The production of metal nanopowders with specified sizes requires an analysis of the relationship between the physical conditions in industrial installations and the geometric properties of nanoparticles. Predictive mathematical modeling is one of the possibilities of investigating this relationship. The use of thermodynamic nucleation models does not allow one to achieve quantitative agreement with experiment results, so the data can differ by a few orders of magnitude. The kinetic Monte Carlo methods popular in recent decades make it possible to approach the scale of a real experiment; however, they consider an initial state of a medium as ideal crystalline nuclei formed in advance and evenly distributed in space, on the surface of which adsorption and desorption processes take place, and monomers diffuse to their surface in the medium. The rates of these processes are determined by coefficients, the values of which can be chosen to best describe experimental data or calculated using MD simulation. These methods do not consider the stage of nucleation and a detailed description of nucleation; therefore, it is impossible to take into account, e.g., the appearance of stable isomeric forms, "magic" clusters, and so on. In this work, we perform a multiscale simulation of the nucleation of copper nanoparticles in a gas-phase condensation chamber. The data used for a mathematical description of the atomic level of nucleation are obtained by analyzing the results of molecular dynamics calculations of copper vapor condensation at various degrees of supersaturation. The microdescription results are used in a macroscopic model, which includes a system of equations of motion, heat conduction, and diffusion. The analysis includes a description of nucleation, unimolecular growth, and coagulation. The concentrations of clusters of various sizes are taken into account on a logarithmic scale. In areas with a temperature close to the boiling point of copper, cluster grow is found to occur mainly via the attachment of monomers; in low-temperature areas, coagulation is the predominant growth mechanism.

The phenomenon of evaporation from the surface of a liquid droplet into a neutral noncondensible gas was numerically studied by taking forced convection gaseous flow into account. The mathematical model considers the effects of surface tension, gravitational force, viscosity of both liquid and gaseous media, as well as the Stefan flow from the droplet surface, possible free gravitational convection, and the Marangoni convection in droplets, and it is designed to describe diffusion-limited evaporation. We consider the diffusion-limited evaporation process when the diffusive gas flux to the droplet surface is compensated by the convective Stefan flow from the surface. The results indicate an interaction of the liquid and gaseous media. Convective gas flows cause the liquid to move and a vortex to occur in the droplet. The flow velocities in a vortex are 103 times less than the characteristic velocity of forced convection flow in air. The droplet surrounded by gaseous flow changes its shape and oscillates, which causes a gas-density wave. Calculations have shown that the diffusion-limited evaporation rate does not change in the presence of forced convection, which contradicts most of the known experimental works. The possible reason for this discrepancy is the presence of non-equilibrium conditions at the liquid–gas interface in experiments. This leads to a consequent change of the evaporation mode to non-diffusive, while the numerical model postulates the Stefan condition and diffusion-limited evaporation.

Experiments on measuring the rate of evaporation of liquid sessile droplets into air show that the rate of evaporation increases in the presence of forced convection flows. However, data on the effect of convection on evaporation are often contradictory and should be clarified. The paper presents a numerical analysis of evaporation from the surface of a water droplet subjected to forced convection in the gas phase. The drop is located on a smooth horizontal isothermal substrate; the mode with constant contact angle is considered. The shape of the drop has axial symmetry, the same for the velocities and pressure. Forced convection compatible with the symmetry conditions are represented by flows directed downward along the axis of the system and diverging along the sides near the drop and the substrate. The mathematical model is constructed for evaporation controlled by diffusion in the gas phase and takes into account surface tension, gravity, and viscosity in both media, buoyancy and Marangoni convection. The results indicate the existence of the mutual influence of liquid and gaseous media. Thus, a drop vibrates under the influence of movements in the atmosphere, which generates a density wave in the gas: the drop «sounds». The magnitude of the velocity in a liquid is 50 times less than the characteristic velocity in air. It is found that the evaporation rate does not change in the presence of forced convection flows, which contradicts most of the experimental works. The reason for the discrepancies is supposed to be the appearance of nonequilibrium conditions at the boundary of the condensed phase: under these conditions, the evaporation regime ceases to be diffusional.

