Random Deposition Model Research Articles

Study of Scaling Exponents of the Surface Evolved in 2 + 1 Dimension through Competitive Growth between Random Deposition and that with Surface Relaxation

AbstractRough surface develops through computer simulation by competition between growth mechanism random deposition (RD) with a probability of occurrence p and growth mechanism random deposition with surface relaxation (RDSR) with a probability of occurrence 1 − p, on L × L square plane for system size L to record the statistical average of time variation of surface roughness W(L, t) and average height H(t) for the model for specific values of L and p. Other than the pure RD model, the entire evolution may be divided into three regions separated by two specific cross‐over times tx and tsat. The value of interface width at saturation Wsat depends on both L and p. The first growth exponent β1 increases exponentially with an increase in p and does not depend on L. The values of the second growth exponent β2, roughness exponent α, dynamic exponent z( = α/β2 ), and α + z are 0.0234 ± 0.0008, 0.0506 ± 0.0065, 2.1577 ± 0.0073, and 2.2083 ± 0.0138 respectively and they show no dependence on L and p values. Value of the first cross‐over time tx increases exponentially with an increase in p and does not depend on L. Value of the second cross‐over time tsat increases with an increase in both p and L values. The average growth velocity is unity for the model and is independent of both L and p. For the model, the growth velocity is unity and the fractional porosity is zero. The scaling exponents show some deviation from the relevant universality classes and depend on competitive growth probability for this model. No finite‐size effect is present in the model.

Numerical simulation and parameter sensitivity analysis of multi-particle deposition behavior in HVAF spraying

Particle deposition behavior is critical to improve coating quality. Based on the Coupled Eulerian and Lagrangian (CEL) method, a three-dimensional multi-particle random deposition model was established, and grain inhomogeneity was introduced into the TC18 substrate. A polycrystalline model of the microstructure for the sprayed substrate was established by the Voronoi method. The evolution of the temperature, strain and stress fields during particle deposition was revealed, and the formation mechanism of pore defects was also analyzed. The results show that the overall strain of small particles is large, the core strain of large particles is small, and the edge strain is significant. The stress value near the large particles in the coating is large, while the stress at the edge of the coating is relatively small. The stress distribution of the TC18 substrate is similar to the uneven random geometric distribution of grains. On this basis, the response surface model of coating characteristics was established, the influence of process parameters on coating characteristics was analyzed, and the sensitivity of process parameters is determined. This research can provide a basis and guidance for engineering applications in related fields.

A Simple Mechanistic Model for Friction of Rough Partially Lubricated Surfaces

We report experimental measurements of friction between an aluminum alloy sliding over steel with various lubricant densities. Using the topography scans of the surfaces as input, we calculate the real contact area using the boundary element method and the dynamic friction coefficient by means of a simple mechanistic model. Partial lubrication of the surfaces is accounted for by a random deposition model of oil droplets. Our approach reproduces the qualitative trends of a decrease of the macroscopic friction coefficient with applied pressure, due to a larger fraction of the micro-contacts being lubricated for larger loads. This approach relates direct measurements of surface topography to realistic distributions of lubricant, suggesting possible model extensions towards quantitative predictions.

Random deposition with spatially correlated noise (RD-SCN) model: Multi-affine analysis

Abstract We study the random deposition model with long-range spatially correlated noise. In this model the particles deposit in a power-law distance of each other as Δ i , j = i n t [ u − 1 2 ρ ] , where u is chosen randomly over the range (0,1) and ρ is the correlation strength. The results show that the enhancement of ρ exponent is accompanied by the appearance of irregularities and jumps in the height fluctuations. In spite of scaling exponents dependent to correlation strength in other linear and non-linear growth equations, enhancement of the correlation strength, does not change the growth exponent β = 1 / 2 . As the short-range correlations in growth equations result in roughness saturation, the results show that the long-range correlations in this growth model does not saturate the interface width for any system size. The fractal analysis of the height fluctuations performed via the multi-fractal detrended fluctuation analysis (MF-DFA) revealed that the synthetic rough surfaces with ρ = 0 are mono-fractal with the Hurst exponent H = 0.5 . It verifies the un-correlated fluctuations in the simple random deposition model. For the correlation strengths in the range [0,1], the Hurst exponent increases in the range [ 1 2 , 1 ) with a mono-fractal behavior. In the critical exponent of ρ c , multi-affinity is occurred. For ρ > ρ c = 1 the mono-fractal feature of the height fluctuations tends to the multi-affine one and the strength of multi-affinity increases by enhancement of ρ exponent. The results show that the observed multi-affinity is because of deviation from the normal distribution and appearance of correlations among small and large fluctuations.

