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Using computer simulations, we have studied the percolation and the electrical conductance of two-dimensional, random percolating networks of curved, zero-width metallic nanowires. We mimicked the curved nanowires using circular arcs. The percolation threshold decreased as the aspect ratio of the arcs increased. Comparison with published data on the percolation threshold of symmetric quadratic B\'{e}zier curves suggests that, when the percolation of slightly curved wires is simulated, the particular choice of curve to mimic the shape of real-world wires is of little importance. Considering the electrical properties, we took into account both the nanowire resistance per unit length and the junction (nanowire/nanowire contact) resistance. Using a mean-field approximation (MFA), we derived the total electrical conductance of the nanowire-based networks as a function of their geometrical and physical parameters. The MFA predictions have been confirmed by our Monte Carlo numerical simulations. For our random homogeneous and isotropic systems of conductive curved wires, the electric conductance decreased as the wire shape changed from a stick to a ring when the wire length remained fixed., Comment: 8 pages, 7 figures, 2 tables, 32 Refs.; Supplemental Material: 9 pages, 2 figures, 2 Refs

Evaporation of a drop, though a simple everyday observation, provides a fascinating subject for study. Various issues interact here, such as dynamics of the contact line, evaporation-induced phase transitions, and formation of patterns. The explanation of the rich variety of patterns formed is not only an academic challenge, but also a problem of practical importance, as applications are growing in medical diagnosis and improvement of coating/printing technology. The multi-scale aspect of the problem is emphasized in this review. The specific fundamental problem to be solved, related to the system is the investigation of the mass transfer processes, the formation and evolution of phase fronts and the identification of mechanisms of pattern formation. To understand these problems, we introduce the important forces and interactions involved in these processes, and highlight the evaporation-driven phase transitions and flows in the drop. We focus on how the deposited patterns are related to and tuned by important factors, for instance substrate properties and contents of the drop. In addition, the formation of crust and crack patterns are discussed. The simulation and modeling methods, which are often utilized in this topic, are also reviewed. Finally, we summarize the applications of drop evaporation and suggest several potential directions for future research in this area. Exploiting the full potential of this topic in basic science research and applications needs involvement and interaction between scientists and engineers from disciplines of physics, chemistry, biology, medicine and other related fields.

The behavior of a system of two-dimensional elongated particles (discorectangles) packed into a slit between two parallel walls was analyzed using a simulation approach. The packings were produced using the random sequential adsorption model with continuous positional and orientational degrees of freedom. The aspect ratio (length-to-width ratio, $\varepsilon=l/d$) of the particles was varied within the range $\varepsilon \in [1;32]$ while the distance between the walls was varied within the range $h/d\in [1;80]$. The properties of the deposits when in the jammed state (the coverage, the order parameter, and the long-range (percolation) connectivity between particles) were studied numerically. The values of $\varepsilon$ and $h$ significantly affected the structure of the packings and the percolation connectivity. In particular, the observed nontrivial dependencies of the jamming coverage $\varphi(\varepsilon)$ or $\varphi(h)$ were explained by the interplay of the different geometrical factors related to confinement, particle orientation degrees of freedom and excluded volume effects., Comment: 8 pages, 7 figures, 52 refs

We mimic random nanowire networks by the homogeneous, isotropic, and random deposition of conductive zero-width sticks onto an insulating substrate. The number density (the number of objects per unit area of the surface) of these sticks is supposed to exceed the percolation threshold, i.e., the system under consideration is a conductor. To identify any current-carrying part (the backbone) of the percolation cluster, we have proposed and implemented a modification of the well-known wall follower algorithm -- one type of maze solving algorithm. The advantage of the modified algorithm is its identification of the whole backbone without visiting all the edges. The complexity of the algorithm depends significantly on the structure of the graph and varies from $O\left(\sqrt{N_\text{V}}\right)$ to $\Theta(N_\text{V})$. The algorithm has been applied to backbone identification in networks with different number densities of conducting sticks. We have found that (i) for number densities of sticks above the percolation threshold, the strength of the percolation cluster quickly approaches unity as the number density of the sticks increases; (ii) simultaneously, the percolation cluster becomes identical to its backbone plus simplest dead ends, i.e., edges that are incident to vertices of degree 1. This behavior is consistent with the presented analytical evaluations., Comment: 9 pages, 9 figures, 78 references

Connectedness percolation phenomena in two-dimensional packings of elongated particles (discorectangles) were studied numerically. The packings were produced using random sequential adsorption (RSA) off-lattice model with preferential orientations of particles along a given direction. The partial ordering was characterized by order parameter $S$, with $S=0$ for completely disordered films (random orientation of particles) and $S=1$ for completely aligned particles along the horizontal direction $x$. The aspect ratio (length-to-width ratio) for the particles was varied within the range $\varepsilon \in [1;100]$. Analysis of connectivity was performed assuming a core-shell structure of particles. The value of $S$ affected the structure of packings, formation of long-range connectivity and electrical conductivity behavior. The effects were explained accounting for the competition between the particles' orientational degrees of freedom and the excluded volume effects. For aligned deposition, the anisotropy in electrical conductivity was observed and the values along alignment direction, $\sigma_x$, were larger than the values in perpendicular direction, $\sigma_y$. The anisotropy in localization of percolation threshold was also observed in finite sized packings, but it disappeared in the limit of infinitely large systems., Comment: 13 pages, 14 figures, 75 references

