Initiation Multiple walkers are randomly initialized within the periodic box. They
may be given a designated maximum length or be deemed free to grow as long as
they can.
Propagation Each walker takes a step to a randomly chosen neighbouring site that
is still available to be visited, extending the polymer with one bead, by placing a
spring of rest length b on the lattice link. We will refine this step in the next
subsection when we discuss how to use the method to generate swollen networks.
Site Marking If the newly visited site is a cross-linking site that has not been visited
before, it will be marked to be available 1 further time. If it is a normal site, or a
cross-linking site that was visited once before already, it will be marked unavailable.
Termination The previous two steps are repeated until the polymers have reached
their designated maximum length or until the propagation step fails because none of
the neighbouring sites are available to visit.
Note that we start all random walkers at the same time, and they propagate in
parallel, so that all chains are statistically equivalent. If, instead, we were to generate
the walks one at a time, the statistics would change during the process because
screening of excluded volume effects would become more and more important as the
process progressed. Note that the networks thus created are generally not spacefilling because the walkers terminate when they get stuck. We will get back to this at
the end of this section. The final result is a cross-linked network, within which all
polymer bridges obey random walk statistics.
3.4.2 Lattice Swelling and Intergenerational Cross-Linking
To make double networks, we follow the experimental procedure in the sense that
we create the generations of the networks one after another in the model. To do this,
we need to extend our method in two ways. Firstly, we need to generate polymer
bridges with statistically larger end-to-end distances, to represent the swollen state of
the first network. Secondly, we need to implement a parameter that governs how
many sites of the first network are available for the walkers that make the second
network to visit, so that we have control over the amount of intergenerational crosslinking. To start with the latter, we consider again the link with experiment, in which
a source of intergenerational cross-linking is the possibility of chain transfer reactions where partially unreacted sites from the first network are available for the
second moiety to react in the polymerization of the second network. In our model,
this means considering the cross-linker sites of the first network that were only
visited once during the first polymerization. At the beginning of the second polymerization, we mark a fraction ξ of these half-reacted sites as available, while the rest
is marked unavailable. Setting ξ ¼ 0 thus prevents all intergenerational cross-linking
(IGC) and corresponds to the experimental situation of when there is no covalent
connection between the two networks. They are entangled via interpenetrating
polymer bridges, but not covalently linked.
90
C. Raffaelli et al.
may be given a designated maximum length or be deemed free to grow as long as
they can.
Propagation Each walker takes a step to a randomly chosen neighbouring site that
is still available to be visited, extending the polymer with one bead, by placing a
spring of rest length b on the lattice link. We will refine this step in the next
subsection when we discuss how to use the method to generate swollen networks.
Site Marking If the newly visited site is a cross-linking site that has not been visited
before, it will be marked to be available 1 further time. If it is a normal site, or a
cross-linking site that was visited once before already, it will be marked unavailable.
Termination The previous two steps are repeated until the polymers have reached
their designated maximum length or until the propagation step fails because none of
the neighbouring sites are available to visit.
Note that we start all random walkers at the same time, and they propagate in
parallel, so that all chains are statistically equivalent. If, instead, we were to generate
the walks one at a time, the statistics would change during the process because
screening of excluded volume effects would become more and more important as the
process progressed. Note that the networks thus created are generally not spacefilling because the walkers terminate when they get stuck. We will get back to this at
the end of this section. The final result is a cross-linked network, within which all
polymer bridges obey random walk statistics.
3.4.2 Lattice Swelling and Intergenerational Cross-Linking
To make double networks, we follow the experimental procedure in the sense that
we create the generations of the networks one after another in the model. To do this,
we need to extend our method in two ways. Firstly, we need to generate polymer
bridges with statistically larger end-to-end distances, to represent the swollen state of
the first network. Secondly, we need to implement a parameter that governs how
many sites of the first network are available for the walkers that make the second
network to visit, so that we have control over the amount of intergenerational crosslinking. To start with the latter, we consider again the link with experiment, in which
a source of intergenerational cross-linking is the possibility of chain transfer reactions where partially unreacted sites from the first network are available for the
second moiety to react in the polymerization of the second network. In our model,
this means considering the cross-linker sites of the first network that were only
visited once during the first polymerization. At the beginning of the second polymerization, we mark a fraction ξ of these half-reacted sites as available, while the rest
is marked unavailable. Setting ξ ¼ 0 thus prevents all intergenerational cross-linking
(IGC) and corresponds to the experimental situation of when there is no covalent
connection between the two networks. They are entangled via interpenetrating
polymer bridges, but not covalently linked.
90
C. Raffaelli et al.