The physical picture of network deformation under a fixed topology has been
revised by the recent development of polymers with dynamic and reversible bonds
[14]. When incorporated into a network, these bonds are able to continuously alter
the network topology by reshuffling the connectivity of chains through bond dissociation and reformation. The molecular events of chain detachment and reattachment
relax the free energy stored in a deformed network, and thus can lead to macroscopic
stress relaxation or creep. Many different mechanisms have been developed to
implement dynamic bonds in polymer networks, either via physical interactions
(e.g., hydrogen bonds [14] or ionic bonds [15]) or chemical reactions (e.g., DielsAlder reaction [16] or transesterification reaction [17]). The practical benefits of
using dynamic bonds are myriad. For example, one can leverage dynamic bonds that
are sensitive to external stimuli such as light, heat, mechanical stress, and pH-value
to obtain stimuli-responsive polymers [14]. Another application is to use dynamic
bonds to engineer self-healing hydrogels [18]. As demonstrated in the pioneering
work of the double network gel [19], it is possible to use a sacrificial network to
substantially enhance the fracture toughness of hydrogels which would be otherwise
brittle. The physical mechanism underlying the toughness enhancement is the energy
dissipation associated with the damage in the sacrificial network [20]. However,
such damage is irreversible if the sacrificial network is crosslinked using covalent
bonds. This limitation has motivated the development of hydrogel networks with
physical bonds [18, 21, 22]. Breaking of the physical bonds can still induce energy
dissipation, but unlike covalent bonds, physical bonds can spontaneously reform and
thus allow self-healing of the hydrogel upon unloading [23, 24].
Polymer networks with dynamic bonds exhibit a time-dependent mechanical
behavior much like a viscoelastic solid, e.g., stress relaxation, creep, and history
dependent stress-strain relation [21, 25]. The macroscopic viscoelastic behavior can
be traced back to the molecular kinetics of bond dissociation and reformation
[25, 26]. Indeed, it has been demonstrated that the viscoelasticity of hydrogels can
be engineered by controlling the type and fraction of dynamic metal-ligand bonds
that serve as crosslinks for the transient hydrogel network [27–29]. This chapter
discusses how to model the time-dependent mechanics due to dynamic bonds.
Unlike conventional phenomenological models that describe viscoelasticity using
springs and dashpots, the focus here is to connect the continuum-level viscoelasticity
to the molecular-level bond kinetics. Such models are valuable in two aspects. First,
they can facilitate the design of applications utilizing polymers with dynamic bonds
by enabling predictive simulations. Second, the model can also help reveal molecular kinetics of the dynamic bonds from macroscopic mechanical tests (e.g., relaxation tests).
Similar to the modeling methodology for rubbery networks with fixed topology,
we will describe two different approaches. The first approach [25, 30], referred to as
the macroscopic deformation theory, treats the network as a collection of polymer
Mechanics of Polymer Networks with Dynamic Bonds
129
revised by the recent development of polymers with dynamic and reversible bonds
[14]. When incorporated into a network, these bonds are able to continuously alter
the network topology by reshuffling the connectivity of chains through bond dissociation and reformation. The molecular events of chain detachment and reattachment
relax the free energy stored in a deformed network, and thus can lead to macroscopic
stress relaxation or creep. Many different mechanisms have been developed to
implement dynamic bonds in polymer networks, either via physical interactions
(e.g., hydrogen bonds [14] or ionic bonds [15]) or chemical reactions (e.g., DielsAlder reaction [16] or transesterification reaction [17]). The practical benefits of
using dynamic bonds are myriad. For example, one can leverage dynamic bonds that
are sensitive to external stimuli such as light, heat, mechanical stress, and pH-value
to obtain stimuli-responsive polymers [14]. Another application is to use dynamic
bonds to engineer self-healing hydrogels [18]. As demonstrated in the pioneering
work of the double network gel [19], it is possible to use a sacrificial network to
substantially enhance the fracture toughness of hydrogels which would be otherwise
brittle. The physical mechanism underlying the toughness enhancement is the energy
dissipation associated with the damage in the sacrificial network [20]. However,
such damage is irreversible if the sacrificial network is crosslinked using covalent
bonds. This limitation has motivated the development of hydrogel networks with
physical bonds [18, 21, 22]. Breaking of the physical bonds can still induce energy
dissipation, but unlike covalent bonds, physical bonds can spontaneously reform and
thus allow self-healing of the hydrogel upon unloading [23, 24].
Polymer networks with dynamic bonds exhibit a time-dependent mechanical
behavior much like a viscoelastic solid, e.g., stress relaxation, creep, and history
dependent stress-strain relation [21, 25]. The macroscopic viscoelastic behavior can
be traced back to the molecular kinetics of bond dissociation and reformation
[25, 26]. Indeed, it has been demonstrated that the viscoelasticity of hydrogels can
be engineered by controlling the type and fraction of dynamic metal-ligand bonds
that serve as crosslinks for the transient hydrogel network [27–29]. This chapter
discusses how to model the time-dependent mechanics due to dynamic bonds.
Unlike conventional phenomenological models that describe viscoelasticity using
springs and dashpots, the focus here is to connect the continuum-level viscoelasticity
to the molecular-level bond kinetics. Such models are valuable in two aspects. First,
they can facilitate the design of applications utilizing polymers with dynamic bonds
by enabling predictive simulations. Second, the model can also help reveal molecular kinetics of the dynamic bonds from macroscopic mechanical tests (e.g., relaxation tests).
Similar to the modeling methodology for rubbery networks with fixed topology,
we will describe two different approaches. The first approach [25, 30], referred to as
the macroscopic deformation theory, treats the network as a collection of polymer
Mechanics of Polymer Networks with Dynamic Bonds
129