We have proposed and implemented a modification of the well-known wall follower algorithm to identify a backbone (a current-carrying part) of the percolation cluster. The advantage of the modified algorithm is identification of the whole backbone without visiting all edges. The algorithm has been applied to backbone identification in networks produced by random deposition of conductive sticks onto an insulating substrate. We have found that (i) for concentrations of sticks above the percolation threshold, the strength of the percolating cluster quickly approaches unity; (ii) simultaneously, the percolation cluster is identical to its backbone plus simplest dead ends, i.e., edges that are incident to vertices of unit degree., 6 pages, 3 figures, 39 references

Using computer simulation, we have studied the random sequential adsorption of stiff linear $k$-mers onto a square lattice. Each such particle occupies $k$ adjacent lattice sites. During deposition, the two mutually perpendicular orientations of the particles are equiprobable, hence, a macroscopically isotropic monolayer is formed. However, this monolayer is locally anisotropic, since the deposited particles tend to form domains of particles with the same orientation. Using the "excluded area" concept, we have classified lattice sites into several types and examined how the fraction of each type of lattice site varies as the number of deposited particles increases. The behaviors of these quantities have allowed us to identify the following stages of domain formation (i) the emergence of domain seeds; (ii) the filling of domains; (iii) densification of the domains., 6 pages, 8 figures, 33 references

Computer simulation was used to study the random sequential adsorption of identical discorectangles onto a continuous plane . The problem was analyzed for a wide range of discorectangle aspect ratios ($\varepsilon \in [1;100]$). We studied anisotropic deposition, i.e., the orientations of the deposited particles were uniformly distributed within some interval such that the particles were preferentially aligned along a given direction. The kinetics of the changes in the packing density $\varphi$ found at different values of $S_0$ are discussed. Partial ordering of the discorectangles significantly affected the packing density at the jamming state, $\varphi_\text{j}$, and shifted the cusps in the $\varphi_\text{j}(\varepsilon)$ dependencies. The structure of the jammed state was analyzed using the adsorption of disks of different diameters into the porous space between the deposited discorectangles. The analysis of the connectivity between the discorectangles was performed assuming a core---shell structure of particles., Comment: 8 pages, 8 figures, 64 references

The effect of defects on the behaviour of electrical conductivity, σ , in a monolayer produced by the random sequential adsorption of linear k -mers (particles occupying k adjacent sites) onto a square lattice is studied by means of a Monte Carlo simulation. The k -mers are deposited on the substrate until a jamming state is reached. The presence of defects in the lattice (impurities) and of defects in the k -mers with concentrations of d l and d k , respectively, is assumed. The defects in the lattice are distributed randomly before deposition and these lattice sites are forbidden for the deposition of k -mers. The defects of the k -mers are distributed randomly on the deposited k -mers. The sites filled with k -mers have high electrical conductivity, σ k , whereas the empty sites, and the sites filled by either types of defect have a low electrical conductivity, σ l , i.e., a high-contrast, σ k ∕ σ l ≫ 1 , is assumed. We examined isotropic (both the possible x and y orientations of a particle are equiprobable) and anisotropic (all particles are aligned along one given direction, y ) deposition. To calculate the effective electrical conductivity, the monolayer was presented as a random resistor network and the Frank–Lobb algorithm was used. The effects of the concentrations of defects d l and d k on the electrical conductivity for the values of k = 2 n , where n = 1 , 2 , … , 5 , were studied. Increase of both the d l and d k parameters values resulted in decreases in the value of σ and the suppression of percolation. Moreover, for anisotropic deposition the electrical conductivity along the y direction was noticeably larger than in the perpendicular direction, x . Phase diagrams in the ( d l , d k )-plane for different values of k were obtained.

The random sequential adsorption (RSA) of identical elongated particles (discorectangles) on a line ("Paris car parking problem") was studied numerically. An off-lattice model with continuous positional and orientational degrees of freedom was considered. The possible orientations of the discorectanles were restricted between $\theta \in [-\theta_\text{m}; \theta_\text{m}]$ while the aspect ratio (length-to-width ratio) for the discorectangles was varied within the range $\varepsilon \in [1;100]$. Additionally, the limiting case $\varepsilon=\infty$ (i.e., widthless sticks) was considered. We observed, that the RSA deposition for the problem under consideration was governed by the formation of rarefied holes (containing particles oriented along a line) surrounded by comparatively dense stacks (filled with almost parallel particles oriented in the vertical direction). The kinetics of the changes of the order parameter, and the packing density are discussed. Partial ordering of the discorectangles significantly affected the packing density at the jamming state, $\varphi_\text{j}$, and shifted the cusps in the $\varphi_\text{j}(\varepsilon)$ dependencies. This can be explained by the effects on the competition between the particles' orientational degrees of freedom and the excluded volume effects., Comment: 10 pages, 12 figures, 32 references

