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

TL;DR

The paper investigates how elasticity and deformation geometry influence liquid-gel phase separation by systematically comparing six elastic models, including a novel AB+RP model that combines finite extensibility with entanglements. It demonstrates that the commonly used Neo-Hookean model fails to admit a common tangent for phase coexistence in 3D deformations, while finite-extensible models like Arruda-Boyce or FENE enable stable coexistence; the ABRP combination captures both softening and hardening phenomena observed in entangled networks. The study further shows that in constrained-dimension swelling, increasing deformation dimension lowers the coexisting polymer fraction and raises the critical temperature, highlighting a strong dimensional dependence that differentiates elastic media from conventional phase separation. The work also emphasizes the significance of the osmotic elastic energy term and discusses how model choice, dimensionality, and geometry shape phase diagrams, providing guidance for modeling LLPS in gels and related soft materials. Overall, finite extensibility and entanglements are essential for accurately describing phase separation in elastic gels, especially under 3D deformation, with practical implications for designing responsive polymer networks and understanding biomaterial phase behavior.

Abstract

We compare six 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 dimensions(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 temperatures 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 elastic media from conventional phase separation.

Figures (10)

• Figure 1: Microscopic representations of (a) 1D, (b) 2D, (c) 3D constrained-dimension deformation. Geometry of the material can coincide with the deformation dimension in ways of: (d) 1D deformation of a semi-infinite slab ($L\gg \ell$) (e) 2D deformation of a semi-infinite rod ($L\gg \ell$) and (f) 3D deformation of an isotropic bulk material.
• Figure 2: Normalization factors $p(N)$ for the finite extensible models.
• Figure 3: Elastic free energy (a) and Mooney ratio (b, c) of an incompressible PDMS network with Young's modulus $E = 40$ kPa undergoing uniaxial tension ($\lambda_z>1$) and compression (($\lambda_z<1$, gray). Parameters are listed in Table. \ref{['tab:fig-parameters']}. The Mooney ratio of the NH model is constant as designed (horizontal black dotted line), which represents chains that behave like harmonic springs.
• Figure 4: Extension ratio upper limit $\lambda^*$ of the linear regimes for the finite extensible models FENE and AB. When the Mooney ratio of the AB and FENE models is $120\%$ that of the NH model, $\lambda^{\ast} \simeq 0.5\:\lambda_\text{max}$ for the AB model and $\lambda^{\ast} \simeq 0.405\:\lambda_\text{max}$ for the FENE model.
• Figure 5: True elastic stresses of a dry network swelling in (a) 1D (c) 2D and (e) 3D; and the elastic stresses of a pre-stretched network with $\lambda_0 = 1.2$ shrinking ($\lambda<1$, in gray) and swelling ($\lambda>1$) in (b) 1D (d) 2D and (f) 3D.