CHANGE OF SHORT-RANGE ORDER IN LIQUID PURE metals and alloys AS FUNCTION OF temperature and COMPOSITION: APPLICATION OF computer simulation and OF THE VORONOY POLYHEDRA METHODs

B.R. Gelchinski, A.A. Mirzoev, N.P. Smolin, G.P. Vyatkin

Chelyabinsk Center of the Ural Brunch of Russian Academy of Sciences,

Southern Ural State University, Chelyabinsk, 454080, Russia

Abstract

The uniqueness of results by the Reverse Monte-Carlo (RMC) method was determining by comparing the characteristics of Voronoi polyhedra in the RMC- model and a model constructed by the classical method of molecular dynamics (MD) on example of the few pure metals. Both models gave pair correlation functions, which practically coincided with the target, which were obtained using the diffraction experiment. The results of the analysis carried out allow us to conclude that the use of a RMC-method for simulating the structure of a liquid metal is justifiable for such temperatures when the melt becomes structurally homogeneous. It is possible to conclude that the RMC-method allows for a rather adequate reproduction of an atomic structure and can be an effective tool for qualitative and semiquantitative studies of the properties of disordered systems.

The equilibrium atomic structure of three liquid Li-Si alloys (Li80Si20, Li65Si35 and Li57Si43) obtained by RMC was analyzed using the Voronoi polyhedron method in order to identify the most typical configuration of the nearest of environment for the atoms of each component. It showed that the change of structure in liquid Li-Si with increasing Li content from 57 to 80 at. % can be represented by a gradual disintegration of a three-dimensional net of silicon atoms to one-dimensional chains, which in turn break up into separate parts, surrounded by lithium cations.

1. Introduction

Due to the occurrence of first-principle molecular dynamics [1], a self-consistent evaluation of a wide range of electronic and atomic properties of topologically random systems can be carried out. However, to employ this method requires large computational capabilities of the most rapid supercomputers. Implemented three-dimensional models contain up to 100 atoms at best. Therefore for systems which contain extended atomic structures, surface phenomena, etc., classical methods of simulation (molecular dynamics (MD) and Monte-Carlo (MC)) are still useful. These methods not only allow for theoretical calculations of properties, but also enable researchers to study the features of atomic structures in the short-range order, which cannot be or are difficult to study using direct experimental methods. To avoid the problem of using pair interatomic potentials within the framework of the classical computer simulation it is possible by employing the so-called Reverse Monte Carlo method (RMC) [2,3].

Shortly after this method was first used, doubts were voiced about its reliability and about the uniqueness of the atomic configurations, which were obtained as the result of such a simulation. The main objection is that the structure factor (SF) and / or pair correlation function (PCF) used for comparison as the goal functions do not determine the structure of a random system in a unique manner. The non-uniqueness follows from the fact that the set of equilibrium configurations of an explored system can be larger than the set of configurations which are generated by the RMC method [4]. The unsuccessful results of simulation for topologically open structures such as liquid and amorphous silicon have proven these fears to be true [5].

Therefore, before using data about the atomic configuration obtained using the RMC-method, it is expedient to clarify the degree of adequacy of a model structure gained by such methods for real liquids. Pure liquid metals are convenient subject for such testing because there are detailed structural data available obtained from diffraction experiments for ceasium [6] and gallium [7]. Recently [8] we have carried out the investigation of caesium melts, having shown that in wide area of temperatures the RMC and MD methods give similar atomic structures. In this work the results of similar research on an example of gallium are considered. Checking the uniqueness of the results gained by the Reverse Monte-Carlo method was carried out by comparing the properties of Voronoi polyhedron (VP) [9] in the RMC- model and a model constructed by the classical method of molecular dynamics.

