molecular dynamics in LAMMPS [75], with a Langevin thermostat set to T ¼ 0.6.
The reversible interactions introduced by replacing covalent bonds are implemented
using stickers with a well depth of 50 (see Sect. 3.2.2 for details).
Once we have a percolating covalent gel (and at least 98% of the bonds have
reacted), we randomly change a fraction (1 À η) of the AB covalently connected
beads to A
0 and B
0 active sticker beads (see Sect. 3.2.2). The parameters of the
stickers are such that they only bind 1-to-1. This effectively changes a portion of the
covalent bonds into physical bonds. We do this over a range of ratios of reversible/
total bonds (from η ¼ 1, fully reversible, to η ¼ 0, fully covalent). We then perform a
long equilibrium simulation for each sample, to obtain statistics on the percolation
properties (which are now fluctuating by virtue of the reversible bonds) and the stress
relaxation modulus. The latter is determined using the correlated fluctuation method
discussed in Sect. 3.5.2.
For all values of η, Fig. 15a, b, respectively, shows the probability (time averaged
over the whole simulation for a range of seeds) to find a percolating network
(i.e. gelation) and the stress relaxation modulus G(t). As a check, the gel’s behaviour
is compared to equivalent gels where the sticker interactions are deactivated
(i.e. fully covalent gels with some of the bonds removed; see Fig. 15c) and to the
initial fully covalent network.
Figure 15a shows the probability to observe a percolating network as a function
of the fraction of covalent bonds (η), both when considering only the covalent bonds
and when considering reversible bonds as well. For intermediate values of η, we see
that the networks reach a state in which their behaviour should be heavily influenced
by the reversible bonds, as they are non-percolating without them but almost always
percolating with them. Indeed, in Fig. 15b, c, we see that we can interpolate
smoothly between the fully covalent solid-like gel and the viscoelastic fluid obtained
when replacing all bonds with reversible ones. Furthermore, comparing the stress
relaxation moduli in (b) to the control experiment (c) in which the reversible links
have been made inactive shows that the reversible cross-links influence the longtime mechanics, particularly in the intermediate η regime. This is in accordance with
the percolation plot; here we see that for these two ratios of reversible bonds, the
system only percolates when the stickers are included.
4.4 Stress Relaxation in Vitrimers
We apply the techniques introduced in Sects. 3.3 and 3.5.2 to characterize a mixture
of stars. We show that a hydrogel can be easily made into a vitrimeric network, thus
achieving the numerous advantages that vitrimers bring.
To emphasize the effect of enriching stars with vitrimeric bonds, we first measure
the stress relaxation modulus via the autocorrelation route (Sect. 3.5.2). Results in
Fig. 16 show that when swaps do not require any external energy (βΔE swap ¼ 0), G
(t) behaves as in a viscous liquid and thus the material flows. In this swapping phase,
the material is self-healing and can be reshaped at will. Then, the swap energy barrier
Rheology, Rupture, Reinforcement and Reversibility: Computational Approaches. . .
113
The reversible interactions introduced by replacing covalent bonds are implemented
using stickers with a well depth of 50 (see Sect. 3.2.2 for details).
Once we have a percolating covalent gel (and at least 98% of the bonds have
reacted), we randomly change a fraction (1 À η) of the AB covalently connected
beads to A
0 and B
0 active sticker beads (see Sect. 3.2.2). The parameters of the
stickers are such that they only bind 1-to-1. This effectively changes a portion of the
covalent bonds into physical bonds. We do this over a range of ratios of reversible/
total bonds (from η ¼ 1, fully reversible, to η ¼ 0, fully covalent). We then perform a
long equilibrium simulation for each sample, to obtain statistics on the percolation
properties (which are now fluctuating by virtue of the reversible bonds) and the stress
relaxation modulus. The latter is determined using the correlated fluctuation method
discussed in Sect. 3.5.2.
For all values of η, Fig. 15a, b, respectively, shows the probability (time averaged
over the whole simulation for a range of seeds) to find a percolating network
(i.e. gelation) and the stress relaxation modulus G(t). As a check, the gel’s behaviour
is compared to equivalent gels where the sticker interactions are deactivated
(i.e. fully covalent gels with some of the bonds removed; see Fig. 15c) and to the
initial fully covalent network.
Figure 15a shows the probability to observe a percolating network as a function
of the fraction of covalent bonds (η), both when considering only the covalent bonds
and when considering reversible bonds as well. For intermediate values of η, we see
that the networks reach a state in which their behaviour should be heavily influenced
by the reversible bonds, as they are non-percolating without them but almost always
percolating with them. Indeed, in Fig. 15b, c, we see that we can interpolate
smoothly between the fully covalent solid-like gel and the viscoelastic fluid obtained
when replacing all bonds with reversible ones. Furthermore, comparing the stress
relaxation moduli in (b) to the control experiment (c) in which the reversible links
have been made inactive shows that the reversible cross-links influence the longtime mechanics, particularly in the intermediate η regime. This is in accordance with
the percolation plot; here we see that for these two ratios of reversible bonds, the
system only percolates when the stickers are included.
4.4 Stress Relaxation in Vitrimers
We apply the techniques introduced in Sects. 3.3 and 3.5.2 to characterize a mixture
of stars. We show that a hydrogel can be easily made into a vitrimeric network, thus
achieving the numerous advantages that vitrimers bring.
To emphasize the effect of enriching stars with vitrimeric bonds, we first measure
the stress relaxation modulus via the autocorrelation route (Sect. 3.5.2). Results in
Fig. 16 show that when swaps do not require any external energy (βΔE swap ¼ 0), G
(t) behaves as in a viscous liquid and thus the material flows. In this swapping phase,
the material is self-healing and can be reshaped at will. Then, the swap energy barrier
Rheology, Rupture, Reinforcement and Reversibility: Computational Approaches. . .
113