Using Monte Carlo simulation, we have studied the percolation of discorectangles. Also known as stadiums or two-dimensional spherocylinders, a discorectangle is a rectangle with semicircles at a pair of opposite sides. Scaling analysis was performed to obtain the percolation thresholds in the thermodynamic limits. We found: (i) for the two marginal aspect ratios $\varepsilon = 1$ (disc) and $\varepsilon \to \infty$ (stick) the percolation thresholds coincide with known values within the statistical error; (ii) for intermediate values of $\varepsilon$ the percolation threshold lies between the percolation thresholds for ellipses and rectangles and approaches the latter as the aspect ratio increases., 4 pages, 6 figures, 1 table, 28 references

We consider $N$ circles of equal radii, $r$, having their centers randomly placed within a square domain $\mathcal{D}$ of size $L \times L$ with periodic boundary conditions ($\mathcal{D} \in \mathbb{R}^2$). When two or more circles intersect each other, each circle is divided by the intersection points into several arcs. We found the exact length distribution of the arcs. In the limiting case of dense systems and large size of the domain $\mathcal{D}$ ($L \to \infty$ in such a way that the number of circle per unit area, $n=N/L^2$, is constant), the arc distribution approaches the probability density function (PDF) $f(\psi) = 4 n r^2\exp(-4 n r^2 \psi)$, where $\psi$ is the central angle subtended by the arc. This PDF is then used to estimate the sheet resistance of transparent electrodes based on conductive rings randomly placed onto a transparent insulating film., Comment: 16 pages, 8 figures, 24 references

Aging in a two dimensional suspension containing rods was studied by means of kinetic Monte Carlo simulations. An off-lattice model with continuous positional and orientational degrees of freedom was considered. The initial state was produced using a random sequential adsorption model. During the aging, the rods underwent translational and rotational Brownian motions. The simulations were run at different values of number density (the number of rods per unit area), $\rho$, and of the initial orientation order parameter, $S_i$. The rods were assumed to have core---shell structures. The evolutions of both the connectivity and the order parameter in the course of the aging were examined., Comment: 8 pages, 8 figures

In our research, we were looking for some criteria detecting incorrect lists of co-authors. We focused our efforts on a gift author problem. The term gift author describes a situation where a person is listed as a co-author but this person has an indirect relationship to the work. Scientometric analysis of the main publications of the Russian dissertations was performed. The open database of dissertations (website http://vak.ed.gov.ru/dis-list, websites of organizations, where dissertations were defended) and CRIS of Astrakhan State University (ASU) http://science.asu.edu.ru were used for the research.

The sedimentation of a two dimensional suspension containing rods was studied by means of kinetic Monte Carlo (MC) simulations. An off-lattice model with continuous positional and orientational degrees of freedom was considered. The initial state before sedimentation was produced using a model of random sequential adsorption. During such sedimentation, the rods undergo translational and rotational Brownian motions. The MC simulations were run at different initial number densities (the numbers of rods per unit area), $\rho_i$, and sedimentation rates, $u$. For sediment films, the spatial distributions of the rods, the order parameters and the electrical conductivities were examined. Different types of sedimentation-driven self-assembly and anisotropy of the electrical conductivity were revealed inside the sediment films. This anisotropy can be finely regulated by changes in the values of $\rho_i$ and $u$., Comment: 11 pages, 11 figures, 68 references

We have studied the electrical conductivity of two-dimensional nanowire networks. An analytical evaluation of the contribution of tunneling to their electrical conductivity suggests that it is proportional to the square of the wire concentration. Using computer simulation, three kinds of resistance were taken into account, viz., (i) the resistance of the wires, (ii) the wire---wire junction resistance, and (iii) the tunnel resistance between wires. We found that the percolation threshold decreased due to tunneling. However, tunneling had negligible effect on the electrical conductance of dense nanowire networks., Comment: 6 pages, 5 figures, 45 references

Four samples of transparent conductive films with different numbers of silver nanorings per unit area were produced. The sheet resistance, transparency, and haze were measured for each sample. Using Monte Carlo simulation, we studied the electrical conductivity of random resistor networks produced by the random deposition of the conducting rings onto the substrate. Both systems of equal-sized rings, and systems with rings of different sizes were simulated. Our simulations demonstrated the linear dependence of the electrical conductivity on the number of rings per unit area. Size dispersity decreased the percolation threshold, but without having any other significant effect on the behavior of the electrical conductance. Analytical estimations obtained for dense systems of equal-sized conductive rings were consistent with the simulations., Comment: 6 pages, 8 figures, 21 refs

