A recent study has demonstrated that phase separation in binary liquid mixtures is arrested in the presence of elastic networks and can lead to a nearly uniformly-sized distribution of the dilute-phase droplets. At longer timescales, these droplets exhibit a directional preference to migrate along elastic property gradients to form a front of dissolving droplets [K. A. Rosowski, T. Sai, E. Vidal-Henriquez, D. Zwicker, R. W. Style, E. R. Dufresne, Elastic ripening and inhibition of liquid-liquid phase separation, Nature Physics (2020) 1-4]. In this work, we develop a complete theoretical understanding of this phenomenon in nonlinear elastic solids by employing an energy-based approach that captures the process at both short and long timescales to determine the constitutive sensitivities and the dynamics of the resulting front propagation. We quantify the thermodynamic driving forces to identify diffusion-limited and dissolution-limited regimes in front propagation. We show that changes in elastic properties have a nonlinear effect on the process. This strong influence can have implications in a variety of material systems including food, metals, and aquatic sediments, and further substantiates the hypothesis that biological systems exploit such mechanisms to regulate important function.
Liquid-liquid phase separation has emerged as a fundamental mechanism underlying intracellular organization, with evidence for it being reported in numerous different systems. However, there is a growing concern regarding the lack of quantitative rigor in the techniques employed to study phase separation, and their ability to account for the complex nature of the cellular milieu, which affects key experimentally observable measures, such as the shape, size and transport dynamics of liquid droplets. Here we bridge this gap by combining recent experimental data with theoretical predictions that …
The damping or attenuation coefficient of sound waves in solids due to impurities scales with the wavevector to the fourth power, also known as Rayleigh scattering. In amorphous solids, Rayleigh scattering may be enhanced by a logarithmic factor although computer simulations offer conflicting conclusions regarding this enhancement and its microscopic origin. We present a tensorial replica field-theoretic derivation based on heterogeneous or fluctuating elasticity (HE), which shows that long-range (power-law) spatial correlations of the elastic constants, is the origin of the logarithmic enhanc…
A nonlinear small-strain elastic theory is constructed from a systematic expansion in Biot strains, truncated at quadratic order. The primary motivation is the desire for a clean separation between stretching and bending energies for shells, which appears to arise only from reduction of a bulk energy of this type. An approximation of isotropic invariants, bypassing the solution of a quartic equation or computation of tensor square roots, allows stretches, rotations, stresses, and balance laws to be written in terms of derivatives of position. Two-field formulations are also presented. Extensio…
We demonstrate, via numerical simulations, that the relaxation dynamics of supercooled liquids correlates well with a plastic length scale measuring a particle's response to impulsive localized perturbations and weakly to measures of local elasticity. We find that the particle averaged plastic length scale vanishes linearly in temperature and controls the super-Arrhenius temperature dependence of the relaxation time. Furthermore, we show that the plastic length scale of individual particles correlates with their typical displacement at the relaxation time. In contrast, the local elastic respon…
In this study we use non-equilibrium thermodynamics to systematically derive a phase-field model of a polyelectrolyte gel coupled to a hydrodynamic model for a salt solution surrounding the gel. The governing equations for the gel account for the free energy of the internal interfaces which form upon phase separation, the nonlinear elasticity of the polyelectrolyte network, and multi-component diffusive transport following a Stefan--Maxwell approach. The time-dependent model describes the evolution of the gel across multiple time and spatial scales and so is able to capture the large-scale sol…
The dynamical properties and mechanical functions of amorphous materials are governed by their microscopic structures, particularly the elasticity of the interaction networks, which is generally complicated by structural heterogeneity. This ubiquitous heterogeneous nature of amorphous materials is intriguingly attributed to a complex role of entropy. Here, we show in disordered networks that the vibrational entropy increases by creating phase-separated structures when the interaction connectivity is close to the onset of network rigidity. The stress-energy, which conversely penalizes the heter…
We develop coarse-grained particle approaches for studying the elastic mechanics of vesicles with heterogeneous membranes having phase-separated domains. We perform simulations both of passive shape fluctuations and of active systems where vesicles are subjected to compression between two plates or subjected to insertion into narrow channels. Analysis methods are developed for mapping particle configurations to continuum fields with spherical harmonics representations. Heterogeneous vesicles are found to exhibit rich behaviors where the heterogeneity can amplify surface two-point correlations,…
We derive a non-linear one-dimensional (1d) strain gradient model predicting the necking of soft elastic cylinders driven by surface tension, starting from 3d finite-strain elasticity. It is asymptotically correct: the microscopic displacement is identified by an energy method. The 1d model can predict the bifurcations occurring in the solutions of the 3d elasticity problem when the surface tension is increased, leading to a localization phenomenon akin to phase separation. Comparisons with finite-element simulations reveal that the 1d model resolves the interface separating two phases accurat…
The equilibrium amorphous solid state -- formed, e.g., by adequately randomly crosslinking the constituents of a macromolecular fluid -- is a heterogeneous state characterized by a universal distribution of particle localization lengths. Near to the crosslink-density-controlled continuous amorphous-solidification transition, this distribution obeys a scaling form: it has a single peak at a lengthscale that diverges (along with the width of the distribution) as the transition is approached. The modulus controlling macroscale elastic shear deformations of the amorphous solid does not depend on t…
Various biological and chemical processes lead to the nucleation and growth of non-wetting fluid bubbles within the pore space of a granular medium, such as the formation of gas bubbles in liquid-saturated lake-bed sediments. In sufficiently soft porous materials, the non-wetting nature of these bubbles can result in the formation of open cavities within the granular solid skeleton. Here, we consider this process through the lens of phase separation, where thermomechanics govern the separation of the non-wetting phase from a fluid-fluid-solid mixture. We construct a phase-field model informed …
Nucleation and growth (N&G) - the emergence of a new phase within an initially homogeneous one - is one of the most important physical phenomena by which gas-liquid, liquid-liquid and solid-liquid phase separation takes place. Accordingly, thermodynamics sets the asymptotic boundaries towards which the system must evolve, while kinetics tries to cope with it by imposing the transport rates at which phase separation is realized. In all heterogeneous colloidal systems observed in nature, the composition, shape, structure and ultimately physical properties result from the trade-off between thermo…
Phase separation has recently emerged as an important organizational principle in the dense and heterogeneous environment within living cells. Here, we use a synthetic system to show that compressive stresses in a polymer network suppress phase separation of the solvent that swells it. These stresses create a barrier to droplet nucleation that leads to robust, stabilized mixtures well beyond the liquid-liquid phase separation boundary. Network stresses not only alter the stability of mixtures, but they also have a dramatic effect on the ripening of droplets. Gradients in network stresses can d…
Surface segregation of the low-molecular weight component in a polymeric mixture leads to degradation of industrial formulations. We report a simultaneous phase separation and surface migration phenomena in oligomer-polymer and oligomer-gel systems following a temperature quench. We compute equilibrium and time varying migrant density profiles and wetting layer thickness using coarse grained molecular dynamics and mesoscale hydrodynamics simulations to demonstrate that surface migration in oligomer-gel systems is significantly reduced due to network elasticity. Further, phase separation proces…
