\ l \ 1 Ml(HIWIWI‘HHKI 1 ‘ Ml NU!HHIU‘IWWl THS 7%)39 _LIBRARY "‘ MlChl an State Un versity This is to certify that the thesis entitled Measuring the Mechanical Response of Swollen Hydrogels presented by ’ Ryan James Monroe has been accepted towards fulfillment of the requirements for the MS. degree in Mechanical Engineering aj r P/rofess’ r’s Signature Z} J“ C, 200? Date MSU is an affirmative-action, equal-opportunity employer PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 5/08 K2/Proi/Acc8rPreleIRCIDateDue,indd MEASURING THE MECHANICAL RESPONSE OF SWOLLEN HYDROGELS By Ryan James Monroe A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Mechanical Engineering 2008 ABSTRACT MEASURING THE MECHANICAL RESPONSE OF SWOLLEN HYDROGELS By Ryan James Monroe A combined experimental and analytical study has been conducted to understand the mechanical response of a swollen poly(vinyl alcohol) (PVA) hydrogel. These PVA hydrogels are formulated using a freeze-thaw crosslinking process, making them suit— able for biomedical applications. In order to predict the mechanical response, a finite elastic constitutive theory is used. Two stress-strain relations are derived to describe the response of hydrogels that are either submerged in water or in air. These stress- strain relations motivate the experimental protocols, which are needed to investigate the applicability of the constitutive theory as a model for the PVA hydrogels used in this study. A uniaxial load frame has been created that performs individual or coupled experiments of stretching and swelling. The apparatus uses a CCD camera to measure finite deformation by tracking spheres attached to a small, central region of the hydrogel, while a load-cell measures the applied traction. Deformation of the hydrogel can be used as feedback to control the rate of deformation during stretching tests. The constitutive parameters can be estimated from the results of stretching specimens to failure, while they are out of water and swelling specimens subject to applied loads. After experimental data are obtained from these two experiments, a nonlinear least squares optimization routine estimates material parameters from the governing stress-strain relations. It is found that a modified Ogden model, which in- corporates the Flory-Huggins theory, is a good predictor of the mechanical response of PVA hydrogels. ll To Anna and my family iii ACKNOWLEDGMENTS I would like to thank Anna, my family, and friends for their patience and support during the past two years. I would like to thank my parents for their financial support during my undergraduate degree which has helped me to get to this point in my career. I would like thank my advisor and friend, Professor Neil Wright, for all of his guidance and support throughout this project. I would also like to thank my co-advisor and friend, Professor Thomas Pence, for his guidance and help with the theoretical aspects of this work. Working under their guidance has allowed me to obtain a wealth of knowledge and for this I am truly grateful. Thanks to the other member of my committee, Professor Seungik Baek, for his advice and suggestions regarding this work. I am thankful for NSF, Professor Wright, Professor Pence, and the Department of Mechanical Engineering at Michigan State University for their financial support while pursuing this degree. Thanks to Brian Olson for his 1AM files that met the graduate school formatting requirements. Thanks to Mike Mclean who helped with the manufacturing associated with this project. Thanks to my friend Jay Joyner for his help on the motion aspect of the project. iv TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES 1 Introduction 2 Finite Deformation Theory 3 Experimental Methods 3.1 Mechanical Testing Apparatus ...................... 3.1.1 Device Measurement System ................... 3.1.2 Testing Capabilities ........................ 3.1.3 Stretch Ratio Measurement ................... 3.1.4 Matlab Routine for data reduction ............... 3.2 Development of Poly(Vinyl Alcohol) Hydrogels ............. 3.2.1 Freeze-Thaw Crosslinking Method ................ 3.2.2 PVA-PAA versus PVA Hydrogel ................. 3.2.3 Specimen Preparation ...................... 3.2.4 Cleansing versus Free Swelling .................. 3.2.5 Forced Swelling .......................... 3.2.6 Preconditioning and Unsaturated Stretch-to—Break ...... 3.3 Parameter Estimation .......................... 3.3.1 Parameter Estimation from Unsaturated Test Results ..... 3.3.2 Parameter Estimation from Saturated Test Results ...... 3.3.3 Combined Parameter Estimation using Saturated and Unsatu- rated Test Results ........................ 3.3.4 Matlab Non-Linear Least Squares Routine for Parameter Esti- mation ............................... 4 Results and Discussion 4.1 Saturated Test Results .......................... 4.2 Unsaturated Test Results ......................... 4.3 Parameter Estimation Results ...................... 4.3.1 Modified Ogden Model with N = 1 ............... 5 Conclusions APPENDICES A Calculating the Cauchy Stress vii viii 1 6 13 14 17 18 20 27 28 31 38 44 60 62 62 65 66 7O 81 83 83 B Calculating the Displacement Gradient C Measurement Uncertainty Calculations BIBLIOGRAPHY vi 86 87 89 ll 4.1 4.2 4.3 4.4 4.5 4.6 LIST OF TABLES Initial A3 and J referred to the dry polymer state before the stretch- to-break experiment. ........................... Material parameters obtained for the separate fitting of the Ogden model with N = 1. ............................ Residual stress in the traction free direction (equation (3.42) with N = 1) for comparison with the assumption of zero applied stress. Residuals 1 through 5 are the five stresses that are determined using the materials parameters (obtained using the separate fitting routine) in Table 4.2 and the five saturated measurements which consists of As and J for each of the applied loads (see Figures 4.1 and 4.2). .......... Material parameters obtained for the combined fitting of the Ogden model with N = 1. ............................ Residual stress in the traction free direction (equation (3.42) with N = 1) for comparison with the assumption of zero applied stress. Residuals 1 through 5 are the five stresses that are determined using the materials parameters (obtained using the combined fitting routine) in Table 4.4 and the five saturated measurements which consists of A3 and J for each of the applied loads (see Figures 4.1 and 4.2). .......... Initial A3 referred to the dry polymer state before the unsaturated stretch-to—break experiment. ....................... vii 68 74 77 2.1 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 LIST OF FIGURES Mapping of volume elements in the homogenous deformed portion of specimen under a uniaxial load in the e2 direction. .......... 7 The experimental apparatus used for the finite strain uniaxial stretch- ing tests .................................. 14 Insertion of hydrogel with cloth and toothpick into gripping mechanism 16 Lengths between tracking markers for real time stretch ratio calculation 21 Mapping of tracking spheres from the X — Y plane to a uniform square in a local 3 — r coordinate system ..................... 22 Initial swelling volume response of PVA and PVA-PAA ......... 30 Three cycle swelling of PVA-PAA hydrogel with water change in between. 31 An isometric view of the aluminum six specimen dumbbell shaped mould. The length of specimen is 38 mm, the thickness is 3 mm, the gage length is 22 mm, and the gage width is 5 mm. Refer to ASTM standard for more detailed dimensions. ................. 34 Front side of specimen after preparation with toothpick, cloth, and spheres glued on. ............................. 36 Back side of specimen after preparation with toothpick, cloth, and spheres attached. ............................. 36 Plateaus observed in the volume swelling ratio .............. 39 This figure shows that the non-uniform swelling ratios during cleansing violate the free swelling assumption in the constitutive model. . . . . 41 This figure shows that a uniform swelling ratio is achieved after the cleansing process which is consistent with the free swelling assumption in the constitutive theory. It also shows that there is a linear decay in the stretch and volume ratios after the cleansing process. ....... 42 Modest values of the Green-Lagrange shear strain, indicating the shear reduction during cleansing ......................... 43 viii In 3.14 Modest values of the Green-Lagrange shear strain, indicating the low shear present during a free swelling test. ................ 3.15 Fishing sinkers used in the forced swelling tests. The sinkers are ar- ranged from heaviest to lightest going left to right. .......... 3.16 The principal stretches and the volume ratio during the 9.77 g forced swelling calibration test. The purpose of this test is to find the duration for future forced swelling tests which is determined by the first plateau region observed. The plateau region in the J response occurs between 10 and 25 hours. ............................. 3.17 The small value of E12, indicating the minimal shear present in a forced swelling test. The initial jump in the shear strain occurs during the initial application of the load and the resulting high rate of swelling. As the rate of swelling slows, the hydrogel becomes equilibrated at the applied load in the water, and the shear strain decreases. After the shear reaches a brief steady-state, the magnitude then slowly increases for the remainder of the test during which volume swelling reduction from possible recrystallization takes place (see Figure 3.16). Had more data been plotted, the initial jump would appear more gradual. 44 45 46 47 3.18 Two cycle loading and unloading after preconditioning at dA/dt = 0.007. 50 3.19 Two-cycle loading and unloading after free swelling at dA/dt = 0.007. 3.20 Two-cycle loading and unloading after preconditioning at three differ— ent stretch rates. ............................. 3.21 Two cycle loading and unloading after preconditioning at three differ- ent stretch rates. ............................. 3.22 Comparing two cycles of loading after preconditioning at dA/dt = 0.005 and dA/dt = 0.009. ............................ 3.23 Comparing two cycles of loading after preconditioning at dA/dt = 0.005 and dA/dt = 0.007. ............................ 3.24 Experimental mapping for all states in the experiments, each referenced to the dry polymer state .......................... 51 54 55 56 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Saturated forced swelling test for three specimens at each of the five different loads, where A3 is referred to the hypothetical dry state. This includes the free swelling result where no load is applied. Error bars are included to show the standard deviation between the three specimens at each load ................................. Volume ratios for saturated forced swelling tests for three specimens at the five different loads, where J is referred to the hypothetical dry state. This includes the free swelling result where no load is applied. Error bars are included to show the standard deviation between the three specimens at each load. ...................... Unsaturated stretch-to-break after preconditioning using three speci- mens. The stretch is referenced to the beginning of the experiment and not to the dry polymer state. Error bars are included to show the uncertainty in the stress measurement; these are smaller than the symbols. .................................. Unsaturated stretch to break after preconditioning using three speci- mens. The stretch is referenced to the dry polymer state according to the experimental mapping (see Figure 3.24). Error bars are included to show the uncertainty in the stress measurement; these are smaller than the symbols .............................. Green-Lagrange shear strain for specimen 2 during the unsaturated stretch-to-break test. The rapid increase in magnitude correlates with the start of failure. E12 and A are referred to the beginning of the stretching test. .............................. Volume ratio during unsaturated stretch-to-break test using specimen 2. This stretch is referenced to the beginning of the stretching test. This shows that specimen 2 had a nearly incompressible response dur- ing the test. ................................ Comparison of the Ogden model and the measurements for the unsat- urated results using the separate fitting with N = 1. This shows good agreement between the model and the measurements for the unsatu- rated results. ............................... Comparison of the Ogden model and the measurements for the satu- rated results using the separate fitting with N = 1. This shows good qualitative agreement between the model and the measurements for the saturated results. ........................... 63 66 67 69 72 4.9 Comparison of the Ogden model and the measurements for the satu- rated results using the separate fitting with N = 1. This shows good qualitative agreement between the model and the measurements for the saturated results ............................ 4.10 Comparison of the Ogden model and the measurements for the un- 4.11 4.12 4.13 4.14 4.15 saturated results using the combined fitting with N = 1. This shows reduced agreement between the model and measurements for the un- saturated results when compared to the separate fitting routine. Comparison of the Ogden model and the measurements for the sat- urated results using the combined fitting with N = 1. This shows better qualitative agreement between the model and measurements for the combined fit as compared to the separate fit ............. Comparison of the Ogden model and the measurements for the sat- urated results using the combined fitting with N = 1. This shows better qualitative agreement between the model and measurements for the combined fit as compared to the separate fit ............. Comparison of Ogden model obtained using specimen 2 and measure- ments for the unsaturated results of specimen 3. The Ogden model with N = 1 displayed is the model obtained using the separate fitting routine which resulted in a better unsaturated response prediction for specimen 2. ................................ Comparison of the unsaturated measurements from all specimens with stretches referred to the beginning of the unsaturated stretch-to-break test. Specimens 1, 2, and 3 results are the same as shown in Figure 4.3. Specimens 4, 5, and 6 are additional unsaturated experiments performed to investigate the repeatability of the stress response. The results show that the response of all six specimen have good agreement until A = 1.4. ............................... Comparison of the Ogden model that is obtained using specimen 2 and the unsaturated measurements from all specimens with stretches referenced to the dry polymer state. Specimens 1, 2, and 3 results are the same as shown in Figure 4.4. Specimens 4, 5, and 6 are additional unsaturated experiments performed to investigate the repeatability of the stress response. The results from specimens 2, 3, 4, and 6 show good agreement with the model, while specimens 1 and 5 do not. 73 75 76 77 78 79 80 CHAPTER 1 Introduction Hydrogels are polymers that can absorb many times their dry weight in water. Hydro- gels are intriguing materials due to their complex nature, and provide many options when designing for applications. They are useful for drug delivery, surgical dressings, contact lenses, artificial tissue (3.9., skin or articular cartilage), and active mate- rials. The simplest elastomeric hydrogels are comprised of high molecular weight, structurally linked elastomers in a low molecular weight solvent. A variety of elec- trochemical effects are possible that play a central role in the resulting mechanical behavior (see, for example Anseth et al. [22]). The manufacture and behavior of hydrogels remain areas of active research, exam- ining polymers and combinations of various polymers to suit various needs. Hydrogels are created through the crosslinking of polymers such as poly(vinyl alcohol) (PVA), polyacrylamide (NIPA or PNIPA), poly(ethylene glycol) (PEG), poly(2-hydroxyethyl methacrylate) (HEMA), among others. Methods of crosslinking include ultraviolet illumination, irradiation with y-rays, and cycles of alternatively freezing and thaw- ing. Some formulations, such as NIPA or HEMA, require additional crosslinking agents and initiators. Other formulations, such as PVA or PEG, can avoid harmful crosslinking agents and initiators which results in greater biocompatibility. The dif- ferent crosslinking methods and protocols of processing lead to various mechanical responses of the final material. Environmental conditions (e.g., temperature, swelling medium, pH) add other variations to the response. Furthermore, even after the ma- terial is made, loss of uncrosslinked polymers into the solvent, possible degradation of crosslinks, and possible recrystallization may alter mechanical responses further [6, 7, 12]. These factors make it challenging to tailor a hydrogel to the requirements of an application. Designing a hydrogel for an application requires understanding the mechanical response of the material as it swells or shrinks subject to changes in mechanical, chemical, and thermal stimuli. A constitutive model is required to predict material response and relate deforma- tions to stresses. Once a constitutive model is chosen, based on prior observation and theoretical development, mechanical testing can validate the model and quantify