-S) At this point, the expression for 115 in the ASOG model, eq 4, is evaluated at infinite dilution and substituted into eq G-5, using the fact that Sl/S2 is close to zero to simplify eq G-8. 254 3 ln 11 - l - R1 + 1n R1 (4) San co on In 11 - l - R1 + ln R1 (G-6) R1 - s1 / (81x1 + 82x2) (5) 0 R1 - s1 / (51(0) + 32(1)) - 31 / $2 (c-7) Sue ln 11 - 1 - 51/32 + ln sl/s2 - 1 + 1n 51/32 (G-8) Sun 71 - e - 51/52 (G-9) n m G” s s M (c-10) 1 ' e 71 ' 1/ 2 ' 2/"1 This expression is rearranged to give the size ratio 82/81 as a function of the other factors and resubstituted into eq 5. Transformations from mole fraction to weight fraction composition are used, and eq 10 is finally produced by assuming Ml/M2 is close to zero. 82/81 - (mom/01‘”) (“z/“1) (G-ll) R1 - s1 / (31x1 + 82x2) (5) R1 - 1 / (x1 + (32/51) x2) (c-12) R1 - 1 / (x1 + (evlc°°/01°°> (Hz/M1) x2) (64» x1 - (142/ml) w1 / “Hz/"1) w1 + up (c-m x2 - w2 / «142/Ml) w1 + "2) - (045) (M /M ) w + w R - 2 1 cl ”2 (G-16) (M2/M1) "1 + (811 /01 ) (Hz/M1) “2 w + (M /M ) w R - 1 g 2 2 (c-17) + (e1 com on) w w1 1 1 2 w1 R ' Ge e (10) w1 + (e1l /01 )w2 Eqs 11 and 12 illustrate how the infinite dilution group interaction 255 (residual) component of the activity coefficient, 116°, is derived from a functional expression for the residual component of the activity coefficient, 110. Eqs 13a and 13b define such a functional expression for an athermal solution, giving activity coefficients of unity at all concentrations. Eq 14a defines a Flory-Huggins type of residual interaction. DERIVATION OF EQUATION 143 To derive eq 14b, take w as zero in eq 10, and use the resulting R in l 2 eq 14a. R1” - <0) / ((0) + (evlcm/nl°> (1)) - o R2 - I - R1 - 1 - 0 - l (G-19) G * 2 In 11 - x R2 (14a) * a: In 110‘” - x (R, )2 - x"'<1>2 - x* Geo * 11 - eXP(x ) (14b) Eqs 15a to 15d are the standard ASOG model equations (Derr and Deal, 1969). Eq lSe merely states that these equations are to be evaluated at x1 - 0 to give the infinite dilution residual component of the activity coefficient. Eq 16 is identical to eq 16 (coincidentally) of "Generalized Correlation for Solvent Activities in Polymer Solutions" and is derived in Appendix F. 256 DERIVATION OF EQUATION 17 Eq 17 is analogous to eq 16, but using the more complex eqs 14a and 14b for the residual component rather than the athermal eqs 13a and 13b. The activity coefficient 01 is given as the product of a size interaction component, 118, and a group interaction component, 116, i.e., eq G—2. If only the expression x17ls/wl in eq G-2 is considered, the derivation of eq 16 in Appendix F applies, with the only difference in the result being the appearance of the additional factor 11cm in eq 10 for R1. Taking the result from Appendix F with the additional factor gives eq G-21. s c 01 ' a1/"1 ' x171 71 /“1 (6’2) w1 R - (10) 1 Goo co w1 + (e71 /01 )w2 Go (e11 /01m) "2 exp Go a: w1 + (e11 /01 ) w2 -11.. G x 7 S/w - 1 1 1 Go a w1 + (e11 /01 ) w2 11 (G-21) The factor 116 in eq G-2 is given by eqs 14a and 10, generating eq G-24. * ln 116 - x R22 (14a) w1 Rl - Gm m (10) w1 + (e71 /01 )w2 (also/015w2 R — 1 - R - 2 1 (G-22) Go on w1 + (e11 /01 )w2 257 Ga 0 2 (e1 /0 )w G * [ l l 2 ] (G-23) w 1n 1 - x Go 1 1 + (e11 /nl°°)w2 Go a 2 (e1 /0 )w 716 ' exp x* 1 1 2 (6'2“) " 2 Gen co 1 + (811 /01 )w Multiplying eqs G-2l and G-24 together produces eq G-25 for 01. Since the product of exponentials equals the exponential of the sum of the arguments, the expression can be simplified into eq G-26. Taking a common factor gives eq G-27, which is identical to eq 17 once the * substitution x - ln 116m from eq 14b is made. 5 c 01 ' x171 11 /V1 (evlcm/ol”) w2 , (8716m/01¢)W2 2 exp Gm @ ' exp X Cw o w1 + (e11 /01 ) w2 w1 + (e11 /01 )w2 Geo co w + (e1 /0 ) w 1 1 1 2 ’(c-25) Gan on Can an 2 exp (811 /01 ) "2 + x* (e11 /01 )w2 w + (e1 Goom do) w w + (e1 G°/n co)w l l l 2 1 1 l 2 w + (e1 /0 on) w l l l 2 [ w, [ , [ w2 H] exp 1 + X w w + (e7 G”/o do) w + (e1 G°/o ”)w l l 1 2 l 1 1 2 01 - Gm m (G-27) w1 + (e71 /O1 ) w2 (e7 Gm/n ”)w (e1 G”/o ”)w l l 2 1 1 2 Gm exp 1 + 1n 11 COD 00 Go: on 0 - 1 + (e11 /01 )w2 + (e71 /01 )w2 (17) 1 w + ( G”/o co)w 1 811 1 2 Eq 18 is a statistical formula for standard error in the expression ln a1 which is generally available in statistics texts discussing 258 analysis of variance. Table G-l. Equations Used in "Prediction of Solvent Activities in Polymer Solutions Using an Empirical Free Volume Correction”. 