THE

STRUCTURE G.

OF L I Q U I D

Z. Pinsker

MEDIA UDC

532.78

A discrete system o f points in space is constructed such that a subdivision L segregates a l i m i t e d volume o f space, invariant r e l a t i v e to a local mapping group. I t is proposed that any condensation of a gas is c h a r a c t e r i z e d by the appearance of a region of space, even if only a finite region invariant relative to a certain group o f motions. A liquid medium represents an infinitely branched part of space f i l l e d with an infinite set of contiguous mirror-symmetric pairs of a characteristic Lpolyhedron. A "liquid model" is constructed graphically from spheres of uniform diameter.

In this work, certain geometric relationships have been e x a m i n e d ; in the opinion of the author, these may c h a r a c t e r i z e liquid (including glassy) media. The distributions of molecules (atoms) in gases and crystals are c h a r a c t e r i z e d by two d i a m e t r i c a l l y opposed distribution laws - statistical disordering in a gas, and invariance relative to Fedorov symmetry groups in a crystal. Under statistical disordering of gas molecules, we understand that this means a discrete distribution such as to allow an arbitrary set of distances between molecules, a random number of closest molecules surrounding a given molecule, and an arbitrary orientation of the molecules in space. If all molecules of a gas are represented by points, and if the entire space is subdivided into regions V (Vorony ~egions [1]), the statistical disordering in the gas will be expressed by an infinite set of different configurations of polyhedrons V. Symmetry transformations in a gas allow the existence of only one finite group - a point group o f symmetry matching a gas m o l e c u l e with itself. It is possible to find motions that will m a t c h up any pair o f gas molecules: translation, translation with rotation, translation with r e flection, translation with rotation and reflection. However, these motions are not ordered for all the gas molecules. In a crystal a Fedorov symmetry group is defined, this is a discrete group of motions with a finite fundamental region. The polyhedron V of all points of the system is one and the same in a crystal. Representing the distribution of molecules (atoms) in a gas and in a crystal as the two limiting cases, we emphasize the difference in principle of the ordering o f motion: ordered motion in a crystal, and motion of individual molecules in a gas. A liquid will be i e g a r d e d as an i n t e r m e d i a t e state between these two extremes. Naturally, our approach to the problem is of the nature of a hypothesis, and the m o d e l that we are proposing here is by no means to be considered as beyond question. The basis of our analysis consists of the following generally known facts about the structure of liquids: the absence of translational invariance, the presence o f short-range order, and statistical disorder in the medium as awhole. We will define a liquid medium as a discrete system o f points that are not bound by a translational lattice, thereby excluding any consideration of " c r y s t a l - l i k e " rnodels. We will assume that the distinction of a liquid from a gas is determined by the short-range order, i.e., by the existence of a l i m i t e d volume of space such that in this volume there is ordered a distribution o f a finite number of molecules (in any case, more than two). We construct our case on an examination of a "product of gas condensation," proposing that any condensed medium must be i n variant r e l a t i v e to some group of motions, even if in a l i m i t e d volume o f space. The discrete system of points o f the liquid medium that is constructed will be analyzed, starting from two dual subdivisions of the s p a c e - bifurcations V and L [1, 2].

N. S. Kurnakov Institute of General and Inorganic Chemistry, A c a d e m y of Sciences of the USSR. Translated from l h u r n a l Stmktumoi Khimii, Vol. 13, No. 6, pp. 985-988, N o v e m b e r - D e c e m b e r , 1972. Original article subm i t t e d May 18, 1972.

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@1973 Consultants Bureau, a division o/ Plenum Publishing Corporation, 227 West 17th Street, York, N. Y. 10011. All rights reserved. This article cannot be reproduced /or any purpose whatsoever I without permission of the publisher. A copy of this article is available from the publisher for $15.00.

