Roughness Exponent Alpha Research Articles

Self-organized criticality in an interface-growth model with quenched randomness

We study a modified model of the Kardar-Paris-Zhang equation with quenched disorder, in which the driving force decreases as the interface rises up. A critical state is self-organized, and the anomalous scaling law with roughness exponent α∼0.63 is numerically obtained.

Proposal and applications of a method for the study of irreversible phase transitions

The gradient method for the study of irreversible phase transitions in far-from-equilibrium lattice systems is proposed and successfully applied to both the archetypical case of the Ziff-Gulari-Barshad model [R. M. Ziff, Phys. Rev. Lett. 56, 2553 (1986)] and a forest-fire cellular automaton. By setting a gradient of the control parameter along one axis of the lattice, one can simultaneously treat both the active and the inactive phases of the system. In this way different interfaces are defined whose study allows us to find the active-inactive phase transition (both of first and second order), as well as the description of the active phase as composed of two further phases: the percolating and the nonpercolating ones. The average location and the width of the interfaces obey standard scaling behavior that is essentially governed by the roughness exponent alpha=1/(1+nu) , where nu is the suitable correlation length exponent.

Equilibrium-restricted solid-on-solid growth model on fractal substrates

The equilibrium-restricted solid-on-solid growth model on fractal substrates is studied by introducing a fractional Langevin equation. The growth exponent beta and the roughness exponent alpha defined, respectively, by the surface width via W approximately t(beta) and the saturated width via W(sat) approximately L(alpha), L being the system size, were obtained by a power-counting analysis, and the scaling relation 2alpha+d(f)=z(RW) was found to hold. The numerical simulation data on Sierpinski gasket, checkerboard fractal, and critical percolation cluster were found to agree well with the analytical predictions of the fractional Langevin equation.

Self-similar roughening of drying wet paper

We studied the kinetic roughening dynamics of drying wet paper. The configurations of dry paper sheets are found to be self-similar, rater than self-affine. Accordingly, the paper roughening dynamics corresponds to the new class of anomalous kinetic roughening [J. J. Ramasco, J. M. López, and M. A. Rodríguez, Phys. Rev. Lett. 84, 2199 (2000)], characterized by the equal local and global roughness exponents zeta = alpha = 1 and the dynamic exponent z = 1.0+/-0.2, whereas the spectral roughness exponent alpha(s) > 1 is determined by the long-range correlations characterized by the fractal dimension of D crumpled sheet.

Depinning transition of the Mullins-Herring equation with an external driving force and quenched random disorder

We study the depinning transition of the quenched Mullins-Herring equation by direct integration method. At critical force Fc, the average surface velocity v(t) follows a power-law behavior v(t) approximately t-delta as a function of time t with delta=0.160(5). The surface width has a scaling behavior with the roughness exponent alpha=1.50(6) and the growth exponent beta=0.841(5). Above the critical force, the steady state velocity v2 follows vs approximately (F - Fc)theta with theta=0.289(8). Finite size scalings of the velocity are also discussed.

The autocorrelation of speckles in deep Fresnel diffraction region and characterizations of random self-affine fractal surfaces

This paper studies the correlation properties of the speckles in the deep Fresnel diffraction region produced by the scattering of rough self-affine fractal surfaces. The autocorrelation function of the speckle intensities is formulated by the combination of the light scattering theory of Kirchhoff approximation and the principles of speckle statistics. We propose a method for extracting the three surface parameters, i.e. the roughness w, the lateral correlation length xi and the roughness exponent alpha, from the autocorrelation functions of speckles. This method is verified by simulating the speckle intensities and calculating the speckle autocorrelation function. We also find the phenomenon that for rough surfaces with alpha = 1, the structure of the speckles resembles that of the surface heights, which results from the effect of the peak and the valley parts of the surface, acting as micro-lenses converging and diverging the light waves.