8 . C 1n a1 - ln x1 + In 11 + 1n 71 S G ln a1 - ln a1 + 1n 11 S S ln a1 - 1n (x171 ) - l - ¢1 + 1n ¢1 G 2 71 - x¢2 S 1n 11 - l - R1 + 1n R1 R1 - S1 / (Slx 82x 1 + 2) m w1 + (e/O1 )w2 w1 Goo co w1 + (e71 /01 )w2 G - f(w c 11 1) Q - £(0) 71 In 116 - In 116” a 0 G * 2 In 11 - x R2 Goo * 11 - eXP (x ) * In 116 - 2 uki (1n Pk - ln Pk ) k (1a) (1b) (2) (3) (4) (S) (6) (7) (8) (9) (10) (11) (12) (13a) (13b) (14a) (14b) (15a) 259 Table G-l (cont'd.). x A ln Pk - - ln 2 XIAk1 + 1 - 2 -l-lE- 1 1 2 x A m In In 1 r * 1 r 1 n k ' “ k(xl ' ) - Z x v / 2 2 x v xk 1 1 ki J 1 j 1) ln 710” - ln 116(x1 - 0) exp ( (e/nl‘”)w2 / [wl + (e/ol°)w2] } 01 - m w1 + (e/fl1 )w2 Go a: Gen co exp (err1 ml )w2 1 + (611 /01 )w2 1n 1 Ga] ] Gm w Go a 1 0 - w1 + (e71 /01 )w2 w1 + (e11 /01 )w2 1 Go a: w1 + (e11 /01 )w2 pred - 1n a exptl 2 2 (ln a ) s _ [ 1 1 11/2 '(n-d) (15b) (15c) (15d) (lSe) (16) (17) (18) APPENDIX B. Derivation of Equations in "Normalization and Bounding Properties Inherent in Solution of Groups Activity Coefficient Models” This appendix contains a more detailed derivation of the equations presented in the manuscript article "Normalization and Bounding Properties Inherent in Solution of Groups Activity Coefficient Models". This article was included as part of Chapter 3 of the dissertation. In this appendix, equation numbers refer to the manuscript article itself, beginning on page xx of the dissertation. New equations not included in the article are numbered with a preceding letter H, e.g., H-1, H-2, etc. Eq 1 is the ASOG definition of group mole fraction Xk taken for a binary solution whose molecules contain two distinct functional groups. Eqs 2 to 4 are the ASOG equations for calculation of the group interaction (residual) component of the activity coefficient, 116, from group mole fractions, Xk’ and group interaction parameters, Akl’ with group activity coefficients, Pk and Pki, as intermediate results. X + X Xk _ ...... 231-}---IB?-? ...... (1) (“11+“21)x1 + (“12+“22)x2 c 1 1 ln 71 - nli(ln P1 - 1n P1 ) + n21(ln F2 - ln P2 ) (2) x A x A In rk - -1n(X1Ak1 + XZARZ) + 1 - ----1-15--- - ----?-?¥--- (3) x A +x A x A +x A 1 ll 2 12 1 21 2 22 260 261 1 1n Pk - ln Pk (xi - l) (A) Eqs 5 and 6 merely define size-weighted fraction composition variables, c1, and group ratio variables, g1, neither of which depends on the size of the unit chosen to measure the functional group composition of a molecule. To derive eq 7, begin with eq 5 for c Eqs H-1 and H-2 are 1. eq 5 with i set to l and 1 set to 2. Similarly, eqs H—3 and H-4 are derived from the group ratio definition of eq 6. When eq H-l is divided by eq H-3, and eq H-2 is divided by eq H-4, eqs H-5 and H-6 result. When eq 1 is written with k equal to l as eq H-7, it is evident that the right hand side of eq 7 equals the sum of the right hand sides of eqs H-5 and H-6. The left hand sides of these equations must follow the same relationship, resulting in eq 7. Ci - ......... lg--g§--; ....... (5) (“11+“21)x1 + (“12+“22)x2 (n +n )x ,1 _ ......... 11--11--1 ....... (1-1) (“11+“21)x1 + (“12+“22)x2 (n +n )x ,2 - ......... 11--11--1 ....... (11-2) (“11+“21)x1 + (“12+“22)x2 g1 ' n21 / n11 (6) 1 + g1 — (n11 + n21) / n11 (H-3) 1 + g2 - (n12 + n22) / n12 (H-4) C n x --1 ............. 11-1 .......... (n-5, l+g1 (n11+n21)x1 + (n12+n22)x2 C n X --1 ............. 11-1 .......... (11-6) 262 x + X xx - ...... 