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I

We will assume that any discrete system of points corresponds to a gas only in that case in which there exist no motions matching up the polyhedrons V of two points of the system. Let us assume further that "condensation" of a gas leads to ordering of the distances between a finite number of points in a l i m i t e d region of the space. That is, of the set of different polyhedrons L o f the system, one polyhedron L is segregated such that its presence will c h a r acterize a discrete system of points. In other words, each p o i n t o f the system, being a vertex of the various polyhedrons L, will always be a vertex of the s e l e c t e d polyhedron L. The polyhedron L characterizing the system will a l ways c u t out such an unchanged part of the polyhedron V of each point of the system that one can aiways find a pair of points such that their c o i n c i d e n c e as a result of motion will be a c c o m p a n i e d by c o i n c i d e n c e of the unchanged part of the polyhedron V. Since the polyhedron L has as its center a vertex of the polyhedron V, then, having m a d e some given poIyhedron L characteristic, we are ordering,even if only one vertex of the polyhedron V of each point of the system. The part of the volume of the polyhedron V included between the faces of the characteristic polyhedron L will be the ordered part of the polyhedron V of each point of the system. Let us define short-range order of a liquid medium as such a l i m i t e d region of space within the limits of which a motion of the second kind is defined that matches up the ordered part of the polyhedron V of two points of the system. We will term the r m p p i n g o f a polyhedron L in any o f its faces a "local group of motion o f the second kind." A region of space of short-range order, m a d e up of a pair of i d e n t i c a l polyhedrons L that are contiguous along entire faces, forms a finite system that is invariant relative to the local group of m a p p i n g Thus, if some arbitrary polyhedron is the characteristic polyhedron L of the system, then the position o f all points o f the system will be determined by the position of the vertices of all such polyhedrons that are connected in pairs by a local mapping group in an entire contiguous face. The polyhedrons L do not intersect; and whenever a s e l e c t e d characteristic polyhedron L does not fill the space without any remainder, there is stii1 possible in the system a set of various other polyhedrons L, the configuration of which in any region of the medium will be random. The characteristic polyhedron L can be m a p p e d in each o f its faces. So that if n is the number of faces of the characteristic polyhedron L, there are possible n local groups of mapping o f the polyhedron L. Hence it follows that a short-range order in different regions of the medium will always contain a contiguous pair of m i r r 0 r - s y m m e t r i c polyhedrons L; however, the local group of mapping r e l a t i v e to which such a pair will be invariant m a y be its own in each such r e g i o n - o n e of n possible local groups. So, our proposed scheme of "condensation of gas" consists of such an ordering of a discrete system of points in space in which the distance between points is ordered so that there appears a characteristic polyhedron L, partially ordering the polyhedron V o f each point o f the system. The number of faces of the characteristic polyhedron L determines the number of its local group of mapping in its faces. The "condensation of gas" is c h a r a c t e r i z e d by the appearance o f a l i m i t e d region of space that is invariant relative to a l o c a l group o f mapping in a plane. It is not difficult to see that the scheme we have defined will exclude those configurations o f the polyhedron L for which para l l e l faces appear for one polyhedron or for two that are directly connected or connected indirectly by one l o c a l group of mapping. Actually, such a situation leads to the appearance of parallel transposition in the m e d i u m - m o tion that is f o ~ i d d e n in a liquid. The system that we have constructed has the following feature. A discrete system of points segregates from the entire space such a part that its subdivision is c h a r a c t e r i z e d only by one polyhedron L. Each point of the system proves to be a boundary between one part o f the space that is filIed by an uncountable set of mappings of the characteristic polyhedron L, and another part of the space that is o c c u p i e d by all other polyhedrons L of the system. We will apply the term "liquid m e d i u m " to that part of the space occupied by an uncountable set of m i r r o r - s y m m e t r i c a l pairs o f characteristic polyhedrons L. From this position, a liquid medium represents an infinitely branched part of space. We will represent as a special case of the polyhedron L a regular tetrahedron and we wili say that this is the characteristic polyhedron L of a "liquid medium" formed by a set of spheres of equal diameter. We will construct a packing of the spheres with centers at the vertices of the smallest regular tetrahedron of packing. Evidently the smallest such tetrahedron will be packing of four spheres according to the principle of tightest packing: One sphere touches three other spheres. Each succeeding sphere in the packing must be at the vertex of a tetrahedron that is m i r r o r - s y m m e t r i c with the given tetrahedron. In this connection, each succeeding sphere in the packing will be set against faces of tetrahedrons made up of the spheres, t t is well known that it is impossible to make up a space from regular tetrahedrons without any gaps, and i t will be simple to m a x i m i z e the filling of the space with spheres of the packing. It is very soon found that spheres in the packing cannot be set against alI faces of the tetrahedrons; the t e l alive location of the closest spheres in the packing leaves an inadequate volume of space for this. As a result, there

