RECENT RESULTS ON THE IDEAL STRUCTURE OF GLASSES

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RECENT RESULTS ON THE IDEAL STRUCTURE OF GLASSES

N. Rivier

To cite this version:

N. Rivier. RECENT RESULTS ON THE IDEAL STRUCTURE OF GLASSES. Journal de Physique

Colloques, 1982, 43 (C9), pp.C9-91-C9-95. �10.1051/jphyscol:1982917�. �jpa-00222446�

JOURNAL DE PHYSIQUE

Colloque C9, supplément au n°12, Tome 43, décembre 1982 page C9-91

RECENT RESULTS ON THE IDEAL STRUCTURE OF GLASSES N. Rivier

Blaokett Laboratory, Imperial College, London SW7 2BZ, U.K.

Résumé. - On définit une structure aléatoire idéale comme la

distribution la plus probable de formes de ses cellules, soumise aux seules contraintes topologiques et de remplissage de l'espace. Il en résulte une relation linéaire entre le volume moyen d'une cellule à f côtés, et f. Une telle relation a été découverte dans des mosaiques bidimensionelles par Lewis. Le nombre moyen de cotés <f> est calculé.

A partir de la valeur <f> = 13.40 dans des tissus non différenciés à 3D, il augmente avec 1'anisotropie des cellules, et diminue si c'est leur volume qui fluctue.

Abstract. - The ideal random structure is defined as the most probable distribution of cell shapes, subject to topological and space-filling constraints. This implies a linear relation between average volume of f-sided cells, and f. This has been observed in 2D tissues (Lewis's law). The average <f> is calculated. From ^f> = 13.40 in undifferenti- ated tissues, it increases with anisotropy of the cells, and decreases if their sizes fluctuate.

1. The random tissue or froth. - The problem of characterizing random space-filling structures like amorphous packings, crystal grain

aggregates, or biological tissues, is of fundamental importance, because any departure from the ideal random structure reveals the existence of a specific physical, chemical or biological interaction, which is of obvious interest to isolate. However, unlike crystallo- graphy, the ideal random structure, being solution of a statistical problem, is not unique, but only .one member of an ensemble of struc- tures corresponding to the most probable distribution of cells subject to topological and space-filling constraints. Criteria for ideality are relations between average properties of the structure (similar to thermodynamical relations), instead of Bragg spots. The botanist, F.T. Lewis, has found empirically one such relation between sizes and shapes of cells in 2D tissues, which will be shown in £ 2 to be a direct consequence of the ideality of the random structure. It is also, like its counterpart in 3D, straightforward to check experimentally.

Examples of the random, space-filling structures discussed in this paper are the Voronoi froth of amorphous packings [2], a network made of the vertices, edges, faces and cells of the Voronoi (Wigner-Seitz) cells of the constituting atoms, or, in the case of multicomponent metallic glasses, the radical froth [3]. In both cases, cells partition space unambiguously, but in the latter the cell volume is roughly proportional to the atom size. Other examples include biologi- cal tissues, and metallurgical aggregates of grains. Cells are then described topologically as polyhedra.

The random tissue in 3D has 4 edges, 6 faces and 4 cells meeting at

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982917

C9-92 JOURNAL DE P H Y S I Q U E

every vertex, every edge shared by 3 faces and cells, and every face by 2 cells. (In 2D, it has vertices with 3 edges and faces, and every edge shared by 2 faces). Exceptional vertices with more than 4 edges are not structurally stable : they can be split into 2 normal vertices by infinitesimal deformation. Probability of their occurrence in a random packing is negligible. The same holds for exceptional edges sharina more than 3 faces. Continuous random networks (models of co- valent glasses like vitreous silica) are excluded from this restricted class of tissues, because their edges may share more than 3 (non-planar) faces, even if their vertices still have all 4 edges.

In

both 2 and 3D, one observes a substantial variety of cell shapes [ I , 41, and statistical analysis is clearly relevant. Let us set the statistical problem : There are three levels of configurations, face, cell and tissue (two in 2D, face and tissue), each controlled by

a

topo- logical Euler equation, and by valence relations. The tissue contains C cells, F faces, E edges and V vertices, a cell has f faces, e edges and v vertices, a face n edges and w vertices. n = w is both Euler equation and valence relation for a face.

In 2D, Euler's theorem reads F

-

E + V = X J 0 , where the ~uler-~oincarg characteristics 3( is an integer of order 1, dependent on the topology of the surface:7(= 1 for a plane and X = 2 for a sphere. 3( is intensive, and therefore negligible by comparison with the extensive quantities F, E l V in large tissues. The valence relations a r e x n ~ ~ = 2E = 3V. Con- sequently, n is the random variable characterizing the shape of a cell, the distribution of cells is given by {p _n

3

^{= {F,/F~}

,

n = 3,4

...,

^and

there is one topological constraint

In 3D, for every cell, Euler's theorem is now f

-

^e + v = 2 and the valence relationsznf, = 2e = 3v, so that the topological relation (1) is

for every cell. There are identities (dramatically exhibited in Matzke's collection of bubbles 141) 3f

-

e = 6 and 2f

-

v = 4 relating e and v to

f.

