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 single chains leads to reduced configurational entropy and hence increased
free energy [1, 2], while the covalent crosslinks preserve the network topology. As a
result, the network can fully recover its original size and shape upon unloading and
thus behaves as an elastic solid. A large volume of literature has been devoted to
modeling the nonlinear elasticity of rubbery networks [3, 4]. These models specify
the network’s Helmholtz free energy as a function of the macroscopic deformation,
from which the stress-strain relation can be derived based on the theoretical framework of continuum mechanics [5]. Depending on how the free energy function is
derived, the nonlinear elasticity models can be divided into two categories. In the
first category, the free energy function is motivated by experimental data, e.g., stressstrain curves from uniaxial tensile tests, and thus is phenomenological in nature.
Examples include the Mooney-Rivlin model [6, 7], the Ogden model [8], and the
Gent model [9]. In the second category, one starts from a molecular model of single
chains, e.g., freely jointed chain with Gaussian or Langevin statistics [1], and links it
to the network free energy through some kinematic assumptions, e.g., the three-chain
[10], four-chain [11], eight-chain [12], or full network [13] model. Examples for this
category include the neo-Hookean model [5] and the Arruda-Boyce model [12].
128
Q. Guo and R. Long
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 single chains leads to reduced configurational entropy and hence increased
free energy [1, 2], while the covalent crosslinks preserve the network topology. As a
result, the network can fully recover its original size and shape upon unloading and
thus behaves as an elastic solid. A large volume of literature has been devoted to
modeling the nonlinear elasticity of rubbery networks [3, 4]. These models specify
the network’s Helmholtz free energy as a function of the macroscopic deformation,
from which the stress-strain relation can be derived based on the theoretical framework of continuum mechanics [5]. Depending on how the free energy function is
derived, the nonlinear elasticity models can be divided into two categories. In the
first category, the free energy function is motivated by experimental data, e.g., stressstrain curves from uniaxial tensile tests, and thus is phenomenological in nature.
Examples include the Mooney-Rivlin model [6, 7], the Ogden model [8], and the
Gent model [9]. In the second category, one starts from a molecular model of single
chains, e.g., freely jointed chain with Gaussian or Langevin statistics [1], and links it
to the network free energy through some kinematic assumptions, e.g., the three-chain
[10], four-chain [11], eight-chain [12], or full network [13] model. Examples for this
category include the neo-Hookean model [5] and the Arruda-Boyce model [12].
128
Q. Guo and R. Long