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 threedimensional 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 selfconsistent 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
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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 (Li 80Si20,
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.
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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/ 2
n
2


1
(1)
Rg  
[ g 0(ri )  g (ri )]2  ,

 (n 2  n1  1) n1

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
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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 shortrange 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 WignerSeitz 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 intercenter 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 4g (R)R2 (here -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 C  36 (V 2 / S 3 ) , where Vvolume, 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
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and metric properties of VP for all temperatures were observed. To illustrate this in fig.23 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.
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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).
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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
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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.
With an increase in the lithium concentration, the environment of the atom of silicon
varies greatly: the number of the nearest neighbors - atoms of Si decrease, and, the
number of atoms if Li increases. So in a line Li57Si43  Li65Si35  Li80Si20 from the
nearest neighbor (Z = 8) atoms Si the number of neighbors of the same sort varies as 2,5
(3,5) 1,88 (2,8)  0.27 (1,6). The average number of neighbor - atoms of Si for
completely random melt are given in brackets as matching values and obtained by us
from the simple ratio XSi Z, where XSi - molar concentration, Z- the coordination number.
The considerable distinction of these two sequences unambiguously testifies to the
presence of a strong interaction between the components, and that the nature of this
interaction varies due to a change in the lithium content from 57 % to 80 %. In fact, in
the Li80Si20 melt the silicon atom is surrounded by extreme atoms of lithium, and all eight
atoms of the nearest environment are approximately located at one distance away (from
2.4 up to 3.0 Å). It is similar to the BCC - cell, where the atom of Si is central.
This point of view is supported by the sphericity coefficient distribution of the Voronoi
polyhedrons (C - distribution), given in fig. 5, where two types of distribution curves for
three compositions of the investigated system are also represented. For the first,
polyhedrons were under construction and all of the atoms of the system were taken into
account (curve 1), for the second (curve 2), other types of atoms were not taken into
account, and the polyhedrons were under construction in the nearest environment of the
same sorts of atoms, as central. The sphericity coefficient for the Li80Si20 composition is
close to the value C = 0.61, reference for an octahedron - the VP for a BCC-cell. In this
case, the character of the inter-atomic interaction is formed due to a charge transfer from
atoms of Li to atoms of Si, however, is most likely not reduced to a purely ionic bond,
and the formation of complexes Li+4Si-4, as the coordination number is rather high.
For the melt Li65Si35, the model of a chain formed by the complexes Li2+1Si-2
arises. In fact, in the nearest environment of the atom Si of this sort of chain structure
there will be just 2 atoms of silicon (more precisely, there can be hardly ess than two
because of the edge effect) and 6 atoms of lithium (two – self-environment and four environments of two neighbor Si atoms). The presence of more than 2 neighbor - atoms
of Si for the composition Li57Si43 is due to the linear chains of complexes Li2+1Si-2 in this
melt that are convoluted in a skein and connect to each other by atoms of Si. Thus, the
atoms of silicon form the net structure with coordinates 2 and 3. This explanation
corresponds with the available models of structure of crystalline phases of the Li-Si
system and results in the 'ab initio' simulation of the melt Li12Si7 [18]. Similar
distributions describing the environment of atoms of Li are given in fig. 6 and 7. They
also correspond to the offered plan. The arrangement of atoms of Li in melts Li57Si43 and
Li65Si35 is less symmetrical, in the melt Li80Si20 the sphericity of the environment of the
atoms of lithium and silicon practically coincide.
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Conclusion
The results of the analysis carried out suggest that the use of the RMC-method for
simulating the structure of a liquid metal is justifiable for such temperatures when the
melt becomes structurally homogeneous. The statistical geometrical properties of such a
constructed model completely coincides with similar properties of a model constructed by
a classical MD-method. In the range of temperatures near the melting point for melts of
pure metals and, apparently, for liquid alloys with a strong coupling between components
when the appearances of clustering can be observed, the RMC method is not as precise as
the MD-method. Nevertheless taking into account the above-mentioned, 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.
It is possible to assert that the Li-Si system as a whole follows the ‘Zintle rule'. In
fact, in the melt Li57Si43, where the ratio of components is almost one to one, the anion of
silicon can receive only one extra electron from each cation of lithium and should
demonstrate valence of about three. In the melt Li65Si35, where the ratio of components is
close to 2:1, the anions of silicon gain 2 electrons and will derive chain structures,
demonstrating valence close to two. In the melt Li80Si20, where the transport of a charge
is greatest, the covalent binding between atoms of Si disappears completely, yielding an
ionic bond place between anions of silicon and cations of lithium. From fig. 2, 4 it
follows, that if on the whole the sphericity of the environment is rather high and is nearly
constant, the distribution of atoms of Si rather, each one has noticeably smaller
sphericity, that this depends on the composition of the melt. Thus, the sphericity of
polyhedrons in the melt Li80Si20 is much higher, than for the two other compositions.
Additionally, the breadth C-distribution is a bit higher for Li65Si35, in the two last
indicated compositions, which is quite natural when in transition from a threedimensional system to a set of chains. Therefore, the modified structure of melts Li-Si, in
accordance with the magnification of the Si content, can be represented as a gradual
decay of a three-dimensional net to one-dimensional chains, which in turn break up into
separate links surrounded by cations of lithium, which is due to the change in the
electronic structure and character of inter-partical interaction.
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