The roles of elasticity and dimension in liquid-gel phase separation

We compare seven elastic models for polymer networks in the context of phase separation within a gel, including a new model that combines the finite extensible Arruda-Boyce model and the slip tube model for entangled chains. We study incompressible uniaxial stretch and compression, and three volume-changing constrained-dimension deformations, in which the material can only deform in the designated dimension(s), while the constrained direction(s) remain(s) the same. Each model responds differently to large deformations, and our proposed model successfully describes both strain softening and strain hardening, which are both present in well-entangled elastomers. When considering phase separation, we show that the commonly used neo-Hookean model fails to admit a common tangent construction for phase coexistence for 3D deformations. This can be resolved by using a model with finite extension, such as the Arruda-Boyce model. In constrained-dimension deformations, where the gel's volume is allowed to change, for elastic models in which phase coexistence is possible, the critical temperature increases and the critical concentration decreases with increasing deformation dimensions. This strong dependence of the phase diagram on spatial dimension and geometry distinguishes phase separation in elastic media from conventional liquid-liquid phase separation.