Edwards-Wilkinson Equation Research Articles

Optical scattering simulation of ice particles with surface roughness modeled using the Edwards-Wilkinson equation

Constructing an appropriate particle morphology model is essential for realistic simulation of optical properties of atmospheric particles. This paper presents a model for generating surface roughness based on a combination of methods from discrete differential geometry combined with a stochastic partial differential equation for surface evolution introduced by Edwards and Wilkinson. Scattering of light by roughened particles is simulated using the Invariant Imbedding T-Matrix (II-TM) method. The effects of surface roughness on the single-scattering properties, namely, the phase matrix, asymmetry factor, and extinction efficiency, are investigated for a single wavelength in the visible range and for a range of size parameters up to x=50. Three different smooth shapes are considered: spherical, spheroidal, and hexagonal, the latter two in just the “compact particle” case of unit aspect ratio. It is shown that roughness has negligible effects on the optical scattering properties for size parameters less than 20. For size parameters ranging from 20 to 50, the phase matrix elements are more sensitive to the surface roughness than are two important integral optical properties, the extinction efficiency and asymmetry factor. As has been seen in studies using other forms of roughening, the phase function is progressively smoothed as roughness increases. The effect on extinction efficiency is to increase it, and on asymmetry factor is to decrease it. Each of these effects is relatively modest in the size range considered, but the trend of results suggests that greater effects will be seen for size parameters larger than ones considered here.

Interface Detection Using a Quenched-Noise Version of the Edwards-Wilkinson Equation.

We report here a multipurpose dynamic-interface-based segmentation tool, suitable for segmenting planar, cylindrical, and spherical surfaces in 3D. The method is fast enough to be used conveniently even for large images. Its implementation is straightforward and can be easily realized in many environments. Its memory consumption is low, and the set of parameters is small and easy to understand. The method is based on the Edwards-Wilkinson equation, which is traditionally used to model the equilibrium fluctuations of a propagating interface under the influence of temporally and spatially varying noise. We report here an adaptation of this equation into multidimensional image segmentation, and its efficient discretization.

Revisiting random deposition with surface relaxation: approaches from growth rules to the Edwards-Wilkinson equation

We present several approaches for deriving the coarse-grained continuous Langevin equation (or Edwards-Wilkinson equation) from a random deposition with surface relaxation (RDSR) model. First we introduce a novel procedure to divide the first transition moment into the three fundamental processes involved: deposition, diffusion and volume conservation. We show how the diffusion process is related to the antisymmetric contribution and the volume conservation process is related to the symmetric contribution, which renormalizes to zero in the coarse-grained limit. In another approach, we find the coefficients of the continuous Langevin equation, by regularizing the discrete Langevin equation. Finally, in a third approach, we derive these coefficients from the set of test functions supported by the stationary probability density function (SPDF) of the discrete model. The applicability of the approaches used to other discrete random deposition models with instantaneous relaxation to a neighboring site is discussed.

Parallel discrete-event simulation schemes with heterogeneous processing elements.

To understand the effects of nonidentical processing elements (PEs) on parallel discrete-event simulation (PDES) schemes, two stochastic growth models, the restricted solid-on-solid (RSOS) model and the Family model, are investigated by simulations. The RSOS model is the model for the PDES scheme governed by the Kardar-Parisi-Zhang equation (KPZ scheme). The Family model is the model for the scheme governed by the Edwards-Wilkinson equation (EW scheme). Two kinds of distributions for nonidentical PEs are considered. In the first kind computing capacities of PEs are not much different, whereas in the second kind the capacities are extremely widespread. The KPZ scheme on the complex networks shows the synchronizability and scalability regardless of the kinds of PEs. The EW scheme never shows the synchronizability for the random configuration of PEs of the first kind. However, by regularizing the arrangement of PEs of the first kind, the EW scheme is made to show the synchronizability. In contrast, EW scheme never shows the synchronizability for any configuration of PEs of the second kind.

Nonsteady dynamic properties of a domain wall for the creep state under an alternating driving field.

With Monte Carlo simulations, the nonsteady dynamic properties of a domain wall have been systematically investigated for the thermally activated creep state under an alternating driving field. Taking the driven random-field Ising model in two dimensions as an example, two distinct growth stages of the domain interface are identified with both the correlation length and roughness function. One stage belongs to the universality class of the random depositions, and the other to that of the quenched Edwards-Wilkinson equation. In the latter case, due to the dynamic effect of overhangs, the domain interface may exhibit an intrinsic anomalous scaling behavior, different from that of the quenched Edwards-Wilkinson equation.

