Annotation of the results obtained in 2022
Recently, using the gradient and integral molecular density functional method, we have shown that not only stable droplets are formed from vapor around solid lyophilic particles, but also thin stable concentric vapor shells-bubbles appear around lyophobic particles in a stretched liquid. Such droplets and bubbles correspond to local minima of the grand thermodynamic potential of the system with the new phase nucleus, the initial phase, and the nucleation seed. One of the objectives of our work within the framework of the RSF grant is to study stable droplets and bubbles on lyophilic and lyophobic nucleation seeds using the modern molecular density functional method with the most complete allowance for hard-sphere correlations in terms of fundamental measure theory. Within this approach, equilibrium three-dimensional density profiles of droplets around solid lyophilic spherical particles in undersaturated and supersaturated vapor and thin concentric vapor shells on lyophobic particles in a stretched and stable liquid were calculated in 2022. The existence of enveloping stable droplets and bubbles on nanosized lyophilic and lyophobic nucleation seeds has been confirmed, the structure of droplets and bubbles has been described, dependences of the chemical potential of molecules in a droplet and in a bubble on the thickness of a liquid or vapor film have been plotted, and the threshold values of the chemical potential of vapor and liquid for barrier-free heterogeneous nucleation have been found. With the help of the expression for a grand thermodynamic potential as a molecular density functional, the disjoining pressures in thin liquid films around nanosized wettable spherical particles and in thin vapor layers around nonwettable particles are calculated depending on the energy parameters of molecular interactions, film thickness, and particle size. Although the results show a qualitative agreement between the calculated dependences of the disjoining pressure and those obtained in the framework of a simpler gradient method of the molecular density functional, the new results differ significantly quantitatively. It has been confirmed that the disjoining pressure in a liquid film around a nanosized lyophilic particle increases with an increase in particle size and lyophilicity. When studying inhomogeneous fluids using the density functional theory in many of its variants, one employs expressions for the local free energy, as well as for the chemical potential in a system of hard spheres without attraction as functions of the concentration of the hard spheres. Exact expressions for these functions are unknown, but there are various approximations that demonstrate different accuracy. In 2022, we have considered and compare the results given by the Carnahan–Starling, the Percus–Yevik, the Kolafa equations of state, and the 6th and the 7th-order Rusanov equations of state for a system of hard spheres without attraction. The dependences of the compressibility factor, chemical potential, and free energy density on the particle number density have been considered and compared with the modified Kolafa–Labík–Malijevský equation describing results of molecular dynamics simulations of the compressibility factor with high accuracy, though in a limited density range. We have calculated, within the integral density functional theory in random-phase approximation, the molecular density profiles in radially inhomogeneous small spherical droplets and bubbles of an argon-like substance representing critical droplets/bubbles in nucleation. We have such calculations for the Carnahan–Starling, Kolafa and truncated 6th-order Rusanov equations of state for hard spheres, as well as for the modified Kolafa–Labík–Malijevský equation. To describe the fluid, all the employed equations of state were supplemented with a term describing the attraction of particles in a mean-field approximation. When comparing the equations of state for a system of hard spheres to describe both a homogeneous fluid and critical droplets/bubbles, the Kolafa equation turned out to be the best of the considered equations of state for a system of hard spheres, repeating the results for the “reference” Kolafa–Labík–Malijevský equation with high accuracy. At the same time, the Kolafa equation is much simpler and resembles the widely used Carnahan–Starling equation, significantly exceeding it in accuracy in describing both homogeneous and inhomogeneous fluids of hard spheres. The increase in performance of computing equipment as well as its accessibility to scientific community inevitably leads to more and more frequent use of molecular modeling methods, such as molecular dynamics and Monte Carlo methods, for tackling the fundamental and applied scientific problems. These problems certainly include the studies of the properties of sessile droplets on the surfaces of different structure and chemical composition. One of the most important properties of the surface is its wettability, because its modification opens up wide opportunities for practical applications. In 2022, using the molecular dynamics method, we simulated the formation of a sessile droplet on a flat surface and calculated its contact angle which is the macroscopic characteristic of the surface wettability. Two methods for calculating the contact angle differing in the clustering procedure (i.e., separation of the molecules constituting the droplet from the molecules of the gas phase) were used. The droplets consisting of argon or water molecules were considered. An impenetrable wall interacting with a drop according to the Steele potential (“3-9” potential) was used as a flat surface, as well as the all-atom models of graphite and cristobalite (SiO2). For all the considered models of the surface we varied the degree of its wettability. The dependences of the contact angle of a droplet on its size and temperature were studied. As is known, the question of rigorous calculation of the stress tensor of a fluctuating electromagnetic field, free energy, and the Casimir–Polder potential in inhomogeneous systems in the presence of thin films, drops, bubbles, vacuum and air slots is of fundamental importance. Simple molecular field integrations for solids can introduce uncontrollable errors. An important task of this project is to study the dependence of free energy and the Casimir–Polder potential on the geometric characteristics of systems and on the physical and chemical properties of materials. In 2022, the theory of the Casimir effect was considered for two diffraction gratings separated by a vacuum slit. Discontinuity in the Casimir energy during rotation of the grating is shown to be the consequence of the fundamental transformation of structure of reciprocal vector space, which is a result of translational symmetry breaking. A novel explicitly gauge-invariant method for derivation of electric and magnetic Green functions in the Casimir effect theory is developed. The method is based on application of Weyl formula and makes it possible to derive the Casimir pressure and the Casimir-Polder potential of an anisotropic atom by a solution of an explicit gauge-invariant system of equations following from transversality of fields and boundary conditions.

