- Email: [email protected]
Topologically disordered networks of rigid polytopes
14
Journal of Non-Crystalline Solids 123 (1990) 14-21 North-Holland
T O P O L O G I C A L L Y D I S O R D E R E D N E T W O R K S O F RIGID P O L Y T O P E S P.K. G U P T A and A.R. C O O P E R
a
Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210, USA a Department of Materials Science and Engineering, Case Western Reserve University, Clevelana~ OH 44106, USA
A condition is derived for the existence of infinitely large d-dimensional topologicallydisordered networks composed of 8-dimensional rigid vertex-sharingpolytopes which is consistent with Zachariasen's first two rules. It generalizesthese rules to polytopes, network dimensions, polytope dimensions, and vertex coordinations not considered previously.
1. Introduction
In 1932, Zachariasen [1], in his rationalization of common experience as to which oxides readily form glasses, postulated " a n ultimate condition for the formation of glass" which is that " t h e substance can form extended three-dimensional networks lacking periodicity with an energy content comparable with that of the corresponding crystal". He then argued that: (a) " t h e energy of the network will in the first place depend upon the nature of the oxygen polyhedra, in the second place upon the way in which the polyhedra are linked together"; (b) "if the energy of the glass is to be comparable with that of the crystal, we must require that oxygen polyhedra in glass and crystal are essentially the same"; and (c) " t h e linking together of the oxygen polyhedra in the crystal lattices is done in such a manner that no two atoms are brought into close contact ... Ions carrying charges of the same sign will tend to be as far apart as possible. We have to apply just the same principles on the network in glass". In other words whether polyhedra in nonperiodic networks share vertices, edges or faces depends on whether they do so in the corresponding crystalline network and the coordination of the polyhedra about an oxygen is the same in crystalline and glassy networks. However, unlike crystals, the relative orientations between adjacent polyhedra " m a y vary within rather wide limits".
This distribution of the relative orientations is the origin of the non-periodicity in the network. By 'extended' Zachariasen clearly meant macroscopically (i.e., infinitely) large for he also states that " t h e network in glass is characterized by an infinitely l a r g e unit cell containing an infinite number of atoms". Following Warren [2], these non-periodic large networks are referred to as random networks. However, we suggest that it is more appropriate to call the Zachaxiasen networks topologically disordered (TD). While finite size T D networks can always be built using any polyhedron as the building block, the conditions under which a particular type of polyhedra can form an infinitely large T D network are not clear. Several attempts have been made to build T D networks by hand [3] or with the aid of computers [4] using vertex-sharing regulax, congruent and rigid (RCR) polyhedra and a set of construction rules which ensure a broad distribution in the relative orientations between adjacent polyhedra. These studies have led to a common realization that the requirements of connectivity at the vertices are rather severe and make it difficult to build large T D networks. The largest model built from R C R tetrahedra has 651 units [5]. Construction of two-dimensional T D networks with RCR triangles have led to similar results [6]. Thus two questions of fundamental importance arise: one about the existence of infinitely large T D networks made of vertex-sharing RCR polyhedra and the second about how to build such
0022-3093/90/$03.50 © 1990 - ElsevierSciencePublishers B.V. (North-Holland)
P.K. Gupta, A.R. Cooper / Topologicallydisorderednetworks of rigidpolytopes
15
networks given their existence. The model building approach is inadequate to answer the first question since all models are necessarily of finite extent. Indeed model building attempts to answer the second question. The question of the existence is related to whether the constraints due to connectivity of the polyhedra exceed the degrees of freedom associated with the vertices. This is strictly a geometrical problem. An answer to this question is provided in this paper using a topological approach. The result is a necessary condition which involves only four parameters: two describing the polyhedron and two associated with the network. The condition derived is sufficiently general and applicable to arbitrary dimension of the polytopes and of the network. To set the stage we explain (and define whenever possible) in section 2 some of the terms frequently used in this paper. In section 3 we state the problem of existence of TD networks in precise mathematical terms and review previous work related to this topic. Section 4 gives our proof of the necessary condition for the existence of infinitely large TD networks. We discuss the consequences of our result and relate them to the work of others, particularly that of Zachariasen, in the last section.
2.3. Regular polytope
2. Basic concepts: definitions and terminology
2.6. Rigid polytopes
2.1. Polytopes
A rigid polytope has fixed shape and size. Throughout this paper, we deal only with rigid polytopes. The fact that in real glasses the coordination polyhedra may not be rigid is not a concern of this paper. However, as our analysis implicitly shows, lack of rigidity of the polyhedra always favors random network formation.
Polytopes are generalizations of polygons and polyhedra to a space of arbitrary dimension. Onedimensional polytopes are called rods. Polygons are two-dimensional polytopes and polyhedra are three-dimensional polytopes. The corresponding geometrical figures in four dimensions are also called polytopes. In this paper the word polytope is used in its general sense.
2.2. Polytope dimension (8) The minimum dimension of Euclidian Space necessary to embed a polytope is called its dimension. Thus 8 (point) = 0, 8 (rod) = 1, 8 (polygon) = 2, and 8 (polyhedron)= 3.
A 8-dimensional polytope is regular if all its faces are alike, all its vertices are alike, and all its faces are regular ( 8 - 1) dimensional polytopes. Thus, a polygon is regular if it is equilateral and equiangular. While there are infinite number of regular polygons, there are only a finite number of regular polytopes in three and higher dimensions [7].
2.4. Characterization of polytopes A complete characterization of a polytope involves both its size and its shape. The shape of a regular polytope is specified by two numbers: its dimension 8 and the number of its vertices, V. For non-regular polytopes, these two parameters specify only the topology. Two polytopes are called topologically equivalent if they have the same values of 8 and V. It is clear that V>_-(8 + 1).
2.5. Congruent polytopes Congruent polytopes are identical in all respects. Congruent regular polytopes have the same dimension, same number of vertices, and same size.
