Ballistic Deposition Model Research Articles

Fractal pore structure of sedimentary rocks: Simulation by ballistic deposition

Experimental studies show that porous sedimentary rocks have a fractal pore‐grain interface. The bidisperse ballistic deposition model (BBDM) generates a porous structure by computer simulation. The model is able to reproduce the behavior of several macroscopic properties of sedimentary rocks quite well. In the present paper we show that the microstructure of the pore space created in BBDM is also realistic. Two‐point correlations of the pore space agree qualitatively with results for real rocks, and the sample spanning pore has a fractal dimension in the range 2.29–2.99 for different porosities. We also study diffusion in the pore space by a random walk and show that it is anomalous at low porosities.

Finite-size scaling study of the ballistic deposition model in (1+1)-dimensions

We performed extensive Monte Carlo simulations of the ballistic deposition model in (1+1)-dimensions for several system sizes up to 1280 lattice constants, on the square lattice. Though the ballistic deposition model is generally accepted to belong to the Kardar–Parisi–Zhang (KPZ) universality class, strong corrections to scaling prevent numerical estimates of the exponents close to the asymptotic values. We obtained α⩾0.40, β⩾0.30, and z⩾1.16, which are consistent with the expected KPZ values of α=1/2, β=1/3, and z=3/2. We found a slow, and even non-monotonic, convergence of the exponents towards the asymptotic values, which corroborates previous claims in the literature of strong corrections to scaling.

Gaussian Fields and Random Packing

Consider sequential packing of unit volume balls in a large cube, in any dimension and with Poisson input. We show after suitable rescaling that the spatial distribution of packed balls tends to that of a Gaussian field in the thermodynamic limit. The results cover related applied models, including ballistic deposition and spatial birth-growth models.

Dynamic rearrangements and packing regimes in randomly deposited two-dimensional granular beds.

We study the structural properties of two-dimensional granular packings prepared by random deposition from a source line. We consider a class of random ballistic deposition models based on single-particle relaxation rules controlled by a critical angle, and we show that these local rules can be formulated as rolling friction in the framework of dynamic methods for the simulation of granular materials. We find that a packing prepared by random deposition models is generically unstable, and undergoes dynamic rearrangements. As a result, the dynamic method leads systematically to a higher solid fraction than the geometrical model for the same critical angle. We characterize the structure of the packings generated by both methods in terms of solid fraction, contact connectivity, and anisotropy. Our analysis provides evidence for four packing regimes as a function of solid fraction, the mechanisms of packing growth being different in each regime.

Scaling behavior of the surface in ballistic deposition.

Using a dynamical scaling form for the surface fractal dimension as well as efficient algorithms for the simulation and analysis of the surface in three-dimensional ballistic deposition, we show that while the top of the surface is self-affine, the complete surface including overhangs has fractal dimension D(f) approximately 3. The existence of such a fractal surface is a consequence of the difficulty of closing off voids in three and higher dimensions. By studying a modified ballistic deposition model in which sticking is allowed with a given probability p, we show that the surface undergoes a phase transition from fractal to compact at a finite value of p. Our results also have implications for understanding the surface morphology in sedimentary rocks and low-temperature thin films.

Limit Theory for Random Sequential Packing and Deposition

Consider sequential packing of unit balls in a large cube, as in the Renyi car-parking model, but in any dimension and with finite input. We prove a law of large numbers and central limit theorem for the number of packed balls in the thermodynamic limit. We prove analogous results for numerous related applied models, including cooperative sequential adsorption, ballistic deposition, and spatial birth-growth models. The proofs are based on a general law of large numbers and central limit theorem for “stabilizing” functionals of marked point processes of independent uniform points in a large cube, which are of independent interest. “Stabilization” means, loosely, that local modifications have only local effects.

Experimental and modeled deposition kinetics of large colloidal particles

The kinetics of the deposition of polymeric microspheres (diameter 3.31– 8.74 μm , density 1.5 g cm −3 ; diameter 4 μm , density 1.055 g cm −3 ), as well as of polymorphonuclear leukocytes (diameter 7 μm , density 1.11 g cm −3 ) on smooth, horizontal silica slides was measured using a vertical cylindrical sedimentation cell. The adsorbing surface was observed with an inverted optical microscope connected to a camera and a micro-computer for the image analysis. The experimental deposition kinetics were first compared to the prediction of the ballistic deposition (BD) model aimed at the description of the sedimentation and adhesion under the influence of a strong gravitational field. This model proves effectively to be appropriate in the case of the largest particles used. However, once the gravitational force ceases to strongly dominate the random thermal force, the diffusion of the particles in the liquid becomes an important feature of the process as far as its kinetic aspect is concerned. To take the diffusion into account accurately, especially in the vicinity of the interface, a non-sequential model is developed. Such a model is needed to avoid the problem raised by the treatment of particles trapped by those near the surface, which cannot be solved in a rigorous way in the framework of a sequential model, even though the bulk concentration is small.