Melts of alkaline metals with elements III, IV, V groups (type Li-Pb, K-Tl etc.), in which the sharp extreme of physical properties, viewed as a function of concentration of components, is observed, have received the title "compound-forming" liquid alloys [10]. To explain the properties of such systems, the ideas first expressed by Zintl were used, that the ions of alkaline metals are the donors of electrons for atoms of the III, IV, V groups that are more electronegative. Thus, not only single ions forming saltlick structures arise, but polyanionic clusters of various compositions [10] do also. In a series of similar alloys, the Li-Si system takes a special position. In this system, the dependence of properties on concentration has a view of a not single extreme, and wide "plateau", surrounding area of concentrations from 20 up to 43 atomic % of Si. To explain the "plateau" it was conjectured that the reorganization of bonds between atoms of Si happens in such a manner that in all the above-stated range of concentrations of the Fermi level appears to be in the field of a minimum of a density of states [11].

Thus, it is possible to expect that the structure of the short-range order of the melt Li-Si will vary depending on the composition. For this reason it is rather interesting to estimate the parameters of the short-range order of these structures and, to test the validity of the above-mention model representations.

The tool of examination that we selected is a method of statistically geometrical analysis using the Voronoi polyhedrons, which permitted us to obtain detailed, unique information on the structure of the short-range order of topologically random systems. The input data used for this investigation are the results of modeling the structure of melts of the Li-Si systems using the Reverse Monte Carlo method (RMC) for three compositions (Li80Si20, Li65Si35 and Li57Si43). The structural factors of Li-Si melts were obtained experimentally by the method of a diffraction of neutrons [12]. To obtain information about the coordinates of atoms of the investigated melts a RMC of model operation and software package, explicitly circumscribed in [2] was used.

2. Construction of computer models of the atomic structure of liquid metals and alloys.

The models of liquid systems (2000 atoms in the basic cube with periodic boundary conditions) were constructed by RMC-simulation using of [3]. For a measure of the accordance of diffraction data ("target" g0(r)) and model g (r) there was a discrepancy

, (1)

where n1 and n2 - are the numbers of initial and terminating points of a histogram, respectively between those a comparison g0(r) and g (r) was carried out. The quantity of the discrepancy estimated by the expression (1) characterizes the degree of proximity of the model and target PCF. For almost all the temperatures it was possible to construct models discrepancy in Rg < 1%. At this discrepancy the diagrams of two PCFs - target and model - are visually indistinguishable. It is possible to consider them practically congruent. For liquid Ga at 323K the target and model PCF are exhibited in fig.1.

Fig.1. A comparison of pair correlation functions of liquid gallium at 323K

1- the diffraction experiment (target) [7], 2 - the RMC model,

3 - the MD- model.

The potential of interatomic interaction for MD-simulation was calculated using the Reatto method [5] from the above mentioned diffraction data. So, in this case the concurrence of the model PCF with the target was ensured. The same results showing of the accordance settlement and the target PCF, are obtained for partial PCF of an Li-Si alloys.

3. The statistical geometrical analysis of structure models using the Voronoi polyhedron method.

The Russian mathematician, G.F. Voronoi, [13] formulated a mathematical procedure for the build-up of polyhedrons. The idea of applying Voronoi polyhedrons (VP) as a tool to study local atomic configurations in topologically disordered systems was proposed by J. Bernal [14]. However, he only used mechanical models. New opportunities to conduct statistically geometrical analyses of the structure of the short-range order of random systems using VP have arisen since computer methods have been developed to construct polyhedrons on the basis of computer simulation of structures in works of J. Finney [15].

The VP is analogous for disordered structure, well known in crystals, the Wigner-Seitz cell. The VP for a particular center atom is defined as the volume of space containing all points closer to this center atom than to other atoms. In a three-dimensional space a VP is a convex polyhedron formed by planes drawn perpendicularly to the inter-center vectors at their midpoints. The neighboring atoms making the VP faces are geometrical neighbors of the given center atom. A set of all the VP of the atomic system forms three-dimensional tessellation, which fills the space with neither gaps nor overlaps. The VP are fundamental geometrical figures, which make it possible to carry out analyses of atomic structure in terms of their unique topological and metric characteristics.

