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TOPOLOGICAL CORRELATIONS IN RANDOM
CELLULAR STRUCTURES
N. Rivier
To cite this version:
TOPOLOGICAL CORRELATLONS
I N
RANDQM CELiüiAR STRUCTURESN . Rivier
Blackett Laboratory, Imperia2 CoZZege, .London, SW7 2BZ, U. K.
Rksumé - On résume nos connaissances sur les formes et dimensions de cellules ou grains remplissant l'espace au hasard, sur leur corrélations, et sur l'équation d'état structurelle de ces mousses.
Abstract - Information on the shape and size of cells or grains filling space at random, on their correlations, and on the structural equation of state of such froths or tissues, is hereby summarized.
Introduction
In this paper, 1 shall discuss spatial correlations which are topological, in that they refer to shapes of the objects making up the structure (cells, grains, etc.), and that neighbourhood is defined according to territorial partition. Apart £rom its obvious relevance to geography, ecology, etc., it is the best one can do when dealing with amorphous materials or glasses, which are homogeneous, not in the sense of exact superimposition of a pattern with its translated counterpart, but because any local difference is not physically relevant or observable.
This type of correlation is immediately noticeable when washing up dishes: Small soap bubbles neighbour large ones. It has been qualitatively summarized by F.T.Lewis as 'random avoidance of the niceties of adjustement". Bubbles may also exchange neighbours or even disappear, but these transformations do not affect in any objective fashion the structure of the froth which is a random, space-filling aggregate of cells (bubbles), homogeneous and indistinguishible (without being superimposable) from other natural or man-made froths like undifferentiated biological tissues, metallurgical aggregates, amorphous packings of, e.g. lead shot, or the structure of atomic liquids or of metallic glasses [il.
Randomness of the structure, the presence of continuous transformations, and the fact that two different structures can be deeemed identical without being superimposable, imply that the relevant geometry is t o ~ o l o w (rather than metric geometry of unit cells or Bragg peaks). Hence the actual size of edges, cells or angles is of no relevance, since they can expand or shrink as space is continuously deformed. Each structure is not an exact copy of a unique original, but only a member of a statistical ensemble of most probable structures under a feu constraints (constant volume and topology of the space which they are filling). Physical or biological forces are irrelevant, to lowest order, in framing that identical and random architecture, which has an equation of state ( 4 - 6 ) .
Randomness also implies that 'defects'
,
i.
e. shape fluctuations (e. g.JOURNAL DE PHYSIQUE
dislocations) are essential constituents of the structure. It is their presence and their motion under topological transformations, which grant froths, foams, ceramics or tissues, their relevant mechanical or geometrical properties [Il.
The topological approach is simpler, poorer but more general than radial distributions, simply because the metric notion of radius or contact is replaced by that of neighbourhood. It also yields non-trivial and measurable information, by using simpler mathematics.
Random froths are cellular structures filling a topological space, where edges, faces and cells can shrink or expand continously as if they were filling rubbery space. This implies a arammar:
1. Elementarv structural transformations (2 in 2D, 3 in 3D): Tl, or switching between neighbouring faces (seen by blowing gently on a soap film formed on a cubic frame); T2, or face disappearance (change of ce11 neighbourhood in 3D) and its inverse; ce11 disappearance and its inverse
(mitosis).
2. Structural stability: Only vkrtices with coordination z=3 (in 2D1 and 2=4 (in 3D) are structurally stable. Vertices of higher 2 can be transformed into these by infinitesimal deformations. A tissue or froth with only structurally stable constituents is called miiximallv random. This will be assumed from now on.
3.
Conservation laws: Euler theorem: F-E+V = O( 11, (2D) ; -C+F-E+V = O ,(30); and continuity of odd lines (lines threading through faces with odd number of edges [2]). Here, the froth has F faces, E edges, C cells, V vertices, and occupies a volume Qo.
Consesuences
4 . The topological random variables are n and f, the numbers of edges on a face, and of faces on a cell. Their expectation values,
are consequences of 52-3. The latter is called Coxeter's relation, valid for every ce11 and for the froth as a whole. For maximally isotropie and isochorous (equal volume) cells, <f> = 13.4 (exhibited in TCP
-
tetrahedrally closed-packed, crystalline structures like A 1 5 or Laves phases [3]), corresponding to <n> = 5.1, the number of regular tetrahedra or milk 'berlingots" sharing one edge in Euclidean space. <f> increases if the cells become more anisotropic; it decreases if fluctuations in their volumes increase 141.5. To~oloaical correlations (Aboav's law) and ' croreversi-
. . .
(first justified by Weaire), where = < (n-<n>)
>,
and n m (n) = 5f-Il-K(f-1-n)9 (3D) (3)
161, where K is a constant of the froth. The limit f=- is the correct procedure for topological stereology (deduct the properties of a random planar section from those of a 3D froth). Then K = 5-(12tua)/f+
...