Healing time for the growth of thin films on patterned substrates

The healing times for the growth of thin films on patterned substrates are studied using simulations of two discrete models of surface growth: the Family model and the Das Sarma–Tamborenea (DT) model. The healing time, defined as the time at which the characteristics of the growing interface are “healed” to those obtained in growth on a flat substrate, is determined via the study of the nearest-neighbor height difference correlation function. Two different initial patterns are considered in this work: a relatively smooth tent-shaped triangular substrate and an atomically rough substrate with single-site pillars or grooves. We find that the healing time of the Family and DT models on a L×L triangular substrate is proportional to Lz, where z is the dynamical exponent of the models. For the Family model, we also analyze theoretically, using a continuum description based on the linear Edwards–Wilkinson equation, the time evolution of the nearest-neighbor height difference correlation function in this system. The correlation functions obtained from continuum theory and simulation are found to be consistent with each other for the relatively smooth triangular substrate. For substrates with periodic and random distributions of pillars or grooves of varying size, the healing time is found to increase linearly with the height (depth) of pillars (grooves). We show explicitly that the simulation data for the Family model grown on a substrate with pillars or grooves do not agree with results of a calculation based on the continuum Edwards–Wilkinson equation. This result implies that a continuum description does not work when the initial pattern is atomically rough. The observed dependence of the healing time on the substrate size and the initial height (depth) of pillars (grooves) can be understood from the details of the diffusion rule of the atomistic model. The healing time of both models for pillars is larger than that for grooves with depth equal to the height of the pillars. The calculated healing time for both Family and DT models is found to depend on how the pillars and grooves are distributed over the substrate.

Long-range spatial correlations and fluctuation statistics of lightning activity rates in Brazil

We report on a statistical analysis of the lightning activity rates in all Brazilian cities. We find out that the average of lightning activity rates exhibits a dependence on the latitude of the cities, displaying one peak around the Tropic of Capricorn and another one just before the Equator. We verify that the standard deviation of these rates is almost a constant function of the latitude and that the distribution of the fluctuations surrounding the average tendency is quite well described by a Gumbel distribution, which thus connects these rates to extreme processes. We also investigate the behavior of the lightning activity rates vs. the longitude of the cities. For this case, the average rates exhibit an approximate plateau for a wide range of longitude values, the standard deviation is an approximate constant function of longitude, and the fluctuations are described by a Laplace distribution. We further characterize the spatial correlation of the lightning activity rates between pairs of cities, where our results show that the spatial correlation function decays very slowly with the distance between the cities and that for intermediate distances the correlation exhibits an approximate logarithmic decay. Finally, we propose to model this last behavior within the framework of the Edwards-Wilkinson equation.

Dynamic scaling behavior of the space-fractional stochastic growth equation with correlated noise

In order to study the dynamic scaling behavior of the space-fractional stochastic growth equation with correlated noise, we simulate numerically the space-fractional Edwards-Wilkinson (SFEW) equation driven by correlated noise in (1+1)-dimensional case based on the Riesz-and the Grmwald-Letnikov-type fractional derivatives. The scaling exponents including growth exponent, roughness exponent and dynamic exponent with different noise correlation factors and fractional orders are obtained, which are consistent with the corresponding scaling analysis. Our results show that the noise correlation factors and fractional orders affect the dynamic scaling behavior of the SFEW equation, which displays a continuous changing universality class.

Equilibrium and non-equilibrium thermodynamics of templating reactions for the formation of nanowires

The thermodynamics of the templating of materials on one-dimensional templates, such as DNA, is modeled by considering two terms: the surface tension of the material (γ) and a line energy (σ = 2πrTγT) that represents the adhesion of the material to the template (radius rT). We show that as long as the reaction stoichiometry does not exceed a certain limit (; v = volume of material per unit length of template) then a sample of smooth, uniform wires is the equilibrium state. If the amount of material exceeds this limit, then the material will comprise a single macroscopic particle at equilibrium. The behavior of the system is similar to a morphological wetting transition and the model can rationalize the available experimental data on the reaction conditions required to form smooth DNA-templated nanowires. Using the framework of linear non-equilibrium thermodynamics, we also show that the model can describe qualitatively the observed evolution of these nanostructures from beads-on-a-string morphologies to smooth nanowires and construct a stochastic differential equation for the process. Numerical simulations and scaling arguments suggest that the same scaling behavior as the Edwards–Wilkinson equation is observed.