2. 7. Network Polytopes can often be connected so as to form extended structures. We call these structures networks if it is possible to make cyclic paths along the polytope edges starting from any vertex (not on the surface) without repeating any polytope. Our definition of networks is more restrictive than a definition, sometimes used in graph theory,
16
P.K. Gupta, A.R. Cooper / Topologicallydisorderednetworks of rigidpolytopes
which includes structures (called trees) which do not have any cyclic paths. Finite size networks are also called clusters. In this paper we are interested in only those networks in which: (a) all polytopes have the same dimension and the same number of vertices (i.e., they are topologically equivalent), and they are rigid, (b) polytopes are connected only at the vertices (i.e., they do not share edges or faces), (c) each vertex (except the vertices on the surface of finite networks) is shared by the same number of polytopes (i.e., it is a regular network), and (d) the join at a vertex is free, like a universal hinge, imposing no constraint on the relative orientations of the polytopes sharing that vertex.
2.8. Network dimension (d) The minimum dimension of the Euclidian space in which a network can be embedded by deforming without severing or adding any connections (i.e., by unfolding or straightening without changing the topology) is called the network dimension, d. This is consistent with Zallen [8].
2.9. Vertex connectivity The number of polytopes connected to a vertex is called its connectivity. Clearly it is a positive integral number, greater than or equal to unity.
2.10. Fully bridging and partially bridging vertices In a regular network, vertices with the highest connectivity (denoted by C) are called fully bridging vertices. Vertices with connectivities less then C are called partially bridging vertices. The average connectivity of a partially bridging vertex is denoted by Cs. Clearly C >I 2 and Cs < C.
2.11. Network connectivity (C) The connectivity of a full-bridging vertex in a regular network will be referred to as the network connectivity.
2.12. Network size (N) A finite network has finite number, N, of vertices. (N - Ns) of the vertices are fully-bridged. Ns vertices are partially bridged. If the partially bridged vertices are only on the surface of a cluster then the fraction of the partially bridged vertices decreases as N increases and goes to zero in the limit of an infinite network.
2.13. Topologically equivalent, toplogically ordered, and topologically disordered (TD) networks The topology of a network characterizes the connectivity of the polytopes and the arrangement of the resulting interstitial regions in the network. In the case of two-dimensional networks, all interstices are polygonal rings and the topology is primarily specified by the distribution of their size R (the number of edges of a ring). On the contrary, in higher dimensions the interstitial regions are connected and the specification of topology is more complex. Two networks are topologically equivalent when one can be converted into the other without severance of any connections. Thus a network is topologically ordered if it is topologically equivalent to a crystalline network. Otherwise a network is called topologically disordered (TD).
2.14. Random networks Networks can be spatially periodic (i.e., crystalline) or aperiodic. An aperiodic network is random (in the sense of Warren [2]) if there is no long range positional or bond orientational order. Thus quasicrystals with long range bond orientational order are aperiodic but not random [9]. A random network may have long range topological order. Such topologically ordered random networks (sometimes termed paracrystals [10]) are unstable with respect to crystaniTation. The necessity to break connections and to form new ones causes TD networks to be resistant to crystani7ation. Stability of real glasses indicates that they are topologically disordered.
P.K. Gupta, A.R. Cooper / Topologically disordered networks of rigid polytopes
3. Statement of the problem and review of the previous work The problem of interest can be stated precisely as follows: given rigid polytopes (all specified by 8 and V), is it possible to form infinitely large topologically disordered networks of connectivity C and dimension d? Zachariasen [1] was the first to deal with this question. He formulated a set of rules about the polyhedra and the ways of linking them which would permit the formation of infinitely large TD networks. For example, according to Zachariasen, the structure of vitreous silica is an infinitely large TD network of corner sharing oxygen tetrahedra with C = 2. Zachariasen used only qualitative arguments to support his rules. Cooper [11] examined the question of the existence of two-dimensional TD networks made of congruent, rigid, regular polygons and showed that the freedom to form a TD network depended upon the existence of free angles between vertex sharing edges of the constituent polygons; angles not fixed by closure constraint (three angles are determined everytime a ring is closed) or by the vertex constraint (the sum of the ring angles at a vertex is determined). His argument can be summarized in general terms as follows. For a two-dimensional TD network made up of regular polygons, each consisting of V vertices, and C polygons sharing each vertex, there are 3/R constraints per angle due to the condition of closure of a ring of size R and 1 / C constraints per angle due to the condition that all angles at a vertex must sum to 2¢r. Thus the degrees of freedom per angle are 1 - (3/R) - 1/C. Since the average ring size is VC/(VC- V - C) and since there are C angles between the polygons at a vertex, the degrees of freedom per vertex, f, are given by f = 2 - 2C[1 - (3/2V)].
(1)
An infinitely large TD network exists only when f is non-negative. This treatment shows that when C = 2, rigid triangles constitute the limiting case of f = 0 and that higher polygons (such as a square) cannot form infinitely large TD networks.
17
Later, for three-dimensional networks, Cooper [12] pursued a somewhat different approach where he considered that all of the vertices of polytopes sharing a specific vertex contributed half a constraint per vertex leading to the expression
f= d - C(V- 1)/2.
(2)
We have shown [13] that this expression is not rigorous and leads to incorrect results in cases where the polytopes are not simplexes (V > (8 + 1)). A more rigorous and more general treatment is presented in this paper. Phillips [14] has proposed that the formation of large TD networks is favored when the degrees of freedom per atom are exactly balanced by the number of constraints per atom imposed by the short range interatomic forces (namely constraints due to nearest neighbor bond lengths and angles between nearest neighbor bonds). Like Phillips, our approach assumed that all long range, interatomic forces are weak allowing constraints to be counted easily.
4. Theory In this section a necessary condition is derived for the existence of infinite TD networks made of vertex sharing rigid polytopes. This condition involves only four variables: two polytope parameters (8, V) and two network variables (C, d). It is useful to make explicit the following three concepts. (1) Coordinate degrees of freedom: a point, free to choose any position in a t-dimensional space, has B-coordinate degrees of freedom. (2) Rigid body degrees of freedom: a rigid body in t-dimensional space has fl(fl + 1)/2 degrees of freedom (/3 translational and/3(/3 - 1)/2 rotational). For example the rigid body degrees of freedom are 1 when /3 = 1, 3 when /3 = 2, and 6 when/3 = 3. Fixing the position and orientation of a cluster of points imposes/3(/3 + 1)/2 rigid body constraints. (3) Internal degrees of freedom, F(N), of a cluster of N points:
F( N) =/3N -/3(/3 + 1)/2 - P( N).
