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Geometrical analysis of thestructure of simple liquids:percolation approachYu. I. Naberukhin a , V.P. Voloshin a & N.N. Medvedev aa Institute of Chemical Kinetics and Combustion,Novosibirsk, 630090, USSRPublished online: 26 Oct 2007.

To cite this article: Yu. I. Naberukhin , V.P. Voloshin & N.N. Medvedev (1991) Geometricalanalysis of the structure of simple liquids: percolation approach, Molecular Physics: AnInternational Journal at the Interface Between Chemistry and Physics, 73:4, 917-936, DOI:10.1080/00268979100101651

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MOLECULAR PHYSICS, 1991, VOL. 73, NO. 4, 917-936

Geometrical analysis of the structure of simple liquids: percolation approach

By YU. I. N A B E R U K H I N , V. P. V O L O S H I N and N. N. M E D V E D E V

Institute of Chemical Kinetics and Combustion, Novosibirsk 630090, USSR

(Received 11 May 1990; revised version accepted 15 January 1991)

The problem of searching for quantitative laws governing the structure of simple liquids is formulated as a site percolation problem on the Voronoi network. The sites of this four-coordinated network correspond to the figures formed by the four neighbouring atoms (Delaunay simplices). Three quantitative characteristics of the form of the Delaunay simplices are introduced to enable one to colour the sites of the Voronoi network corresponding to the simplices of a specific form and to study the percolation of colouring through the network sites. The clusters of contiguous Delaunay simplices of the specific form have been studied and the percolation thresholds for various colouring types have been obtained for instantaneous configurations of the Lennard-Jones liquid (obtained by the Monte Carlo procedure) as well as for the configurations with removed thermal excitations (F structure). Percolation of all the types of colour- ing introduced turns out to be correlated, i.e., the Delaunay simplices of a given form are situated on the network, not at random. The Delaunay simplices in the form of good tetrahedra tend to join in long branched chains with built-in five-membered rings. The simplices resembling a quarter of a perfect octahedron group into shorter chains and sometimes associate in semioctahedra and full octahedra. In the structure of liquids there exists regions of low local density which unite the simplices with large circumradii.

1. Introduction

Despite considerable progress in the computer simulation of disordered systems (liquids and amorphous solids) it can hardly be claimed that any methods have become common in describing their structure, i.e., the regularities of the mutual spatial arrangement of particles. Pair (or even triplet) correlation functions, i.e., a standard tool of statistical physics, are insufficient for describing computer models. They provide too averaged a description and fail to realize the whole of structural information obtained with a computer. More suitable are the methods of 'statistical geometry' proposed by Bernal [1].

The most widespread is the method ofVoronoi polyhedra, which reflect topologi- cal and metric properties of the nearest environment of each particle. At present, the specific features of the Voronoi polyhedra are determined for both phase and thermo- dynamic states of disordered systems, namely, overheated crystals, liquids at various temperatures and densities, supercooled liquids, and glasses [2-5]. The second subject of statistical geometry, the Delaunay simplices (tetrahedra with vertices at the centres of four nearest particles), is closely related to the Voronoi polyhedra. The properties of separate Delaunay simplices have been studied fairly well in computer models of dense monoatomic liquids and amorphous substances [7-13]. In these systems most Delaunay simplices were found to belong to atomic configurations which are close in form to a regular tetrahedron or to one-fourth of a regular octahedron, the so-called

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quartoctahedron (see references [10, 12 and 13] and section 3). Thus the structural elements of two types are predominant here. This makes the formalism of the Delaunay simplices preferable to the language of the Voronoi polyhedra where no single form predominates within a great variety of types.

It is typical of modern studies of liquid structure to use the above methods to describe local order, i.e., either the nearest coordination shell of an atom or the configuration of a few particles. However, along with the local order the structural laws must also define the principles of its atomic arrangement at large distances, and formulate the rules of spatial organization of structural elements throughout the whole volume of a specimen. The methods of statistical geometry are suitable for this purpose. The Voronoi polyhedra and Delaunay simplices fill the space, sharing their faces. As in crystallography, the characteristics of the mutual arrangement of poly- hedra at fairly large distances may be determined by describing their face-to-face packing. It is the essence of Bernal's approach to the structure of liquids: reduce the laws of the arrangement of particles to the rules of the packing of geometric figures.

In the present paper the methods of percolation theory have been employed to solve the problem of the mutual arrangement of the basic structural elements of liquids, i.e., Delaunay simplices of a certain form. The idea of this approach and the first results have been reported by us [14]. In this paper they are presented in more detail.

The approach proposed is based on the fact that the assemblage of the Voronoi polyhedron edges for a given atomic configuration in the liquid forms a four- coordinated network (the Voronoi network), each site of which corresponds to an individual Delaunay simplex. A traditional two-colour site percolation problem may be formulated on this network by colouring only the simplices of a certain form, e.g., those which are close in their form to a regular tetrahedron. By increasing the number of coloured sites we can study the cluster form of the contiguous simplices until, at a certain concentration (percolation threshold), the cluster spreads over the whole of the network. This idea is given in section 2 along with the related concepts, To realize it in a computer experiment we need suitable characteristics of the forms of the Delaunay simplex; they have been developed earlier [10, 1 I] and are discussed in section 2.2. Their properties for the computer models studied are presented in section 3.

The present percolation problem has two aspects. The first, given in section 4, is directly related to the structural problem and concerns the study of clusters of coloured Delaunay simplices. The second is indirectly associated with the structural problems and treats the distribution functions of various cluster properties, the percolation thresholds of networks at different colourings, etc. Our methods of colouring the Voronoi networks lead to a correlated percolation problem and the correlated arrangement of coloured simptices is displayed distinctly in the statistics of colouring. All these problems are considered in section 5.

The subject of our study is the model of the Lennard-Jones liquid near the triple point obtained by Monte Carlo simulation. For comparison, the crystal model has been studied near the melting point. A quantitative formulation of the structural laws is obviously hindered by thermal chaos involved in the instantaneous atomic con- figurations generated by a computer. Therefore, along with these instantaneous structures (I structures), we have studied so-called proper or inherent structures in which thermal excitations are removed. Here the inherent structure of one type, named F structure (see reference [15] and section 2.4), has been considered, and all the properties of the models for the I and F structures have been compared.

