This book elucidates in detail the physics of polymer gels and reviews their unique properties that make them attractive for innumerable applications. Geared towards experienced researchers and entrants to the field, it covers rubber elasticity, swelling and shrinking, deformation and fracture of as well as mass transport in polymer gels, enabling the readers to purposefully design polymer gels fit for specific purposes.
Divided into two parts, Physics of Polymer Gels starts by explaining the statistical mechanics and scaling of a polymer chains, and that of polymer solutions. It then introduces the structure of polymer gels and explains the rubber elasticity, which predicts the solid-like nature of polymer gels. Next, it describes swelling/deswelling, which can be understood by combining the rubber elasticity and the osmotic pressure of a polymer solution. Large deformation and fracture, and the diffusion of substances in polymer gels, which are essential for practical applications, are also introduced. The last half of the book contains the authors' experimental results using Tetra-PEG gels and provides readers with the opportunity to examine and compare it with the first half in order to understand how to utilize the models to experiments. This title:
Is the first book dedicated to the physics of polymer gels
Describes in detail the properties of polymer gels and their underlying physics, facilitating the development of novel, polymer gel-based applications
Serves as a reference for all relevant polymer gel properties and their underlying physics
Provides a unified treatment of the subject, explaining the physical properties of polymer gels within a common nomenclature framework
Physics of Polymer Gels is a must-have book for experienced researchers, such as polymer chemists, materials scientists, organic chemists, physical chemists, and solid-state physicists, as well as for newcomers to the field.
Table of Contents
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
