Evaporation and coarsening dynamics with open boundaries

Phase separation in binary systems is an interesting example of pattern formation in nonequilibrium systems @1–8#. The system, placed in an unstable state, evolves spontaneously towards the equilibrium by generating domains rich in each of the two components. When the volume fraction of one of the components is sufficiently small, the domains of the minority phase coarsen to form circular domains ~droplets! immersed in the bulk of the majority phase. In its late stages, this process can be described by Lifshitz-Slyosov theory @3#, which assumes that the larger droplets grow at the cost of smaller ones, which are thermodynamically less stable due to their higher surface energy. From a theoretical and practical point of view, phase separation processes are usually modeled by time-dependent Ginzburg-Landau equations for the local concentration, with a conserved order parameter. When the system is supposed to be closed, the average of the order parameter over the whole system ~mean density! is conserved, but the existence of open boundaries through which the droplet phase evaporate once converted into the bulk phase leads to a decrease of this quantity. We are interested in the latter situation. We have previously studied the evaporation of periodic arrays of initially equal droplets in two-dimensional systems with open ~absorbing! boundaries @9#. In the present work we extend the study by analyzing the evaporation of a set of initially randomly located identical droplets in a system with open boundaries. One of the simplest models to describe heterogeneous systems showing phase coarsening is formulated through the Cahn-Hilliard equation @4,6#. In dimensionless form it reads