material parameters that it may contain. Mechanical testing of hydrogels typically apply traction (biaxially or uniaxially) or investigate their swelling response in var- ious low molecular weight liquids. Some studies investigate the combined effects of applying tractions during swelling. Specimens typically are designed as cylinders [13, 20, 33, 35, 40] or standard “dog-bone” shapes, with reference to ASTM D638 [5, 17, 21]. Most experiments focus on uniaxial stretching of hydrogels, likely due to the ready availability of uniaxial load frames, and the relative ease of gripping hydrogels for uniaxial testing in contrast to biaxially applied tractions. Hydrogels are typically more fragile than rubbers making rubber biaxial gripping techniques [9, 36, 44, 48] difficult. Most uniaxial experiments have measured deformation from a grip to grip measurement during stretching [3, 13, 17, 20, 21, 25, 33, 35, 40, 46, 47]. One exception is Johnson et al. [5], who used small tracking particles mixed in their prepolymer solution. Deformations in this case are measured from a smaller central region of the specimen, where the deformation is assumed to be homogeneous, rather than over the whole length of the hydrogel. This is preferred because the compli- ance of the typical hydrogels suggests that grip-to-grip measures may inaccurately reflect the deformation in the central, presumably homogeneous, gage region of the specimen. Johnson et al. found only small errors between inferring strains from grip- to-grip and with the imbedded particles, but this result may not reflect the general circumstance. Likewise, Marra et al. have performed biaxial [38] and uniaxial exper- iments [37] which track small particles in the central and usually homogenous region to measure strain. Shear strain in the region of measurement is also often neglected. The method of data analysis may produce error if shear is present, which may occur even during uniaxial stretching if there is misalignment between the grips. This can lead to an incorrect measurement of strain and calculation of stress. Here we measure the strain field in a manner to account for shear, which allows examination of the assumptions made with respect to deformation in the theory. One focus of the swelling experiments examined here is the mechanically induced transition between saturated and unsaturated states, which has received little at- tention. The saturated state is defined here as the one for which the hydrogel is surrounded by abundant water; all other states are unsaturated, including the bone- dry state. In the bone-dry state, it is assumed that the hydrogel is essentially void of unbound water and consists of polymer alone, becoming quite brittle. Typically, the hydrogel can no longer be reswollen to its useful testing state. This state is useful to determine the mass of the dry polymer, which is needed to calculate the water content of a given formulation of hydrogel. Thus, once a specimen is removed from the liquid bath, it is defined as being unsaturated even if no evaporation has occurred. The mechanical response changes dramatically from the bone dry state to other unsatu- rated states. At less than some level of saturation, the process may be irreversible. Takigawa et al. found the Young’s modulus is enhanced when going from the swollen state to the collapsed state in PN IPA hydrogels [41]. They defined the collapsed state for PNIPA as when the hydrogel exceeds a temperature above 35 °C and loses “much of the solvent present in the network.” The observations by Takigawa at al. indicate a stiffening stress response as the gel drys by losing solvent in the network. Similarly, Johnson et al. found decreasing the pH of the solvent results in decreased swelling and an increase in the Young’s modulus of HEMA hydrogels [5]. However, Marra et al. found hydrogels of PVA combined with poly(acrylic acid) (PVA-PAA) to be stiffer in the swollen state. Similar to HEMA hydrogels, PVA-PAA hydrogels are found to swell in basic solutions and contract in acidic solutions. The saturated state may be also a function of the mechanical deformation. As a porous network deforms due to swelling, external traction, or other stimuli it tends to open or close its porous regions allowing for possible changes in its water content [32]. In addition, transient effects including water diffusion through the hydrogel may cause spatial gradients of the solvent that must be considered. Most constitutive models are based on equilibrium equations that do not account for time-dependant deformations. We are therefore motivated to minimize these transient effects, as well as quantify the saturation state of a hydrogel as referenced to its dry polymer configuration. The results of mechanical testing of hydrogels is typically described using linear elastic theories of deformation [3, 5, 13, 17, 20, 21, 33, 35, 41, 46, 47], despite typically large deformations. There have been some exceptions [11, 25, 37, 38]. Constitutive relations are valid within the range of their underlying assumptions. Thus, linear elas- tic descriptions are well suited for describing mechanical behavior for circumstances without large deformation, instability, or transition from saturation to unsaturation. Furthermore, small deformation theories can describe these phenomena by allowing for parametric dependence of constitutive parameters (6.9., elastic constants, osmotic pressure) on the underlying electrochemistry (6.9., concentrations, charge density). However, owing to their large extensibility and rapid volume changing characteristics due to swelling, water evaporation, and mechanically induced traction, finite defor- mation theories are better equipped to describe the mechanical response of hydrogels. Furthermore, there is a pressing need for a more complete description of hydrogel be- havior for design scenarios where such phenomena are unavoidable, or even desirable. The purpose of this paper is to investigate experimentally a finite deformation theory that accounts for hydrogel saturation. Consequently, the goal will be to obtain material parameters present in the theory, thereby aiding in future numerical studies, as well providing a means to quantify these high degree of freedom systems. CHAPTER 2 Finite Deformation Theory Finite elasticity is used to model the mechanical response of hydrogels used in this study. The purpose of this section is first to introduce the constitutive theory and then apply the theory to the case of a uniaxially loaded hydrogel. Most importantly, we are motivated to formulate the definitions to be used in relating a total deformation his— tory throughout testing, rather than a specific test. The dry polymer state, where no water is present, is chosen as the reference configuration for all future testing. Three equations of particular interest describing the stress state of a hydrogel will result from this analysis. These equations are investigated further from an experimental point of view. Starting from a standard continuum mechanics framework [14], consider a map- ping for a material particle from its reference position vector X to the current position vector a: relative to a fixed coordinate system. This deformation takes X -—> a: a: = :i:(X). (2.1) The deformation gradient F 3a: F = —_ 2.2 8X ’ ( ) describes the transformation of line segments that connect two particles during de- formation from a reference configuration (dry gel) to a current configuration. Con- sidering a specimen loaded in tension in such a way as to cause no shear during deformation, then the mapping becomes F A Homogenoue Deformation Reglon Reference Current Uniaxial Loading Configuration Configuration Figure 2.1. Mapping of volume elements in the homogenous deformed portion of specimen under a uniaxial load in the 83 direction. [- xi = AiXi => Xi = fl , i=1,2,3; (2.3) 2 where A1, A2, and A3 are the principal stretches in the 1, 2, and 3 directions, respec- tively. The deformation gradient for this circumstance is F = /\1e1 (8) el + Agez (8) e2 + A3e3 (8) e3, (2.4) where ei are the unit vectors along the 1, 2, and 3 directions. Using the deformation gradient defined in (2.2), the left Cauchy-Green B and the right Cauchy-Green C deformation tensors are defined as B = FFT and C = FTF , with C = B = [\fel ® el + A382 ® e2 + A363 ® e3. (2.5) The Cauchy (current configuration) stress a is related to the first Piola-Kirchhoff (reference configuration) stress P by 1 T = —PF 0' J ’ where P = — 2.6 6F 7 ( ) W is the strain energy density of the material, and J =2 det F is ratio of the volume change in the material. In general, W is a function of the deformation gradient (i.e., W = W(F)). Because invariance under rigid body rotations must hold for W, it can be expressed as a function of the right Cauchy-Green deformation tensor, W = W(C) [31]. For W = W(C), then W can also be expressed in terms of the three principal invariants of C, 11, I2, and I3 [31, 39]. Then, ~ W(C) = W(Ili12a13) where 11 = tr(C), 12 = gum)? — we?» . and I3 = detC = J2 (2.7) The strain energy can be expressed as two parts. One part, ()5, accounts for the elastic energy of the crosslinked polymer and the other, H (J), accounts for the free energy due to mixing of the surrounding fluid and the polymer [42]. Thus, W(I1,I2,J) =¢(11,12)+H(J), (2-8) where c5 can be the, for example, well-known Mooney-Rivlin or neo—Hookean strain energy density function and H (J) is consistent with the Flory-Huggins theory [42, 43, 44], which is defined H(J) = M[(J — 1) ln(1 — J—1)+ x(1 — J-1)], (2.9) where M and x are material parameters to be determined from measurements. A useful strain energy function for this purpose has the form (101.12) = gm — ea. — 3) + «I2 — 3)], (2.10) where p is the shear modulus of the material and 5 is a parameter obeying 0 S 5 S 1. When 6 = 0 or 5 = 1, the neo-Hookean or Mooney-Rivlin are recovered, respectively. An alternative strain energy equation available to characterize material response is the Ogden model [28, 30]. The Ogden strain energy function is typically written in terms of the principal stretches q5(/\1,)\2,)13), that is N - ¢(A1,A2, A3) = 2 E193” + Ag“ + A? —- 3), (2.11) i=1 02' where p,- and a,- are material parameters obtained from fitting data. The initial shear modulus of the material in the undeformed configuration is related to p,- and 0:,- and is N 2 mar = 2M, (2-12) i=1 where N is a positive integer used in the limit of the summation for the Ogden model. The Ogden model also recovers the neo—Hookean model (N = 1, al = 2) and Mooney-Rivlin model (N = 2,a1 = 2,a2 = —2). Using equation (2.6) and considering one of the strain energy functions previously introduced, stress-strain relationships can be derived. The stress-strain relationship one obtains will depend on the type of loading and consequently, the resulting defor- mation from the loading. In addition, the stress-strain relationship will also depend on if the hydrogel is submerged or not. Therefore, one must distinguish between the two possible states of the hydrogel as discussed in [42]. First, a hydrogel that has available ambient water is termed saturated. Second, a hydrogel with no ambient water is considered unsaturated. An unsaturated hydrogel may be fully swollen, but if there is no more water in the surrounding bath to accommodate further volume expansion, then the hydrogel is defined as unsaturated. For the experiments per- formed here, a stress-strain relationship for a uniaxially loaded hydrogel undergoing homogenous deformation is derived for the saturated, as well as the unsaturated case. The loading and deformation scenario is shown in Figure 2.1 where the e2 direction is that of the applied load. Using the Ogden model and the Flory-Huggins theory, the saturated Cauchy stress can be calculated (see Appendix A for details) N 02 = le-sz-Agi + h(J) (saturated), (2.13) i=1 in the direction of loading and for the traction-free directions N 0 = 1 2: MA? + h(J) (saturated), (2.14) J i=1 and 1 N a. 0 = 7 22—; p,x\3' + h(J) (saturated). (2.15) Here, h(J) is the reduced Flory-Huggins term h(J) : M[ln(1— J—1)+ J-1 + xJ"2] (2.16) A consequence of being saturated is the lack of a constraint on the deformation al- lowing J to change when the hydrogel is submerged in a solvent. If the hydrogel is unsaturated, then a constant reaction stress —pI is introduced to satisfy the incom- pressibility constraint restricting J to be constant during testing [42]. The Cauchy stress of the unsaturated hydrogel is therefore 1 N 02 = — 2 11,213" + h(J) — p (unsaturated), (2.17) J i=1 in the direction of loading and for the traction-free directions N 1 a- 0 = 7 £1 iii/\I' + h(J) — p (unsaturated), (2.18) 10 and N 0 = 7 22—1 mA3 + h(J) — p (unsaturated). (2.19) The condition of J being a constant implies that the volume in the current config- uration equals that of the configuration at which the transition occurred between saturation and unsaturation. If voids are not present in the specimen in its loaded or unloaded state, incompressibility assumes the hydrogel does not gain or lose water during an unsaturated test. Experimentally, J is calculated by dividing the current volume dv by the original volume dV (see Figure 2.1). That is _fl_qm3 _ dV ‘ L1L2L3 = A1A2A3 (2.20) This constant volume constraint restricts the duration of an experiment because wa- ter may evaporate from these hydrogels when they are not submerged. Assuming isotropic response, A1 = A3, J is written J = A22] (2.21) Using equation (2.21), equations (2.13), (2.14), and (2.15) become N 1 . 02 = 7 2 MA? + h(J), (2.22) i=1 and l N A2 0 = 7 [wire/2 + h(J). (2.23) 221 respectively, for the saturated case. Likewise, for the unsaturated case, equations (2.17), (2.18), and (2.19) become N 1 . 02 = 7 Z 11,-)? + h(J) - p, (2.24) i=1 and 1 N A2 0 = 7 Z ,1,-(7)"0'2‘/2 + h(J) — p. (2.25) i=1 11 for {q I) 1-.- Can it» 3.. .11]: l For the unsaturated case, it is convenient to subtract equation (2.24) from equation (2.25), which eliminates h(J) and p. This results in an equation for the unsaturated Cauchy stress that can be used to estimate p,- and a,- from experiment results. That is _1N .O‘i A2 ——a«/2 02 — 7 122.02 — (7) . ), (2.25) where N dictates the number of parameters to be used in the Ogden model. For the saturated case, equations (2.22) and (2.23) may be used to fit experimental results and estimate material parameters M and X with known values of p,- and (1,, which have been obtained from previous unsaturated experiments. Another option is to find all parameters simultaneously using both the unsaturated and saturated test results. There is also an equation that describes free swelling, this is when the hydrogel is free of external tractions and immersed in a bath. With no external tractions, free swelling implies the deformation is identical in all directions, thus A1 = A2 = A3 = A. (2.27) With no tractions in free swelling, 0': 0, and making use of the uniform deformation assumption, equations (2.22) and (2.23) reduce to N Z midi—3 + h(J) = 0. (2.28) i=1 Therefore, useful equations for parameter estimation by analysis of data based on the Ogden model, modified by incorporating the Flory-Huggins theory, are equations (2.22), (2.23), (2.26), and (2.28). The next section will show the experimental protocol used to obtain material parameters using these equations. 12 CHAPTER 3 Experimental Methods This chapter describes the experimental protocol followed to measure parameters in the constitutive model discussed in Chapter 2. First, a mechanical testing apparatus is described in Section 3.1. This apparatus performs individual or coupled experi— ments of stretching and swelling. The two types of experiments are saturated tests during swelling and unsaturated stretching tests. The stretching may be cyclic or stretch-to-break experiments and occur outside of solvent, which are defined as un- saturated tests. The swelling experiments, forced and free swelling, are performed in water with and without loading respectively, and thus defined as saturated. The distinction of saturated or unsaturated results in two diflerent stress-strain relations as shown in Chapter 2. These measurements will be discussed in detail in Section 3.2. To correspond with the complementary theoretical developments, the mechani- cal testing apparatus must be capable of applying uniaxial loads and equipped with an environmental chamber for swelling studies. The device must also incorporate a measurement system to record the applied forces and resulting deformations during loading or swelling. The applied forces are measured using a load—cell and the de- formations are measured Optically with a CCD camera. From these measurements, stress and strain can be obtained and used in the constitutive model to find material parameters. The parameter estimation method will be discussed in Section 3.3. 13 Envlronmental \ Stepper Motor Chamber \ Controller “HR ]|]||Ill|| I p. I. I -" Unlaxlal Tenslon Devlce Figure 3.1. The experimental apparatus used for the finite strain uniaxial stretching tests 3.1 Mechanical Testing Apparatus The test apparatus, illustrated in Figure 3.1 is designed to stretch a specimen uniaxi- ally and perform swelling studies in a sealed environmental chamber. The mechanical structure of the device consists of four 1.27 cm diameter by 45.72 cm length optical bars connected by a flat rectangular aluminum plate on top and mounted on an opti— cal bench. The stepper motor is secured to the top of this aluminum plate and turns a lead screw to move vertically the platform with the load—cell. The load-cell is directly connected to the specimen inducing deformation via a unique gripping system. The load-cell and stepping motor are protected by the two limit switches, which stop the motion of the device in the event of loss of control. 