Using Monte Carlo simulation, we studied the percolation of sticks, i.e. zero-width rods, on a plane paying special attention to the effects of stick alignment and their length dispersity. The stick lengths were distributed in accordance with log-normal distributions, providing a constant mean length with different widths of distribution. Scaling analysis was performed to obtain the percolation thresholds in the thermodynamic limits for all values of the parameters. Greater alignment of the sticks led to increases in the percolation threshold while an increase in length dispersity decreased the percolation threshold. A fitting formula has been proposed for the dependency of the percolation threshold both on stick alignment and on length dispersity., Comment: 6 pages, 6 figures, 2 tables, 36 references

The experiments conducted by various scientific groups indicate that, in dense two-dimensional systems of elongated particles subjected to vibration, the pattern formation is possible. Computer simulations have evidenced that the random walk of rectangular particles in a discrete two-dimensional space can lead to their self-organisation. We propose a technique for calculating the entropy characteristics of a two-dimensional system in a discrete two-dimensional space consisting of rectangular particles of two mutually perpendicular orientations, and a change in these characteristics for a random walk of particles is investigated., 6 pages, 5 figures, presented at 7th International Conference on Mathematical Modeling in Physical Sciences, August 27-31, 2018, Moscow, Russia; to be published in J.Phys.Conf.Ser

We simulated random walk of rectangular particles on a square lattice with periodic boundary conditions. Two kind of particles were investigated, viz., so-called `blind' and `myopic' particles. We found that steady state patterns occurred only for some values of the ratio $L_x/L_y$ where $L_x$ and $Ly$ are the linear sizes of the system and only for `needles', i.e., the particles of size $1 \times k$. Different patterns were observed for `blind' and `myopic' particles., Comment: 6 pages, 9 figures; to be presented at International Conference on Computer Simulation in Physics and beyond, September 24-27, 2018, Moscow, Russia http://csp2018.ac.ru/ and to be submitted to J.Phys.Conf.Ser

Numerical simulations by means of the Monte Carlo method have been performed to study the electrical properties of a two-dimensional composite filled with rodlike particles. The main goal was to study the effect of the alignment of such rods on the anisotropy of its electrical conductivity. A continuous model was used. In this model, the rods have zero-width (i.e., infinite aspect ratio) and they may intersect each other. To involve both the low conductive host matrix and highly conductive fillers (rods) in the consideration, a discretization algorithm based on the use of a supporting mesh was applied. The discretization is equivalent to the substitution of rods with the polyominoes. Once discretized, the Frank-Lobb algorithm was applied to evaluate the electrical conductivity. Our main findings are (i) the alignment of the rods essentially affects the electrical conductivity and its anisotropy, (ii) the discrete nature of computer simulations is crucial. For slightly disordered system, high electrical anisotropy was observed at small filler content, suggesting a method to enable the production of optically transparent and highly anisotropic conducting films.

The electrical conductivity of two-dimensional films filled with rodlike particles (rods) was simulated by the Monte Carlo method. The main attention has been paid to the investigation of the effect of the rod alignment on the electrical properties of the films. Both continuous and lattice approaches were used. Intersections of particles were forbidden. Our main findings are (i) both models demonstrate similar behaviors, (ii) at low concentration of rods, both approaches lead to the same dependencies of the electrical conductivity on the concentration of the rods, (iii) the alignment of the rods essentially affects the electrical conductivity, (iv) at some concentrations of partially aligned rods, the films may be conducting only in one direction, and (v) the films may simultaneously be both highly transparent and electrically anisotropic.

The paper simulated the spread of certain ideas in a professional virtual group. We consider the propagation of excitation in an inhomogeneous excitable medium of high connectivity. It is assumed that the network elements form a complete graph. Parameters of the elements are normally distributed. The simulation showed that interest in the idea can fade or fluctuate depending on the settings in the virtual group. The presence of a permanent excited element with relatively high activity leads to chaos - the fraction of members of the community actively interested in an idea varies irregularly.

By means of computer simulation, we examined effect of dispersity of filler length on electrical conductivity of two-dimensional (2D) composites with rod-like fillers. Continuous approach has been used. Highly conductive zero-width rod-like particles were deposited uniformly with given anisotropy onto a poorly conductive substrate. Length of particles varied according to the lognormal distribution. Our simulation evidenced that slightly disordered systems may have both the high figure of merit, i.e., both high electrical conductivity and the optical transmission simultaneously, and the high electrical anisotropy., Comment: 6 pages, 7 figures; to submitted to J.Phys.Conf.Ser, Proceedings of CSP2018

Using Monte Carlo simulation, we studied the electrical conductivity of two-dimensional films. The films consisted of a poorly conductive host matrix and highly conductive rod-like fillers (rods). The rods were of various lengths fitting a log-normal distribution. They could be aligned along a direction. Special attention was paid to films having completely aligned rods. The impact of length dispersity and the extent of rod alignment on the insulator-to-conductor phase transition was studied. The greater the length dispersity the smaller the critical concentration. The anisotropy of the electrical conductivity was more pronounced in the vicinity of the phase transition. A finite size effect was found to be significant only in the vicinity of the phase transition., Comment: 10 pages, 9 figures, 1 table, 51 references; submitted to Journal of Applied Physics

