Phase-ordering Kinetics Research Articles

Domain Growth Scaling at the Isotropic-to-Cholesteric Liquid Crystal Transition

The phase-ordering kinetics for the growth of the ordered cholesteric liquid crystalline phase from the disordered isotropic melt is investigated at isothermal conditions after a temperature quench. The growth of a characteristic length, here the nucleus diameter, follows a universal growth law L(t) ≈ tn with the growth exponent changing from n = 1/2 in the vicinity of the transition to n = 1 for large quench depths. The short-term nucleus growth process is found to be dependent on the sample confinement with increasing growth exponents as the cell gap is increased. Nevertheless, no crossover from three-dimensional to two-dimensional growth could be observed as the nucleus diameter reached the dimension of the cell gap. A long-term coarsening process was found to be independent of the quench depth and confinement dimension, obeying a square-root growth law L(t) ≈ t1/2.

Cell sorting is analogous to phase ordering in fluids.

Morphogenetic processes, like sorting or spreading of tissues, characterize early embryonic development. An analogy between viscoelastic fluids and certain properties of embryonic tissues helps interpret these phenomena. The values of tissue-specific surface tensions are consistent with the equilibrium configurations that the Differential Adhesion Hypothesis predicts such tissues reach after sorting and spreading. Here we extend the fluid analogy to cellular kinetics. The same formalism applies to recent experiments on the kinetics of phase ordering in two-phase fluids. Our results provide biologically relevant information on the strength of binding between cell adhesion molecules under near-physiological conditions.

Nonmonotonic roughness evolution in unstable growth

The roughness of vapor-deposited thin films can display a nonmonotonic dependence on film thickness, if the smoothening of the small-scale features of the substrate dominates over growth-induced roughening in the early stage of evolution. We present a detailed analysis of this phenomenon in the framework of the continuum theory of unstable homoepitaxy. Using the spherical approximation of phase-ordering kinetics, the effect of nonlinearities and noise can be treated explicitly. The substrate roughness is characterized by the dimensionless parameter ${Q=W}_{0}{/(k}_{0}{a}^{2}),$ where ${W}_{0}$ denotes the roughness amplitude, ${k}_{0}$ is the small-scale cutoff wave number of the roughness spectrum, and a is the lattice constant. Depending on Q, the diffusion length ${l}_{D}$ and the Ehrlich-Schwoebel length ${l}_{\mathrm{ES}},$ five regimes are identified in which the position of the roughness minimum is determined by different physical mechanisms. The analytic estimates are compared by numerical simulations of the full nonlinear evolution equation.

Short-time behavior of the kinetic spherical model with long-ranged interactions

The kinetic spherical model with long-ranged interactions and an arbitrary initial order m_{0} quenched from a very high temperature to T < T_{c} is solved. In the short-time regime, the bulk order increases with a power law in both the critical and phase-ordering dynamics. To the latter dynamics, a power law for the relative order m_{r} ~ -t^{-k} is found in the intermediate time-regime. The short-time scaling relation of small m_{0} are generalized to an arbitrary m_{0} and all the time larger than t_{mic}. The characteristic functions $\phi (b,m_{0})$ for the scaling of m_{0} and $\epsilon (b,T')$ for T'=T/T_{c} are obtained. The crossover between scaling regimes is discussed in detail.

Pinning of phase separation in a model of binary polymer blends

The phase-ordering kinetics of a homopolymeric binary mixture is studied numerically in the framework of the Flory--Huggins--de Gennes continuum model. For deep quenches the suppression of the evaporation-condensation mechanism leads to slower growth, due to transport of molecules along the interfaces. When the concentration of the minority phase is low all growth mechanisms are suppressed and the system exhibits a temporary stop in the separation kinetics.

