Phase-field modeling of interfacial dynamics in emulsion flows: Nonequilibrium surface tension

Abstract

a b s t r a c tA weakly nonlocal phase-field model is used to define the surface tension in liquid binary mixtures in terms of the composition gradient in the interfacial region so that, at equilibrium, it depends linearly on the characteristic length that defines the interfacial width. Contrary to previous works suggesting that the surface tension in a phase-field model is fixed, we define the surface tension for a curved interface and far-from-equilibrium conditions as the integral of the free energy excess (i.e., above the thermodynamic component of the free energy) across the interface profile in a direction parallel to the composition gradient. Consequently, the nonequilibrium surface tension can be widely different from its equilibrium value under dynamic conditions, while it reduces to its thermodynamic value for a flat interface at local equilibrium. In nonequilibrium conditions, the surface tension changes with time: during mixing, it decreases as the inverse square root of time, while in the linear regime of spinodal decomposition, it increases exponentially to its equilibrium value, as nonlinear effects saturate the exponential growth. In addition, since temperature gradients modify the steepness of the concentration profile in the interfacial region, they induce gradients in the nonequilibrium surface tension, leading to the Marangoni thermocapillary migration of an isolated drop. Similarly, Marangoni stresses are induced in a composition gradient, leading to diffusiophoresis. We also review results on the nonequilibrium surface tension for a wall-bound pendant drop near detachment, which help to explain a discrepancy between our numerically determined static contact angle dependence of the critical Bond number and its sharp-interface counterpart from a static stability analysis of equilibrium shapes after numerical integration of the Young-Laplace equation. Finally, we present new results from phase-field simulations of the motion of an isolated droplet down an incline in gravity, showing that dynamic contact angle hysteresis can be explained in terms of the nonequilibrium surface tension.

Non-local phase field revisited

“…where ϑ = ϑ − T is an O(1) coefficient representing the ratio between the surface tension and its limit value (41), with ϑ → 1 as − T → 0. The coefficient ϑ is very similar to the analogous function plotted in figure 2 of Lamorgese and Mauri [26], representing the surface tension between the coexisting phases of a partially miscible, regular and symmetric binary mixture.…”

Diffusion-Driven Dissolution or Growth of a Liquid Drop Embedded in a Continuous Phase of Another Liquid via Phase-Field Ternary Mixture Model

“…Instead, it treats the interfaces as having nonzero thickness and, as a result, all physical quantities vary continuously across the interfacial volume, a layer of finite (sub-micron-scale) thickness. Another advantage of the diffuse-interface model is that its governing equations are valid for equilibrium and general nonequilibrium conditions alike, while inherent in its formulation is a (nonequilibrium) surface tension which arises as a result of weakly nonlocal effects in the thermodynamic potentials , [i.e., such potentials also depend on the spatial gradients of the order parameter(s)]. This is in contrast to a sharp-interface description wherein both assumptions of local thermodynamic equilibrium and constant surface tension (that are normally incorporated into sharp-interface-based numerical models) are expected to fail in far-from-equilibrium situations.…”

Triphase Separation of a Ternary Symmetric Highly Viscous Mixture

“…This dimensionless group can be introduced as an inverse capillary number [ 43 ], while it can also be interpreted as a Peclet number [ 33 , 34 , 38 , 39 , 40 , 41 ], i.e., the ratio of convective to diffusive mass fluxes in the species balance equations. In fact, in previous works [ 33 , 34 , 44 , 45 , 46 , 47 , 48 , 49 ], we have noted that, in low-viscosity systems, is usually of order , while highly viscous mixtures (e.g., polymer melts and alloys) correspond to a vanishing fluidity coefficient. In the latter case (which is the focus of the work reported herein), the diffuse-interface model describes a diffusive (or antidiffusive) separation process in the absence of flow, and the species balance equations assume the particularly simple form seen earlier [ 26 , 30 ]: …”