Physics of Polymer Gels

Preface xi Acknowledgements xiii Part I Theories 1 1 Single Polymer Chain 3 Takamasa Sakai 1.1 General Features 3 1.1.1 Conformation of a Polymer Chain 3 1.1.2 Coarse-Graining of a Polymer Chain 4 1.1.3 Free Rotation Model 5 1.2 Statistics of a Single Polymer Chain 7 1.2.1 End-to-End Distance of a 1D Random Walk 7 1.2.2 End-to-End Distance of a 3D Random Walk 10 1.2.3 Force Needed to Stretch an Ideal Chain 12 1.3 Scaling of a Single Polymer Chain 15 1.3.1 Stretching of an Ideal Chain 17 1.3.2 Real Chains 18 1.3.3 Stretching of a Real Chain 19 Column 1: Miscible Gels and Immiscible Gels 21 References 22 2 Polymer Solution 23 Takamasa Sakai 2.1 Polymer Chains in Solution 23 2.1.1 Chain Swelling in a Good Solvent 23 2.1.2 Existing Conditions of an Ideal Chain and a Real Chain 25 2.2 Effect of Concentration on the Polymer Conformation 26 2.2.1 Overlapping Concentration 26 2.2.2 Semidilute Solution 28 2.2.3 Blobs in Semidilute Solution 29 2.3 Osmotic Pressure of a Polymer Solution 32 2.3.1 Entropy Change in Mixing 33 2.3.3 Basic Equation of Osmotic Pressure 36 2.3.4 Phase Separation of the Polymer Solution 37 2.3.5 Scaling of Osmotic Pressure 40 Column 2: Blob Size of a Polymer Gel 42 References 43 3 Definition of Polymer Gels and Rubber Elasticity 45 Takamasa Sakai 3.1 Elasticity of Gels 45 3.2 Definition of Polymer Gels 46 3.2.1 Criterion for Gelation by Rheology 47 3.2.2 Criterion for Gelation by Scattering 48 3.3 Mesh Size of a Polymer Gel 49 3.4 Elastic Modulus 51 3.4.1 Affine Network Model 51 3.4.2 Phantom Network Model 54 3.5 Network Strands and Crosslinks 60 3.5.1 Percolate Network Model 62 3.5.2 Bethe Approximation 63 3.6 Topological Interaction 67 3.7 Sol–Gel Transition 69 3.7.1 Gelation Threshold of Bethe Approximation 69 3.7.2 Gelation Threshold from the Percolation Model 70 3.8 Heterogeneity of Polymer Gels 71 Column 3: Elastic Deformation and Plastic Deformation 73 References 74 4 Swelling and Deswelling 77 Takamasa Sakai 4.1 Changes in the Elastic Modulus Due to Swelling/Deswelling 77 4.1.1 Statistical Model for Networks Consisting of Ideal Chains 78 4.1.2 Scaling for Networks Consisting of Nonideal Chains 79 4.1.3 Scaling for Highly Deswollen Networks 82 4.2 Equilibrium Swelling 85 4.2.1 Scaling Prediction of the Equilibrium Swelling 86 4.2.2 Statistical Mechanics of Equilibrium Swelling 87 4.3 Volume Phase Transition 91 4.3.1 Electrically Neutral Gels 91 4.3.2 Electrically Charged Gels 94 4.4 Swelling/Shrinking Kinetics 95 4.5 Degradation of Polymer Gels 102 4.5.1 Degradation by Cleavage of Specific Bonds 102 4.5.2 Degradation by Cleavage of Nonspecific Bonds 104 Column 4: Diffusions of Polymer Network During Swelling 105 References 106 5 Deformation and Fracture 109 Takuya Katashima and Takamasa Sakai 5.1 Description of Deformation 109 5.1.1 Displacement Vector 109 5.1.2 Strain Tensor 110 5.1.2.1 Normal Strain 110 5.1.2.2 Shear Strain 111 5.1.3 Principal Direction and Strain 113 5.2 Phenomenological Description of the Strain Energy Density Function 115 5.2.1 Estimation of the Strain Energy Density Function 116 5.3 Molecular Models for the Strain Energy Density Function 120 5.3.1 Neo-Hookean Model 120 5.3.2 Inverse Langevin Model 121 5.4 Scaling for Large Deformation 125 5.5 Fracture Behavior of Polymer Gels 126 5.5.1 Griffith Model 127 5.5.2 Lake–Thomas Model 128 5.6 Mesh Size Estimated from Elastic Modulus and Finite Extensibility 130 Column 5: Linear Viscoelasticity and Nonlinear Viscoelasticity 134 References 134 6 Mass Transport in Polymer Gels 137 Xiang Li and Takamasa Sakai 6.1 Thermal Motion and Brownian Motion 137 6.1.1 Diffusion Coefficient and Relaxation Time 138 6.1.2 Diffusion and Migration 139 6.2 Diffusion in Dilute Polymer Solutions 139 6.2.1 Diffusion of a Hard Sphere 139 6.2.2 Rouse Model 140 6.2.3 Zimm Model 141 6.3 Diffusion in Semidilute Polymer Solutions and Polymer Gels 142 6.3.1 Obstruction Model 142 6.3.2 Hydrodynamic Model 144 6.3.3 Free Volume Model 145 6.3.4 Reptation Model 146 6.3.5 Entropic Trapping Model 147 Column 6: Effects of Mesh Sizes on Mass Transport 149 References 149 7 Tetra Gel as a Near-Ideal Polymer Network 153 