14 load-cell and stepping motor are protected by the two limit switches, which stop the motion of the device in the event of loss of control. A gripping system is designed for ease of specimen insertion and removal. Due to their fragile nature and slippery surface, hydrogels are difficult to grip. Methods of gripping discussed in the literature can be complex. For instance, Hirotsu et al. use a gripping system relying on volume expansion of the gel during hydration to produce a force fitting within a tube [13]. Sasaki and Koga use a silk string wrapped around the specimen to facilitate gripping [35]. Others directly grip the specimen with a clamp type grip [5, 34, 37, 38], while others use an adhesive (i.e., cyanoacrylate) [17] or a combined adhesive and clamp-type grip [21]. The gripping mechanism used in this study uses adhesive, but is unique in the manner of inserting the specimen into the device using a double gripping mechanism. Cyanoacrylate is used to glue the hydrogel to a swatch of cotton and polyester fabric. Cyanoacrylate holds well even after long periods of swelling and subsequent loading. The cotton and polyester fabric is glued to both the front and back of the hydrogel to ensure a uniform strain field is imposed through the thickness upon loading. The fabric forms a loop that is useful for the second gripping mechanism that allowed the specimen to be attached inside the testing apparatus. This mechanism, is built from a hollow garolite tube with wall thickness of 0.635 cm and outside diameter of 1.746 cm. Two tapped holes are machined in the tops of the tubes for a nylon threaded rod to be attached to both grips. One end of the nylon rod is attached to the load-cell and the other to the base of the testing apparatus. The nylon rods allowed the garolite tube to rotate inside the apparatus so that the specimen could be positioned correctly before testing. This also helped to reduce the shearing induced by the grips as fine-tuning adjustments can be made until the specimen is loaded uniformly. This adjustment also made specimen thickness measurements possible before and after testing. In each of the grips a slot is machined to allow the fabric (glued to the hydrogel) to slide in. A toothpick is then 15 glued to the cloth in order to hold the cloth in the gripper when loading is applied. The toothpick sits in the hollow part and is slightly larger than the slot (see Figure 3.2). Figure 3.2. Insertion of hydrogel with cloth and toothpick into gripping mechanism To facilitate physical measurements, the apparatus is equipped with a computer controlled stepper motor, load cell, and a video deformation measurement and feed- back system. The computer controlled stepper motor system consists of a DC stepper motor (SloSyn), 36 volt (3.6 amp) DC power supply (Standa), and a stepper motor controller (Standa). The stepper motor controller allowed for acceleration and decel— eration ramps while providing full, 1 / 2, 1 / 4, and 1/8 step increments and speeds of up to 5000 steps per second. The stepper motor has a full step of 1.8 degrees allowing for a maximum of 1600 and a minimum of 200 steps per revolution. The stepping motor turned a 0.635 cm threaded lead screw with 20 threads per inch. Therefore, the resolution in stretching is 7.94 x 10‘5 cm. 16 .l 3.1.1 Device Measurement System A IOO-gram load-cell (Transducer Techniques) was chosen after initial calibration (40 g maximum failure load) using a store bought hydrogel (Puronyx). Additional components needed to measure the load-cell output voltage included a strain gage amplifier (National Instruments [SCC-SG-24]), signal conditioning connector block (National Instruments [SC-2345]), and a data acquisition card (National Instruments [POI-6221]). The strain gage amplifier provided the needed 10 volt excitation to the load-cell resulting in an output of 10 millivolts when loaded at full capacity. The amplifier reduced the noise in this 10 millivolt output signal coming from electromag- netic interference with nearby electronic devices. The load-cell produced a repeatable signal to within 5%, giving a resolution in load of 0.05 g. Video strain measurements are made with a monochrome CCD camera capable of capturing images at 30 Hz. A 200 mm Nikon micro lens is used to track microspheres during the stretch-to—break tests. The large field of view is needed because of the quasi-static loading system. Since the bottom grippers are fixed, the field of view must be large enough so that the tracking markers can be imaged throughout the stretching. This camera also must be positioned far from the specimen thereby decreasing the resolution. A 500—600 pm diameter microsphere provides adequate resolution. A 105 mm Nikon micro lens is used for tracking spheres during swelling experiments. The camera with the smaller lens is mounted on a scissorjack to adjust its height during swelling experiments while maintaining the distance from the specimen. The fixed distance from the specimen is required for consistency in measuring specimen dimensions. The acquired images are fed through a single channel video framegrabber board (National Instruments POI-1405) to be analyzed in LabVIEW 8.2. LabVIEW Vision Assistant 7.1 is used to design a filter that resolves only the glued tracking spheres on the specimen and computes coordinate locations of the center of mass of each sphere. Additionally, this program converts the filter into LabVIEW code to be 17 incorporated into the overall data acquisition program. The video system is capable of computing real—time principal stretch ratios and saving the sphere positions to a computer file at the 30 Hz rate of the CCD camera. 3.1.2 Testing Capabilities Two LabVIEW Virtual Instruments (VI’s) provide interfaces for the device compo— nents during swelling and stretching tests. One V1 is written to run during swelling tests while another is written for stretching tests. Both VI’s capture images, perform real-time calculations of deformation, and write to measurement files, simultaneously. A frame-grabbing acquisition is used in the main LabVIEW program to store images in a buffer during program execution. This allows capture of images at 30 Hz and performance of calculations on the images simultaneously during testing. As images are being captured and saved in a buffer, the program extracts images from the buffer, performs real—time calculations, and force measurements (stretch-to—break only), and writes the result to files. For synchronization of images with corresponding calcula- tions and measurements performed, the program is required to run at the same speed as the frame acquisition. This is achieved by using timers in the program where ad- justments are then made until the program executes at a rate of 30 Hz. Adjustments included finding the most efficient method of programming the real-time calculations in both VI’s and adjusting the rate and number of load-cell measurements in the stretching program. There is an initial synchronization problem at the beginning of image capture, before the program executes. Due to the overhead of loading the program, an initial 12 images on average (about 0.4 seconds) must be deleted at the beginning of the tests. The images after this time then correspond to the calculations and measurements made in the program. For example, during a stretching test, the program pulls the first image from the buffer and takes a load-cell reading with this image. This load-cell reading does not correspond to the first image after the initial 18 DEE the time delay as it should. The VI written for the swelling tests includes an interface allowing the gel to be easily positioned into the view of the camera, or vice versa. Once in position, the initial specimen cross-sectional area can be measured Optically. The gripping mechanism discussed above is advantageous here because it allows the specimen to be turned sideways for a thickness measurement. Once the specimen area is measured (forced swelling test only), the reference tracking sphere positions are recorded and the program is started. The program then tracks the sphere positions at a rate of 30 Hz during swelling. Because swelling tests generally run for at least 24 hours, the positions are saved at only 1 Hz to reduce the file size. The VI saves two user named files of the sphere positions and corresponding time during the test. Other optional files may be saved in the beginning and end of a test for initial and final dimensions between tracking spheres in pixels. The other V1 is written to run two types of uniaxial stretching tests. The first is a preconditioning test having the same interface as the swelling test for initial placement of specimen and cross-sectional area measurements. In general, the preconditioning VI allows the user to select the stretch ratio rate (dA/dt) and maximum stretch ratio of the test. It then cyclically stretches to the maximum stretch ratio chosen and returns to the reference position. This can be repeated for as many times as desired by the user, but must be manually stopped at completion. The user must also specify the initial start speed of the motor and the stepping increment. A stepping increment of eight steps per full step provides good resolution in positioning and smooth motion. The initial start speed is adjusted depending on the stretch rate chosen. For instance, the experimenter may need to run tests at different speeds to calibrate the initial crosshead speed of the grips to insure that the specimen stretches at the chosen dA/dt. This results in less buffer time for the control algorithm and more consistency between various dA/dt by avoiding the initial speed differing greatly from the initial 19 rate to which the material responds. This program then computes a running average of dA/dt and compares this rate to the chosen rate and adjusts the motor speed, accordingly. This type of control is necessary because the motor speed is nonlinearly related to the stretch ratio rate in the central portion of the specimen. Tests indicate the motor speed increases throughout the stretch to maintain the desired stretch ratio rate. For uniaxial stretching-to-break tests, the same program is used, and the user chooses a large stretch ratio that is beyond the failure point of the specimen. After specimen failure, the user manually stops the program. This program writes four main files at a rate of 30 Hz during testing. These files are the real—time stretch ratios, particle histories, load-cell readings and corresponding time of measurement. Similar to the swelling VI, initial and final cross-sectional dimensions, in pixels, can be measured optically and saved. Although final specimen dimensions can only be measured if the test had not run to failure. 3.1.3 Stretch Ratio Measurement Stretch ratios are estimated in real-time, for control of the motor, and then calculated more accurately after the test, using a bilinear interpolation algorithm. The real- time stretch ratio is estimated by averaging the change in the horizontal and vertical positions of the spheres. That is _ 31 _ 52 A1 — L1, A2 — L2’ (3-1) and 231 = (X2 - X1) + (X4 - X3), 232 = (Y4 - Y2) + (Y3 — Y1), (3-2) where X,- and Y,- are defined in Figure 3.3 and L1, L2 are the reference configuration lengths taking the place of £1 and 82 (current configuration lengths) in equation (3.2), respectively, before testing begins. It is important to note the manner in which the spheres are staggered in Figure 3.3. This pattern is necessary due to the manner 20 Ya—Y. k——> Y4-Y2 Y v Figure 3.3. Lengths between tracking markers for real time stretch ratio calculation in which LabVIEW’s particle analysis program labels the spheres. The first sphere is assigned to the particle with the lowest Y—coordinate. The second sphere having the second lowest Y-coordinate and similarly for the other two spheres. Therefore, the spheres are staggered in order to avoid the case for which spheres could switch positions relative to the axes during stretching, as might happen if they are aligned more evenly. If the spheres switch positions, LabVIEW computes the stretch ratios using incorrect lengths between spheres and the particle histories fail to correspond to the same sphere. The real-time stretch ratio measurement is incapable of capturing possible shear- ing that may take place during stretching. A bilinear interpolation analysis [19], inspired by the finite element method, calculates a more accurate stretch ratio mea- surement. The bilinear interpolation algorithm defines functions to approximate the displacement field of the tracked spheres, namely u(7‘,3)= Z fi(r,s)u,-, (3.3) i=1,2,3,4 v(r,s)= Z f,(r,s)v,, (3.4) i=1,2,3,4 21 (3.5) X035) = fZ(T13)X21 i=1,2,3,4 YUM?) = fz(7", 5W2, (3-6) z=1,2,3,4 where, I fi(r, S) = 1(1 + 887;)(1-1- T‘T‘i) , i: 1, 2, 3, 4 (3.7) and u,- and v,- are the displacements of the four particles in the x and y directions, respectively. These displacements can be calculated using 2': 1,2,3,4 (3.8) Ui=$t-Xt vizyi‘Yi , The interpolation function maps the particles locations from the (:r,y) plane to a new Figure 3.4. Mapping of tracking spheres from the X — Y plane to a uniform square in a local 3 — r coordinate system. set of local coordinates (3,7) The locations of the points mapped to (s,r) are chosen to be a uniform square, shown in Figure 3.4. Bilinear interpolation is then used to calculate the displacement gradient H (see Appendix B for details). _ Bit/82: Bit/59y H _ ( 612/82: 622/83; ) (3.9) 22 With thU the it .;.. rurdf Tod sheer With the displacement gradient, the deformation gradient F is then calculated _ _ 8u/8x+1 8u/8y _ A1 3% F—H+I_( 8v/83: 8v/8y+1)_(g% A2 ’ (310) where A1 and A2 are the bilinear interpolated stretch ratios. As may be expected, the interpolated stretch ratios are the same as the real-time stretch ratios when the shearing terms are zero (i.e., 8u/8y = 812/82: = O) [19]. Ortt [29] suggests that if the shear strain is small, the interpolated stretch ratios can be corrected using the shearing terms [45] such that 811 = 2 __ 2 , and A2 = A2 + (g)? (3.12) 2 8:1: To determine the magnitude of shear strain in the specimen, the Green-Lagrange shear strain may be written as E _1(Qg+§3+@@+_8_v2q) 12_2 8y 8:1: 8x83; 82:83]. Ortt observed a maximum of E12 = 0.07 in biaxial stretching measurements of Neo- (3.13) prene rubber. In the current work, dynamic uniaxial stretching experiments produced the largest shear as expected. The maximum shear strain observed was E12 = 0.40, although E12 = 0.07 is the maximum shear strain accepted for parameter estima- tion. Therefore, the corrected bilinear interpolated stretches (equations (3.11) and (3.12)) are used for the uniaxial stretching experiments in this study. On the other hand, the swelling experiments are found to have less shear strain with a maximum of E12 = 0.024. The small shear strain in these tests suggest that the use of real—time principal stretches with associated maximum error of less than 3%. The use of real- time principal stretches is convenient, because it allows multiple swelling tests to be run simultaneously without constant camera monitoring. After the stretch ratios are obtained, the volume swelling ratio J can be calculated according to equation (2.21) and used in the constitutive model developed in Chapter 2. 