A deposition process with particles having realistic intermediate stickiness is studied in 2+1 dimensions. At each stage of the deposition process, for any given configuration, a newly depositing particle gives rise to allowed set of configurations that are vastly larger than those for deposition of a mixture of purely non-sticky (random like) and purely sticky (ballistic like) particles. We obtain scaling behavior and demonstrate collapse of scaled data for surface width and porosity. Scaling of conductivity, when a porous structure thus formed, is saturated with conductive fluid, e.g. brine, is studied. The results obtained are in good agreement with Archie's law for porous sedimentary rocks.
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 inter-particle 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 inter-particle force. However, t…
We study roughness scaling of the outer surface and the internal porous structure of deposits generated with the three-dimensional bidisperse ballistic deposition (BBD), in which particles of two sizes are randomly deposited. Systematic extrapolation of roughness and dynamical exponents and the comparison of roughness distributions indicate that the top surface has Kardar-Parisi-Zhang scaling for any ratio F of the flux between large and small particles. A scaling theory predicts the characteristic time of the crossover from random to correlated growth in BBD and provides relations between the…
Ballistic deposition (BD) is considered to be a paradigmatic discrete growth model that represents the Kardar-Parisi-Zhang (KPZ) universality class. In this paper we question this connection by rigorously deriving a formal continuum equation from the BD microscopic rules, which deviates from the KPZ equation. In one dimension these deviations are not important in the presence of noise, but for higher dimensions or when considering deterministic evolution they are very relevant.
The surface exponents, the scaling behavior and the bulk porosity of a generalized ballistic deposition (GBD) model are studied. In nature, there exist particles with varying degrees of stickiness ranging from completely non-sticky to fully sticky. Such particles may adhere to any one of the successively encountered surfaces, depending on a sticking probability %should have the possibility of sticking to any of the %allowed points of contact on the surface with a sticking probability that is governed by the underlying stochastic mechanism. The microscopic configurations possible in this model …
Loose granular structures stabilized against gravity by an effective cohesive force are investigated on a microscopic basis using contact dynamics. We study the influence of the granular Bond number on the density profiles and the generation process of packings, generated by ballistic deposition under gravity. The internal compaction occurs discontinuously in small avalanches and we study their size distribution. We also develop a model explaining the final density profiles based on insight about the collapse of a packing under changes of the Bond number.
We consider a (1+1)-dimensional ballistic deposition process with next-nearest neighbor interaction, which belongs to the KPZ universality class, and introduce for this discrete model a variational formulation similar to that for the randomly forced continuous Burgers equation. This allows to identify the characteristic structures in the bulk of a growing aggregate ("clusters" and "crevices") with minimizers and shocks in the Burgers turbulence, and to introduce a new kind of equipped Airy process for ballistic growth. We dub it the "hairy Airy process" and investigate its statistics numerical…
We consider a deposition model in which balls rain down at random towards a 2-dimensional surface, roll downwards over existing adsorbed balls, are adsorbed if they reach the surface, and discarded if not. We prove a spatial law of large numbers and central limit theorem for the ultimate number of balls adsorbed onto a large toroidal surface, and also for the number of balls adsorbed on the restriction to a large region of an infinite surface.