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Structure of simple liquids by percolation method 919

2. Basic concepts and ideas

2.1. Voronoi network The Voronoi polyhedron of a given atom is a space region all points of which are

closer to the centre of this atom than to the centres of others. The Voronoi polyhedron describes the coordination of the nearest particles, and atoms sharing its face can be named geometrical neighbours. A set of all Voronoi polyhedra of the atomic system forms a three-dimensional tesselation which fills the space by convex polyhedra without gaps and overlaps. The vertices and edges of the Voronoi tesselation comprise a space-filling network, the Voronoi network, which exhibits the following properties.

First, four edges (network bonds) almost always meet at each vertex (network site) in disordered atomic systems because in the Voronoi tesselation four polyhedra (the minimum possible number) meet at any vertex. Note that this property may fail for the degenerate atomic configurations with a special atomic arrangement so that more than four polyhedra can meet at some vertex and, hence, more than four bonds can meet at this network site. For example, in the FCC crystal the Voronoi tesselation involves the vertices at which six polyhedra meet; the corresponding six atoms form an octahedral configuration. However, already at infinitely small crystal pertur- bations the degeneracy vanishes. A few nondegenerate vertices (usually four) being the sites of the Voronoi network with four bonds replace the degenerate one. In this case the octahedron decomposes, as a rule, into four (sometimes five [13]) Delaunay simplices of a special shape close to one-fourth of the regular octahedron in which one of the edges (octahedron diagonal) is ~ 2 '/2 times as large as five of the rest. The simplices of such a form are named quartoctahedra [13, 14]. In disordered systems, degeneracy is hardly probable. In our models it was not observed.

Second, each vertex of the Voronoi tesselation is equidistant from the centres of four atoms whose Voronoi polyhedra meet at this vertex, i.e., each vertex is the centre of the circumsphere of its four atoms. The centres of these atoms form vertices of a tetrahedron which is called the Delaunay simplex [6], the four atoms being called a simplicial configuration (see figure 1). As is known, a simplex is the simplest figure in the space of a given dimensionality (i.e., a triangle in a plane (figure 1)) or a tetrahedron of a general form in three-dimensional space. The Delaunay simplex differs from all the other simplices in that it is formed not by any four atoms but by a quadruplet of geometrical neighbours.

Third, any bond of the Voronoi network connecting two neighbouring sites shows that the corresponding Delaunay simplices are contiguous, i.e., share one face or, in other words, these simplicial configurations have three atoms in common.

Finally, since there is one-to-one correspondence between the sites of the Voronoi network and the simplicial configurations (Delaunay simplices) we can assign to each network site a numerical parameter characterizing a certain physical or geometrical property of the corresponding Delaunay simplex. Such an arithmeticized Voronoi network offers a possibility to use not ony topological but also percolation analysis of the network. Now we may reduce the problem of the liquid structure formulated as regularities in the mutual arrangement of the Delaunay simplices of a particular form to the problem of studying the clusters of the neighbouring sites on the Voronoi network that are selected (coloured) according to the numbers assigned to them, i.e. reduce it to the percolation problem (see section 2.3).

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Figure 1. Two-dimensional illustration of the Voronoi-Delaunay division of space into polyhedra (here into polygons). Large circles represent atoms. Thick lines represent edges of the Voronoi polygons, small filled circles are their vertices. A set of these edges and vertices comprises the Voronoi network. Triangles of thin lines which join the neighbour- ing atom centres are the Delaunay simplices. The dashed circle with centre at the vertex A is circumscribed around one of the Delaunay simplices.

2.2. Form characteristics of the Delaunay simplices

All simplicial atomic configurations are topologically equivalent; these are tetra- hedra of a general form. They can differ only in metric characteristics (size and shape). In the general case of a random system of points (ideal gas) there are no reasons for any preferable shape for the Delaunay simplices. This not so for the dense packing of particles. Two preferable configuration types were found for several models of amorphous solids and liquids [10-13]: the Delaunay simplices close in form to a regular tetrahedron ('good' tetrahedra) or to one-fourth of a regular octahedron (quartoctahedra), A great number of good tetrahedral configurations in dense dis- ordered systems have been predicted by Bernat [1] and confirmed more than once by computer simulation (see, e.g., reference [16]). These correspond to packing of spheres with the maximum possible local density, i.e., to the absolute minimum of the local energy. The reason for the appearance of the second class of the Delaunay simplices, quartoctahedra, is less clear. In crystals with the densest packing (cubic or hexagonal) these are the necessary elements of the structure. The liquid is quite different struc- turally from these crystals, and its quartoctahedra are unlikely to form octahedral configurations. Nevertheless, here quartoctahedra are a necessary addendum to fill the space together with tetrahedra.

The quantitative characteristics of simplex shapes were introduced to distinguish the Delaunay simplices of the prevailing types [10, l 1]. To distinguish good tetrahedra we propose the specific measure--tetrahedricity:

T = y. (t~- ~)~/15(,t) ~'. (1) i> j

Here li is the edge length and ( l ) is the average edge length of a given simplex. For a perfect tetrahedron the value of T is zero. For small distortions of tetrahedral configurations T remains small, and vice versa: only the Delaunay simplices close in

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Structure of simple liquids by percolation method 921

form to the perfect tetrahedra correspond to small T values. Thus, good tetrahedral simplices may be distinguished by a small value of T. However, the high T values are ambiguously related to the simplex shape. Therefore, in order to reveal the quartoctahedral simplices, a special measure is proposed---octahedricity:

0 = ~ (l~ - lj)2/lO(l) 2 + ~ (Z, - 2 '/'-Zm)"/5(l)'-. (2) i > j i ~ m

i,j r ra

Here l,, is the length of the longest simplex edge. For the perfect quartoctahedron O = 0, and low O values indicate this simplex to be close in form to the perfect quartoctahedron.

Note that the definition of octahedricity differs here from that in references [10] and [11] in a normalization factor. (There it was 15(1) 2 for both sums.) A new normalization appears more convenient as it emphasizes the role of the maximum edge and leads to clearer separation of quartoctahedra on histograms.

Another quantitative characteristic of the Delaunay simplices is the circumradius R. The structural meaning of a circumsphere is that there may be no atomic centre inside it [17]. Thus, the circumsphere of the Delaunay simplex may be said to represent the cavity between the four atoms; its radius gives an efficient size to this cavity.

