Random sequential adsorption of lattice animals on a three-dimensional cubic lattice.

The properties of the random sequential adsorption of objects of various shapes on simple three-dimensional (3D) cubic lattice are studied numerically by means of Monte Carlo simulations. Depositing objects are "lattice animals," made of a certain number of nearest-neighbor sites on a lattice. The aim of this work is to investigate the impact of the geometrical properties of the shapes on the jamming density θ_{J} and on the temporal evolution of the coverage fraction θ(t). We analyzed all lattice animals of size n=1, 2, 3, 4, and 5. A significant number of objects of size n⩾6 were also used to confirm our findings. Approach of the coverage θ(t) to the jamming limit θ_{J} is found to be exponential, θ_{J}-θ(t)∼exp(-t/σ), for all lattice animals. It was shown that the relaxation time σ increases with the number of different orientations m that lattice animals can take when placed on a cubic lattice. Orientations of the lattice animal deposited in two randomly chosen places on the lattice are different if one of them cannot be translated into the other. Our simulations performed for large collections of 3D objects confirmed that σ≅m∈{1,3,4,6,8,12,24}. The presented results suggest that there is no correlation between the number of possible orientations m of the object and the corresponding values of the jamming density θ_{J}. It was found that for sufficiently large objects, changing of the shape has considerably more influence on the jamming density than increasing of the object size.

Particle morphology effects in random sequential adsorption.

The properties of the random sequential adsorption of objects of various shapes on a two-dimensional triangular lattice are studied numerically by means of Monte Carlo simulations. The depositing objects are formed by self-avoiding lattice steps, whereby the size of the objects is gradually increased by wrapping the walks in several different ways. The aim of this work is to investigate the impact of the geometrical properties of the shapes on the jamming density θ_{J} and on the temporal evolution of the coverage fraction θ(t). Our results suggest that the order of symmetry axis of a shape exerts a decisive influence on adsorption kinetics near the jamming limit θ_{J}. The decay of probability for the insertion of a new particle onto a lattice is described in a broad range of the coverage θ by the product between the linear and the stretched exponential function for all examined objects. The corresponding fitting parameters are discussed within the context of the shape descriptors, such as rotational symmetry and the shape factor (parameter of nonsphericity) of the objects. Predictions following from our calculations suggest that the proposed fitting function for the insertion probability is consistent with the exponential approach of the coverage fraction θ(t) to the jamming limit θ_{J}.

Adsorption-desorption processes of polydisperse mixtures on a triangular lattice.

Adsorption-desorption processes of polydisperse mixtures on a triangular lattice are studied by numerical simulations. Mixtures are composed of the shapes of different numbers of segments and rotational symmetries. Numerical simulations are performed to determine the influence of the number of mixture components and the length of the shapes making the mixture on the kinetics of the deposition process. We find that, above the jamming limit, the time evolution of the total coverage of a mixture can be described by the Mittag-Leffler function θ(t)=θ∞-ΔθE[-(t/τ)ß] for all the mixtures we have examined. Our results show that the equilibrium coverage decreases with the number of components making the mixture and also with the desorption probability, via corresponding stretched exponential laws. For the mixtures of equal-sized objects, we propose a simple formula for predicting the value of the steady-state coverage fraction of a mixture from the values of the steady-state coverage fractions of pure component shapes.

Structural characterization of submerged granular packings.

We consider the impact of the effective gravitational acceleration on microstructural properties of granular packings through experimental studies of spherical granular materials saturated within fluids of varying density. We characterize the local organization of spheres in terms of contact connectivity, distribution of the Delaunay free volumes, and the shape factor (parameter of nonsphericity) of the Voronoï polygons. The shape factor gives a clear physical picture of the competition between less and more ordered domains of particles in experimentally obtained packings. As the effective gravity increases, the probability distribution of the shape factor becomes narrower and more localized around the lowest values of the shape factor corresponding to regular hexagon. It is found that curves of the pore distributions are asymmetric with a long tail on the right-hand side, which progressively reduces while the effective gravity gets stronger for lower densities of interstitial fluid. We show that the distribution of local areas (Voronoï cells) broadens with decreasing value of the effective gravity due to the formation of lose structures such as large pores and chainlike structures (arches or bridges). Our results should be particularly helpful in testing the newly developed simulation techniques involving liquid-related forces associated with immersed granular particles.