In this work, we set out to develop a model of gas-phase nucleation in a mixture of copper and argon atoms, which can be further used for analysing macro-systems. Processes occurring at the atomic level are described using coefficients obtained by statistical analysis of molecular dynamic (MD) data on interactions of metal clusters with metal and argon atoms. The MD simulation results are compared with those obtained using the proposed macroscopic model. It is found that the coefficients obtained by averaging the interaction data suitably represent the integral value of the heat of condensation, although result in the smoothing of the energy distribution functions of the clusters. Analysis of the evolution of the number of clusters has shown that the values of their increase rate were lower than those obtained by MD simulation. The conclusion is made, that in order to improve the precision of the developed gas-phase condensation model, it should be supplemented by cluster coagulation.

The results of the molecular-dynamic calculations of gas-phase nanocluster “self-assembly” during metal vapor condensation have been statistically analyzed parting terms of a computational scheme suitable for describing condensation. The laws of collisions and the growth of small copper clusters are revealed. The parameters that describe the interaction between clusters and metal atoms and are used to transfer information to a macroscopic nucleation model are determined. The results can be used for describing nucleation for predicting a nanoparticle size distribution in the production of metal nanopowders.

A molecular dynamics study was carried out on the dependence of the radii of small metal clusters that form in the process of gas-phase copper condensation on the number of atoms in them. The radii are calculated with the molecular-kinetic theory formulas from molecular-dynamic simulating data on the number of collisions between clusters. It was found that the dependence of the radii on the number of particles has a form close to the classical n1/3. However, the average cluster radii are about 40% larger than the case in which the cluster is considered a drop of a bulk fluid.

A computing scheme for a mesoscopic description of a nucleation process has been built using a statistical analysis of the results of molecular dynamics simulation of a supersaturated metallic vapor. The parameters that enter into equations for cluster growth rate and heat release during condensation are found by a direct statistical computation in a molecular dynamics method. The rates of cluster formation calculated by a molecular dynamics method agree well with the numerical model developed in this work.

A statistical analysis of the results of molecular-dynamic calculations of metal (Cu or Ti) vapor condensation in an inert gas (Ar) medium has been performed. Condensation proceeds with the formation of small particles consisting of metal atoms—clusters. The internal energy of the metal cluster—the sum of the kinetic energy of the atoms in the center of mass system and potential energy—is chosen as the key characteristic describing its state. It is shown that the internal energy value gives the possibility for the prediction of the duration of the cluster existence, from birth to decay, i.e., it describes the ability of the cluster to grow. The temporal evolution of the distribution function of the clusters over the internal energy values is presented.

We discuss the problem of dimer formation during the homogeneous nucleation of atomic metal vapor in an inert gas environment. We simulated nucleation with molecular dynamics and carried out the statistical analysis of double- and triple-atomic collisions as the two ways of long-lived diatomic complex formation. Close pair of atoms with lifetime greater than the mean time interval between atom–atom collisions is called a long-lived diatomic complex. We found that double- and triple-atomic collisions gave approximately the same probabilities of long-lived diatomic complex formation, but internal energy of the resulted state was essentially lower in the second case. Some diatomic complexes formed in three-particle collisions are stable enough to be a critical nucleus.

The effects of the spin-orbit interaction of light are observed at the wavelength scale and can be used in nanophotonics. The computer simulation of these effects is a necessary tool for creating devices, based on the spin-orbit interaction of light. We have developed a software package in Python to simulate two known effects, namely, transverse and longitudinal shifts of the center of gravity of beams at the total internal reflection. The package is based on the algorithm for the numerical solution of Maxwell's equations by the finite difference time domain method. We have used the modification of PML - UPML as absorbing boundary conditions. We simulated the refraction and total internal reflection of an electromagnetic wave at the plane interface between two transparent dielectrics and calculated the depth of penetration of a non-uniform electromagnetic wave, occurring at total 1 internal reflection in the optically less dense medium. To test the software package, we have compared the reflection coefficients of the electromagnetic wave for the normal and oblique incidence on the plane interface, calculated by the simulation with the coefficients, obtained by Fresnel formulas for the s-polarization of the wave. We have numerically simulated the shifts of the center of gravity of beams of finite size in the plane of incidence and perpendicular to it, namely, the effects of Goos-Hänchen and Fedorov. The dependence of the direction of the transverse shift on the sign of the circular polarization of the incident beam was demonstrated. The dependence of the longitudinal shift on the direction of linear polarization is also demonstrated. It should be stressed that the developed software package can be applied for simulating of the longitudinal and transverse shifts in anisotropic media, media with optical activity, as well as thin films on the dielectric surface.

