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Glass structure
Journal of Non-Crystalline Solids 52 (1982) 31-43 North-Holland Publishing Company
31
GLASS STRUCTURE J. Z A R Z Y C K I Laboratory of Materials Science and CNRS Glass Laboratory, University of Montpellier, France A review of recent advances in the determination of the structure of glasses is presented. Statistical information on the short range order is usually obtained by combining X-ray and neutron diffraction with spectroscopic methods. Diffraction methods have benefited from new powerful sources which have extended the Q-range available and have resulted in a better resolution as well as a (limited) possibility of extracting partial pair-distribution functions. Among the spectroscopic methods, EXAFS and FLN are particularly promising. Modelling is shown to be indispensable for the detailed interpretation of the experimental results. In spite of progress made, the present methods are, however, still insufficient to determine the "middle-range" order actually present in glasses.
1. Introduction Glasses m a y be defined as supercooled, frozen-in liquids which have progressively a t t a i n e d the characteristics of a solid without crystallizing. T h e y are essentially non-crystalline m a t e r i a l s which possess structural features a n a l o g o u s to those of liquids, minus the s h o r t - t i m e mobility. Because of their technological i m p o r t a n c e the field of inorganic n o n - c r y s t a line solids has long been d o m i n a t e d b y oxide glasses a n d m o r e specifically, silicate glasses. Interest in vitreous chalcogenides was s t i m u l a t e d b y their infrared transmission a n d s e m i - c o n d u c t i n g properties. The latest n e w c o m e r s in this field are metal glasses which reveal special mechanical, m a g n e t i c a n d chemical p r o p e r ties. Q u e n c h i n g of a supercooled liquid constitutes one way of o b t a i n i n g noncrystalline solids; one should n o t forget, however, that there are o t h e r ways (e.g. v a p o u r deposition, chemical precipitation, r a d i a t i o n d a m a g e of crystals etc.) which can also lead to materials i n d i s t i n g u i s h a b l e from classical glasses o b t a i n e d b y quenching.
2. Description of the structure All these m a t e r i a l s have one feature in c o m m o n : they are solids with a disordered structure. By structure we refer here to a precise d e s c r i p t i o n of the s u b s t a n c e in terms of a t o m i c positions, b o n d lengths a n d b o n d angles. 0022-3093/82/0000-0000/$02.75
© 1982 N o r t h - H o l l a n d
32
J. Zarzycki / Glass structure
In the case of a crystal the arrangement of atoms is periodic in three directions of space. The detailed description of such a structure is complete once the dimensions and content of the unit cell are specified. The disposition of all the atoms is then determined by translation of this cell along the three directions of space. Crystals are said to possess both a short-and long-range order and the radiocrystallographic methods which have been developed are based on the properties of point groups and translational groups which characterise a given structure. The case of disordered materials such as glasses (and liquids) is more complex. Only short-range order is present and the unit cell can no longer be defined. The absence of the long-range order imposes other means of description; (a) The coordinates and the nature of all the atoms have to be specified the material being considered as a giant molecule - a detailed model of structure is required. (b) The structure is described statistically by means of distribution functions expressing the probability of finding an atom at a certain distance from another atom. For a material containing only one type of atom a function p(r) or radial density may be defined; it represents the average number of atoms per unit volume situated at a distance r from any atom of the structure taken as the origin (fig. la). This function expresses the short-range order. When r ~ 0¢, p(r) tends to P0, the average density of the material, p(r)/po = g(r) is then the pair-distribution function and G( r )= g( r ) - 1 is the pair correlation function (fig. lb). The function 4~rr2p(r) is the radial distribution function (fig. lc). It shows maxima which correspond to the interatomic distances most frequently found in the structure, while the areas subtended by the successive peaks are related to average coordination numbers associated with these distances. When in the disordered structure several different atomic species i, j, are present it is necessary to consider radial distibution functions corresponding to different atomic pairs i.e. to define partial pair-distribution functions g(j(r).
p(r)
pl
a '
r
D
G(r)
0
~...---~_
C
b __ r
14
rI r 2
r
Fig. 1. Radial description of a disordered structure: (a) radial density; (b) pair-correlation function; (c) radial distribution function.
J. Zarzycki / Glass structure
33
For a structure formed by two atomic species A and B, 3 pair distribution functions relative to pairs A - A , A - B and B - B have to be considered. For three atomic species 6 functions are required. The number of partial functions increases rapidly; for n species it is: n(n + 1)/2. This sets a practical limit to the determination of polyatomic disordered structures because as is well known, from a scattering experiment only a weighted sum of the pair-distribution function may be obtained, which brings about the superposition of various peaks and reduces the resolution.