PARTICLE SHAPE-INDUCED CORRELATION EFFECT IN RANDOM DEPOSITION IN 1+1 DIMENSION AND RELATED EFFECT IN BALLISTIC DEPOSITION

This work is a systematic computational study of random and ballistic deposition (BD) in 1+1 dimension for deposition of two differently shaped particles. The effect of particle shape and size on different growth parameters has been reported. It has been observed that when the shape of the particle gets changed, the surface that evolves following the mechanism of random deposition (RD) model shows acute correlation effect. The growth appeared with two regions along with saturation with two acutely different slopes, one of which is around 0.5 and, other is nearly 0.33. This shows that initially, the deposition obeys the RD model, while when particle shape changes correlation gets introduced, the deposition mechanism obeys the law of BD. Same when applied in the case of BD, it had been observed that introduction of this artificial correlation introduces another region in the growth regime separating the entire regime into three parts along with saturation. In each case, different critical exponents were calculated along with the other associated parameters like porosity, growth velocity, etc. Also, it has been shown that porosity has profound effect on size and shape of the particle deposited and it varies between 0 and 0.5 for the surfaces evolved by RD model whereas the same has value between 0.75 and 0.6 in the case of BD.

NUMERICAL AND THEORETICAL STUDY OF THE CASIMIR FORCE CORRECTIONS BETWEEN TWO ROUGH SURFACES

The Casimir force between two rough surfaces grown through statistical growth models is studied, both theoretically and numerically. The models include the random deposition (RD), the random deposition with surface relaxation (RDSR), and the restricted solid-on-solid growth (RSOS). To calculate the Casimir force, the pairwise summation (PWS) method in [Formula: see text] dimensions is used. First, the force between two growing surfaces with RD equations, Edwards–Wilkinson, (EW) and Kardar–Parisi–Zhang (KPZ) equations is theoretically investigated. Then, using the Monte Carlo simulation, the rough surfaces with RD, RDSR, and RSOS models in [Formula: see text] dimensions are simulated in different growth times, and the Casimir force between each pair of rough surfaces in each time step is calculated. The results show that the force between the surfaces increases as the roughness increases.

Random deposition with a power-law noise model: Multiaffine analysis.

We study the random deposition model with power-law distributed noise and rare-event dominated fluctuation. In this model instead of particles with unit sizes, rods with variable lengths are deposited onto the substrate. The length of each rod is chosen from a power-law distribution P(l)∼l^{-(μ+1)}, and the site at which each rod is deposited is chosen randomly. The results show that for μ<μ_{c}=3 the log-log diagram of roughness, W(t), versus deposition time, t, increases as a step function, where the roughness in each interval acts as W_{loc}(t)≈t^{β_{loc}}. The local growth exponent, β_{loc}, is zero for μ=1. By increasing the μ exponent, the value of β_{loc} is increased. It tends to the growth exponent of the random distribution model with Gaussian noise, β=1/2, at μ_{c}=3. The fractal analysis of the height fluctuations for this model was performed by multifractal detrended fluctuation analysis algorithm. The results show multiaffinity behavior for the height fluctuations at μ<μ_{c} and the multiaffinity strength is greater for smaller values of the μ exponent.

Random deposition model with surface relaxation in higher dimensions

A random deposition model with surface relaxation, so-called the Family model is studied in higher dimensions. In three dimensions, the surface width W(t) characterizing the roughness of a surface grows as 2blogt at the beginning and becomes saturated at 2alogL for t≫Lz, where L is the system size. The dynamic exponent z=1.99(2) is estimated from the relation z=a∕b and a nice data collapse of the scaling plot W2(L,t)∼logL2agt∕Lz is given with z=2. In four dimensions, the surface width approaches an intrinsic width Wint with a small correction term W2(L,t)=Wint2−L2αft∕Lz, where z≈1.97 and negative exponent α≈−0.52 are obtained. Our results support that the Family model belongs to the Edwards–Wilkinson universality class even in higher dimensions.

Random deposition of particles with different angles

In this work we study numerically the effects of the angle of deposition of particles in the growth process of a thin-film generated by aggregation of particles added at random. The particles are aggregated in a random position of an initially flat surface and with a given angle distribution. This process gives rise to a rough interface after some time of deposition. We performed Monte Carlo simulations and, by changing the angle of deposition, we observed different results from the random deposition (RD) model. We measured the usual scaling exponents, namely, the roughness () and the growth () exponents. Our results show that the particles added non-perpendicularly to the substrate, can change the behavior in a discrete atomistic random deposition model. When particles are deposited with an angle of 45° in relation to the surface, the values of and are observed in the Random Deposition model. We also propose an analytic approach, using a differential stochastic equation to analyze the growth process evolution, and our theoretical results corroborate the computer simulations.

