Phase segregation via Vlasov-Boltzmann particle dynamics

In order to better understand and model the phase segregation of binary fluids we opted for a mesoscopic description that proves to be simplifying both conceptually and computationally. The system that we studied is a mixture of two kinds of particles. All particles interact with each other through strong short-range interactions modeled by hard spheres with the same mass and diameter. There is also a smooth long-range repulsion between particles of different kinds. At low overall densities and weak enough repulsion the natural dynamical description for this system is given in terms of two coupled, energy and momentum conserving Vlasov- Boltzmann equations, making it what we call a dynamical mean-field model. The computational scheme that we used is a combination of direct sim- ulation Monte Carlo (DSMC) and particle-in-the-cell (PIC) evolution, that inherits the efficiency and robustness of these two algorithms. The DSMC is a stochastic algorithm due to Bird that consistently incorporates the as- sumptions behind the Boltzmann equation into the particle dynamics. The method is essentially the following: the physical space is divided into a net- work of cells containing typically tens of particles and the free flow of the particles over a small time interval {Delta}t is followed by representative collisions among pairs of particles sharing the same cell. The typical linear dimension of a cell is a fraction of the mean free path between collisions. The PIC method for integrating the equations of motion was first used to deal with the l/r potential in plasma physics. It takes advantage of the simple form of the Vlasov potential, which is a product in Fourier space, by calculating the densities on a grid through some weighting, then the potentials and forces on the same grid, and finally interpolating the forces at the position of each particle. These two methods can be naturally brought together by replacing the free flow of the DSMC procedure by motion in the background, Vlasov potential. If the repulsive potential between the two species is sufficiently weak and long ranged (so no new inter-particle correlations are introduced), such an algorithm contains the essential ingredients of the Vlasov-Boltzmann kinetics. The structure of the interface separating the two phases coexisting inside the miscibility gap is related to the dominating coarsening mechanism. We compared the equilibrium interface profiles that result directly from the Vlasov-Boltzmann equations with the profiles obtained in simulations and found very good agreement. Our model and computational scheme provide a convenient framework for the study of another important problem, the influence of phase segrega- tion on an initially prescribed hydrodynamical flow. The approach to phase segregation kinetics described here takes advan- tage of an important analytical tool available in nonequilibrium physics, the Boltzmann equation, and has a computational simplicity that should make it useful for other interesting applications