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I. E. Ivanov et al.
to achieve this goal, it is necessary to separate the clusters using a skimmer. This
study examined the nozzle-skimmer system, which is used to obtain Argon clusters.
Using bombardment of inert gas clusters in the future, it is planned to get integrated
circuits to a higher order of smoothness on an industrial scale.
A cluster is a system consisting of atoms or molecules combined by the forces
of van der Waals into a single whole. The number of atoms or molecules in such a
system can vary from units to hundreds of thousands [4, 5].
The properties of molecular clusters depend on their size. The properties of small
clusters are determined by the structure of the molecules, while the properties of large
clusters (droplets) approach the properties of a liquid medium. Clusters are formed
as a result of collisions of particles; the following processes are taken into account:
elastic collisions of molecules, recombination of molecules, cluster and monomer
associations, cluster associations, and monomer evaporation from a cluster [6, 7].
The process of cluster formation begins with a combination of two atoms or
molecules (monomers) and obtaining dimers, then trimers, etc. As a rule, small
clusters (dimers, trimers, etc.) are obtained as a result of three-particle collisions
in which excess energy (the excess energy of colliding particles over the energy of
the formed cluster) is carried away by one of the primary particles. A similar threeparticle process continues until the formed cluster has a size sufficient to absorb
excess energy in the internal degrees of freedom. After this, more often in the process
of cluster formation, two-particle reactions are realized [6, 7].
Kinetic models are currently used to study cluster formation. The main method in
this direction is statistical modeling using the Monte Carlo method [8–10]. The quasichemical model of condensation has become widespread [9, 11, 12]. Simulation of
the process of volumetric gas condensation can be carried out based on solving
the kinetic equation for the size distribution function of droplets, which describes
the evolution of the spectrum of clusters in time and space [13]. However, a direct
solution of the kinetic equation is possible only for relatively simple model problems.
The moment method considers two processes that affect the mass fraction of the
liquid component: cluster formation (nucleation) and growth of formed clusters due
to condensation. Equations describing an evolution of clusters (droplets) in the gas
medium flow are obtained as moments from the general equation of dynamics of the
size distribution function of cluster droplets. In this work, the system of equations uses
the equation for the mass fraction of the liquid phase and the equation for the mass
fraction of the condensing phase at the beginning of the calculation (the sum of the
mass fractions of the liquid and vapor phases) [14]. Using the equation for the mass
fraction of the condensing phase is a new element in the model of moments that allows
us to significantly expand the range of problems to be solved, for example, to simulate
the outflow of a jet with the condensation of vapor of a condensing substance into a
space filled with medium without a condensing substance. Significant development
of the method of moments is its generalization to the case of droplet evaporation
[15, 16].
In mathematical modeling of the process of homogeneous condensation including
moment methods, the results of classical nucleation theory (CNT) are used [16–19].
I. E. Ivanov et al.
to achieve this goal, it is necessary to separate the clusters using a skimmer. This
study examined the nozzle-skimmer system, which is used to obtain Argon clusters.
Using bombardment of inert gas clusters in the future, it is planned to get integrated
circuits to a higher order of smoothness on an industrial scale.
A cluster is a system consisting of atoms or molecules combined by the forces
of van der Waals into a single whole. The number of atoms or molecules in such a
system can vary from units to hundreds of thousands [4, 5].
The properties of molecular clusters depend on their size. The properties of small
clusters are determined by the structure of the molecules, while the properties of large
clusters (droplets) approach the properties of a liquid medium. Clusters are formed
as a result of collisions of particles; the following processes are taken into account:
elastic collisions of molecules, recombination of molecules, cluster and monomer
associations, cluster associations, and monomer evaporation from a cluster [6, 7].
The process of cluster formation begins with a combination of two atoms or
molecules (monomers) and obtaining dimers, then trimers, etc. As a rule, small
clusters (dimers, trimers, etc.) are obtained as a result of three-particle collisions
in which excess energy (the excess energy of colliding particles over the energy of
the formed cluster) is carried away by one of the primary particles. A similar threeparticle process continues until the formed cluster has a size sufficient to absorb
excess energy in the internal degrees of freedom. After this, more often in the process
of cluster formation, two-particle reactions are realized [6, 7].
Kinetic models are currently used to study cluster formation. The main method in
this direction is statistical modeling using the Monte Carlo method [8–10]. The quasichemical model of condensation has become widespread [9, 11, 12]. Simulation of
the process of volumetric gas condensation can be carried out based on solving
the kinetic equation for the size distribution function of droplets, which describes
the evolution of the spectrum of clusters in time and space [13]. However, a direct
solution of the kinetic equation is possible only for relatively simple model problems.
The moment method considers two processes that affect the mass fraction of the
liquid component: cluster formation (nucleation) and growth of formed clusters due
to condensation. Equations describing an evolution of clusters (droplets) in the gas
medium flow are obtained as moments from the general equation of dynamics of the
size distribution function of cluster droplets. In this work, the system of equations uses
the equation for the mass fraction of the liquid phase and the equation for the mass
fraction of the condensing phase at the beginning of the calculation (the sum of the
mass fractions of the liquid and vapor phases) [14]. Using the equation for the mass
fraction of the condensing phase is a new element in the model of moments that allows
us to significantly expand the range of problems to be solved, for example, to simulate
the outflow of a jet with the condensation of vapor of a condensing substance into a
space filled with medium without a condensing substance. Significant development
of the method of moments is its generalization to the case of droplet evaporation
[15, 16].
In mathematical modeling of the process of homogeneous condensation including
moment methods, the results of classical nucleation theory (CNT) are used [16–19].