Towards finite-dimensional gelation

Abstract

We consider the gelation of particles which are permanently connected by random crosslinks, drawn from an ensemble of finite-dimensional continuum percolation. To average over the randomness, we apply the replica trick, and interpret the replicated and crosslink-averaged model as an effective molecular fluid. A Mayer-cluster expansion for moments of the local static density fluctuations is set up. The simplest non-trivial contribution to this series leads back to mean-field theory. The central quantity of mean-field theory is the distribution of localization lengths, which we compute for all connectivities. The highly crosslinked gel is characterized by a one-to-one correspondence of connectivity and localization length. Taking into account higher contributions in the Mayer-cluster expansion, systematic corrections to mean-field can be included. The sol-gel transition shifts to a higher number of crosslinks per particle, as more compact structures are favored. The critical behavior of the model remains unchanged as long as finite truncations of the cluster expansion are considered. To complete the picture, we also discuss various geometrical properties of the crosslink network, e.g. connectivity correlations, and relate the studied crosslink ensemble to a wider class of ensembles, including the Deam-Edwards distribution.

Finitely coordinated models for low-temperature phases of amorphous systems

“…These results have in fact been obtained earlier via different routes [51]. A zoom into the low x 2 (large ξ) region reveals significant substructure which shows that correlation lengths of particles depend in fact also on coordinations of their neighbours (and on coordinations of next neighbours and so on).…”

“…Models of this type have been used to describe gels; typically, uniform distributions of cross-link strengths P (k ij ) = δ(k ij − k) where chosen in that context [49,50,51]. Alternatively models with random harmonic couplings on regular lattices [33] have been looked at in connection with Bose-peak phenomena.…”

“…Alternatively models with random harmonic couplings on regular lattices [33] have been looked at in connection with Bose-peak phenomena. Note that [51], which -unlike our present modelling -provides a description preserving information about spatial structure, has recently been used to provide a microscopic underpinning of a phenomenological model of an elasticity theory for soft random solids based on randomly varying elastic constants [53]. In the present investigation, we combine elements of random structure with random values for existing couplings.…”

Cavity Approach to the Random Solid State

“…Cavity method for randomly bonded Brownian particles at arbitrary bonding densities-The cavity approach can be extended to address the chemical gelation transition, i.e., the transition triggered by the introduction of random covalent bonds between atoms or small molecules (rather than macromolecules) in the liquid state. We shall address the model studied previously by Broderix et al [16], which consists of a collection of point particles undergoing Brownian motion at a certain temperature. Permanent bonds are then introduced at random between nearby particles, so that pairs of bonded particles become constrained softly (i.e.…”

“….). The distribution of localization lengths for the randomly bonded Brownian particle model was studied previously by Broderix et al [16] using the replica method and a Mayer cluster expansion. To see that the cavity approach result (19) recovers their result, we take the limit N c → ∞ whilst keeping finite the mean number of bonds from the new particle (either to the infinite cluster or to delocalized particles), i.e.…”