Non-equilibrium effects in profile evolution measurements of surface diffusion

In this work, we use the Boltzmann–Matano method to determine the collective diffusion coefficient, D C( θ), as a function of coverage, θ, from scaled coverage profiles obtained from Monte Carlo simulations using a lattice-gas model of O/W(110). We focus on the temporal behavior of D C( θ) as the system approaches equilibrium. We demonstrate that the effective D C( θ) obtained in this fashion depends strongly on the time regime chosen for analysis of the density profiles, and thus may differ significantly from results obtained from equilibrium simulations within ordered phases and close to phase boundaries. This is due to the interplay between spreading and phase-ordering kinetics and is reflected in enhanced particle number fluctuations with respect to the equilibrium case. Also, both the effective diffusion barriers of D C and the location of phase boundaries, as extracted from the data, depend on time.

Perturbation expansion in phase-ordering kinetics. II.n-vector model

The perturbation theory expansion presented earlier to describe the phase-ordering kinetics in the case of a nonconserved scalar order parameter is generalized to the case of the n-vector model. At lowest order in this expansion, as in the scalar case, one obtains the theory due to Ohta, Jasnow, and Kawasaki (OJK). The second-order corrections for the nonequilibrium exponents are worked out explicitly in d dimensions and as a function of the number of components n of the order parameter. In the formulation developed here the corrections to the OJK results are found to go to zero in the large n and d limits. Indeed, the large-d convergence is exponential.

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.

Phase-ordering dynamics of the Gay-Berne nematic liquid crystal.

Phase-ordering dynamics in nematic liquid crystals has been the subject of much active investigation in recent years in theory, experiments, and simulations. With a rapid quench from the isotropic to nematic phase, a large number of topological defects are formed and dominate the subsequent equilibration process. Here we present the results of a molecular dynamics simulation of the Gay-Berne model of liquid crystals after such a quench in a system with 65,536 molecules. Twist disclination lines as well as type-1 lines and monopoles were observed. Evidence of dynamical scaling was found in the behavior of the spatial correlation function and the density of disclination lines. However, the behavior of the structure factor provides a more sensitive measure of scaling, and we observed a crossover from a defect dominated regime at small values of the wave vector to a thermal fluctuation dominated regime at large wave vector.

Crossover from multiscaling to standard scaling in systems with conserved scalar order parameter

The phase-ordering dynamics with conserved scalar order parameter is studied simulating the Cahn–Hilliard equation and the kinetic Ising model. In both cases a preasymptotic multiscaling regime is found revealing that the late stage of phase-ordering is always approached through a crossover from multiscaling to standard scaling, independently from the nature of the microscopic dynamics.

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.

Curvature autocorrelations in domain growth dynamics

We show how the interface curvature autocorrelation function (ICAF) and associated structure factor (ICSF), of relevance in non-equilibrium pattern-formation problems where sharp interfaces are present, provide new and interesting information on domain structure, as yet not visible via the order-parameter structure factor (OPSF). This is done by discussing numerical simulations of model A (non-conserved relaxational phase-ordering kinetics) in two-dimensional systems. The ICAF is Gaussian over short distances and exhibits dynamical scaling and $t^{1/2}$ power-law growth. We use it to show what the typical length-scale in the model A dynamics corresponds to physically and how it can be obtained uniquely, rather than simply within a multiplicative constant. Experimental methods to measure the ICAF and/or ICSF are still needed at this point.

Anomalous phase-ordering kinetics in the Ising model

Using Monte Carlo simulations we show that for 2D and 3D Ising models at zero temperature the average life-time τ of a square/cubic-like domain of the linear size L is proportional to L 2. However, for certain initial configurations τ increases much faster: it scales as L 3 for the 2D model and for the 3D case the increase is even faster. We suggest that such a scaling is responsible for some anomalies of the phase-ordering kinetics in this model.

Phase-ordering dynamics of binary mixtures with field-dependent mobility in shear flow

The effect of shear flow on the phase-ordering dynamics of a binary mixture with field-dependent mobility is investigated. The problem is addressed in the context of the time-dependent Ginzburg-Landau equation with an external velocity term, studied in self-consistent approximation. Assuming a scaling ansatz for the structure factor, the asymptotic behavior of the observables in the scaling regime can be analytically calculated. All the observables show log-time periodic oscillations which we interpret as due to a cyclical mechanism of stretching and break-up of domains. These oscillations are dumped as consequence of the vanishing of the mobility in the bulk phase.