We study the surface growth generated by the random deposition of particles of different sizes. A model is proposed where the particles are aggregated on an initially flat surface, giving rise to a rough interface and a porous bulk. By using Monte Carlo simulations, a surface has grown by adding particles of different sizes, as well as identical particles on the substrate in (1 + 1) dimensions. In the case of deposition of particles of different sizes, they are selected from a Poisson distribution, where the particle's sizes may vary by one order of magnitude. For the deposition of identical p…
We formulate a model for a cooperative ballistic deposition (CBD) process whereby the incoming particles are correlated with the ones already adsorbed via attractive force. The strength of the correlation is controlled by a tunable parameter $a$ that interpolates the classical car parking problem at $a=0$, the ballistic deposition at $a=1$ and the CBD model at $a>1$. The effects of the correlation in the CBD model are as follows. The jamming coverage $q(a)$ increases with the strength of attraction $a$ due to an ever increasing tendency of cluster formation. The system almost reaches the close…
We investigate the origin of the scaling corrections in ballistic deposition models in high dimensions using the method proposed by Alves \textit{et al}. [Phys Rev. E \textbf{90}, 052405 (20014)] in $d=2+1$ dimensions, where the intrinsic width associated with the fluctuations of the height increments during the deposition processes is explicitly taken into account. In the present work, we show that this concept holds for $d=3+1$ and 4+1 dimensions. We have found that growth and roughness exponents and dimensionless cumulant ratios are in agreement with other models, presenting small finite-ti…
The growth of a rough and porous thin surface by deposition of randomly shaped clusters with different sizes over an initially flat linear substrate is simulated, using Monte Carlo technique. Unlike the ordinary Random Deposition, our approach results in aggregation of clusters which produces a porous bulk with correlation along the surface and the surface saturation occurs in long enough deposition times. The scaling exponents; the growth, roughness, and dynamic exponents are calculated based on the time scale. Moreover, the porosity and its dependency to the time and clusters size are also c…
We compute exactly the asymptotic distribution of scaled height in a (1+1)--dimensional anisotropic ballistic deposition model by mapping it to the Ulam problem of finding the longest nondecreasing subsequence in a random sequence of integers. Using the known results for the Ulam problem, we show that the scaled height in our model has the Tracy-Widom distribution appearing in the theory of random matrices near the edges of the spectrum. Our result supports the hypothesis that various growth models in $(1+1)$ dimensions that belong to the Kardar-Parisi-Zhang universality class perhaps all shar…
Ballistic deposition is one of the many models of interface growth that are believed to be in the KPZ universality class, but have so far proved to be largely intractable mathematically. In this model, blocks of size one fall independently as Poisson processes at each site on the $d$-dimensional lattice, and either attach themselves to the column growing at that site, or to the side of an adjacent column, whichever comes first. It is not hard to see that if we subtract off the height of the column at the origin from the heights of the other columns, the resulting interface process is Markovian…
In order to estimate roughness exponents of interface growth models, we propose the calculation of effective exponents from the roughness fluctuation (sigma) in the steady state. We compare the finite-size behavior of these exponents and the ones calculated from the average roughness <w_2> for two models in the 2+1-dimensional Kardar-Parisi-Zhang (KPZ) class and for a model in the 1+1-dimensional Villain-Lai-Das Sarma (VLDS) class. The values obtained from sigma provide consistent asymptotic estimates, eventually with smaller finite-size corrections. For the VLDS (nonlinear molecular beam epit…
We study the ballistic deposition and the grain deposition models on two-dimensional substrates. Using the Kardar-Parisi-Zhang (KPZ) ansatz for height fluctuations, we show that the main contribution to the intrinsic width, which causes strong corrections to the scaling, comes from the fluctuations in the height increments along deposition events. Accounting for this correction in the scaling analysis, we obtained scaling exponents in excellent agreement with the KPZ class. We also propose a method to suppress these corrections, which consists in divide the surface in bins of size $\varepsilon…
We study a thin film growth model with temperature activated diffusion of adsorbed particles, allowing for the formation of overhangs and pores, but without detachment of adatoms or clusters from the deposit. Simulations in one-dimensional substrates are performed for several values of the diffusion-to-deposition ratio R of adatoms with a single bond and of the detachment probability epsilon per additional nearest neighbor (NN), respectively with activation energies are Es and Eb. If R and epsilon independently vary, regimes of low and high porosity are separated at 0.075 < epsilon_c < 0.09, w…
The ballistic deposition on a one dimensional substrate of grains with one degree of freedom, called spin, is studied with respect to relaxation processes during deposition. The "spin" represents the grain anisotropy, e.g. its longest axis with respect to the vertical. The grains interact through some contact energy (J) and are allowed to flip with a probability q during deposition and relaxation. Different relaxation processes are investigated. The pile structure is investigated, i.e. the density and "magnetisation", as a function of q and J. A percolation transition is found across which the…
We introduce a new class of growth models, with a surface restructuring mechanism in which impinging particles may dislodge suspended particles, previously aggregated on the same column in the deposit. The flux of these particles is controlled through a probability $p$. These systems present a crossover, for small values of $p$, from random to correlated (KPZ) growth of surface roughness, which is studied through scaling arguments and Monte Carlo simulations on one- and two-dimensional substrates. We show that the crossover characteristic time $t_{\times}$ scales with $p$ according to $t_{\tim…