2.3. Percolation problem and colouring of the Voronoi network

The problem of searching for the rules of the structure of atomic configurations is reduced, in terms of statistical geometry, to the study of the connectivity of the corresponding Voronoi networks. The well-developed methods of percolation theory are quite suitable for this purpose [18]. From a mathematical viewpoint the perco- lation problem is concerned with the study of the properties of connected components of random graphs [19]. Its initial object is a network (graph) consisting of the sites (vertices) connected by edges (bonds). The network may be either regular (crystalline lattice) or random. For our case this object is represented by the Voronoi network. A certain numerical value of the definite quantity given by the specificity of the problem is assigned by some law to each site. In our case this quantity may be any geometric characteristic of the Delaunay simplex corresponding to a given site of the Voronoi network. In the classical (two-coloured) formulation of the percolation problem all the sites of the network are divided into two classes depending on the assigned value. For example, we may colour those sites of the Voronoi network that correspond to the Delaunay simplices with characteristic T being below the fixed boundary value Tb (T ~< Tb). The rest of the sites are uncoloured.

The connected coloured sites form the clusters (connected graph components) the study of which is the main subject of percolation theory. When the fraction p of coloured sites is small there are only isolated clusters in the network. However, a critical concentration Pc always exists when the 'infinite' cluster appears spanning the basic cube in at least one coordinate direction: the colouring is said to percolate through the whole volume. With random colouring of the sites the percolation threshold value Pc is defined mainly by the coordination number of sites in the network, and is practically independent of its topology [20]. If the arrangement of coloured sites on the network is correlated (as in our case), the percolation threshold differs from Pc in the randomly coloured network; the magnitude of this difference may be a useful structural characteristic of the system.

We propose three methods ofcolouring the sites of the Voronoi network. The first one, named T colouring, implies the colouring of the most tetrahedral Delaunay

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simplices, i.e., those with the lowest values of tetrahedricity parameter T. Similarly, in O colouring the sites are coloured in the order of increasing values of parameter O starting with the simplices which resemble perfect quartoctahedra most of all (with the lowest octahedricity value O). Finally, in the colouring of the third type, R colouring, the simplices with the highest values of circumradii are the first to be coloured. Studying the clusters of the connected coloured sites with the fractions of T, O or R coloured Delaunay simplices increasing from p = 0 to the corresponding percolation thresholds Pc, we obtain a complete picture of the mutual arrangement of the simplices of three characteristic types: those close in form to regular tetrahedra, to regular quartoctahedra or conforming to the regions of low local density.

Thus, if the Delaunay simplices are taken as structural elements of the liquid, the problem of the mutual spatial arrangement of the elements of a certain form is reduced to the problem of percolation of the characteristics of this form (T, O or R) through the coloured sites of the four-coordinated Voronoi network. Different methods of colouring of the sites of one and the same Voronoi network allow one to reveal the connectivity laws of different structural elements.

2.4. Structural models

This paper deals with a computer model of the system of particles interacting with the Lennard-Jones potential. Atom coordinated were generated by Monte Carlo simulation in the NVT ensemble (Metropolis algorithm) for 108 particles in a box with periodic boundary conditions at the reduced density p* = 0.9 and temperature T* = 0.719, corresponding to the liquid near the triple point. The parameters of argon were taken for the LJ potential: e/k = 119.8 K, ~ = 0.34nm.

The computer generates instantaneous atom coordinates at some moment of time. However, this instantaneous structure (I structure) gives inconvenient material for determining the structural rules of the liquid. Indeed, for a crystal, the I structure has no translational symmetry and, hence, this peculiarity of a crystalline state can hardly be identified by the Monte Carlo model at finite temperatures. By crystal structure one traditionally means the configuration of the equilibrium positions of the constituting particles rather than the instantaneous positions. To generalize this consideration we have proposed a special notion proper structure [15], which means the particle arrangement in the absence of thermal excitation. The structure of the crystal in the conventional sense is just its proper structure rather than I structure.

The same must be valid for the liquid. Before determining the rule of particle packing it is reasonable to remove the superfluous thermal chaos by passing from the instantaneous particle positions to the nearest local minima of the potential energy surface, i.e., to the proper structure of the liquid. Various methods are available for realizing the proper structures in the computer experiment [I 5]. Here the simplest one has been used which leads to the so-called frozen (F) structure [15]. Each F structure corresponds to the definite I structure and is obtained from it by using the same Monte Carlo algorithm but with T = 0K. We have employed such an algorithm which ensures a constant acceptance rate, approximately 50%. In this case, on freezing, the length of the maximum step rapidly decreases so that in 500-600 moves (per one atom) it practically vanishes. The resulting F structure corresponds to a random descent of each particle to its local energy minimum. This way of 'quenching' is much quicker than the steepest descent method along the potential energy gradient resulting in 'hidden' or 'inherent' structures (after Stillinger and Weber [21]) which represent another type of the proper structure. Acceleration is due to the fact that in

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Structure of simple liquids by percolation method 923

I I I " | I I I

.1 . I

i . o8 o8

06 ~ 06 a

�9 8 4 n~ 0 4 '

�9 0 2 0 2 . . . . . , . . ~ - ~

�9 0 3 . 0 6 . O g . 0 3 . 0 6 . Og

TETRAHEDRICITY, T OCTkHEDRICITY, 0

(a) (b)

I . . . . I

"li 1

�9 0 5

, [ .55 .5 .85 .7 .75

CIRCUMRADIUS, R

(c)

Figure 2. Distributions of the form characteristics of the Delaunay simplices for the I struc- ture of the FCC crystal: (a) tetrahedricity; (b) octahedricity; and (c) circumradius. The dashed lines mark boundaries of good tetrahedra and good quartoctahedra.

the F structure the particle fails to reach the exact local minimum; here the main thermal excitations are removed. The atomic shifts for such quenching are below 0.2 of the distance between the atoms.

It is also our aim to compare the properties of I and F structures of the model. All the quantitative characteristics given below were averaged over ten realizations. The main cube for the I structure contained 667.6 Delaunay simplices in average (the sites of the Voronoi network) and, for the F structure, 661-2 simplices.

3. Distribution of the characteristics of Delaunay simplex shapes

Before studying mutual arrangement of the Delaunay simplices, let us consider the statistical distributions of various characteristics of the shape. Figure 2 presents these distributions for the FCC crystal at T* = 0.719 and p* = 1. Under these conditions the crystal remains stable and the structure, consequently, preserves tetrahedral and octahedral atomic configurations although of a slightly distorted form due to thermal

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Table 1. Fractions of the Delaunay simplices in the form of slightly distorted tetrahedra (0 ~< T ~< 0-018), nr, and quartoctahedra (0 ~< O ~< 0.03), no, in the I and F structures of the crystal and liquid.

nr no nr + no nr/no

FCC ~I 0.314 0.626 0.940 0'50 crystal ~F 0.333 0-667 1 0.5

Liquid {I 0.184 0.295 0.474 0.62 0-292 0.404 0.696 0-72

perturbations. Therefore the bimodal character of the histograms in figure 2 reflects the existence of two simplex types in the I structure of the crystal. The maxima at small Tand O values belong to 'good' (i.e., slightly distorted) tetrahedra and quartoctahedra, respectively. The positions of the deep minima on the histograms in figure 2(a, b) allow us to indicate approximate values of the shape characteristics which limit the classes of good tetrahedra and quartoctahedra: Tb = 0"018 and O b = 0.030. The areas under the first peaks in the T and O distributions from zero to these boundary values give the fractions of these classes of simplices: nr and no. Table i shows that this way of determining the amount of these simplices in the I structure of the crystal is in good agreement with the theoretical data for the F structure, i.e., for a perfect crystal.

