Diffusiophoresis of two-dimensional liquid droplets in a phase-separating system.
The motion of phase-separating liquid drops was simulated in two dimensions following the model H, where convection and diffusion are coupled via a body force, expressing the tendency of demixing systems to minimize their free energy. This driving force depends on the capillary number, i.e., the ratio of viscous to thermal forces, which in a typical case is of order 10(-4), inducing a convective material flux much larger than its diffusive counterpart. Three problems were considered. In the first, we studied the motion of a single drop immersed in a continuum field with constant concentration gradient, finding that the drop speed is proportional to the concentration gradient and inversely proportional to the capillary number. In the second problem, we found that the motion of a single drop immersed in a homogeneous concentration field depends on the difference (Delta(phi))(0) between the initial concentration of the continuum phase and its equilibrium value. In fact, when (Delta(phi))(0)<0, the drop shrinks without moving, while when (Delta(phi))(0)>0, the drop consumes material from the surrounding field and moves randomly, propelled by the induced capillary driving force. During its movement, the drop grows linearly in time, with a growth rate proportional to the ratio between molecular diffusivity and interface thickness. In addition, during its random motion, the drop mean square displacement grows linearly with time, with an effective diffusion coefficient which is of the same magnitude as the molecular diffusivity. The predicted drop growth rate and mean velocity are in good agreement with experimental observations. Finally, the motion of two drops is studied, showing that the capillary forces induce a mutual attraction between the two drops. When (Delta(phi))(0)<0, the attractive force is unchallenged, thus leading always to coalescence, while when (Delta(phi))(0)>0 a screening effect arises which may keep the two drops apart from each other.