(3)
P.K. Gupta, A.R. Cooper / Topologically disordered networks of rigid polytopes
18
"----q 4
0.
/
6
(a)
I
(b)
Fig. 1. Two possible sets of constraints for a cluster of six vertices in two dimensions to be rigid. The dashed lines indicate bond-length constraints. The marked angles represent the bond-angle constraints. (a) The five distances from the central point to the five boundary points and four of the internal angles are fixed. (b) The six distances between the boundary points and three of the six internal angles are fixed.
The first term on the right hand side gives the coordinate degrees of freedom of the cluster in a fl-dimensional space. The second term represents the loss in the degrees of freedom due to fixing the position and the orientation of the system as a whole. This is equivalent to fixing the origin and the orientation of the coordinate system. The loss of degrees of freedom caused by the last term is due to constraints on the distances and angles between the points in the cluster. F ( N ) = 0 implies a rigid cluster. This is seen by considering an example where N is sufficiently small that the constraints can be readily counted. Let N = 6 , f l = 2 , F = 0 , then P ( N ) = 9 . Two possible configurations for these six points are shown in fig. 1. In case (a), rigidity is achieved if the five distances from the central point to the five boundary points and four of the internal angles are fixed. In case (b), fixing the six distances between the boundary points and three of the six internal boundary angles are sufficient to make the duster rigid. When F(N) is positive, the cluster is non-rigid (i.e., floppy). F(N) < 0 implies that either a cluster does not exist that satisfies all the constraints or that some of the constraints are not independent. For example in fig. l(b) if, in addition to the nine constraints already mentioned, one specifies the distance between points 1 and 4 which is different than the one shown in the figure then no cluster can exist that satisfies the ten independent constraints. On the other hand, if one specifies this distance to be exactly the same as given in the figure, then this constraint is redundant.
By setting F(N) = 0 in eq. (3), one can obtain the number, P0(N), of independent constraints necessary to make a duster rigid:
Po(N) = fiN - fl(fl + 1 ) / 2 .
(4)
Equation (4) can be used to determine the number of independent constraints due to the rigidity of an isolated polytope ha~ng V vertices and of dimension 8. Here fl = 8 and N = V. Equation (4) gives
Po(8, V) = 8 V - 8(8 + 1)/2.
(5)
For example according to eq. (5) a rigid square has five internal constraints. The lengths of the four edges and a diagonal (or lengths of three edges and specification of two of the internal angles) make up the five constraints. We now calculate the degrees of freedom, F ( N ) , for a d-dimensional finite network of N vertices made up of vertex-sharing rigid polytopes (8, V). The total number of constraints due to the rigidity of the polytopes can be expressed as
P( N ) = [ ( N - Ns)( C / V ) + Ns( Cs/V)] ×P0(8,V).
(6)
First notice that C / V is the number of polytopes per fully bridging vertex. The first term on the right hand side gives the number of constraints due to the rigidity of the polytopes connecting to the fully bridging vertices. Similarly, the second term gives the number of constraints due to the rigidity of the polytopes connecting to the partially bridging vertices. Substitution of eqs. (5) and (6) and fl = d in eq. (3) gives
F ( N ) = N [ d - ( C / V ) { S V - 8(8 + 1)/2}] + ( U s / V ) ( C - C s ) [ S V - 8(8 + 1)/2] - d(d+
1)/2.
(7)
F(N) can be used to test two characteristics of a network: (1) Existence. If F(N)>~ 0, the finite network can exist. If F ( N ) < 0 then the finite network cannot exist because it is overconstrained (unless the constraints are not independent). (2) Rigidity. If F(N) > 0, the network is floppy. If F(N) <~0 then the network is rigid.
P.K. Gupta, A.R. Cooper / Topologically disorderednetworks of rigid polytopes
Since the second term on the right hand side of eq. (7) is always positive, it is clear that F ( N ) > 0 for sufficiently large values of N provided that
[d- (C/V){SV, 8(8 + 1)/2}]
lim
N---~ oo
(F/N).
Degrees of freedom f per vertex for d-dimensional infinite TD networks of connectivity C composed of 8-dimensional polytopes of V vertices each Polytope
8
V
d
C
f
Rod
1
2
1 2 2 2 2 3
2 2 3 4 6 6
0 1 0.5 0 -1 0
2 1 1 1 1 1
2 2 2 2 2 2
Triangle
2
3
2 2 3 3 3
2 3 2 3 4
0 -1 1 0 - 1
2 3 4/3 2 8/3
2 2 2 2 2
Square
2
4
2 3
2 2
- 0.5 0.5
2 4/3
8/3 8/3
Hexagon
2
6
2 3
2 2
- 1 0
6 4/3
4 4
Circle
2
oo
2 3 4
2 2 2
- 2 - 1 0
2 4/3 1
oo oo
(8)
Dividing eq. (7) by N, taking the limit as N oo, and using the fact that Ns/N vanishes in this limit one obtains
f= a - ( c / v ) [ s v -
Table 1
> 0.
The expression on the left hand side can be identified as the degrees of freedom f per vertex of an infinite fully bridged network defined as f=
19
8(3 + 1)/2].