Mathematics of random growing interfaces

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.

Universality and corrections to scaling in the ballistic deposition model

In order to analyze some controversies on the equivalence between ballistic deposition (BD) and the Kardar-Parisi-Zhang (KPZ) theory, we simulated the BD model in one and two dimensions. Effective exponents betaL were obtained in the growth regions, which were rigorously determined for various lengths L. Effective exponents alphaL were obtained from saturation widths in the steady-state regimes. In d=1 we found betaL=beta+AL(-lambda) and alphaL=alpha+BL(-delta), with asymptotic exponents consistent with the KPZ values beta=1/3 and alpha=1/2, and correction-to-scaling exponents 0.2 < or approximately = lambda < or approximately = 0.4 and 0.6 < or approximately = delta < or approximately = 0.8. These strong finite-size corrections explain the previous discrepancies between numerical estimates for BD and the exact KPZ results. In d=2 we could only obtain reliable estimates of alphaL, which are consistent with KPZ values if finite-size corrections with delta approximately 0.4 are considered.

Roughness scaling and sensitivity to initial conditions in a symmetric restricted ballistic deposition model

In this work, we introduce a restricted ballistic deposition model with symmetric growth rules that favors the formation of local finite slopes. It is the simplest model which, even without including a diffusive relaxation mode of the interface, leads to a macroscopic groove instability. By employing a finite-size scaling of numerical simulation data, we determine the scaling behavior of the surface structure grown over a one-dimensional substrate of linear size L. We found that the surface profile develops a macroscopic groove with the asymptotic surface width scaling as , with . The early-time dynamics is governed by the scaling law , with . We further investigate the sensitivity to initial conditions of the present model by applying damage spreading techniques. We find that the early-time distance between two initially close surface configurations grows in a ballistic fashion as , but a slower Brownian-like scaling ( ) sets up for evolution times much larger than a characteristic time scale .

A discrete surface growth model for two components

We present a ballistic deposition model for the surface growth of a binary species A and C. Numerical simulations of the growth kinetics show a deviation from the Kardar–Parisi–Zhang universality class, model valid for only one kind of deposited particles. The study also shows that when the deposition of particles with less active bonds occurs more frequently the voids under the surface become relevant. However, the increase in overhang/voids processes under the moving interface does not strengthen greately the local surface gradient.

Dynamic scaling in a ballistic deposition model for a binary system

A ballistic deposition model for the kinetics of surface growth for two species is introduced as a description of the evolution of a surface under vapor deposition. We used a tunable parameter P to control the deposition of the particles such that one type is deposited with probability P while the other is deposited with 1-P. Simulations in 2+1 dimensions using local surface diffusion lead to the formation of a rough surface whose dynamical evolution is not that of the Kardar-Parisi-Zhang universality class. Also, when surface diffusion becomes dominant, the model moves away from the Edwards-Wilkinson universality.

Growth of amorphous films at low temperatures: the ice model

We present Monte Carlo simulations of the growth of an amorphous ice film on the uniform substrate at low temperatures. The employed 1+1 lattice model takes into account anisotropy of interactions between water molecules. In addition, the newly arrived molecules are allowed to relax. The time dependence of the interface width is found to be described by the power law with exponent β=0.28–0.30, typical for the conventional ballistic deposition models. This finding supports the concept of universality of the growth of amorphous interfaces.

From Random Sequential Adsorption to Ballistic Deposition: A General View of Irreversible Deposition Processes

Irreversible adsorption/deposition processes are defined as processes in which, once adsorbed, the particles can neither desorb from nor diffuse along the surface. These processes are commonly encountered when macromolecules or colloidal particles deposit on solid surfaces. This article presents a general and unified picture of these processes based mainly on experimental, simulation, and theoretical results accumulated by the authors over recent years. We describe first the influence of surface exclusion effects on the adsorption process under negligible influence of gravity. Essential results relative to the random sequential adsorption (RSA) model are given. The influence of particle diffusion and hydrodynamic interactions during the adsorption process in the deposition zone near the deposition plane on the structure of the assembly of deposited particles and on the adsorption kinetics are analyzed. It is shown that hydrodynamic interactions have the tendency to randomize the position of the particles over the available surface and thus to render the adsorption process RSA-like. Model predictions are confirmed by experimental data. For the deposition of larger particles gravity plays an important role, and the deposition process is characterized by a dimensionless particle radius R*. The case of an infinite value of R* is first discussed and the ballistic deposition (BD) model is described. The deposition processes characterized by R* values ranging from 0.6 to 3.3 are then analyzed. For such R* values one observes a smooth transition from the RSA to the BD behavior when R* is increased.

