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Communication

Structure of Argon Solid Phases Formed from the Liquid State at Different Isobaric Cooling Rates

by
Eugeny I. German
1,
Shulun B. Tsydypov
1,
Michael I. Ojovan
2,* and
Migmar V. Darmaev
1,3
1
Institute of Mathematics, Physics and Computer Science, Banzarov Buryat State University, Smolina Str., 24a, 670000 Ulan-Ude, Russia
2
Department of Radiochemistry, Lomonosov Moscow State University, 119991 Moscow, Russia
3
Institute of Physical Materials Science, Siberian Branch of the Russian Academy of Sciences, 670047 Ulan-Ude, Russia
*
Author to whom correspondence should be addressed.
Appl. Sci. 2024, 14(3), 1295; https://doi.org/10.3390/app14031295
Submission received: 27 November 2023 / Revised: 27 December 2023 / Accepted: 29 December 2023 / Published: 4 February 2024
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Abstract

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Featured Application

The research presented in this paper can be used to develop methods for determining optimal conditions for thermal and baric synthesis of materials.

Abstract

By the method of molecular dynamics, computer simulation of the processes of isobaric cooling of argon particle systems under initial conditions with a temperature of 150 K at pressure values from 0.1 to 4 MPa to a temperature of 40 K with cooling rates of 108, 109, 1010, 1011 and 1012 K/s was performed. As a result of a computer experiment, coordinate arrays of particles were obtained, which were subjected to the procedure of three-dimensional Voronoi partitioning to identify and calculate the number of elementary cells of the crystal structure. Analysis of the structure of argon solid phases formed during isobaric cooling allowed us to deduce an estimated pattern between the concentration of FCC (face-centered cubic) cells in solid argon and the cooling rate from the liquid state. The evaluation of the orientation of the axes of translation of crystal cells in the array of particle coordinates made it possible to classify the solid phases formed as a result of cooling as single crystals, glassy media with the inclusion of clusters and single cells of FCC structures. It was revealed that during isobaric cooling at a rate not exceeding 108 K/s, argon completely crystallizes, at isobaric cooling rates of 109–1010 K/s, the union of elementary cells of the crystal structure into clusters is observed in glassy argon, and at rates of 1011 K/s and higher at pressures of 1 MPa and lower, solid vitreous phases of argon are formed in which no crystal structure cells are detected.

1. Introduction

The molecular structure of solids makes a significant contribution to the manifestation of certain anisotropic rheological, optical, and even electrical and magnetic properties in them. For example, in the development of integrated circuits, the priority is to obtain single crystals of silicon of macroscopic dimensions, but an even more promising task may be the synthesis of a crystal with a uniform spatial distribution of Schottky defects, which would allow the valence impurities to be fully replaced and the purest semiconductor material to be obtained. The selection of conditions for the synthesis of materials with a given structure, as well as the development of methods for quantifying the structure of solids, are priority tasks of materials science and nanotechnology.
Full-scale research methods, such as X-ray diffraction, allow us to quantify the structure only in a small surface layer of the sample under study, currently tomographic methods for studying bulk crystal structures are still at the level of the concept. In this regard, numerical experiments by the method of molecular dynamics are a kind of “panacea” in the problems of determining quantitative structural characteristics, because in the process of modeling, the researcher has full access to the array of coordinates of the particles of the system, which allows operating this array, subjecting it to processing by clustering algorithms and structure recognition [1,2,3].
Methods of molecular dynamics allow performing computer experiments in which it is possible to simulate thermodynamic processes of molecular systems undergoing phase transitions. This approach can be useful for the selection of conditions for the thermal and baric synthesis of new materials.
Modeling of thermodynamic processes in particle systems with complex chemical bonds is accompanied with great algorithmic and computational difficulties. Therefore, it remains relevant to conduct computer experiments on simple models such as monoatomic particle systems with spherically symmetric interaction potentials in order to develop and test new methods and algorithms and the prospect of their further transfer to more complex models [4,5,6,7,8,9,10].
In this paper, we propose methods for quantitative analysis of coordinate arrays of particles for the presence of elementary cells of a face-centered cubic structure in them, which were tested on the results of numerical experiments on cooling argon from liquid to solid under various conditions. Of course, the above model experiments are quite difficult to repeat in situ, but the calculations performed correspond to the physical nature of the crystallization and glass transition processes and predict the behavior of argon under conditions of intense thermodynamic processes.
The performed studies on the argon model allow us to assert that when choosing the values of the applied pressure and the cooling rate, materials with different structures can be obtained from the liquid phase: single crystals, single crystals with lattice defects, polycrystals, composite substances representing clusters and crystal cells distributed in a glassy mass, as well as glassy media devoid of crystal nuclei.

