Index Experiments Theory Simulation Drops Conclusions
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When we quench a partially miscible mixture from its one-phase to its two-phase region, it phase separates. In this research effort, we investigate the dynamics of phase separation of deeply quenched liquid mixtures, showing both experimentally and theoretically that it is driven by convection, and therefore it is intrinsecally different than that of traditional nucleation-and-growth processes.

Theoretical Model

We have modeled a phase separating liquid mixture using the model H, developed by Kawasaki and by Hohenberg and Halperin. We assume that:

Buoyancy forces are negligible.
Inertial terms are negligible.
Density differences are small.
Viscosity is uniform.
Molecular weight M_W is uniform.

If the flow is assumed to be slow enough to neglect the dynamic terms in the Navier-Stokes equation, conservations of mass and momentum lead to the following system of equation,

where

is the local compositon, v is the average local fluid velocity, rho

is the density of the system, j is the diffusion flux, and F_phi

is a body force. The diffusion flux and the body force are determined through the relations,

where D is the temperature-dependent diffusion coefficient, and

is the generalized chemical potential defined as,

Here g denotes the molar Gibbs free energy,

where g_A and g_B are the molar free energies of the pure species A and B, respectively, at temperature T and pressure P, R is the gas constant, Psi

is the Flory parameter, and a is a characteristic microscopic length. As shown by Van der Waals, a is proportional to the surface tension at equilibrium sigma

, as

is the energy stored in the unit interfacial area at equilibrium,

Now, we restrict our analysis to two-dimensional systems, so that the velocity v can be expressed in terms of a stream function psi

. Since the main mechanism of mass transport at the beginning of the separation process is diffusion, the lengthscale of the process is the microscopic length a. Therefore, using the scaling,

we obtain,

where

The non-dimensional number alpha

is the ratio between convective and diffusive mass fluxes in the convection-diffusion equation and can be interpreted as the Peclet number. At the same time, at the later stages of phase separation, when the system is composed of patches of almost constant compositions, separated by sharp interfaces, this parameter can also be interpreted as the inverse of the capillary number. For systems with very large viscosity, alpha

is small, so that the model describes the diffusion-driven separation process of polymer melts and alloys. For most liquids, however, alpha

is very large, with typical values ranging from 10³ to 10⁵. Therefore, it appears that diffusion is important only at the very beginning of the separation process, in that it creates a non-uniform concentration field. Then, the concentration-gradient-dependent capillary force induces the convective material flux which is the dominant mechanism for mass transport. At no time, however, the diffusive term can be neglected, as it stabilizes the interface and saturates the initial exponential growth.

Index Experiments Theory Simulation Drops Conclusions
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By: Roberto Mauri, Natalia Vladimirova
Last modified: 10 January 1999