(9)
One should note that F/N, for finite N, is always greater than f. In other words a finite network because of the partially bridged vertices on its surface has always more degrees of freedom per vertex than those in the corresponding infinite network. The expression f > 0 implies that a floppy infinite network can exist. When f= 0, this implies (as can be seen from eq. (7)) that a network can exist with limited (surface) degrees of freedom. When f < 0, a network cannot exist unless some of the constraints are no longer independent. Constraints due to the rigidity of the polytopes do become interdependent in the case of large crystalline networks. For example an infinite two-dimensional crystalline network of corner-sharing regular, congruent, and rigid squares has f < 0 (see table 1). Yet, this infinite network (the extended chess board) not only exists but is floppy since it can be sheared without changing the topology. This is because, in this case, one degree of freedom survives in the limit of an infinite network. The interdependence of constraints arises due to the long range order in the crystalline networks. For TD-networks since there is no long range order, the constraints due to the rigidity of the polytopes remain independent. Therefore the condition for the existence of infinitely large TD networks is that f must be non-negative. It should be emphasized that this is only a necessary condition. Equation (9) is our main result giving the degrees of freedom per vertex in an infinitely large
(~)
( 1~)
Tetrahedron
3
4
3 3
2 3
0 - 3/2
2 3
2 2
Octahedron
3
6
3 4
2 2
- 1 0
2 3/2
3 3
Cube
3
8
3
2
- 3/2
2
4
Dodecahedron
3
20
3
2
- 2.4
2
10
Icosahedron
3
12
3 5
2 2
- 2 0
2 6/5
Sphere
3
o¢
3 6
2 2
- 3 0
2 1
6 6 oe
TD-network of vertex-sharing rigid polytopes where the constraints arise only from the rigidity of the polytopes. More specifically, no orientational constraints between neighboring polytopes have been considered in the derivation. The derivation presented does not require polytopes to be regular or congruent but only that they are rigid and belong to the same family (i.e., topologically equivalent). It clearly includes the case when the polytopes are also regular and congruent-a situation which approximates the real glasses.
20
P.K. Gupta, A.R. Cooper / Topologically disordered networks of rigid polytopes
While our main concern is in the application to real glasses in one, two, and three dimensions, the theory is not limited to these cases. Others have suggested that treatment in higher dimensional spaces may provide new insights into the glass structure [15].
5. D i s c u s s i o n
Equation (9) can be rewritten as f~= 1 - d [ 1 - ( 1 / 1 7 ) ] ,
(10)
where f = f / d , C = CS/d, and V= 2V/(8 + 1) are appropriately normalized expressions for the degrees of freedom, the connectivity, and the number of vertices of a p olytope. It is clear that I7">t 2. The larger C and V, the greater the number of connections in the network. 17 is a parameter determined by the polytope alone. C also involves information about the network. Figure 2 is a plot of C versus 17. Only for values of 17 and
0 "
"
I
"
"
I
"
I
"
"
"
0
o
~"
<
o
"I:
o o
,
2.
I
.
5.
.
.
.
I
.
.
10.
.
.
I
.
.
15.
.
corresponding to )¢>~0, can infinite T D networks exist. It is evident that infinite network formation is favored by low values of C and 17. This is qualitatively consistent with Zachariasen's first two rules which can be stated: 'V is small ( < 4) and C cannot exceed 2'. The region corresponding to ~< 2 and 17 = 2, represents the region where infinite T D network formation is possible according to the Zachariasen's rules (assuming 8 = d = 3). Table 1 lists values of f based on eq. (9) for a variety of polytopes. For d = 2 and C = 2, only rods and triangles can form infinitely large T D networks. For d = 3 and C = 2, only rods, triangles and tetrahedra can form large T D networks. The condition f = 0, i.e., the boundary between freedom and overconstraint, appears to be associated with easy glass formation. This is evidenced by the fact that good glass formers, such as BeF2, SiO 2 and GeO 2, are composed of tetrahedra sharing corners. Also, random close packing of equal spheres, the prototype for metallic glasses, has an average contact coordination number of 6 [16]. This arrangement is equivalent to that of line segments with C = 6 and d = 3 (the line segments join centers of neighboring spheres). As seen in table 1 for all of these cases, f = 0. This perhaps is related to the fact that in this case the T D network is least floppy and hence less susceptible to crystallization. A similar rationalization has been suggested by Phillips [14]. Equation (9) reduces to eq. (1) when d = 8 = 2. This suggests that eq. (9) is a generalization of Cooper's first approach to higher dimensions. It differs from his later approach (eq. (2)) when V is not equal to (8 + 1) showing the incorrect nature of eq. (2). The present approach also differs from Phillips's in an important way. It considers the rigidity of polytopes as the only constraint. This permits distinction between the polytope dimension and the network dimension.
.
20.
V
Fig. 2. A diagrana showing region of existence of infinite TD network in the C, V space. Only the part I ~ 2 and C ~ l / d has physical meaning. The curve represents equation 10 for / = 0. Infinite TD networks can exist in the region wheref'> 0. The region I~= 2 and C g 2 corresponds to the Zachariasen's regime.
One of the authors (P.K.G.) thanks National Science Foundation for supporting this work under grant # DMR-8617916. He also wishes to thank Nick Rivier for valuable discussions. A.R.C. thanks Franz Spaepen for similarly valuable discussions.
P.K. Gupta, A.R. Cooper / Topologically disorderednetworks of rigid polytopes
References [1] [2] [3] [4] [5] [6] [7] [8]
W.H. Zachariasen, J. Awl. Chem. Soc. 54 (1932) 3841. B.E. Warren, J. Am. Cer. Soc. 17 (1934) 249. R.J. Bell and P. Dean, Nature 212 (1966) 1354. L. Guttman and S.M. Rahman, Phys. Rev. B37 (1988) 2657. D.L. Evans and S.V. King, Nature 212 (1966) 1353. J.F. Shackelford, J. Non-Cryst. Solids 49 (1982) 19. H.S.M. Coxeter, Regular Polytopes (Dover, New York, 1973) p. 293. R. Zallen, J. Non-Cryst. Solids 75 (1985) 3.
21
[9] M. Widom, Aperiodicity and Order, Vol. 1, ed. J. Varic (Academic Press, New York, 1988) p. 59. [10] J.M. Ziman, Models of Disorder (Cambridge University Press, Cambridge, 1979) p. 71. [11] A.R. Cooper, Phys. Chem. Glasses 19 (1978) 60. [12] A.R. Cooper, J. Non-Cryst. Solids 49 (1982) 1. [13] P.K. Gupta and A.R. Cooper, in: Proc. XVth Int. Congress on Glass, Leningrad, Vol. la, (1989) p. 13. [14] J.C. Phillips, Physics Today 35 (1981) 27. [15] J. Sadoc and R. Mosserri, Philos. Mag. 1345 (1982) 467. [16] C.H. Bennett, J. Appl. Phys. 43 (1972) 2727.
