resides within the spherical volume of radius r ¼ 10D around a given permanent
cross-link. For comparison, the black dashed lines are the radial distribution of
reversible cross-links (either bound or unbound) in the E rev ¼ 0k B T reference system.
Before strain, we see that both the permanent and reversible bridge distributions
have distinct maxima near r ¼ 2D. This suggests two structural properties. First, it
means that the permanent cross-links tended to cluster during the in situ network
formation phase of the simulation described above. This suggests that the polymer
network is heterogeneous, with ‘bundled’ regions rich in permanent cross-links and
inter-connected polymer chains, separated by less dense regions with sparse
connecting polymers. This can be seen in the network snapshots in Fig. 5a, c.
Second, these distributions indicate that bound reversible cross-links are localized
near permanent cross-links. This is consistent with the theoretical arguments and
simple simulation results presented in Sects. 3.2.1 and 3.2.2; that is, by localizing
near permanent cross-links, bound reversible cross-links minimize the entropy
penalty to the polymer network.
The same trend is observed in the strained network at equilibrium, albeit both
distributions are actually sharper, indicating two features. First, the bundled parts of
the network rich in permanent cross-links remain largely intact, while the
‘unbundled’ parts of the network in between have done most of the sacrificial
stretching in order to satisfy the simulation box strain. This can be seen in the
snapshot of the strained network in Fig. 5b, d. And second, it suggests that the
bridging reversible cross-links are strongly localized to these bundled regions around
the permanent cross-links. In fact, the bridging reversible cross-links are more
sharply peaked than the permanent cross-links.
In contrast, the radial distribution of reversible cross-links around permanent
cross-links at zero binding energy (black dashed line) exhibits a local depletion
until approximately r ¼ 3D, after which the uniform background density is reached.
This depletion effect is due to crowding near the permanent cross-links, where
polymer segments and additional permanent cross-links are more likely to be
found at equilibrium. Thus, the reversible cross-links with a non-zero bonding
energy must overcome this steric effect in order to form bonds near permanent
cross-links; apparently, this steric penalty is less significant than the entropy cost that
would be entailed in forming the reversible bond further away from permanent crosslinks, where the polymer chains have more a priori configurational freedom.
4.1.7 Enhanced Clustering of Reversible Cross-Links Around
Permanent Cross-Links in the Strained Network
The density of bridging reversible cross-links in the gel scales with the reversible
cross-link binding strength E rev . Figure 8c, d plots the local number density of
bridging reversible cross-links as a function of radial distance around a permanent
cross-link. These distributions are obtained by multiplying the radial distribution
function by the bulk average density of bridging reversible cross-links before and
after strain (given as coloured horizontal lines in the two panels).
104
C. Raffaelli et al.
cross-link. For comparison, the black dashed lines are the radial distribution of
reversible cross-links (either bound or unbound) in the E rev ¼ 0k B T reference system.
Before strain, we see that both the permanent and reversible bridge distributions
have distinct maxima near r ¼ 2D. This suggests two structural properties. First, it
means that the permanent cross-links tended to cluster during the in situ network
formation phase of the simulation described above. This suggests that the polymer
network is heterogeneous, with ‘bundled’ regions rich in permanent cross-links and
inter-connected polymer chains, separated by less dense regions with sparse
connecting polymers. This can be seen in the network snapshots in Fig. 5a, c.
Second, these distributions indicate that bound reversible cross-links are localized
near permanent cross-links. This is consistent with the theoretical arguments and
simple simulation results presented in Sects. 3.2.1 and 3.2.2; that is, by localizing
near permanent cross-links, bound reversible cross-links minimize the entropy
penalty to the polymer network.
The same trend is observed in the strained network at equilibrium, albeit both
distributions are actually sharper, indicating two features. First, the bundled parts of
the network rich in permanent cross-links remain largely intact, while the
‘unbundled’ parts of the network in between have done most of the sacrificial
stretching in order to satisfy the simulation box strain. This can be seen in the
snapshot of the strained network in Fig. 5b, d. And second, it suggests that the
bridging reversible cross-links are strongly localized to these bundled regions around
the permanent cross-links. In fact, the bridging reversible cross-links are more
sharply peaked than the permanent cross-links.
In contrast, the radial distribution of reversible cross-links around permanent
cross-links at zero binding energy (black dashed line) exhibits a local depletion
until approximately r ¼ 3D, after which the uniform background density is reached.
This depletion effect is due to crowding near the permanent cross-links, where
polymer segments and additional permanent cross-links are more likely to be
found at equilibrium. Thus, the reversible cross-links with a non-zero bonding
energy must overcome this steric effect in order to form bonds near permanent
cross-links; apparently, this steric penalty is less significant than the entropy cost that
would be entailed in forming the reversible bond further away from permanent crosslinks, where the polymer chains have more a priori configurational freedom.
4.1.7 Enhanced Clustering of Reversible Cross-Links Around
Permanent Cross-Links in the Strained Network
The density of bridging reversible cross-links in the gel scales with the reversible
cross-link binding strength E rev . Figure 8c, d plots the local number density of
bridging reversible cross-links as a function of radial distance around a permanent
cross-link. These distributions are obtained by multiplying the radial distribution
function by the bulk average density of bridging reversible cross-links before and
after strain (given as coloured horizontal lines in the two panels).
104
C. Raffaelli et al.