Optimization of the monolayer growth in adsorption-desorption processes.

Kinetics of the deposition process of dimers in the presence of desorption is studied by Monte Carlo method on a one-dimensional lattice. The aim of this work is to investigate how do various temporal dependencies of the desorption rate hasten or slow down the deposition process. The growth of the coverage θ(t) above the jamming limit to its steady-state value θ(∞) is analyzed when the desorption probability P(des) decreases both stepwise and linearly (continuously) over a certain time domain. We report a numerical evidence that the time needed for a system to reach the given coverage θ can be significantly reduced if P(des) decreases in time. Finally, a self-consistent optimization procedure, when the probability P(des) depends on the current coverage density θ(t), is formulated and tested. The present model reproduces qualitatively the densification kinetics and the memory effects of vibrated granular materials. Our results suggest that the process of vibratory compaction of granular materials can be optimized by using a time dependent intensity of external excitations.

Percolation in random sequential adsorption of extended objects on a triangular lattice.

The percolation aspect of random sequential adsorption of extended objects on a triangular lattice is studied by means of Monte Carlo simulations. The depositing objects are formed by self-avoiding lattice steps on the lattice. Jamming coverage θ{jam}, percolation threshold θ{p}, and their ratio θ{p}/θ{jam} are determined for objects of various shapes and sizes. We find that the percolation threshold θ{p} may decrease or increase with the object size, depending on the local geometry of the objects. We demonstrate that for various objects of the same length, the threshold θ{p} of more compact shapes exceeds the θ{p} of elongated ones. In addition, we study polydisperse mixtures in which the size of line segments making up the mixture gradually increases with the number of components. It is found that the percolation threshold decreases, while the jamming coverage increases, with the number of components in the mixture.

Simulation study of anisotropic random sequential adsorption of extended objects on a triangular lattice.

The properties of the anisotropic random sequential adsorption (RSA) of objects of various shapes on a two-dimensional triangular lattice are studied numerically by means of Monte Carlo simulations. The depositing objects are formed by self-avoiding lattice steps, whereby the first step determines the orientation of the object. Anisotropy is introduced by positing unequal probabilities for orientation of depositing objects along different directions of the lattice. This probability is equal p or (1-p)/2, depending on whether the randomly chosen orientation is horizontal or not, respectively. Approach of the coverage θ(t) to the jamming limit θ(jam) is found to be exponential θ(jam)-θ(t)is proportional to exp(-t/σ), for all probabilities p. It was shown that the relaxation time σ increases with the degree of anisotropy in the case of elongated and asymmetrical shapes. However, for rounded and symmetrical shapes, values of σ and θ(jam) are not affected by the presence of anisotropy. We finally analyze the properties of the anisotropic RSA of polydisperse mixtures of k-mers. Strong dependencies of the parameter σ and the jamming coverage θ(jam) on the degree of anisotropy are obtained. It is found that anisotropic constraints lead to the increased contribution of the longer k-mers in the total coverage fraction of the mixture.

Relaxation properties in a diffusive model of k-mers with constrained movements on a triangular lattice.

We study the relaxation process in a two-dimensional lattice gas model, based on the concept of geometrical frustration. In this model the particles are k-mers that can both randomly translate and rotate on the planar triangular lattice. In the absence of rotation, the diffusion of hard-core particles in crossed single-file systems is investigated. We monitor, for different densities, several quantities: mean-square displacement, the self-part of the van Hove correlation function, and the self-intermediate scattering function. We observe a considerable slowing of diffusion on a long-time scale when suppressing the rotational motion of k-mers; our system is subdiffusive at intermediate times between the initial transient and the long-time diffusive regime. We show that the self-part of the van Hove correlation function exhibits, as a function of particle displacement, a stretched exponential decay at intermediate times. The self-intermediate scattering function (SISF), displaying slower than exponential relaxation, suggests the existence of heterogeneous dynamics. For each value of density, the SISF is well described by the Kohlrausch-Williams-Watts law; the characteristic timescale τ(q(n)) is found to decrease with the wave vector q(n) according to a simple power law. Furthermore, the slowing of the dynamics with density ρ(0) is consistent with the scaling law 1/τ(q(n);ρ(0))∝(ρ(c)-ρ(0))(Ï°), with the same exponent Ï°=3.34±0.12 for all wave vectors q(n). The density ρ(c) is approximately equal to the closest packing limit, θ(CPL)≲1, for dimers on the two-dimensional triangular lattice. The self-diffusion coefficient D(s) scales with the same power-law exponent and critical density.