Scaling properties of fractional continuous growth equations

The fractional Langevin equation is obtained by generalization of the Edward–Wilkinson equation in order to describe various growth processes. In order to solve the fractional Langevin equation analytically, due to the linearity, Fourier transforms and the Riesz derivative is implemented. It is shown that the surface width in very short-times behaves as W∼t(α−d)/2α and in long-times behaves as W∼L(α−d)/2. The mean-square surface width, the height–height correlation function, and the auto-height correlation function are calculated and discussed for arbitrary dimensions. Also, the results where checked for known cases: the Edwards–Wilkinson equation (α=2) and the Mullins–Herring equation (α=4) justifying the calculations and results.

Dynamic scaling behaviors of the discrete growth models on fractal substrates

The dynamic scaling behaviors of the Family model and the Etching model on different fractal substrates are studied by means of Monte Carlo simulations, so as to discuss the microscopic mechanisms influencing the dynamic behavior of growth interfaces by changing the structure of the substrates. The Sierpinski arrowhead, crab lattice and dual Sierpinski gasket are employed as the substrates of the growth. These substrates have same fractal dimensions (df ≈ 1.585), but with different morphologies. It is shown that the structure of the substrates can affect the dynamic scaling properties of the surfaces and interfaces. Although the standard Family–Vicsek scaling is still satisfied in describing the scaling behavior of the growth on fractal substrates, the original continuum equations are invalid. The dynamic behavior of the Family model satisfies the fractional Edwards–Wilkinson equation introduced by Lee and Kim, and the dynamic behavior of the Etching model implies that α + z > 2, which is different from the analytical results of the Kardar–Parisi–Zhang equation. The fractal character of the substrate also affects the lateral behavior of the Etching growth. Interestingly, the same fractal dimensions lead to different scaling exponents. The scaling exponents of the growth models on fractal substrates are determined by not only the fractal dimensions of the substrates, but also the spectral dimensions. Fortunately, it seems that the fractal dimension and the spectral dimension are sufficient to determine the scaling exponents of the growth model on fractal substrates.

Dynamics of radial fractional growing surfaces

Fractional derivatives in the form of a Laplacian operator −(−∇2)ζ/2 appear in the stochastic differential equations of many physical systems. For a stochastically growing interface, the evolution of the height of the interface is usually written as the Langevin form of a fractional diffusion equation. This fractional Langevin equation has been recently generalized to radial geometries, and this has raised questions regarding the validity of the application of the standard scaling analysis to the study of radially growing interfaces. Here, we study the fractional Langevin equation in polar coordinate and calculate the correlation functions. We show that, for ζ > 1, the correlation increases as t1−(1/ζ) for small angles and in the long-time limit. This means that the interface width grows as t1/4 for ζ = 2 (radial Edwards-Wilkinson equation) and as t3/8 for ζ = 4 (radial Mullins-Herring equation). These results are the same as those obtained in the corresponding planar geometry. Finally, we obtain a logarithmic behavior with time for ζ ⩽ 1. In addition, we numerically solve the fractional stochastic equation and obtain the interface width for different values of ζ, arriving at results which confirm our analytical calculations.

Surface morphology control of Transparent Conducting Oxide layers for improved light trapping using wafer grating and feedback control

This work first introduces a kinetic Monte-Carlo simulation model for a two-species thin film deposition process on a grated (patterned) wafer, and then utilizes a model predictive controller, which manipulates a spatially distributed deposition rate, to produce thin films whose surface morphology has a structure that improves light trapping. This approach to thin film surface morphology control can be applied to many deposition processes, and in particular, in the context of Transparent Conducting Oxide (TCO) thin film manufacturing processes used in thin film solar cells where it is desirable to produce thin films with precisely tailored surface morphology. Specifically, a two species thin film deposition process involving atom adsorption, surface relaxation and surface migration is considered and is modeled on a large-scale lattice (lattice size=40,000) via kinetic Monte-Carlo methods and aggregate surface roughness and slope are used to describe the surface morphology. Subsequently, multiple sets of simulations are carried out to understand the process dynamics dependence on wafer grating parameters, such as magnitude and period of grating, and other process parameters, such as temperature and deposition rate. From these simulations, it is concluded that a spatially distributed deposition rate profile is necessary to be utilized across the lattice in order to induce desirable surface morphology at light wavelength spatial scales that lead to desired thin film solar cell performance. Then, an Edwards–Wilkinson-type equation is utilized to predict the surface evolution and forms the basis for the design of a predictive feedback controller. The model parameters of the Edwards–Wilkinson equation are identified from kinetic Monte-Carlo open-loop simulations. Analytical solutions of the expected surface roughness and surface slope at the visible light wavelength spatial scale are obtained by solving the Edwards–Wilkinson equation and are used in the predictive controller formulation and in the control action calculation. The controller is applied to the kinetic Monte-Carlo simulation of the deposition process taking place on a sinusoidal grated wafer. Simulation results demonstrate that the proposed controller, wafer grating and patterned actuator design successfully regulate aggregate surface roughness and slope to desirable set-point values at the end of the deposition.

