Coarsening in Two-Dimensional Soap Froths and the Large-Q Potts Model
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COARSENING IN TWO-DIMENSIONAL SOAP FROTHS AND THE LARGE-Q POTTS MODEL GARY S. GREST*, JAMES A. GLAZIER#t, MICHAEL P. ANDERSON*, HOLM+ AND DAVID J. SROLOVITZ+ *Corporate Research Science Laboratory, Exxon Research
ELIZABETH
and
A.
Engineering
Company, Annandale, NJ 08801 #Research Institute of Electrical Communications, Tohoku University, Sendai 980, Japan +Department of Material Science and Engineering, The University of Michigan, Ann Arbor, Michigan 48109 ABSTRACT A detailed comparison between the experimental evolution of a twodimensional soap froth and the large Q state Potts model is presented. The pattern evolution starting from identical initial conditions will be compared as well as a variety of distribution functions and correlations of the two systems. Simulations on different lattices show that the discrete lattice of the Potts model causes deviations from universal domain growth by weakening the vertex angle boundary conditions that form the basis of von Neumann's law. We show that the anisotropy inherent in a discrete lattice simulation, which masks the underlying 'universal' grain growth, can be overcome by increasing the range of the interaction between spins or increasing the temperature. Excellent overall agreement between the kinetics, topological distributions and domain size distributions between the low lattice anisotropy Potts-model simulations and the soap froth suggests that the Potts model is useful for studying domain growth in a wide variety of physical systems. Introduction There are many experimental systems which exhibit grain growth or coarsening in time, including metallic and ceramic recrystallization, magnetic materials, lipid monolayers, biological aggregates and soap froths [1-3]. There are a similar variety of models for coarsening including simple mean field theories which look only at distributions [4,5], mean field theories which include topology [6-8], "exact" models which calculate the motions of boundaries or vertices [9-12], and the Potts model which takes a microscopic approach to modeling [13-17]. In all such systems surface energy driven diffusion leads to the motion of curved boundary walls causing certain grains to grow while others shrink and disappear. The result is a gradual increase in the overall length scale of the grains. This basic dynamics is influenced by the geometrical constraint that vertices are three-fold connected to produce a family of typical patterns of coordination number three. Independent of the initial configuration of domains or bubbles, these systems gradually become disordered with time-invariant distributions of the number of sides per bubble and bubble areas (a scaling state). Besides the intrinsic interest of the transition from order to disorder, coarsening has many technological applications in metallurgy, for example in the design of materials with particular mechanical properties. The coarsening properties of metals have therefore been widely studied [2]. Unfortunately, in many cases, secondary effects such as impurity segregation
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