Simulating fluctuating mesoscopic dynamics using three dimensional Voronoi cells

Materials such as micro-phase separated liquids, colloidal suspensions and polymeric materials have structure on a mesoscopic length scale. The relevant length scales are well beyond the microscopic level. Therefore the materials are, locally, well characterized by a thermodynamic state and by the macroscopic transport properties such as the viscosity. Thermal fluctuations are, however, still of significant importance for the dynamics of the structure of such a liquid. The presented method describes blobs of material by dividing space in discrete volumes by means of a three-dimensional Voronoi tessellation. Each discrete volume obeys local thermal equilibrium. Because the volume of a Voronoi cell is well defined it is easy to impose a local equation of state. The time evolution of the volumes obeys conservation of energy and momentum, and is thermodynamically consistent. Furthermore thermal fluctuating and dissipative forces are imposed that obey fluctuation-dissipation. This technique has significant benefits compared to other mesoscopic techniques such as DPD. In these other techniques fluctuations obey fluctuation-dissipation relations, but coarse-grained degrees of freedom are not modeled by means of a local entropy (or free energy). Another drawback of DPD is that transport coefficients such as the viscosity are not primary input variables but have to be determined by means of a "numerical experiment". A major advantage of the new method is that locally the equation of state of a cell can be imposed. Therefore any kind of matter can be modeled. Also complicated systems such as liquid-vapor mixture fall into the range of possibilities. Transport coefficients such as viscosity can be given any value. The dynamics obeys 'fluctuating hydrodynamics'. Fluxes based on chemical potential differences can be introduced such that mixtures can be studied. In the case of demixing initially diffuse boundaries that are a few cells in width will evolve. When the width becomes much smaller than the cell size sharp interfaces are formed. These interfaces are naturally localized at the boundaries between cells. In this case the boundaries between different species can be assigned thermodynamic properties such as an interfacial energy. In this talk I will discuss some numerical details that make the three dimensional simulation of by means of Voronoi cells possible. Next, I will discuss a few modeling issues and will finish with the presentation of interesting results of three dimensional simulations.