Unstable Nonlocal Interface Dynamics

Abstract

Nonlocal effects occur in many nonequilibrium interfaces, due to diverse physical mechanisms like diffusive, ballistic, or anomalous transport, with examples from flame fronts to thin films. While dimen sional analysis describes stable nonlocal interfaces, we show the morphologically unstable condition to be nontrivial. This is the case for a family of stochastic equations of experimental relevance, paradigmatically including the Michelson Sivashinsky system. For a whole parameter range, the asymptotic dynamics is scale invariant with dimension independent exponents reflecting a hidden Galilean symmetry. The usual Kardar Parisi Zhang nonlinearity, albeit irrelevant in that parameter range, plays a key role in this behavior. DOI: 10.1103/PhysRevLett.102.256102 PACS numbers: 68.35.Ct, 05.45. a, 47.54. r The interface dynamics of many nonequilibrium systems arises from the interplay between nonlocal interactions and morphological instabilities. Examples range from flame front propagation to thin film growth [1]. Often, although the basic physical interactions are short-ranged and local, the evolution is driven by nonlocal effects implicitly (via projection of the overall dynamics on the interface) or explicitly (as in elastic media or viscous flow) [1]. These nonlocalities appear in many fields, diffusion-limited growth being a prominent example. Here, although diffusion events of aggregating units are local, the morphology of a growing cluster is dominated by shadowing of the most prominent surface features over less exposed ones. Hence, the local growth velocity depends on the global surface shape. This moreover leads to the classic Mullins-Sekerka (MS) morphological instability [1], whereby prominent features grow faster. Still, nonlocality and morphological stability are independent properties. Thus, we may consider diffusion-limited erosion (DLE) [2,3], qualitatively relevant to experiments on, e.g., ion irradiation smoothing [4]. Here the most exposed surface features are eroded faster, leading to nonlocal stable interface evolution in which differences in surface height are smoothed out.Often [3], the nonequilibrium dynamics of these interfaces can be cast into a stochastic equation for the surface height hðr; tÞ at time t and point r on a d-dimensional substrate. Assuming translational and rotational symmetry, we propose the following equation (after space Fourier transform F )Here, , m, n, K, and N are positive constants (0 < 2, m ! 2, and n > m; see below), the linear dispersionproviding the amplification rate for periodic disturbances (with wave vector k) of a planar interface. The term proportional to is the Kardar-Parisi-Zhang (KPZ) nonlinearity, generically expected whenever the interface evolves in the absence of conservation laws [3], while k ðtÞ is Gaussian uncorrelated noise. Indeed, equations like (1) have been derived from constitutive laws, and from symmetry arguments, as weakly nonlinear, long wavelength (k ¼ jkj ( 1) descriptions of a variety of systems [1,3,6]. Locality holds whenever ! ...

A Novel Approach to the KPZ Dynamics

“…The exactness of this relation has been traditionally attributed to the Galilean invariance of the KPZ equation (related to the tilting of the interface). The conjectured central role of this symmetry has however been challenged in this as well as in different nonequilibrium models both from a theoretical [21][22][23][24] and a numerical [25][26][27] point of view. The other one is the generally accepted lack of existence of a suitable functional that allowed formulating the KPZ equation as a gradient flow.…”

Faceted anomalous scaling in the epitaxial growth of semiconductor films

“…However, self-affine (FV scaling) unstable growth was recently found in stochastic equations related to nonlocal interface dynamics [3,4].…”

“…The kinetic roughening of interfaces is an outstanding topic of nonequilibrium Statistical Physics which has been intensively investigated in both theoretical [1][2][3][4][5] and experimental [6][7][8][9][10] frontlines. A generic dynamical scaling theory (GDST) for evolving interfaces includes both interface fluctuations and power spectra (structure factor) in the real and momentum spaces, respectively [2].…”

“…The number in parenthesis represents the uncertainty in last digit meaning β = 1.11 ± 0.02. The mentioned unstable SMS equation, written in the momentum space with suitable rescaled parameters, reads as [3] …”

Erratum: Dynamic effects induced by renormalization in anisotropic pattern forming systems [Phys. Rev. E84, 015202(R) (2011)]

“…(1) within a one loop Dynamical Renormalization Group (DRG) approach. After its application to fluctuating hydrodynamics [26], this method has recently shown a large explanatory power in related contexts, like a multiscale descriptions of fluctuating interfaces [27] or morphological instabilities mediated by non-local interactions [28]. Following the standard approach [26], we arrive at the following RG parameter flow [22],…”