The vertical drying of a two dimensional colloidal film containing zero-thickness sticks (lines) was studied by means of kinetic Monte Carlo (MC) simulations. The continuous two-dimensional problem for both the positions and orientations was considered. The initial state before drying was produced using a model of random sequential adsorption with isotropic orientations of the sticks. During the evaporation, an upper interface falls with a linear velocity in the vertical direction and the sticks undergo translational and rotational Brownian motions. The MC simulations were run at different initial number concentrations (the numbers of sticks per unit area), $p_i$, and solvent evaporation rates, $u$. For completely dried films, the spatial distributions of the sticks, the order parameters and the electrical conductivities of the films in both the horizontal, $x$, and vertical, $y$, directions were examined. Significant evaporation-driven self-assembly and stratification of the sticks in the vertical direction was observed. The extent of stratification increased with increasing values of $u$. The anisotropy of the electrical conductivity of the film can be finely regulated by changes in the values of $p_i$ and $u$., 10 pages, 8 figures, 47 references, submitted to PRE

In this work, a model is developed for investigating the redistribution of colloidal particles in the film of an aqueous solution evaporating on a solid horizontal substrate under a mask with holes. Considering the characteristic horizontal film size as large and taking into account the symmetry in the arrangement of the holes in the mask the problem is solved for one film cell under a mask with a hole in its center. It is believed that vapour passes into the atmosphere only through the hole in the mask, the vapor flux density is calculated on the basis of the equation of steady-state diffusion of vapor in the atmosphere. The height-averaged velocity and volume fraction of colloidal particles are calculated using the conservation of mass, taking into account diffusion and deposition of particles onto the substrate. The calculation is performed using FlexPDE. We study the effect of the ratio of hole radius to hole spacing, the distance between the film and the mask, the diffusion, the deposition, the initial volume fraction of particles on the redistribution of particles in solution and on the substrate at the initial stage of film drying. These studies have shown that the behavior of the redistribution of particles in the solution and on the substrate depends primarily on the distance between the film and the mask, on the ratio of hole radius to hole spacing, on the diffusion coefficient of the particles. These parameters determine whether the particles will accumulate under the holes in the mask or will be distributed uniformly. The results agree with the experimental data.

The diffusion-driven self-assembly of rod-like particles was studied by means of Monte Carlo simulation. The rods were represented as linear $k$-mers (i.e., particles occupying $k$ adjacent sites). In the initial state, they were deposited onto a two-dimensional square lattice of size $L\times L$ up to the jamming concentration using a random sequential adsorption algorithm. The size of the lattice, $L$, was varied from $128$ to $2048$, and periodic boundary conditions were applied along both $x$ and $y$ axes, while the length of the $k$-mers (determining the aspect ratio) was varied from $2$ to $12$. The $k$-mers oriented along the $x$ and $y$ directions ($k_x$-mers and $k_y$-mers, respectively) were deposited equiprobably. In the course of the simulation, the numbers of intraspecific and interspecific contacts between the same sort and between different sorts of $k$-mers, respectively, were calculated. Both the shift ratio of the actual number of shifts along the longitudinal or transverse axes of the $k$-mers and the electrical conductivity of the system were also examined. For the initial random configuration, quite different self-organization behavior was observed for short and long $k$-mers. For long $k$-mers ($k\geq 6$), three main stages of diffusion-driven spatial segregation (self-assembly) were identified: the initial stage, reflecting destruction of the jamming state, the intermediate stage, reflecting continuous cluster coarsening and labyrinth pattern formation and the final stage, reflecting the formation of diagonal stripe domains. Additional examination of two artificially constructed initial configurations showed that this pattern of diagonal stripe domains is an attractor, i.e., any spatial distribution of $k$-mers tends to transform into diagonal stripes. Nevertheless, the time for relaxation to the steady state essentially increases as the lattice size growth., 12 pages, 14 figs, 3 videos, 53 refs. Submitted to Phys. Rev. E

Formation of the ring-like pattern by nano-particles of the desiccated droplet of a laponite-based aqueous nanofluid (the well known ‘coffee-ring effect’) is studied. The temperature was controlled and fixed (T = 300 K). The laponite concentration was controlled and varied in order to investigate its effect on the pattern formation. It was observed that the coffee-ring effect vanishes if concentration of laponite exceeds 5%. A simple model for analysis of colloidal sessile droplet desiccation is proposed. The model describes correctly dynamics of the drying droplet profile and the final shape of the dried film (deposit). The model is based on the mass conservation law and reasonable assumption that deposit (gel phase) prevents the flows.