3. Diffraction studies
Scattering of radiation is the usual experimental way of obtaining the pair-distribution functions. Generally speaking a scattering event is accompanied by an energy change h~0 and a momentum change hQ, where Q is the scattering vector Q = K - K 0 , the difference of the wave vectors of the scattered and the incident wave. IQI = Q = (41r sin O)/X for a scattering angle 20 and wavelength X. Elastic scattering (when he0 = 0) is related to the spatial distribution Gij(r ) of the scattering centres. The coherent scattering intensity I(Q) is then:
I(O) = E Ecicjf G,j(r) exp(iQ.r)dr, i j
where c~ and cj are scattering lengths of atomic species i and j. It can be showen in particular that for an isotropic medium: icOh(Q)
= E c2i + ~ ciCJfo°°4~rr2gij(r ),----~dr.SQr in i
i~j
Methods involving Fourier transformation have been developed by means of which a weighted sum of gij(r) or a total radial distribution function may be obtained from the experimentally measured F°h(Q). The representation thus obtained is a one-dimensional function of a single radial parameter r. This sets up a serious limitation as a three-dimensional disordered structure is thus insufficiently defined. For a complete description a correlation function of a higher order would be required, (taking into account the simultaneous positions of three atoms and more). Unfortunately these functions cannot be obtained from present experimental methods. In the case of glasses, scattering of X-rays and neutrons, and very occasionally of electrons, has been used. The initial methods developed by Warren et al. [1] have recently been improved [2] thanks to the rigorous formalism developed. The main efforts in the last years have been mainly directed towards two goals: (1) Improvement of the resolution of the method by extending the range of Q measured experimentally.
J. Zarzycki / Glass structure
34
(2) Determination of the partial distributions g i / o f the structures. The classical experiments with X-rays and neutrons used monochromatic sources and Q = (4~r sin 0 ) / ~ was varied by modifying the scattering angle 20. This conventional diffractometer technique limited the range of Q to less than 20 ,~- i as sources with ~ < 0.5 ,~ were not available. With the advent of powerful synchrotron X-ray sources and pulsed neutron sources of high intensity as well as suitable solid state detectors it has become possible to use continuous radiation with wavelengths as low as 0.2 ,~ for the structural determination of glasses. Measurements can then be made at a fixed scattering angle as a function of X. This permits, in principle, the extension of the Q range to 60 ,~-1, but in most cases measurements of intensity modulations will be difficult to observe beyond 30 to 40 ,~-1. Even so the doubling of the Q range considerably improves the resolution of the radial distribution functions. Fig. 2 shows the improved determinations of S i O 2 structure, [3] the partial pair-distribution functions were calculated assuming different models in order
2~
// t
/
s
//
TglX - x.i
T
t o~
i
2
3
r(A)
~
5
Fig. 2. Pair distribution functions for vitreous silica. The partial functions have been separated below. After ref. 3.
J. Zarzycki / Glass structure
35
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36
J. Zarzycki / Glass structure
to obtain the best fit for the total distribution function. It would be desirable to determine these partial functions directly without any hypothesis. In principle, for a biatomic structure requiring 3 partial gu, it is possible to determine them separately if 3 independent total scattering functions I(Q) are available. These can be obtained using: (1) Three different radiations (X-rays, neutrons, electrons); (2) Isotopic substitution (for neutrons); (3) Anomalous dispersion (for X-rays); (4) Polarized neutrons (for magnetic materials); (5) Isomorphous substitution. The last method is always uncertain as there is no guarantee as to how the structure is changed by the substitution. The first method requires different samples which may have dissimilar structures; There is no known example for oxide glasses. An attempt has been made on PdsoSi2o glass [4]. Fig. 3 shows the use of isotopic substitution for K20, TiO 2, 2SiO 2 glass where the isotopes 46Ti (c i = 0.48) and 48Ti (c, = -0.58) have been used [5]. This change of contrast enables the positions of Ti ions to be situated on the difference curve. Fig. 4 shows the improved results obtained for LiO2, 2SiO2 and Na20, 2SiO 2 glasses [6]. L i - O and N a - O distances could be resolved. Fig. 5 shows the partial distribution curves obtained for the Co31P19 disordered alloy using polarized neutrons [7]. The variation of the scattering amplitude of X-rays due to anomalous dispersion close to the absorption edge of an element in the structure has been
8
s
r L~
i t
I
n
,P,
f,
li/" s i
!
,
,i, ~ ,!
Z
Fig. 5. Partial distribution functions for amorphous Co81P19 alloy obtained using polarized neutrons. Full curves, Co-Co; dotted curve, Co-P; dashed curve P-P. From ref. 7.
J. Zarzycki / Glass structure
37
used in the case of amorphous GeSe [8]; the precision of the results is, however, insufficient. It can be seen that the extraction of partial distribution functions has only been attempted on very limited occasions.
