an existing cross-link minimizes the length of the resulting loop, and therefore the
entropy cost to the two polymer chains involved.
In this section, we take a detailed look at the binding statistics and spatial
distribution of freely diffusing reversible cross-links in full three-dimensional polymer gel simulations. Spatial structure is examined in the swollen gel at equilibrium,
during isotropic strain, and at equilibrium in the strained sample.
To bring our gel simulations a bit closer to experiment, we choose some of the
design parameters based on the reversibly cross-linked networks studied experimentally in Kean et al. [53]. These choices are discussed at length where relevant.
A hybrid molecular dynamics/Monte Carlo approach is used to simulate the gels
with reversible cross-links, following the strategies outlined in Sects. 3.1, 3.2.2 and
3.2.3. Simulation units are expressed as D for the length unit, M for the mass unit
and E for the energy unit. Equations of motion are integrated with a time step size of
δt ¼ 0:001
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
MD
2
=E
q
, where
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
MD
2
=E
q
e t is the simulation time unit. Simulations are carried out in the HOOMD-Blue molecular dynamics package [93, 94].
4.1.1 Model Ingredients, Interactions and Permanent Network
Formation
All networks are constructed in a three-dimensional cubic box of volume of %
320, 000D
3 , periodic in all three dimensions. To construct a network, linear polymer
strands and permanent cross-links are placed into the box.
To connect explicitly to experiments, we focus on the gel described in [53]: a
material composed of poly(4-vinylpyridine) polymers and hexyl chain permanent
cross-links. The polymers have on average 314 chemical monomers per chain, and
permanent cross-links are formed at a ratio of 1 per 50 chemical monomers, or
approximately 6.3 per polymer chain.
In simulation, these polymers are represented as coarse-grained bead-spring
freely jointed chains. A polymer segment in the simulation is formally a ‘statistical
segment’ representing a given number n of chemical monomers, set by the persistence length (in monomers) of the polymer. The chemical structure of poly
(4-vinylpyridine) is nearly identical to that of poly(styrene), with the exception of
the para nitrogen in the aryl ring. The persistence length of poly(styrene) is
approximately 6 chemical monomers. Therefore, an average poly(4-vinylpyridine)
polymer from the experiment is represented in the simulation by a bead-spring
polymer of 314/6 % 52 statistical segments. For every coarse-grained polymer, we
add %6.3 permanent cross-link monomers.
In the present simulations, we utilize 705 polymer strands of 52 segments each,
along with N perm ¼ 4, 440 permanent cross-link monomers (initially unbound)
following the experimental ratio noted above.
Particles interact via the pairwise intermolecular Weeks-Chandler-Andersen
(WCA) non-bonded potential (Eq. (10)), where r ij is the distance between any two
particles, ε is the strength of the potential and σ is the range. This potential is just the
96
C. Raffaelli et al.
entropy cost to the two polymer chains involved.
In this section, we take a detailed look at the binding statistics and spatial
distribution of freely diffusing reversible cross-links in full three-dimensional polymer gel simulations. Spatial structure is examined in the swollen gel at equilibrium,
during isotropic strain, and at equilibrium in the strained sample.
To bring our gel simulations a bit closer to experiment, we choose some of the
design parameters based on the reversibly cross-linked networks studied experimentally in Kean et al. [53]. These choices are discussed at length where relevant.
A hybrid molecular dynamics/Monte Carlo approach is used to simulate the gels
with reversible cross-links, following the strategies outlined in Sects. 3.1, 3.2.2 and
3.2.3. Simulation units are expressed as D for the length unit, M for the mass unit
and E for the energy unit. Equations of motion are integrated with a time step size of
δt ¼ 0:001
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
MD
2
=E
q
, where
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
MD
2
=E
q
e t is the simulation time unit. Simulations are carried out in the HOOMD-Blue molecular dynamics package [93, 94].
4.1.1 Model Ingredients, Interactions and Permanent Network
Formation
All networks are constructed in a three-dimensional cubic box of volume of %
320, 000D
3 , periodic in all three dimensions. To construct a network, linear polymer
strands and permanent cross-links are placed into the box.
To connect explicitly to experiments, we focus on the gel described in [53]: a
material composed of poly(4-vinylpyridine) polymers and hexyl chain permanent
cross-links. The polymers have on average 314 chemical monomers per chain, and
permanent cross-links are formed at a ratio of 1 per 50 chemical monomers, or
approximately 6.3 per polymer chain.
In simulation, these polymers are represented as coarse-grained bead-spring
freely jointed chains. A polymer segment in the simulation is formally a ‘statistical
segment’ representing a given number n of chemical monomers, set by the persistence length (in monomers) of the polymer. The chemical structure of poly
(4-vinylpyridine) is nearly identical to that of poly(styrene), with the exception of
the para nitrogen in the aryl ring. The persistence length of poly(styrene) is
approximately 6 chemical monomers. Therefore, an average poly(4-vinylpyridine)
polymer from the experiment is represented in the simulation by a bead-spring
polymer of 314/6 % 52 statistical segments. For every coarse-grained polymer, we
add %6.3 permanent cross-link monomers.
In the present simulations, we utilize 705 polymer strands of 52 segments each,
along with N perm ¼ 4, 440 permanent cross-link monomers (initially unbound)
following the experimental ratio noted above.
Particles interact via the pairwise intermolecular Weeks-Chandler-Andersen
(WCA) non-bonded potential (Eq. (10)), where r ij is the distance between any two
particles, ε is the strength of the potential and σ is the range. This potential is just the
96
C. Raffaelli et al.