2. Data and Methods

2.1. Calculated Data

Using the molecular dynamics method, three series of computer experiments that simulated the processes of isobaric cooling of argon particle systems initiated in initial states at a temperature of 150 K and pressure values of 0.5 MPa, 1 MPa, 2 MPa, 3 MPa and 4 MPa. Cooling of the systems was carried out to a temperature of 40 K, with such cooling, gaseous argon first condensed, and after liquid argon passed into a solid crystalline or glassy state. Cooling of the systems was carried out at rates of 108, 109, 1010, 1011 and 1012 K/s.
The simulation was carried out in the NanoSim2020 code [11], in which the numerical solution of the dynamics equations for particle systems was performed using the Verlet velocity form algorithm [12].
In numerical experiments, systems consisting of 4096 argon particles were used, the interaction of which was described by the Lennard-Jones potential [13]:
Uij = 4ε((σ/rij)12 − (σ/rij)6),
where ε/kb = 121 K is the depth of the potential well, σ = 3.4 Å is the effective particle size [14]. This form of interaction potential has proven itself well in numerical experiments on the study of glass transition and crystallization of argon [15,16,17,18].
Periodic boundary conditions were used. The time step at the moment of initialization of the systems was 10−15 s. After initialization, the system was brought to an equilibrium state for 10,000 iterations of modeling with the barostat procedure enabled.
The cooling of the system was modeled by proportionally changing the particle velocities by the kv coefficient, which provides the necessary rate of temperature change dT/dt
kv = (1 − (3kb/2Eki)(dT/dt)nmidΔt)1/2,
where Eki is the kinetic energy of the system at the i-th iteration of the simulation, nmid = 100 is the number of iterations for subsequent relaxation of the system after the act of temperature change.
When simulating cooling at a rate of 1011 K/s and 1012 K/s, the time step was reduced to 10−16 s.
To maintain the set pressure value, a barostat was used, which allows modeling the NPT ensemble described in [2], according to which the linear dimensions of the system and the distances between the particles were adjusted by a factor of µ
µ = (1 + (P(V) − P)/(3P(V) + ∑Uij(rij)2/3V),
where P(V) is the pressure at the current volume V, P is the pressure set by the experimental conditions. The barostat procedure was performed cyclically after each act of temperature change until the required pressure was reached.
During the cooling process, according to the temperature dependence of density (Figure 1), the gas-liquid and liquid-solid transition regions were qualitatively distinguished, at cooling rates above 1011 K/s, the temperature dependence of density did not undergo characteristic differences in the transition region from liquid to solid, which is characteristic of the process glass transition. In general, the density dynamics is comparable to the results of numerical experiments of the predecessors [16].
Upon completion of the cooling procedure, the coordinates of the system particles were stored in text files. These coordinate arrays were used as input data for quantitative structural analysis of argon solid phases formed as a result of computer experiments.