Thermal rounding of the depinning transition in ultrathin Pt/Co/Pt films

We perform a scaling analysis of the mean velocity of extended magnetic domain walls driven in ultrathin Pt/Co/Pt ferromagnetic films with perpendicular anisotropy, as a function of the applied external field for different film-thicknesses. We find that the scaling of the experimental data around the thermally rounded depinning transition is consistent with the universal depinning exponents theoretically expected for elastic interfaces described by the one-dimensional quenched Edwards-Wilkinson equation. In particular, values for the depinning exponent $\beta$ and thermal rounding exponent $\psi$ are tested and the present analysis of the experimental data is compatible with $\beta=0.33$ and $\psi=0.2$, in agreement with numerical simulations.

Asymptotic dynamic scaling behavior of the (1+1)-dimensional Wolf-Villain model

Extensive kinetic Monte Carlo simulations are presented for the Wolf-Villain model in (1+1) dimensions. Asymptotic dynamic scaling is found for lattice sizes L≥2048. The exponents obtained from our simulations, α=0.50±0.02 and β=0.25±0.02, are in excellent agreement with the exact values α=1/2 and β=1/4 for the one-dimensional Edwards-Wilkinson equation. Our findings explain the widespread discrepancies of previous reports for exponents of the Wolf-Villain model in (1+1) dimensions, and the results are also consistent with the theoretical predictions of López et al. [J. M. López, M. Castro, and R. Gallego, Phys. Rev. Lett. 94, 166103 (2005)].

Modeling and control of Transparent Conducting Oxide layer surface morphology for improved light trapping

This work first introduces a kinetic Monte–Carlo simulation model for a two species thin film deposition process and demonstrates the use of feedback control, coupled with a suitable actuator design, in manufacturing thin films whose surface morphology has a structure that improves light trapping. This work is relevant in the context of a Transparent Conducting Oxide (TCO) thin film layer manufacturing used in thin film solar cells where it is desirable to produce thin films with precisely tailored surface morphology. Specifically, a two species thin film deposition process involving atom adsorption, surface relaxation and surface migration is initially considered and is modeled using a large-lattice (lattice size=40,000) kinetic Monte–Carlo simulation. Subsequently, thin film surface morphology characteristics like roughness and slope are computed with respect to different characteristic length scales ranging from atomic to the ones corresponding to visible light wavelength and it is found that a patterned actuator design is needed to induce thin film surface roughness and slope at visible light wavelength spatial scales, that lead to desired thin film solar cell performance. Then, an Edwards–Wilkinson type equation is used to model the surface evolution at the visible light wavelength spatial scale and form the basis for the design of a predictive feedback controller whose objective is to manipulate the deposition rate across the spatial domain of the process. The model parameters of the Edwards–Wilkinson equation are estimated from kinetic Monte–Carlo simulations and their dependence on the deposition rate is used in the formulation of the predictive controller to predict the influence of the control action on the surface roughness and slope throughout the thin film growth process. Analytical solutions of the expected surface roughness and surface slope at the visible light wavelength spatial scale are obtained by solving the Edwards–Wilkinson equation and are used in the predictive controller formulation and in the control action calculation. The controller is applied to the large-lattice kinetic Monte–Carlo simulation. Simulation results demonstrate that the proposed controller and patterned actuator design successfully regulate aggregate surface roughness and slope to set-point values at the end of the deposition.