Using Monte Carlo simulation, we studied the electrical conductance of two-dimensional films. The films consisted of a poorly conductive host matrix and highly conductive rodlike fillers (rods). The rods were of various lengths, obeying a log-normal distribution. They were allowed to be aligned along a given direction. The impacts of length dispersity and the extent of rod alignment on the insulator-to-conductor phase transition were studied. Two alternative computational approaches were compared. Within Model I, the films were transformed into resistor networks with regular structures and randomly distributed conductances. Within Model II, the films were transformed into resistor networks with irregular structures but with equal conductivities of the conductors. Comparison of the models evidenced similar behavior in both models when the concentration of fillers exceeded the percolation threshold. Some analytical results were obtained: (i) the relationship between the number of fillers per unit area and the transmittance of the film within Model I, (ii) the electrical conductance of the film for dense networks within Model II., Comment: 8 pages, 9 figures, 58 references

Using lubrication theory, drying processes of sessile droplets of blood serum (i.e. colloids with the salt admixture) on a solid hydrophilic horizontal substrate are studied. The concentration of the solute is supposed to be high enough to ensure good adhesion and strong anchoring (pinning) of the triple line, thereby a drop is desiccating with constant base. The simulation of spatial distribution of the components in the droplet of biological fluid drying on a substrate is performed using advection–diffusion equation. The proposed model explains the redistribution of components in a sessile drop of biological fluid when it dries. Competition between advection and diffusion leads to large particles (protein) accumulating at the edge of the drop (well-known coffee-ring effect), while solutes (salts) are distributed more uniformly across the diameter of the sample.

Diffusion in a two-species 2D system has been simulated using a lattice approach. Rodlike particles were considered as linear $k$-mers of two mutually perpendicular orientations ($k_x$- and $k_y$-mers) on a square lattice. These $k_x$- and $k_y$-mers were treated as species of two kinds. A random sequential adsorption model was used to produce an initial homogeneous distribution of $k$-mers. The concentration of $k$-mers, $p$, was varied in the range from 0.1 to the jamming concentration, $p_j$. By means of the Monte Carlo technique, translational diffusion of the $k$-mers was simulated as a random walk, while rotational diffusion was ignored. We demonstrated that the diffusion coefficients are strongly anisotropic and nonlinearly concentration-dependent. For sufficiently large concentrations (packing densities) and $k \geq 6$, the system tends toward a well-organized steady state. Boundary conditions (BC) predetermine the final state of the system. When periodic BCs are applied along both directions of the square lattice, the system tends to a steady state in the form of diagonal stripes. The formation of stripe domains takes longer time the larger the lattice size, and is observed only for concentrations above a particular critical value. When insulating (zero flux) BCs are applied along both directions of the square lattice, each kind of $k$-mer tries to completely occupy a half of the lattice divided by a diagonal, e.g., $k_x$-mers locate in the upper left corner, while the $k_y$-mers are situated in the lower right corner ("yin-yang" pattern). From time to time, regions built of $k_x$- and $k_y$-mers exchange their locations through irregular patterns. When mixed BCs are used (periodic BCs are applied along one direction whereas insulating BCs are applied along the other one), the system still tends to form the stripes, but they are unstable and change their spatial orientation., Comment: 18 pages, 9 figures, 24 references

The vertical drying of a colloidal film containing rod-like particles was studied by means of kinetic Monte Carlo (MC) simulation. The problem was approached using a two-dimensional square lattice and the rods were represented as linear $k$-mers (i.e., particles occupying $k$ adjacent sites). The initial state before drying was produced using a model of random sequential adsorption (RSA) with isotropic orientations of the $k$-mers (orientation of the $k$-mers along horizontal $x$ and vertical $y$ directions are equiprobable). In the RSA model, overlapping of the $k$-mers is forbidden. During the evaporation, an upper interface falls with a linear velocity of $u$ in the vertical direction and the $k$-mers undergo translation Brownian motion. The MC simulations were run at different initial concentrations, $p_i$, ($p_i \in [0, p_j]$, where $p_j$ is the jamming concentration), lengths of $k$-mers ($k \in [1, 12]$), and solvent evaporation rates, $u$. For completely dried films, the spatial distributions of $k$-mers and their electrical conductivities in both $x$ and $y$ directions were examined. Significant evaporation-driven self-assembly and orientation stratification of the $k$-mers oriented along the $x$ and $y$ directions were observed. The extent of stratification increased with increasing value of $k$. The anisotropy of the electrical conductivity of the film can be finely regulated by changes in the values of $p_i$, $k$ and $u$., 11 pages, 10 figures, submitted to Phys. Rev. E

Using lubrication theory, drying processes of sessile colloidal droplets on a solid substrate are studied. A simple model is proposed to describe temporal dynamics of both the shape of the drop and the volume fraction of the colloidal particles inside the drop. The concentration dependence of the viscosity is taken into account. It is shown that the final shapes of the drops depend on both the initial volume fraction of the colloidal particles and the capillary number. The results of our simulations are in a reasonable agreement with the published experimental data. Computations for the drops of aqueous solution of human serum albumin are presented.

