Filamentous networks in phase-separating two-dimensional gels

We introduce a toy model that contains the basic features of microphase separation in polymer gels: a stretched elastic network of Lennard-Jones particles, studied in two dimensions. When temperature is lowered below some value T*, attraction between particles dominates over both thermal motion and elastic forces, and the network separates into dense domains of filaments connected by three-fold vertices, surrounded by low-density domains in which the network is homogeneously stretched. The length of the filaments decreases and the number of domains increases with decreasing temperature. The system exhibits hysteresis characteristic of first-order phase transitions: pre-formed filaments thin upon heating and eventually melt at a temperature T^** (>T*). Although details may vary, the above general features are independent of network topology (square or hexagonal), system size, distribution of spring constants, and perturbations of initial conditions.