Coarsening of random interfaces in the spinodal decomposition of a binary fluid

The growth of random interfaces during the late stage spinodal decomposition for a near symmetric quench of a binary fluid is analyzed. Inertial effects are neglected, and the motion of the interface is determined by a balance between the surface tension, which tends to reduce the curvature, and the viscous stresses in the fluid. The interface is described by an ‘‘area distribution function’’ A(K,t), defined so that A(K,t)dKdx is the area of the interface with curvature in the interval dK about K in the volume dx at time t. Here, $K=(K^2_1+K^2_2 )^{1/2}$ is the magnitude of the curvature, and K1 and K2 are the principal curvatures. There is a change in the area distribution function due to a change in the curvature, and due to the tangential compression of the interface. Phenomenological relations for the change in curvature and surface area are obtained using the assumption that the only length scale affecting the dynamics of the interface at a point is the radius of curvature at that point. These relations are inserted in the conservation equation for the interface, and a similarity solution is obtained for the area distribution function. This solution indicates that the area of the interface decreases proportional to $t^{-1}$ in the late stages of coarsening, and the mean curvature also decreases proportional to $t^{-1}$. The effect of the motion of the interface on the interfacial concentration profile and interfacial energy is analyzed using a perturbation analysis. The diffusion equation for the concentration in the interfacial region contains an additional source term due to the convective transport of material caused by the motion of the interface, and this causes a correction to the equilibrium concentration profile of the interface. The excess interfacial energy due to the nonequilibrium motion of the interface is calculated using the Cahn–Hilliard square gradient free energy for a near-critical quench. It is found that the variation in the concentration causes an increase in the interfacial energy which is proportional to the curvature K of the interface