4. Spectroscopic methods The alternative way of obtaining information on the structure of glasses is to use spectroscopic methods which analyse the various individual atomic sites. Among the different methods used, infrared and Raman spectroscopy, Mrssbauer spectroscopy, ligand-field-theory applied to the spectroscopy in the visible, EPR and N M R are among those used most frequently. They have permitted the determination of the symmetry of some particular atomic sites or the number of neighbours (coordination number) surrounding a given atomic species. The classic example is that of the determination of the fraction of borons in fourfold coordination in B203-X20 glasses (X = Li, Na, K, Cs) fig. 6, [9]. This was possible using the difference in N M R spectra for ' 1B atoms respectively in triangular and tetrahedral coordination [9]. Recently the use of the ~°B isotope, which gives a spectrum richer in detail, has enabled a whole range of structures in borate glasses [10] to be determined. Generally the presence of a whole range of sites broadens the spectra, compared to those of corresponding crystals. The spectra in glass results from a superposition of those related to individual sites. This is particularly clear in the case of the fluorescence spectra of rare earth ions where this gives rise to inhomogeneously broadened fluorescence. The
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A
o.2 A
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20
30
40
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~
70
Idol. % alkali modifier Fig. 6. The fraction N 4 of four-coordinated boron atoms in alkali borate glasses. O, Na; O, K, zx, Li: + , Rb, × , Cs. From ref. 9.
J. Zarzycki / Glass structure
38
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0.$
~
iAil
llill
0.6
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!11
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GeSe-C
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Fig. 7. Radial correlation functions for crystalline or a m o r p h o u s GeSe: (a) relative to a Ge atom; (b) relative to a Se atom. F r o m ref. 11.
excitation of spectra by laser leads to resonant line-narrowed fluorescence. Using a tunable laser it is possible to explore the various sites in a disordered structure which give different responses. Laser-induced fluorescence line narrowing (FLN) provides a unique microscopic probe of the local fields at an ion site and has proved to be a powerful tool for investigating the local structure surrounding optically active ions in glass. Another particularly fruitful method proved to be the study of Extended X-ray Absorption Fine Structure (EXAFS) which consists of recording the fine structure of an X-ray absorption spectrum of an absorbing element. The variations of the absorption coefficient are measured on the high-energy side close to the absorption edge. The oscillations observed are due to the diffraction of the ejected photoelectron by the atoms surrounding the absorbing atom. A Fourier inversion analysis then leads to a radial distribution function relative to a given atomic pair. The information is limited to the first neighbours but the advantage of the
J. Zarzycki / Glass structure
39
method lies in the fact that the local structures in the vicinity of various elements may be studied separately, without fear of superpositions which are inevitable in X-ray diffraction. EXAFS thus constitutes a method intermediate between diffraction and spectroscopy. Fig. 7 refers to amorphous GeSe and shows partial distributions around a Se or a Ge atom.
5. Modelling The interpretation of the results of either diffraction or spectroscopic methods inevitably requires the adoption of a structural model to compare the calculated results with those given by experiment. Models were first constructed by manual methods e.g. the ball-and-spike model of the SiO 2 network [12] or automatically, as the random packing of hard spheres [13]. Today the use of high speed computers permits the refinement of mechanically built models by structural relaxation methods. What is even more important, Monte-Carlo and molecular dynamics methods enable the direct construction of disordered structures from atomic systems subjected to given constraints: interatomic potentials, density and temperature, for example. The computer randomly generates millions of allowable
22
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Itl s$" O(ll
I0 :N(r)
l 7 6
5 4 3 [2) 2
Si-So $i-012) I
2
3
4
5
Fig. 8. Partial radial distribution functions obtained by molecular dynamics modelling of vitreous SiO 2 at 300 K. Dashed lines represent results deduced from X-rays. N(r) is the average number of ions around Si atoms. From ref. 14.
40
J. Zarzycki / Glass structure 0.25
~gj ~
2.35
12
///nI
0.2
10
0,1G
8
;t
0.1
0.05
¢
/\.,.x,,
6
4
/
•
0
0
2.0
2
2.5
3.0
3.5
4.0
r,A
4.5
5.0
5.5
Fig. 9. Partial pair distribution functions calculated by Monte-Carlo methods for a Eu doped BeF2 glass, g - , E u - F pairs; g+, Eu-Be pairs; n, average coordination number relative to F. From ref. 15.
