A Closed Form Simulation of a Coarsening Analog System Vaughan Voller, University of Minnesota

1. A Closed Form Simulation of a Coarsening Analog System Vaughan Voller, University of Minnesota a·nal·o·gy http://www.thefreedictionary.com/ 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 • May provide physical insight into your process of interest • Allows for the development and testing of cross-cutting modeling technologies

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

3. 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 276 30 (2000).

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

5. When two points approach within A distance “d” they combine into one point A simple conceptual model for soap froth coarsening Model each bubble in 2-D domain as a point undergoing a random walk 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 282 50 (2000) Could develop a direct simulation but prefer to develop a “conceptual” solution

6. 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 A conceptual solution of random walk model: Basic Let—assuming multiple realizations—the average time for the destructive meeting of two particles to be Domain area A “diffusivity” If this holds for any number of particles k—the mean time to go from an initial particle (bubble) count of N0 to N particles is Matches long-term bubble coarsening dynamics derived from Dim. Anal.

7. A conceptual solution of random walk model: Extension Is meeting time valid if bubble count k is large 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 ~ dmean And the time to go from N0 to N (>>1) may be better given by “velocity”

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

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

10. Start with the Array version of von-Neumann-Mullins Growth law Integrate to Set So that Where 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 2. The rate of change of the area of the average n-sides bubble <an>

11. Value measured in experiment D= 2.742 Glazier et al Phys Rev A 1987

12. Variance of bubble sides 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

13. Summary Key features of soap froth coarsening can be recovered with simple closed form models • 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

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

15. 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