23 1". iii 9L1 3.1.4 Matlab Routine for data reduction After the experiments, the measurement files are imported into the Matlab workspace for further calculations. Separate Matlab routines are programmed for each of the three types of experiments, these being stretch-to-break, free swelling, and forced swelling. The routines compute the stretch ratio by bilinear interpolation, Cauchy stress (for stretch—to—break and forced swelling tests only), and the uncertainties in the measurements. The bilinear interpolation algorithm uses the particle history file written in LabVIEW containing coordinate locations of the four spheres. For the forced swelling and stretch-to-break experiments, the Cauchy stresses are calculated according to the methods discussed in Appendix A. Three subroutines are used by the three main routines. One subroutine computes the particle displacements frOm the particle history file for the bilinear interpolation. Another subroutine averages the force measurements from the load-cell file for the stretch-to—break experiments. As describe above, measurements are taken at 30 Hz corresponding to 30 iterations of the LabVIEW program. In each iteration of the program, ten force measurements are recorded. This subroutine then averages the ten force measurements for each iteration of the program. The last subroutine computes the uncertainty of the opti- cally measured stretches for each of the three experiments, while the stretch-to-break and forced swelling experiments also calculate the combined uncertainty of the stress measurements. See Appendix C for uncertainty calculations. The Matlab files are written to extract the appropriate measurements correspond- ing to the initial synchronization problem discussed previously. For synchronization of images with load-cell measurements, the first 0.4 seconds of images are deleted. This is determined using a timer in LabVIEW showing two time delays before the program starts executing at 30 Hz. The first delay is the time difference between when the frame capture starts and the program begins and is 0.21 seconds on av- erage. This time corresponds to seven captured frames (rounded up to the nearest 24 whole number). The second time delay is the time it takes to run the first iteration of the program and is about 0.14 seconds on average. This corresponds to five cap- tured frames before the second iteration of the program. Subsequent iterations after the initial time delay then maintain the 30 Hz execution, staying in-sync with the camera. The data deleted includes the first seven data points of the stretch ratios during testing. Then, skip one data point, and delete the remaining five data points. Similarly, the last 12 measurements of the stress calculations are deleted to obtain a plot of stress as a function of strain. Other files using images or force measurements for calculations are deleted in the same manner. Lastly, the 0.4 second total time de— lay also required a test to be run for at least an extra half a second after completion. This happens naturally due to the small time, of course. Similarly, for free and forced swelling experiments, the initial seven stretch ratio measurements are deleted. Then, one stretch ratio measurement is skipped and the following three measurements are deleted. Before running the Matlab routine, the user also has an option of how many data points to print for each measurement in the Matlab workspace for later plotting. As mentioned above, experiments can also be run without continuous camera monitoring, as long as the initial and final specimen dimensions and sphere positions are measured. This is possible because of the small shearing strain observed in both the free-swelling and forced-swelling experiments for which the real-time stretch ratio can be used. For these experiments, separate Matlab files are written to compute one real time stretch for the experiment using the averaged initial and final lengths between between spheres. Also, for the forced swelling test, the cross-sectional area is measured at the beginning of the experiment for the stress calculation at the end. Note that the specimen is measured in the beginning and the end of experiment in the mechanical testing device. During the experiment, the specimen is typically placed in a auxiliary testing chamber, allowing the device to be used for another experiment. A bias error is possible here if the specimen is not inserted back into the device grippers 25 Wf Dd in the exact spot it was previously. To minimize this bias error an alignment line is drawn on the cloth (see Figure 3.9) and the gripper to keep the specimen centered, and the grips are perturbed using small twists during the measurements. 3.2 Development of Poly(Vinyl Alcohol) Hydrogels To aid in construction of the mechanical testing apparatus and preliminary studies, we used a commercially available hydrogel (Puronyx). Synthesizing hydrogels in house then became desired because it allowed control of the specimen properties, as well as an understanding of the components of the hydrogel (i.e., molecular weights, concen- trations, etc), used in manufacturing and their effect on the response of the material during swelling or loading. The hydrogel in this study needed to be relatively com- plaint and be able to swell in water. In addition, a biologically compatible hydrogel would be beneficial for further application studies and safety in manufacturing meth- ods due to the harmless constituents. These requirements led to the development of a poly(vinyl alcohol) (PVA) hydrogel specimen. PVA hydrogels are known to be hydrophilic with a significant swelling response in water. This swelling response can be further enhanced through combination of PVA with other polymers during crosslinking. One combination investigated here involves crosslinking PVA and poly(acrylic acid) (PAA) using a freeze-thaw process [2, 4, 26]. Freeze-thaw crosslinking is a safe and simple process avoiding dangerous chemical agents that are unsuitable for biomedical applications. Therefore, freeze- thaw crosslinking is the chosen method of synthesis for our hydrogels. After determining a specimen synthesis protocol, experiments are designed to investigate the unsaturated and saturated stress equations obtained in Chapter 2. In general, these experiments consist of an initial process of soaking in water to remove unentangled PVA [6, 7, 12], followed by free swelling and forced swelling experiments. The two swelling experiments will be used in an attempt to obtain material parameters 26 from the saturated stress equations. On the other hand, a stretch-to—break experiment is performed out of the water and will be used to obtain material parameters from the unsaturated stress equation. With stretch-to—break experiments, preconditioning the specimen must be considered and this will be discussed below. Finally, this chapter ends discussing how the designed experiments relate the deformation during a test to the dry polymer reference state for the parameter estimation process. 3.2.1 Freeze-Thaw Crosslinking Method The method used to crosslink PVA involves cycles of freezing and thawing until a solid network results upon thawing. Before this process, the PVA must be dissolved in water. The dissolution process, while not described in detail in the literature, is a intricate process. Dissolving must be done carefully as it may adversely affect the final structure of the hydrogel if not performed appropriately. The method of dissolving PVA particles in water will be discussed in further detail in the specimen preparation subsection below. A freeze-thaw PVA hydrogel is formed from a “physical” crosslink— ing due to crystallite formation [12]. Stammen et al. [18] suggest the freeze-thaw cycle promotes a mesh entanglement between molecules of PVA to create mechanical strength [23]. After freezing solutions of PVA, the crystalline nature of the structure is visible to the eye and becomes less visible as the hydrogel thaws. On a molecular level, Hassan suggests [12], the crystallite can be described as a layered structure consisting of a double layer of molecules held together by hydroxyl bonds and weaker Van Der Waals forces operating between the layers. For describing semicrystalline polymers, models have been proposed by Flory [10] and Keller [24]. Hassan [l2] cites the Flory model that proposed the presence of small ordered regions called crystal- lites scattered in an unordered, amorphous polymer matrix. From results obtained through X-ray diffraction, Hassan [12] cites the Keller model that considers polymer crystals to be in the form of a lamellae, or thin platelets, from the folding of polymer 27 chains. Hassan agrees that Keller’s model is more likely to describe crystallite for- mation because polymer chains prefer to be in a folded state, unless exposed to some stress or force [12]. As mentioned above, the freeze-thaw type of crosslinking has the benefits of avoid- ing harmful chemicals. In addition, the mechanical properties can be adjusted by the number and duration of freeze-thaw cycles performed, as well as the molecular weight and concentration of PVA used. This is also one of the drawbacks because the crosslinking agents typically produce stronger bonds resulting in crosslinks that are more stable which make hydrogel degradation less of a problem. Similar to the mechanical properties, the stability of PVA freeze-thaw hydrogels depends on the molecular weight, concentration of PVA in solution, and the number and duration of freeze-thaw cycles [6, 7, 12]. Furthermore, the method of synthesis determines the type of crystalline structure (size, quantity, and distribution) formed and this also con- tributes to the stability and mechanical properties of the hydrogel. In general, more cycles of freezing and thawing are found to produce a more physically crosslinked, stable hydrogel, whereas fewer freeze-thaw cycles leads to a hydrogel that is more susceptible to secondary crystallization. Secondary crystallization is the formation of additional crystallites, thereby excluding more of the amorphous regions and altering the mechanical properties of the hydrogel [6, 7, 12]. These phenomena dictate the re- sulting experimental approach, which avoids their time dependant effects that alters the response of the hydrogel. 3.2.2 PVA-PAA versus PVA Hydrogel An interesting hybrid hydrogel is obtained from a freeze-thaw process of both PVA and PAA. This results in a pH-sensitive hydrogel having an increased swelling response over the standard PVA hydrogel. Marra et al. used a method to make PVA-PAA hydrogel involving a heating and boiling process [34, 37, 38]. With the resulting 28 hydrogels, they observed an equilibrium volume swelling response in five minutes using various pH solutions [34]. They found the pH to dictate the magnitude of the swelling as well as a hystersis in the swelling response as the pH is increased from 2 to 8 and then similarly decreased. In general, they observed more swelling at a greater pH (about 6) and reduced swelling until complete contraction at a pH of 2, which is the most acidic that they tested. Here, the time to reach maximum swelling volume upon entering water is approximately five hours with PVA-PAA and this maximum is not steady-state as indicated in Figure 3.5 below. In comparison, the standard PVA took about 10 hours to reach this maximum swelling volume and is similarly not steady-state. The PVA specimen appears to be more stable than the PVA-PAA specimen because of the less steep SIOpe after the peak indicating less reduction in volume swelling in the PVA hydrogel. Note that the initial dip shown for the PVA specimen is a result of the initial inhomogeneous swelling response that will be discussed ahead. The maximum swelling volume is found to be greater for the PVA-PAA hydrogel and increased as more PAA is used in the specimen preparation. As the water bath in which the hydrogel swells is replaced with fresh water, the volume swelling response increases for the PVA-PAA specimen as shown in Figure 3.6. On the other hand, the standard PVA specimen’s volume swelling response is unaffected by changing the bath water. It is suggested [6, 7, 12] that these “phys- ically” crosslinked systems have an unstable swelling response due to secondary re- crystallization and uncrosslinked polymer leeching out, effects that are present upon swelling. These phenomena happen in both the PVA—PAA and PVA hydrogels due to the freeze-thaw crosslinking process. Therefore, the increased instability of the PVA-PAA can be attributed to the uncrosslinked PAA chains leaving the network. As mentioned above, Marra et al. observed their PVA-PAA hydrogel to contract at pH’s between 2 through 6 with slight differences due to the direction of changing pH 29 2.6 . . . T 2.4 are Tl < > 2.2 .............. 2 b .. ................... -( 1.8 ’3 1.6 l I 1.4 1.2 ° 1m] lg. _ 0.8 ‘ ' L ‘ 0 5 1 0 1 5 20 25 Time [h] *i' i' *- . *- .*. - i- *- *- *i' *- Figure 3.5. Initial swelling volume response of PVA and PVA-PAA. (hysteresis). This contraction progressed as the pH decreased from about 6 to 2. The PAA polymer (Scientific Polymer Products, Inc.) used in this study here has a pH between 2.2-3.0, as indicated by the material safety data sheet. As the PAA chains leeched into solution upon swelling, the initial pH (about 7) of the water bath de- creases. This decrease in pH results in further reduction in swelling and may explain the faster volume reduction indicated in Figure 3.5. Measurement of the water baths PH level after testing indicated pH levels between 5 and 6.5 depending on the size 0f the swelling chamber. As suspected, the smaller the chamber, the smaller the pH level at the end of test. It should be emphasized that pH measurements are recorded after testing without the specimen in the bath and therefore the PAA chains would have had more time to disperse into the solution. In addition, it seems reasonable 30 3.5 2.5 1.5 Figure 3.6. Three cycle swelling of PVA-PAA hydrogel with water change in between. to suspect that during initial leeching of PAA chains the local pH at the hydrogel surface would be less than the measurement after testing leading to increased vol- ume swelling reduction. In conclusion, a PVA-PAA specimen has the shared unstable swelling response of the PVA specimen, but has an additional swelling instability due to contamination of the water bath. To insure repeatable experiments, this additional instability led to the decision to use standard PVA for further testing and parameter estimation. First Swell 0 Second Swell 1* Third Swell x Final Swell Xxx Xxx Xxxx Xxx Changed Swelling Bath Water 1 1 0 Time [h] 3.2.3 Specimen Preparation Three grams of 99% hydrolyzed poly(vinyl alcohol) (Scientific Polymer Products, inc.) with an average molecular weight of 86,000 is dissolved in 50 grams of deionized water 31 15 20 25 '“W -. fl." . using a 150 mililiter beaker. The PVA is slowly dispersed into a beaker under constant, but gentle stirring, with no heating. It is important to emphasize “slow” dispersion because better dissolution results are obtained if the PVA particles are individually wetted and clumping is avoided [16]. After all of the PVA has been dispersed into the water, an additional ten minutes of stirring with no heating follows. After the PVA particle “wetting” stage, the hot plate is powered and allowed to heat the solution under constant light stirring for ten minutes (79 °C). Note that the temperatures reported here are calibrated without using PVA and with no stirring, while the amount of water and duration of each process is consistent. The temperatures are measured using a thermocouple and they are reported at the end of each process. This is so the process can be replicated on any hot plate stirrer, where the temperature settings need to be calibrated to the temperatures reported here and the speed should be adjusted to avoid clumping (hardening) of PVA near the surface during the dissolving process. Near the end of the ten minutes of heating with light stirring, a metal spatula is used to push PVA particles that may be stuck to the side of the beaker back into the mixture. This helps to maintain a consistent fraction of PVA after dissolution and ensures that all of the PVA particles dissolve. The stirring speed is then increased with the same amount of heating for an additional five minutes (84 °C), after which the stirring is increased again with the same amount of heating for another five minutes (78 °C). The heating is then increased and stirred at the same rate for another five minutes (90 °C). Finally, the last process involves fast stirring followed by no stirring until all of the PVA is dissolved. No stirring allows the partially dissolved particles to fall to the bottom, thereby heating them and aiding in dissolving. The subsequent fast stirring helps reduce clumps that form near the surface from over-heating and helps force the particles into solution. This stage is repeated until no PVA particles are observed in the solution. After the last PVA particle is dissolved, the beaker is weighed to determine the 32 difference between the final weight and initial weight. More specifically, before the dissolution process the beaker, magnetic stirring rod, and 50 g of water are weighed on scale (AND FR—300) and then the scale is tared. After dissolving, the difference in weight is recorded accounting for the 3 g of PVA not measured in the initial zeroing of the scale. A simple calculation yields the mass fraction of PVA in the final solution _ mPVA , (3.14) mPVA+mw where C is the mass fraction of PVA, mPVA is the mass of PVA polymer (3 g), mw is the mass of water present after dissolving. To find the mass of water lost during dissolving, the following calculation is used mw = mW,o — mW,e = mW,o — (mscale “ mPVA): (3-15) where mwp is the mass of water used before dissolving begins and is 50 g. The scale is tared with the water and magnetic stirring rod in a beaker with no PVA before dissolving starts. At the end of the dissolving, the beaker with water, magnetic stirring rod, and PVA are weighed. Assuming that all 3 g of PVA dissolved into solution, the mass the scale reads after dissolving, mscale, less mpVA yields a negative number indicating the mass of the water evaporated mw,e during dissolving. The 3 g of PVA mass is subtracted because it was omitted in the initial measurement before dissolution. It has been found that it is better to mix the PVA particles into the solution during constant stirring. On average, the mass of water evaporated during the dissolving process is approximately 14 g. Also, C of the solutions prepared with this process varied between about 7 to 8%. Upon dissolving, the heated mixture cools for at least one hour in the ambient air before the freeze-thaw crosslinking process. After one hour, approximately one milliliter of polymer solution is measured into each of six individual dumbbell shaped moulds that are designed following ASTM D1708-02a standard for micro—tensile plas- tic specimen. After the six moulds have been filled, the mould is inserted into a 33 O O O O O O O O O O O O O O O O O O o O O O Figure 3.7. An isometric view of the aluminum six specimen dumbbell shaped mould. The length of specimen is 38 mm, the thickness is 3 mm, the gage length is 22 mm, and the gage width is 5 mm. Refer to ASTM standard for more detailed dimensions. standard household freezer allowing the solution to cool to a temperature of approx- imately -11 °C. This initial freezing process continues for 90 minutes and