Surface growth by cluster particles: Effects of diffusion and cluster’s shape

We use Monte Carlo simulation to study the spatiotemporal behavior of thin films generated by low kinetic energy cluster particles. To this end, two models are employed to investigate the effects of clusters’ shapes and their diffusion ability on the surface. In the first model, clusters are considered as a string of particles with unit height and non-unit length. The growth process in this model is implemented based on Random Deposition with Surface Relaxation (RDSR) model where particles can diffuse on the surface. The second model studies deposition of porous clusters with different shapes according to Random Deposition (RD) model. According to previous studies, the deposition of unit size particles based on RD model leads to a random growth. Using the RDSR model the phenomenon is observed to be under the Edward-Wilkinson universality class. Our results for both models reveal that using cluster particles with non-unit size gives rise to a porous medium which changes the universality class of models to Kardar-Parizi-Zhang (KPZ) by scaling exponents β=0.3 and α=0.5.

Electrical conductivity of the films grown by ballistic deposition of rodlike particles

Using extensive Monte Carlo simulations, we study the growth of films by ballistic deposition of rodlike particles with various sizes on a one-dimensional substrate. Particles are deposited over an initially flat substrate, which leads to the formation of a porous film with rough surface. The surface width and the corresponding scaling exponents α and β and, hence, the dynamic exponent z are calculated. Also studied is the time evolution of the porosity of the material and its dependence on the particles' size. The frequency-dependent electrical conductivity of the film and its dependence on the size of the particles and the porosity are also studied. The morphology of the films, as characterized by its surface width, follows three types of evolution before reaching its ultimate structure. At short times, film growth is close to the random deposition model with the growth exponent β1 ≈ 1/2. At intermediate times, the surface width grows more slowly with a growth exponent of β2 ≈ 1/3. Finally, at long times, the width saturates and is characterized by a roughness exponent α ≈ 1/2. The results also indicate that even if the film is grown with particles of various sizes, the universality class of the model remains unchanged. The films' porosity grows rapidly with the time, before eventually saturating. As the size of the particles increases, the saturation porosity ultimately attains a value of ϕs=0.5. The frequency-dependent effective conductivity σe is a decreasing function of the deposited particles' size, as well as the porosity. The dc conductivity depends on the particle size through a power law. As is the case with a wide variety of disordered materials, the effective conductivity depends on the frequency through a power law.

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.

Surface morphology of a modified ballistic deposition model.

The surface and bulk properties of a modified ballistic deposition model are investigated. The deposition rule interpolates between nearest- and next-nearest-neighbor ballistic deposition and the random deposition models. The stickiness of the depositing particle is controlled by a parameter and the type of interparticle force. Two such forces are considered: Coulomb and van der Waals type. The interface width shows three distinct growth regions before eventual saturation. The rate of growth depends more strongly on the stickiness parameter than on the type of interparticle force. However, the porosity of the deposits is strongly influenced by the interparticle force.

Interface dynamics of immiscible two-phase lattice-gas cellular automata: A model with random dynamic scatterers and quenched disorder in two dimensions

We use a lattice gas cellular automata model in the presence of random dynamic scattering sites and quenched disorder in the two-phase immiscible model with the aim of producing an interface dynamics similar to that observed in Hele-Shaw cells. The dynamics of the interface is studied as one fluid displaces the other in a clean lattice and in a lattice with quenched disorder. For the clean system, if the fluid with a lower viscosity displaces the other, we show that the model exhibits the Saffman-Taylor instability phenomenon, whose features are in very good agreement with those observed in real (viscous) fluids. In the system with quenched disorder, we obtain estimates for the growth and roughening exponents of the interface width in two cases: viscosity-matched fluids and the case of unstable interface. The first case is shown to be in the same universality class of the random deposition model with surface relaxation. Moreover, while the early-time dynamics of the interface behaves similarly, viscous fingers develop in the second case with the subsequent production of bubbles in the context of a complex dynamics. We also identify the Hurst exponent of the subdiffusive fractional Brownian motion associated with the interface, from which we derive its fractal dimension and the universality classes related to a percolation process.

Lagrangian/Eulerian model of coagulation and deposition of inhaled particles in the human lung