Anomalous roughness with system-size-dependent local roughness exponent

We note that in a system far from equilibrium the interface roughening may depend on the system size which plays the role of control parameter. To detect the size effect on the interface roughness, we study the scaling properties of rough interfaces formed in paper combustion experiments. Using paper sheets of different width lambda L0, we found that the turbulent flame fronts display anomalous multiscaling characterized by nonuniversal global roughness exponent alpha and the system-size-dependent spectrum of local roughness exponents zeta(q) (lambda) = zeta(1) (1) q(-omega) lambda(phi) <alpha, whereas the burning fronts possess conventional multiaffine scaling. The structure factor of turbulent flame fronts also exhibit unconventional scaling dependence on lambda. These results are expected to apply to a broad range of far from equilibrium systems, when the kinetic energy fluctuations exceed a certain critical value.

Numerical study of discrete models in the class of the nonlinear molecular beam epitaxy equation

We study numerically some discrete growth models belonging to the class of the nonlinear molecular beam epitaxy equation, or the Villain-Lai-Das Sarma (VLDS) equation. The conserved restricted solid-on-solid model (CRSOS) with maximum height differences Delta H(max)=1 and Delta H(max)=2 was analyzed in substrate dimensions d=1 and d=2 . The Das Sarma and Tamborenea (DT) model and a competitive model involving random deposition and CRSOS deposition were studied in d=1. For the CRSOS model with Delta H(max)=1, we obtain the more accurate estimates of scaling exponents in d=1:roughness exponent alpha=0.94+/-0.02 and dynamical exponent z=2.88+/-0.04. These estimates are significantly below the values of one-loop renormalization for the VLDS theory, which confirms Janssen's proposal of the existence of higher-order corrections. The roughness exponent in d=2 is very near the one-loop result alpha=2/3, in agreement with previous works. The moments W(n) of orders n=2 , 3, 4 of the height distribution were calculated for all models, and the skewness S triple bond W3/W(3/2)(2) and the kurtosis Q triple bond W4/W(2)2-3 were estimated. At the steady states, the CRSOS models and the competitive model have nearly the same values of S and Q in d=1, which suggests that these amplitude ratios are universal in the VLDS class. The estimates for the DT model are different, possibly due to their typically long crossover to asymptotic values. Results for the CRSOS models in d=2 also suggest that those quantities are universal.

Evidence for power-law dominated noise in vacuum deposited CaF2.

We have studied the surface roughness of CaF2 vacuum deposited on glass using atomic force microscopy for film coverages spanning an order of magnitude. We find the roughness exponent alpha=0.88+/-0.03, the growth exponent beta=0.75+/-0.03, and the dynamic exponent z=alpha/beta=1.17+/-0.06. Multifractality is also present, along with power-law behavior in the nearest neighbor height difference probability distribution. The results indicate noise dominated by a power-law distribution with exponent micro+1 approximately 4.6.

Magnetic-field generation in Kolmogorov turbulence.

We analyze the initial, kinematic stage of magnetic field evolution in an isotropic and homogeneous turbulent conducting fluid with a rough velocity field, v(l) approximately l(alpha), alpha<1. This regime is relevant to the problem of magnetic field generation in fluids with small magnetic Prandtl number, i.e., with Ohmic resistivity much larger than viscosity. We propose that the smaller the roughness exponent alpha, the larger the magnetic Reynolds number that is needed to excite magnetic fluctuations. This implies that numerical or experimental investigations of magnetohydrodynamic turbulence with small Prandtl numbers need to achieve extremely high resolution in order to describe magnetic phenomena adequately.

Universality in two-dimensional Kardar-Parisi-Zhang growth.

