Scaling in late stage spinodal decomposition with quenched disorder

Abstract

We study the late stages of spinodal decomposition in a Ginzburg-Landau mean field model with quenched disorder. Random spatial dependence in the coupling constants is introduced to model the quenched disorder. The effect of the disorder on the scaling of the structure factor and on the domain growth is investigated in both the zero temperature limit and at finite temperature. In particular, we find that at zero temperature the domain size, R(t), scales with the amplitude, A, of the quenched disorder as R(t) = A −β f (t/A −γ ) with β ≃ 1.0 and γ ≃ 3.0 in two dimensions. We show that β/γ = α, where α is the Lifshitz-Slyosov exponent. At finite temperature, this simple scaling is not observed and we suggest that the scaling also depends on temperature and A. We discuss these results in the context of Monte Carlo and cell dynamical models for phase separation in systems with quenched disorder, and propose that in a Monte Carlo simulation the concentration of impurities, c, is related to A by A ∼ c 1/d .

Lattice-gas simulations of domain growth, saturation, and self-assembly in immiscible fluidsand microemulsions

“…(The final point, corresponding to a total reduced surfactant density of 0.09, lies below this line probably because the simulation had not fully equilibrated.) It is worth noting that the result shown in Fig. 14 is also consistent with the relationship found between the final domain size and the amplitude of disorder in systems with quenched impurities, as determined by Gyure et al 20 . 15, which is a plot of domain size versus ln t for the case of surfactant density 0.14, indicates that in this case the slow domain growth may go as (ln t) θ with θ ≃ 0.5 over the dominant timescale of the simulations (beyond the initial transient region) and before the domain size saturates completely.…”

“…9) so that we are able to observe any algebraic exponent for the ln t growth. If the slow growth in these systems can indeed be related in some way to that in systems with quenched impurities 20 , then we would expect to find some power θ for the growth function (ln t) θ , which decreases as the amount of surfactant in the system is further increased. In this initial case we find a value θ ≃ 3.0 for the timescale of the simulation beyond the very early-time transient behavior.…”

“…BMB thanks Gene Stanley, Steve Harrington, and Francis Starr for pointing out the existence and relevance of one of the references 20 , and for helpful discussions. ANE is grateful to EPSRC and Schlumberger Cambridge Research for funding his research.…”

Coarsening and percolation in a disordered ferromagnet

“…(1) is reported in some experiments [16,17], the asymptotic growth-law R(t) is much slower than in a clean system [16][17][18][19][20]. Similar conclusions are reached in the numerical studies of models of disordered ferromagnets, such as the kinetic Ising model in the presence of random external fields [21][22][23][24][25][26][27][28][29][30][31][32], with varying coupling constants [32][33][34][35][36][37][38][39], or in the presence of dilution [33,38,[40][41][42][43][44][45][46][47]49]. A natural question is, then, whether the percolation effects observed in bi-dimensional clean systems show up also in the disordered ones.…”

Lattice-Boltzmann model for interacting amphiphilic fluids

“…8 this form fits our results extremely well across the full time scale of the simulations and for all surfactant concentrations considered. We also investigated a fit of the logarithmic form R(t) = a + b(ln t) c , which describes the phase-separation of binary alloys in the presence of impurities [37]. Obviously, this form is unable to describe the late time saturation of the domain size and we found that the root-mean-square errors using this form to describe the early times of domain formation are also an order of magnitude larger than that of the stretched exponential form.…”