Scaling and multiscaling in the ordering kinetics of a conserved order parameter

Radiation-Induced Patterning at the Nanometric Scale: A Phase Field Approach

The phase field approach was developed in the last 20 years to handle radiation damage in materials. This approach bridges the gap between atomistic simulations extensively used to model first step of radiation damage at short time and continuum approach at large time. The main advantage of such an approach lies in its ability to compute not only the microstructure at the nanometric scale but also to calculate generalized susceptibilities such as elastic constants under irradiation. After a brief description of the rate theory, used to model the microstructure induced by irradiation, we briefly discuss the foundation of the phase field method, highlighting not only its advantages, but also its limitations in comparison with the rate theory. We conclude this presentation by proposing future orientations for computing the microstructure in irradiated materials.

Coarsening dynamics of phase-separating systems

When a system such as a binary liquid is cooled rapidly from a homogeneous phase into a two-phase region, domains of the two equilibrium phases form and grow ('coarsen') with time. In the absence of an external drive, such as gravity or an imposed shear flow, a dynamical-scaling regime emerges in which the domain morphology is statistically self-similar at different times, up to an overall length-scale (coarsening scale) that grows with time. In the first part of the paper, the scaling phenomenology will be reviewed and the time-dependence of the coarsening scale will be discussed in the context of a number of different physical systems and scaling regimes. In the second part, the influence of an external drive, in particular a shear flow, will be addressed and recent developments reviewed. Interesting open questions include the late-time behaviour under shear and whether the coarsening continues indefinitely or is ultimately arrested by the shear flow.

Structure and rheology of binary mixtures in shear flow

Results are presented for the phase separation process of a binary mixture subject to a uniform shear flow quenched from a disordered to a homogeneous ordered phase. The kinetics of the process is described in the context of the time-dependent Ginzburg-Landau equation with an external velocity term. The large-n approximation is used to study the evolution of the model in the presence of a stationary flow and in the case of an oscillating shear. For stationary flow we show that the structure factor obeys a generalized dynamical scaling. The domains grow with different typical length scales Rx and R( perpendicular), respectively, in the flow direction and perpendicularly to it. In the scaling regime R( perpendicular) approximately t(alpha( perpendicular)) and Rx approximately gammat(alpha(x)) (with logarithmic corrections), gamma being the shear rate, with alpha(x)=5/4 and alpha( perpendicular)=1/4. The excess viscosity Deltaeta after reaching a maximum relaxes to zero as gamma(-2)t(-3/2). Deltaeta and other observables exhibit logarithmic-time periodic oscillations which can be interpreted as due to a growth mechanism where stretching and breakup of domains occur cyclically. In the case of an oscillating shear a crossover phenomenon is observed: Initially the evolution is characterized by the same growth exponents as for a stationary flow. For longer times the phase-separating structure cannot align with the oscillating drift and a different regime is entered with an isotropic growth and the same exponents as in the case without shear.

Effect of shear on phase-ordering dynamics with order-parameter-dependent mobility: the large-n limit

The effect of shear on the ordering kinetics of a conserved order-parameter system with O(n) symmetry and order-parameter-dependent mobility Gamma(straight phi-->) approximately (1-straight phi-->(2)/n)(alpha) is studied analytically within the large-n limit. In the late stage, the structure factor becomes anisotropic and exhibits multiscaling behavior with characteristic length scales (t(2alpha+5)/ln t)(1/2(alpha+2)) in the flow direction and (t/ln t)(1/2(alpha+2)) in directions perpendicular to the flow. As in the alpha=0 case, the structure factor in the shear-flow plane has two parallel ridges.

Corrections to scaling in the phase-ordering dynamics of a vector order parameter

Corrections to scaling, associated with deviations of the order parameter from the scaling morphology in the initial state, are studied for systems with O(n) symmetry at zero temperature in phase-ordering kinetics. Including corrections to scaling, the equal time pair correlation function has the form C(r,t)=f0(r/L)+L(-omega)f1(r/L)+., where L is the coarsening length scale. The correction-to-scaling exponent omega and the correction-to-scaling function f1(x) are calculated for both nonconserved and conserved order parameter systems using the approximate Gaussian closure theory of Mazenko. In general omega is a nontrivial exponent which depends on both the dimensionality d of the system and the number of components n of the order parameter. Corrections to scaling are also calculated for the nonconserved one-dimensional XY model, where an exact solution is possible.

Dynamical scaling: the two-dimensional XY model following a quench

To sensitively test scaling in the two-dimensional XY model quenched from high temperatures into the ordered phase, we study the difference between measured correlations and the (scaling) results of a Gaussian-closure approximation. We also directly compare various length scales. All of our results are consistent with dynamical scaling and an asymptotic growth law L approximately (t/ln[t/t(0)])(1/2), though with a time scale t(0) that depends on the length scale in question. We then reconstruct correlations from the minimal-energy configuration consistent with the vortex positions, and find them significantly different from the "natural" correlations - though both scale with L. This indicates that both topological (vortex) and nontopological "spin-wave" contributions to correlations are relevant arbitrarily late after the quench. We also present a consistent definition of dynamical scaling applicable more generally, and emphasize how to generalize our approach to other quenched systems where dynamical scaling is in question. Our approach directly applies to planar liquid-crystal systems.