Takamasa Sakai 7.1 Ideal Polymer Network 153 7.2 Tetra-PEG Gel 155 7.3 Structure Tuning of Tetra-PEG Gels 155 References 158 8 Sol-Gel Transition 161 Takamasa Sakai 8.1 Determination of Sol–Gel Transition by Rheometry 161 8.2 Phase Diagram 161 8.3 Fractal Dimension at the Critical Point 163 8.4 Critical Behavior of Elastic Modulus 165 8.5 Reaction Kinetics of a Gelling System 166 8.5.1 Hydrolysis Kinetics of Tetra-PEG–OSu 167 8.5.2 Gelation Kinetics of Tetra-PEG Gel 167 References 169 9 Structural Analysis by Light and Neutron Scattering 173 Takamasa Sakai and Xiang Li 9.1 Scattering Curves of Tetra-PEG Gels 173 9.2 Scattering Curves of Stretched Tetra-PEG Gels 176 References 177 10 Elastic Modulus 179 Takamasa Sakai and Yuki Yoshikawa 10.1 Effect of Connectivity 179 10.2 Effect of the Polymer Concentration and Network Strand Length 180 References 182 11 Large Deformation 183 Takamasa Sakai and Takuya Katashima 11.1 Estimation of Strain Energy Density Function 183 11.1.1 Applicability of Neo-Hookean Model 184 11.1.2 Finite Extensibility Effect 185 11.1.3 Coupling Between Different Principal Axes 186 11.1.4 Extended Gent Model 187 11.2 Cross-Coupling 189 11.2.1 Effects of the Fraction of Elastically Effective and Ineffective Chains 190 11.2.2 Effects of Polymer Volume Fraction and Network Strand Length 191 11.2.3 Effect of the Fraction of Guest Chains 193 11.2.4 Conjecture on Origin of Cross-Coupling 196 11.3 Stretchability in Uniaxial Stretching 196 11.3.1 Kuhn Model 197 11.3.2 Effect of Connectivity 197 11.3.3 Effect of Polymer Concentration and Network Strand Length 198 11.3.4 Semiempirical Model Based on Experiments 200 References 201 12 Fracture 205 Takamasa Sakai and Takeshi Fujiyabu 12.1 Estimation of Fracture Energy 205 12.2 Conversion-Tuned Tetra-PEG Gels 207 12.3 Effects of Network Concentration and Strand Length 208 12.4 Bimodal Tetra-PEG Gels 209 12.5 Summary 210 References 211 13 Mass Transport 213 Takamasa Sakai and Takeshi Fujiyabu 13.1 Diffusion of Water Molecules 213 13.1.1 Estimation of Diffusion Coefficient of Water Molecules 213 13.1.2 Effect of Structural Parameters 214 13.1.3 Applicability of Theoretical Models 215 13.1.4 Effect of Correlation Length on Diffusion 216 13.2 Migration of Water Molecules in Hydrogels 217 13.2.1 Water Permeation Through Hydrogel 217 13.2.2 Effect of Structural Parameters on Friction Coefficient 219 13.2.3 Effect of Correlation Length on Friction Coefficient 220 13.3 Electro-Osmotic Flow in Electrically Charged Gels 221 13.3.1 Electro-Osmosis in an Electrically Balanced System 221 13.3.2 Electro-Osmosis in an Electrically Imbalanced System 222 13.3.3 Sum Rule of Electro-Osmotic Flow and Electrophoretic Motion 224 13.4 Migration of Small Double-Stranded DNAs 225 13.4.1 Electrophoresis of dsDNA in Tetra-PEG Gels and Solutions 225 13.4.2 Semiempirical Model 226 13.4.3 Effect of Correlation Length on Electrophoretic Mobility 228 13.4.4 Interaction Between Elastic Blobs and Contour of dsDNA 229 13.5 Migration of Large Double-Stranded DNAs 229 13.5.1 Electrophoresis of Large dsDNA in Tetra-PEG Gels and Solutions 230 13.5.2 Transition of the Migration Mechanism 231 References 233 14 Osmotic Pressure 235 Takamasa Sakai 14.1 Osmotic Pressure of Gels and Prepolymer Solutions 235 14.2 Change in Osmotic Pressure During Gelation 235 14.3 c* Theorem at the Gelation Threshold 237 References 239 15 Swelling 241 Takamasa Sakai and Takeshi Fujiyabu 15.1 Elastic Modulus of Swollen and Highly Deswollen Gels 241 15.2 Equilibrium Swelling 243 15.3 Swelling Kinetics 244 15.3.1 Examination of Swelling Equation 244 15.3.2 Cooperative Diffusion Coefficient 245 References 246 16 Degradation 249 Takamasa Sakai and Takeshi Fujiyabu 16.1 Cleavage of a Specific Site 249 16.2 Cleavage of Nonspecific Sites 253 16.2.1 Initial Swelling Equilibrium 254 16.2.2 Degradation Behavior of Tetra-PEG Gels 254 16.2.3 A Model for Degradation 255 16.2.4 Estimation of Degradation Rate Constants 257 References 258 17 Control Over Swelling of Injectable Gel 26 Takamasa Sakai and Takeshi Fujiyabu 17.1 Nonswellable Gels 261 17.2 Nonosmotic Gel 265 17.3 Oligo-Tetra-PEG Gel 269 References 275 Index 277