The distribution of circumradii (figure 2(c)) is less resolved than the T and O distributions. However, it is also clear that the left maximum corresponds to the tetrahedral and the right one to quartoctahedral simplices. Their positions are close to the values of the radii in the perfect crystal: Rr = (3/8) '/,- = 0"612 for tetrahedra and Ro = 2- u2 = 0-707 for octahedra. The shift of the right peak to the left versus Ro shows that the thermal distortions of quartoctahedra decrease mostly the length of the maximum edge compared to its length 2 '/2 in the perfect quartoctahedron.

Figure 3 depicts the T and O distributions for the liquid. These are very different from those for the crystal in figure 2. In the liquid no minimum is observed to divide

�9 OB

. 0 4 t.--I

�9 0 2

I I , , I I

! .... " t ! 1 : . . .

. . j " ~ L~

i I : '"

": I ! I " i

: 7., t I ":7

I " ,

�9 03 . OB . og .12

TETRAHEORICITY, T

(a)

�9 O B

. 0 4

O2

i '" : ' : ~ i

: " i :-.: :J ',

~ t "" i b " :

" '-

i " 7 , . .

�9 0 3 . OB , Og . 1 2

OCT^HEORICITY, 0

(b) Figure 3. Distributions of (a) tetrahedricity and (b) octahedricity of the Delaunay simplices

in the liquid. The full line corresponds to the F structure, the dotted line to the I structure. The vertical line shows the same boundaries for good tetrahedra and good quartoctahedra as in figure 2.

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Structure of simple liquids by percolation method 925

!

.16

i -i �9 0 4

.05 .1

Z ~ . t2 I,.--

. - - I

. 0a

I !

�9 1 2

. i

~ . 08 ~ . 06

~-. o4

.02 i "

.15 .2 .25 DELAUNAY SIMPLEX VOLUME

(a)

.6

i I I i

. . |

.7 .8 ,9 CIRCUMRADIUS, R

(b)

Figure 4. Distributions of (a) the volumes and (b) circumradii for the Delaunay simplices in the F structure (full line) and the I stucture (dotted line) of the liquid.

the Delaunay simplices of good tetrahedra and good quartoctahedra. This suggests that many simplices of intermediate form occur which cannot be assigned to either of these classes. Although these classes are indistinct in the histograms, the number of simplices belonging to them may be counted using the property of our shape charac- teristics that small T or O values correspond to the simplices close in form to a regular tetrahedron or quartoctahedron, respectively. Therefore, integrating our histograms in the range from zero to the same boundary values, Tb and Ob, established in the crystal, we derive the relative quantities of good tetrahedra and quartoctahedra, nr and no (see table 1). The total amount of the simplices of these two classes, ny + no, is here below 100%. This indicates the presence of Delaunay simplices of intermediate forms in the liquid.

Figure 3 demonstrates very well the differences in the distributions for the I and F structures of the liquid, especially that in the I structure the histograms quickly tend to zero at small T and O values. This is easily understood. Thermal fluctuations destroy any order that could exist in the atomic configurations of the liquid. There- fore, the number of regular simplices which have approximately equal lengths of six (as in good tetrahedra) or five edges (as in good quartoctahedra) decreases drastically in the I structure (see table 1).

The decrease of the chaos in the F structure of the liquid affects the other properties as well. It makes the maxima of the radial distribution function sharper, leads to a decrease in the number of various topological types of the Voronoi polyhedra [12, 15], and to the narrowing of the volume distribution of the Delaunay simplies (figure 4(a)). Of interest also is the behaviour of the circumradius distribution (figure 4(b)). For the F structure it becomes bimodal, resembling that of the crystal (see figure 2(c)) which illustrates clearly the existence of the Delaunay simplices of two classes in the liquid (for further arguments see reference [13]).

4. Coiouring the Voronoi network

In order to study the mutual arrangement of the main structural elements of the liquid one could choose (colour) in computer realizations only 'good tetrahedra'

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Figure 5. T colouring of the Voronoi network for one of the realizations of the F structure of the liquid. The circles represent the sites of the Voronoi network (the vertices of the Voronoi polyhedra) but not atom centres. Bonds connect the coloured sites which are the neighbours on the network. The bonds sticking out of the cube are directed to the images of sites. 150 network sites are coloured (among the total number of 660 sites) correspond- ing to the Delaunay simplices with the smallest values of tetrahedricity: T < 0-0165.

(i.e., the Delaunay simplices for which T ~< Tb) or 'good quartoctahedra ' (with O ~< Ob) and investigate the form of the resulting clusters. This approach is, however, insufficient as the boundary values T b and O b cannot be indicated unambiguously. Besides, the connectivity of the clusters on the Voronoi network is defined primarily by the fraction p of coloured sites but not by Tb or Ob.

The percolation approach provides a more convenient and universal way for investigating the mutual arrangement of structural elements. It implies the colouring of an arbitrary number of sites corresponding to any concentration p, and the successive consideration of the cluster picture of the contiguous sites within the whole of the range from p = 0 to p = 1. Then in T colouring we will add the coloured sites in the order of increasing T measure starting with the most regular tetrahedral simplex (that has the minimum T value in the given ensemble of the Delaunay simplices). Although the quality of newly added simplices becomes increasingly worse we obtain as a result a complete picture of spatial arrangement of the Delaunay simplices depending on the deviations of their shape from the perfect tetrahedron.

In the percolation approach a great number of colourings of one and the same model is necessary at various p values. Convenient in this case is the following method. A special array is created in the program in which the numbers assigned to all N sites of the network are listed in the order they will be coloured with increasing p, i.e., in the order of the increasing T measure. Thus, the first Np elements of our array give the numbers of sites which must be coloured at any given p.