Pattern formation in a two-dimensional system of rod-like particles has been simulated using a lattice approach. Rod-like particles were modelled as linear $k$-mers of two mutually perpendicular orientations ($k_x$- and $k_y$-mers) on a square lattice with periodic boundary conditions (torus). Two kinds of random sequential adsorption model were used to produce the initial homogeneous and isotropic distribution of $k$-mers with different values of packing fraction. By means of the Monte Carlo technique, translational diffusion of the $k$-mers was simulated as a random walk, while rotational diffusion was ignored, so, $k_x$- and $k_y$-mers were considered as individual species. The system tends toward a well-organized nonequilibrium steady state in the form of diagonal stripes for the relatively long $k$-mer ($k \geq 6$) and moderate packing densities (in the interval $p_{down} < p < p_{up}$, where both the critical packing fractions $p_{down}$ and $p_{up}$ are depended on $k$)., 10 pages, 10 figures, 20 references; XXIX IUPAP Conference in Computational Physics (CCP2017) https://ccp2017.sciencesconf.org/; submitted to J.Phys.Conf.Ser

A convection-diffusion equation is used to study the effect of the character of evaporation on the substance redistribution in a drying drop of a multicomponent fluid. Diffusion is shown to hinder the motion of substances with a high diffusion coefficient toward the periphery of the evaporating drop. Qualitatively, the model law describing the vapor flux density does not affect the character of substance redistribution.

Flow field and height averaged radial velocity inside a droplet evaporating in an open circular well were calculated for different modes of liquid evaporation.

The electrical conductivity of a monolayer produced by the random sequential adsorption (RSA) of linear $k$-mers onto a square lattice was studied by means of computer simulation. Overlapping with pre-deposited $k$-mers and detachment from the surface were forbidden. The RSA continued until the saturation jamming limit, $p_j$. The isotropic and anisotropic depositions for two different models: of an insulating substrate and conducting $k$-mers (C-model) and of a conducting substrate and insulating $k$-mers (I-model) were examined. The Frank-Lobb algorithm was applied to calculate the electrical conductivity in both the $x$ and $y$ directions for different lengths ($k=1$ -- $128 $) and concentrations ($p=0$ -- $p_j$) of the $k$-mers. The `intrinsic electrical conductivity' and concentration dependence of the relative electrical conductivity $\Sigma (p)$ ($\Sigma=\sigma/ \sigma_m$ for the C-model and $\Sigma=\sigma_m /\sigma$ for the I-model, where $\sigma_m$ is the electrical conductivity of substrate) in different directions were analyzed. At large values of $k$ the $\Sigma (p)$ curves became very similar and they almost coincided at $k=128$. Moreover, for both models the greater the length of the $k$-mers the smoother the functions $\Sigma_{xy}(p)$, $\Sigma_{x}(p)$ and $\Sigma_{y}(p)$. For the C-model, the other interesting findings are: for large values of $k$ ($k=64, 128$), the values of $\Sigma_{xy}$ and $\Sigma_{y}$ increase rapidly with the initial increase of $p$ from 0 to 0.1; for $k \geq 16$, all the $\Sigma_{xy}(p)$ and $\Sigma_{x}(p)$ curves intersect with each other at the same iso-conductivity points; for anisotropic deposition, the percolation concentrations are the same in the $x$ and $y$ directions, whereas, at the percolation point the greater the length of the $k$-mers the larger the anisotropy of the electrical conductivity, i.e., the ratio $\sigma_y/\sigma_x$ ($>1$)., Comment: 11 pages, 12 figures, 2 tables, 44 references, submitted to Phys.Rev.E

The influence of disorder in the cation sublattice on the magnetic properties of 1: 1 double perovskites is investigated in the framework of the Heisenberg model. It is revealed that the dependence of the sublattice magnetization on the degree of atomic ordering for strontium ferromolybdate exhibits a linear behavior. The results of simulation allow the assumption that disordered lead ferroniobate has a spin-glass structure.

It is shown that, when a drop of a biological fluid dries out on a substrate, diffusion processes to a great extent suppress the transfer of the salt toward the periphery of the drop by capillary flows and still have a minor effect on the motion of colloid particles.

Monte Carlo simulation studies were performed to examine the implications of octahedral cation antisite disorder for magnetization in the double 1:1 perovskites A 2+ (B1 3+ 1/2 B2 5+ 1/2 ) O 3 : Sr 2 FeMoO 6 (SFMO) and Pb 2 FeNbO 6 (PFN) at finite temperature. The percolation approach was applied at 0 K. We found that about 14.5 % of antisite defects is a critical value for the properties of the 1:1 double perovskites.