131-1---131-1 ...... (1) (“11+“211x1 + (“12+“221x2 n 2‘ +11 x x1 - ....... 11-1----11-1 ...... (1-7) (“11+“211x1 1 (“12+“221x2 C C x1 - "-1- + ---1- (7) Eq 8 follows immediately from eq 2 when both sides are divided by n 11 and the ratio “Zi/nli is replaced by g1. In G - n (ln F - l P i) + n (ln F — In P 1) (2) 11 11 1 n 1 21 2 2 1“ 11 1 1 ------ - (ln P1 - 1n F1 ) + gi(ln F2 - ln F2 ) (8) n11 The derivation of eq 9 is algebraically lengthy, but follows directly from combining eqs 3, 4, 7, and 8. Begin with the expression for group activity coefficient given by eq 3, and substitute the expression given by eq 7 for group mole fraction X and use l-X for X . 1’ 1 2 X A X A m - 1M“, .XAk).1-----1-1*s ........ 1-21:--- (3) k 1 1 2 2 X A +X A X A +X A 1 ll 2 12 1 21 2 22 C C x1 - ---}-- + ---g-- (7) 1 + g1 l + g2 C C l 2 x2 - 1 - x1 - 1 - ------ + ------ (H-8) C1 c2 c1 c2 1n r - - 1n [ ( ------ + ...... ) Ak + (1 - ------ + ------ ) AR 1 k 1 + 1 + 1 1 + 1 + g 2 g1 g2 81 2 C C ("'}" + ___g_-) A1k l + g l + g l 2 + 1 - ------------------------------------------------- C C C C 1 2 l 2 ( ------ + ------ ) All + (1 - ------ + ------ ) A12 1 + g1 l + g2 l + g1 l + g2 C C (1 - --1-- + ---1--, 12k 1 + g1 l + g2 - ------------------------------------------------- (H-9) C C C C (---1-- . ---1--, A +11- "1--.---1-“ 21 22 l + g1 l + g2 l + g1 1 + g2 The first step in reducing the complexity of H-9 is the calculation of the pure component 1 basis group activity coefficient, ln Pkl, defined as the group activity coefficient ln P taken with the mole fraction of k component 1 equal to one. Substituting x1 - l and x2 - 0 into eqs H-1 and E~2 gives results for the size-weighted fractions c1, which can then be used in eqs 7 and H-8 for Xk. These group mole fractions can be substituted into eq 3, giving ln P 1 which simplifies to eq H-lS. k ,1 _ --------f?11f?113513 ....... - 1 (1.10) (n11+n21)(l) + (n12+n22)(0) ,2 _ __---_--fi‘11‘f‘-‘1135‘33------- _ o (p-11) (n11+n21)(l) + (n12+n22)(0) 1 o 1 x1 - i-+--- +1 ------------ (ii-12) 51 + g2 1 + g1 1 g1 x2-1-x1-1- ------ - ------ (ll-13) l g1 ln Pk - ' ln [ (l 1---) Akl + (i';-") Ak2 ] g1 81 1 < ------ >A 1k 1 + g +1- ............. 1.’ ............. l g1 ( """ ) A11 + ( """ 1 A12 1 + g1 l + g1 g1 ( """ ) A2k l + g1 - ........................... (3-14) (1 (g1) ------ ) A + --—--- A 21 22 l + g1 1 + g1 + g A A 3 ln rkl - - ln [ 111---111-1 1 + 1 - ----- 15 --------- 11-1--- (H-lS) 1 1 81 A11 + A1251 A21 + A2251 Since eq 2 for the calculation of the activity coefficient requires the difference ln Pk - ln Pkl, not the individual terms, this difference can be calculated by combining eqs H-9 and H-lS, cancelling and combining several terms in the process. 265 + 3 1n rk - ln rkl - 1n [ ------------------- f¥}---é¥?-} -------------------- 1 l + g1 1 + g1 (°1 1 """ c2) Akl 1 (1 1 81 ‘ c1 ' """ °2> AkZ l + g2 l + g2 1+g1 A1k (C1 1 i """ C2) A1k +82 + ................................................................ 1 + g1 l + g1 A11 1 A1231 (c1 1 """ °2) A11 1 (1 1 81 ' °1 ' """ c2) A12 1 + g2 l + g2 1 + g1 A2k31 (1 1 81 ' °1 ' """ C2) A2k l + g2 + ................................................................ l + g1 1 + g1 A21 1 A2231 (cl 1 """ c2) A21 1 (1 1 31 ' °1 ' """ c2) A22 1 + g2 l + g2 (H-l6) Multiply out terms in the expressions so that the composition variables are principal factors rather than the interaction parameters, giving eq H-l7. Then eliminate c1 for c2, using c2 - l - CI, to give eq H-18. 266 1 1n Pk - ln Pk - “1:1 1 A1:231 ln [ ------------------------------------------------------- ] 1 + g1 Akz (1 1 31) 1 (Akl ‘ Ak2) c1 1 (Akl ‘ ARZ) <1 + A1231) 1 + 31 In t ------------------------- 1 (1-A )(g -g)c 1 1 ------ Z}---?----l---? (1 + g2)(A21 + g1) + (1 + 31)(32 - 31) c2 ' 1 112-{-Eff-1-§z?ff-f-fze§13I (1 + g2)<1 + A1231) A / [(1 + g )(A + 8 )1 - -2} .......... 2---?