Journal of Non-Crystalline Solids 123 (1990) 14-21 North-Holland
T O P O L O G I C A L L Y D I S O R D E R E D N E T W O R K S O F RIGID P O L Y T O P E S P.K. G U P T A and A.R. C O O P E R
a
Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210, USA a Department of Materials Science and Engineering, Case Western Reserve University, Clevelana~ OH 44106, USA
A condition is derived for the existence of infinitely large d-dimensional topologicallydisordered networks composed of 8-dimensional rigid vertex-sharingpolytopes which is consistent with Zachariasen's first two rules. It generalizesthese rules to polytopes, network dimensions, polytope dimensions, and vertex coordinations not considered previously.
1. Introduction
In 1932, Zachariasen [1], in his rationalization of common experience as to which oxides readily form glasses, postulated " a n ultimate condition for the formation of glass" which is that " t h e substance can form extended three-dimensional networks lacking periodicity with an energy content comparable with that of the corresponding crystal". He then argued that: (a) " t h e energy of the network will in the first place depend upon the nature of the oxygen polyhedra, in the second place upon the way in which the polyhedra are linked together"; (b) "if the energy of the glass is to be comparable with that of the crystal, we must require that oxygen polyhedra in glass and crystal are essentially the same"; and (c) " t h e linking together of the oxygen polyhedra in the crystal lattices is done in such a manner that no two atoms are brought into close contact ... Ions carrying charges of the same sign will tend to be as far apart as possible. We have to apply just the same principles on the network in glass". In other words whether polyhedra in nonperiodic networks share vertices, edges or faces depends on whether they do so in the corresponding crystalline network and the coordination of the polyhedra about an oxygen is the same in crystalline and glassy networks. However, unlike crystals, the relative orientations between adjacent polyhedra " m a y vary within rather wide limits".
This distribution of the relative orientations is the origin of the non-periodicity in the network. By 'extended' Zachariasen clearly meant macroscopically (i.e., infinitely) large for he also states that " t h e network in glass is characterized by an infinitely l a r g e unit cell containing an infinite number of atoms". Following Warren [2], these non-periodic large networks are referred to as random networks. However, we suggest that it is more appropriate to call the Zachaxiasen networks topologically disordered (TD). While finite size T D networks can always be built using any polyhedron as the building block, the conditions under which a particular type of polyhedra can form an infinitely large T D network are not clear. Several attempts have been made to build T D networks by hand [3] or with the aid of computers [4] using vertex-sharing regulax, congruent and rigid (RCR) polyhedra and a set of construction rules which ensure a broad distribution in the relative orientations between adjacent polyhedra. These studies have led to a common realization that the requirements of connectivity at the vertices are rather severe and make it difficult to build large T D networks. The largest model built from R C R tetrahedra has 651 units [5]. Construction of two-dimensional T D networks with RCR triangles have led to similar results [6]. Thus two questions of fundamental importance arise: one about the existence of infinitely large T D networks made of vertex-sharing RCR polyhedra and the second about how to build such
0022-3093/90/$03.50 © 1990 - ElsevierSciencePublishers B.V. (North-Holland)
P.K. Gupta, A.R. Cooper / Topologicallydisorderednetworks of rigidpolytopes
15
networks given their existence. The model building approach is inadequate to answer the first question since all models are necessarily of finite extent. Indeed model building attempts to answer the second question. The question of the existence is related to whether the constraints due to connectivity of the polyhedra exceed the degrees of freedom associated with the vertices. This is strictly a geometrical problem. An answer to this question is provided in this paper using a topological approach. The result is a necessary condition which involves only four parameters: two describing the polyhedron and two associated with the network. The condition derived is sufficiently general and applicable to arbitrary dimension of the polytopes and of the network. To set the stage we explain (and define whenever possible) in section 2 some of the terms frequently used in this paper. In section 3 we state the problem of existence of TD networks in precise mathematical terms and review previous work related to this topic. Section 4 gives our proof of the necessary condition for the existence of infinitely large TD networks. We discuss the consequences of our result and relate them to the work of others, particularly that of Zachariasen, in the last section.
2.3. Regular polytope
2. Basic concepts: definitions and terminology
2.6. Rigid polytopes
2.1. Polytopes
A rigid polytope has fixed shape and size. Throughout this paper, we deal only with rigid polytopes. The fact that in real glasses the coordination polyhedra may not be rigid is not a concern of this paper. However, as our analysis implicitly shows, lack of rigidity of the polyhedra always favors random network formation.
Polytopes are generalizations of polygons and polyhedra to a space of arbitrary dimension. Onedimensional polytopes are called rods. Polygons are two-dimensional polytopes and polyhedra are three-dimensional polytopes. The corresponding geometrical figures in four dimensions are also called polytopes. In this paper the word polytope is used in its general sense.
2.2. Polytope dimension (8) The minimum dimension of Euclidian Space necessary to embed a polytope is called its dimension. Thus 8 (point) = 0, 8 (rod) = 1, 8 (polygon) = 2, and 8 (polyhedron)= 3.
A 8-dimensional polytope is regular if all its faces are alike, all its vertices are alike, and all its faces are regular ( 8 - 1) dimensional polytopes. Thus, a polygon is regular if it is equilateral and equiangular. While there are infinite number of regular polygons, there are only a finite number of regular polytopes in three and higher dimensions [7].
2.4. Characterization of polytopes A complete characterization of a polytope involves both its size and its shape. The shape of a regular polytope is specified by two numbers: its dimension 8 and the number of its vertices, V. For non-regular polytopes, these two parameters specify only the topology. Two polytopes are called topologically equivalent if they have the same values of 8 and V. It is clear that V>_-(8 + 1).
2.5. Congruent polytopes Congruent polytopes are identical in all respects. Congruent regular polytopes have the same dimension, same number of vertices, and same size.