configurations in such a way that the more probable structures appear more often than the less probable ones. Then the structural and thermodynamic properties of the fluid may be computed by averaging over all the configurations. To simulate a glass, high-temperature configurations are quenched to temperatures so low that no diffusion occurs. The result is a model of a rigid but disordered solid. Each configuration may be thought of as a different microscopic region of a large glass sample. Fig. 8 shows the partial radial distribution of interatomic distances in a SiO 2 glass obtained by molecular dynamics methods [14]. Fig. 9 refers to the distribution of Eu ions in a BeF2 glass used in laser techniques. Partial interatomic distribution functions which are impossible to extract from diffraction methods have been obtained. Fig. 10 even shows the probability of distribution of various coordinations of F ions surrounding the active Eu ions. These results are in excellent agreement with experimental results obtained from fluorescence line narrowing (FLN) spectra. 0.5
i-
>- 0,~ 0.3
~ 02 °-0.1 2.5
3 RADIUS r ( ,~ )
35
Fig. 10. The fraction of Eu ions that have exactly n neighbours within a radius r calculated from the model of fig. 9. From ref. 16.
41
J. Zarzycki / Glass structure
6. Middle-range order As we have seen, the diffraction and spectroscopic methods permit the study of the short-range order which in a great majority of cases is found to be identical or similar to that of related crystalline phases. This local order is known up to 6-8 ,~ in most favourable cases. On the other hand the textures resulting from phase-separation phenomena are known to arise in particular cases, the finest are of the order of 30-50 and are perfectly identified by small angle X-ray scattering (SAXS) methods or by electron microscopy. Between these two limits lies the zone of the "middle-range order" [17] which are difficult to attain by present investigation methods. Various expressions such as "microheterogeneous structure", "domain structure", "local fluctuations" etc .... have been used. All these expressions show the difficulty of defining quantitatively a disordered structure in this intermediate region. Radial distribution curves are insensitive to local variations of the short-range order beyond the first neighbours. Progressive variations of order (fig. 11) would disappear in the averaging process. High resolution electron microscopy could in principle give additional two-dimensional information. In practice the calculation of the contrast of images directly from models does not lead to unambiguous results due to the superposition of atomic positions along° the direction parallel to the observation axis. Very thin specimens ( - 10 A) are required to test the results of calculations: it is not clear whether under these conditions the structure would correspond to that of the massive specimens.
C
/, /
1(0) luc 200
/
/
/
/P
II
BzO3/ 7 / /
100
50
/
/
/
/
/
25 Toc
0 2~0 4~0' 860'
~2'00' ~6'00
Fig. 11. Example of a network with a progressively increasing degree of order. Fig. 12. Limiting X-ray intensity I(0) diffused at zero angle for vitreous and liquid B203; + , vitreous GeO2; zx, vitreous SiO 2. From ref. 18.
42
J. Zarzycki / Glass structure
Fig. 13. Hypotheses on middle range order: (a) crystallites; (b) random network; (c) paracrystalline arrangement.
The limiting intensity of the small-angle scattering of X-rays I(0), extrapolated to zero angle may be used to evaluate the density fluctuation level in glasses [18]. Measurements of I(0) in molten and glassy B203 (fig. 12) show that only thermal density fluctuations are present: these are quenched in for temperatures lower than the transition temperature Tg corresponding to glass formation. In the past different hypotheses on the structure of glasses were proposed. In the "crystallite" hypothesis due to Lebedev, glass was considered as an assemblage of "microcrystallites" sufficiently small not to give rise to a diffraction spectrum (fig. 13a). On the other hand the "perfectly disordered network" of Zachariasen-Warren (fig. 13b) postulates a disordered network of glass-formers for oxide glasses and a random filling of holes by modifying cations. Between the two lies the "paracrystalline" structure proposed by Porai-Koshits (fig. 13c) which implies locally variable degree of order. The latest improved X-ray correlation functions of vitreous SiO2 [19] seem to be consistent with a random network structure of the type modelled by Bell and Dean [12]. For vitreous B203 the best fit is obtained for a mixed random network of boroxol groups and B O 3 triangles [19]. Taking into account the relative insensitivity of these methods to local variation of order a paracrystalline arrangement cannot be excluded. The present structural methods are not capable of fully discriminating between these hypotheses which crudely express our ignorance of the middlerange order actually present in glasses. Even the modelling techniques are of little help as the number of atoms on which they generally operate is insufficient to correctly simulate a domain of adequate extension.
References [1] [2] [3] [4] [5]
B.E. Warren, H. Krutter and O. Morningstar, J. Am. Ceram. Soc. 19 (1936) 202. B.E. Warren, X-ray Diffraction (Addison Wesley, New York, 1969). R.L. Mozzi and B.E. Warren, J. Appl. Cryst. 2 (1969) 164. T. Masumoto, T. Fukunaga and K. Suzuki, Bull. Am. Phys. Soc. 23 (1978) 467. P.A.V. Johnson, A.C. Wright, C.A. Yarker and R.N. Sinclair, cited by C.N.J. Wagner, J. Non-Crystalline Solids 42 (1980) 3. [6] M. Misawa, D.L. Price and K. Suzuki, J. Non-Crystalline Solids 37 (1980) 85.