2.2. The Method of Calculating the Concentration of FCC Cells

To explain the method of quantitative calculation of structural characteristics, we present the basic concepts of solid state physics. To calculate, we need a certain unit of measurement, with the help of which it is possible to assess the degree of ordering of the arrangement of atoms. The definition of an elementary cell of a crystal lattice is suitable for this concept—the minimum cell corresponding to a single lattice point of a structure with translational symmetry [22]. The main requirements for the construction of an elementary cell: the volume of the cell must be equal to the volume per one lattice node (the space inside the cell must be the geometric location of the points closest to the lattice node); the cells put together must occupy the entire space of the system.
To construct an elementary cell, they resort to the Voronoi partitioning procedure [23,24]: in the two-dimensional case, segments are drawn from a particle that is a lattice node to neighboring particles, which are split by perpendicular lines in the middle, the figure formed by these lines will be an elementary cell or a Voronoi polygon, in three-dimensional space, segments to the nearest neighbors are split by perpendicular planes and the geometric body formed by these planes will be an elementary cell or a Wigner-Seitz cell [22].
The particles of crystalline argon line up in the nodes of a face-centered cubic lattice [24,25], each node of which, with a three–dimensional Voronoi partition, forms a Wigner-Seitz cell in the form of a rhombododecahedron (Figure 2a)—a 12-sided composed of identical rhombuses. The rhombodecahedron has three axes of symmetry passing through three mutually perpendicular long diagonals of the body, these axes of symmetry coincide with the translation vectors of the FCC lattice.
Particle configurations (arrays of particle coordinates) obtained as a result of numerical experiments on argon cooling to a temperature of 40 K at different cooling rates and pressure values were used as input arguments for Delaunay triangulation procedures [26] and subsequent Voronoi partitioning. For this purpose, the standard Matlab R2013 software package functions Delaunay Triangulation and Voronoi [27] were used.
The geometric bodies obtained at the output of the procedures were compared for compliance with the Wigner-Seitz reference cell of the FCC structure. The topological task of comparing two geometric bodies for similarity is mathematically difficult to implement, so, for example, a geometric body formed as a result of a Voronoi partition for one of the argon particles (Figure 2b) cooled at a speed of 108 K/s from the liquid state to a temperature of 40 K is visually similar to the reference cell, but due to fluctuation distortions in the dislocation of neighboring particles minor deviations from the standard appeared in this cell: additional edges, stretching of the faces. A programmatically implemented recognition algorithm would mark such a cell as non-conforming to the FCC, but such a cell is close to the reference both in shape and size.
Therefore, when implementing the comparative analysis procedure programmatically, the following criteria were selected: with a 5% tolerance, the equality of the volume of the cell under study and the reference cell and with a 6% tolerance, the equality of the longest diagonals. Deviations in linear cell sizes were selected taking into account the critical values of the deviations of atoms from the locations of dislocation in the nodes of the crystal lattice during glass transition in the excited state model [28,29].
The elements of the particle array whose Wigner-Seitz cells met the entered criteria were marked with a logical change as “true”, and those that did not meet the criteria were marked as “false”. The calculation did not involve particles located closer than the value of the lattice period of the FCC to the boundaries of the system, since the Voronoi partition for such particles gives open figures, by default not similar to the reference cell. The concentration of the FCC cells of the system was determined as the ratio of the number of particles marked with a logical variable as “true” to the total number of particles involved in the comparison procedure.