Optical phenomena connected with the influence of the longitudinal component (i.e. parallel to the wave vector) of the electric field in electromagnetic wave are considered. The topicality of the study is connected with the analysis of the objects of subwave scales and additional degrees of freedom of spatially inhomogeneous optical fields in new areas of optics; they are photonics, plasmonics and nanooptics. The numerical analysis is carried out for plane waves with Gauss intensity profile and homogeneous (linear or circular) polarization. The solution of Maxwell equations is carried by the method of finite differences. To test the accuracy of the program the reflection coefficients of an electromagnetic wave on the boundary of a transparent dielectric medium with analytic Fresnel solution are compared. The distribution of the electromagnetic field at focusing with a thick symmetrical lens is calculated. The increase in the intensity of the longitudinal component in the focal plane of the lens is found to be ~ 25 fold compared with the value of E || in the incident radiation. A numerical analysis of the geometric spin Hall effect is carried out for the first time when focusing an asymmetrically converging light beam. The effect is in shifting the center of gravity of the intensity distribution of the longitudinal component of the light beam for different states of circular polarization. The calculated shift is ~0,5 λ for right-handed and left polarized light with a focal spot diameter of ~2 λ which agrees with experimental data well. Carried out research has shown that the developed package can be used to calculate field distribution in the propagation of electromagnetic waves of arbitrary configuration in optically inhomogeneous media

Knowing the mechanisms of birth, growth and development of nanoparticles is important for optimization of their production techniques. The majority of nanoparcticles production methods implies self-assembly from the liquid or gas phase. The initial stage of homogeneous nucleation from atomic vapor to a considerable degree specifies the ultimate size distribution of particles, which determines the topicality of its study. The paper presents a statistical analysis of the results of molecular dynamics simulation of metal (Cu) vapor nucleation in the inert gas atmosphere (Ar). The peculiar features of the initial stage of nucleation to estimate probability of diatomic molecule growth (Cu2) in the supersaturated nonequilibrium medium are considered. It’s shown that as a result of collision of two metal atoms an unstable dimer can be formed, which lifetime is comparable to time between metal atom collisions. Based on the results of the statistical analysis we assess the differential probability of forming the long-lived unstable dimer in two-particle interactions depending on the energy value of colliding particles in the system of their mass center. The integration of differential probability in terms of all energies with regard to theoretical energy distribution of copper atoms at the given temperature has allowed us to arrive at an integral estimate of the probability of forming the long-lived dimer in the Cu–Cu collision. It’s been discovered that when increasing the temperature in the range of 300–1500 K the probability of formation is decreased from 0,86 % to 0,16 %.

Numerical simulation of the metallic nanoparticles synthesis in a chamber with an arc discharge were elaborated. The efficiency of the vapor phase condensation of metallic nanoclusters and nanopowders was determined by setting optimum process parameters, the possibilities of experimental estimation of which is limited. Mathematical and physical models were developed to perform computer analysis of the vapor phase condensation to describe the macroscopic characteristics of the process (temperature regime, gas mixture dynamics, diffusion and convective transport of clusters) with allowance for the properties of the components on a microscopic level. The classical Becker-Döring‑Folmer‑Weber thermodynamic nucleation theory was used for the description of the probability of atom-cluster aggregation. The distributions of macroscopic (temperature, pressure, velocity field) and microscopic (cluster size distribution) values in the chamber were obtained. It is found that the size distribution function of clusters deposited on the chamber walls has two peaks, the first – in the region of small clusters (1–50 atoms) and the second – for the clusters containing more than 10 4 atoms. Based on the results of numerical calculations the assumptions are made about the relationship between the type of size distribution function and characteristics of the process.

Statistical analysis of the results of molecular-dynamic modeling for nucleation of metal (Cu) vapors in inert gas atmosphere (Ar) is carried out. Features of the initial stage of nucleation are considered for estimation of the probability of the growth of small atomic clusters within a supersaturated nonequilibrium medium and the efficiency of heat removal from them during collisions with inert gas atoms and other metal atoms.