Figure 5 presents the typical result of T colouring of the Voronoi network. Depicted there are only the sites which correspond to the 150 most tetrahedral Delaunay simplices for the F structure of the liquid. The concentration of the coloured sites p = 150/660 = 0-227 and the values of tetrahedricity of the represented simplices are limited by T = 0.0165 ~ Tb. Thus, almost all good tetrahedral configurations

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Structure of simple liquids by percolation method 927

Figure 6. O colouring of the Voronoi network for the F structure of the liquid. 100 sites are depicted corresponding to the Delaunay simplices with the smallest values of octahedricity (O < 0.013) for the same realization as in figure 5. Arrows A and B indicate octahedral atomic configurations, other arrows mark semioctahedral configurations.

are shown in this figure. The picture of the connectivity of the Delaunay simplices is specific. The good tetrahedra produce the clusters in the form of branching and interweaving chains with the built-in five-membered rings. The large number of 5-rings is of course the most intriguing feature of the T colouring. These rings in our picture are nothing but the pentagonal faces of the Voronoi polyhedra which prevail among the faces of another shape in liquids [2, 3, 5]. Studying the properties of the separate Voronoi polyhedra we have already mentioned that the pentagonal faces often unite the Delaunay simplices of a good tetrahedral shape [12]. The percolation approach reveals this feature in a more constructive way. Figure 5 shows clearly that 5-rings tend to condense, sharing their edges. However, this condensation does not lead to the creation of closed pentagondodecahedra| cavities (corresponding to the icosahedral atomic configurations). Such cavities are absent at all p values up to Pc: that is in accordance with our earlier observations [12, 22].

For the I structure of the FCC crystal the picture of clusters of the T coloured sites of the Voronoi network is quite different from that in the liquid: only isolated sites are observed in the crystal. Thus, in this case the tetrahedra contact via the edges and the vertices as in a perfect FCC crystal and have no faces in common.

The spatial arrangement of the quartoctahedral atomic configurations is demon- strated using the O colouring of the Voronoi network (figure 6). The resulting picture of the connectivity is strongly different from that for the T colouring of the same network. The most probable associates of quartoctahedral simplices in both I and F structures are the pairs with very short bonds corresponding to semioctahedral atomic configurations (see arrows in figure 6). The number of such dimers increases with the rise in the concentration of coloured sites. We see also some rings of four sites connected by short bonds (arrows A and B in figure 6) which represent good octahedral configurations. Some realizations display the trimers with short bonds. These types of associates show that the liquid contains slightly distorted octahedral and semioctahedral cavities between the neighbouring atoms.

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@

t

Figure 7. R colouring of the Voronoi network for the same realization of the liquid F structure as in figures 5 and 6. 100 sites are depicted which correspond to the Delaunay simplices with the largest values of circumradii (R > 0.64).

The appearance of many long bonds in O colouring is a less trivial fact: they demonstrate that quartoctahedra contact by the faces which lack the longest simplex edge (see figure 4 in reference [14]). Such clusters cannot be a part of the single octahedral atomic configuration but are formed rather by quartoctahedra from different octahedral holes. When the number o fO coloured sites increases, the amount of long bonds increases as well; they join into the chains which involve octahedral and semioctahedral aggregates.

In the I structure of the FCC crystal no long bonds were observed on the Voronoi network among all good quartoctahedra (with O < Ob). It is quite clear because in this structure octahedra have no faces in common. In the hexagonal close packed crystal containing the pairs of contiguous octahedra, the long bonds between the quartoctahedra along axis C will occur on the Voronoi network. This has nothing in common with a chaotic orientation of the long bonds in the liquid (see figure 6).

The R colouring shown in figure 7 is quite different from the T and O colourings. Here 100 sites of the Voronoi network are depicted which correspond to the Delaunay simplices with the largest circumradii (with R > 0.0726; the coordinate of LD poten- tial minimum is taken as a length unit). Unlike the other two, this colouring has practically no single sites demonstrating that the local rarefaction (large inter- atomic cavity) in a dense packing is formed usually by more than four atoms. Hence, many strongly distorted Delaunay simplices appear to be contiguous. Already at low concentrations of the sites in R colouring the close clusters appear that reveal the regions of lowest local density. In the I structure of the FCC crystal the clusters in R colouring are mainly of a tetragonal form with short bonds, i.e., the rarefaction is due to fluctuation-enlarged octahedral cavities. In the liquid the number of such rings is much less and the clusters organization is quite different from that in the crystal.

It is interesting to compare the pictures of connectivity for the I and F structures of the liquid. On passing from the I to the F structure a reorganization of the Voronoi

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Structure of simple liquids by percolation method 929

Figure 8.

o ~176

~ o

m~ o ~o~

o 0 ~176 0

0 o

o

o %

O~ o

Random colouring of the Voronoi network of the F structure of the liquid. 150 sites are coloured (p = 0-219).

network obviously takes place. It is seen in particular in a change in total number of sites in the basic cube: for a realization which is depicted in figures 5-8 the I structure had 665 sites whereas the F structure had 660 sites. However, an analysis shows that most coloured sites in the I structure are also observed among the coloured sites in the F structure. Thus the same good tetrahedra are observed in both structures, i.e., the tetrahedral simplicial configurations are preserved on transition from the I to F structure, and only slightly change their forms. This is also confirmed by the fact that an average picture of clusters for ten realizations is practically the same in both structures. In particular, they have almost the same number of five-membered rings and some other characteristics (see section 5). Hence, the reorganization of the Voronoi network when passing from the I to F structure does not involve the simplicial configurations in the form of good tetrahedra. Another situation takes place for O colouring. Here a small number of sites remains invariant on transition from the I to F structure. The cluster shape is systematically different for the I and F structures; this is manifested most vividly in the larger probability of the appearance of octahedral configurations of particles in the F structure. Nevertheless, the mean characteristics of clusters of the O coloured sites are hardly different in two structural types (see section 5). Thus, despite reorganization of the Voronoi network in the regions of quartoctahedral simplicial configurations, the statistical laws of the network in the I and F structures are close.

For comparison let us consider the result of random colouring of the Voronoi network (figure 8). The picture of connectivity in this case is drastically different from that for T and R colourings. This testifies that good tetrahedral Delaunay simplices and the simplices of large volume are not randomly placed on the network. Note that the five-membered rings are frequently observed at random colouring due to a high probability of the appearance of pentagonal faces of the Voronoi polyhedra for dense packings. Smaller differences are observed for the pictures of random and O colour- ings, although they are not identical. Thus the T, O and R colourings of the Voronoi network lead to the correlated percolation problem.

