Scaling and Crossover in the Large-N Model for Growth Kinetics

The dependence of the scaling properties of the structure factor on space dimensionality, range of interaction, initial and final conditions, presence or absence of a conservation law is analysed in the framework of the large-N model for growth kinetics. The variety of asymptotic behaviours is quite rich, including standard scaling, multiscaling and a mixture of the two. The different scaling properties obtained as the parameters are varied are controlled by a structure of fixed points with their domains of attraction. Crossovers arising from the competition between distinct fixed points are explicitely obtained. Temperature fluctuations below the critical temperature are not found to be irrelevant when the order parameter is conserved. The model is solved by integration of the equation of motion for the structure factor and by a renormalization group approach.Comment: 48 pages with 6 figures available upon request, plain LaTe

Multiscaling to Standard Scaling Crossover in the Bray-Humayun Model for Phase Ordering Kinetics

The Bray-Humayun model for phase ordering dynamics is solved numerically in one and two space dimensions with conserved and non conserved order parameter. The scaling properties are analysed in detail finding the crossover from multiscaling to standard scaling in the conserved case. Both in the nonconserved case and in the conserved case when standard scaling holds the novel feature of an exponential tail in the scaling function is found.Comment: 21 pages, 10 Postscript figure

Growth Kinetics in a Phase Field Model with Continuous Symmetry

We discuss the static and kinetic properties of a Ginzburg-Landau spherically symmetric $O(N)$ model recently introduced (Phys. Rev. Lett. {\bf 75}, 2176, (1995)) in order to generalize the so called Phase field model of Langer. The Hamiltonian contains two $O(N)$ invariant fields $\phi$ and $U$ bilinearly coupled. The order parameter field $\phi$ evolves according to a non conserved dynamics, whereas the diffusive field $U$ follows a conserved dynamics. In the limit $N \to \infty$ we obtain an exact solution, which displays an interesting kinetic behavior characterized by three different growth regimes. In the early regime the system displays normal scaling and the average domain size grows as $t^{1/2}$, in the intermediate regime one observes a finite wavevector instability, which is related to the Mullins-Sekerka instability; finally, in the late stage the structure function has a multiscaling behavior, while the domain size grows as $t^{1/4}$.Comment: 9 pages RevTeX, 9 figures included, files packed with uufiles to appear on Phy. Rev.

The 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({\vec\phi}) \propto (1- {\vec\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^{2\alpha+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.Comment: 6 pages, 2 figure

Scaling properties in off equilibrium dynamical processes

In the present paper, we analyze the consequences of scaling hypotheses on dynamic functions, as two times correlations $C(t,t')$. We show, under general conditions, that $C(t,t')$ must obey the following scaling behavior $C(t,t') = \phi_1(t)^{f(\beta)}{\cal{S}}(\beta)$, where the scaling variable is $\beta=\beta(\phi_1(t')/\phi_1(t))$ and $\phi_1(t')$, $\phi_1(t)$ two undetermined functions. The presence of a non constant exponent $f(\beta)$ signals the appearance of multiscaling properties in the dynamics.Comment: 6 pages, no figure

Early stage scaling in phase ordering kinetics

A global analysis of the scaling behaviour of a system with a scalar order parameter quenched to zero temperature is obtained by numerical simulation of the Ginzburg-Landau equation with conserved and non conserved order parameter. A rich structure emerges, characterized by early and asymptotic scaling regimes, separated by a crossover. The interplay among different dynamical behaviours is investigated by varying the parameters of the quench and can be interpreted as due to the competition of different dynamical fixed points.Comment: 21 pages, latex, 7 figures available upon request from [email protected]

Dynamical Scaling: the Two-Dimensional XY Model Following a Quench

To sensitively test scaling in the 2D 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 $L \sim (t/\ln[t/t_0])^{1/2}$, 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 non-topological (``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.Comment: 10 pages, 10 figure

Coarsening of Surface Structures in Unstable Epitaxial Growth

We study unstable epitaxy on singular surfaces using continuum equations with a prescribed slope-dependent surface current. We derive scaling relations for the late stage of growth, where power law coarsening of the mound morphology is observed. For the lateral size of mounds we obtain $\xi \sim t^{1/z}$ with $z \geq 4$. An analytic treatment within a self-consistent mean-field approximation predicts multiscaling of the height-height correlation function, while the direct numerical solution of the continuum equation shows conventional scaling with z=4, independent of the shape of the surface current.Comment: 15 pages, Latex. Submitted to PR

A Soluble Phase Field Model

The kinetics of an initially undercooled solid-liquid melt is studied by means of a generalized Phase Field model, which describes the dynamics of an ordering non-conserved field phi (e.g. solid-liquid order parameter) coupled to a conserved field (e.g. thermal field). After obtaining the rules governing the evolution process, by means of analytical arguments, we present a discussion of the asymptotic time-dependent solutions. The full solutions of the exact self-consistent equations for the model are also obtained and compared with computer simulation results. In addition, in order to check the validity of the present model we confronted its predictions against those of the standard Phase field model and found reasonable agreement. Interestingly, we find that the system relaxes towards a mixed phase, depending on the average value of the conserved field, i.e. on the initial condition. Such a phase is characterized by large fluctuations of the phi field.Comment: 13 pages, 8 figures, RevTeX 3.1, submitted to Physical Review

Scaling in Late Stage Spinodal Decomposition with Quenched Disorder

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 $R(t) = A^{-\beta} f(t/A^{-\gamma})$ with $\beta \simeq 1.0$ and $\gamma \simeq 3.0$ in two dimensions. We show that $\beta/\gamma = \alpha$, where $\alpha$ 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 $A \sim c^{1/d}$.Comment: RevTex manuscript 5 pages and 5 figures (obtained upon request via email [email protected]