2. 7. Network Polytopes can often be connected so as to form extended structures. We call these structures networks if it is possible to make cyclic paths along the polytope edges starting from any vertex (not on the surface) without repeating any polytope. Our definition of networks is more restrictive than a definition, sometimes used in graph theory,
16
P.K. Gupta, A.R. Cooper / Topologicallydisorderednetworks of rigidpolytopes
which includes structures (called trees) which do not have any cyclic paths. Finite size networks are also called clusters. In this paper we are interested in only those networks in which: (a) all polytopes have the same dimension and the same number of vertices (i.e., they are topologically equivalent), and they are rigid, (b) polytopes are connected only at the vertices (i.e., they do not share edges or faces), (c) each vertex (except the vertices on the surface of finite networks) is shared by the same number of polytopes (i.e., it is a regular network), and (d) the join at a vertex is free, like a universal hinge, imposing no constraint on the relative orientations of the polytopes sharing that vertex.
2.8. Network dimension (d) The minimum dimension of the Euclidian space in which a network can be embedded by deforming without severing or adding any connections (i.e., by unfolding or straightening without changing the topology) is called the network dimension, d. This is consistent with Zallen [8].
2.9. Vertex connectivity The number of polytopes connected to a vertex is called its connectivity. Clearly it is a positive integral number, greater than or equal to unity.
2.10. Fully bridging and partially bridging vertices In a regular network, vertices with the highest connectivity (denoted by C) are called fully bridging vertices. Vertices with connectivities less then C are called partially bridging vertices. The average connectivity of a partially bridging vertex is denoted by Cs. Clearly C >I 2 and Cs < C.
2.11. Network connectivity (C) The connectivity of a full-bridging vertex in a regular network will be referred to as the network connectivity.
2.12. Network size (N) A finite network has finite number, N, of vertices. (N - Ns) of the vertices are fully-bridged. Ns vertices are partially bridged. If the partially bridged vertices are only on the surface of a cluster then the fraction of the partially bridged vertices decreases as N increases and goes to zero in the limit of an infinite network.
2.13. Topologically equivalent, toplogically ordered, and topologically disordered (TD) networks The topology of a network characterizes the connectivity of the polytopes and the arrangement of the resulting interstitial regions in the network. In the case of two-dimensional networks, all interstices are polygonal rings and the topology is primarily specified by the distribution of their size R (the number of edges of a ring). On the contrary, in higher dimensions the interstitial regions are connected and the specification of topology is more complex. Two networks are topologically equivalent when one can be converted into the other without severance of any connections. Thus a network is topologically ordered if it is topologically equivalent to a crystalline network. Otherwise a network is called topologically disordered (TD).
2.14. Random networks Networks can be spatially periodic (i.e., crystalline) or aperiodic. An aperiodic network is random (in the sense of Warren [2]) if there is no long range positional or bond orientational order. Thus quasicrystals with long range bond orientational order are aperiodic but not random [9]. A random network may have long range topological order. Such topologically ordered random networks (sometimes termed paracrystals [10]) are unstable with respect to crystaniTation. The necessity to break connections and to form new ones causes TD networks to be resistant to crystani7ation. Stability of real glasses indicates that they are topologically disordered.
P.K. Gupta, A.R. Cooper / Topologically disordered networks of rigid polytopes
3. Statement of the problem and review of the previous work The problem of interest can be stated precisely as follows: given rigid polytopes (all specified by 8 and V), is it possible to form infinitely large topologically disordered networks of connectivity C and dimension d? Zachariasen [1] was the first to deal with this question. He formulated a set of rules about the polyhedra and the ways of linking them which would permit the formation of infinitely large TD networks. For example, according to Zachariasen, the structure of vitreous silica is an infinitely large TD network of corner sharing oxygen tetrahedra with C = 2. Zachariasen used only qualitative arguments to support his rules. Cooper [11] examined the question of the existence of two-dimensional TD networks made of congruent, rigid, regular polygons and showed that the freedom to form a TD network depended upon the existence of free angles between vertex sharing edges of the constituent polygons; angles not fixed by closure constraint (three angles are determined everytime a ring is closed) or by the vertex constraint (the sum of the ring angles at a vertex is determined). His argument can be summarized in general terms as follows. For a two-dimensional TD network made up of regular polygons, each consisting of V vertices, and C polygons sharing each vertex, there are 3/R constraints per angle due to the condition of closure of a ring of size R and 1 / C constraints per angle due to the condition that all angles at a vertex must sum to 2¢r. Thus the degrees of freedom per angle are 1 - (3/R) - 1/C. Since the average ring size is VC/(VC- V - C) and since there are C angles between the polygons at a vertex, the degrees of freedom per vertex, f, are given by f = 2 - 2C[1 - (3/2V)].
(1)
An infinitely large TD network exists only when f is non-negative. This treatment shows that when C = 2, rigid triangles constitute the limiting case of f = 0 and that higher polygons (such as a square) cannot form infinitely large TD networks.
17
Later, for three-dimensional networks, Cooper [12] pursued a somewhat different approach where he considered that all of the vertices of polytopes sharing a specific vertex contributed half a constraint per vertex leading to the expression
f= d - C(V- 1)/2.
(2)
We have shown [13] that this expression is not rigorous and leads to incorrect results in cases where the polytopes are not simplexes (V > (8 + 1)). A more rigorous and more general treatment is presented in this paper. Phillips [14] has proposed that the formation of large TD networks is favored when the degrees of freedom per atom are exactly balanced by the number of constraints per atom imposed by the short range interatomic forces (namely constraints due to nearest neighbor bond lengths and angles between nearest neighbor bonds). Like Phillips, our approach assumed that all long range, interatomic forces are weak allowing constraints to be counted easily.
4. Theory In this section a necessary condition is derived for the existence of infinite TD networks made of vertex sharing rigid polytopes. This condition involves only four variables: two polytope parameters (8, V) and two network variables (C, d). It is useful to make explicit the following three concepts. (1) Coordinate degrees of freedom: a point, free to choose any position in a t-dimensional space, has B-coordinate degrees of freedom. (2) Rigid body degrees of freedom: a rigid body in t-dimensional space has fl(fl + 1)/2 degrees of freedom (/3 translational and/3(/3 - 1)/2 rotational). For example the rigid body degrees of freedom are 1 when /3 = 1, 3 when /3 = 2, and 6 when/3 = 3. Fixing the position and orientation of a cluster of points imposes/3(/3 + 1)/2 rigid body constraints. (3) Internal degrees of freedom, F(N), of a cluster of N points:
F( N) =/3N -/3(/3 + 1)/2 - P( N).