Adsorption, desorption, and diffusion of k-mers on a one-dimensional lattice.

Kinetics of the deposition process of k -mers in the presence of desorption or/and diffusional relaxation of particles is studied by Monte Carlo method on a one-dimensional lattice. For reversible deposition of k-mers, we find that after the initial "jamming," a stretched exponential growth of the coverage theta(t) toward the steady-state value theta(eq) occurs, i.e., theta(eq)-theta(t) is proportional to exp[-(t/tau)(beta)]. The characteristic time scale tau is found to decrease with desorption probability P(des) according to a power law, tau is proportional to P(des)(-gamma), with the same exponent gamma=1.22+/-0.04 for all k-mers. For irreversible deposition with diffusional relaxation, the growth of the coverage theta(t) above the jamming limit to the closest packing limit (CPL) theta(CPL) is described by the pattern theta(CPL)-theta(t) is proportional to E(beta)[-(t/tau)(beta)], where E(beta) denotes the Mittag-Leffler function of order beta(0,1) . Similarly to the reversible case, we found that the dependence of the relaxation time tau on the diffusion probability P(dif) is consistent again with a simple power-law, i.e., tau is proportional to P(dif)(-delta). When adsorption, desorption, and diffusion occur simultaneously, coverage always reaches an equilibrium value theta(eq), which depends only on the desorption/adsorption probability ratio. The presence of diffusion only hastens the approach to the equilibrium state, so that the stretched exponential function gives a very accurate description of the deposition kinetics of these processes in the whole range above the jamming limit.

Upward penetration of grains through a granular medium.

We study experimentally the creeping penetration of guest (percolating) grains through densely packed granular media in two dimensions. The evolution of the system of the guest grains during the penetration is studied by image analysis. To quantify the changes in the internal structure of the packing, we use Voronoï tessellation and a certain shape factor which is a clear indicator of the presence of different underlying substructures (domains). We first consider the impact of the effective gravitational acceleration on upward penetration of grains. It is found that the higher effective gravity increases the resistance to upward penetration and enhances structural organization in the system of the percolating grains. We also focus our attention on the dependence of the structural rearrangements of percolating grains on some parameters like polydispersity and the initial packing fraction of the host granular system. It is found that the anisotropy of penetration is larger in the monodisperse case than in the bidisperse one, for the same value of the packing fraction of the host medium. Compaction of initial host granular packing also increases anisotropy of penetration of guest grains. When a binary mixture of large and small guest grains is penetrated into the host granular medium, we observe size segregation patterns.

Random sequential adsorption of polydisperse mixtures on discrete substrates.

We study random sequential adsorption of polydisperse mixtures of extended objects both on a triangular and on a square lattice. The depositing objects are formed by self-avoiding random walks on two-dimensional lattices. Numerical simulations were performed to determine the influence of the number of mixture components and length of the shapes making the mixture on the kinetics of the deposition process. We find that the late stage deposition kinetics follows an exponential law theta(t) approximately theta_{jam}-Aexp(-tsigma) not only for the whole mixture, but also for the individual components. We discuss in detail how the quantities such as jamming coverage theta_{jam} and the relaxation time sigma depend on the mixture composition. Our results suggest that the order of symmetry axis of the shape may exert a decisive influence on adsorption kinetics of each mixture component.

Reversible random sequential adsorption of mixtures on a triangular lattice.