}----}-- ) (H-28) 312 - """""""""" (10) 1231) 521 - """"""""" (11) (1 + 32)(A21 + 31) 273 In G 71 ...... - - 1n (1 + 3 “11 1n (1 + B 12°21 ' g1 21°21 + (1 + 81)(82 ° 81) c2 . A12 / [(1 + 82)(1 + A12g1)] ( ........................... l + 312c2 A / [(1 + g )(A + g )1 - -2} .......... 3---?}----!-- 1 (H-29) Further rearrangement of the final terms of eq H-29 results in eq H-30. Eqs H-31 and H-32 indicate how the final terms can be rewritten to become eq 12. G In 11 ~----- - - 1n (1 + BIZCZ) - g1 1n (1 + B “11 + c2 / (l + g2) 21c2) 1 112.5? 1 113512-: §1Z-{-f}-1-112§13 1 + Blzc2 A (1 + g )(g - g ) / (A + g ) - -?} ------- 1---?----} ...... g}----}- ) (H-30) (g - g )(A - 1) (g - g )(1 + A g ) (1 + g2)312 + (g2 - g1) :- --g----1----]:g ..... + --g---_].' ....... 12-]:- (l + A1281) (1 + A (g - g )(1 + g ) A - --?----} ....... 1---}? (H-3l) 274 (82 ' 817(A21 + 81) 81(82 ' gl)(1 ' A21) (32 - gl) - (1 + 82781812 - ----------------------------- (g g )(1 + g ) A - --?----} ....... 1---?! (3-32) (A21 + $1) In 116 --;--' - ' 1n (1 + BIZCZ) ' 81 In (1 + 821C2) 11 C (82’81) + (1+82)812 (32-81) - (1+32)81321 1 --?- 1 ----------------------------------------- ) <12) l+g2 l + Blzc2 l + 821c2 Eq 13 results from taking c2 equal to one in eq 12. Eqs 14 and 15 define a second set of transformed interaction parameters which are used in deriving eq 16. Steps of this derivation are given as eqs H-33 to H-36. In 116 m (--;---) - - 1n (1 + 312) - g1 1n (1 + 321) 11 1 (g -g ) + (1+3 )3 (g -g ) - (1+8 )8 B 1 1-3--1 ........ 2112 - "2-1 ........ 1-1-211 1131 1+g2 1 + 312 1 + 321 c12 - 1 + 312 (14) c21 - 1 + 321 (15) In 71G m (--;---) - - 1n C12 g1 ln C21 11 1 (g -g ) + (1+3 )8 (g -g ) - (1+3 )8 B 1 ---- 1--?--1 ........ 2..1? - --?--1 -------- 2--1-¥11 (H-331 1132 C12 c21 275 In 1 1 co ( ------ ) - - 1n C12 - g1 ln C21 “11 1 --T- 1f??2§13-T-f}f%zif?1z-Z-TZ - 5121113-:-511123115121-2-131 1132 C12 C21 (3-34) In 7 (g -g ) (1+5 ) (1+8 )C (----1-) - - In 012 g1 1n C21 + -_?__} -------- g ----- g--1g “11 (1132)C12 (11g2)c12 (g 'g ) + (1+3 )3 (1+3 )8 C - "2-1 -------- 2-11 "-2-1-21 111-351 (11321021 (1132)021 1n 1 (1+g ) g + g g (____1_)m - - 1n c12 - g1 1n c21 - ----- 1--- + 1 - —?----}-z + g1 n (1+3 )c (1+g )c 11 2 12 2 21 (H-36) c In 11 w l+g1 l g2 ( ------ ) - - In 012 - g1 1n c21 - ---- (--- + ---) + (1+g1) (16) “11 1132 c12 C21 A sufficient condition for eq 16 to result in a value of zero is given by eq 17. This is proved by substitution in eq H-37 below. Eqs 18 to 20 result when g1 or g2 or both take on specific values. Eqs H-38 to H-ha show these derivations from eq 16. C12 1 C21 ‘ 1 (17) In 11G m l+g1 l g2 ( ------ ) - - 1n 1 - g 1n 1 - ---- (- + --) + (1+g ) n 1 1+ 1 l 1 11 82 1+g1 - - 0 - s (0) - ---- (1+3 ) + (1+3 ) l 2 l 1+g2 - - (1+gl) + (1+g1) - O (H-37) 12 12 12 12 12 12 12 12 276 1+0 1 g - 0 In C - ---- (--- + -9-) + (1+0) 21 1+ C C 82 12 21 l l g2 - ---- (--- + ---) + 1 1152 C12 C21 1+g1 1 0 - g ln c - ---- (--- + ---) + (1+g ) 1 21 1+0 C C 1 12 21 1 - g1 1n C21 + (1+g1)(1 - ---) C12 1+0 1 g2 - 0 1n C21 - ---- (--- + ---) + (1+0) 1132 c12 “21 1 l g2 l - (---) - <---> + 1 l+g2 C12 1+g2 021 1 1 - <0) <---> - (1) (---> + 1 C12 C21 1 - --- + 1 C21 (H-38) (H-39) (18) (H-AO) (H-Al) (19) (H-38) (H-39) (H-42) (H-43) (H-Ah) (20) Eq 21 defines the interaction parameter in terms of molar volumes and interaction energies. differentiated with respect to A and H-46. Eqs 22 and 23 result when eqs 14 and 15 are and A 12 21 respectively to give eqs H-AS 277 (l + g1)(l + A C - ------------------- (14) 1231 1232) - .............................. (3-45) 0 - ------------------ (15) — ............................ (H-46) Since A12, A21, g1, and g2 are all nonnegative, eqs H-hS and H-46 show that Ck1 are monotone increasing (or decreasing) functions of Ak1 dependent upon the sign of g2 - g This implies that the minima and 1. maxima of Ak1 are also the minima and maxima of the functions Ckl(Akl)' Eq 21 restricts Ak1 to take on positive values. Therefore, limits on the values of Ck1 can be given by taking Ak1 equal to zero and approaching infinity in eqs 14 and 15. A12 - 0 (H-47) A21 -> o (a-aa) u+gpu+ u+gp c12 - --------------------------- (n-49) U+8QU+(mfi) a+gp (1 + g )(A + s ) (1 + s ) (A + a ) (1 + g ) c _ ...... 1---?}----? ....... l- --?}----? ....... l- (1) (H-SO) (1 + 32)(A21 + 31) (1 + 32) (A21 + 31) (1 + 32) 278 A12 -> on (PI-51) A - o (H-SZ) C _ f?-1-§135?-1-?12§23 _ f?-1-§135?(§12-1-§23 _ 51-1-é13-éz 11-531 12 (1 + g2)<1 + A1281) <1 + 82)(1/A12 + 31) (1 + 32> 21 (1 + g )((0) + g ) (1 + g ) g 021 - ...... 1 ......... 2 ........ 1---? (H-54) (1 + g2)((0) + 31) (1 + 82) 81 Eqs H-49, H-SO, H-53, and H-54 express the limits on C which can be kl’ succintly written as eqs 22 and 23. l+g1 1+g1 g2 ---- < c12 , 021 < ------ when g2 > g1 (22) l+g 1+8 8 2 2 1 1+g1 g2 l+g1 ------ < C , C < ---- when g < g (23) 12 21 , 2 1 l+g2 g1 l+g2 Eqs H-SS and H-56 are derived by differentiating eq 16 with respect to one of the Ck1 while the other is held constant. Since the function will increase when its derivative is positive, eqs 24 and 25 result when the right hand sides of eqs H-SS and H-S6 are set greater than zero. 1n ‘71 00 1+3]. 1 $2 ( ------ ) - - In C - g In c - ---- (--- + ---) + (1+g ) (16) n 12 1 21 1+ 6 c 1 11 82 12 21 6 1n 1 | l l+g -l 1 l+g 1 1 1 ----(----}-)”| - - --- - ---- 1---?) - --- (---- --- - 1) (H-SS) 5C12 “11 lc21 C12 1132 C12 C12 1132 C12 6 1n 1 I 8 1+8 3 1 1+8 8 1 1 1 2 1 2 ----( ------ )°| - - --- - ---- <---§) - --- (---- --- - g1) (H-ss) 5°21 “11 lc12 c21 1132 c21 C21 1132 C21 279 1 l+g1 1 --. (---- --- - 1) > 0 (3-57) C12 1132 c12 l+g1 l ---- --- - l > 0 (H-53) l+g2 012 l+g1 l 1 (H 59) ....... > - 1132 C12 l+g1 24 1:11;" “12 ‘ ’ 2 1 1+g1 g2 --- 1---- --- - g1) > o (H-eo) C21 1132 C21 1+g g ---1 -2- - 31 > o (H-61) 1132 C21 1+g g --.l -3- > 31 (H-62) 1132 c21 1+g g ---1 -3 > c (25) 21 l+g2 g1 The permissible domain of Ck1 values given by eqs 22 and 23 is such that either eq 24 will always be satisfied and eq 25 will never be satisfied, or the opposite will occur. When g2 > g1, bounds are given by eq 22. Eq 24 will never be satisfied so the function takes on its maximum at the minimum possible 012 value; at the same time, eq 25 will always be satisfied so the function takes on its maximum at the maximum possible 021 value. These values where the function takes on its maximum are given by eqs 26 and 27. When g2 < g1, bounds are given by eq 23. Eq 24 will always be satisfied so the function takes on its maximum at the 280 maximum possible C12 value; at the same time, eq 25 will never be satisfied so the function takes on its maximum at the minimum possible C21 value. This is the opposite of the case when g2 > g1. However, the bounds of eq 23 are also the opposite of the bounds of eq 22, so eqs 26 and 27 hold regardless of the sign of g2 - g When the objective is to 1. minimize the function, the logic reverses and eqs 28 and 29 must hold. 1+g1 C12 - ---- (26) 1+g2 1+g g c - ---1 -2 (27) 21 1+ g2 g1 1+g g c12 - ---} -1 (23) 1+g2 g1 1+g1 C21 - ---- (29) 1+g2 Eqs 30 and 31 result from substitution of either eqs 26 and 27, or eqs 28 and 29, into eq 16. C In 11 1+g 1 g m 1 2 1 ...... ) - - 1n c12 - g1 1n c21 - ---- 1--- + ---) + (1+g1) (16) n 1+g C C 11 2 12 21 G In 1 1+3 1+8 5 l ( nnnnnn ) - C 1n --.1 o g 1 (.n-} u?) n11 max 1+g2 1+82 81 1+g 1+g 1+g g 1 2 2 - ---- (---- + g2 ..... l) + (1+81) (H-63) C 1n 1 1+8 8 (mi)co - (1+g1) ln ---3 + 31 In -1 n11 max 1+g1 g2 ' (1181) + (1181) G In 1 1+g g (----1'-)co - (1+g1) 1n ---g + g1 1n -} n11 max 1+g1 g2 C 1n 1 1+g g 1+g <----1->” - - 1n <---1 -?) - g1 1n ---1 n11 min 1+g2 g1 1+g2 1+g 1+g g 1+g2 - ---1 (---g -1 + g ----) + (1+g1) 1+g2 1+g1 g2 1+g1 C 1n 1 1+8 8 1+8 (----1-)” - - 1n (---1 -3> - 31 In ---1 n11 min 1+g2 g1 1+g2 g1 - -- - g2 + (1+g1) 82 ln 716 m 1+g2 g1 < ------ ) - (1+31) 1n ---- + 1n -- + (82'81)(" - 1) n11 min 1+g1 g2 281 (H-64) (30) (H-65) (H-66) (31) Eqs 32 and 33 are generated by inverting eqs 14 and 15 to express the Ak1 as functions of C . k1 112 - f?