Adv Polym Sci (2020) 285: 127–164
https://doi.org/10.1007/12_2020_60
© Springer Nature Switzerland AG 2020
Published online: 12 May 2020
Mechanics of Polymer Networks
with Dynamic Bonds
Qiang Guo and Rong Long
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
2 Continuum Mechanics and Thermodynamics of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
2.1 Kinematics . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 131
2.2 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
2.3 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3 Macroscopic Deformation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.1 Continuum Model to Capture Chain Detachment and Reattachment . . . . . . . . . . . . . . . 137
3.2 Kinetics of Chain Detachment and Reattachment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.3 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3.4 Steady-State Kinetics . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . 144
4 Transient Network Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.1 Statistical Description of Polymer Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.2 Evolution of the Chain Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.3 Macroscopic Constitutive Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.4 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5 Summary and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Abstract Incorporation of dynamic, reversible bonds into the polymer network of
soft gels has been exploited as a strategy to enhance fracture toughness and to enable
self-healing. Gels with dynamic bonds often exhibit macroscopic viscoelasticity
which can be traced back to the kinetics of bond dissociation and reformation.
This chapter discusses recent efforts in developing constitutive models to connect
the molecular-level bond kinetics to the continuum-level viscoelasticity. Two different modeling approaches are described using a model system, i.e., hydrogel with
dynamic physical crosslinks and static chemical crosslinks. Both approaches are
based on the theoretical framework of continuum mechanics and thermodynamics
Q. Guo and R. Long (*)
Department of Mechanical Engineering, University of Colorado Boulder, Boulder, CO, USA
e-mail: rong.long@colorado.edu
https://doi.org/10.1007/12_2020_60
© Springer Nature Switzerland AG 2020
Published online: 12 May 2020
Mechanics of Polymer Networks
with Dynamic Bonds
Qiang Guo and Rong Long
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
2 Continuum Mechanics and Thermodynamics of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
2.1 Kinematics . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 131
2.2 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
2.3 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3 Macroscopic Deformation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.1 Continuum Model to Capture Chain Detachment and Reattachment . . . . . . . . . . . . . . . 137
3.2 Kinetics of Chain Detachment and Reattachment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.3 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3.4 Steady-State Kinetics . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . 144
4 Transient Network Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.1 Statistical Description of Polymer Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.2 Evolution of the Chain Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.3 Macroscopic Constitutive Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.4 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5 Summary and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Abstract Incorporation of dynamic, reversible bonds into the polymer network of
soft gels has been exploited as a strategy to enhance fracture toughness and to enable
self-healing. Gels with dynamic bonds often exhibit macroscopic viscoelasticity
which can be traced back to the kinetics of bond dissociation and reformation.
This chapter discusses recent efforts in developing constitutive models to connect
the molecular-level bond kinetics to the continuum-level viscoelasticity. Two different modeling approaches are described using a model system, i.e., hydrogel with
dynamic physical crosslinks and static chemical crosslinks. Both approaches are
based on the theoretical framework of continuum mechanics and thermodynamics
Q. Guo and R. Long (*)
Department of Mechanical Engineering, University of Colorado Boulder, Boulder, CO, USA
e-mail: rong.long@colorado.edu