Void structure in models of vitreous silica

] O U R N A L OF

ELSEVIER

Journal of Non-CrystallineSolids 192&193 (1995) 263-266

Void structure in models of vitreous silica S.N. Taraskin, S.R. Elliott *, M.I.K.linger Departmentof Chemistry, Universityof Cambridge, LensfieldRoad, CambridgeCB2 1EW, UK

Abstract The interstitial-void statistics of two models of vitreous silica have been investigated, with particular regard to the shape characteristics of aggregates of overlapping voids (trees). The longest branches of such trees are somewhat linear in shape and seem to have a universal local dimensionality of 4/3, irrespective of the model or overlap criterion used.

I. Introduction Structural models of amorphous materials are normally characterised in terms of atom-atom pair correlations. However, an equally valid description is in terms of the interstitial voids in the structure. Such an approach has often been employed in the case of dense random packings of spheres, representing the structure of amorphous metals [1,2] but some continuous random network models, simulating the structure of network glasses, have also been analysed in this way [3,4]. In addition to providing a complementary description of the structure, there are two other reasons for performing an analysis of the void network: a recent model for the so-called 'first sharp diffraction peak' (FSDP) in the structure factor of network glasses ascribes it to a prepeak in the concentration-concentration partial structure factor for an {atom, void} packing, associated with the ordering of interstitial voids [5,6]; in addition, a proper microscopic understanding of atomic/ionic diffusion in amorphous materials requires a knowledge of the

* Corresponding author. Tel: +44-1223 336 525. Telefax: +44-1223 336 362.

void network through which the diffusing species move. Previously, the voids in structural models of amorphous materials have been described in terms of their number and size distribution [1,4], as well as in terms of the diffusion doorways connecting the voids and the overall percolative behaviour [2,4]. However, there has been no previous analysis of the shapes adopted by aggregates of overlapping voids, specifically in continuous random network models. Such a statistical description is potentially important for an understanding of the dynamics associated with the amorphous structure, both of atoms/ions diffusing through the glassy matrix and of the (low-frequency) vibrational behaviour of the framework itself. The cluster statistics of interstitial void aggregates in models simulating the structure of vitreous silica have therefore been investigated.

2. The models The void statistics have been analysed for two structural models of v-SiO 2 constructed by means of molecular-dynamics simulation, both containing 648

0022-3093/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0022-3093(95)00361-4

264

S.N. Taraskin et al. /Journal of Non-Crystalline Solids 192& 193 (1995) 263-266

atoms (216 Si, 432 O) in a cubic box of length 21.397 ,~ subject to periodic boundary conditions. The first was constructed by Feuston and Garofalini [7] using a three-body potential, the second by Uhlherr and Elliott [8] using 'charge-transfer molecular dynamics' [9].

3. Void-related definitions

An elementary void is defined [2,4] as the largest spherical volume that can be inserted into sets of four oxygen atoms, touching all of them, such that no other atom (Si or O) lies within the spherical region. Open void-like regions not satisfying the relation [2,4] 4

E aiRi= O,

(1)

i=1

where all coefficients, a~, have the same sign and R~ are the vectors connecting centres of the sphere and the surrounding oxygen atoms, i, are not taken into account. The size of a void is characterized by the radius, R v = [Ril-Rox, where Rox is the atomic radius of oxygen, taken to be Rox = 1.51 .~ [4]. The total number of voids, Nv, so defined, is found to be approximately equal to the number of atoms in the models; for the Feuston-Garofalini model [7], N v = 609, for example. The radii of the voids fall within the range 0.18 < R < 1.83 ,~. Such voids can overlap with each other. Two voids with radii R 1 and R 2 (
(2)

where a is a parameter having a value in the range of 0 < a < l . 'Weak' overlap occurs for a = l [2,4,10] and 'strong' overlap occurs for a = 0 [11,12]. The following analysis concerns mainly the cases of strong (a = 0) and 'intermediate' (a = 0.5) overlap. A set of overlapping voids is denoted as a 'tree' of voids. The shape of a tree can be characterized in terms of the branches of the tree. A branch of voids with its origin at void i is defined in terms of a sub-set of the overlapping voids comprising a tree when it satisfies the following conditions. A void at

