200 units are added into the swollen box. This leads to a reversible cross-link volume
fraction of ϕ rev % 0.02.
Reversible cross-links in the simulation comprise the host bead of diameter
d rev ¼ d seg along with two connected sticker binding sites. The latter have a diameter
of d sticker ¼ 0:5D , held at a fixed angle of 180
∘ to each other to resemble the van
Koten complexes used in [53]. A reversible cross-link binds to a polymer segment
via one such sticker. The stickers have Lennard-Jones parameters of E ¼ 1E and
σ sticker ¼ d sticker þ d seg
À
Á =2 ¼ 0:75D with polymer segments, permanent cross-links
and the host beads of other reversible linkers. (The stickers themselves do not
interact with each other.)
Reversible cross-link stickers form and unform bonds with polymer segments via
the Monte Carlo move defined in Sect. 3.2.3, Eq. (42). To choose the attempt interval
N MC for MC moves, the timescale τ LJ must be defined as given by Eq. (38). In that
equation we take E ¼ 1E, σ ¼ σ sticker ¼ 0:75D, from the Lennard-Jones parameters
for the stickers. All particle masses in the simulation are taken to be unity,
i.e. m ¼ 1M. Thus, MC moves are attempted every N MC ¼ 750 integration steps.
On a given MC move, a bind/unbind move is attempted for every reversible crosslink sticker (i.e. ξ ¼ 1).
Reversible cross-link stickers bound to polymer segments have a bond
potential of
U
2
ð Þ
REV r ij
À Á ¼ U
2
ð Þ
FENE r ij
À Á À U
2
ð Þ
FENE r min
ð
ÞÀE rev ,
ð64Þ
where E rev is a parameter to control the strength of reversible cross-link binding and
r min % 0.96σ eye is the minimum of the total potential U
2
ð Þ
WCA r ij
À Á þ U
2
ð Þ
FENE r ij
À Á
for a
sticker bound to a polymer segment. The potential U
2
ð Þ
REV is therefore just the standard
FENE potential shifted by a constant (ÀE rev À U
2
ð Þ
FENE r min
ð
Þ). A polymer segment
may only be bound to one partner at any given time, and this is enforced in the MC
moves.
In order to evaluate the Monte Carlo success probabilities given in Eq. (42), the
energy change ΔE(r( j)) must be calculated upon addition or deletion of the reversible bond. Since the only difference between the bound and unbound state of a
sticker is the presence of the FENE bond connecting the sticker to polymer segment
j, then
ΔE r j
ð Þ
ð
Þ ¼ U
2
ð Þ
REV r j
ð Þ
ð
Þ:
ð65Þ
4.1.3 Simulation Protocol, Strain and Sampling
Once reversible cross-links are added, the network is equilibrated in the NVT
ensemble with a Langevin thermostat for the implicit solvent, for 1 Â 10
7 time
98
C. Raffaelli et al.
fraction of ϕ rev % 0.02.
Reversible cross-links in the simulation comprise the host bead of diameter
d rev ¼ d seg along with two connected sticker binding sites. The latter have a diameter
of d sticker ¼ 0:5D , held at a fixed angle of 180
∘ to each other to resemble the van
Koten complexes used in [53]. A reversible cross-link binds to a polymer segment
via one such sticker. The stickers have Lennard-Jones parameters of E ¼ 1E and
σ sticker ¼ d sticker þ d seg
À
Á =2 ¼ 0:75D with polymer segments, permanent cross-links
and the host beads of other reversible linkers. (The stickers themselves do not
interact with each other.)
Reversible cross-link stickers form and unform bonds with polymer segments via
the Monte Carlo move defined in Sect. 3.2.3, Eq. (42). To choose the attempt interval
N MC for MC moves, the timescale τ LJ must be defined as given by Eq. (38). In that
equation we take E ¼ 1E, σ ¼ σ sticker ¼ 0:75D, from the Lennard-Jones parameters
for the stickers. All particle masses in the simulation are taken to be unity,
i.e. m ¼ 1M. Thus, MC moves are attempted every N MC ¼ 750 integration steps.
On a given MC move, a bind/unbind move is attempted for every reversible crosslink sticker (i.e. ξ ¼ 1).
Reversible cross-link stickers bound to polymer segments have a bond
potential of
U
2
ð Þ
REV r ij
À Á ¼ U
2
ð Þ
FENE r ij
À Á À U
2
ð Þ
FENE r min
ð
ÞÀE rev ,
ð64Þ
where E rev is a parameter to control the strength of reversible cross-link binding and
r min % 0.96σ eye is the minimum of the total potential U
2
ð Þ
WCA r ij
À Á þ U
2
ð Þ
FENE r ij
À Á
for a
sticker bound to a polymer segment. The potential U
2
ð Þ
REV is therefore just the standard
FENE potential shifted by a constant (ÀE rev À U
2
ð Þ
FENE r min
ð
Þ). A polymer segment
may only be bound to one partner at any given time, and this is enforced in the MC
moves.
In order to evaluate the Monte Carlo success probabilities given in Eq. (42), the
energy change ΔE(r( j)) must be calculated upon addition or deletion of the reversible bond. Since the only difference between the bound and unbound state of a
sticker is the presence of the FENE bond connecting the sticker to polymer segment
j, then
ΔE r j
ð Þ
ð
Þ ¼ U
2
ð Þ
REV r j
ð Þ
ð
Þ:
ð65Þ
4.1.3 Simulation Protocol, Strain and Sampling
Once reversible cross-links are added, the network is equilibrated in the NVT
ensemble with a Langevin thermostat for the implicit solvent, for 1 Â 10
7 time
98
C. Raffaelli et al.