-1-§1351-1-§12§z? (1 + 82)(1 + A1281) 51-1-??? 1 $1-1-f1zézi (1 + 31) 12 (1 + A1231) 51-1-92? 1 1 A g 51.1.??? c - 1 1 A g (1 + 81) 12 12 1 (1 + 81) 12 12 2 ff-1-§23 c - 1 - A g - A g 1 53-1-??? (1 + 81) 12 12 2 12 l 12 (1 + 81) (14) (H-67) (H-68) (H-69) 282 (1 + 32) C12 ‘ (1 + 51) ' A12 32‘1 + 51) ’ A12 31‘1 + 52) C12 A12 - -------------------------- 82(1 + $1) ' 81(1 + 82)C12 (1 + gl)(A21 + 82) 021 - ------------------ (1 + 32)(A21 + 31) f}-f-§23 C ffzz-f-§23 (1 + g1) 21 (A21 + 81) A 5?-T-§23 c + 53-T-§23 c _ (A + g ) 21 (1 + $1) 21 1 (1 + 81) 21 21 2 g f}-f-§23 C - g _ A - A f}-f-§23 C 1 (1 + 81) 21 2 21 21 (1 + 31) 21 21 (1 + 81) ' (1 + g2)C21 (H-70) (32) (15) (H-71) (H-72) (H-72) (H-73) (33) Table H-l. Equations Used in "Normalization and Bounding Properties Inherent in Solution of Groups Activity Coefficient Models" Xk - ...... T32T1-T-?32T2 ...... (“11+“21)x1 + (“12+“22)x2 c 1 1 In 11 - nli(1n F1 - 1n F1 ) + n21(1n F2 - 1n F2 ) x A x A In Pk - -1n(XlAk1 + XZAkZ) + 1 - ----}-}¥ -------- 2-25--- XA HA XA mA 1 11 2 12 1 21 2 22 i In Pk - 1n Pk (x1 - 1) (“11+“21)x1 + (“12+“22)x2 (l) (2) (3) (4) (5) 283 X1 - ---}-- + ---g-- 1 + g1 1 + g2 1“ 11 1 1 ------ - (1n P1 - 1n P1 ) + gi(1n P2 - 1n F2 ) n11 In 116 (1 + g1><1 + A1231) ...... - 1n ----------------- ------------------------ (1 + 31)(A21 + 31) + gi 1n ------------------- . --------------------- + (1 + gi)(gJ - 31)cj ' 1 .................. §1z ..................... (1 + gj)(1 + A1231) + (gj - 81)(A12 1)cj - ................. §z1 ..................... , (1 + gj)(A21 + 31) + (gj 31)(1 A21)cJ (32 81)(A12 - 1) B - ................... 12 (1 + g2)(1 + A12g1) (82 - 31)(1 - A21) B - .................. 21 (1 + g2>(A21 + 81) 1n 1 G 1 ------ - - 1n (1 + 31202) - g1 1n (1 + 321c2) n 11 1 .f2. 1f§22§13-f-fff§23?12 _ f§22§13-2-fo§23§1?21) 1+g2 1 + Blzc2 1 + 821c2 1n 1 1 co (--;---) - - 1n (1 + 312) - 31 In (1 + 321) 11 1 ..T. 15§22§13-f-fo§zZ?12 - 5§22513-2-5}f§23§1?21, 1+g2 1 + 312 1 + 321 (6) (7) (8) (9) (10) (11) (12) (13) (1 + 31)(1 + A1232) (1 + 32)(1 + A1231) (1 + gl)(A21 + 82) 21 21 (1 + 82)(A21 + 31) G In 11 co 1+g]. 1 82 ( ------ ) - - In C - g In C - ---- (--- + ---) + (1+g ) n 12 1 21 1+ C C 1 11 82 12 21 C12 ' C21 ' 1 In 116 m 1 1 g2 ( ------ ) - - In C - ---- (--- + ---) + 1 n 12 1+ C C 11 32 12 21 In 11G m 1 (--;---) - - 1n C12 - g1 1n C21 - (1+g1)(1 - 6--) 11 12 In 11C 1 ( ------ ) - - In C - --- + 1 n 12 C 11 21 v (A - A ) Aij - -1 exp [ .. --!'J----I'¥- ] vi RT 1+g 1+g1 g ---- < C , C < ------ when g > g 12 21 2 1 1+g2 1+g2 g1 1+g1 g2 1+g ------ < C12 , C21 < ---- when g2 < g1 1+g2 g1 1+g2 1+g ---- > c 12 1+g2 1+g1 g2 ------ > C 1+g g 21 2 1 1+g1 C12 ' "" (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) 285 1+g g 021 - ---1 -3 (27) 1+g2 g1 1+g g C12 - ---l -g (28) 1+g2 gl 1+g1 C21 - ---- (29) 1+g2 G In 11 w 1+g2 g1 ( ------ ) - (1+g1) 1n ---- + g1 1n -- (30) n11 max 1+g1 g2 G In 11 m 1+g2 g1 1 < ------ ) - (1+g1) 1n ---- + 1n -- + (gz-gl><-- - 1) (31) 1111 min 1+g1 g2 g2 A12 - """"""""""""""""""" (32) A21 - """"""""""""""" (33) (1 + 31) - (1 + 32)021 APPENDIX 1. Derivation of Equations in "A Diffusion Coefficient Model for Polymer Devolatilization" This appendix contains a more detailed derivation of the equations presented in the reprint article ”A Diffusion Coefficient Model for Polymer Devolatilization". This article was included as part of Chapter A of the dissertation. In this appendix, equation numbers refer to the manuscript article itself, beginning on page 21 of the dissertation. New equations not included in the article are numbered with a preceding letter I, e.g., I-l, I-2, etc. Eqs 1 to 5 were taken from previously published work (Duda, Vrentas, Ju, and Liu, 1982) and are not