Dissolution or Growth of a Liquid Drop via Phase-Field Ternary Mixture Model Based on the Non-Random, Two-Liquid Equation

Abstract

We simulate the diffusion-driven dissolution or growth of a single-component liquid drop embedded in a continuous phase of a binary liquid. Our theoretical approach follows a diffuse-interface model of partially miscible ternary liquid mixtures that incorporates the non-random, two-liquid (NRTL) equation as a submodel for the enthalpic (so-called excess) component of the Gibbs energy of mixing, while its nonlocal part is represented based on a square-gradient (Cahn-Hilliard-type modeling) assumption. The governing equations for this phase-field ternary mixture model are simulated in 2D, showing that, for a single-component drop embedded in a continuous phase of a binary liquid (which is highly miscible with either one component of the continuous phase but is essentially immiscible with the other), the size of the drop can either shrink to zero or reach a stationary value, depending on whether the global composition of the mixture is within the one-phase region or the unstable range of the phase diagram.

Triphase Separation of a Ternary Symmetric Highly Viscous Mixture

“…Below, we provide a brief summary of the governing equations for a diffuse-interface description of partially miscible ternary liquid mixtures, given that a detailed derivation of those equations has been presented elsewhere [ 2 , 26 , 30 ]. Consider a regular ternary mixture, whose component liquids have the same molar density, .…”

“…Consider a regular ternary mixture, whose component liquids have the same molar density, . This mixture can be modeled within a diffuse-interface description by assuming that its free energy is the sum of a thermodynamic part and a nonlocal contribution [ 26 , 30 , 31 , 32 , 33 , 34 ], i.e., where g is the (dimensionless) thermodynamic (i.e., coarse-grained) bulk free energy density, T the absolute temperature, V the volume, R the universal gas constant, while a is a characteristic length. Note that, since Equation ( 1 ) represents a coarse-grained expression of the free energy, the characteristic length is in no way equal to the actual interfacial thickness (still a is representative, on the mesoscale, of the typical interface thickness at local equilibrium).…”

“…[Obviously, since we are considering an ideally, perfectly symmetric ternary mixture, the Margules parameter for the ( ) component pair must be equal to those for the remaining pairs. Note that, for strongly nonideal systems (particularly for partially immiscible components when there is no symmetry among the pure species), we have shown a derivation of the species balance equations that incorporates NRTL as an excess Gibbs energy submodel [ 26 ], in place of the standard one-parameter Margules correlation employed in Equation ( 2 ).] When the mixture finds itself in a state of nonequilibrium, it evolves according to the equations governing dynamic processes, expressing standard conservation principles [ 36 ].…”

“…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 ]: …”

“…In contrast, in some cases, more complex excess Gibbs energy submodels are required for modeling nonideal mixtures having a strong asymmetry among the component species (e.g., when two or more components are partially immiscible). That was specifically the case with the benzene-acetonitrile-water mixture considered in our previous work [ 26 ], where the phase-field ternary mixture model presented herein was generalized to incorporate the nonrandom, two-liquid (NRTL) equation [ 27 , 28 , 29 ] as an excess Gibbs energy submodel. Finally, in Section 3 , using different initial concentration field perturbations, we discuss simulation results of a symmetric ternary mixture in terms of the integral scale of the pair-correlation function and the separation depth for each component.…”