Mechanics of Polymer Networks with Dynamic Bonds
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 trac
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Mechanics of Polymer Networks with Dynamic Bonds Qiang Guo and Rong Long
Contents 1 Introduction 2 Continuum Mechanics and Thermodynamics of Solids 2.1 Kinematics 2.2 Stress 2.3 Thermodynamics 3 Macroscopic Deformation Theory 3.1 Continuum Model to Capture Chain Detachment and Reattachment 3.2 Kinetics of Chain Detachment and Reattachment 3.3 Constitutive Equations 3.4 Steady-State Kinetics 4 Transient Network Theory 4.1 Statistical Description of Polymer Network 4.2 Evolution of the Chain Distribution Function 4.3 Macroscopic Constitutive Relationship 4.4 Special Cases 5 Summary and Discussions References
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: [email protected]
Q. Guo and R. Long
and aim to quantify how the total network free energy is governed by macroscopic deformation and molecular kinetics. In the first approach, the network is treated as a collection of polymer chains formed at different instants along the loading history. These chains experience different extent of deformation and thus carry different free energy. The total free energy is the sum of contributions from all chains. The second approach considers a statistical distribution of the chain end-to-end vectors, which evolves upon macroscopic deformation and reaction of dynamic bonds. The total free energy is calculated by integrating the single-chain free energy over the chain distribution space. These two approaches, capable of capturing the time-dependent mechanical behaviors of hydrogels with reversible crosslinks, can be extended to model the macroscopic mechanics induced by other molecular mechanisms such as bond exchange and chain scission. Keywords Dynamic bonds · Continuum mechanics · Molecular kinetics · Polymer network · Viscoelasticity
1 Introduction The molecular structure of soft polymeric materials, e.g., hydrogels or unfilled elastomers, can be represented by an amorphous network of crosslinked flexible polymer chains. How the network responds to mechanical loading depends on the force-extension behavior of individual chains [1] as well as on the spatial localization of the crosslinks connecting these chains to the network. For example, when a rubbery network crosslinked by covalent bonds is subjected to external loading, the stretch of
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