Scaling functions in phase-ordering dynamics: A comparison of theory and simulations

Phase-ordering kinetics in the Allen-Cahn (Model A) class: Universal aspects elucidated by electrically induced transition in liquid crystals

The two-dimensional (2D) Ising model is the statistical physics textbook example for phase transitions and their kinetics. Quenched through the Curie point with Glauber rates, the late-time description of the ferromagnetic domain coarsening finds its place at the scalar sector of the Allen-Cahn (or Model A) class, which encompasses phase-ordering kinetics endowed with a nonconserved order parameter. Resisting exact results sought for theoreticians since Lifshitz's first account in 1962, the central quantities of 2D Model A-most scaling exponents and correlation functions-remain known up to approximate theories whose disparate outcomes urge experimental assessment. Here we perform such assessment based on a comprehensive study of the coarsening of 2D twisted nematic liquid crystals whose kinetics is induced by a superfast electrical switching from a spatiotemporally chaotic (disordered) state to a two-phase concurrent, equilibrium one. Tracking the dynamics via optical microscopy, we first show the sharp evidence of well-established Model A aspects, such as the dynamic exponent z=2 and the dynamic scaling hypothesis, to then move forward. We confirm the Bray-Humayun theory for Porod's regime describing intradomain length scales of the two-point spatial correlators and show that their nontrivial decay beyond the Porod's scale can be captured in a free-from-parameter fashion by Gaussian theories, namely the Ohta-Jasnow-Kawasaki (OJK) and Mazenko theories. Regarding time-related statistics, we corroborate the aging hypothesis in Model A systems, which includes the collapse of two-time correlators into a master curve whose format is, actually, best accounted for by a solution of the local scaling invariance theory: the same solution that fits the 2D nonconserved Ising model correlator along with the Fisher-Huse conjecture. We also suggest the true value for the local persistence exponent in Model A class, in disfavor of the exact outcome for the diffusion and OJK equations. Finally, we observe a fractal morphology for persistence clusters and extract their universal dimension. Given its accuracy and possibilities, this experimental setup may work as a prototype to address further universality issues in the realm of nonequilibrium systems.

Direct test of the Gaussian auxiliary field ansatz in nonconserved order parameter phase ordering dynamics

The assumption that the local order parameter is related to an underlying spatially smooth auxiliary field, u(r[over ⃗],t), is a common feature in theoretical approaches to non-conserved order parameter phase separation dynamics. In particular, the ansatz that u(r[over ⃗],t) is a Gaussian random field leads to predictions for the decay of the autocorrelation function which are consistent with observations, but distinct from predictions using alternative theoretical approaches. In this paper, the auxiliary field is obtained directly from simulations of the time-dependent Ginzburg-Landau equation in two and three dimensions. The results show that u(r[over ⃗],t) is equivalent to the distance to the nearest interface. In two dimensions, the probability distribution, P(u), is well approximated as Gaussian except for small values of u/L(t), where L(t) is the characteristic length-scale of the patterns. The behavior of P(u) in three dimensions is more complicated; the non-Gaussian region for small u/L(t) is much larger than that in two dimensions but the tails of P(u) begin to approach a Gaussian form at intermediate times. However, at later times, the tails of the probability distribution appear to decay faster than a Gaussian distribution.

Curvature-driven coarsening in the two-dimensional Potts model

We study the geometric properties of polymixtures after a sudden quench in temperature. We mimic these systems with the q -states Potts model on a square lattice with and without weak quenched disorder, and their evolution with Monte Carlo simulations with nonconserved order parameter. We analyze the distribution of hull-enclosed areas for different initial conditions and compare our results with recent exact and numerical findings for q=2 (Ising) case. Our results demonstrate the memory of the presence or absence of long-range correlations in the initial state during the coarsening regime and exhibit superuniversality properties.

Scaling anomalies in the coarsening dynamics of fractal viscous fingering patterns

We analyze a recent experiment of Sharon et al. (2003) on the coarsening, due to surface tension, of fractal viscous fingering patterns (FVFPs) grown in a radial Hele-Shaw cell. We argue that an unforced Hele-Shaw model, a natural model for that experiment, belongs to the same universality class as model B of phase ordering. Two series of numerical simulations with model B are performed, with the FVFPs grown in the experiment and with diffusion limited aggregates as the initial conditions. We observed Lifshitz-Slyozov scaling t(1/3) at intermediate distances and slow convergence to this scaling at small distances. Dynamic scale invariance breaks down at large distances.

Phase ordering with a global conservation law: Ostwald ripening and coalescence

Globally conserved phase ordering dynamics is investigated in systems with short range correlations at t=0. A Ginzburg-Landau equation with a global conservation law is employed as the phase field model. The conditions are found under which the sharp-interface limit of this equation is reducible to the area-preserving motion by curvature. Numerical simulations show that, for both critical and off-critical quench, the equal-time pair correlation function exhibits dynamic scaling, and the characteristic coarsening length obeys l(t) approximately t(1/2). For the critical quench, our results are in excellent agreement with earlier results. For off-critical quench (Ostwald ripening) we investigate the dynamics of the size distribution function of the minority phase domains. The simulations show that, at large times, this distribution function has a self-similar form with growth exponent 1/2. The scaled distribution, however, strongly differs from the classical Wagner distribution. We attribute this difference to coalescence of domains. A theory of Ostwald ripening is developed that takes into account binary coalescence events. The theoretical scaled distribution function agrees well with that obtained in the simulations.

Normal scaling in globally conserved interface-controlled coarsening of fractal clusters

We find that globally conserved interface-controlled coarsening of diffusion-limited aggregates exhibits dynamic scale invariance (DSI) and normal scaling. This is demonstrated by a numerical solution of the Ginzburg-Landau equation with a global conservation law. The general sharp-interface limit of this equation is introduced and reduced to volume preserving motion by mean curvature. A simple example of globally conserved interface-controlled coarsening system: the sublimation/deposition dynamics of a solid and its vapor in a small closed vessel, is presented in detail. The results of the numerical simulations show that the scaled form of the correlation function has a power-law tail accommodating the fractal initial condition. The coarsening length exhibits normal dynamic scaling. A decrease of the cluster radius with time, predicted by DSI, is observed. The difference between global and local conservation is discussed.