then the mould is removed to allow thawing for 30 minutes. This freeze-thaw process is re- peated once more to enhance the overall toughness of the network. Note that after the last freezing cycle, the hydrogels are removed from the mould while frozen and allowed to thaw for approximately 30 minutes, until no ice crystals can be seen. Re- moving the specimens while frozen reduces damage to the hydrogels as compared to removing in the more fragile thawed state. Typically, a specimen sticks to the sides of the mould and requires “peeling” from the mould. The two freeze-thaw cycles of 90 minutes of freezing followed by 30 minutes of thawing produces the appropriate gel to fail at less than 100 grams of loading, which is the maximum capacity of the load cell. Calibration experiments have indicated that stiffer hydrogels can be made by increasing the duration of the freezing and the number of freeze-thaw cycles, while the number of freeze-thaw cycles has been observed to contribute more to the stiffness than does the duration of each freezing. After the final freeze, the hydrogel thaws at room temperature. Before it has had time to thaw, the frozen hydrogel is less dense than water and will float in water. After the hydrogel has thawed (approximately 15 to 30 minutes), the hydrogel becomes 34 denser than water and sinks upon being placed in water. The frozen and thawed hydrogel are observed to be neutrally buoyant in the residual polymer solution, which suggests that the frozen and thawed hydrogel have the same density. This seemingly contradictory behavior may result from the relatively small difference in densities between the frozen and thawed states and the greater viscosity of the residual polymer solution. The net buoyancy may be unable to overcome the viscous resistance. After the final thaw, the mass of the hydrogel is recorded to be used later to account for diflerences in the amount of PVA between specimens, assuming the same water. The last step before the pre-test cleansing process is to glue the tracking spheres along with the toothpicks and cloth to be used by the gripping mechanism. As mentioned above, the spheres are used for the stretch calculations during defor- mation and cross-sectional area calculations for stress. For the stretch calculations, four spheres are glued on the front in a centered (away from the edges) presumably homogenous region of the specimen. For the stress calculations, two spheres are glued on the edges in the front above the other four spheres. Also, one additional sphere is glued directly behind one of the front edge spheres, on the back of the hydrogel. Dur- ing the freeze-thaw process, a small ridge forms along the middle of the specimen on the side exposed to the air during freezing, while the hard side facing the aluminum mould remains smooth. The four spheres to be tracked during experiments are then glued to the smooth side of the hydrogel as well as the two spheres positioned on the edges. Also, a permanent marker is used to make alignment lines on the cloth as shown in Figure 3.9. The purpose of these lines is to ensure the center placement of the hydrogel in the grips to reduce error in measuring specimen cross-sectional area, non-axial loading, and measuring sphere locations when experiments are run in a separate chamber. Finally, the toothpick and cloth are glued to the specimen as shown in Figure 3.8 and the specimen is weighed a final time before testing, which accounts for water loss between weighing after the freeze-thaw process and gluing the 35 Figure 3.8. Front side of specimen after preparation with toothpick, cloth, and spheres glued on. Figure 3.9. Back side of specimen after preparation with toothpick, cloth, and spheres attached. tracking spheres along with the toothpick and cloth. The weight of the toothpick and cloth must therefore be weighed before being glued to the specimen. Weighing the specimen before further testing is important to be able to relate the hydrogel 36 deformation to its dry polymer state in the experimental testing protocols to follow. The dry polymer state is obtained by letting one of the six freeze—thaw specimens dry. The leftover polymer with no unbound water is then defined to be the dry poly- mer reference state. To be consistent, the specimen to be dried out will undergo the same swelling routines discussed below, to rid out the uncrosslinked PVA. Using the mass of the dry polymer along with the mass of the hydrogel with water before testing one can calculate the volume ratio after the freeze-thaw and specimen preparation pl'OCCSS th, l’ p m H20 1 dp( H20) Vdp pH20 (mdp) , where V1120, Vdp, pH20, pdp, m H20, mdp are the volume, density, and mass of the water in the hydrogel and the dry polymer, respectively. The density of water at the ambient air temperature of 26 °C is pH20 = 0.997 g/cm3. The density of the dry polymer is found to be pdp = 1.268 g/cm3. Hassan and Peppas note that Sakurada et al. [15] found 100% amorphous PVA to be 1.269 g/cm3 and 100% crystalline PVA to be 1.345 g/cm3 [8]. The measured value here is similar to the 100% amorphous value reported in the literature which is expected because the hydrogels constructed here are “loosely” crosslinked using only two freeze-thaw cycles. The mass of the water in the hydrogel after specimen preparation, m H201 is calculated mH20 = mg — mdp, (3.17) where mg is the mass of the hydrogel after specimen preparation. Note that the mass of the toothpick and cloth glued to the hydrogel is used to calculate mg. The mass of the dry polymer for the specimen used in testing, mdp, is . )mdg,d1 (3.18) where mm and mdgfl- are the mass of the hydrogel to be tested and to be dried, respectively, after the freeze-thaw process. The mass of the dried hydrogel, mdgfl, is 37 determined after being exposed to the ambient air until its mass becomes constant, which takes about a week. Note that only the volume ratio can be measured here, where it is assumed the hydrogel consists of only water and polymer. The deformation gradient F ft can not be measured at this state, but this will become useful in the parameter estimation when the experiments are referenced to the dry polymer state to then be used in the constitutive model. With the volume ratio referenced to the dry polymer state (J ft) known, the spec- imen is ready to begin a cleansing process. 3.2.4 Cleansing versus Free Swelling Cleansing, or soaking to remove uncrosslinked PVA, is necessary before testing to reduce material inhomogeneities resulting from the freeze-thaw crosslinking process and diffusion at surface during the specimen preparation. Specifically, this process rids the specimen of uncrosslinked polymer [6, 7, 12] and allows for uniform distribution of water within the network. Cleansing involves swelling the hydrogel in deionized water for 60 hours, which is suflicient based on results from the literature and preliminary studies. Hassan and Peppas [6, 7, 12] demonstrate, by spectroscopic examination of the solution, that after 4 to 5 days all of the uncrosslinked polymer exits the PVA network. After this time, no more polymer left the network during an additional six months of swelling. Their results show that after approximately 60 hours, within the uncertainty of the measurement, all of the uncrosslinked polymer had exited the network. Likewise, initial swelling of PVA after specimen preparation produced a repeatable plateau at approximately 60 hours of testing in preliminary tests here. According to Hassan and Peppas, at this time all uncrosslinked polymer has exited the network and the physical mechanism left working is the further crystallization of PVA chains in the network. This further crystallization enhances the stiffness of the hydrogel by forming more crosslinks and thereby excludes amorphous regions, 38 resulting in water being excluded from the network. 2 . . . . . . s 1'8 ' Plateaus ‘ c1..- _ 1.2 - 0'80 2'0 410 ah 8b 160 120 1:10 160 Time [h] Figure 3.10. Plateaus observed in the volume swelling ratio. As may be seen in Figure 3.10, after an initial onset of swelling, the volume ratio reaches a peak at which crystallization is assumed to dominate while some uncrosslinked polymer still leaves the network. As 60 hours of swelling is approached, the hydrogel loses less of the uncrosslinked polymer until all of the uncrosslinked polymer has left and the first plateau is observed. A steady-state does not exist for these hydrogels that are crosslinked without a chemical agent. According to Hassan and Peppas, even the most stable freeze-thaw PVA hydrogels show water loss during the first two months of testing. After the first plateau, Hassan and Peppas suggest that only further crystallization is taking place, being a naturally occurring process and unavoidable. Thus, 60 hours is chosen for cleansing the hydrogel. 39 Although the cleansing process is a flee swelling process, it violates the uniform deformation assumption discussed in Chapter 2 for saturated free swelling. It is therefore considered independent of flee swelling. The cleansing process prepares the hydrogel for flee swelling by allowing the hydrogel to develop a uniform water distribution. Achieving a uniform water distribution can be thought of as undoing the non-uniform water distribution developed by a diffusion process during the fleeze— thaw cycle along with the other preparatory steps. Figure 3.11 illustrates this by showing the stretch ratios during 60 hours of cleansing. There is an initial decrease in A1, indicating non-uniform swelling which appears to have been caused by greater swelling in the vertical direction. As seen in Figure 3.11, the stretch in the vertical direction jumps flom 1 to 1.02 and continues to increase at a steeper slope until the swelling in the horizontal direction hits a minimum and starts to increase. As A1 starts to increase, the slope becomes less steep in the vertical swelling direction. Therefore, as the hydrogel develops a uniform water distribution, a Poisson-type thinning effect occurs in the vertical direction, after which the water distribution tends to equilibrate allowing both directions to expand. This illustrates a difference between a cleansing and a flee swelling test. The constitutive model assumes that flee swelling suggests equal expansion (or contraction) in all directions. This assumption is not observed until a uniform water distribution is present. Figure 3.12 shows a swelling study performed after a 60-hour cleansing and therefore referenced to the hydrogel configuration at the end of cleansing. Figure 3.12 shows that after cleansing, the hydrogel exhibits isotropic deformation. For the first 40 hours, the flee swelling assumptions holds with a slight deviation near the end. This is acceptable here because 24 hours has been determined to be the duration of the flee swelling test. Therefore, not only does the cleansing reduce material inhomogeneities by excluding uncrosslinked polymer flom the network, but it also prepares the hydrogel for a flee swelling test by satisfying the model assumptions of uniform deformation. Another 40 1.3 . . . . n T 0 A2 1.25‘ O /\1 ‘ .' O 12- - - CD 0 .2 ' . 331.15 ‘ :3 U) 1.05 . - 37’ 1 ,' initial dip from non—uniform swelling _ :/ 0.95 l 1 1 1 m l 0 1O 20 30 40 50 60 Time [h] Figure 3.11. This figure shows that the non-uniform swelling ratios during cleansing violate the free swelling assumption in the constitutive model. indication of how cleansing provides for a uniform water distribution is shown in Figure 3.13. During the non-uniform swelling process, E12 increases until a maximum is reached which is followed by reduction of shear as a uniform water distribution is achieved. An important characteristic of the flee swelling and the cleansing tests is the low shear strain involved during swelling, which is typically E12 < 4.6 x 10‘3 and E12 < 0.02, respectively. Note that the values of shear strain indicated here are the maximum values during the test that are typically greater than the last measurement of the test. This low shear allows for a simpler calculation of the stretch ratio using the real-time method, as discussed above. Using the real-time calculation for A2 and J after cleansing, there was found to be less than 1% and 2.5% error, respectively, 41 .2 0.98 7 E 0.96 - E 0.94 - , ° .. . i; 51* _ ale 9.8 0.92 ”5|!” 0 A2 3% 0 9 ' fin ' A1 :3 o 88 - l” a“ J g . “*3?“ 4‘ as ,93’ 0.86 - ”m ~ U) “*1.“ 0 84 ' *fis ‘ . l l l l l l l l * O 1 0 20 30 40 50 60 70 80 90 Time [h] Figure 3.12. This figure shows that a uniform swelling ratio is achieved after the cleansing process which is consistent with the flee swelling assumption in the constitutive theory. It also shows that there is a linear decay in the stretch and volume ratios after the cleansing process. when compared to the bilinear interpolated values. Using the real-time calculation for A2 and J after flee swelling, there was found to be less than 1% when compared to the bilinear interpolated values. Using the real-time method allows one to perform a flee swelling experiment in a separate chamber and avoid tracking the spheres throughout the entire test. This reduces the random error flom variations between batches of specimen because one batch of hydrogels could be used to run a full set of flee swelling tests. A batch is defined as a single specimen preparation process. Due to the small specimen thickness (3 mm), rough edges, and tracking sphere size (500—600 pm), it is difficult to obtain accurate stretch ratio measurements in the thickness direction. Assuming isotropy, that is A1 = A3, which allows stretches to 42 0‘ I I I I I I . '2 —o.005- '; ~ ‘ . ° W .. . W N . . [:3 -0.01 . . 3" ] ° I —o.015- . .o,‘ . ‘ . If, -o.02 . 1 . . . . o 10 20 so 40 50 60 70 Time [h] Figure 3.13. Modest values of the Green-Lagrange shear strain, indicating the shear reduction during cleansing. be determined flom the in—plane, presumed homogenous region, is found to produce more repeatable and realistic measurements. Since uniform deformation is absent from the cleansing process, the deformation is calculated as Fe = A1,c91 ‘8 91 + A2282 ® 62 + Ances ® 83 (3-19) and Jc = Amigo, (3.20) where ’\1,c and Au are the principal stretches measured in the 1 and 2 directions re- spectively, during the cleansing experiment. For the free swelling experiment, uniform deformation is observed and thus Ff=Afe1®e1+Afe2®e2+Afe3®e3 (3.21) 43 16x10 . . , . I M 14 12» - 0 E12 0 O O O O l l l 0 5 10 1 5 20 25 Time [h] Figure 3.14. Modest values of the Green-Lagrange shear strain, indicating the low shear present during a flee swelling test. and Jf = A3, (3.22) where Af is the measured principal stretch which is the same in all three directions during a flee swelling experiment. These deformations will be related to the dry polymer state in the parameter estimation section. 3.2.5 Forced Swelling The forced swelling test is designed to find material parameters in the saturated stress equation of the constitutive model. Similar to flee swelling described above, this pro— tocol follows cleansing and is performed for the same duration as the flee swelling protocol. The same testing duration is required because of the time dependent reduc- 44 tion in swelling due to possible recrystallization. The forced swelling experiments are performed using four different loads, which are applied to the hydrogel using tungsten or lead weights (fishing sinkers). The fishing sinkers are convenient due to their inert composition, small size, assorted masses, and inexpensive cost. Figure 3.15 shows the weights used in this experiment. The lighter weights (i.e., 4.57 and 2.93 g) appear Figure 3.15. Fishing sinkers used in the forced swelling tests. The sinkers are arranged flom heaviest to lightest going left to right. larger than the heavier sinkers because they are made flom lead, whereas the heavier sinkers are made flom tungsten, which is denser. It is noted that these sinkers apply loads to the hydrogel under water and so there is a small buoyancy force resisting their weight. To account for this, the sinkers weights are measured under water using the load-cell. Similar to the cleansing test, a time flame for the forced swelling test had to be determined according to time at which the first plateau region is reached. Pilot ex— Periments indicate that the heavier the weight, the longer it takes to reach the first Plateau. Therefore, the heaviest weight (9.77 g) is used to determine the duration of this experiment. The results are shown in Figure 3.16, where the reference configura- 45 tion is defined as the end of the cleansing test. The plateau in J appears to start just ".25I d r I I I I I _I <59 00 O J 12- (96% 0&0 ° "2 : CW x A1 1.15- (“22.9 - o (269,. m» .....--~w W 1 .1 ' ”00"....“W m o K my. 1.05 ”'5 XWW i Stretch ratios & Volume ratio l M l 0.951 . . . . . o 10 20 30 40 50 60 70 80 90 Time [h] Figure 3.16. The principal stretches and the volume ratio during the 9.77 g forced swelling calibration test. The purpose of this test is to find the duration for future forced swelling tests which is determined by the first plateau region observed. The plateau region in the J response occurs between 10 and 25 hours. after 10 hours and proceeds until about 25 hours, when recrystallization effects may begin dominating the response as the hydrogel begins to lose water volume. There- fore, any duration between 10 to 25 hours is sufficient and 24 hours is chosen for convenience. So, to eliminate time dependant water volume decay in comparisons, 24 hours is chosen for both flee swelling tests and forced swelling tests at the four different loads. This test also benefits flom the manner of loading which resulted in little shear strain, as shown in Figure 3.17. Similar to flee swelling, the low shear strain provides the convenience of measuring the microsphere positions only at the beginning and end of the experiment, using the real-time principal stretch calculation. 