3. Results

Figure 3 shows the results of calculating the concentration of FCC cells in argon solid phases at a temperature of 40 K, obtained as a result of isobaric cooling with different rates dT/dt at different pressure values.
It has been established that no crystal structure cells are formed in solid argon during isobaric cooling at speeds of 1012 K/s and higher. At pressure values of 0.5 MPa and 1 MPa, no FCC structure cells were found in systems formed by cooling at a rate of 1011 K/s, but at high pressure values, a small concentration of crystal structure cells is observed from 2.3% at a pressure of 2 MPa to 5.5% at a pressure of 4 MPa.
In molecular systems formed at a cooling rate of 1010 K/s, the content of crystal structures is estimated at 7–8% at all pressure values maintained during numerical experiments. In systems obtained by cooling at a speed of 109 K/s, the content of 15% to 18% of particles located in the nodes of FCC structures was observed, while the correlation between the concentration of the formed crystal structures and the applied pressure ceases to be traced.
As a result of cooling at a rate of 108 K/s, solid argon with a high degree of crystallization is obtained, the results of numerical calculations show the concentration of FCC cells from 27% to 41%, while there is also no correlation with pressure. It is worth noting that when one particle is shifted to a critical distance from the dislocation position in the node, the Wigner-Seitz cells of the neighboring 12 particles are distorted and cease to meet the selection criteria, therefore, the concentration values of the crystal structure cells from 20% should be considered a sufficiently large indicator, in volume ratio clusters of crystal structure in such a system can occupy about 60% spaces.
The spatial orientation of the Wigner-Seitz cells may reflect the uniformity of the crystal structure and influence the manifestation of anisotropic properties of the solid phase. Of course, the anisotropy of noble gas cryocrystals is rather weakly expressed in comparison with more complex substances, but at the same time, the study of the orientation of the crystal lattice cells at the micro level allows us to trace possible structural defects in ordered media or the unification of individual cells into clusters in less ordered media.
Figure 4a shows the configuration of argon particles at a temperature of 40 K and a pressure of 3 MPa, obtained as a result of a computer experiment on cooling from the liquid phase at a speed of 108 K/s, black lines represent the long diagonals of polyhedra calculated by Voronoi partitioning (particles close to the boundaries of the system were not included in the calculation).
The orthogonality and parallelism of the axes of symmetry of the Wigner-Seitz cells indicate a high degree of ordering of the particles of the system. The phase shown in Figure 3a is a single crystal with local defects.
As a result of cooling at a rate of 109 K/s at 3 MPa (see Figure 4b), a solid representing a large cluster of crystalline FCC structure immersed in an unordered amorphous medium in which smaller crystal clusters are present.
At 1010 K/s cooling (see Figure 4c), the system is an amorphous medium in which differently oriented cells of the crystal structure are evenly distributed, the unification of cells into small clusters of 3–5 elements is noticeable.
Rapid cooling (1011 K/s) leads to a more unstructured system (see Figure 4d), in a loosely packed matrix there are rare single cells of the FCC structure whose spatial orientation is chaotic.

4. Discussion

The study of crystallization and glass transition processes on models of argon atoms has been conducted for more than half a century. Thus, in [17], the results of molecular dynamics experiments on isochoric cooling of particle systems with a Lennard-Jones potential are described, in which signs of crystallization are observed at cooling rates from 1012 K/s and below. In [16], a simulation of isobaric cooling of argon with different cooling rates was performed, while signs of crystallization were observed at cooling rates from 1010 K/s, which is confirmed by the results of our calculations. In contrast to the above noted very fast cooling rates characteristic for molecular dynamic simulations the non-computer experimental fast cooling techniques are typically characterized by much lower rates which result in complete crystallization rather than glass formation in both inorganic [30,31,32,33] and biological systems [34]. Unfortunately, in relation to argon, there are no results of empirical experiments in the literature that investigated the processes of rapid cooling although methods of rapid quenching of melts have been practiced for a long time, such as the common rapid solidification processing (RSP) method [35,36], which provides cooling at rates up to 105–106 K/s, and the fastest picosecond pulsed laser quenching method [37], which provides a cooling rate of up to 1014 K/s. When using such methods, the formation of metallic glasses has been experimentally confirmed [30,31,32,33,38,39].
The most common practice of studying the structure of molecular media in a computer experiment is the calculation and analysis of the shape of the functions of the radial distribution of particles. Such a study was carried out by us in [40]. It is worth noting that the analysis of the radial distribution functions allows us to give a qualitative assessment of the structure [41]. The method proposed in this article provides a quantitative assessment of the concentration of crystal structures in solid phases. This approach opens up prospects for calculating the conditions of thermobaric synthesis of materials with specified structural characteristics: crystals with the required concentration of crystal lattice defects, composite materials representing a glassy medium with clusters of crystals of specified sizes immersed in it e.g., liquid crystalline and glassy systems [10].