We analyze simulation results of a model proposed for etching of a crystalline solid and results of other discrete models in the (2+1)-dimensional Kardar-Parisi-Zhang (KPZ) class. In the steady states, the moments W(n) of orders n=2,3,4 of the height distribution are estimated. Results for the etching model, the ballistic deposition model, and the temperature-dependent body-centered restricted solid-on-solid model suggest the universality of the absolute value of the skewness S identical with W(3)/W(3/2)(2) and of the value of the kurtosis Q identical with W(4)/W(2)(2)-3. The sign of the skewness is the same as of the parameter lambda of the KPZ equation which represents the process in the continuum limit. The best numerical estimates, obtained from the etching model, are absolute value of S=0.26+/-0.01 and Q=0.134+/-0.015. For this model, the roughness exponent alpha=0.383+/-0.008 is obtained, accounting for a constant correction term (intrinsic width) in the scaling of the squared interface width. This value is slightly below previous estimates of extensive simulations and rules out the proposal of the exact value alpha=2/5. The conclusion is supported by results for the ballistic deposition model. Independent estimates of the dynamical exponent and of the growth exponent are 1.605< or =z< or =1.64 and beta=0.229+/-0.005, respectively, which are consistent with the relations alpha+z=2 and z=alpha/beta.

Near-field speckles produced by random self-affine surfaces and their contrast transitions.

By use of a numerical calculation based on Green's integral equation, we study the near-field speckles produced by the random self-affine fractal surfaces of a dielectric medium. The speckle intensities evolve considerably in the near-field region, and the local fluctuations in them disappear before they have traversed the distance of a wavelength. The transition of the speckle contrast either on the surface or in the near field and in the neighborhood non-near-field regions depends on lateral correlation length xi and roughness exponent alpha of the random surfaces.

Partition functions and metropolis-type evolution rules for surface growth models with constraints.

We study dynamical scaling properties of the surface growth model with the Metropolis-type evolution rule from a partition function Z= sum ([h(r)])II (h(max))(h=h(min))1/2(1+z(n(h))), where z is a fugacity-like quantity and n(h) is the number of sites with height h in a surface configuration [h(r)]. The partition function describes a 2-particle correlated growth model when z=-1 and a self-flattening growth model when z=0. For one-dimensional equilibrium surfaces, the scaling properties for z>or=-1 except z=1 are all one phase with roughness exponent alpha=1/3 and growth exponent beta approximately equal 0.22. For the growing (eroding) surfaces, there exists a phase transition at z=0 from the grooved phase (alpha=1) for -1<or=z<0 to the ordinary Kardar-Parisi-Zhang phase (alpha=1/2) for z>0.

Avalanche dynamics, surface roughening, and self-organized criticality: Experiments on a three-dimensional pile of rice.

We present a two-dimensional system that exhibits features of self-organized criticality. The avalanches that occur on the surface of a pile of rice are found to exhibit finite size scaling in their probability distribution. The critical exponents are tau=1.21(2) for the avalanche size distribution and D=1.99(2) for the cutoff size. Furthermore, the geometry of the avalanches is studied, leading to a fractal dimension of the active sites of d(B)=1.58(2). Using a set of scaling relations, we can calculate the roughness exponent alpha=D-d(B)=0.41(3) and the dynamic exponent z=D(2-tau)=1.56(8). This result is compared with that obtained from a power-spectrum analysis of the surface roughness, which yields alpha=0.42(3) and z=1.5(1) in excellent agreement with those obtained from the scaling relations.

Restricted curvature model with suppression of extremal height.

A discrete growth model with a restricted curvature constraint is investigated by measuring both the surface width and the height difference correlation function. In our model, where an extremal height is suppressed, the surface width W shows the roughness exponent alpha approximately 0.561 and the dynamics exponent z approximately 1.69 in one substrate dimension. However the correlation function has an unusual scaling behavior and produces different wandering exponent alpha(') approximately 1.33 and its dynamic exponent z(') approximately 4. The discrepancy is due to the fact that the correlation length increases with a power law t(1/z(')) until it reaches the value proportional to Ldelta at time t(s) approximately L(z), where L is the system size and delta is the "window exponent" satisfying the relation delta=z/z(')=alpha/alpha('). delta is a new exponent to characterize the window size of the system.

