Exact Results for Two-Dimensional Coarsening

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• Prof. Alan Bray, University of Manchester
• Thursday 21 February 2008, 14:15-15:45
• TCM Seminar Room, Cavendish Laboratory.

If you have a question about this talk, please contact Joe Bhaseen.

A system which is rapidly cooled from a homogeneous high-temperature phase into a two-phase region forms domains of the two low-temperature phases, with a characteristic length scale that grows (the domain structure “coarsens”) with time. The evolution of the domain structure is well described by a dynamical scaling hypothesis, according to which the domain morphology is statistically time-independent apart from one overall length scale (the “domain scale”) that grows with time.

In this talk, exact results pertaining to the domain morphology are obtained for the case where a system described by a two-dimensional, non-conserved scalar field (e.g. a twisted nematic liquid crystal), starting from disordered initial condition appropriate to the state immediately after the quench, evolves according to the curvature-driven motion of its domain walls. In particular, the number density of “hulls” (the outer boundaries of domains) with area A, at time t after the quench, is given by

n(A,t) = 2c/(A+lambda t)²,

where c = 1/(8 pi sqrt(3)) = 0.023 is a universal constant associated with the initial condition. This result validates (at least one aspect of) the scaling hypothesis. Exploiting the smallness of c, approximate results are obtained for the number density of domains of area A at time t. The result has, for large t, the scaling form

n_d(A,t) = 2 c_d (lambda_d t)^tau-2/ (A + lambda_d t)^tau,

where c_d = c + O(c²), lambda_d = lambda(1+ O(c¹)) and tau = 187/91. The corresponding results for the case where the initial state is the critical state will also be presented.

This talk is part of the TCM Blackboard Series series.

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