derived here. DERIVATION 0F EQUATION 6 The first new equation is eq 6. It was derived from the expression for the activity of solvent developed in Chapter 2 of this dissertation. Definitions from Chapter 2 are given as eqs 1-1 and I-2, and are differentiated with respect to mole fraction to give eqs I-3 and 1-4. The second factor on the right hand side of eq I-3 is given as a function of the size ratio 82/81 and mole fraction x in eq I-S, which 1 286 287 is combined with eq 1-4 to give eq I-6 for the derivative of the logarithm of activity coefficient, In 11, with respect to mole fraction x Eq I-7 uses the chain rule is used to express the derivative 1. d ln 11 / d In x in terms of the derivative in eq I-6, which is l substituted to give eq I-8. The definition of activity in used in eq I-9 to produce an expression for the derivative d In a1 / d In x1 in eq I-lO. In 11 - l - R1 - 1n R1 (I-l) R1 - $1 / (Slx1 + 82x2) - l / (x1 + (82/81) x2) (I-2) d In 1 dR l 1 l —————-— - --- <1 - -> (1 - —3><1 - x1) dx1 (x1 + (82/81) x2) S1 81 2 [1 - (S /s )1 <1 - x ) [1 - (s /8 >12<1 - x > _ 2 1 l _ 2 1 1 (I 6) 2 2 - (x1 + x2) [(82/81) + [1 - (sz/sl>1x11 d In 11 d In 11 dx1 d In 11 --———— - -—-——- - ———-—-— - x1 ———-——— (I-7) d In x1 dx1 d In x1 dx1 2 d 1n 1 [1 - (S /S )] x (1 - x ) l _ 2 l 1 I (I-8) d In X1 [(32/31) + [1" (82/81)]xllz 288 3.33.31 - 3.33.3131 - 3.33.31 1 3.33.31 - 1 1 3.33.31 (1-9) d In x1 d 1n x1 d 1n x1 d In x1 d 1n x1 [1 - (sz/sl>12x1<1 - x1) [(82/51) + [1 - (82/8911512 [1 - (sz/slnzx1 - [1 - (82/8913:1 (82/81)2 + 2(82/81111 - (sz/slnx1 + [1 - (sz/slnzx1 2 _ + [1 + (82/81)1[1 - (s2/81nx1 (1-10) <32/8112 + 2(82/sl>11 - (s2/8111x1 + 11 - (82/81)]2x12 - l + 2 - 1 + 2 When eq I-lO is multiplied through by x to give eq I—ll, the left hand 2 side matches eq 6. The right hand side must be transformed from size ratio and mole fraction variables to infinite dilution activity coefficient 01do and weight fraction variables. Eqs I-12 and I-13 provide the concentration variable transformations, giving eq I-14 when the substitutions are made in eq I-11 and numerator and denominator are multiplied through by the square of the denominator of I-12. Inspection of the denominator of eq I-lh shows it to be a perfect square as written in eq I-15 and simplified in eq I-l6, while the numerator is simplified by multiplying out some terms in eq I-lS, then cancelling in eq I-16. 2 x d ln a1 _ (52/51) x2 + [1 + (32/51)][1 - (82/81)]x1x2 2 d 1n x1 (32/31)2 + 2(32/51)[1 - (82/81)]x1 + [1 - (52/51)]2x12 (1-11) X1 " (Hz/H1)V1 / [(M2ml)wl + V2] (1'12) x2 - w2 / [(MZ/M1)w1 + w2] (I-13) 289 2 (32/51) w2[(M2/M1)W1 + W2] x d 1n a1 - + [1 + (82/81)][1 - (82/81)](M2/M1)w1w2 2 (1 ln x1 (sz/sl)2[(112/111)w1 + 11212 + 2(32/81)[1 - (Sz/Sl)](HZ/M1)w1[(M2/M1)w1 + W2] 2 2 (I-lA) 2 2 2 (32/51) w2""2/1“1)"1 ‘3 (32(31) w2 2 x d In a1 - + (HZ/M1)w1w2 - (82/81) (Hz/M1)w1w2 2 d In X1 [(82/31)[(M2/M1)W1 + wzl + [1 - (32/31)](M2/M1)W1]2 (I-lS) d In a (S /S )2w 2 + (M /M )w w 1 2 1 2 2 l 1 2 x2 —-———- - q 2 (I-l6) d In x1 [(82/S1)w2 + (Hz/M1)w1] When the result for the size ratio from Chapter 2, eq I-l7, is substituted into eq I-16, eq I-18 results. Multiplication of numerator and denominator by (Ml/M2)2 gives eq I-l9. When the assumption M << M l 2 is made in eq I-19, eq 6 results. 32/81 - («e/01‘”) (112/111) <1-17) d ln al - (e/Olm)2(M2/M1)2w22 + (Hz/M1)w1w2 2 d In x1 Ice/01‘”) (112/111w2 + (”z/"1"‘112 d In a1 (e/Olw)2w22 + (Ml/M2)w1w2 __ - 2 (1-19) 11 X (I-18) X 2 a d ln x1 [(e/fl1 )w2 + w r e . 1 2 A _; w2 p2V2p1 apl d In a1 01 <—>T - x2 ————- - (6) RT apl ,p d 1n x1 e w + -— w l w 2 L 01 J Eq 7 results directly from substitution of eq 6 into eq 2, then substitution of that result into eq 1. DERIVATION OF EQUATION 10 Eqs 8 and 9 define parameter groups which appear in eq 3. When eqs 8 and 9 are used in eq 3, it becomes eq 1-20. substitution of eqs I-20 and 4 into eq 7 results in eq 10. v K K FH 11 12 .. 