transient cross-links on the shear modulus. At short timescales, a fraction of the
bound transient cross-links act as effectively permanent bonds. These lead to an
increase in modulus around their unbinding frequency 2π/ω. For moderate values of
the binding energy, the second plateau is preempted by the Rouselike ω
3/4 regime.
Combining permanent and transient cross-links in materials can result in various
highly functional properties. In addition to the greatly enhanced toughness reported
in literature [97], we show that the reversible links may separately serve to implement a graded dynamic response, leading effectively to two distinct rubberlike
phases in dynamical rheology. By virtue of the time-temperature superposition
principle, moreover, we predict a transition between two differentially elastic solid
phases to occur at a critical temperature determined by the reversible linker
energetics.
4.3 Effect of Reversible Cross-Links Near Percolation
In this section, we apply the concepts explained in Sects. 2.2.1 and 3.2.2 to study the
mechanics of hybrid hydrogels. These gels are employed for their improved toughness and their ability to serve as responsive gels. Their mechanics is crucially
determined by the ratio of physical (reversible) to chemical (permanent) crosslinks. Here, we use MD simulations to systematically assess the rigidity and stress
relaxation of these hybrid hydrogels.
We focus our efforts on the mechanically nontrivial regime where the gel is very
soft or even fluid-like if only the chemical cross-links are considered, but where the
presence of physical cross-links can make for a significantly stiffer material. In the
simulations shown here, we focus on the effect of the relative concentration of
reversible cross-links, defined as the ratio of the number of reversible cross-links
to the total number of cross-links. The concentration of polymer as well as the total
amount of cross-links and the strength of the reversible interactions remain fixed.
The model we employ is a coarse-grained description of the tetraPEG hydrogels
of Sakai et al. [98]. Starting from a solution of four-arm star polymers with
functionalized ends, we first model the formation of a fully covalent gel via a click
reaction. Hybrid gels are then obtained from these click gels by replacing a fraction
of the click bonds by reversible bonds.
More specifically, we mix equal amounts of functionalized four-arm star polymers, denoted as A 4 and B 4 . The polymers are coarse-grained to strings of N ¼ 10
beads connected by springs of unit length, with reactive A and B beads at the ends of
the arms. The click reaction is implemented by adding a spring between an A-bead
and a B-bead when they are within a cut-off distance during the simulation. After
reacting this way, the A-bead and B-bead are changed to an inert bead type, so that
each reactive group forms at most one bond.
All beads have a purely repulsive WCA potential (see Eq. 10), with σ ¼ 1.3 and
ε ¼ 1, the latter defining our units of energy and (with k B ¼ 1) temperature. Both the
gelation process and the mechanical testing are performed using coarse-grained
112
C. Raffaelli et al.
bound transient cross-links act as effectively permanent bonds. These lead to an
increase in modulus around their unbinding frequency 2π/ω. For moderate values of
the binding energy, the second plateau is preempted by the Rouselike ω
3/4 regime.
Combining permanent and transient cross-links in materials can result in various
highly functional properties. In addition to the greatly enhanced toughness reported
in literature [97], we show that the reversible links may separately serve to implement a graded dynamic response, leading effectively to two distinct rubberlike
phases in dynamical rheology. By virtue of the time-temperature superposition
principle, moreover, we predict a transition between two differentially elastic solid
phases to occur at a critical temperature determined by the reversible linker
energetics.
4.3 Effect of Reversible Cross-Links Near Percolation
In this section, we apply the concepts explained in Sects. 2.2.1 and 3.2.2 to study the
mechanics of hybrid hydrogels. These gels are employed for their improved toughness and their ability to serve as responsive gels. Their mechanics is crucially
determined by the ratio of physical (reversible) to chemical (permanent) crosslinks. Here, we use MD simulations to systematically assess the rigidity and stress
relaxation of these hybrid hydrogels.
We focus our efforts on the mechanically nontrivial regime where the gel is very
soft or even fluid-like if only the chemical cross-links are considered, but where the
presence of physical cross-links can make for a significantly stiffer material. In the
simulations shown here, we focus on the effect of the relative concentration of
reversible cross-links, defined as the ratio of the number of reversible cross-links
to the total number of cross-links. The concentration of polymer as well as the total
amount of cross-links and the strength of the reversible interactions remain fixed.
The model we employ is a coarse-grained description of the tetraPEG hydrogels
of Sakai et al. [98]. Starting from a solution of four-arm star polymers with
functionalized ends, we first model the formation of a fully covalent gel via a click
reaction. Hybrid gels are then obtained from these click gels by replacing a fraction
of the click bonds by reversible bonds.
More specifically, we mix equal amounts of functionalized four-arm star polymers, denoted as A 4 and B 4 . The polymers are coarse-grained to strings of N ¼ 10
beads connected by springs of unit length, with reactive A and B beads at the ends of
the arms. The click reaction is implemented by adding a spring between an A-bead
and a B-bead when they are within a cut-off distance during the simulation. After
reacting this way, the A-bead and B-bead are changed to an inert bead type, so that
each reactive group forms at most one bond.
All beads have a purely repulsive WCA potential (see Eq. 10), with σ ¼ 1.3 and
ε ¼ 1, the latter defining our units of energy and (with k B ¼ 1) temperature. Both the
gelation process and the mechanical testing are performed using coarse-grained
112
C. Raffaelli et al.