Journal of Applied Mathematics

Volume 2019, Article ID 7915780, 8 pages

https://doi.org/10.1155/2019/7915780

## Statistical Distributions of Physical Characteristics of Molecules with Casimir Force in the Transfer of Admixtures in Nanoscale Volumes

Correspondence should be addressed to Liudmila Uvarova; ur.xednay@11ravu and Sergey Babarin; moc.liamg@nirababs

Received 25 March 2019; Revised 31 May 2019; Accepted 17 June 2019; Published 14 July 2019

Academic Editor: Syed Abdul Mohiuddine

Copyright © 2019 Liudmila Uvarova and Sergey Babarin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this work, we study the statistical properties of molecular systems of admixtures, which are placed in nanoscale volumes like cube or parallelepiped with impermeable surfaces on their walls. We simulate interactions of free motion of molecules of admixtures with molecules on the surfaces of walls: we modeled them in the form of atomic structure grid using SiO_{2} as a material. This type of substance allows us to take into account the manifestation of one of the important quantum effects: Casimir force. We used its general expression in terms of interaction energy, with the assumption of dependence on distance and projected area between atoms of the wall and atoms of the admixtures. To model surface roughness we used uniform random distribution function for surface heights (z coordinates). The results of computational experiments can be used to estimate the distribution of chemical bond lengths, valence, and dihedral angles lengths deviations in polyatomic molecules. Our model can be useful to determine the stable configuration properties of the system, to solve practical problems in the conditions of physical limitations of nanoscale devices, filtration of admixtures in highly dispersed systems, in the development of mechanisms in structures that have parallel plates or membranes, including porous structures. Also we compute radial distribution function for multicomponent admixtures including atoms of inert gases, water vapor, and ethanol. And we took into account the influence of Casimir force on admixtures for cube and parallelepiped type volumes. In results, we showed distributions of deviations of lengths from their equilibrium values.

#### 1. Introduction

One of the most relevant studies of dynamic systems in the nanoscale is modeling of properties of molecular structures that are necessary for the production of materials with new or improved characteristics as well as determining equilibrium states and defining deviation intervals that are critical parameters in medicine and biotechnology related to gene modification of proteins and viruses. In technological sphere that could be engines for nanoscale robots, filtration of admixtures for air purification systems, adsorption of bacteria, and other applications of natural science and technical disciplines, where there exists production of highly efficient nanoscale systems, resistant to environmental influences are of importance.

In particular, there is class of problems related to the modeling of molecular structures [1, 2]; clusters, admixtures, agglomerates in technical systems [3–5], comparative analysis of models [6] and calculation of the distribution of important parameters [7] are of fundamental interest. Due to use of mathematical models with high parallelization calculations, it becomes possible to carry out intensive computer work with taking into account many features of physical systems with constraints. Methods of numerical simulation had a great influence in the development and solutions of such problems. However, there are a number of technical difficulties when distances between surfaces in real systems are considered less than ten nanometers. This leads to the need of additional consideration of quantum effects contributions and strong interactions. Thus, flat parallel surfaces produce activation of the Casimir force that generates an additional contribution to the net interaction energy. It can cause deformations, structural surface instabilities [8].

In our work, we consider the use of Casimir force as an additional term in the processes of molecular admixtures transfer in nanoscale volumes [9, 10]. And we consider the effect of volumes produced on statistical properties of the system, in particular, the distribution of equilibrium values of chemical bonds in polyatomic molecules. In the existing mathematical models the main types of boundary conditions are periodic [11–13], for example, in film models [14] and in the sphere-plate interactions [15], and that cannot be used in our case to solve problems in confined volumes [16–21]. We propose a model based on interaction potentials, like AMBER and CHARMM.

#### 2. Mathematical Model

Casimir force plays an important role in our model. As noted earlier, it has a quantum nature, but it can also be represented in the classical approximation in the form of the interaction energy potential [22–24]. Description of the evolution of the system is based on the classical system of Newton’s equations. Solution is based on the method of molecular dynamics, which allows us to describe system under the assumption of additivity of potential energy terms in force field. The initial conditions for the model are velocities and coordinates of all atoms of admixture’s molecules, the initial temperature of the system, the size and type of the simulated volume, and the functional description of interactions with system’s boundaries. We do not consider any special conditions for the distribution of velocities and coordinates for the atoms of admixtures; therefore, their values at initial time t=0 are determined by the quantiles of the normal distribution with the most probable value of the speed corresponding to the preset temperature. The coordinates of the atoms inside the volume can be set in free intersections of inner volume grid with a step of 0.5 nm, or by getting real coordinates from measurements of surfaces using the atomic force microscope [25, 26]. We will set the admixtures molecules on virtual inner volume’s grid randomly and distributed uniformly, taking into account the physical constraints and conditions imposed on the distances of intermolecular interaction to eliminate bursts of energy values and instabilities of the numerical simulation. This makes possible to get a minimum of net potential energy at the initial time. If necessary, a preliminary relaxation of the system can be carried out by methods of mathematical optimization.with initial conditions:

##### 2.1. Casimir Force and Interactions with Inner Walls of Modeled Volume

Casimir force (not for the case of quantum calculations) is regarded as a type of van der Waals interaction [19–21] without dipole moment fluctuation. This is a semiclassical approach. To use Casimir force in our model we integrated a well-known expression for the force [28] that can be used as an approximation function of potential energy and depends on distance and interacting area of corresponding atoms (wall-admixtures). Since the surfaces of the box are modeled as molecular structures, we used the same type of “atom-atom” interaction for the calculations for every atom in molecules of admixtures. Further assumption we made is that we consider average of four neighbor atoms that covers projection of admixture’s atoms on the wall surface. As atoms projections are much smaller than the area of the walls and taking into account orthogonal direction of the Casimir force vector, the energy of the interaction is conveniently represented schematically and is shown on Figure 1.