J. Zarzycki / Glass structure
43
[7] J.F. Sadoc and J. Dixmier, in: The Structure of Non-Crystalline Materials, ed. P,H. Gaskell (Taylor and Francis, London, 1977) p. 85. [8] P.H. Fuoss, W.K. Warburton and A. Bienenstock, J. Non-Crystalline Solids 35-36 (1980) 1233. [9] P.J. Bray and J.G. O'Keefe, Phys. Chem. Glasses 4 (1963) 37. [10] P.J. Bray, S.A. Feller, G.E. Jellinson ans Y.H. Yun, J. Non-Crystalline Solids 38-39 (1980) 93. [11] D.E. Sayers, F.W. Lytle and E.A. Stern, J. Non-Crystalline Solids 8 (1972) 401. [12] R.F. Bell and P. Dean, Phil. Mag. 25 (1972) 1381. [13] J.D. Bernal, Proc. Roy. Soc. 280A, (1964) 299. [14] C.A. Angell, P.A. Cheseman, J.H.R. Clarke and L.V. Woodcock, in ref. 7, p. 194. [15] S. Brawer and M.J. Weber, J. Non-Crystalline Solids, 38-39 (1980) 9. [16] S. Brawer and M.J. Weber, Energy and Technology Review (Lawrence Livermore Laboratory, April 1980) p. 9. [17] J. Zarzycki, Proc. X Int. Congr. Glass, Kyoto, Japan, No. 12 (1974) p. 28. [18] J. Zarzycki, Les Verres et l'Etat Vitreux (Masson, Paris, 1982) p. 172. [19] R.N. Sinclair, J.A. Desa, G. Etherington and A.C. Wright, J. Non-Crystalline Solids 42 (1980) 107.
31
GLASS STRUCTURE J. Z A R Z Y C K I Laboratory of Materials Science and CNRS Glass Laboratory, University of Montpellier, France A review of recent advances in the determination of the structure of glasses is presented. Statistical information on the short range order is usually obtained by combining X-ray and neutron diffraction with spectroscopic methods. Diffraction methods have benefited from new powerful sources which have extended the Q-range available and have resulted in a better resolution as well as a (limited) possibility of extracting partial pair-distribution functions. Among the spectroscopic methods, EXAFS and FLN are particularly promising. Modelling is shown to be indispensable for the detailed interpretation of the experimental results. In spite of progress made, the present methods are, however, still insufficient to determine the "middle-range" order actually present in glasses.
1. Introduction Glasses m a y be defined as supercooled, frozen-in liquids which have progressively a t t a i n e d the characteristics of a solid without crystallizing. T h e y are essentially non-crystalline m a t e r i a l s which possess structural features a n a l o g o u s to those of liquids, minus the s h o r t - t i m e mobility. Because of their technological i m p o r t a n c e the field of inorganic n o n - c r y s t a line solids has long been d o m i n a t e d b y oxide glasses a n d m o r e specifically, silicate glasses. Interest in vitreous chalcogenides was s t i m u l a t e d b y their infrared transmission a n d s e m i - c o n d u c t i n g properties. The latest n e w c o m e r s in this field are metal glasses which reveal special mechanical, m a g n e t i c a n d chemical p r o p e r ties. Q u e n c h i n g of a supercooled liquid constitutes one way of o b t a i n i n g noncrystalline solids; one should n o t forget, however, that there are o t h e r ways (e.g. v a p o u r deposition, chemical precipitation, r a d i a t i o n d a m a g e of crystals etc.) which can also lead to materials i n d i s t i n g u i s h a b l e from classical glasses o b t a i n e d b y quenching.