Surface Morphology Dynamics in ITO Thin Films

In this study, indium tin oxide (ITO) thin films were prepared by electron beam evaporation method on float glass substrates at room temperature (RT). The surface morphology and dynamic scaling behavior of the films were studied by atomic force microscopy (AFM). It was found that average surface roughness values decreased as the film thickness increased from 100 nm to 350 nm. Fractal geometry and statistical physics techniques have been used to study a variety of irregular films within a common framework of the variance thickness. The Hurst exponent H and growth exponent ? for ITO thin films were determined to be 0.73 ? 0.01 and 0.078, respectively. Based on these results, we suggest that the growth of ITO thin films can be described by the combination of the Edwards-Wilkinson equation and Mullins diffusion equation.

Simulation and control of aggregate surface morphology in a two-stage thin film deposition process for improved light trapping

This work focuses on the development of a model predictive control algorithm to simultaneously regulate the aggregate surface slope and roughness of a thin film growth process to optimize thin film light reflectance and transmittance. Specifically, a two-stage thin film deposition process, which involves two microscopic processes: an adsorption process and a migration process, is modeled based on a one-dimensional solid-on-solid square lattice. The first stage of this process utilizes a uniform deposition rate profile to control the thickness of the thin film and the second stage of the process utilizes a spatially distributed deposition rate profile to control the surface morphology of the thin film. Kinetic Monte Carlo (kMC) methods are used to simulate this two-stage thin film deposition process. To characterize the surface morphology and to evaluate the light trapping efficiency of the thin film, aggregate surface roughness and slope corresponding to length scale of visible light are introduced as the root-mean squares of the aggregate surface height profile and aggregate surface slope profile. An Edwards–Wilkinson (EW)-type equation with appropriately computed parameters is used to describe the dynamics of the surface height profile and predict the evolution of the aggregate root-mean-square (RMS) roughness and aggregate RMS slope. A model predictive control algorithm is then developed on the basis of the EW equation model to regulate the aggregate RMS slope and the aggregate RMS roughness at desired levels. Closed-loop simulation results demonstrate the effectiveness of the proposed model predictive control algorithm in successfully regulating the aggregate RMS slope and the aggregate RMS roughness at desired levels that optimize thin film light reflectance and transmittance.

Pinning of interfaces in random media

For a model for the propagation of a curvature sensitive interface in a time independent random medium, as well as for a linearized version which is commonly referred to as Quenched Edwards–Wilkinson equation, we prove existence of a stationary positive supersolution at non-vanishing applied load. This leads to the emergence of a hysteresis that does not vanish for slow loading, even though the local evolution law is viscous (in particular, the velocity of the interface in the model is linear in the driving force).

Dynamics and control of aggregate thin film surface morphology for improved light trapping: Implementation on a large-lattice kinetic Monte Carlo model

This work demonstrates the use of feedback control, coupled with a suitable actuator design, in manufacturing thin films whose surface morphology is characterized by a desired visible light reflectance/transmittance level. The problem is particularly important in the context of thin film manufacturing for thin film solar cells where it is desirable to produce thin films with precisely tailored light trapping characteristics. Initially, a thin film deposition process involving atom adsorption and surface migration is considered and is modeled using a large-lattice (lattice size=40,000) kinetic Monte Carlo simulation. Subsequently, thin film surface morphology characteristics like roughness and slope are computed with respect to different characteristic length scales ranging from atomic to the ones corresponding to visible light wavelength and it is found that a patterned actuator design is needed to induce thin film surface roughness and slope at visible light wavelength spatial scales, which lead to desired thin film reflectance and transmittance levels. Then, an Edwards–Wilkinson-type equation (a second-order stochastic partial differential equation) is used to model the surface evolution at the visible light wavelength spatial scale and form the basis for the design of a feedback controller whose objective is to manipulate the deposition rate across the spatial domain of the process. The model parameters of the Edwards–Wilkinson equation are estimated from kinetic Monte Carlo simulations and their dependence on the deposition rate is used in the formulation of the predictive controller to predict the influence of the control action on the surface roughness and slope throughout the thin film growth process. Analytical solutions of the expected surface roughness and surface slope at the visible light wavelength spatial scale are obtained by solving the Edwards–Wilkinson equation and are used in the control action calculation. The cost function of the controller involves penalties on both surface roughness and slope from set-point values as well as constraints on the magnitude and rate of change of the control action. The controller is applied to the large-lattice kinetic Monte Carlo simulation. Simulation results demonstrate that the proposed controller and patterned actuator design successfully regulate aggregate surface roughness and slope to set-point values at the end of the deposition that yield desired levels of thin film reflectance and transmittance.