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

Peleg, Avner; Conti, M.; Meerson, Baruch

doi:10.1103/physreve.64.036127

Cited by 10 publications

(23 citation statements)

References 36 publications

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“…Our present results, combined with those of Ref. [7], indicate "normal" scaling properties of globally conserved systems for any generic initial conditions.…”

Section: Discussionsupporting

confidence: 83%

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

et al. 2002

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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) ∼ 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 new 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.

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“…Our present results, combined with those of Ref. [7], indicate "normal" scaling properties of globally conserved systems for any generic initial conditions.…”

Section: Discussionsupporting

confidence: 83%

“…(7) in the general sharp-interface formulation of the problem. In some important cases this formulation can be simplified further [5,7,23]. When the two terms on the right hand side of Eq.…”

Section: Phase Field Model and Sharp-interface Limitmentioning

confidence: 99%

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

et al. 2002

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“…Most remarkable of them is the predicted decrease of the cluster radius with time [8]. As of present, only the systems where the conservation law is imposed globally, rather than locally, have been found to indeed show this effect [9]. On the contrary, the "frozen" structure of fractal clusters at large distances, observed in simulations of locally conserved (diffusion-controlled) fractal coarsening [10,11,12,13] implies breakdown of DSI in these systems [11,13].…”

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confidence: 79%

Scaling anomalies in the coarsening dynamics of fractal viscous fingering patterns

2004

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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 very slow convergence to this scaling at small distances. Dynamic scale invariance breaks down at large distances.PACS numbers: 61.43. Hv, 64.75.+g, 47.54.+r Coarsening is an important paradigm of emergence of order from disorder. It has been extensively studied in two-phase systems quenched from a disordered state into a region of phase coexistence [1,2,3]. In another class of systems, disordered configurations are generated by an instability of growth in combination with noise, and they often exhibit long-range correlations and fractal geometry [4]. Examples include fractal clusters developing in the process of solidification from an under-cooled liquid [5], fractal clusters on a substrate grown by deposition [6] and fractal viscous fingering patterns (FVFPs) formed by the Saffman-Taylor instability in the radial Hele-Shaw cell [5]. When the driving stops, the fractal clusters coarsen by surface tension, and the coarsening dynamics provide a valuable characterization of these systems.An important simplifying factor in the analysis of coarsening dynamics is dynamic scale invariance (DSI): the presence of a single, time-dependent length scale L(t), so that a normalized pair correlation function C(r, t) depends, at long times, only on r/L(t). The coarsening length scale L(t) often exhibits a power law in time [3]. For systems with short-range correlations there is a lot of evidence, from experiments and numerical simulations, in favor of DSI [3]. For systems with long-range correlations the situation is more complicated. In the case of a non-conserved order parameter DSI was established in particle simulations following a quench from T = T c to T = 0 [7]. Implications of mass conservation in DSI were addressed more recently, in the context of coarsening of fractal clusters. Most remarkable of them is the predicted decrease of the cluster radius with time [8]. As of present, only the systems where the conservation law is imposed globally, rather than locally, have been found to indeed show this effect [9]. On the contrary, the "frozen" structure of fractal clusters at large distances, observed in simulations of locally conserved (diffusion-controlled) fractal coarsening [10,11,12,13] implies breakdown of DSI in these systems [11,13]. The frozen structure is due to Laplacian screening of transport at large distances [13].An additional scaling anomaly, observed in the numerical simulations of diffusion-controlled fractal coarsening [10,11,12,13], was the...

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“…Even bigger differences have been found in the phase ordering dynamics of locally and globally conserved systems with long-range correlations. Coarsening of fractal clusters shows dynamic scale invariance and "normal" scaling in the case of global conservation [6], and breakdown of scale invariance and anomalous scaling in locally conserved systems [7,8].…”

Section: Introductionmentioning

confidence: 99%

“…In this work we investigate the globally conserved dynamics of a long slender finger. An additional motivation for studying the finger dynamics is a recent observation that, at a late stage of coarsening of fractal clusters, the cluster morphology shows long branches, or fingers, in both locally [7,8,11], and globally [6,12] conserved systems.…”

Section: Introductionmentioning

confidence: 99%

Area-preserving dynamics of a long slender finger by curvature: A test case for globally conserved phase ordering

et al. 2001

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A long and slender finger can serve as a simple "test bed" for different phase ordering models. In this work, the globally-conserved, interface-controlled dynamics of a long finger is investigated, analytically and numerically, in two dimensions. An important limit is considered when the finger dynamics are reducible to the area-preserving motion by curvature. A free boundary problem for the finger shape is formulated. An asymptotic perturbation theory is developed that uses the finger aspect ratio as a small parameter. The leading-order approximation is a modification of "the Mullins finger" (a well-known analytic solution) which width is allowed to slowly vary with time. This time dependence is described, in the leading order, by an exponential law with the characteristic time proportional to the (constant) finger area. The subleading terms of the asymptotic theory are also calculated. Finally, the finger dynamics is investigated numerically, employing the Ginzburg-Landau equation with a global conservation law. The theory is in a very good agreement with the numerical solution.

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

Cited by 10 publications

References 36 publications

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

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

Scaling anomalies in the coarsening dynamics of fractal viscous fingering patterns

Area-preserving dynamics of a long slender finger by curvature: A test case for globally conserved phase ordering

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