(3)
P.K. Gupta, A.R. Cooper / Topologically disordered networks of rigid polytopes
18
"----q 4
0.
/
6
(a)
I
(b)
Fig. 1. Two possible sets of constraints for a cluster of six vertices in two dimensions to be rigid. The dashed lines indicate bond-length constraints. The marked angles represent the bond-angle constraints. (a) The five distances from the central point to the five boundary points and four of the internal angles are fixed. (b) The six distances between the boundary points and three of the six internal angles are fixed.
The first term on the right hand side gives the coordinate degrees of freedom of the cluster in a fl-dimensional space. The second term represents the loss in the degrees of freedom due to fixing the position and the orientation of the system as a whole. This is equivalent to fixing the origin and the orientation of the coordinate system. The loss of degrees of freedom caused by the last term is due to constraints on the distances and angles between the points in the cluster. F ( N ) = 0 implies a rigid cluster. This is seen by considering an example where N is sufficiently small that the constraints can be readily counted. Let N = 6 , f l = 2 , F = 0 , then P ( N ) = 9 . Two possible configurations for these six points are shown in fig. 1. In case (a), rigidity is achieved if the five distances from the central point to the five boundary points and four of the internal angles are fixed. In case (b), fixing the six distances between the boundary points and three of the six internal boundary angles are sufficient to make the duster rigid. When F(N) is positive, the cluster is non-rigid (i.e., floppy). F(N) < 0 implies that either a cluster does not exist that satisfies all the constraints or that some of the constraints are not independent. For example in fig. l(b) if, in addition to the nine constraints already mentioned, one specifies the distance between points 1 and 4 which is different than the one shown in the figure then no cluster can exist that satisfies the ten independent constraints. On the other hand, if one specifies this distance to be exactly the same as given in the figure, then this constraint is redundant.
By setting F(N) = 0 in eq. (3), one can obtain the number, P0(N), of independent constraints necessary to make a duster rigid:
Po(N) = fiN - fl(fl + 1 ) / 2 .
(4)
Equation (4) can be used to determine the number of independent constraints due to the rigidity of an isolated polytope ha~ng V vertices and of dimension 8. Here fl = 8 and N = V. Equation (4) gives
Po(8, V) = 8 V - 8(8 + 1)/2.
(5)
For example according to eq. (5) a rigid square has five internal constraints. The lengths of the four edges and a diagonal (or lengths of three edges and specification of two of the internal angles) make up the five constraints. We now calculate the degrees of freedom, F ( N ) , for a d-dimensional finite network of N vertices made up of vertex-sharing rigid polytopes (8, V). The total number of constraints due to the rigidity of the polytopes can be expressed as
P( N ) = [ ( N - Ns)( C / V ) + Ns( Cs/V)] ×P0(8,V).
(6)
First notice that C / V is the number of polytopes per fully bridging vertex. The first term on the right hand side gives the number of constraints due to the rigidity of the polytopes connecting to the fully bridging vertices. Similarly, the second term gives the number of constraints due to the rigidity of the polytopes connecting to the partially bridging vertices. Substitution of eqs. (5) and (6) and fl = d in eq. (3) gives
F ( N ) = N [ d - ( C / V ) { S V - 8(8 + 1)/2}] + ( U s / V ) ( C - C s ) [ S V - 8(8 + 1)/2] - d(d+
1)/2.
(7)
F(N) can be used to test two characteristics of a network: (1) Existence. If F(N)>~ 0, the finite network can exist. If F ( N ) < 0 then the finite network cannot exist because it is overconstrained (unless the constraints are not independent). (2) Rigidity. If F(N) > 0, the network is floppy. If F(N) <~0 then the network is rigid.
P.K. Gupta, A.R. Cooper / Topologically disorderednetworks of rigid polytopes
Since the second term on the right hand side of eq. (7) is always positive, it is clear that F ( N ) > 0 for sufficiently large values of N provided that
[d- (C/V){SV, 8(8 + 1)/2}]
lim
N---~ oo
(F/N).
Degrees of freedom f per vertex for d-dimensional infinite TD networks of connectivity C composed of 8-dimensional polytopes of V vertices each Polytope
8
V
d
C
f
Rod
1
2
1 2 2 2 2 3
2 2 3 4 6 6
0 1 0.5 0 -1 0
2 1 1 1 1 1
2 2 2 2 2 2
Triangle
2
3
2 2 3 3 3
2 3 2 3 4
0 -1 1 0 - 1
2 3 4/3 2 8/3
2 2 2 2 2
Square
2
4
2 3
2 2
- 0.5 0.5
2 4/3
8/3 8/3
Hexagon
2
6
2 3
2 2
- 1 0
6 4/3
4 4
Circle
2
oo
2 3 4
2 2 2
- 2 - 1 0
2 4/3 1
oo oo
(8)
Dividing eq. (7) by N, taking the limit as N oo, and using the fact that Ns/N vanishes in this limit one obtains
f= a - ( c / v ) [ s v -
Table 1
> 0.
The expression on the left hand side can be identified as the degrees of freedom f per vertex of an infinite fully bridged network defined as f=
19
8(3 + 1)/2].
