Stability of the one-dimensional kink solution to a general Cahn-Hilliard equation

Linear stability of a horizontal phase boundary subjected to shear motion

We investigate the stability of slowly smearing phase boundary that appears at the contact of two miscible liquids. A hydrodynamic flow is imposed along the boundary. The aim is to find out whether the slow diffusive smearing of a boundary can be overrun by faster mixing. The phase-field approach is used to model the evolution of the binary mixture. The linear stability in respect to 2D perturbations is studied. If the heavier liquid lies above the lighter liquid, the interface is unconditionally unstable due to the Rayleigh-Taylor and Kelvin-Helmholtz instabilities. The imposed flow accelerates the growth of the long-wave modes and suppresses the growth of the short-wave perturbations. Viscosity, diffusivity and capillarity reduce the growth of perturbations. If the heavier liquid underlies the lighter one, the interface can be stable. The stability boundaries are defined by the strength of gravity (density contrast) and the intensity of the imposed flow. Thinner interfaces are usually characterised by larger zones of instability. The thermodynamic instability, identified for the thicker interfaces with the thicknesses greater than the thickness of a thermodynamically equilibrium phase boundary, makes such interfaces unconditionally unstable. The zones of instability are enlarged by diffusive and capillary terms. Viscosity plays its stabilising role.

Linear stability analysis of a horizontal phase boundary separating two miscible liquids

The evolution of small disturbances to a horizontal interface separating two miscible liquids is examined. The aim is to investigate how the interfacial mass transfer affects development of the Rayleigh-Taylor instability and propagation and damping of the gravity-capillary waves. The phase-field approach is employed to model the evolution of a miscible multiphase system. Within this approach, the interface is represented as a transitional layer of small but nonzero thickness. The thermodynamics is defined by the Landau free energy function. Initially, the liquid-liquid binary system is assumed to be out of its thermodynamic equilibrium, and hence, the system undergoes a slow transition to its thermodynamic equilibrium. The linear stability of such a slowly diffusing interface with respect to normal hydro- and thermodynamic perturbations is numerically studied. As a result, we show that the eigenvalue spectra for a sharp immiscible interface can be successfully reproduced for long-wave disturbances, with wavelengths exceeding the interface thickness. We also find that thin interfaces are thermodynamically stable, while thicker interfaces, with the thicknesses exceeding an equilibrium value, are thermodynamically unstable. The thermodynamic instability can make the configuration with a heavier liquid lying underneath unstable. We also find that the interfacial mass transfer introduces additional dissipation, reducing the growth rate of the Rayleigh-Taylor instability and increasing the dissipation of the gravity waves. Moreover, mutual action of diffusive and viscous effects completely suppresses development of the modes with shorter wavelengths.

Phase separation in the presence of spatially periodic forcing

We have investigated theoretically phase separation in the presence of spatially periodic forcing. From the analytic and numeric study of a suitably generalized Cahn-Hilliard equation in one and two dimensions, we find that the forcing amplitude necessary to generate a periodic kink-type state from small random initial conditions depends weakly on wave number. This amplitude is much larger than the one necessary to stabilize the periodic state, i.e., to prevent late-stage coarsening, once it is established. Surprisingly, the destabilizing mode is of long-wave type, which is in contrast to the well-known most rapidly growing coarsening mode in the unforced system. In the Allen-Cahn equation with nonconserved order parameter the relevant modes are always long wave. It appears feasible to observe these effects by imposing a temperature modulation by optical grating which then couples to concentration modulation via the (Ludwig-)Soret effect.