923

Fig. 1 . Fragment of packing of spheres of equat di.ameter according to law of mirror-symmetric tetrahedrons, where 12 spheres surround one sphere: a, b) View of loose part of packing, where sphere with coordination number of 3 occupies one of two possible positions; c) view of densest part of packing. is formed a finite-infinite packing of spheres, where each sphere and a tetrahedron face unoccupied by a sphere form a highly branched surface of the densest part of the packing. Considering the statistical character of the distribution o f the spheres of the packing at a distance exceeding the dimensions of the region of short-range order of the initial tetrahedmn, let us determine the smallest and the largest coordination numbers (CN) and the mean CN for spheres in the given packing. According m the condition of sphere packing, the smallest possible will be CN = 3, where one sphere touches only three spheres in the packing. Further, around a sphere of radius r, let us distribute 12 spheres of radiu~ r at the vertices of an icosahedron, at a distance of 2r., By subsequent convergence of these 12 peripheral spheres, one can obtain a packing of 10 of the smallest tetrahedrons with a common vertex in the central sphere (Fig. 1). Thus is obtained the maximum CN =12 and the m e a n CN = (3 + 12)/2 = 7.5. Let us represent the given packing of spheres as the model of a liquid, w here the packing sphere is a molecule (atom) of the liquid. Here we do not insist that a vertex of a polyhedron L is the sole possible position of the centers of the atoms or molecules: The volume of the polyhedron L, its faces, and its edges constitute the region of distribulion of centers of the atoms or molecules. The proposed model of sphere packing represents a somewhat simplified model of a liquid. So, let us return to the model in Fig. 1. Four of the peripheral spheres of this model have CN = 3. Having rolled one of these spheres to the closest free face of a neighboring tetrahedron, and repeating this operation with the other spheres of the model having CN = 3, we will change the spatial orientation while having preserved the packing. In an infinite packing of spheres, such a rollaway o f spheres with CN = 3 from one position to another position identical with the original does reflect, in our view, the scheme of such highly important properties of liquids as their incapability of forming their own shape in a macrovolume, and the process of flow. Such a liquid medium does not fill space without a remainder. A "characteristic liquid medium" in our m o del is represented by a set of the smallest tetrahedrons of packing. The faces of the tetrahedtons that prove to be at the boundary of this medium represent "aggressive" sections of the surface, tending to attract to themselves a molecule, i.e., a sphere of the packing, thereby changing the surface configuration in the given region Such an "aggressive" boundary of a tetrahedron follows from the formulation of the medium, the properties of which are de termined by a local group of mapping of the characteristic polyhedron L in its faces. As it appears to the author, the proposed model of a "liquid space" gives a graphic geometric image that permits an examination o f the atomic structure and many characteristics of liquids and g l a s s e s - for example, flow, dissolution, diffusion and self-diffusion, dissociation, etc. The author expresses deep gratitude to Z. G. Pinsker, R. V. Galiulin, N. P. Dolbilin, S. A. Dembovskii, and O. Ya. Samoiiov for very useful discussions on a number of the questions set forth in this paper. LITERATURE 1. 2.

924

CITED

B. Delone, N. Padurov, and A. Aleksandrov, Mathematical Principles of Structural Analysis of Crystals [in Russian], ON TI-GTTI, Leningrad-Moscow (1984). B.N. Delone, Usp. Mat. SeL, No. 3, 16-62 (1937).