2. Description of the structure All these m a t e r i a l s have one feature in c o m m o n : they are solids with a disordered structure. By structure we refer here to a precise d e s c r i p t i o n of the s u b s t a n c e in terms of a t o m i c positions, b o n d lengths a n d b o n d angles. 0022-3093/82/0000-0000/$02.75
© 1982 N o r t h - H o l l a n d
32
J. Zarzycki / Glass structure
In the case of a crystal the arrangement of atoms is periodic in three directions of space. The detailed description of such a structure is complete once the dimensions and content of the unit cell are specified. The disposition of all the atoms is then determined by translation of this cell along the three directions of space. Crystals are said to possess both a short-and long-range order and the radiocrystallographic methods which have been developed are based on the properties of point groups and translational groups which characterise a given structure. The case of disordered materials such as glasses (and liquids) is more complex. Only short-range order is present and the unit cell can no longer be defined. The absence of the long-range order imposes other means of description; (a) The coordinates and the nature of all the atoms have to be specified the material being considered as a giant molecule - a detailed model of structure is required. (b) The structure is described statistically by means of distribution functions expressing the probability of finding an atom at a certain distance from another atom. For a material containing only one type of atom a function p(r) or radial density may be defined; it represents the average number of atoms per unit volume situated at a distance r from any atom of the structure taken as the origin (fig. la). This function expresses the short-range order. When r ~ 0¢, p(r) tends to P0, the average density of the material, p(r)/po = g(r) is then the pair-distribution function and G( r )= g( r ) - 1 is the pair correlation function (fig. lb). The function 4~rr2p(r) is the radial distribution function (fig. lc). It shows maxima which correspond to the interatomic distances most frequently found in the structure, while the areas subtended by the successive peaks are related to average coordination numbers associated with these distances. When in the disordered structure several different atomic species i, j, are present it is necessary to consider radial distibution functions corresponding to different atomic pairs i.e. to define partial pair-distribution functions g(j(r).
p(r)
pl
a '
r
D
G(r)
0
~...---~_
C
b __ r
14
rI r 2
r
Fig. 1. Radial description of a disordered structure: (a) radial density; (b) pair-correlation function; (c) radial distribution function.
J. Zarzycki / Glass structure
33
For a structure formed by two atomic species A and B, 3 pair distribution functions relative to pairs A - A , A - B and B - B have to be considered. For three atomic species 6 functions are required. The number of partial functions increases rapidly; for n species it is: n(n + 1)/2. This sets a practical limit to the determination of polyatomic disordered structures because as is well known, from a scattering experiment only a weighted sum of the pair-distribution function may be obtained, which brings about the superposition of various peaks and reduces the resolution.
3. Diffraction studies
Scattering of radiation is the usual experimental way of obtaining the pair-distribution functions. Generally speaking a scattering event is accompanied by an energy change h~0 and a momentum change hQ, where Q is the scattering vector Q = K - K 0 , the difference of the wave vectors of the scattered and the incident wave. IQI = Q = (41r sin O)/X for a scattering angle 20 and wavelength X. Elastic scattering (when he0 = 0) is related to the spatial distribution Gij(r ) of the scattering centres. The coherent scattering intensity I(Q) is then:
I(O) = E Ecicjf G,j(r) exp(iQ.r)dr, i j
where c~ and cj are scattering lengths of atomic species i and j. It can be showen in particular that for an isotropic medium: icOh(Q)
= E c2i + ~ ciCJfo°°4~rr2gij(r ),----~dr.SQr in i
i~j
Methods involving Fourier transformation have been developed by means of which a weighted sum of gij(r) or a total radial distribution function may be obtained from the experimentally measured F°h(Q). The representation thus obtained is a one-dimensional function of a single radial parameter r. This sets up a serious limitation as a three-dimensional disordered structure is thus insufficiently defined. For a complete description a correlation function of a higher order would be required, (taking into account the simultaneous positions of three atoms and more). Unfortunately these functions cannot be obtained from present experimental methods. In the case of glasses, scattering of X-rays and neutrons, and very occasionally of electrons, has been used. The initial methods developed by Warren et al. [1] have recently been improved [2] thanks to the rigorous formalism developed. The main efforts in the last years have been mainly directed towards two goals: (1) Improvement of the resolution of the method by extending the range of Q measured experimentally.
J. Zarzycki / Glass structure
34
(2) Determination of the partial distributions g i / o f the structures. The classical experiments with X-rays and neutrons used monochromatic sources and Q = (4~r sin 0 ) / ~ was varied by modifying the scattering angle 20. This conventional diffractometer technique limited the range of Q to less than 20 ,~- i as sources with ~ < 0.5 ,~ were not available. With the advent of powerful synchrotron X-ray sources and pulsed neutron sources of high intensity as well as suitable solid state detectors it has become possible to use continuous radiation with wavelengths as low as 0.2 ,~ for the structural determination of glasses. Measurements can then be made at a fixed scattering angle as a function of X. This permits, in principle, the extension of the Q range to 60 ,~-1, but in most cases measurements of intensity modulations will be difficult to observe beyond 30 to 40 ,~-1. Even so the doubling of the Q range considerably improves the resolution of the radial distribution functions. Fig. 2 shows the improved determinations of S i O 2 structure, [3] the partial pair-distribution functions were calculated assuming different models in order
2~
// t
/
s
//
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T
t o~
i
2
3
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~
5
Fig. 2. Pair distribution functions for vitreous silica. The partial functions have been separated below. After ref. 3.