 

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Annotation of the results obtained in 2023
When describing nucleation of droplets in supersaturated vapor, it is usually assumed that a single nucleated droplet is very small compared to the size of the system and cannot noticeably affect the vapor properties. However, a confined system in the form of a small-sized cell is often used in nucleation modeling. In this case, there is an effect of a significant decrease in the concentration of the vapor molecules in the cell during the formation of a new phase embryo. A similar effect exists when calculations are carried out in the framework of, for example, the density functional method using the canonical ensemble. We have considered thermodynamics of droplet nucleation on a solid spherical particle in a supersaturated vapor first at macroscopic level of description using the disjoining pressure in the liquid film, and then using two variants of the classical density functional theory on lyophilic and lyophobic molecular-sized particles. Two modes, depending on the parameters, are found: (1) with an only stable droplet and (2) with three equilibrium droplets, the smallest and largest one being stable and the third one being critical. The appearance of the second (larger) stable droplet is a consequence of the confinement. At a small total number of molecules on the lyophobic particle, a sessile-droplet shaped solution is obtained. It is stable in the canonical ensemble but can correspond to a critical one in the grand canonical ensemble. This makes it possible to calculate the density distributions and characteristics of critical droplets or bubbles, including sessile ones, by minimizing a functional in the canonical ensemble instead of searching for a saddle point in the grand canonical ensemble. This significantly accelerates the calculations and increases their accuracy. In the study of the diffusive growth of gas bubbles and the evolution of their size distribution in gas-saturated solution, the role of the joint effects of capillarity and viscosity has been revealed, that, via the pressure inside the bubble, slow down the transition to stationary or self-similar regimes and may even prevent their establishment at the nucleation stage, making the bubble growth rate essentially non-stationary. They can also noticeably affect the distribution of supercritical gas bubbles during degassing of the solution at the nucleation stage. Thus, an increase in the solution viscosity provides a slowdown in the growth of bubbles of maximum and average size, as well as in the growth rate of the swelling coefficient of the whole solution. Thus, an increase in the solution viscosity provides a slowdown in the growth of bubbles of maximum and average size, as well as in the growth rate of the swelling coefficient of the whole solution. When the solution viscosity is increased by five decimal orders of magnitude at a fixed diffusion coefficient, viscosity has a damping effect on the growth rate of the average bubble radius and the swelling rate by a factor of two and, accordingly, doubles the time of the nucleation stage. Taking into account the feedback between the diffusion coefficient and viscosity, we found a much stronger inhibition by a decimal order when the solution viscosity is changed by three decimal orders. In spite of the fact that the self-similar theory of diffusional growth of a bubble cannot be directly applied at large viscosities, many of its features are preserved. The results obtained allow us to consider any specific system for which the physicochemical parameters, such as gas solubility, surface tension, diffusion and viscosity coefficients, nucleation rate, and work of formation are self-consistent. At present, such a set of values can only be obtained experimentally. Several methods based on density profiles and on Sobel filters have been applied to calculate contact angles of sessile droplets on planar surfaces using the data obtained from molecular dynamics simulations. Using these methods, the contact angles of argon and water droplets with sizes ranging from 1 to 10 nm have been determined. The dependencies of the contact angle on the droplet size were analyzed using the generalized Young equation, the values of both the line tension and the macroscopic contact angle were obtained. Molecular dynamics simulations of the nucleation of argon molecules on solid spherical particles with atomistic structure have been carried out. By varying the parameters of interaction between molecules of the fluid and these molecules with the spherical particle, we were able to adjust the degree of lyophilicity/lyophobicity of the particle and to observe both spherically symmetric liquid films and sessile droplets on the surface of the spherical particle. This important result, obtained by molecular dynamics modeling, confirmed the possibility of observing spherically asymmetric configurations of the “spherical particle – droplet” systems predicted in this project via the density functional method. All-atom molecular dynamics simulations of water films of different thicknesses on a nanoscale hydrophilic spherical particle consisting of silver atoms were also carried out, and the dependence of the chemical potential of a water molecule on the number of water molecules in the liquid film was obtained. The article [V.N. Marachevsky, A.A. Sidelnikov, “Gauge-invariant derivation of the Casimir–Lifshitz pressure”, Phys. Part. Nuclei Lett., 2023, v. 20, No. 5, pp. 1114–1116] presents a new gauge-invariant derivation of electrical Green’s functions in the gap between two dielectric half-spaces. The article also provides a derivation of the Casimir–Lifshitz pressure in this system. In the article [V.N. Marachevsky, A.A. Sidelnikov, “Casimir–Polder interaction with Chern–Simons boundary layers”, Phys. Rev. D, 2023, v. 107, 105019] published in a 1st quartile journal, the case of rotation of polarizations upon reflection of an electromagnetic field from systems with Chern–Simons flat boundary layers on a dielectric substrate is considered. In the article, a new analytical result for the Casimir–Polder potential of an anisotropic atom (molecule) in the presence of a flat Chern–Simons layer on the boundary of a dielectric half-space obtained, generalizing the result for the Casimir–Polder potential of an anisotropic atom in the presence of a flat Chern–Simons layer in vacuum, obtained in the work [V.N. Marachevsky, Yu.M. Pis’mak, “Casimir–Polder effect for a plane with Chern–Simons interaction”, Phys. Rev. D, 2010, v. 81, 065005]. Further in the article, new analytical results are obtained for the Casimir–Polder potential of an anisotropic atom (molecule) located in a vacuum gap between two dielectric half-spaces with boundary plane-parallel Chern–Simons layers. The results obtained for the Casimir–Polder potential have a remarkable property discovered for the first time: when one of the Chern–Simons layers is rotated by 180 degrees, the expression for the Casimir–Polder potential of an atom takes on a different value. We emphasize that in the calculation of the Casimir–Polder potential, the neutral atom (molecule) is described in the point dipole approximation within the framework of quantum electrodynamics.

 

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