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930 Yu. I. Naberukh in et al.

>., E--

O

I

}--

uJ E--

1.0

0.8

0.6

0.4

0.2

0.0

0.8 >,, E--

- - 0 . 0 L_)

OC

0 . 4 I <

t - 0 . 2 0

0 . 0

0 . 0

0.8 09

o.o , < t Y

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A

\ ~ A

o �9 o

0.2 0.40.O 0.8 1.0

P

1.0

0.8 >~

0.0 O

n,

c~ 0.4 tad I

n, 0.2 uJ I--

0.0

> , , I-"-

U

U.I

<[ i--

O O

0 3

Q

e l ,

:!2 D r r

t . .)

0.0

"2 \ / /

0 . 2 0.40.B 0.8 t.0

P

Figure 9. Dependence of the fraction of the coloured sites having j coloured neigh- bours on the total fraction p of coloured sites for the I structure of the FCC crystal: (a) T colouring; (b) O colour- ing; and (c) R colouring. The full lines show the binomial distributions according to formula (3). The colouring of the Voronoi network is for: zx,j = 0; O , j = 1; O , j = 2; ~>,j = 3; and v, j = 4 .

Figure 10. The same as in figure 9 but for the F structure of the liquid.

5. The properties of the Voronoi networks with correlated site colouring

5.1. Fraction of the sites with j coloured neighbours

A correlated character o f our percolat ion problems may also be demonstrated by considering various staistical regularities o f colouring the Vorono i network. Figures 9 and 10 s h o w the p-dependence o f the fraction o f co loured sites having j co loured ne ighbours in the nearest network sites, fj ( j = 0, 1, 2, 3, 4). The T and O colourings are compared here to the random colouring o f the sites for which t h e ~ values in the

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Structure of simple liquids by percolation method 931

four-coordinated network are determined by the binomial distribution [23, 24]

Needless to say that the strongest deviations from random colouring are observed for the Voronoi network of the crystal I structure (figure 9). In Tcolouring only good tetrahedra are coloured a t p < 1/3 (that g ivesf = 0 fo r j ~> 1) and in O colouring good octahedral configurations are coloured at p < 2/3 in which each quartoctahedron has two contiguous neighbouring quartoctahedra (that leads to more pronounced f2 distribution and to increase in f3 and f4 only when all octahedra are coloured).

The pictures of T, O and R colouring for the liquid are also considerably different from the random distributions but differencies o f f ( p ) from fB(p) are here quite different from those in the crystal. Figure 10 contains the pictures only for the F structure of the liquid; the pictures for its I structure are the same in the range of statistical fluctuations. The T colouring demonstrates that from the very beginning (from small p values)f0 is less and the rest o f f ( j >~ 1) are larger than the correspond- ing binomial distributions. This means that in the liquid, in contrast to the crystal, the good tetrahedral simplices from the very beginning join each other by faces. For example, a nonrandom character of the appearance of four neighbouring tetrahedral Delaunay simplices around good tetrahedral configurations (large ~ values) can be related to the cluster type in the form of three crossed five-membered rings which are characteristic for liquids (see figure 5).

The picture of O colouring is also different from that in the crystal. The deviations off~, f2 and f3 from binomial at small p are even larger than in the crystal. With p /> 0.5 all t h e f functions approach random behaviour, i.e., the colouring fails to distinguish the specificity of the quartoctahedral simplex shape. Hence, at p >/ 0.5 no good quartoctahedral configurations occur in the liquid, which is in agreement with the decrease of their number compared to the crystal (see table 1).

The only feature in common for R coiouring in the liquid and crystal is the absence of single coloured sites at any values o fp (f0 ~ 0) and the difference offl and f2 from zero already at very small p (figure 10(c)). However, in the liquid at small p, thef~ and f4 values exceed the random ones whereas in the crystal they were practically zero. This means that the regions of local rarefaction are formed by the association of a few Delaunay simplices of the specific shape. If in the crystal these were localized in the distorted octahedral configurations, in the liquid they are related to the configurations of a more complex form. The corresponding associate of the Delaunay simplices may be represented as, e.g., octahedron (that yields f2 > f~) to the faces of which other nontetrahedral simplices with large circumradii are added (that provides f3 > f3 a and L > f4").

All this testifies that the Delaunay simplices of different forms are placed non- randomly on the Voronoi network. The rules of their arrangement are substantially different from those in the crystal.

5.2. Mean cluster size The number of sites in the clusters resulting from a step-by-step colouring of the

Voronoi network sites is their simplest quantitative characteristic. Figure 11 presents the behaviour of these values, averaged over the realizations studied, against the fraction p of coloured sites. The first aspect that attracts attention in all types of colouring is the coincidence (within the limits of statistical scattering) between the

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932 Yu. I. Naberukhin et al.

8

- 6 Y

~4

�9 i ~ ! I I I

I

/Z : "~

I .' !

f / I �9 ." ] IX "1

. . . ' " / / m

.3 .6 .9 1,2 P I P e

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~4

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.3 .g P \ Po

(a) (b)

I

-- ~ . : .

1.2

16 /

," ~, 12 / /

N e ~n 8 . /..I

2 ........ """ ! ........ =

, I , i I

.3 .6 I. PIPe

(c}

Figure 11. Average number of sites in the finite clusters: (a) T colouring; (b) O colouring; and (c) R colouring. The full line depicts the F structure of the liquid, the short dashed line, the I structure of the liquid, the long dashed line, the I structure of the FCC crystal, and the dotted line the random colouring of the Voronoi network for the F structure of the liquid.

curves for the I and F structures of the liquid. The other details of the behaviour of the mean cluster size depend on the type of colouring. In T colouring (figure 1 l(a)) the good tetrahedral simplices in the crystal are observed to be isolated, and in the liquid the cluster size grows with increasing p faster than at random colouring.

In O colouring the cluster size increases atp/pc < 0-5 with equal rate in both liquid and crystal (figure ll(b)). Thus, in octahedral cages of the crystal, the separate quartoctahedra are distorted under thermal excitation in the same independent manner as in the liquid; in this case, no more than two Delaunay simplices preserve their good quartoctahedral shape. The restrictions imposed by the connectivity of the quartoctahedra in the octahedral cages in the crystal are observed only at P/Pc > 0.5, where the cluster size increases more slowly than in the liquid and at random colouring.