The most widely applicable, of the statistically-geometrical analysis, of random systems using VP are topological (distribution of the number of faces and the number of edges per face and topological types) and metric (distribution of VP volumes, distribution of VP face areas, the radial distribution of geometric neighbors, the sphericity coefficient etc.) characteristics. There are some algorithms of construction and calculation of VP parameters (see for example, [15-17]). We used the software package created on algorithm, circumscribed in [17]. All above mentioned metric and topological characteristics were obtained, but for discussion two metric characteristics have been selected:

- the radial distribution of neighboring atoms from the central atom (G (R) - distribution);

the distribution here coincides with the function 4prg (R)R2 (here r-atomic density, g (R) - pair correlation function) inside of the VP, and points to zero outside of it, taking into account only the atom - neighbors forming a VP.

- the distribution of sphericity coefficient (C-distribution).

The sphericity coefficient of VP- is defined by the ratio , where V- volume, and S- the surface area of a polyhedron. Characteristic values of the coefficient of sphericity: C = 1 - for sphere, C = 0.8288 - for a regular icosahedron, C = 0,6046 for an octahedron, C = 0.5236 - for a cube, C = 0.3023 for a tetrahedron.

4. The results and discussion.

The statistical geometrical analysis of the Voronoi polyhedra of both models was carried out for liquid Ga. The analysis showed that practically total concurrence of all topological and metric properties of VP for all temperatures were observed. To illustrate this in fig.2-3 the distributions of volume of the Voronoi polyhedra (V-distribution) and of the coefficient of the sphericity (C – distribution) are given. From this figures the perfect accordance of the C – and V – distributions obtained both from MD and from RMC models is well visible. From the results of our investigation of Cs melts [8], it might by possible to conclude that the simulation by the MD-method should more precisely reproduce a structure of a melt near the melting point.

Fig.2. C - distribution for melts of gallium.

Solid line - the MD-model, point - the RMC – model

Fig.3. V-distribution for gallium melts.

Solid line - the MD-model, point - the RMC - model.

The VP method provides rather complete information on the geometry of the short-range order. The partial G(R)-distribution functions for various compositions of melt (GLi, GLi-Li and GSi, GSi-Si) of the Li-Si system are represented in fig. 4,6 and the partial C-distribution functions of Li-Si melt (CLi, CLi-Li and CSi, CSi-Si) are represented in fig. 5, 7. Where GMe and CMe means that we take into account both components of the alloy which surrounded the Si atom or the Li atom (quasi-partial functions), and GMe-Me CMe-Me means that we take into account only those neighbors, that surround the Si or Li atoms, that are the same sort of atoms (true partial functions).

Fig. 4 Distribution of the nearest neighbor around atoms of Si in melts Li-Si:

1 - taking into account both components of the atoms (GSi);

2 - taking into account only Si atoms as neighbors (GSi-Si).

Fig. 5 The C - distribution of the VP around atoms of Si in melts Li-Si:

1 - taking into account both components of theatoms (CSi);

2 - taking into account only Si atoms as neighbors (CSi-Si).

Fig. 6 Distribution of the nearest neighbor around atoms of Li in melts Li-Si:

1 - taking into account both of the components of atoms (GLi);

2 - taking into account only Li atoms as neighbors (GLi-Li).

Fig. 7 The C - distribution of the VP around atoms of Li in melts Li-Si:

1 - taking into account both of the components of atoms (CLi)

2 - taking into account only Li atoms as neighbors (CLi-Li).

From fig. 4 it follows, that in melts of the Li-Si system from a complete environment of atoms of silicon (consisting of approximately 14 to 15 atoms) it is possible to select the atoms of the first and second coordination spheres (boundary radius Rb). The partitioning border is determined by a drastic change in the composition of the environment (minimum on a curve 1 and the beginning growth of a curve 2). For all the investigated melts we accepted Rb = 3.3 Å. This makes it possible to estimate the number of neighbors of different types in the first coordination sphere by integration of the relevant function G (R) up to R = Rb. So, for example, the number of atoms of both components in the first coordination sphere for all alloys is equal Z = 8.