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ABSTRACT: We study the domain geometry during spinodal decomposition of a 50:50 binary mixture in two dimensions. Extending arguments developed to treat nonconserved coarsening, we obtain approximate analytic results for the distribution of domain areas and perimeters during the dynamics. The main approximation is to regard the interfaces separating domains as moving independently. While this is true in the nonconserved case, it is not in the conserved one. Our results can therefore be considered as a "first-order" approximation for the distributions. In contrast to the celebrated Lifshitz-Slyozov-Wagner distribution of structures of the minority phase in the limit of very small concentration, the distribution of domain areas in the 50:50 case does not have a cutoff. Large structures (areas or perimeters) retain the morphology of a percolative or critical initial condition, for quenches from high temperatures or the critical point, respectively. The corresponding distributions are described by a cA-tau tail, where c and tau are exactly known. With increasing time, small structures tend to have a spherical shape with a smooth surface before evaporating by diffusion. In this regime, the number density of domains with area A scales as A1/2 , as in the Lifshitz-Slyozov-Wagner theory. The threshold between the small and large regimes is determined by the characteristic area A approximately t2/3. Finally, we study the relation between perimeters and areas and the distribution of boundary lengths, finding results that are consistent with the ones summarized above. We test our predictions with Monte Carlo simulations of the two-dimensional Ising model.
Physical Review E 09/2009; 80(3 Pt 1):031121. · 2.26 Impact Factor
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ABSTRACT: We study electric field driven deracemization in an achiral liquid crystal through the formation and coarsening of chiral domains. It is proposed that deracemization in this system is a curvature-driven process. We test this prediction using the recently obtained exact result for the distribution of hull-enclosed areas in two-dimensional coarsening with nonconserved scalar order parameter dynamics [J. J. Arenzon et al., Phys. Rev. Lett. 98, 145701 (2007)]. The experimental data are in very good agreement with the theory. We thus demonstrate that deracemization in such bent-core liquid crystals belongs to the Allen-Cahn universality class, and that the exact formula, which gives us the statistics of domain sizes during coarsening, can also be used as a strict test for this dynamic universality class.
Physical Review Letters 12/2008; 101(19):197801. · 7.37 Impact Factor
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ABSTRACT: We study the distribution of domain areas, areas enclosed by domain boundaries ("hulls"), and perimeters for curvature-driven two-dimensional coarsening, employing a combination of exact analysis and numerical studies, for various initial conditions. We show that the number of hulls per unit area, n_{h}(A,t)dA , with enclosed area in the interval (A,A+dA) , is described, for a disordered initial condition, by the scaling function n_{h}(A,t)=2c_{h}(A+lambda_{h}t);{2} , where c_{h}=18pi sqrt[3] approximately 0.023 is a universal constant and lambda_{h} is a material parameter. For a critical initial condition, the same form is obtained, with the same lambda_{h} but with c_{h} replaced by c_{h}2 . For the distribution of domain areas, we argue that the corresponding scaling function has, for random initial conditions, the form n_{d}(A,t)=2c_{d}(lambda_{d}t);{tau'-2}(A+lambda_{d}t);{tau'} , where c_{d} and lambda_{d} are numerically very close to c_{h} and lambda_{h} , respectively, and tau'=18791 approximately 2.055 . For critical initial conditions, one replaces c_{d} by c_{d}2 and the exponent is tau=379187 approximately 2.027 . These results are extended to describe the number density of the length of hulls and domain walls surrounding connected clusters of aligned spins. These predictions are supported by extensive numerical simulations. We also study numerically the geometric properties of the boundaries and areas.
Physical Review E 01/2008; 76(6 Pt 1):061116. · 2.26 Impact Factor
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ABSTRACT: The domain morphology of weakly disordered ferromagnets, quenched from the high-temperature phase to the low-temperature phase, is studied using numerical simulations. We find that the geometrical properties of the coarsening domain structure, e.g., the distributions of hull enclosed areas and domain perimeter lengths, are described by a scaling phenomenology in which the growing domain scale R(t) is the only relevant parameter. Furthermore, the scaling functions have forms identical to those of the corresponding pure system, extending the 'super-universality' property previously noted for the pair correlation function. Comment: 6 pages, 6 figures
11/2007;
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ABSTRACT: We consider the statistics of the areas enclosed by domain boundaries ("hulls") during the curvature-driven coarsening dynamics of a two-dimensional nonconserved scalar field from a disordered initial state. We show that the number of hulls per unit area that enclose an area greater than A has, for large time t, the scaling form Nh(A,t)=2c/(A+lambdat), demonstrating the validity of dynamical scaling in this system, where c=1/8pisquare root 3 is a universal constant. Domain areas (regions of aligned spins) have a similar distribution up to very large values of A/lambdat. Identical forms are obtained for coarsening from a critical initial state, but with c replaced by c/2.
Physical Review Letters 05/2007; 98(14):145701. · 7.37 Impact Factor