The primary objectives of the present study were to mathematically describe particle coagulation within the human respiratory tract and to analyze its impact on local particle deposition patterns for high concentrations of inhaled cigarette smoke particles. Coagulation mechanisms simulated were thermal motion, gravitational settling, laminar shear, turbulences, electrical charges, and inertial effects at airway bifurcations. To implement concentration dependent coagulation processes, the Lagrangian random walk deposition model IDEAL, simulating the paths of an individual particle, was modified by tracking the random path of an elemental air volume containing the full size distribution, thus adding an Eulerian element to the Lagrangian random path model. In this combined Lagrangian/Eulerian deposition model, the initial size distribution is continuously modified by coagulation, hygroscopic growth and deposition. For the specific inhalation conditions and the cigarette smoke size distribution assumed in this study, thermal diffusion is by far the greatest source of coagulation, with a minor contribution of laminar shear in peripheral airway generations.Number deposition, which comprises the loss of inhaled particles not only by deposition but also by coagulation, and mass deposition fractions were computed for different puff and breath-hold scenarios as a function of lung generation numbers. The main reduction of the number of inhaled particles in the respiratory tract occurs in the mouth due to coagulation, accompanied by a significant shift of the size distribution to larger particle diameters. In the lung, particle loss is caused primarily by deposition due to diffusion with a preferential deposition of smaller particle diameters. In the size range from about 250 to 500nm, the number concentration of particles in the exhaled air is higher than that in the inhaled air, indicating that the production of particles in this size range by coagulation is greater than the loss by coagulation to larger sizes and deposition.

Random Deposition Model for Bonds with Relaxation

Random Deposition Model for Bonds with Relaxation

Characterization and growth modeling of ash deposits in coal fired boilers

An ash deposit growing model based on surface growing theory is developed in order to represent the coal fired ash deposits. The predicted characteristics are compared to physical and chemical measurements of a typical ash deposit collected from an existing coal fired power plant located in the south of Brazil. Several structures were generated and compared to scanning electron images of ash deposits collected from the steam superheater section. The physical and chemical characteristics of the ash deposits, such as porosity, pore size distribution and fractal dimension of a typical ash deposit were measured using SEM image. All the images were analyzed using Imago software. Also, the existing ceramic phases were quantified using EDX detection and X-ray diffraction. The proposed random trajectory particle deposition model led to good agreement when compared to the macro quantities such as porosity and fractal dimension. It is expected that the modeled porous structure can be used for further heat transfer analysis and evaluation of coal-fired steam boiler designs.

Revisiting surface diffusion in random deposition

An investigation of the effect of surface diffusion in random deposition model is made by analytical methods and reasoning. For any given site, the extent to which a particle can diffuse is decided by the morphology in the immediate neighbourhood of the site. An analytical expression is derived to calculate the probability of a particle at any chosen site to diffuse to a given length, from first principles. Using the method, the probabilities for different diffusion lengths are calculated and their dependence on system size and the number of deposited layers is studied. Numerical simulation of surface diffusion in random deposition model with varying extents of diffusion are performed and their results are interpreted in the light of the analytical calculations. Thus, a clearer understanding of the diffusion process and the effect of diffusion length on surface roughness is obtained. Systems with surface diffusion show nearly random deposition-like behaviour upto monolayer deposition. Their interface widths, in a logarithmic plot, are initially linear, as in random deposition. With increase in the number of layers, correlation effects between neighbouring columns become dominant. The interface deviates from its initial linear growth and eventually becomes saturated. An explanation for this behaviour is discussed and the point of departure from the linear form is estimated analytically.

Random nanoparticle deposition: inter-particle distances in 2D, 3D, and multilayer samples

We consider diluted assemblies of particles randomly distributed on a surface (2D samples), in a volume (3D samples) or in 2D layers separated with a spacer of thickness t (multilayer samples). Among various considerations on the inter-particle separations, a special interest is given to the nearest-neighbor distance distribution. We also discuss the influence of the particle size, by looking at the edge-to-edge distance, and analyze the particle distribution beyond the nearest-neighbor. Moreover, we investigate the particular case of multilayer samples, which to our knowledge has never been discussed. Finally, we experimentally illustrate the applicability of the random deposition model.

Thin‐film growth by random deposition of rod‐like particles on a square lattice

AbstractMonte Carlo simulations are employed to investigate the surface growth generated by deposition of particles of different sizes on a substrate, in one and two dimensions. The particles have a linear form, and occupy an integer number of cells of the lattice. The results of our simulations have shown that the roughness evolves in time following three different behaviors. The roughness in the initial times behaves as in the random deposition model, with an exponent β1 ≈ 1/2. At intermediate times, the surface roughness depends on the system dimensionality and, finally, at long times, it enters into the saturation regime, which is described by the roughness exponent α.The scaling exponents of the model are the same as those predicted by the Villain‐Lai‐Das Sarma equation for deposition in one dimension. For the deposition in two dimensions, we show that the interface width in the second regime presents an unusual behavior, described by a growing exponent β2, which depends on the size of the particles added to the substrate. If the linear size of the particle is two, we found that β2 &lt; β1, otherwise it is β2 &gt; β1, for all particles sizes larger than three. While in one dimension the scaling exponents are the same as those predicted by the Villain‐Lai‐Das Sarma equation, in two dimensions, the growth exponents are nonuniversal. (© 2011 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)