J. Zarzycki / Glass structure
35
,,6
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36
J. Zarzycki / Glass structure
to obtain the best fit for the total distribution function. It would be desirable to determine these partial functions directly without any hypothesis. In principle, for a biatomic structure requiring 3 partial gu, it is possible to determine them separately if 3 independent total scattering functions I(Q) are available. These can be obtained using: (1) Three different radiations (X-rays, neutrons, electrons); (2) Isotopic substitution (for neutrons); (3) Anomalous dispersion (for X-rays); (4) Polarized neutrons (for magnetic materials); (5) Isomorphous substitution. The last method is always uncertain as there is no guarantee as to how the structure is changed by the substitution. The first method requires different samples which may have dissimilar structures; There is no known example for oxide glasses. An attempt has been made on PdsoSi2o glass [4]. Fig. 3 shows the use of isotopic substitution for K20, TiO 2, 2SiO 2 glass where the isotopes 46Ti (c i = 0.48) and 48Ti (c, = -0.58) have been used [5]. This change of contrast enables the positions of Ti ions to be situated on the difference curve. Fig. 4 shows the improved results obtained for LiO2, 2SiO2 and Na20, 2SiO 2 glasses [6]. L i - O and N a - O distances could be resolved. Fig. 5 shows the partial distribution curves obtained for the Co31P19 disordered alloy using polarized neutrons [7]. The variation of the scattering amplitude of X-rays due to anomalous dispersion close to the absorption edge of an element in the structure has been
8
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r L~
i t
I
n
,P,
f,
li/" s i
!
,
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Z
Fig. 5. Partial distribution functions for amorphous Co81P19 alloy obtained using polarized neutrons. Full curves, Co-Co; dotted curve, Co-P; dashed curve P-P. From ref. 7.
J. Zarzycki / Glass structure
37
used in the case of amorphous GeSe [8]; the precision of the results is, however, insufficient. It can be seen that the extraction of partial distribution functions has only been attempted on very limited occasions.
4. Spectroscopic methods The alternative way of obtaining information on the structure of glasses is to use spectroscopic methods which analyse the various individual atomic sites. Among the different methods used, infrared and Raman spectroscopy, Mrssbauer spectroscopy, ligand-field-theory applied to the spectroscopy in the visible, EPR and N M R are among those used most frequently. They have permitted the determination of the symmetry of some particular atomic sites or the number of neighbours (coordination number) surrounding a given atomic species. The classic example is that of the determination of the fraction of borons in fourfold coordination in B203-X20 glasses (X = Li, Na, K, Cs) fig. 6, [9]. This was possible using the difference in N M R spectra for ' 1B atoms respectively in triangular and tetrahedral coordination [9]. Recently the use of the ~°B isotope, which gives a spectrum richer in detail, has enabled a whole range of structures in borate glasses [10] to be determined. Generally the presence of a whole range of sites broadens the spectra, compared to those of corresponding crystals. The spectra in glass results from a superposition of those related to individual sites. This is particularly clear in the case of the fluorescence spectra of rare earth ions where this gives rise to inhomogeneously broadened fluorescence. The
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J. Zarzycki / Glass structure
38
1.0 _(o) ;~
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Fig. 7. Radial correlation functions for crystalline or a m o r p h o u s GeSe: (a) relative to a Ge atom; (b) relative to a Se atom. F r o m ref. 11.
excitation of spectra by laser leads to resonant line-narrowed fluorescence. Using a tunable laser it is possible to explore the various sites in a disordered structure which give different responses. Laser-induced fluorescence line narrowing (FLN) provides a unique microscopic probe of the local fields at an ion site and has proved to be a powerful tool for investigating the local structure surrounding optically active ions in glass. Another particularly fruitful method proved to be the study of Extended X-ray Absorption Fine Structure (EXAFS) which consists of recording the fine structure of an X-ray absorption spectrum of an absorbing element. The variations of the absorption coefficient are measured on the high-energy side close to the absorption edge. The oscillations observed are due to the diffraction of the ejected photoelectron by the atoms surrounding the absorbing atom. A Fourier inversion analysis then leads to a radial distribution function relative to a given atomic pair. The information is limited to the first neighbours but the advantage of the
J. Zarzycki / Glass structure
39
method lies in the fact that the local structures in the vicinity of various elements may be studied separately, without fear of superpositions which are inevitable in X-ray diffraction. EXAFS thus constitutes a method intermediate between diffraction and spectroscopy. Fig. 7 refers to amorphous GeSe and shows partial distributions around a Se or a Ge atom.