1110121 + 1‘ - 131) + w2 (K22 + T - ng) (3) 1 1 1 1.311111 11.1, (a) 1 1 21 gl A-lflgw(l( +T-T) (9) 2 2 22 g2 1 v FH —;— - Alw1 + A2w2 (1'20) D01 - Do exp (-E/RT) (4) 291 r 1 2 e —— W A A * 01cc 2 -1(w1V1* + w2£V2 ) D - D exp [ ] (7) 01 e A W1 + -—; W2 VFH 0 L l J r e 1 2 —- W A A 01” 2 w1V1* + wzév; E D - Do exp [- - -—] (10) e w + A w RT w + _ w A1 1 2 2 1 w 2 L 01 J DERIVATION OF EQUATIONS 12, 12A, 123, 12C Eq 11 is a Taylor (or Maclaurin) expansion of the function D(w1) about the point w1 equal to zero. Eq 12c results when w1 is taken as zero (w2 will then equal one) in eq 10, as shown in eq I-21. For simplicity of derivation, define F1 and F2 to be the two factors in eq 10 which are functions of wl, allowing eq 10 to be rewritten as eq I-24, and eq 11, the partial derivative of eq 10 with respect to w to be written as eq 1. I-25. The derivatives of F1 and F2 themselves are given and simplified in eqs I-26 and I-27, and substituted back into eq I-25 to give eq I-28. Removing common factors results in eq I-29, and recognition of the leading factor as the right hand side of eq I-24 gives eq I-30. When W1 is taken as zero, eqs 1-31 and 1-32 result, simplifying the expression for the derivative to eq I-33. Comparison of eq I-33 with eqs 11, 12, 12a, 12b, and 12c shows the set of equations to be identical. 292 r e 12 —w A A n” 2 o.v +1.ev* 1: 1 1 2 D-Do exp[- -_] e Al-O + A2-1 RT 0+7]. L 01 1 “ * E 5V2 D(O) - Do exp [ - (-- + ----) ] RT A 2 1 e ‘ F32 F- 1 l e w + -—- w 1 m 2 1 01 1 6* 6* E w1 1 + w2£ 2 F2 - exp [- - __] Alw1 + Azw2 RT 2 D - D0 F1 (wl) F2(w1) 8D 3F 6F —--D [112—3+? -21-' —1] 6w 0 1 6w 2 law 1 1 1 31’ 37* 131*A A A A {7* 1* __g_F .(w11 +"’2‘52)(1' 2"(1w1+ 2"2)(1 -€2) 2 2 awl (Alw1 + A2w2) A G * Ahv * _F 152 ("1+"2" 21 ("1‘”‘2’ 2 2 (Alw1 + A2w2) AVA}. Av” 152 ‘ 21 - F2 . 2 (Alw1 + A2w2) (I-21) (12c) (I-22) (I-23) (I-24) (I-25) (I-26) 293 aFI (w1 + (e/0131w2) <- m 2 awl (w1 + (e/O1 )w2) o o 2 - -(e/01 )(w1 + wz) - -(e/01 ) - - F1 °° 2 O 2 an (w1 + (e/n1 )"2) (w1 + (e/o1 )Vz) (e/n1 ) an A 3 * A G * 2F 3 2 1€ 2 ‘ 2 1 1 “‘ ' Do [F1 F2 ' 2 ' F2 ' m ] 6w1 (Alw1 + A2w2) (e/O1 ) an A 6 * 3 A 6 * 2? 2 15 2 ' 2 1 1 ' Do F1 F2 ° [ 2 ' a 1 awl (Alw1 + A2w2) (e/O1 ) an 9 * 6 * 2F A16 2 ’ A2 1 1(V1) -- - D(W1) ° [ 2 ' -—-—;-l 6w1 (Alw1 + A2w2) (e/O1 ) an A 9 * A G * 2F 0 ._ -D1o).[1‘2'21__1‘_’_] 2 a: awl wl-O (Al-O + A2-1) (e/O1 ) F1<0) - («e/01°)2 / [o + (e/01°)-1]2 - (e/ol‘”)2 / (ea/01‘”)2 - 1 an A G * 9 * 2 1e 2 ' A2 1 _ - 13(0) ° [ 2 ' Q ] aw1 wl-O A2 (e/fl1 ) a n(wl) - D + --- (w1 - 0) wl-O aw1 T,w1-0 D(w1) - 0(0) [1 + (x1 - x2) wl] E ev2* D(0) - Do exp [ - (-- + ----) ] RT A 2 G A G * ‘3‘1’E 2 ' 2 1 K1 """" 2 """ A (I-27) (I-28) (I-29) (I-30) (1-31) (I-32) (I-33) (11) (12) (12c) (12a) 294 e x2 - 2 / --5 <12b) 01 Eq 13 is the commonly used WLF equation for viscosity, as applied by Duda, Vrentas, Ju, and Liu (1982). Table I-1. Equations Used in "A Diffusion Coefficient Model for Polymer Devolatilization". ' 9 V p an 2 2 l 1 D - I)1 ------ (---).1. (1) RT apl ,p (w 6 + w 16 *) -1 l l 2 2 'D1 - D01 exp [ ------------------ l (2) VFH v K K FH 11 12 -;- - w1(x21 + 'r - 1'81) + w2 (K22 + '1‘ - 1'82) (3) D01 - Do exp (-E/RT) (4) p V p 3p 2 2 l l 2 ------ <---)T - (1 - ¢1) <1 - 2x¢1) (5) RT 6p ,p 1 r e W 2 Q a d 1 nmwz P2 2p1 “1 3 a1 1 ( )1 - x2 —-—-- - (6) RT ap1 '9 d In x1 e W + -— w 1 w 2 L 0 J 295 f 12 e _w A A 0 Q 2 -1(w V * + w {V *) D-D 1 ex[ 11 2 2] 01 P e A W1+—;W2 VFH 0 L 1 J K 11 Al - --- w1 (K21 + T - Tgl) ‘7 K 12 A2 - --- w2 (K22 + T - T 2) '7 r e 12 —w A A 0 2 * * o n 01 "1V1 3 "25"2 E - o epr- --—1 e Alw1 + A2w2 RT w + —-— w 1 m 2 L 01 J a D(w1) - D + --- (w1 - 0) wl-O awl T,w1-0 D(wl) - D(O) [l + (K1 - K2) wl] AG 16* 152 ' 21 K .............. l A 2 2 e K2-2/‘3-; l A * E 6V2 D(O) - Do exp [ - (-- + ----) ] RT A 2 ‘ * 5V1/K11 1n "1 - 1n A1 + ............. 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