(9)
One should note that F/N, for finite N, is always greater than f. In other words a finite network because of the partially bridged vertices on its surface has always more degrees of freedom per vertex than those in the corresponding infinite network. The expression f > 0 implies that a floppy infinite network can exist. When f= 0, this implies (as can be seen from eq. (7)) that a network can exist with limited (surface) degrees of freedom. When f < 0, a network cannot exist unless some of the constraints are no longer independent. Constraints due to the rigidity of the polytopes do become interdependent in the case of large crystalline networks. For example an infinite two-dimensional crystalline network of corner-sharing regular, congruent, and rigid squares has f < 0 (see table 1). Yet, this infinite network (the extended chess board) not only exists but is floppy since it can be sheared without changing the topology. This is because, in this case, one degree of freedom survives in the limit of an infinite network. The interdependence of constraints arises due to the long range order in the crystalline networks. For TD-networks since there is no long range order, the constraints due to the rigidity of the polytopes remain independent. Therefore the condition for the existence of infinitely large TD networks is that f must be non-negative. It should be emphasized that this is only a necessary condition. Equation (9) is our main result giving the degrees of freedom per vertex in an infinitely large
(~)
( 1~)
Tetrahedron
3
4
3 3
2 3
0 - 3/2
2 3
2 2
Octahedron
3
6
3 4
2 2
- 1 0
2 3/2
3 3
Cube
3
8
3
2
- 3/2
2
4
Dodecahedron
3
20
3
2
- 2.4
2
10
Icosahedron
3
12
3 5
2 2
- 2 0
2 6/5
Sphere
3
o¢
3 6
2 2
- 3 0
2 1
6 6 oe
TD-network of vertex-sharing rigid polytopes where the constraints arise only from the rigidity of the polytopes. More specifically, no orientational constraints between neighboring polytopes have been considered in the derivation. The derivation presented does not require polytopes to be regular or congruent but only that they are rigid and belong to the same family (i.e., topologically equivalent). It clearly includes the case when the polytopes are also regular and congruent-a situation which approximates the real glasses.
20
P.K. Gupta, A.R. Cooper / Topologically disordered networks of rigid polytopes
While our main concern is in the application to real glasses in one, two, and three dimensions, the theory is not limited to these cases. Others have suggested that treatment in higher dimensional spaces may provide new insights into the glass structure [15].
5. D i s c u s s i o n
Equation (9) can be rewritten as f~= 1 - d [ 1 - ( 1 / 1 7 ) ] ,
(10)
where f = f / d , C = CS/d, and V= 2V/(8 + 1) are appropriately normalized expressions for the degrees of freedom, the connectivity, and the number of vertices of a p olytope. It is clear that I7">t 2. The larger C and V, the greater the number of connections in the network. 17 is a parameter determined by the polytope alone. C also involves information about the network. Figure 2 is a plot of C versus 17. Only for values of 17 and
0 "
"
I
"
"
I
"
I
"
"
"
0
o
~"
<
o
"I:
o o
,
2.
I
.
5.
.
.
.
I
.
.
10.
.
.
I
.
.
15.
.
corresponding to )¢>~0, can infinite T D networks exist. It is evident that infinite network formation is favored by low values of C and 17. This is qualitatively consistent with Zachariasen's first two rules which can be stated: 'V is small ( < 4) and C cannot exceed 2'. The region corresponding to ~< 2 and 17 = 2, represents the region where infinite T D network formation is possible according to the Zachariasen's rules (assuming 8 = d = 3). Table 1 lists values of f based on eq. (9) for a variety of polytopes. For d = 2 and C = 2, only rods and triangles can form infinitely large T D networks. For d = 3 and C = 2, only rods, triangles and tetrahedra can form large T D networks. The condition f = 0, i.e., the boundary between freedom and overconstraint, appears to be associated with easy glass formation. This is evidenced by the fact that good glass formers, such as BeF2, SiO 2 and GeO 2, are composed of tetrahedra sharing corners. Also, random close packing of equal spheres, the prototype for metallic glasses, has an average contact coordination number of 6 [16]. This arrangement is equivalent to that of line segments with C = 6 and d = 3 (the line segments join centers of neighboring spheres). As seen in table 1 for all of these cases, f = 0. This perhaps is related to the fact that in this case the T D network is least floppy and hence less susceptible to crystallization. A similar rationalization has been suggested by Phillips [14]. Equation (9) reduces to eq. (1) when d = 8 = 2. This suggests that eq. (9) is a generalization of Cooper's first approach to higher dimensions. It differs from his later approach (eq. (2)) when V is not equal to (8 + 1) showing the incorrect nature of eq. (2). The present approach also differs from Phillips's in an important way. It considers the rigidity of polytopes as the only constraint. This permits distinction between the polytope dimension and the network dimension.
.
20.
V
Fig. 2. A diagrana showing region of existence of infinite TD network in the C, V space. Only the part I ~ 2 and C ~ l / d has physical meaning. The curve represents equation 10 for / = 0. Infinite TD networks can exist in the region wheref'> 0. The region I~= 2 and C g 2 corresponds to the Zachariasen's regime.
One of the authors (P.K.G.) thanks National Science Foundation for supporting this work under grant # DMR-8617916. He also wishes to thank Nick Rivier for valuable discussions. A.R.C. thanks Franz Spaepen for similarly valuable discussions.
P.K. Gupta, A.R. Cooper / Topologically disorderednetworks of rigid polytopes
References [1] [2] [3] [4] [5] [6] [7] [8]
W.H. Zachariasen, J. Awl. Chem. Soc. 54 (1932) 3841. B.E. Warren, J. Am. Cer. Soc. 17 (1934) 249. R.J. Bell and P. Dean, Nature 212 (1966) 1354. L. Guttman and S.M. Rahman, Phys. Rev. B37 (1988) 2657. D.L. Evans and S.V. King, Nature 212 (1966) 1353. J.F. Shackelford, J. Non-Cryst. Solids 49 (1982) 19. H.S.M. Coxeter, Regular Polytopes (Dover, New York, 1973) p. 293. R. Zallen, J. Non-Cryst. Solids 75 (1985) 3.
21
[9] M. Widom, Aperiodicity and Order, Vol. 1, ed. J. Varic (Academic Press, New York, 1988) p. 59. [10] J.M. Ziman, Models of Disorder (Cambridge University Press, Cambridge, 1979) p. 71. [11] A.R. Cooper, Phys. Chem. Glasses 19 (1978) 60. [12] A.R. Cooper, J. Non-Cryst. Solids 49 (1982) 1. [13] P.K. Gupta and A.R. Cooper, in: Proc. XVth Int. Congress on Glass, Leningrad, Vol. la, (1989) p. 13. [14] J.C. Phillips, Physics Today 35 (1981) 27. [15] J. Sadoc and R. Mosserri, Philos. Mag. 1345 (1982) 467. [16] C.H. Bennett, J. Appl. Phys. 43 (1972) 2727.