46 (1015 . . . as . . . . . 0.01 "g. l ' 0.005 - J N as 01* - C . . . - O -0.005‘ g . ... .C. . . .. 0 ~ 0. .. .0 Q 0 . O. 0.” ~ . ... . o . ' _ _0.01 7 . ... ... ... .‘. .. '0'. . o ' . . ’ _0.015 1 l_ 1 1 1 1 1 L O 2 4 6 8 10 12 14 16 18 Time [h] Figure 3.17. The small value of E12, indicating the minimal shear present in a forced swelling test. The initial jump in the shear strain occurs during the initial application of the load and the resulting high rate of swelling. As the rate of swelling slows, the hydrogel becomes equilibrated at the applied load in the water, and the shear strain decreases. After the shear reaches a brief steady-state, the magnitude then slowly increases for the remainder of the test during which volume swelling reduction flom possible recrystallization takes place (see Figure 3.16). Had more data been plotted, the initial jump would appear more gradual. From experiments, E12 is typically found to be less than 0.024. Using the real-time calculation for A2 and J after forced swelling, there was found to be less than 1% and 2.6% error, respectively, when compared to the bilinear interpolated values. Pilot experiments indicated the isotropy assumption, A1 = A3, to be reasonable. Using the isotropy assumption, the deformation during a forced swelling test is Ili‘fs = )‘1,fsel ‘8 81 + )‘2,fseZ ® ‘32 + )‘1,fse3 ® e3 (3'23) 47 and J,, = 123,13”, (3.24) where )‘1,fs and )‘2.fs are the measured principal stretches during the forced swelling experiment. 3.2.6 Preconditioning and Unsaturated Stretch-to-Break The unsaturated stretch-to—break is a simple uniaxial stretching experiment that is continued until specimen failure. As previously stated, unsaturated is defined as the hydrogel being removed flom the water bath during this experiment and therefore the material is assumed to be incompressible. This is an important distinction because the governing stress-strain equation is different flom the saturated stress equation govern- ing a forced swelling test. In the forced swelling test, the incompressibility constraint is not present and so the volume can change. As in previous swelling experiments, we obtained stretch and, depending on the experiment, stress measurements at equilib- rium points to be used in the constitutive model. These equilibrium points occur after the response stretch reached a maximum and the plateaus became noticeable. Con- versely, in unsaturated experiments the duration of the experiment is critical. When removed flom water, these materials lose water flom evaporation at the surface and diffusion which makes them unusable if allowed to dry for a day. To minimize the evaporation and loss of water content, measurements must be made in the shortest time possible for the incompressibility assumption to hold. This requires running the dynamic stretching experiments in attempts to satisfy incompressibility in the unsaturated theory. In addition, according to the constitutive modeling, the dynamic stretching experiments must measure equilibrium stress and strains throughout the deformation. To obtain nearly equilibrium measurements in a dynamic stretching ex- periment, one must reduce the viscoelastic hysteresis between loading and unloading and precondition the specimen to obtain repeatable loading and unloading responses, 48 that is reduce the Mullin’s effect. Additionally, the speed of stretching (dA/dt) must be chosen such that the material has stress relaxed at each measurement. Viscoelastic hysteresis and the Mullin’s effect are both well-known characteristics of elastomeric materials [14, 44]. Although reducing hysteresis is desired, the Mullin’s effect is more significant for this experiment because the unloading portion of the curve will not effect our results during a stretch-to-break test. Generally speaking, the Mullin’s effect suggests that the mechanical response of an elastomer will change between subsequent loadings. To reduce the Mullin’s effect and viscoelastic hysteresis, a preconditioning protocol is run before the actual stretching test. After precondi— tioning (either static or dynamic), the mechanical response will repeat between two consecutive loadings. The preconditioning protocol applies dynamic or static loading close to the failure point of the material. Applying this to our materials, requires the preconditioning protocol to be performed in water to prevent evaporation of water flom the specimen. Dynamic preconditioning is avoided because experiments have indicated that PVA specimens have a swelling response that depends on the applied load. With dynamic loading, the material would not reach an equilibrium volume state as water would be driven in and out of the network during the cyclic loading. To avoid these issues, a saturated static preconditioning test before an unsaturated stretch-to—break experiment is used. This is a forced swelling experiment with the largest weight of 9.77 g. As discussed above, the same duration of 24 hours for forced swelling is used for our saturated static preconditioning protocol. To investigate the effectiveness of the proposed preconditioning protocol, two-cycle loading and unloading tests are run to measure the material response. In addition, a flee swelling (no load) test is run for 24 hours to compare its response with the saturated static preconditioning test. Lastly, the two—cycle loading and unloading for both static preconditioning and flee swelling tests are run at three different dA/dt, to determine the stretching rate allowing the material to remain in a stress relaxed 49 state. Figure 3.18 shows results of a two-cycle loading and unloading at a stretch rate of dA/dt = 0.007, after static preconditioning. For comparison, Figure 3.19 18000 T . 1 . r 1 16000 : 14000 ' 12000 10000 8000- I I I I 0' [Pa] 6000- 4000 2000 I I "20 1 1 1 1 1 1 00 1 1.1 “L2 ‘L3 'L4 “L5 Figure 3.18. Two cycle loading and unloading after preconditioning at dA/dt = 0.007. shows results of a two—cycle loading and unloading at the same stretch rate, but with no preconditioning (free swelling). These results clearly demonstrate that an initially preconditioned specimen exhibits negligible Mullin’s effect. In addition, the viscoelastic hysteresis between loading and unloading for the first cycle is reduced for the preconditioned specimen. After the first cycle of loading, the flee swelling specimen seems to show a similar trend of negligible Mullin’s effect and viscoelastic hysteresis. Therefore, the results show a forced swelling test helps to precondition the specimen before an unsaturated stretch-to—break experiment. 50 Figure 3.19. Two-cycle loading and unloading after flee swelling at dA/dt = 0.007. With this preconditioning protocol, it is now important to determine the value of dA/dt at which to run an unsaturated stretch-to—break test. The optimal rate is the one at which each measurement is taken when the material has stress relaxed at the current deformation. Figure 3.20 shows results of a two-cycle loading and unloading test at three different stretch rates after preconditioning. This figure shows the stress response to be stiffer as the rate is decreased. This seems counterintuitive because it is often thought that the slower the rate gives a material more time to stress relax. The results could suggest that the stiffer response at slower rate is due to increased water evaporation. Evaporation is not the cause because the second loading curves coincide with the first loading curves, as can be seen in Figure 3.18. Note that the response for dA/dt = 0.007 in Figure 3.20 is the same test as shown in Figure 3.18. 51 16000 . . . . , , 0 d). dt = 0.005 .6 14000- / ~ 0 dA/dt=0.007 3> 12000“ x dA/dt=0.009 '3' ' 10000~ , - as 8000- , 8 8" - ‘—' 9 8.... b 6000- .. O - .. “”611 4000- @1990 - ,. 5a 2000- e36 959 5%? "‘ . 0- wwm” - -200 . . . . . . 8.9 1 1.1 1.2 1.3 1.4 1.5 16 Figure 3.20. Two-cycle loading and unloading after preconditioning at three different stretch rates. Conversely, the force response shown in Figure 3.21 shows no stiffening trend as is seen in the stress response. All force responses are nearly indistinguishable during the consecutive loadings. As originally expected, the force plots show a slightly stiffer response for the fastest rate. Therefore, the slower rate appears to be stiffer in the stress response because the stress accounts for both force and cross-sectional area. Where the force follows the trend shown in Figure 3.21, the cross-sectional area is then the additional factor. Since the slower test gives the specimen more time to relax, its cross-sectional area appears smaller than the faster test, which increases the stress. Using the stress plots, the stretch rate for future unsaturated stretch-to-break tests can be determined. The rate should be chosen such that the cross-sectional area has relaxed. This would be the value for which a change in the rate results in no 52 0.09 . I I I I dA dt = 0.005 ’30 0.08 - / o . o dA/dt = 0.007 , o.07~ _ x dA/dt = 0.009 g 0.06- Isa-z: . a 0.05 ' x Q. @ J § 0.04- O " — p2 *' 8. O 0.03I (e 0‘“ - x6 ‘ O 0.02 — . “5859 ° - @90' 0.01 - “Sb £996 ' . saw 899’ O *' xc% q 1 1.1 1.2 1.3 1.4 1.5 1.6 A Figure 3.21. Two cycle loading and unloading after preconditioning at three different stretch rates. further stiffening of the stress plot. This is investigated with the uncertainty in the measurement below. Figure 3.22 shows the stress response of the fast and slow rate tests. After A = 1.3, within the uncertainty of the measurement, the slower rate is stiffer. Therefore, dA/dt = 0.009 is too fast for testing because the material shows further cross-sectional area relaxation when stretched at dA/dt = 0.005. Comparing dA/dt = 0.005 and dA/dt = 0.007, Figure 3.23 shows no significant difference in the stress response. Therefore, the rate decrease here resulted in no further cross-sectional area relaxation and 0.007 is a sufficient rate at which to run a stretch-to-break test. Assuming isotropy, the deformation gradient for the unsaturated stretch-to—break experiment is A Fsb = ’\1,sbel ® 91 1' A2,3592 ® e2 + )‘1,sbe3 ‘8 e31 (325) 53 1 6000 dA/dt = 0.005 I: 14000- . o dA/dt=0.009 12000- I: 5 - 10000- - It in} '6' 8000- . P_-I. :1 e s 6000- I. - 0 4000— g 11,, - 2000 I; I J1 51 5 0- 9?qu - —200 I I I I I I 8.9 1 1.1 1.2 1.3 1.4 1.5 16 Figure 3.22. Comparing two cycles of loading after preconditioning at dA/dt = 0.005 and dA/dt = 0.009. where )‘1,sb and A23), are the corrected bilinear interpolated stretch ratios that are calculated according to equations (3.11) and (3.12), respectively. 3.3 Parameter Estimation At this point, the experimental protocols discussed above will be mapped with the total deformation. This will characterize the deformation history of the hydrogel to be used for parameter estimation. This map will quantify the water content of the hydrogel at any state during the experimental protocol. As mentioned above, this is possible because the dry polymer will be used as the reference state. In general, two methods will be used to find the material parameters. Each method 54 16000 T. ' ' ' ' ' é dA/dt=0.005 14000" O dA/dt=0.007 J 12000- 1? ° — 10000I . E 8000- I; :5 - b 6000- If: . ‘5 8 tea a. as O M *8 1 1.1 1.2 1.3 1.4 1.5 1.6 Figure 3.23. Comparing two cycles of loading after preconditioning at dA/dt = 0.005 and dA/dt = 0.007. will involve minimizing a penalty function consisting of the difference between the experimental measurements and the model in a least squares sense. One method involves separate penalty functions for both the unsaturated and saturated stress equations, while the other method involves a combined penalty function where both the unsaturated and the saturated stress equations are fit to simultaneously. The minimization of the penalty functions is done using a non-linear least squares routine in Matlab which will be discussed after the penalty functions are introduced. Figure 3.24 shows the mapping of all experiments with the dry polymer state serving as the reference state, where the deformation gradients measured during the cleansing FC, flee swelling F f, forced swelling F f3, and stretch-to—break F sb tests (re- fer to equations (3.19), (3.21), (3.23), and (3.25) for details). After referencing these 55 Unsaturated Stretch-to—Break 533, f Saturated Free Swelling Dry c Polymer Fft Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q. Q. Unsaturated Stretch-to-Break 1633 Figure 3.24. Experimental mapping for all states in the experiments, each referenced to the dry polymer state. swelling mapping, F F, is FF = f" chF ft = )‘1,F91 (8 e1 + A2,F92 ® 92 + )‘3,Fe3 ® 63, (3-25) where /\2,F = A f/\2,c)‘2, ft (3-27) and /\1,F = ’\3,F = AfA1,c/\1,fta (338) where the deformation gradient F ft and principal stretch ratio Ai, ft during the fleeze- thaw cycle, which cannot be measured. However, F ft becomes useful in that it will be 56 and ’\1,F = ’\3,F = AfA1,c>I1,ft, (3-28) where the deformation gradient F ft and principal stretch ratio A,-, ft during the fleeze— thaw cycle, which cannot be measured. However, F ft becomes useful in that it will be used to make the flee swelling mapping (equation (3.26)) consistent with the uniform deformation assumption in the constitutive model. Uniform deformation for the total flee swelling map requires F ft to be II“ ft = A1, ftel 99 e1 + Amez 99 e2 + A1, fte3 99 es, (329) where det F ft = J ft: which was determined earlier using a mass measurement (see equation (3.16)). Using equations (3.16) and (3.29), the principal stretch ratios A1, ft and A2, ft can be obtained upon assuming uniform deformation, thus ALF = A2,}: = AF (3.30) leads to J13 2, )‘1,ft=th/ ( —-:-’)1/3 (3.31) and J1 3 A1, .1, ft: ,/ (— ——:>2/3 (3.32) Using equation (3.31) or (3.32), equation (3.28)2 or (3.27) gives the free swelling stretch ratio 2/3 1/3 1/3 where the volume ratio is J F = At}. After determining the hypothetical deformation state F ft: the total forced swelling and stretch-to—break mappings can be calculated. Using equations (3.19), (3.23), and (3.29), the total forced swelling mapping becomes FFS = F fsfi‘ cf‘ ft = A1,Fse1 ® 61 + )‘2,FS‘32 <8 (32 + )‘1,FSe3 ® 83, (3-34) 57 where 2/3 1/3 1/3 A1 F5: A1 7,1,, 126.17, (3.35) and 2 3 1 3 1 3 A2193: 127,114 A2 / th/ , (3.36) and the volume ratio is JFS = )‘2.FS)‘f,FS' Using equations (3.19), (3.23), (3.29), and (3.25) the total stretch-to—break mapping becomes FSB = Ifsbf“ stcF ft = Arsaei ® 81 + A2,SBez 69 82 + Aisses ® 83. (337) where A1 33 A1 ,3, 7312/33/31}? (3.38) and 1253123,1273A2/3AI/3J]{3, (3.39) and because of the incompressibility constraint, J33 = 1, and the volume ratio is JSB = #5- It is convenient to let ’\2,FS = A3, A253 = Au, JFS = J53 = J, and 02 = 0‘. Using equation (2.21), we can also let Ag’ps = (/J/A3 and A253 = (/J/Au. Using equation (2.26), the Cauchy stress in an unsaturated hydrogel can then be written as = _ 1:77:91] Aai __ 7")—az°/2) (3.40) whereas for a saturated hydrogel, equations (2.22) and (2.23) become 1 N . = .j Z “,1?“ + h(J), (3.41) and =7 211.873 VIII/2 + W) (3.42) respectively. For a flee swelling hydrogel, equation (2.28) becomes N 211,123” + h(J) = 0. (3.43) 58 3.3.1 Parameter Estimation from Unsaturated Test Results Parameter estimation flom data taken during unsaturated stretch-to-break experi- ments attempts to find 11,- and 0,- using equation (3.40). To fit these data, a penalty function II.) is minimized, that is P 11,, = [(0, — 6(Au, J, 11,, 6.))2, (3.44) i=1 where cu is the Cauchy stress measured during the experiment according to the method discussed in Appendix A and 0(Au, J, 11,-, (1,) is given by equation (3.40), where Au and J are the measured stretch and volume ratio referenced to the dry polymer state. Note that p is the number of data points obtained flom the experiment. One must choose the number of parameters to use in the Ogden model, as determined by choice of N in equation (3.40). Afterwards, material parameters pi and a,- are then used in analysis of the data during flom the saturated tests, which are used to determine M and X flom equation (3.41). 3.3.2 Parameter Estimation from Saturated Test Results Estimating M and x flom measurements of saturated hydrogels requires elimination of the flee equation (3.43) flom the parameter estimation routine. Notice, one can solve equation (3.43) for the flee swelling value of A F and J F = A35. with knowledge of the material parameters 11,-, 01,-, M, and x. Also, recall equation (3.43) is a special case of the saturated stress equation (3.41). Namely, if one makes the flee swelling assumption, A2 = J, equation (3.41) results in a = 0, which is the flee swelling state. As long as we include the stress flee condition, corresponding to the measured flee swelling values of A p and J F = At}, then the flee swelling equation will be identically satisfied during our fitting using equations (3.41) and (3.42). Then, using [1,- and a, flom the unsaturated tests, the saturated penalty function H3 to be minimized is 11,— _ 2K6, — 0( (A3, J, X,M +(0 — — 111:”, -’\7( )W2 + h(J))Z], (3.45) 59 where p is the number of experimental measurements, and as, A3, and J are the measured Cauchy stress, stretch ratio, and volume swelling ratio, respectively. See Appendix A for calculation of the saturated Cauchy Stress. Due to the flagile nature of the hydrogel and the limited number of available weights, only five data points are measured here. 3.3.3 Combined Parameter Estimation using Saturated and Unsaturated Test Results After estimating parameters for each of the saturated conditions, a combined routine is used to examine the parameters found by simultaneously fitting to both the unsat- urated and saturated results. The combined penalty function (H) to be minimized is H = Hu + H3. (3.46) Here, all of the material parameters #2” 04,-, M, and X are found simultaneously. 