Fluctuations of self-flattening surfaces.

We study the scaling properties of self-flattening surfaces under global suppression on surface fluctuations. Evolution of self-flattening surfaces is described by restricted solid-on-solid type monomer deposition-evaporation model with reduced deposition (evaporation) at the globally highest (lowest) site. We find numerically that equilibrium surface fluctuations are anomalous with roughness exponent alpha approximately equal to 1/3 and dynamic exponent z(W) approximately equal to 3/2 in one dimension (1D) and alpha=0 (log) and z(W) approximately 5/2 in 2D. Stationary roughness can be understood analytically by relating our model to the static self-attracting random walk model and the dissociative dimer-type deposition-evaporation model. In case of nonequilibrium growing-eroding surfaces, self-flattening dynamics turns out to be irrelevant and the normal Kardar-Parisi-Zhang universality is recovered in all dimensions.

Slow crossover to Kardar-Parisi-Zhang scaling.

The Kardar-Parisi-Zhang (KPZ) equation is accepted as a generic description of interfacial growth. In several recent studies, however, values of the roughness exponent alpha have been reported that are significantly less than that associated with the KPZ equation. A feature common to these studies is the presence of holes (bubbles and overhangs) in the bulk and an interface that is smeared out. We study a model of this type in which the density of the bulk and sharpness of the interface can be adjusted by a single parameter. Through theoretical considerations and the study of a simplified model we determine that the presence of holes does not affect the asymptotic KPZ scaling. Moreover, based on our numerics, we propose a simple form for the crossover to the KPZ regime.

Roughness exponent in two-dimensional percolation, Potts model, and clock model.

We present a numerical study of the self-affine profiles obtained from configurations of the q-state Potts (with q=2,3, and 7) and p=10 clock models as well as from the occupation states for site percolation on the square lattice. The first and second order static phase transitions of the Potts model are located by a sharp change in the value of the roughness exponent alpha characterizing those profiles. The low temperature phase of the Potts model corresponds to flat (alpha approximately 1) profiles, whereas its high temperature phase is associated with rough (alpha approximately 0.5) ones. For the p=10 clock model, in addition to the flat (ferromagnetic) and rough (paramagnetic) profiles, an intermediate rough (0.5<alpha<1) phase-associated with a soft spin-wave one-is observed. Our results for the transition temperatures in the Potts and clock models are in agreement with the static values, showing that this approach is able to detect the phase transitions in these models directly from the spin configurations, without any reference to thermodynamical potentials, order parameters, or response functions. Finally, we show that the roughness exponent alpha is insensitive to geometric critical phenomena.

Roughening, deroughening, and nonuniversal scaling of the interface width in electrophoretic deposition of polymer chains

Growth and roughness of the interface of deposited polymer chains driven by a field onto an impenetrable adsorbing surface are studied by computer simulations in (2+1) dimensions. The evolution of the interface width W shows a crossover from short-time growth described by the exponent beta(1) to a long-time growth with exponent beta(2) (>beta(1)). The saturated width increases, i.e., the interface roughens, with the molecular weight L(c), but the roughness exponent alpha (from W(s) approximately Lalpha) becomes negative in contrast to models for particle deposition; alpha depends on the chain length-a nonuniversal scaling with the substrate length L. Roughening and deroughening occur as the field E and the temperature T compete such that W(s) approximately (A+BT)E(-1/2).

Kinetic roughening in electrodissolution of copper.

We studied kinetic roughening of copper surfaces electrochemically dissolved at constant current densities by atomic force microscopy. The surface was found to roughen with time and the surface width increased with the length scale with the roughness exponent alpha of 0.73+/-0.05 in the stationary state. This value is different from that in electrochemical deposition, 0.87, under the stable growth condition [A. Iwamoto et al., Phys. Rev. Lett. 72, 4025 (1994)]. The observed roughening in the dissolution process is discussed in terms of nonlocal effects.