We have studied the percolation behaviour of deposits for different (2+1)-dimensional models of surface layer formation. The mixed model of deposition was used, where particles were deposited selectively according to the random (RD) and ballistic (BD) deposition rules. In the mixed one-component models with deposition of only conducting particles, the mean height of the percolation layer (measured in monolayers) grows continuously from 0.89832 for the pure RD model to 2.605 for the pure RD model, but the percolation transition belong to the same universality class, as in the 2- dimensional ran…
The growth of ballistic aggregates on deterministic fractal substrates is studied by means of numerical simulations. First, we attempt the description of the evolving interface of the aggregates by applying the well-established Family-Vicsek dynamic scaling approach. Systematic deviations from that standard scaling law are observed, suggesting that significant scaling corrections have to be introduced in order to achieve a more accurate understanding of the behavior of the interface. Subsequently, we study the internal structure of the growing aggregates that can be rationalized in terms of th…
We study surface and bulk properties of porous films produced by a model in which particles incide perpendicularly to a substrate, interact with deposited neighbors in its trajectory, and aggregate laterally with probability of order $a$ at each position. The model generalizes ballistic-like models by allowing attachment to particles below the outer surface. For small values of $a$, a crossover from uncorrelated deposition (UD) to correlated growth is observed. Simulations are performed in 1+1 and 2+1 dimensions. Extrapolation of effective exponents and comparison of roughness distributions co…
Langevin equations for several competitive growth models in one dimension are derived. For models with crossover from random deposition (RD) to some correlated deposition (CD) dynamics, with small probability p of CD, the surface tension νand the nonlinear coefficient λof the associated equations have linear dependence on p due solely to this random choice. However, they also depend on the regularized step functions present in the analytical representations of the CD, whose expansion coefficients scale with p according to the divergence of local height differences when p->0. The superposition …
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 a transition from the random deposition (RD) universality class to the Kardar-Parisi-Zhang (KPZ) universality class. We measured the usual …
We establish a thermodynamic limit and Gaussian fluctuations for the height and surface width of the random interface formed by the deposition of particles on surfaces. The results hold for the standard ballistic deposition model as well as the surface relaxation model in the off-lattice setting. The results are proved with the aid of general limit theorems for stabilizing functionals of marked Poisson point processes.
A generalized so called magnetically controlled ballistic rain-like deposition (MBD) model of granular piles has been numerically investigated in 2D. The grains are taken to be elongated disks whence characterized by a two-state scalar degree of freedom, called ''nip'', their interaction being described through a Hamiltonian. Results are discussed in order to search for the effect of nip flip (or grain rotation from vertical to horizontal and conversely) probability in building a granular pile. The characteristics of creation of + (or $-$) nip's clusters and clusters of holes (missing nips) ar…
We simulated a growth model in 1+1 dimensions in which particles are aggregated according to the rules of ballistic deposition with probability p or according to the rules of random deposition with surface relaxation (Family model) with probability 1-p. For any p>0, this system is in the Kardar-Parisi-Zhang (KPZ) universality class, but it presents a slow crossover from the Edwards-Wilkinson class (EW) for small p. From the scaling of the growth velocity, the parameter p is connected to the coefficient of the nonlinear term of the KPZ equation, lambda, giving lambda ~ p^gamma, with gamma = 2.1…
The properties of a wide variety of growing models, generically called $X/RD$, are studied by means of numerical simulations and analytic developments. The study comprises the following $X$ models: Ballistic Deposition, Random Deposition with Surface Relaxation, Das Sarma-Tamboronea, Kim-Kosterlitz, Lai-Das Sarma, Wolf-Villain, Large Curvature, and three additional models that are variants of the Ballistic Deposition model. It is shown that after a growing regime, the interface width becomes saturated at a crossover time ($t_{x2}$) that, by fixing the sample size, scales with $p$ according to …
We study the short time properties of a two-dimensional film growth model in which incident particles execute advective-diffusive motion with a vertical step followed by $D$ horizontal steps. The model represents some features of the deposition of anisotropic colloidal particles of the experiment in Phys. Rev. Lett. {\bf 110}, 035501 (2013), in which wandering particles are attracted to particle-rich regions in the deposit. Height profiles changing from rough to columnar structure are observed as $D$ increases from $0$ (ballistic deposition) to $8$, with striking similarity to the experimental…
The article [Phys. Rev. E {\bf 73}, 031111 (2006)] by Horowitz and Albano reports on simulations of competitive surface-growth models RD+X that combine random deposition (RD) with another deposition X that occurs with probability $p$. The claim is made that at saturation the surface width $w(p)$ obeys a power-law scaling $w(p) \propto 1/p^δ$, where $δ$ is only either $δ=1/2$ or $δ=1$, which is illustrated by the models where X is ballistic deposition and where X is RD with surface relaxation. Another claim is that in the limit $p \to 0^+$, for any lattice size $L$, the time evolution of $w(t)$…
We use the model of ballistic deposition as a simple way to establish cooperation among the columns of a growing surface, \emph{the single individual of the same society}. We show that cooperation generates memory properties and at same time non-Poisson renewal events. The variable generating memory can be regarded as the velocity of a particle driven by a bath with the same time scale, and the variable generating renewal processes is the corresponding diffusional coordinate.
