Order and localization in randomly cross-linked polymer networks

We study, by molecular dynamics, the onset of order and localization in randomly cross-linked polymer networks as the number n of cross links is increased. We find a well-defined critical number of cross links ${\mathit{n}}_{\mathit{c}}$ above which the order parameter q=1/${\mathit{Ntsum}}_{\mathit{i}}$〈‖expik⋅${\mathbf{r}}_{\mathit{i}}$‖${\mathrm{〉}}^2$ increases as q∼(n-${\mathit{n}}_{\mathit{c}}$ ${)}^{\mathrm{\beta }}$, with β≊0.5 for ‖k‖=2π/L, where L is the length of the computational cell. At the same critical number of cross links, particles in the network become localized around their mean positions. We find that the distribution of localization lengths P(ξ) is a universal function when plotted in terms of a suitable scaled variable. © 1996 The American Physical Society.