Irreversible adsorption of colloidal particles on solid substrates

In this article, we summarize the basic results relative to the field of irreversible deposition processes of colloidal particles on solid surfaces. An irreversible deposition process is defined as a process in which, once adsorbed, a particle can neither diffuse along, nor desorb from the surface. However, some extensions leading to reversible adsorption models are also outlined. First the basic tools used in these studies are introduced, notably the concept of available surface function. General results relative to these processes are then presented. We discuss, in particular, the connection between the reduced variance of the number density fluctuations of adsorbed particles and the available surface function. We then review the main models which were introduced in the literature to account for these processes. They can be divided in two classes: (i) the models which are based entirely on statistical and geometrical grounds. The most widely studied of them is the Random Sequential Adsorption (RSA) model which is discussed in detail. For the processes in which gravity plays an important role one uses the Ballistic Deposition (BD) model. We also present models which are aimed at accounting for the behavior lying between the ballistic deposition and the RSA. (ii) The second type of models corresponds to those which take explicitly the diffusion of the particles in the vicinity of the adsorption plane into account. The results relative to these models, called diffusional models, are discussed in detail. Finally, the last part of the article is devoted to experimental results. We present and discuss in a critical way experimental evidence which seems to indicate the validity of the RSA and BD models.

Correlation between porosity, conductivity and permeability of sedimentary rocks — a ballistic deposition model

We contruct a porous rock structure by ballistic deposition of grains of two different sizes. The conductivity ( σ) of the rock with brine-filled pores is calculated using a simple parallel resistor model, taking the tortuosity of the pore channels into account. The results are used to calculate the permeability ( K) from an equivalent channel model. Variation of σ and K with porosity are compared with geological field data. The agreement is quite satisfactory.

Particle dynamics in a mass-conserving coalescence process

We consider a fully asymmetric one-dimensional diffusion model with mass-conserving coalescence. Particles of unit mass enter at one edge of the chain and coalesce 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 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 A process without mass. We introduce a comprehensive scaling theory for this process. The exact analytical and numerical results confirm its validity.

Growth kinetics and morphology of a ballistic deposition model that incorporates surface diffusion for two species.

We introduce a ballistic deposition model for two kinds of particles (active and inactive) in (2+1) dimensions upon introducing surface diffusion for the inactive particles. A morphological structural transition is found as the probability of being the inactive particle increases. This transition is well defined by the change in the behavior of the surface width when it is plotted as a function of time and probability. The exponents alpha and beta calculated for different values of probability show the same behavior. The presence of both types of particles gives rise to three different processes that control the growing surface: overhanging, nonlocal growth, and diffusion. It finally leads to a morphological structural transition where the universality changes away from that of Kardar, Parisi, and Zhang, in (2+1) dimensions, but not into that of Edwards and Wilkinson.

Investigation of the formation mechanism of Spindt-type cathode by simulation and experiments

The formation mechanism of Spindt-type metal emitter is investigated by the simulation and the fabrication. There are several factors in this mechanism. The aspect ratio (emitter height/bottom diameter of emitter) is the major factor determined mainly by the physical properties of evaporation materials and process conditions. On the emitter tip, the emission current depends on the aspect ratio that affects the suitable current density for field emission display. In the fabrication of metal emitter by electron beam evaporator, the dependence of the aspect ratio on materials and process conditions is analyzed by the particle simulation using the modified ballistic deposition model. From these results, the formation mechanism of the metal field emitter is analyzed with experimental data, therefore the optimized process condition is obtained for a suitable aspect ratio.

A discrete model for particle deposition

Simulation of deposit growth on a two-dimensional substrate was studied based on a new model that tracks individual cubic particles as they form a deposit structure. The present model is an extension of the classical ballistic deposition model. Effects of three different parameters were studied. These include an attraction parameter that is a measure of the particle to particle attractions, an interaction length within which the particles are assumed to influence and be influenced by surrounding particles, and allowed sticking positions (face-face, edge-edge and corner-corner) that favor particular growth directions. Structures with widely varying properties were obtained using this model. The three parameters were found to have considerable effect on the structure including indications of morphological phase transformations. A new property of the system (saturated roughness/deposit growth rate) was identified that can classify the different types of growth into a single type.