Microreversibility, and factor 5 in (2) depend on assuming no topological correlation beyond nearest neighbours (apart from those arising from Coxeter's sum rule (1) in 3D). Some 2D systems (e.g. random Voronoi polygons) have F fixed and cannot undergo T2 transformations. Then there is only one (Tl) recursion relat'on, microreversibility is automatic, and factor 5 in (2) is replaced by &ni-1, where a n ) is the same as m(n), but refers to second neighbours to ce11 n. For random Voronoi polygons, Boots [81 observes 5.12-5.46 in his simulations.
6. Statistical eauilibriu: (~ustified by microreversibility). Its properties (distributions p(n) or pff), and of ce11 sizes, equation of state) are obtained by maximum entropy formalism (MEF) [9], that is by maximizing the arbitrariness (cf. $1) subject to a few constraints (space-filling, topology (1)).
7.
Structural eauation of state (Lewis's lawl for ideal froths: LetA,,
y*,
denote the average area, volume of n-sided faces, f-faceted cells, respectively. Then,
[10,4]. Equation of state (4) has been discovered empirically by Lewis [Il] in undifferentiated biological tissues. Here A is a Lagrange multiplier enforcing the constraints. It increases (linearly if Ro/F is constant) with time [12], and the slow evolution of cells is ruled by von Neumann's law [131. Distributions of cells shapes and sizes can be found in [6,1].
8. Structural eauation of state for non-ideal aaareaates: Lewis's law is not obeyed by metallurgical grains, where it is the average perimeter or radius Rn of n-sided faces , rather than the area, which is proportional to n, as observed experimentally [14] or in simulations 1151. This fact, according to MEF methodology [9], betrays the presence of (at least) another constraint, which is, obviously, the energy carried by interfaces between grains, i.e. their perimeter. The system increases its entropy by selecting a size-shape relation which could be either Lewis's or the perimeter law:
The maximum entropy associated with the perimeter law is larger than for Lewis's [6], so that the former alternative is selected whenever interfacial energy is relevant.
JOURNAL DE PHYSIQUE
cells, either materially (grains or bubbles vs. pores or holes) or only distinct in their sizes. Statistical equilibrium, defined by MEF, still applies with similar equations, suitably modified to account for the presence of pores. The main interest consists in predicting the final evolution of a sintering aggregate or a soap bubble froth [Il. In the absence of pores, shape fluctuations saturate to ps (t=-) = (6-a)(7-a), where a is the number of sideS of the smallest cells [6]. Finite concentration of pores increases pP(t=-) [16].
Two remarks in conclusion: The problem of finding a probability distribution (of ce11 shapes or sizes) is not well defined: It is only by maximizing the entropy (minimizing any biais) that one grants uniqueness to an otherwise highly underconstrained problem. If the predictions of MEF (the structural equation of state) disagree with experiment, this indicates the presence of hitherto unsuspected constraints (cf.87). Thus;the theory of $6-9 is not falsifiable, since MEF does not contain within itself any criteria for its own valid application.
Nevertheless, MEF yields two important results: i) The structural equations of state (4-6)are precise, necessary and sufficient criteria for the structure to be random (given the constraints). ii) Because the entropy is a convex function of the constraints, its maximization has one solution, or only a few, discrete alternatives, each of which corresponds to a different equation of state. By contrast, the potential or free energy has many valleys (minima) in configuration space. These configurations are linked by elementary structural transformations as in the soap film of $1. But they al1 belong to the statistical ensemble.
The original contributions of my coauthors, and the hospitality of the Institute of Theoretical Physics, Lausanne University, are gratefully acknowledged.
REFERENCES
111
Review: D.Weaire and N.Rivier, C0ntemp.Phys.a (1984) 59 [2] N.Rivier, Phi1.Mag.m (1979) 859[3] J.F.Sadoc, J.Physique Lettres 44 (1983) L-707 [4] N.Rivier, 3.Physique Col1.U (1982) Cg-91 [5] M.Blanc and A.Mocellin, Acta Met.22 (1979) 1231 [6] N.Rivier, Phil.Mag.2 (19851, to appear
[7] D.A.Aboav, Metal1ogr.Q (1970) 383 [8] B.N.Boots, Metal1ogr.x (1982) 53
[9] E.T.Jaynes, Phys.Rev.m (1957) 620; 1QB (1957) 171 [lO] N.Rivier and A.Lissowski, J.Phys.A
15
(1982) LI43 [ll] F.T.Lewis, Anat.Records 3 (1928) 341[12] N.Rivier, Phi1.Mag.m (1983) L45
LI31 3.von Neumann, in Metal Interfaces, Amer.Soc.Metals (1952) 108 [14] C.H.Desch, J.Inst.Metals