Reversible random sequential adsorption of binary mixtures of extended objects on a two-dimensional triangular lattice is studied numerically by means of Monte Carlo simulations. The depositing objects are formed by self-avoiding lattice steps. We concentrate here on the influence of the symmetry properties of the shapes on the kinetics of the adsorption-desorption processes in two-component mixtures. We provide a detailed discussion of the significance of collective events for governing the time coverage behavior of component shapes with different rotational symmetries. We also investigate the role that the mixture composition plays in the deposition process. For the mixtures of equal sized objects, we propose a simple formula for predicting the value of the steady-state coverage fraction of a mixture from the values of the steady-state coverage fractions of pure component shapes.

Simulation study of random sequential adsorption of mixtures on a triangular lattice.

Random sequential adsorption of binary mixtures of extended objects on a two-dimensional triangular lattice is studied numerically by means of Monte Carlo simulations. The depositing objects are formed by self-avoiding random walks on the lattice. We concentrate here on the influence of the symmetry properties of the shapes on the kinetics of the deposition processes in two-component mixtures. Approach to the jamming limit in the case of mixtures is found to be exponential, of the form: theta(t) approximately theta jam - Deltatheta exp(- t/sigma), and the values of the parameter sigma are determined by the order of symmetry of the less symmetric object in the mixture. Depending on the local geometry of the objects making the mixture, jamming coverage of a mixture can be either greater than both single-component jamming coverages or it can be in between these values. Results of the simulations for various fractional concentrations of the objects in the mixture are also presented.

Simulation study of granular compaction dynamics under vertical tapping.

We study, by numerical simulation, the compaction dynamics of frictional hard disks in two dimensions, subjected to vertical shaking. Shaking is modeled by a series of vertical expansion of the disk packing, followed by dynamical recompression of the assembly under the action of gravity. The second phase of the shake cycle is based on an efficient event-driven molecular-dynamics algorithm. We analyze the compaction dynamics for various values of friction coefficient and coefficient of normal restitution. We find that the time evolution of the density is described by rho(t)=rho{infinity}-DeltarhoE{alpha}[-(ttau){alpha}], where E{alpha} denotes the Mittag-Leffler function of order 0

Symmetry effects in reversible random sequential adsorption on a triangular lattice.

Reversible random sequential adsorption of objects of various shapes on a two-dimensional triangular lattice is studied numerically by means of Monte Carlo simulations. The growth of the coverage rho(t) above the jamming limit to its steady-state value rho(infinity) is described by a pattern rho(t) = rho(infinity - deltarhoE(beta)[-(t/tau)beta], where E(beta) denotes the Mittag-Leffler function of order beta element of (0, 1). The parameter tau is found to decay with the desorption probability P_ according to a power law tau = AP_(-gamma). The exponent gamma is the same for all shapes, gamma = 1.29 +/- 0.01, but the parameter A depends only on the order of symmetry axis of the shape. Finally, we present the possible relevance of the model to the compaction of granular objects of various shapes.

Memory effects in vibrated granular systems: response properties in the generalized random sequential adsorption model.

We investigate, by numerical simulation, the dynamical response of a granular system to an abrupt change in shaking intensity within the framework of the reversible random sequential adsorption models. We analyse the two-dimensional lattice model in which, in addition to the adsorption-desorption process, there is diffusion of the adsorbed particles on the surface. Our model reproduces qualitatively the densification kinetics and the memory effects of vibrated granular materials. An interpretation of the simulation results is provided by the analysis of the insertion probability function. The importance of the diffusional relaxation is discussed. We conclude that a complex time-evolution of the density could be explained as a consequence of the variation of the diffusion rate during the compaction. We study the nonequilibrium time-dependent density-density autocorrelation function and show that the model displays out-of-equilibrium dynamical effects such as aging.

Transport theory of granular swarms.

The transport of trace granular gas (swarm) in a carrier granular fluid is studied by means of the Boltzmann-Lorentz kinetic equation. Time-dependent perturbation theory is used to follow the evolution of the granular swarm from an arbitrary initial distribution. A nonhydrodynamic extension of the diffusion equation is derived, with transport coefficients that are time dependent and implicitly depend on the wave vector. Transport coefficients of any order are obtained as velocity moments of the solutions of the corresponding kinetic equations derived from the Boltzmann-Lorentz equation. For the special case of the initial distribution of swarm particles, transport coefficients are identified as time derivatives of the moments of the number density. Finally the granular particle transport theory is extended by the introduction of the concept of non-particle-conserving collisions.