The distribution {fnS is not normalized. At the tissue level, Euler's theorem is C

-

^{F +}

^E -

^{V = 0} (including, if necessary, the cell at infinity), with valence relations 2E = 6V = Z n F n , 2F = ZfCf, 3E = Z e c e , 4V = Z v C v . All these equations are equivalent to con-

straint (2) and the identities given above, except for 2F = Z f c f , which is

a

definition of F in terms of the distribution

ip f

^{= {C} /Cj

,

with

f f

f = 4

...,

the random variable at this level. (The complete information for the tissue is given by the multivariable distribution p(f), with f = ( { fnS ) a vector of positive components and Manhattan metric f =

-

Zfn). We have therefore two statistical problems, one at the tissue (f), the other at cellular (n) level (even at fixed f, a considerable variety of cell shapes is observed [41), related through (2) by

which is the sole topological constraint in 3D (cf. eq. (1) in 2D).

2. The ideal random tissue : Lewis's relation between sizes and shapes of cells.- The statistical information on the tissue is given by the cellular distribution

IPf5 ,

^{f = 4,}

. ^{. .N}

⁺

³

in 3D and by

{ plf ,

n = 3,

...

^{N + 2 in 2D.} A tissue containing a large number of cells

will take up the most probable distribution compatible with the con- straints, as is well known in statistical thermodynamics [5]. There are, in addition to possible, specific physical, chemical or biological laws, three obvious structural constraints : Normalization ( Z p = I), the topological constraints ( 1 ) in 2D and (3) in 3D, and the space filling requirement,

ZAnp, = A o I k C 39

^{( 4 )}

where An is the average area of a n-sided cell, A the available area, Vf the average volume of a f-faceted cell, and Vo the available volume 0

for the tissue.

The constraints constitute a system of 3 linear, inhomogeneous equations

M p

=

q

for the N unknowns = p. The solutions of the system lie in

-

-

a hyperplane of dimension d = N

-

r, where r is the rank of the matrix M. In general, that is for arbitrary area (An) or volume (Vf) laws,

1C- r = 3.

The most probable distribution maximizes the entropy, or arbitrariness, here the dimensionality d of the hyperplane of solutions. This is achieved if r = 2, i.e. when there is a linear relation between the constraints, so that An is a linear function of n [6]

and Vf, a linear function of f

v$ , ^(Vole) p f f - ~ l l / ( b - < n 9 ) - ~ - ' ~ J (3~)

₍₆₎ where

A

and are arbitrary. Equation (5) has been observed by Lewis

[I 1 in cucumger epidermis and pigmented epithelium of the retina. In that case

A-J

1/4 so that A, =

0

which seems natural enough. However, Voronoi polygons of Poisson-distributed seeds also follow Lewis's law

( 5 ) , but with 1 / 6 [ 7 ]

.

The theoretical derivation of eq. (5) is due

to Lissowski and the author, who demonstrated that the entropy - z p In p is indeed maximized by eq. (5) 161

.

Equation (6) is new, and awalts experimental verification.

Both equations ( 5 ) and (6) are the analogue of thermodynamical

relations : they are direct consequences of the system taking up a most probable distribution. If they are not obeyed, the average cell size is not regulated simply by space-filling requirement, but by a specific law, or by the fact that space-filling constraints act in a higher dimension (e.g. plane sections of 3D crystal aggregates should

not

^obey

( 5 ) 1 .

3.

Average number of faces per cell, or average coordination of a

randompacking. -

The quantity to evaluate is < f 7 ( = < z X > by duality).

It is known empirically that<f>CJ14 [41, and that it is also weakly dependent on the network or packing. It is not a universal quantity, unlike its 2D counterpart (n7 = 6.

The simplest estimate of <f) goes as follows : Consider a random, undifferentiated froth, where 4 cells (and faces) meet at every vertex, and 3 on every edge. Let the angles between edges at a given vertex i be

ocP

i ⁽

p

^{= 1}

. .

^-6)

.

Coqsider a particular cell

r.