We simulated an experiment in which a thin colloidal sessile droplet is allowed to dry out on a horizontal hydrophilic surface when a mask just above the droplet predominantly allows evaporation from the droplet free surface directly beneath the holes in the mask [Harris D J, Hu H, Conrad J C and Lewis J A 2007 \textit{Phys. Rev. Lett.} \textbf{98} 148301]. We considered one particular case when centre-to-centre spacing between the holes is much less than the drop diameter. In our model, advection, diffusion, and sedimentation were taken into account. FlexPDE was utilized to solve an advection-diffusion equation using the finite element method. The simulation demonstrated that the colloidal particles accumulate below the holes as the solvent evaporates. Diffusion can reduce this accumulation., Comment: 6 pages, 5 figs

The jamming and percolation for two generalized models of random sequential adsorption (RSA) of linear $k$-mers (particles occupying $k$ adjacent sites) on a square lattice are studied by means of Monte Carlo simulation. The classical random sequential adsorption (RSA) model assumes the absence of overlapping of the new incoming particle with the previously deposited ones. The first model LK$_d$ is a generalized variant of the RSA model for both $k$-mers and a lattice with defects. Some of the occupying $k$ adjacent sites are considered as insulating and some of the lattice sites are occupied by defects (impurities). For this model even a small concentration of defects can inhibit percolation for relatively long $k$-mers. The second model is the cooperative sequential adsorption (CSA) one, where, for each new $k$-mer, only a restricted number of lateral contacts $z$ with previously deposited $k$-mers is allowed. Deposition occurs in the case when $z\leq (1-d)z_m$ where $z_m=2(k+1)$ is the maximum numbers of the contacts of $k$-mer, and $d$ is the fraction of forbidden NN contacts. Percolation is observed only at some interval $k_{min}\leq k\leq k_{max}$ where the values $k_{min}$ and $k_{max}$ depend upon the fraction of forbidden contacts $d$. The value $k_{max}$ decreases as $d$ increases. A logarithmic dependence of the type $\log(k_{max})=a+bd$, where $a=-4.03 \pm 0.22$, $b=4.93 \pm 0.57 $, is obtained., Comment: 7 pages, 8 figures, 32 references

The effect of defects on the percolation of linear $k$-mers (particles occupying $k$ adjacent sites) on a square lattice is studied by means of Monte Carlo simulation. The $k$-mers are deposited using a random sequential adsorption mechanism. Two models, $L_d$ and $K_d$, are analyzed. In the $L_d$ model, it is assumed that the initial square lattice is non-ideal and some fraction of sites, $d$, is occupied by non-conducting point defects (impurities). In the $K_d$ model, the initial square lattice is perfect. However, it is assumed that some fraction of the sites in the $k$-mers, $d$, consists of defects, i.e., are non-conducting. The length of the $k$-mers, $k$, varies from 2 to 256. Periodic boundary conditions are applied to the square lattice. The dependencies of the percolation threshold concentration of the conducting sites, $p_c$, vs the concentration of defects, $d$, were analyzed for different values of $k$. Above some critical concentration of defects, $d_m$, percolation is blocked in both models, even at the jamming concentration of $k$-mers. For long $k$-mers, the values of $d_m$ are well fitted by the functions $d_m \propto k_m^{-\alpha}-k^{-\alpha}$ ($\alpha = 1.28 \pm 0.01$, $k_m = 5900 \pm 500$) and $d_m \propto \log (k_m/k)$ ($k_m = 4700 \pm 1000$ ), for the $L_d$ and $K_d$ models, respectively. Thus, our estimation indicates that the percolation of $k$-mers on a square lattice is impossible even for a lattice without any defects if $k\gtrapprox 6 \times 10^3$., Comment: Submitted to Physical Review E

Calculation of the electronic structure as well as of optical and electric properties of alternate crystals is discussed in terms of the tight-binding method. The crystals in question are of the A, B type where the electron state bands arise as a result of interaction between nearest neighbors. General formulas have been derived for the calculation of dielectric functions. A simple and effective scheme of parameterization of optical transition integrals has been developed.

The processes of formation of electrically conductive films filled by aligned elongated nanoparticles, i.e. nanotubes, nanowires or fibers attract great attention in nanotecnological applications. The alignment can be controlled by external electric fields, evaporation-driven self-assembly and assisted by different other techniques. This work studies the impact of defects on electrical connectivity of ideally aligned monolayer of rods (k-mers). By means of Monte Carlo simulation the problem of percolation for conductive rods on a discrete insulating substrate (square lattice) is analyzed. The aspect ratio of the particles changes within the interval 1-64, the insulating defects were distributed both on the lattice and on the particles. We found that even a very small amount of the insulating defects on the particles can destroy the electrical connectivity. The critical concentration of the defects decreases as the aspect ratio of the particles increases.

We simulated the spread of an activity in a virtual group using the model of excitable medium. We assumed that the structure of the virtual group corresponds to a scale- free network. In our simulation, the network consists of 100 nodes, the average degree of the nodes is 1.98. We considered the propagation of excitation both in a homogeneous and an inhomogeneous excitable medium. The simulation showed that the initial conditions have a little effect on the behaviour of the model. In inhomogeneous medium, fraction of the excited nodes increases, when permanent excited elements ('active' centres) appear in the network. The fraction of the excited nodes increases, when we increase the number of the permanent excited elements. Locations of the active centres do not affect at the level of excitation. External source of activator increases the fraction of the excited nodes in the scale-free network with distribution of parameters.