3.3.4 Matlab Non-Linear Least Squares Routine for Parameter Estima- tion The three penalty functions, II, Hu, and H3, will be minimized using a non-linear least-squares algorithm (LSQNONLIN) that is available in the Matlab optimization toolbox. The algorithm uses a subspace trust region method along with a interior- reflective Newton method [1]. The input to this algorithm is the penalty function, stopping criteria, and initial starting guess for each parameter. The penalty function makes use of the experimental data that is imported into the Matlab workspace. The main stopping criteria is the Newton step (TolX) and change in the penalty function after the step is taken (TolFun). Additional stopping criteria consisted of the maximum number of function evaluations and maximum iterations. The same stopping criteria that is used by Ogden [30] is used here, that is TolX=1 x 10'8 and 60 TolFun=1 x 10"”. Minor corrections are used when the algorithm stops too early or fails to converge. The residual sum of the squares (RSS) is computed flom the penalty functions above using the obtained material parameters and the experimental data sets used in the fitting which indicates how well the model fits the experimental data. The Ogden model requires, for physically realistic material response and stability [30], that ma,- > 0, (3.47) for each value of i flom 1 to N (no sum over i). According to Ogden [30], however, if N 2 3 it is unnecessary that equation (3.47) holds for every 2'. In addition to equation (3.47), Pence and Deng [42] note constraints on M and x that are M > 0, (3.48) and O S X S 1. (3.49) These constraints can be set within the search routine. Another method for parame- ter estimation uses the same separate and combined fitting routines described except it uses equation (3.42) as a constraint instead of incorporating this equation into the penalty function. This alternative method requires the saturated stress equation in the traction flee directions to be zero within a specified tolerance. The param- eter estimation is performed using a constrained minimization algorithm in Matlab (FMINCON) and was found to fail in searching for a minimum since every initial parameter guess returned an optimal solution being the same as the initial guess. This issue may have been because the problem was too constrained for the Matlab algorithm. 61 CHAPTER 4 Results and Discussion This chapter presents the experimental results based on the protocols that are shown in Figure 3.24. Results are presented for a saturated forced swelling test at four separate loads and a unsaturated stretch-to—break test after preconditioning. These are the two tests needed to obtain the material parameters required by the consti- tutive model. Both tests will use the experimental map to relate deformations to the hypothetical dry polymer state, thus allowing a global comparison between the experiments. After the results are shown, the unsaturated and saturated data are used to obtain material parameters from an Ogden model modified by incorporating the Flory-Huggins theory. 4.1 Saturated Test Results The results of the saturated forced swelling tests are shown in Figures 4.1 and 4.2. This forced swelling experiment was performed three times at the five loads. In each run, the complete deformation mapping is calculated for each specimen and the values of J and As reported are the average over three runs. The average values are shown in Figure 4.1 with the error bars indicating the standard deviation of the three runs. As expected, the forced swelling plot shows the length of the specimen to increase as the load is increased. Shown in Figure 4.2 are the volume swelling ratios with 62 18000 I . I I I T 16000 14000 12000 10000 8000 6000 4000 2000- a I 1—0—1 1 l 1 I l l I T l 0' [Pa] I H-l 1 l l l l l l 2 4 6 8 1o 12 14 16 )‘3 Figure 4.1. Saturated forced swelling test for three specimens at each of the five different loads, where As is referred to the hypothetical dry state. This includes the flee swelling result where no load is applied. Error bars are included to show the standard deviation between the three specimens at each load. the associated stress flom the same forced swelling experiments. The results suggest that there is a threshold load at which the hydrogel achieves maximum swelling. This load is less than the maximum load applied. After this maximum swelling, one may conclude that further increasing the load causes water to be exuded. However, continuing to run a test beyond this load will cause additional water loss flom possible recrystallization, as observed in forced swelling tests. Therefore, one cannot conclude that the water loss is solely due to the change in load. This presents an issue with using the constitutive theory to predict experiments that are not run for the same duration. Figure 4.2 demonstrates how loading affects the amount of swelling, but says nothing about water loss. Further constitutive development of the natural degrading process 63 100 I I I T I I I I I l—O—l 4 l 70 40- i - 3O 1 1 1 1 1 1 1 1 0 2000 4000 6000 8000 10000 12000 14000 16000 0' [Pa] Figure 4.2. Volume ratios for saturated forced swelling tests for three specimens at the five different loads, where J is referred to the hypothetical dry state. This includes the flee swelling result where no load is applied. Error bars are included to show the standard deviation between the three specimens at each load. of these hydrogels could then allow comparisons on how loading affects water loss or gain. Comparison of the stiffness in the response between the unsaturated and saturated tests is impractical because they have different initial global stretches before testing. The starting stretch for the saturated forced swelling test is the stretch at the end of flee swelling, AF, which is referred to the dry polymer state. Conversely, the starting stretch for the unsaturated stretch—to—break test is the residual stretch leftover at the end of a forced swelling experiment using the maximum load of 9.77 g. The stretch at the end of forced swelling with the maximum load of 9.77 g is the Asmax, which corresponds to the largest measured stess shown in Figure 4.1. At the end of the 64 forced swelling test, a permanent deformation of the hydrogel results that is less than Amman, (when the load is removed), but greater than AF, which is why the initial stretches flom the saturated and unsaturated experiments are different. 4.2 Unsaturated Test Results Unsaturated stretch-to—break tests were performed with three separate specimens af- ter the forced swelling preconditioning routine. All three specimens tested came from the same preparation of hydrogels to reduce variability between batches, which may result because of the complicated dissolving process. Figure 4.3 shows the response of these unsaturated specimens. Note that here the reference configuration is taken as their common post cleansing state, rather than the dry polymer state. All three specimens shown follow the same response curve until A = 1.2. The response curves for specimen 1 and 2 follow the same trend until about /\ = 1.3. Specimens 1 and 3 did not break in the gage section of the specimen, as desired. Specimen 1 failed at the grip where the cloth is glued to the specimen. Specimen 3 started to fail at the grip, but the test was stOpped before complete failure. This resulted in the drOp in stress at the end of this test. Figure 4.4 shows the unsaturated stretch-to—break test referenced to the dry polymer state according to the experimental mapping. After referencing to the dry polymer state, the deviation between specimen 1 and the other two specimen increases. This is expected because during the forced swelling precon- ditioning routine this specimen stretched to A3 = 10.1 before the stretch-to—break experiment, as compared to specimens 2 and 3 with )‘3 = 7.98 and As = 8.47, respec- tively. With a larger starting As, the data for specimen 1 stretches further during the test as compared to the other two tests. Conversely, notice that the response curves for specimens 2 and 3 show better agreement after referencing their stretches to the dry polymer state. Specimen 2 and 3 have the same response until Au = 12, where specimen 2 starts to show a stiffer response until failure. This may indicate 65 x10 5_ v Specimen1 x 1 O Specimen2 i I * Specimen3 a: 4- 3" - I G 33‘ .. v ' .__. a: 0 b — 4 2 g a v? x o v {:96 V 1- I a) v' 4 £33393'V @ 0- vtvfi‘m - 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 A Figure 4.3. Unsaturated stretch-to—break after preconditioning using three specimens. The stretch is referenced to the beginning of the experiment and not to the dry polymer state. Error bars are included to show the uncertainty in the stress measurement; these are smaller than the symbols. how referring back to the dry polymer state can be useful because it allows for more consistent comparison during testing, which allows the response of the hydrogel to be independent of the initial water content. Specimen 2 is the specimen that failed in the gage region, as designed. 4.3 Parameter Estimation Results Using the results flom specimen 2 in the unsaturated stretch-to—break test, along with the data obtained flom the forced swelling tests, material parameters can be determined using the methods discussed in Section 3.3. Additionally, the unsaturated 66 x10 5- v Specimen1 * - O Specimen2 i i 4_ *1 Specimen3 , 5‘ _ i G '53" ‘ _. cl; b 2_ G... v; _ 9% V 1- “6?; VV ~ a v' '0‘, V VVVV O-®@9§e.ed¥vvvvvvvvv 4 8 1O 12 14 16 11. Figure 4.4. Unsaturated stretch to break after preconditioning using three specimens. The stretch is referenced to the dry polymer state according to the experimental mapping (see Figure 3.24). Error bars are included to show the uncertainty in the stress measurement; these are smaller than the symbols. stretch-to—break data set for specimen 2 will be modified slightly because of increased shear near failure. Figure 4.5 shows the shear strain for specimen 2. At the end of the experiment, the Green-Lagrange shear strain increases rapidly during specimen failure which fails at E12 = —0.14, where the negative sign indicates the direction of shearing. Referring back to Figures 4.3 and 4.4, this rapid increase in shear occurs when the response curve has an inflection point. Only the data up to this inflection point will be used in the parameter fitting routines because the constitutive model assumes hyper-elastic behavior. Ortt [29] suggested maintaining the shearing strain on the order of E12 2 0.07. For specimen 2, the shear strain at the inflection point is approximately E12 = —0.07 which is the same as Ortt observed. The assumption of 67 Table 4.1. Initial A 3 and J referred to the dry polymer state before the stretch-to-break experiment. Specimen 1 Specimen 2 Specimen 3 J = 73.8 J = 48.6 J = 58.2 0.02 . i l I T I I I F " 1: . ° ‘ > °-. 3' i . 0 A ‘ . ,-",x- f “ (“5‘ .. I" .’ . ‘ b . . " '° 5 ,thy‘i \ o. 4' ‘. ‘ o ’ -002 ~ . t In 33% 5.3:: .,.. _ o .8”. ‘ £\ ’7ny . : ‘zoi -o.04 - ° .. ’ :.-° - a. f g‘: .3 a -0.06 - 'syfi ] m -o.oaL ’ f}, - —O.1 . . ‘ -o.12 ~ 3: -o.14 - 1 _O.16 1 1 1 1 1 1 4 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 A Figure 4.5. Green-Lagrange shear strain for specimen 2 during the unsaturated stretch- to—break test. The rapid increase in magnitude correlates with the start of failure. E12 and A are referred to the beginning of the stretching test. incompressibility was also checked. Shown in Figure 4.6 is the volume ratio during stretching with deformation referenced to the beginning of this test. Figure 4.6 shows specimen 2 to be nearly incompressible during the test with an average volume ratio of J = 0.99, using all of the data points. After removal of volume ratios corresponding to [En] _<_ 0.07, the average volume ratio is J = 0.98. During parameter estimation, 68 1.1 I I I I I I I 1.08 1.06 1.04 :.,- 1.02 0.98 0.96 0.94 0.92 Figure 4.6. Volume ratio during unsaturated stretch-to—break test using specimen 2. This stretch is referenced to the beginning of the stretching test. This shows that specimen 2 had a nearly incompressible response during the test. specimen 2 will be assumed incompressible with J = I assumed constant throughout the experiment. As observed in initial parameter estimation trials, the few data points obtained in the forced swelling tests make it difficult to find Ogden model parameters because many local minima exist when using multiple parameters to fit a few data points. Dif- ferent initial guesses have been found to estimate different parameters. In addition, some of these parameters produce Ogden model predictions containing curvature in between the measurements that fails to represent the curve if more data had been measured. To assist in finding initial guesses closer to a global minimum and in maintaining the trend in the data, additional data points are generated between the 69 measured data points and used in the fit. The parameters flom these fits then be- come the initial guess to be used with the five measured data in the separate fitting technique. Recall that the estimated parameters flom the unsaturated results are ob- tained first and then used in the parameter estimation for the saturated results which is called the separate fitting here. There is also a combined fitting case where all parameters are estimated simultaneously using both the saturated and unsaturated results. To estimate parameters for the combined fitting, first, the unsaturated mea- surements are used with the saturated measurements which include the additional synthetic data points between the saturated measurements. After parameters are obtained, these parameters become the initial guess to be used in the final parameter estimation using the unsaturated measurements and the five saturated measurements. 4.3.1 Modified Ogden Model with N = 1 The unsaturated results for N = 1 are used to estimate material parameters [11 and a1, and subsequently the saturated results are used to estimate M and x. Table 4.2 shows the estimated parameters for the modified Odgen model with N = 1, using the separate fitting technique. Figures 4.7 through 4.9 show comparisons of the unsaturated and saturated response results with the modified Ogden model with N = 1 using the parameters in Table 4.2. Figure 4.7 shows the Ogden model with N = 1 provides a good fit to the unsaturated stretch-to—break data. Conversely, the estimated parameters using the saturated results do not fit the data as well. Recall that equation (3.42), which is the stress response in the traction flee directions, is contained in the saturated penalty function (equation (3.45)). When this penalty function is minimized, it does not guarantee that equation (3.42) is identically zero for all As and J measurements, which is assumed. Using the parameters in Table 4.2 and the A3 and J measurements, the residual stresses for equation (3.42) are shown in Table 4.3. The largest residual shown in Table 4.3 is approximately 15% of the 70 largest stress measured in this experiment and is approximately 88% of the first stress measurement, which is obtained using the smallest load. Table 4.2. Material parameters obtained for the separate fitting of the Ogden model with N = 1. 111 011 M X RSS(unsaturated) RSS(saturated) 3.594 x 10-5 9.333 7.56 x 106 2.44 x 10-14 4.258 x 106 5.79 x 108 4 x 10 3 I I T I I I 0 Data 0 2.5» —N = 1 Ogden ] 2 - - 7:? &. 1 5- - b 1 __ a 0.5 - - 0r .. 