Velocity distribution for strings in phase-ordering kinetics

The continuity equations expressing conservation of string defect charge can be used to find an explicit expression for the string velocity field in terms of the order parameter in the case of an O(n) symmetric time-dependent Ginzburg-Landau model. This expression for the velocity is used to find the string velocity probability distribution in the case of phase-ordering kinetics for a nonconserved order parameter. For long times $t$ after the quench, velocities scale as $t^{-1/2}$. There is a large velocity tail in the distribution corresponding to annihilation of defects which goes as $V^{-(2d+2-n)}$ for both point and string defects in $d$ spatial dimensions.

Phase-ordering dynamics with an order-parameter-dependent mobility: The large-nlimit

The effect of an order-parameter dependent mobility (or kinetic coefficient), on the phase-ordering dynamics of a system described by an n-component vector order parameter is addressed at zero temperature in the large-n limit. We consider cases in which the mobility or kinetic coefficient vanishes when the magnitude of the order parameter takes its equilibrium value. In the large-n limit, the system is exactly soluble for both conserved and non-conserved order parameter. In the non-conserved case, the scaling form for the correlation function and it's Fourier transform, the structure factor, is established, with the characteristic length growing as a power of time. In the conserved case, the structure factor is evaluated and found to exhibit a multi-scaling behaviour, with two growing length scales differing by a logarithmic factor. In both cases, the rate of growth of the length scales depends on the manner in which the mobility or kinetic coefficient vanishes as the magnitude of the order parameter approaches its equilibrium value.

Overall time evolution in phase-ordering kinetics

The phenomenology from the time of the quench to the asymptotic behavior in the phase-ordering kinetics of a system with a conserved order parameter is investigated in the Bray-Humayun model [Phys. Rev. Lett. 68, 1559 (1992)] and in the Cahn-Hilliard-Cook model [J. Chem. Phys. 28, 258 (1958); Acta Metall. 18, 297 (1970)]. From the comparison of the structure factor in the two models the generic pattern of the overall time evolution, based on the sequence ``early linear--intermediate mean field--late asymptotic regime'' is extracted. It is found that the time duration of each of these regimes is strongly dependent on the wave vector and on the parameters of the quench, such as the amplitude of the initial fluctuations and the final equilibrium temperature. The rich and complex crossover phenomenology arising as these parameters are varied can be accounted for in a simple way through the structure of the solution of the Bray-Humayun model.

Perturbative corrections to the Ohta-Jasnow-Kawasaki theory of phase-ordering dynamics

A perturbation expansion is considered about the Ohta-Jasnow-Kawasaki (OJK) theory of phase-ordering dynamics, which is an approximate theory describing the coarsening dynamics of a system with a nonconserved scalar order parameter. In this calculation the nonlinear terms neglected in the OJK equation for the evolution of the auxiliary field are reinstated and treated as a perturbation to the linearized equation. The first order correction term to the pair correlation function is calculated in the large-$d$ limit, and found to be of order ${1/d}^{2}.$

Perturbation expansion in phase-ordering kinetics: I. Scalar order parameter

A consistent perturbation theory expansion is presented for phase-ordering kinetics in the case of a nonconserved scalar order parameter. At zeroth order in this expansion one obtains the theory due to Ohta, Jasnow and Kawasaki (OJK). At the next nontrivial order in the expansion, worked out in d dimensions, one has small corrections to the OJK result for the nonequilibrium exponent $\lambda$ and the introduction of a new exponent $\nu$ governing the algebraic component of the decay of the order parameter scaling function at large scaled distances.

Phase-ordering kinetics of cemetery organization in ants

The clustering of dead bodies by ants is simulated, using a cellular automaton model, the rules of which are carefully derived from experiments. Starting from a random spatial distribution of corpses, a cemetery organizes itself into clusters of corpses. The dynamics of clustering can be compared to the phase-ordering kinetics of a bidimensional idealized magnetic system with a scalar conserved order parameter. In particular, scaling relations are found for the structure factor and the dynamics of cluster growth, which can be compared with those predicted by the theory of phase-ordering kinetics. Observed exponents are consistent with those expected in early stage phase-ordering kinetics.