Parameter-free scaling relation for nonequilibrium growth processes

Abstract

We discuss a parameter free scaling relation that yields a complete data collapse for large classes of nonequilibrium growth processes. We illustrate the power of this new scaling relation through various growth models, as for example the competitive growth model RD/RDSR (random deposition/random deposition with surface diffusion) and the RSOS (restricted solid-on-solid) model with different nearest-neighbor height differences, as well as through a new deposition model with temperature dependent diffusion. The new scaling relation is compared to the familiar Family-Vicsek relation and the limitations of the latter are highlighted.PACS numbers: 64.60.Ht,68.35.Ct,05.70.Np The study of growing interfaces has been a very active field for many years [1,2,3]. Many studies focus on the technologically relevant growth of thin films or nanostructures, but growing interfaces are also encountered in various other physical, chemical, or biological systems, ranging from bacterial growth to diffusion fronts. Over the years important insights into the behavior of nonequilibrium growth processes have been gained through the study of simple model systems that capture the most important aspects of real experimental systems [4,5].In their seminal work, Edwards and Wilkinson investigated surface growth phenomena generated by particle sedimentation under the influence of gravity [6]. They proposed to describe this process in (d + 1) dimensions by the following stochastic equation of motion for the surface height h(x, t), now called the Edwards-Wilkinson (EW) equation,where ν is the diffusion constant (surface tension), whereas η(x, t) is a Gaussian white noise with zero mean and covariance η(x, t)η(y, s) = Dδ d (x − y)δ(t − s). Since Eq. 1 is linear, it can be solved exactly by Fourier transformations [2,4,6]. Later, Family [7] discussed the random deposition (RD) and random deposition with surface relaxation (RDSR) processes. RD [3,7] is one of the simplest surface growth processes. In this lattice model particles drop from randomly chosen sites over the surface and stick directly on the top of the selected surface site. Since there is no surface diffusion, the independently growing columns yield an uncorrelated and never-saturated surface. The RDSR process is realized by adding surface diffusion which allows particles just deposited on the surface to jump to the neighboring site with lowest height. This diffusion step smoothes the surface and limits the maximum interface width W (t), defined at deposition time t as the standard deviation of the surface height h from its mean value h: W (t) = h − h 2 . Starting from an initially flat surface, RDSR yields at very early times, with t < t 1 ∼ 1 (we assume here that one layer is deposited per unit time), a surface growing in the same way as for the RD process since no (or only very few) diffusion steps occur in that regime. For t > t 1 the width increases as a power law of time with a growth exponent β before entering a saturation regime after a crossover time t 2 , see Fig. 1. Both...

Feedback control of surface roughness in a one-dimensional Kardar-Parisi-Zhang growth process

“…2 differs drastically from that obtained from the scheme in Ref. [8], where the dynamics is separated into a linear and a non- The variety of multiple temporal regimes in the controlled growth process is reminiscent of the appearance of different regimes in competitive growth models [10]. Examples include the (random deposition / random deposition with surface relaxation (RD / RDSR) model [24], where the deposition of particles happens with probability p respectively 1 − p following the RDSR respectively RD rules, which results in a crossover between these two distinct regimes before the system settles into its steady state; or the restricted solid-on-solid (RSOS) model [25] that displays a crossover from a random-deposition to a KPZ regime, followed by the final crossover to saturation.…”

“…Examples include the (random deposition / random deposition with surface relaxation (RD / RDSR) model [24], where the deposition of particles happens with probability p respectively 1 − p following the RDSR respectively RD rules, which results in a crossover between these two distinct regimes before the system settles into its steady state; or the restricted solid-on-solid (RSOS) model [25] that displays a crossover from a random-deposition to a KPZ regime, followed by the final crossover to saturation. For this type of competitive growth systems, a generalized scaling law has been demonstrated [10]. Using rescaled variables W n = W/W 1 and t n = t/t 1 where t 1 and W 1 are the (system parameter-dependent) time and surface height at the crossover location between the first two regimes (e.g., between random deposition and KPZ for the RSOS model), the following scaling relation yields a complete data collapse for such competitive growth models:…”

“…where h(L, t) = L/2 −L/2 h(x, t)dx/L denotes the mean height at time t. In a finite system this quantity displays for many growth processes the Family-Vicsek scaling behavior [9,10] growth exponent and is given by β = α/z. It follows that the time until saturation scales as L z , and that the saturation width is proportional to L α .…”

Cyclic competition of four species: domains and interfaces

“…We observe three different regimes, as expected for the width of an interface that initially was given by a straight line: uncorrelated fluctuations prevail at early times, followed by a correlated fluctuation regime, before the fluctuations saturate at a level that depends on the length H of the interface. The two last regimes can be summarized by the Family-Vicsek scaling relation [64,65] …”

One-dimensional diffusion: Discrepancy between exact results and Monte Carlo calculations

“…[34], and by using the technique developed in Ref. [35], it is possible by using similar scaling arguments to obtain the collapse of the MSD curves. To do that, the two crossover times, t 1 and t 2 , and the corresponding values of the MSD, S T (t 1 ) and S T (t 2 ), are determined for each curve.…”