8 9 1O 11 12 13 14 )‘u Figure 4.7. Comparison of the Ogden model and the measurements for the unsaturated results using the separate fitting with N = 1. This shows good agreement between the model and the measurements for the unsaturated results. To improve the overall fit of the model to both the saturated and unsaturated data sets, a combined fitting is performed. The combined fit uses both sets of data in the combined penalty function. For the combined fit, the parameters for the modified 71 _L h o o O —L—L 08 so 00 N-b 00 OO 00 Figure 4.8. Comparison of the Ogden model and the measurements for the saturated results using the separate fitting with N = 1. This shows good qualitative agreement between the model and the measurements for the saturated results. Ogden model with N = 1 are shown in Table 4.4. Figures 4.10 through 4.12 show the comparisons of the unsaturated and saturated results with the modified Ogden model with N = 1 using the parameters in Table 4.4. The results indicate improved agreement for the saturated case between the model and measurements. This is at the expense of a reduced agreement for the unsaturated fit in the combined fitting as compared with the separate fitting done previously. As expected, 01 and (11 have changed between the separate and the combined fitting because the unsaturated results agreement with the model is worse, which is independent of M and x for the unsaturated case. The largest change occurred in the material parameter 111, which is about 3255 times greater in the combined fitting as compared to the separate 72 90 ' I I T Q \ 80 r — o o 70 - 0 ~ '5 60 - - 50 - 1 0 Data — N = 1 Ogden 4O - ‘13 . . . - 0 5000 1 0000 1 5000 0' [Pa] Figure 4.9. Comparison of the Ogden model and the measurements for the saturated results using the separate fitting with N = 1. This shows good qualitative agreement between the model and the measurements for the saturated results. fitting. Conversely, the material parameter a1 changed little and is approximately 66% of the value obtained flom the separate fitting. The material parameter x was approximately zero in both fitting techniques and the material parameter M decreased by 77% between the separate and combined fitting. If one uses the estimated material parameters flom the separate fitting and computes the RSS of the combined penalty function, one obtains RSS(combined)= 5.83 x 108, which is larger than the RSS using the parameters flom the combined fitting (see Table 4.4). For comparison to the separate fitting routine, Table 4.5 shows the residuals (stresses in the traction flee direction) of equation (3.42) using the five saturated measurements and the material parameters in Table 4.4. The largest residual shown in Table 4.5 is approximately 73 Table 4.3. Residual stress in the traction flee direction (equation (3.42) with N = l) for comparison with the assumption of zero applied stress. Residuals 1 through 5 are the five stresses that are determined using the materials parameters (obtained using the separate fitting routine) in Table 4.2 and the five saturated measurements which consists of A3 and J for each of the applied loads (see Figures 4.1 and 4.2). Residual 1 Residual 2 Residual 3 Residual 4 Residual 5 —2338 Pa —753 Pa —683 Pa —524 Pa —717 Pa 4% of the largest stress measured in this experiment and is approximately 20% of the first stress measurement which is obtained using the smallest load. Therefore, the combined fitting routine resulted in a better agreement with the stress flee assumption in the traction flee directions when compared to the separate fitting routine. Although it cannot be distinguished flom the figures for the Ogden model with N = 1, the combined fitting routine is a better parameter search routine in that it provides more freedom in searching for the best fit parameters for both the saturated and unsaturated results. Table 4.4. Material parameters obtained for the combined fitting of the Ogden model with N = 1. 111 (11 M X RSS(Combined) 0.117 6.126 1.752x106 2x10-13 1.69x108 Table 4.5. Residual stress in the traction flee direction (equation (3.42) with N = 1) for comparison with the assumption of zero applied stress. Residuals 1 through 5 are the five stresses that are determined using the materials parameters (obtained using the combined fitting routine) in Table 4.4 and the five saturated measurements which consists of A3 and J for each of the applied loads (see Figures 4.1 and 4.2). Residual 1 Residual 2 Residual 3 Residual 4 Residual 5 -536 Pa —174 Pa —158 Pa -121 Pa —166 Pa The results obtained for the unsaturated Ogden model thus far will be compared 74 x104 0 Data 0 2.5. ———N = 1 Ogden , .0 01 14 Figure 4.10. Comparison of the Ogden model and the measurements for the unsaturated results using the combined fitting with N = 1. This shows reduced agreement between the model and measurements for the unsaturated results when compared to the separate fitting routine. to see how they predict the unsaturated response of specimen 3, which had a similar stress response as specimen 2 when referred to the dry polymer state (see Figure 4.4). Figure 4.13 shows the unsaturated stretch-to—break results of specimen 3 compared to the Ogden model with N = 1 (separate fit routine) that was obtained using the unsaturated stretch-to-break results of specimen 2. The Ogden model obtained flom specimen 2 predicts the unsaturated response of specimen 3 until Au = 11, after which the Ogden model predicts a stiffer response then was measured for specimen 3. This is expected because Figure 4.4 shows the same trend where specimen 2, the specimen used in obtaining the Ogden model parameters, shows a stiffer response than specimen 3. 75 _1. .5 O o O _L_s ON 00 CO CO [0 8 O 4 6 8 10 12 14 16 Figure 4.11. Comparison of the Ogden model and the measurements for the saturated results using the combined fitting with N = 1. This shows better qualitative agreement between the model and measurements for the combined fit as compared to the separate fit. To investigate the repeatability of the stress response, Figure 4.14 shows all un- saturated stretch-to-break experiments performed. Specimens 4, 5, and 6 are three additional experiments performed after the first three experiments whose results were displayed in Figures 4.3 and 4.4. Using stretches that are referred to the beginning of the unsaturated stretch-to—break test, Figure 4.14 indicates that the response of all specimens is qualitatively reasonable, especially up to A = 1.4. When stretches are referred to the hypothetical dry polymer state, Figure 4.15 shows improved agreement between specimen 2, 3, 4, and 6. Specimens 1 and 5, however, fail to follow the same trend. In the case of specimen 1, it was observed to be a different material at the start of testing. There was no such apparent anomaly with specimen 5. Figures 4.14 76 90 I I I I 80 - 70 - H 60 r 50 - 0 Data — N = 1 Ogden 40 *- 0 - 0 5000 1 0000 1 5000 0' [Pa] Figure 4.12. Comparison of the Ogden model and the measurements for the saturated results using the combined fitting with N = 1. This shows better qualitative agreement between the model and measurements for the combined fit as compared to the separate fit. and 4.15 suggest that the dry reference state gives a more consistent description of the response and allows easy identification of possible material inconsistencies. Table 4.6. Initial As referred to the dry polymer state before the unsaturated stretch-to- break experiment. Specimen 1 Specimen 2 Specimen 3 Specimen 4 Specimen 5 Specimen 6 77 4 45x10 . . . . . , o Specimen 3 Data ——N = 1 Ogden o 15 Figure 4.13. Comparison of Ogden model obtained using specimen 2 and measurements for the unsaturated results of specimen 3. The Ogden model with N = 1 displayed is the model obtained using the separate fitting routine which resulted in a better unsaturated response prediction for specimen 2. 78 x 104 5 _ o Specimen 1 - x Specimen 2 v D Specimen 3 D an 4 4 b 1* Specimen 4 V ,, v Specimen 5 0 v ,1 F 3 - - Specimen 6 D :7 ,, l 91- a 3. * b 2 ~ 2 6 ale . 0 o a Z O *‘ [j Vx O .‘X' 1 ' El %XV;O-‘p°* - x O . - NM 45535513- ‘°' 1 1.2 1.4 1.6 1.8 2 A Figure 4.14. Comparison of the unsaturated measurements from all specimens with stretches referred to the beginning of the unsaturated stretch-to-break test. Specimens 1, 2, and 3 results are the same as shown in Figure 4.3. Specimens 4, 5, and 6 are ad- ditional unsaturated experiments performed to investigate the repeatability of the stress response. The results show that the response of all six specimen have good agreement until A = 1.4. 79 x 10 5 I I I I I 4.5 _ O Specimen 1 V _ 4 _ x Specimen 2 D * _ 1:1 Specimen 3 V ale 3.5 ' . v D " as Spec1men 4 x16 3 ' V Specimen 5 V x fil‘ ‘ . X 0c: 2.5[ - Spec1men 6 v 51" _ "" _ -————N = 1 Ogden V _ b 2 g," 08 o _ o O .. o o O O - 1 6 1 8 Figure 4.15. Comparison of the Ogden model that is obtained using specimen 2 and the unsaturated measurements from all specimens with stretches referenced to the dry polymer state. Specimens 1, 2, and 3 results are the same as shown in Figure 4.4. Specimens 4, 5, and 6 are additional unsaturated experiments performed to investigate the repeatability of the stress response. The results from specimens 2, 3, 4, and 6 show good agreement with the model, while specimens 1 and 5 do not. 80 CHAPTER 5 Conclusions In conclusion, it has been observed that the stretch of an unsaturated and satu- rated hydrogel increases as the loading is increased. Conversely, the volume swelling response increased and then decreased during the loading for the saturated hydro- gel, while the unsaturated hydrogel remained nearly incompressible during loading. Comparison of the stiffness in the response between the unsaturated and saturated tests is not possible because they do not share the same initial global stretch before testing. This presents an issue between the interpretation of the constitutive model and experiments. Given material parameters, the constitutive model can predict a flee swelling response that becomes the reference configuration for all additional tests following flee swelling. In the experiments, this scenario only holds for the saturated forced swelling experiment but fails for the unsaturated stretch-to—break experiment. This is because the unsaturated stretch-to—break test requires an initial precondition- ing process to minimize the Mullins effect and viscoelastic hysteresis. Whether it is a saturated forced swelling or dynamic stretching preconditioning process, this test imparts a residual deformation that differs flom the flee swelling deformation. In addition, the constitutive model used here is an equilibrium theory that fails to account for the time-dependant mechanical responses that are observed during experiments with PVA hydrogels. Due to the nature of the crosslinking, these physi— cally entangled PVA hydrogels are unstable, which causes them to degrade over time. 81 Although hydrogels exist that can be formulated to have more stable crosslinks, they are typically less applicable for biomedical uses. A constitutive theory accommodat- ing the time-dependant degradation response would be useful for making comparison of experiments that may run for different durations. This could make possible con- stitutive model interpretations such as water loss associated with changing the load during a forced swelling test. The modified Ogden model with the Flory-Huggins theory is found to be a good predictor for the mechanical response of unsaturated and saturated PVA hydrogels. The separate fitting routine for the Ogden model with N = 1 provided the best prediction of the unsaturated response, while the combined fitting routine provided the best prediction of the saturated response. The combined fitting for the Ogden model with N = 1 also resulted in smaller residuals for the stress in the traction flee directions. Ogden models with N > 1 are not found to improve the prediction of the hydrogel response. As a whole, the best prediction to both the saturated and unsaturated measurements is found using the combined fitting routine. 82 APPENDIX A Calculating the Cauchy Stress The purpose of this appendix is to show how the Cauchy stress, which is related through equation (2.6), is derived for the case of uniaxial tension. After the resulting stress-strain relation is derived, the experimental method of measuring the Cauchy stress is explained for both the unsaturated stretch-to-break and the saturated forced swelling experiments. To begin, using equation (2.6) and evaluating the partial derivative, BW/BF, yields 8W _qu 8H(J) __ 9? Mil 6F 6F 6F ‘ 6F W 01" Using the relation dJ/BF = J F‘T and defining h( J) = 6H (J) / BJ , the Cauchy stress (A.1) becomes _1 805 T+ =J —a—FF +.h(J)I (A2) Using the chain rule and symmetry of the stress tensor, (A2) becomes 20¢ _ _ _ T —JFBCF +h(J)I (A3) for the saturated case, and _2 aquT F ac +(h(J)- I’ll (A4) for the unsaturated case, where p is the incompressibility constraint. Using equation (2.5) and evaluating the dependence 805/80 has on A1, A2, and A3 in equations (A3) 83 and (AA), yields 1 8 3 8 a = —(A1-2e1 (8) el + Az—(Eeg (8 e2 + A3—¢e3 (8) e3) + h(J)I, (A.5) J A1 >12 A3 1 8 3 3 0 = —(/\1-281 ‘8 61 + /\2—¢62 ® G2 + A3393 ® 63) + (h(J) — PH, (A5) J 11 A2 A3 for saturated and the unsaturated cases, respectively. Using Cauchy’s theorem to relate the stress tensor flom (A5) or (A6) to the traction vector on the surface of a body, 01' 02' = Uijnj. (A7) If the load is being applied uniaxially in the 2 direction of the specimen as shown in Figure 2.1, the result is F2 02 = 762 = 02191 + 02282 + 02383 01‘ F2 0 = — 22 A with 021 = 023 = 0 (A8) where F2 is the applied force in the 2 direction which is measured with a load-cell during the unsaturated stretch-to—break experiments or is the applied sinker weight in the saturated forced swelling experiments. Using equation (2.3), the current cross- sectional area A of the specimen perpendicular to the loading direction (13 plane) is A = €163 = A1A3L1L3, (A.9) where 3,; and L,- are the specimen dimensions in the 1 and 3 directions for the current and reference configurations, respectively. The specimen dimensions, L1 and L3, 84 are measured optically before testing which determine the cross—sectional area of the specimen in the reference configuration. Assuming isotropy, A1 = A3, A = L1L3,\§, (A.10) where A1 is the measured stretch ratio in the horizontal direction. Since no tractions are applied on surfaces in the 1 and 3 directions, one finds the remaining six stress components flom equation (A8) to be zero. For the case of uniaxial tension and assuming isotropy, equations (A5) and (A6) reduce to _ A1645 0 — J A1 +h(J) (All) and _ /\2 395 02 _ J 12 +h(J) (A.12) for saturated case in the 1 and 2 directions, respectively. Similarly, the unsaturated case reduces to _ A1 34> 0— J A1 +h(J) p (A.13) and _ A2 79¢ 02—— J A2 +h(J) p (A.14) for the 1 and 2 directions, respectively. Lastly, one must choose the appropriate strain energy function (0 and compute the derivatives in equations (A.11) through (A.14), which will result in the final stress-strain relation for a saturated and unsaturated hydrogel subject to uniaxial tension. 85 APPENDIX B Calculating the Displacement Gradient For calculation of the displacement gradient for the bilinear interpolation algorithm, one must write out the interpolated functions for X, Y, 11, and '0 using equations (3.3) through (3.6) and evaluate these functions at the points mapped to the uniform square in the s-r coordinate system. Then one calculates the partial derivatives of X, Y, 11, and v with respect to s and 1'. One can then write 590_609_4+6024 8s — 5173-38 83:83 (13.1) and <19. 8992 we 87' — 81: 87 8y 87" where () corresponds to u or v. Rewriting equations (BI) and (B2) in matrix form, (3.2) 8()/8s : 8x/8s 8y/8s 8()/8:L' (B 3) 8()/8r 8:2/87‘ 8y/8r 8()/8y ° ' Solving for 8()/8z and 8() / 8y in (B3) by computing the inverse, yields 60 _ (3()/Bs)(3y/6r) — (6()/6r)(8y/88) (B4, 527 “ (ax/asxay/ar) — (Byway/as) ’ and 84 = (ax/asxawar) — (ax/arxay/as) ' The displacement gradient, H, is then __ 811/856 811/83; H .- ( 811/82: 8v/8y ) ' (13.6) _<9_(_) (BO/arXBIE/BS) - (3()/37‘)(3$/38) (B5) 86 APPENDIX C Measurement Uncertainty Calculations Measurement uncertainty of the experiments performed in this work are quantified in two ways. The two ways are dictated by whether the experiment is performed in the device with constant monitoring or if the experiment is run on the side where mea- surements are taken at only the beginning and end. If measurements are done with constant monitoring, the deformation is then constantly changing with each measure- ment. Therefore, the stretch uncertainty calculation is inferred flom an initial sample of length measurements between spheres taken before an experiment is started. Using the NIST definition for combined uncertainty [27], the stretch ratio uncertainties are 2 a L 0&2 = Elm + A3), (c.1) 0% Oil = L21 (n + AT): (0.2) l 03 = Alia/2“2 + 4AgAgoil, (C.3) where 0A210A1a and OJ are the uncertainties of the stretch ratio and volume ratio measurements, respectively. In addition, L1 and L2 are the averaged reference lengths between tracking spheres measured n times and calculated in LabVIEW according to Figure 3.3 with their associated uncertainties 0L1 and 0L2. To obtain equations Cl and C2, equation (2.3) is used and it is assumed the uncertainty in lengths before and during stretching is 031. = naLi, for i = 1,2. This is assumed because it is not possible to measure a population of lengths during stretching. Since a population of 87 measurements are taken for the reference lengths, their uncertainty is n i ' 2 2 _ i=1(L1 — L1) 0L1 — n(n — 1) (0'4) and Tl i " 2 2 _ 25:1(112 " L2) where n is the population of lengths measured before stretching begins, L] and Lg is the 71 reference measurements taken in the 1 and 2 directions, and [1, £2 is the mean of those measurements, respectively. Similarly, it is also possible to measure a population of force measurements during stretching. The uncertainty in the force is ?=1(fi - 192 n(n - 1) ’ a} = (C6) where f, are the ten force measurements taken at a time during stretching in Lab- VIEW, and f is the mean of those measurements. From the force and initial cross- sectional area measurements, one can obtain the stress. The uncertainty in the stress is 0.2 = ( 1 f f f " A§L1L3 A§L1L3 A§L§L3 A§L1L§ where L3 is the thickness direction and is calculated in the same way the in—plane 1%} + 4( Vail +< M, + < >201, (9.7) direction lengths are in equation C4 and C5. If tests do not involve constant monitoring, uncertainty is calculated differently than above. The experiments performed on the side included flee swelling and forced swelling. These experiments are performed at least three times for each testing pro- tocol as discussed in Chapter 3. The number of times the experiment is performed results in the population of measurements for the stretch ratios and volume ratios to be used to calculate uncertainty. 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