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Mogadalai Gururajan's picture

Going beyond 2D Neumann-Mullins (or, what is popularly known as, solving the beer froth structure)


The blogosphere is abuzz with the latest report of the generalisation of the von Neumann-Mullins grain growth relation to 3 (and N) dimensions by MacPherson and Srolovitz (As an interesting aside, almost all the reports say mathematical structure of beer foam structure resolved, or words to that effect --hence, I also decided to join the bandwagon on that one). I heard Prof. Srolovitz describe the work in a seminar nearly six months ago. Based on my notes of the talk, I would like the explain their work in this post. Curvature in the following refers to mean curvature (and not Gaussian).

Dynamics of wrinkle growth and coarsening in stressed thin films

Rui Huang and Se Hyuk Im, Physical Review E 74, 026214 (2006).

A stressed thin film on a soft substrate can develop complex wrinkle patterns. The onset of wrinkling and initial growth is well described by a linear perturbation analysis, and the equilibrium wrinkles can be analyzed using an energy approach. In between, the wrinkle pattern undergoes a coarsening process with a peculiar dynamics. By using a proper scaling and two-dimensional numerical simulations, this paper develops a quantitative understanding of the wrinkling dynamics from initial growth through coarsening till equilibrium. It is found that, during the initial growth, a stress-dependent wavelength is selected and the wrinkle amplitude grows exponentially over time. During coarsening, both the wrinkle wavelength and amplitude increases, following a simple scaling law under uniaxial compression. Slightly different dynamics is observed under equi-biaxial stresses, which starts with a faster coarsening rate before asymptotically approaching the same scaling under uniaxial stresses. At equilibrium, a parallel stripe pattern is obtained under uniaxial stresses and a labyrinth pattern under equi-biaxial stresses. Both have the same wavelength, independent of the initial stress. On the other hand, the wrinkle amplitude depends on the initial stress state, which is higher under an equi-biaxial stress than that under a uniaxial stress of the same magnitude.

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