Each vertex imparts a curvature 2

a - ( ~ f

.t di-tci; ) = 2~

-

G L to the surface of cell

P

bordered by edges 1, 2 and 3 (Gauss-Bonnet theorem). The total curva- ture of the cell is 4 n

,

^thus,

C9-94 JOURNAL DE PHYSIQUE

thereby defining a local relation between number of vertices per cell v and curvature

2 ~ - 0

at a vertex, averaged over a given cell. Euler's and valence relations for a cell

( f l )

yield f

= 2 +

v, whence

An undifferentiated froth with isotropic cells of equal volume (not- withstanding small fluctuations necessary to ensure that each cell has

integer number of vertices) has all

6

w's nearly equal at every vertex, thus a= ^1/3

⁼

cos-'(- 1/31

=

109.47°, and

=

13.40, which is exactly the value obtained by using Schlaefli's trigonometric criterion for regular honeycombs [8]. Coxeter himself introduces a different, alge- braic formula derived again for regular honeycombs, which yields f

=

13.56 [81. A similar derivation of f

=

13.40, albeit for a somewhat artificial model, has been given by Dodds

[ 9 J .

The main advantage of eq.

( 7 )

is that it enables us to investigate the effect of fluctuations in sizes and shapes of cells. Consider

fluctuations in&, described by the probability distribution

P ( d ) ,

with mean

<w> = j d w

?(dl

^Gi

. ^Then,

If the fluctuations are small, so that<'G))d, corresponding to cells -

of roughly the same size but anisotropic, f

(<a)

) y f and <f> > T . Thus, anisotropy of the cells increases the average number-of their faces. Uniaxially compressed lead-shot of identical volume yields

(f72 14.17 [lo], and betrays anisotropy of the compressed shot due to uniaxial stress

(D.E.G.

~illiams, private communication).

Conversely, for-isotropic cells of different sizes, one has

(a)

<

and f

( <a> ) <

f. To prove this, it is simplest (for

2

different sizes) to argue by vertex decoration. In a first stage, construct-a space- filling tissue of large cells only. Then, ( f >

^{= 2% /C =}

^{f. In a}

second stage, introduce the smaller cells by Odecorating i of

the V

vertices. The smaller cells are, at first, all tetrahedra, and

F =O

Fo

f

4i,

C = C +

i,

so

that

0

where<£)

=

2F/C and x

=

i/C

= 1

- C /C is the concentration of smaller cells. The case (8) where all smalle? cells are tetrahedra gives the smallest value of ( f ) , which increases again if some tetrahedra start to overlap and make up penta, etc. -hedra. It is straishtforward to generalize eqn. (8) to include such overlap. ~ o n s e ~ u e n t l ~ , fluctuations in sizes sf cells decreases the average number of faces per cell. This is indeed as observed [3,4,111. The qeneral trend was noted, and made plausible by analyzing the- regular cube-octahedric honeycomb; by

Meijering [121. The present, statistical analysis is new.

To summarize, various-types of tissue have average numbers of faces per cell departing from f

=

13.40. The deviation is small

( N 5%)

(13.36<<$>

44.25 for such diverse systems as compressed lead shot, soap bubbles, varlous undifferentiated cellular tissues, or random close packings of hard spheres with minimal, or no fluctuation in sizes [2,4]). Indeed,

"the average 14-hedral shape observed in massed bodies of diverse surface

tension may be due to random avoidance of the niceties of adjustment.

Failure to arrange the bodies so that 5 or

6

meet at a mathematical point, or form intersection where 4 meet along a mathematical line is sufficient to account for promiscuous, unoriented polyhedra having an average of 14 facets" (Lewis [I]). This failure gives rise to a substantial variety of cell shapes, and was at the basis of eq. (7).

I have benefited from specific suggestions by A. Lissowski, R. Mosseri, G. Giraud and

D.

Williams, and acknowledge helpful discussions with

D.

Duffy, S. Steinemann and

D.

Weaire.

[I] LEWIS F.T., Anat. R e c . 3 (1928) 341;

Am.

J. Bot. 30 (1943) 74 [2] ZALLEN R., in Fluctuation Phenomena, E.W. ont troll and J.L. Lebowitz

(North Holland, Amsterdam 1979) ~ . 1 7 7

131 GELLATLY B.J. and FINNEY J.L.,

J:

non-Cryst. Sol. (1982), to appear

[ 4 ]

MATZKE E.B., Am. J. Bot. 33 (1946) 58: Bull. Torrey Bot. Club 2

(1950) 222

151 JAYNES E.T., Phys. Rev. 106 (1957) 620

[ 6 1

RIVIER N. and LISSOWSKI

A.,

J. Phys.

A

15 (1982) L 143

[71 CRAIN I.K., Comput. and Geosc. 4 (1978) 131 181 COXETER H.S.M., Ill.

J.

Math. 2-(1958) 746 [91 DODDS J.A., J. Coll. Interf. ~ c i . 77 (1980) 317 [I01 MARVIN J.W., Am. J. Bot. 26 (1939) 280

[Ill MATZKE E.B. and NESTLER

,J Am.

J. Botany 33 (1946) 130

[I21 MEIJERING J.L., Philips Res. Rep. 8 (1953)770.