the origin has, in general, n shells of neighbouring voids each containing Jn voids. Only one void from each shell is included in the branch, subject to it being connected to the origin by the minimum number of voids from the same branch. Such a definition does not exclude the splitting, and subsequent joining, of branches comprising the same number of voids. A branch can be characterized in terms of the following quantities: (i) the length of the branch, given by Nb-1

l~= v~ IRj,j+,l,

(3)

j=l where Rj, j+ 1 = R j+ 1 - R j is the vector connecting the centres of void j and the neighbouring void j + 1, and N b is the number of voids in the branch, with its origin at void i; (ii) the distance, R~, between the ends of the branch, given by Nb-1

Ri=

~., Rj,j+ 1

(4)

j=l

(iii) the average angle, 0~, between intercentre vectors in the branch, where N~2

Oi= ~_, arccos[Rj,j+l"Rj+l.j+2 j=l

/I Rj,j+I IIRj+,,y+21].

(5)

The longest branch in the tree is defined as the branch having the maximum length l = max{li}, with i running over all possible branches.

4. Results

Many of the voids in the models that were investigated were found to be overlapping and form trees comprising varying numbers, Nt, of elementary voids. The distribution of trees of different size is shown in Fig. 1. This is a steeply decreasing function; most trees contain just a single void. However, there exist, even in the case of the strong-overlap criterion, some rather large trees composed of ~ 10-20 elementary voids. It was found that the biggest trees result mainly from the overlapping of several smaller trees,

S.N. Taraskin et aL /Journal of Non-Crystalline Solids 192& 193 (1995) 263-266 30

/-

~

~

/

25

t

!~ ................... ...........

20

..............

....................

o [] oX

FG FG UE1 UE1

+ &

uE2 a--O UE2 a=0.5

265

(a)

a=0 a=0.5 a---O a=0.5

1

UE3 a--O

z

15

P

.........:............................

10

.....................................................................

!

~

uE3~=0.s

M0.1 o

i.............

:

~-

k

~-

F

0 4 0

. ~. 5

~

* 10

Bg!~ 15

20 N

I X [Z]JX 25

30

i

i

35

40

I

.,~

/

I

j-

i

I ;

,

,i

0.01

_ , , ,~

0.1

and that they have a rather irregular, non-spherical shape. In order to investigate quantitatively the shape of the trees, the number of voids in the longest branch in every tree, N~b, was calculated as a function of number of voids in the tree, Nt (see Fig. 2). A power-law behaviour was found for the dependence Nlb = Nt a ,

100

100

z

UF-2 a---0

~,

uE2 a=O.S I

[] []

10

1

/.

|

UE3 a=O l UE3 a=O 5 | ' I

/

x /

^/ ~:]~("

.....~. . . . . . . . . . . . . . . . . . . . . . .

~

..............................

r 1

10

100

Iog(N t) Fig. 2. Dependence of the average number of voids in the longest branch of a tree, Nlb, on the total number of voids in a tree, Nt, plotted double-logarithmically. (The averaging was made over different trees with a fixed number of voids in the trees.)

10

r ....

© I

5

I

o X

~

FG FG UE1 UE1

a=0 J a=0.5 | a=O I a=0-5 !

80 60 '

40

X

20 0

with the exponent taking the apparently universal value a -= 0.73; this value was found to be character-

UE1 a=0.5 |

UE3 a=0 UE3 a=0.5

(b)

(6)

+

II;

UE2a;0.S

1

~-r

x

[] ~

UE1 a=0

log(I/L)

120

FG a=0.5 | UE1 a=0 |

o

FG a--0 FG a=0.5

0.01

t

Fig. 1. Distribution of number of trees of voids, N, with number of voids in a tree, Nt, for various overlap criteria (a = 0, 0.5) and for different models: FG, Feuston-Garofalini model [7]; UEn, Uhlherr-Elliott model [8], where n denotes different snap-shot configurations of the model.

