steps. Equilibration includes the MC reversible cross-link bind/unbind moves. The
following protocol is then used to examine the equilibrium and strain response of the
network:
1. Record equilibrium statistics over 2 Â 10
7 time steps.
2. Do an isotropic strain experiment over 3 Â 10
7 time steps, in which the simulation
box is expanded at a constant rate to double its initial length, width and height.
3. Stop straining, and record new equilibrium statistics over 2.5 Â 10
7 time steps in
the strained configuration.
In all cases the simulation proceeds via the NVT ensemble with a Langevin
thermostat.
Simulation results as a function of time are output every 50,000 integration steps.
Recall that the Monte Carlo sweep interval is N MC ¼ 750 steps and every reversible
cross-link sticker is tested for a bind or unbind move (depending on its current state)
on each MC sweep. Therefore, %67 MC sweep are attempted per reversible crosslink sticker during each of these output simulation frames.
During the strain time of 3 Â 10
7 integration steps, the simulation box dimensions
are doubled in length, corresponding to an engineering strain of 1.0 in x, y and z. The
strain rate is therefore % 3:3 Â 10
À5 1= e t (recalling the simulation time step size of
dt ¼ 0:001 e t ). For 1 unit of engineering strain in the simulation, there are 40,000
attempted MC moves per reversible cross-link sticker. The strain rate is therefore
2.5 Â 10
À5 units per MC sweep.
4.1.4 Reversible Cross-Link Binding Statistics
Figure 6 presents the bulk statistics of reversible cross-link binding as a function of
time, expressed in units of 50,000 integration steps. From t ¼ 1 to 400, the simulation is at the initial swollen volume of V swollen . A snapshot of the entire network with
reversible cross-links is shown in Fig. 5a. Isotropic strain begins at t ¼ 400 and
continues at a constant rate until t ¼ 1,000. After that time the simulation is
continued at the new strained box size until t ¼ 1,500; a snapshot of the simulation
in this state is given in Fig. 5b.
Panel (a) in Fig. 6 presents the number of bound stickers (relative to the total
number of stickers) through the simulation, for four choices of sticker bond strength
E rev . Panel (c) shows the fraction of reversible cross-links that have formed bridges –
i.e. that have both of their stickers bound to polymer segments. Panel (d) shows the
number of reversible bridges relative to the number of permanent cross-links in the
system, with close-ups for the behaviour of the E rev ¼ 10k B T and 15k B T systems in
panels (e) and (f).
Increasing the binding strength results in a larger fraction of stickers bound to
polymer segments at equilibrium, during and after strain. The reversible cross-links
are, however, more susceptible to strain when their binding strength grows smaller.
For example, when E rev ¼ 15k B T, the fraction of bound stickers and fraction of
reversible link bridges remain relatively constant with strain. On the other hand, for
Rheology, Rupture, Reinforcement and Reversibility: Computational Approaches. . .
99
following protocol is then used to examine the equilibrium and strain response of the
network:
1. Record equilibrium statistics over 2 Â 10
7 time steps.
2. Do an isotropic strain experiment over 3 Â 10
7 time steps, in which the simulation
box is expanded at a constant rate to double its initial length, width and height.
3. Stop straining, and record new equilibrium statistics over 2.5 Â 10
7 time steps in
the strained configuration.
In all cases the simulation proceeds via the NVT ensemble with a Langevin
thermostat.
Simulation results as a function of time are output every 50,000 integration steps.
Recall that the Monte Carlo sweep interval is N MC ¼ 750 steps and every reversible
cross-link sticker is tested for a bind or unbind move (depending on its current state)
on each MC sweep. Therefore, %67 MC sweep are attempted per reversible crosslink sticker during each of these output simulation frames.
During the strain time of 3 Â 10
7 integration steps, the simulation box dimensions
are doubled in length, corresponding to an engineering strain of 1.0 in x, y and z. The
strain rate is therefore % 3:3 Â 10
À5 1= e t (recalling the simulation time step size of
dt ¼ 0:001 e t ). For 1 unit of engineering strain in the simulation, there are 40,000
attempted MC moves per reversible cross-link sticker. The strain rate is therefore
2.5 Â 10
À5 units per MC sweep.
4.1.4 Reversible Cross-Link Binding Statistics
Figure 6 presents the bulk statistics of reversible cross-link binding as a function of
time, expressed in units of 50,000 integration steps. From t ¼ 1 to 400, the simulation is at the initial swollen volume of V swollen . A snapshot of the entire network with
reversible cross-links is shown in Fig. 5a. Isotropic strain begins at t ¼ 400 and
continues at a constant rate until t ¼ 1,000. After that time the simulation is
continued at the new strained box size until t ¼ 1,500; a snapshot of the simulation
in this state is given in Fig. 5b.
Panel (a) in Fig. 6 presents the number of bound stickers (relative to the total
number of stickers) through the simulation, for four choices of sticker bond strength
E rev . Panel (c) shows the fraction of reversible cross-links that have formed bridges –
i.e. that have both of their stickers bound to polymer segments. Panel (d) shows the
number of reversible bridges relative to the number of permanent cross-links in the
system, with close-ups for the behaviour of the E rev ¼ 10k B T and 15k B T systems in
panels (e) and (f).
Increasing the binding strength results in a larger fraction of stickers bound to
polymer segments at equilibrium, during and after strain. The reversible cross-links
are, however, more susceptible to strain when their binding strength grows smaller.
For example, when E rev ¼ 15k B T, the fraction of bound stickers and fraction of
reversible link bridges remain relatively constant with strain. On the other hand, for
Rheology, Rupture, Reinforcement and Reversibility: Computational Approaches. . .
99