Of great interest are the pictures of R colouring (figure 1 l(c)). When p ~ 0 the mean size of clusters in the crystal is 3, and in the liquid it ranges from 4 to 5. This

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Structure of simple liquids by percolation method 933

Table 2. Percolation thresholds, critical values of the Delaunay simplex characteristics at the percolation threshold and the relative number of sites in the backbone of percolation clusters for various types of colourings of the Voronoi network of liquid and crystal.

y Pb(?~) --

Type ofcolouring Pc T a Y~ = Y(Pc) P~(Pc) + a

Liquid

T(1) 0.311 -T- 0-022 0-026 0.612 T 0-030 T(F) 0-287 -T- 0-029 0-018 0.627 -T- 0.090 R(I) 0-340 -T- 0-035 0-705 0.899 T- 0.026 R(F) 0.351 ~ 0-049 0.695 0.917 T- 0.033 O(I) 0.429 ~ 0-024 0.037 0-589 -T- 0.066 O(F) 0.423 -T- 0.038 0.031 0-594 T- 0.088 Random (I) 0-456 T- 0-028 0-456 0-542 T- 0-042 Random (F) 0.464 T- 0.036 0.464 0.521 -T- 0-068

FCC crystal

T 0-489 -T- 0-029 0-034 0.617 -T- 0-049 R 0-637 T 0-034 0-651 0.989 -T- 0-007 O 0"685 -T- 0-012 0.040 0-960 T 0.014 Random 0-448 -T 0.019 0.55 0-529 -T- 0.044

indicates that the local rarefactions in the crystal originate from the octahedral cages, and therefore consist of no more than four Delaunay simplices. In the liquid, on the other hand, the empty regions of large volume do not relate to the octahedral configurations; these consist of more than four Delaunay simplices. The appearance of large clusters in R colouring from the very beginning (with p ~ 0) may be accounted for in the following way. A certain simplex may have a large circumradius providing at least one edge is much longer than the rest. Since this edge is common to several contiguous simplices, all of them will have large R, and hence belong to one coloured cluster.

With increasing p the mean cluster size increases for all types ofcolouring, the rate being increased near the percolation threshold Pc. After the threshold (PIPe > 1) all the pictures in figure 11 display a sharp decay since we count only finite clusters.

5.3. Percolation thresholds

The percolation thresholds determined for all the types of the Voronoi network colouring are collected in table 2. As a percolation threshold, a concentration of coloured sites has been taken such that at least one cluster crosses the opposite faces of the basic cube. The Pc values given in table 2 as well as the mean-square deviations, o, result from the averaging over ten independent realizations of the Voronoi network of the liquid (I and F structures) and of the I structure of the FCC crystal.

Consider now the values of the percolation threshold for the liquid. Note first the coincidence (within the statistical errors) of these values for the I and F structures within any colouring type. This demonstrates a topological similarity between the Voronoi networks for these structures. Second, the Pc values for all three of our colourings are lower than the percolation threshold for random colouring of the networks, which reveals a correlated character of these colourings. The lowest thres- hold is observed for T colouring. It is clear that the Delaunay simplices of a good tetrahedral shape are situated on the network non-randomly. It is their specific aggregation in branched chains and five-membered rings that gives rise to 'infinite'

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934 Yu. I. Naberukhin et al.

clusters even at low concentration of coloured sites. For the percolation through O coloured sites the threshold is much closer to that at random colouring. This may be attributed to the fact that quartoctahedra do not tend to form long chains, and in this respect their positions are more random than those of tetrahedra.

Thepc value for R colouring as well as for Tis very low which indicates to a strong correlation in the arrangement of the Delaunay simplices with large circumradii. However, the origin of correlation are quite different in the two cases which is reflected in the cluster shape. In Tcolouring the chains with built-in 5-membered rings have more free ends; in R colouring the clusters are more compact involving numerous small rings (corresponding to the local aggregates of small faces of the Voronoi polyhedra). This difference is seen clearly for the percolation cluster for which the ratio between the fraction of sites belonging to the cluster backbone (i.e, cluster without free ends), Pb(P~), and to 'infinite' cluster, P~(p~), is given in table 2. The backbone of the 'infinite' clusters in T colouring is seen to contain about 62% of sites and in R colouring it has about 90% of sites. Even the larger value of this ratio is obtained for the R and O colouring of the crystal where the 'infinite' clusters consist mainly of the four-membered rings of octahedral cages bound in quadruplets via the sites corre- sponding to the strongly distorted tetrahedral cages.

It is obvious that our methods of Voronoi network colouring lead to a correlated percolation problem for the I structure of the crystal, too, which is clearly seen in table 2. Since the tetrahedral and octahedral cages in the crystal are isolated, the percolation thresholds through Tand O coloured sites cannot be less than the fraction of tetrahedral and quartoctahedral simplices in the FCC lattice, i.e., 1/3 and 2/3, respectively. This is confirmed by table 2. Note that for random colouring the percolation threshold for the FCC crystal (table 2) is statistically the same as Pc for the diamond lattice: 0-437 T 0.018 by our estimate [14] and 0-428 -T- 0-004 according to reference [25]. For random colouring of the Voronoi network of the liquid, Pc is a bit higher than these crystalline values. This means that the value of the percolation threshold depends to some extent on the network topology.

6. Conclusion

The problem of describing the structure of liquids involves two aspects: (i) the choice of basic structural elements, and (ii) the determination of the rules governing their mutual arrangement. The latter is undoubtedly the main one. However, it may be solved successfully providing a suitable choice of the main structural elements is made. It is the choice of the Delaunay simplices as the basic structural elements that results in the possibility to formulate the structural laws in terms of percolation theory developed in this paper. The key point here is in the correspondence of these simplices to the sites of the Voronoi network determined throughout the space where a mol- ecular object occurs. This allows one to formulate the rules of the mutual arrangement of the Delaunay simplices at any given distance as long as it is along the network. It is this feature that makes percolation theory attractive as a means of formulation of structural laws.

It is shown that it is possible to recognize the rules of the arrangement of the main structural elements of simple liquids at large distances. The Delaunay simplices of the specific shape are not randomly situated on the sites of the Voronoi network. The rules are illustrated in figures 5-7. The tetrahedral simplicial atomic configurations on the network generate clusters in the form of ramified chains interspersed with five-

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Structure of simple liquids by percolation method 935

membered rings. The quartoctahedral atomic arrangements join sometimes into semioctahedra, or full octahedra but more often they line up in chains which are not part of octahedral configurations. The clusters of the Delaunay simplices with large circumradii which exhibit the structural defects with a reduced local density are a characteristic feature of atomic packings in liquids.

