Dynamics of driven interfaces with a conservation law

Sun, Tao; Guo, Hong; Grant, Martin

doi:10.1103/physreva.40.6763

Cited by 192 publications

(161 citation statements)

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“…(5), is always conserved by construction. This immediately leads to the hyperscaling relationẑ = 2α+d+2 for the slope critical exponents, which is an exact outcome from the non-renormalization of the noise intensity D for systems with conserved dynamics [25,27]. This hyperscaling relation is exact, independent of the detailed form of G, and allows us to directly link the anomalous time exponent to the dynamics of the local slopes by 2κ = 1 − (d + 2)/ẑ.…”

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confidence: 96%

Scaling of Local Slopes, Conservation Laws, and Anomalous Roughening in Surface Growth

2005

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We argue that symmetries and conservation laws greatly restrict the form of the terms entering the long wavelength description of growth models exhibiting anomalous roughening. This is exploited to show by dynamic renormalization group arguments that intrinsic anomalous roughening cannot occur in local growth models. However some conserved dynamics may display super-roughening if a given type of terms are present. PACS numbers: 81.15.Aa,64.60.Ht,05.70.Ln Recent theoretical and experimental studies of self-affine kinetic roughening have uncovered a rich variety of novel features [1]. In particular, the existence of anomalous roughening has received much attention. Anomalous roughening refers to the observation that local and global surface fluctuations may have distinctly different scaling exponents. This leads to the existence of an independent local roughness exponent α loc that characterizes the local interface fluctuations and differs from the global roughness exponent α. More precisely, global fluctuations are measured by the global interface width, which for a system of total lateral size L scales according to the Family-Vicsek ansatz [2] aswhere the scaling function f (u) behaves asThe roughness exponent α and the dynamic exponent z characterize the universality class of the model under study. The ratio β = α/z is the time exponent. In contrast, local surface fluctuations are given by either the height-height correlation function,, where the average is calculated over all x (overline) and noise (brackets), or the local width,where · · · l denotes an average over x in windows of size l. For growth processes in which an anomalous roughening takes place these functions scale as w(l, t) ∼ G(l, t) = t β f A (l/t 1/z ), with an anomalous scaling function [3,4] given byinstead of Eq.(2). The standard self-affine Family-Vicsek scaling [2] is then recovered when α = α loc . This singular phenomenon was first noticed in numerical simulations of both continuous and discrete models of ideal molecular beam epitaxial growth [3,4,5,6,7,8,9,10,11]. Anomalous roughening has later on been reported to occur in growth models in the presence of disorder [12,13] Nowadays it has become clear [3,23] that anomalous kinetic roughening is related to a non-trivial dynamics of the average surface gradient (local slope), so that (∇h) 2 ∼ t 2κ . Anomalous scaling occurs whenever κ > 0, which leads to the existence of a local roughness scaling with exponent α loc = α − zκ [23]. Also, it has recently been shown that the existence of power-law scaling of the correlation functions (i.e. scale invariance) does not determine a unique dynamic scaling form of the correlation functions [24]. On the one hand, there are super-rough processes, α > 1, for which α loc = 1 always. On the other hand, there are intrinsically anomalous roughened surfaces, for which the local roughness α loc < 1 is actually an independent exponent and α may take values larger or smaller than one depending on the universality class (see [4,24] and references therein). T...

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confidence: 96%

Scaling of Local Slopes, Conservation Laws, and Anomalous Roughening in Surface Growth

2005

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“…The simplest form for A: A = (λ/2)m 2 has been introduced by Sun et al [19]. It is also called "conserved KardarParisi-Zhang term", because in Eq.…”

Section: B Mullins-like Currentmentioning

confidence: 99%

Kink dynamics in a one-dimensional growing surface

Politi

1998

Phys. Rev. E

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A high-symmetry crystal surface may undergo a kinetic instability during the growth, such that its late stage evolution resembles a phase separation process. This parallel is rigorous in one dimension, if the conserved surface current is derivable from a free energy. We study the problem in presence of a physically relevant term breaking the up-down symmetry of the surface and which can not be derived from a free energy. Following the treatment introduced by Kawasaki and Ohta [Physica 116A, 573 (1982)] for the symmetric case, we are able to translate the problem of the surface evolution into a problem of nonlinear dynamics of kinks (domain walls). Because of the break of symmetry, two different classes (A and B) of kinks appear and their analytical form is derived. The effect of the adding term is to shrink a kink A and to widen the neighbouring kink B, in such a way that the product of their widths keeps constant. Concerning the dynamics, this implies that kinks A move much faster than kinks B. Since the kink profiles approach exponentially the asymptotical values, the time dependence of the average distance L(t) between kinks does not change: L(t) ∼ ln t in absence of noise, and L(t) ∼ t 1/3 in presence of (shot) noise. However, the cross-over time between the first and the second regime may increase even of some orders of magnitude. Finally, our results show that kinks A may be so narrow that their width is comparable to the lattice constant: in this case, they indeed represent a discontinuity of the surface slope, that is an angular point, and a different approach to coarsening should be used.

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“…One such model starts with a non-linear chemical potential introduced by Sun, Guo, and Grant [15], resulting in…”

Section: Conservative "Mbe" Modelsmentioning

confidence: 99%

Dynamic scaling phenomena in growth processes

Kardar

1996

Physica B: Condensed Matter

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Inhomogeneities in a deposition process may lead to formation of rough surfaces. Fluctuations in the height h(x, t), of the surface (at location x and time t) can be probed directly by scanning microscopy, or indirectly by scattering. Analytical or numerical treatments of simple growth models suggest that, quite generally, the height fluctuations have a self-similar character; their average correlations exhibiting a dynamic scaling form,. The roughness and dynamic exponents, α and z, are expected to be universal; depending only on the underlying mechanism that generates self-similar scaling. Despite its ubiquitous occurrence in theory and simulations, experimental confirmation of dynamic scaling has been scarce. In some cases where such scaling has been observed, the exponents are different from those expected on the basis of analysis or numerics. I shall briefly review the theoretical foundations of dynamic scaling, and suggest possible reasons for discrepancies with experimental results.

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Dynamics of driven interfaces with a conservation law

Cited by 192 publications

References 13 publications

Scaling of Local Slopes, Conservation Laws, and Anomalous Roughening in Surface Growth

Scaling of Local Slopes, Conservation Laws, and Anomalous Roughening in Surface Growth

Kink dynamics in a one-dimensional growing surface

Dynamic scaling phenomena in growth processes

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