A Closed Form Simulation of a Coarsening Analog System Vaughan Voller, University of Minnesota a·nal·o·gy Similarity in some respects between things that are otherwise dissimilar. E.g., the coarsening of a froth and grain growth in material microstructure MORRIS COHEN—grains in a thin film Two Uses of analogy 1.May provide physical insight into your process of interest 2.Allows for the development and testing of cross-cutting modeling technologies

N 6 A Fundamental Coarsening Law: The Von-Neumann-Mullins Growth law For an individual isolated 2-D bubble A balance of pressure and surface-tension forces shows that n-number of sides, D- diffusivity a n area of bubble with n -sides But in array of bubbles topological change will create new n < 6 bubbles Propose modified array version of von-Neumann-Mullins Growth law Rate of change of Average n-sided bubble area Rate of change of average area

Experimental Verification of array form of von-Neumann-Mullins Growth law coarsening soap-froth structure formed by colloidal particles. Mejía-Rosales, et al Physica A (2000).

What might we want to know 1. The rate of change of the average area (t) ~ 1/N(t), N(t) number of bubbles 2. The rate of change of the area of the average n-sides bubble

A simple conceptual model for soap froth coarsening Model each bubble in 2-D domain as a point undergoing a random walk When two points approach within A distance “d” they combine into one point In this way the bubbles will reduce over time Can visualize the bubble array at a point In time by creating a Voronoi diagram around the remaining “bubble points” Similar to the colloidal aggregation model of Moncho-Jordá, et al Physica A (2000) Could develop a direct simulation but prefer to develop a “conceptual” solution

A conceptual solution of random walk model: Basic Let—assuming multiple realizations—the average time for the destructive meeting of two particles to be “diffusivity” Domain area A With 3-particels it is reasonable to project that-- since there will possible meetings – the average time-from multiple realizations—will be With 4-particels With k particles If this holds for any number of particles k—the mean time to go from an initial particle (bubble) count of N 0 to N particles is Matches long-term bubble coarsening dynamics derived from Dim. Anal.

Is meeting time valid if bubble count k is large A conceptual solution of random walk model: Extension With many of bubbles (k>>1) the distance between will become relatively uniform—i.e., the variance about the mean distance will be small The mean meeting time ~ d mean And the time to go from N 0 to N (>>1) may be better given by “velocity”

Compare with experiments of Glazier et al Phys Rev A 1987 A simple Linear Combination Of the time scales

Compare with experiments of Glazier et al Phys Rev A 1987 A three parameter Fit Note in long time limit the average area

2. The rate of change of the area of the average n-sides bubble Start with the Array version of von-Neumann-Mullins Growth law Integrate to Where Set So that Choice of justified by noting that in long time limit —as full disorder is reached the Lewis Law Is recovered—consistent with theoretical Result of Rivier

Value measured in experiment D= Glazier et al Phys Rev A 1987

Other work At a point in the bubble coarsening model Visualization of the bubble froth can be obtained using a Voronoi Diagram How do the statistics of this Visualization compare with real bubble froths =0.222 Variance of bubble sides

Summary 1.Based on a conceptual random walk model the mechanisms for an early and late time-scale for froth coarsening have been hypothesized. A simple linear combination provides excellent agreement with experiments Comparison with more cases is needed Key features of soap froth coarsening can be recovered with simple closed form models

2. From the Proposed Von Neumann-Mullins Modification Best fit value consistent with independently measured D A more general version of the Lewis law since slope depends on time coarsening dependent equation for relationship between avaerage bubble area With different sides

3. Voronoi Visualization exhibits features of Coarsening systems BUT NOT with the same coefficients 4. The Open Question—How is this related to Metals ? MORRIS COHEN—grains in a thin film