[] O

E

L _ _ ~

0

I

5

10

15

20

Nib

Fig. 3. Characterization of the geometry of the longest branches of trees of voids. (a) Average distance, R, between ends of the longest branches as a function of average length, l, of the longest branch, both normalized to the model box length, L, and plotted double-logarithmically. (The averaging was made over the longest branches of trees with fixed number of voids in a longest branch, Nlb.) (b) Dependence of the average angle, 0, for the longest branches as a function of number of voids in a longest branch, Nlb.

istic of all configurations of both models [7,8] studied, and for both overlap criteria used. The relationship has also been found between the distance, R, between the ends of the longest branches and the average length, l, of the branches, and this is shown in Fig. 3(a). Again, a power-law behaviour for branches comprising a minimum of four voids is found: R ~ l t~, (7)

266

S.N. Taraskin et aL /Journal of Non-Crystalline Solids 192& 193 (1995) 263-266

with the exponent taking the apparently universal value /3 ~ 0.82. Another parameter characterizing the degree of curvature of the branches is the average angle, 0, between vectors connecting the centres of two neighbouring voids (Eq. (5)). The dependence of 0 on the number of voids in the longest branch, Nab, is shown in Fig. 3(b), whence it can be seen that the average value is 0 = 30 ° with considerably greater fluctuations for smaller branches than for longer branches. This shows that an absence of sharp kinks between links formed by overlapping voids is characteristic of the longest branches so that they look like rather smooth lines not forming rings (R 4: 0). The latter conclusion follows directly from the dependence R(l) presented in Fig. 3(a).

6. Conclusions

The interstitial-void statistics of two moleculardynamics models of v-SiO 2 have been analysed with particular regard to the shape characteristics of aggregates of overlapping voids (trees). The longest branches of such trees are somewhat linear and have a seemingly universal local dimensionality of 4/3, irrespective of model or overlap criterion employed.

The authors are grateful to Drs S.L. Chan and A. Uhlherr for supplying some void-analysis programmes and model coordinates. M.I.K. highly appreciates the hospitality and financial support of Trinity College, Cambridge; S.N.T. is grateful to The Royal Society of London for financial support.

5. Discussion

The apparent universality of the exponent a (Eq. (6)) is perhaps the most significant result reported in this paper. It should be noted that for a regular 3D tree, o~= ½, for a 2D tree, a = 71 and for a linear (1D) tree, a = 1. Thus, the finding that a = ~3 for trees of voids in models of v-SiO 2 implies that they can be characterized by a local dimensionality of 4/3. At present, the authors have no explanation for this apparently universal behaviour. A value of unity for the exponent fl (Eq. (7)) would imply perfectly linear chains of voids forming the longest branches. The value of /3 slightly less than unity (Fig. 3(b)) indicates that the longest branches have only a small degree of curvature. It is significant also that the same power-law behaviour linking R and l is found for all configurations of both models and for both void-overlap criteria.

References [1] J.D. Bernal, Proc. R. Soc. London A280 (1964) 299. [2] S.L Chan and S.R. Elliott, J. Non-Cryst. Solids 124 (1990) 22. [3] S.K. Mitra and R.W. Hockney, J. Phys. C13 (1980) L739. [4] S.L. Chart and S.R. Elliott, Phys. Rev. B43 (1991) 4423. [5] S.R. Elliott, Phys. Rev. Lett. 67 (1991) 711. [6] S.R. Elliott, J. Phys.: Condens. Matter 4 (1992) 7661. [7] B.P. Feuston and S.H. Garofalini, J. Chem. Phys. 89 (1988) 5818. [8] A. Uhlherr and S.R. Elliott, unpublished. [9] A. Alavi, L.J. Alvarez, S.R. Elliott and I.R. McDonald, Philos. Mag. B65 (1992) 489. [10] J. Shao and Z-Q. Tang, Chin. Sci. Bull. 36 (1991) 911. [11] L.T. Hamill and J.M. Parker, Phys. Chem. Glasses 26 (1985) 52. [12] A. Uhlherr, D.R. MacFarlane and T.J. Bastow, J. Non-Cryst. Solids 123 (1990) 42.