The percolation behaviour during the deposit formation, when the spanning cluster was formed in the substrate plane, was studied. Two competitive or mixed models of surface layer formation were considered in (1+1)-dimensional geometry. These models are based on the combination of ballistic deposition (BD) and random deposition (RD) models or BD and Family deposition (FD) models. Numerically we find, that for pure RD, FD or BD models the mean height of the percolation deposit $\bar h$ grows with the substrate length $L$ according to the generalized logarithmic law $\bar h\propto (\ln (L))^γ$, w…
A competitive (2+1)-dimensional model of deposit formation, based on the combination of random sequential absorption deposition (RSAD), ballistic deposition (BD) and random deposition (RD) models, is proposed. This model was named as RSAD$_{1-s}$(RD$_f$BD$_{1-f}$)$_s$. It allows to consider different cases of interparticle interactions from complete repulsion between near-neighbors in the RSAD model ($s=0$) to sticking interactions in the BD model ($s=1, f=0$) or absence of interactions in the RD model ($s=1$, $f=0$). The ideal checkerboard ordered structure was observed for the pure RSAD mode…
We simulated models of diffusion and aggregation in long pores of small widths in order to represent the basic mechanisms of coke deposition in catalysts' pores. Coke precursors are represented by particles injected at the pore entrance. Knudsen diffusion, which is usually expected inside the pores, is modeled by ballistic motion of those particles. The regime of molecular diffusion is also analyzed via models of lattice random walks biased along the pores. The aggregation at the surface or near previously aggregated particles was modeled by different probabilistic rules, accounting for the po…
We have studied the iso-height lines on the $\mathrm{WO_3}$ surface as a physical candidate for conformally invariant curves. We have shown that these lines are conformally invariant with the same statistics of domain walls in the critical Ising model. They belong to the family of conformal invariant curves called Schramm-Loewner evolution (or $SLE_κ$), with diffusivity of $κ\sim 3$. This can be regarded as the first experimental observation of SLE curves. We have also argued that Ballistic Deposition (BD) can serve as a growth model giving rise to contours with similar statistics at large sca…
We show numerically that the roughness and growth exponents of a wide range of rough surfaces, such as random deposition with relaxation (RDR), ballistic deposition (BD) and restricted solid-on-solid model (RSOS), are independent of the underlying regular (square, triangular, honeycomb) or random (Voronoi) lattices. In addition we show that the universality holds also at the level of statistical properties of the iso-height lines on different lattices. This universality is revealed by calculating the fractal dimension, loop correlation exponent and the length distribution exponent of the indiv…
We consider a fully asymmetric one-dimensional model with mass-conserving coalescence. Particles of unit mass enter at one edge of the chain and coalescence while performing a biased random walk towards the other edge where they exit. The conserved particle mass acts as a passive scalar in the reaction process $A+A\to A$, and allows an exact mapping to a restricted ballistic surface deposition model for which exact results exist. In particular, the mass- mass correlation function is exactly known. These results complement earlier exact results for the $A+A\to A$ process without mass. We introd…
We study the random growth of surfaces from within the perspective of a single column, namely, the fluctuation of the column height around the mean value, y(t)= h(t)-< h(t)>, which is depicted as being subordinated to a standard fluctuation-dissipation process with friction gamma. We argue that the main properties of Kardar-Parisi-Zhang theory, in one dimension, are derived by identifying the distribution of return times to y(0) = 0, which is a truncated inverse power law, with the distribution of subordination times. The agreement of the theoretical prediction with the numerical treatment of …
To construct continuum stochastic growth equations for competitive nonequilibrium surface-growth processes of the type RD+X that mixes random deposition (RD) with a correlated-growth process X, we use a simplex decomposition of the height field. A distinction between growth processes X that do and do not create voids in the bulk leads to the definition of the {\it effective probability} $p_{\mathrm{eff}}$ of the process X that is a measurable property of the bulk morphology and depends on the {\it activation probability} $p$ of X in the competitive process RD+X. The bulk morphology is reflecte…