Data presented show that the pictures of the connectivity of our structural elements in T colouring are statistically identical in I and F structures for the same fraction p of coloured sites, i.e., they remain unchanged when thermal excitations are removed. Indeed, the same values of p~ thresholds and the coincidence between the distributions in figures 10 and 11 for Tcolouring in the I and F structures indicate that the topology of the Voronoi network is practically equivalent for these two structural types.

What does it mean? Obviously, the liquid has a specific organization of structural elements. First of all, this organization involves simplicial configurations close in form to regular tetrahedra. The atoms in these polytetrahedral clusters ('pseudonuclei' after Bernal [1]) are in deeper potential wells that the other atoms of the system. When thermal excitations are removed during transition to the F structure, they remain in the same wells, i.e., no significant reorganization of atoms occurs in these clusters and the character of their connectivity remains the same. Studying the I structure, it is impossible to reveal a true sense of the T coloured clusters due to random thermal excitations. Only in the F structure are these clusters observed to involve the Delaunay simplices of a good tetrahedral form. Thus, one of the structural laws for a dense simple liquid is that the whole of its volume is permeated with clusters of tetrahedral simplicial configurations of the above form. This regularity in the arrangement of the tetrahedral Delaunay simplices predetermines a correlated character of percolation in T colouring.

In O colouring the situation is somewhat different. Many coloured sites fail to coincide here in the I and F structures. In other words, the particles in the quar- toctahedral configurations are situated in less deep minima of the local potnetial and often change their local environment under thermal excitation. However, the statistics of the O coloured dusters remain very close in the I and F structures (see section 5). This means that the clusters of good quartoctahedral configurations, despite their labile nature, may also be considered as the characteristic formations in the structure of simple liquids. The polytetrahedral aggregates are likely to cause around them a certain packing of quartoctahedral configurations that are the necessary additional elements to fill space.

Preservation of connectivity in the I and F structures justifies the very ideology of the proper structures in the liquid. Removing thermal excitations does not change actually the laws of 'topological disorder' in the liquid, i.e., preserves the character of connectivity of its structural elements. The quantitative rules may, however, be sought more successfully in the proper structure since the basic structural elements acquire here a more regular form.

Percolation analysis offers a way to reduce mobility to structure--an idea which had also been proclaimed by Bernal [1]. Indeed, the clusters of good tetrahedra represent 'rigid' constructional elements in which the atom mobility is minimal. In the case when they are spanning the whole of the specimen, i.e., when good tetrahedra form a percolative cluster, macroscopic mobility is impossible and a liquid transforms into an amorphous solid. This idea is verified for the molecular dynamics models of rubidium [26].

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Finally, it should be noted that the percolation approach to the structure of simple liquids based on the three types of the Voronoi network colourings enables one to connect the structural problems of dense packing with the widely discussed class of percolation problems, and thus to use verified research methods and accumulated experience. In particular, the general method brings together such different objects as the real four-coordinated networks of the water hydrogen bonds [23, 24] and abstract four-coordinated Voronoi networks describing the structure of simple liquids.

References [1] BERNAL, J. D.., 1964, Proc. R. Soc. A, 280, 299. [2] FINNEY, J. L., 1970, Proc. R. Soc. A, 319, 479; 319, 495. [3] POLUCUIN, V. A., DZUGUTOV, M. M., Uctiov, V. F., and VATOLIN, N. A., 1980, J. de

Physique C, 8, 284. [4] TANAKA, M., 1986, J. phys. Soc. Japan, 55, 3108; 55, 3428. [5] MEDVEDEV, N. N., VOLOSHIN, V. P., and NABERUKHIN, Yu. I., 1986, Materiats Chem.

Phys., 14, 533. [6] ROGERS, C. A., 1964, Packing and Covering (Cambridge University Press). [7] K1MURA, M., and YONEZAWA, F., 1984, J. non-cryst. Solids, 61-62, 535. [8] HIWATARI, Y., SAITO, T., and UEDA, A., 1984, J. chem. Phys., 81, 6044. [9] MEDVEOEV, N. N., and NABERUKHIN, YU. I., 1986, Dokl. Akad. Nauk SSSR (in Russian),

288, 1104. [10] MEDVEOEV, N. N., VOLOSHIN, V. P., and NABERUKHIN, YU. I., 1987, Zh. Strukt. Khim.

(in Russian), 28, No. 2, 62. [11] MEDVEDEV, N. N., and NABERUKHIN, Yu. I., 1987, J. non-cryst. Solids, 94, 402. [12] MEDVEDEV, N. N., and NABERUKH1N, YU. 1., 1987, Zh. Strukt. Khim. (in Russian), 28,

No. 3, 117. [13] VOLOSHIN, V. P., NABERUKHIN, YU. I., and MEDVEDEV, N. N., 1989, Molec. Simulation, 4,

209. [14] MEDVEt)EV, N. N., VOLOSHI~, V. P., and NABERUKHIN, YU. I., 1988, J. Phys. A, 21, L247. [15] NABERt;KmN, YU. I., VOLOSHIN, V. P., and MEDVEDEV. N. N., 1987, Melts (in Russian),

1, No. 2, 71. [16] FINNEY, J. L., and WALLACE, J., 1981, J. non-cryst. Solids, 45, 165. [17] TANEMURA, M., OGAWA, T., and OGITA, N., 1983, J. eomp. Phys., 51, 191. [18] STAUVEER, D., 1985, Introduction to Percolation Theory (Taylor & Francis). [19] ZUBKOV, A. M., 1986, Introduction to the Russian translation of the book: KESTEN, H.,

Percolation Theory for Mathematicians (Mir, Moscow). [20] SHANTE, V. K., and KIRKPATRICK, S., 1971, Adv. Phys., 20, 325. [21] STILLINGER, F. H., and WEBER, T. A., 1982, Phys. Rev. A, 25, 978; 1984, J. chem. Phys.,

80, 4434. [22] MEDVEDEV, N. N., VOLOSHIN, V. P., and NABERUKHI~, YU. I., 1986, Zh. Strukt. Khim.

(in Russian), 27, No. 4, 91. [23] STANLEY, H. E., and TEIXEmA, J., 1980, J. ehem. Phys., 73, 3403. [24] BLUMBERG, R. L., STANLEY, H, E., GEIGER, A., and MAUSBACH, P., 1984, J. chem. Phys.,

80, 5230. [25] SYKES, M. F., GAUNT, D. S., and GLEN, M., 1976, J. Phys. A, 9, 1705. [26] MEDVEDEV, N. N., GHGER, A., and BROSTOW, W., 1990, J. chem. Phys., 93, 8337.

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