(321f) Using Voronoi Tessellations to Measure Coexisting Densities for Molecular Dynamics Simulations

There are a variety of different methods available to simulate the properties of coexisting phases via molecular simulation. One of the advantages of using Molecular Dynamics (MD) simulation to study phase equilibrium is that two phase simulations allow one to study not only the bulk phases but the interface as well. One of the most important bulk properties that one can obtain from two-phase MD simulations is the density of each phase. There are a variety of techniques in the literature by which one can obtain vapor and liquid densities for vapor-liquid equilibrium (VLE)[1-3]. All of these methods require bins, either planar or cubic, which divide the simulation volume into sub-volumes in order to obtain either a 1-D or 3-D distribution of the local density. However, there is a fundamental problem with the use of bins in these simulations, stemming from a desire for small bins, in order to have fine spatial resolution, and large bins, each containing many molecules, in order to obtain reasonable statistics. One element of this problem can be seen by imagining two limits in terms of the number of spatial bins. At one limit, we have one very large bin, which delivers good statistics on the density within that bin, but delivers no spatial resolution distinguishing between the vapor and the liquid phases. At the other limit, we can have an infinite number of bins, in which N of the bins each contain 1 molecule and all the other bins are empty. Again, this situation has fine spatial resolution but provides no information, when examined from the point of view of a density histogram, regarding the density of the coexisting phases. Therefore, choosing bin sizes between these two unacceptable limits requires a careful balance between spatial resolution and statistical reliability, which may not exist. Furthermore, even if a reasonable bin size is identified for a specific thermodynamic state point, it may not apply to another state point and one is still required to introduce arbitrary ?cut-offs? in the density distribution in order to define the densities of each bulk phase. In the literature, these problems have been partially overcome by the use of very large (500,000 particle simulations)[2]. In this work, we present a new method for the determination of coexisting densities in multiple phase MD simulations that is completely free of all bins, both spatial bins and bins in the density histogram, and is also completely free of any arbitrary ?cut-offs?. This method uses voronoi tessellations to obtain the molar volume of every particle in the simulation cell. Voronoi tessellations have been used for centuries to study the solar system and were first proposed by R. Descartes in 17th century[4, 5]. Dirichlet and Voronoi later formally introduced the concept of partitioning space into convex regions[4-8]. To obtain the volume of each particle in the simulation volume by voronoi tessellation one needs to use a periodic system then perform a triangulation in space of the neighbors around a given point. The triangulation is performed by fitting a sphere around three of the neighbors and the molecule of interest. The center of that sphere is a vertex of the voronoi cell if no other neighbors are inside. This analysis is performed for all of the neighbors around the molecule of interest. The resulting vertices are then used to calculate the volume of the resulting voronoi cell, thus giving the density of every molecule in the simulation volume. The summation of the individual volumes equals the total simulation volume to machine precision. By calculating several thermodynamic properties based upon the two phase simulation result and comparing them with the same properties from the single phase simulations at the same density we have a self-consistent method for determining the ?cut-offs? which removes all arbitrary choice from the procedure.

1. Gelb, L.D. and E.A. Muller, Location of phase equilibria by temperature-quench molecular dynamics simulations. Fluid Phase Equilibria, 2002. 203(1-2): p. 1-14. 2. Martinez-Veracoechea, F. and E.A. Muller, Temperature-quench Molecular Dynamics Simulations for Fluid Phase Equilibria. Molecular Simulation, 2005. 31(1): p. 33-43. 3. Pamies, J.C., et al., Coexistence densities of methane and propane by canonical molecular dynamics and Gibbs ensemble Monte Carlo simulations. Molecular Simulation, 2003. 29(6-7): p. 463-470. 4. Aurenhammer, F. and R. Klein, Voronoi Diagrams. Handbook of Computational Geometry, ed. I.J.-R.S.a. J.Urrutia. 2000, North-Holland, Amsterdam. Elsevier Science Publishers. 201-290. 5. Descartes, R., Principia Philosophiae. 1644, Amsterdam: Ludovicus Elzevirius. 6. O'Rourke, J., Computational Geometry in C. 1994, New York: Cambridge University Press. 7. Voronoi, G.M., Nouvelles applications des paramèters continues à la théorie des formes quadratiques. deuxième Memoire: Recherches surles parallélloèdres primitives. J. Reine Angew. Math., 1908. 134: p. 198-287. 8. Dirichlet, G.L., Über die Reduktion der positiven quadratischen Formen mit drieunbestimmten ganzen Zahlen. J. Reine Angew. Math, 1850. 40: p. 209-227.