5. Conclusions

To study the structure of solids formed as a result of cooling from the liquid phase with different cooling rates, a quantitative approach based on the identification and calculation of the number of elementary cells of the crystal structure can be used.
As a result of the analysis of the structure of the solid phases of argon formed during isobaric cooling, an estimated pattern was established between the concentration of FCC cells in solid argon and the cooling rate from the liquid state.
When analyzing coordinate arrays of particles, patterns were revealed, the tracing of which is practically impossible with a qualitative study of the structure by determining radial distribution functions: with isobaric cooling at a speed not exceeding 108 K/s solid phases are formed with a high degree of crystallization, with isobaric cooling speeds of 109–1010 K/s in glassy argon, the union of elementary cells of the crystal structure into clusters is observed, and with at speeds of 1011 K/s and above at pressures of 1 MPa and below, solid phases of argon are formed in which cells of the crystal structure are not detected.

Author Contributions

Conceptualization, E.I.G. and S.B.T.; methodology, E.I.G. and S.B.T.; software, E.I.G.; formal analysis, M.I.O. and M.V.D.; writing—original draft, E.I.G., S.B.T., M.I.O. and M.V.D.; writing—review and editing, E.I.G., S.B.T., M.I.O. and M.V.D.; visualization, E.I.G. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data are contained within the article.

Acknowledgments

This study was supported by the Banzarov Buryat State University.

Conflicts of Interest

The authors declare no conflicts of interest.

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Applsci 14 01295 g001
Figure 1. Temperature dependence of argon density at a pressure of 4 MPa: blue line—results of modeling cooling at a rate of 108 K/s; red line—results of modeling cooling at a rate of 1011 K/s; circles—reference data [19]; triangles—experimental studies [20,21].
Figure 1. Temperature dependence of argon density at a pressure of 4 MPa: blue line—results of modeling cooling at a rate of 108 K/s; red line—results of modeling cooling at a rate of 1011 K/s; circles—reference data [19]; triangles—experimental studies [20,21].
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Figure 2. The Wigner-Seitz cell of the FCC crystal (a) and the geometric body obtained by spatial Voronoi partitioning for one of the particles of the argon system cooled at a rate of 108 K/s from the liquid state to a temperature of 40 K at a pressure of 4 MPa (b).
Figure 2. The Wigner-Seitz cell of the FCC crystal (a) and the geometric body obtained by spatial Voronoi partitioning for one of the particles of the argon system cooled at a rate of 108 K/s from the liquid state to a temperature of 40 K at a pressure of 4 MPa (b).
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Figure 3. Concentration of FCC cells in solid argon phases formed by isobaric cooling from the liquid phase at speeds of 108–1011 K/s and at pressure values of 0.5 MPa (a), 1 MPa (b), 2 MPa (c), 3 MPa (d), 4 MPa (e).
Figure 3. Concentration of FCC cells in solid argon phases formed by isobaric cooling from the liquid phase at speeds of 108–1011 K/s and at pressure values of 0.5 MPa (a), 1 MPa (b), 2 MPa (c), 3 MPa (d), 4 MPa (e).
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Figure 4. Configurations of particles and axes of symmetry of FCC cells in solid phases of argon formed by cooling from the liquid phase at rates: (a) 108 K/s, (b) 109 K/s, (c) 1010 K/s and (d) 1011 K/s at a pressure value of 3 MPa.
Figure 4. Configurations of particles and axes of symmetry of FCC cells in solid phases of argon formed by cooling from the liquid phase at rates: (a) 108 K/s, (b) 109 K/s, (c) 1010 K/s and (d) 1011 K/s at a pressure value of 3 MPa.
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German, E.I.; Tsydypov, S.B.; Ojovan, M.I.; Darmaev, M.V. Structure of Argon Solid Phases Formed from the Liquid State at Different Isobaric Cooling Rates. Appl. Sci. 2024, 14, 1295. https://doi.org/10.3390/app14031295

AMA Style

German EI, Tsydypov SB, Ojovan MI, Darmaev MV. Structure of Argon Solid Phases Formed from the Liquid State at Different Isobaric Cooling Rates. Applied Sciences. 2024; 14(3):1295. https://doi.org/10.3390/app14031295

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German, Eugeny I., Shulun B. Tsydypov, Michael I. Ojovan, and Migmar V. Darmaev. 2024. "Structure of Argon Solid Phases Formed from the Liquid State at Different Isobaric Cooling Rates" Applied Sciences 14, no. 3: 1295. https://doi.org/10.3390/app14031295

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