5. Modelling The interpretation of the results of either diffraction or spectroscopic methods inevitably requires the adoption of a structural model to compare the calculated results with those given by experiment. Models were first constructed by manual methods e.g. the ball-and-spike model of the SiO 2 network [12] or automatically, as the random packing of hard spheres [13]. Today the use of high speed computers permits the refinement of mechanically built models by structural relaxation methods. What is even more important, Monte-Carlo and molecular dynamics methods enable the direct construction of disordered structures from atomic systems subjected to given constraints: interatomic potentials, density and temperature, for example. The computer randomly generates millions of allowable
22
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Fig. 8. Partial radial distribution functions obtained by molecular dynamics modelling of vitreous SiO 2 at 300 K. Dashed lines represent results deduced from X-rays. N(r) is the average number of ions around Si atoms. From ref. 14.
40
J. Zarzycki / Glass structure 0.25
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12
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Fig. 9. Partial pair distribution functions calculated by Monte-Carlo methods for a Eu doped BeF2 glass, g - , E u - F pairs; g+, Eu-Be pairs; n, average coordination number relative to F. From ref. 15.
configurations in such a way that the more probable structures appear more often than the less probable ones. Then the structural and thermodynamic properties of the fluid may be computed by averaging over all the configurations. To simulate a glass, high-temperature configurations are quenched to temperatures so low that no diffusion occurs. The result is a model of a rigid but disordered solid. Each configuration may be thought of as a different microscopic region of a large glass sample. Fig. 8 shows the partial radial distribution of interatomic distances in a SiO 2 glass obtained by molecular dynamics methods [14]. Fig. 9 refers to the distribution of Eu ions in a BeF2 glass used in laser techniques. Partial interatomic distribution functions which are impossible to extract from diffraction methods have been obtained. Fig. 10 even shows the probability of distribution of various coordinations of F ions surrounding the active Eu ions. These results are in excellent agreement with experimental results obtained from fluorescence line narrowing (FLN) spectra. 0.5
i-
>- 0,~ 0.3
~ 02 °-0.1 2.5
3 RADIUS r ( ,~ )
35
Fig. 10. The fraction of Eu ions that have exactly n neighbours within a radius r calculated from the model of fig. 9. From ref. 16.
41
J. Zarzycki / Glass structure
6. Middle-range order As we have seen, the diffraction and spectroscopic methods permit the study of the short-range order which in a great majority of cases is found to be identical or similar to that of related crystalline phases. This local order is known up to 6-8 ,~ in most favourable cases. On the other hand the textures resulting from phase-separation phenomena are known to arise in particular cases, the finest are of the order of 30-50 and are perfectly identified by small angle X-ray scattering (SAXS) methods or by electron microscopy. Between these two limits lies the zone of the "middle-range order" [17] which are difficult to attain by present investigation methods. Various expressions such as "microheterogeneous structure", "domain structure", "local fluctuations" etc .... have been used. All these expressions show the difficulty of defining quantitatively a disordered structure in this intermediate region. Radial distribution curves are insensitive to local variations of the short-range order beyond the first neighbours. Progressive variations of order (fig. 11) would disappear in the averaging process. High resolution electron microscopy could in principle give additional two-dimensional information. In practice the calculation of the contrast of images directly from models does not lead to unambiguous results due to the superposition of atomic positions along° the direction parallel to the observation axis. Very thin specimens ( - 10 A) are required to test the results of calculations: it is not clear whether under these conditions the structure would correspond to that of the massive specimens.
C
/, /
1(0) luc 200
/
/
/
/P
II
BzO3/ 7 / /
100
50
/
/
/
/
/
25 Toc
0 2~0 4~0' 860'
~2'00' ~6'00
Fig. 11. Example of a network with a progressively increasing degree of order. Fig. 12. Limiting X-ray intensity I(0) diffused at zero angle for vitreous and liquid B203; + , vitreous GeO2; zx, vitreous SiO 2. From ref. 18.
42
J. Zarzycki / Glass structure
Fig. 13. Hypotheses on middle range order: (a) crystallites; (b) random network; (c) paracrystalline arrangement.
The limiting intensity of the small-angle scattering of X-rays I(0), extrapolated to zero angle may be used to evaluate the density fluctuation level in glasses [18]. Measurements of I(0) in molten and glassy B203 (fig. 12) show that only thermal density fluctuations are present: these are quenched in for temperatures lower than the transition temperature Tg corresponding to glass formation. In the past different hypotheses on the structure of glasses were proposed. In the "crystallite" hypothesis due to Lebedev, glass was considered as an assemblage of "microcrystallites" sufficiently small not to give rise to a diffraction spectrum (fig. 13a). On the other hand the "perfectly disordered network" of Zachariasen-Warren (fig. 13b) postulates a disordered network of glass-formers for oxide glasses and a random filling of holes by modifying cations. Between the two lies the "paracrystalline" structure proposed by Porai-Koshits (fig. 13c) which implies locally variable degree of order. The latest improved X-ray correlation functions of vitreous SiO2 [19] seem to be consistent with a random network structure of the type modelled by Bell and Dean [12]. For vitreous B203 the best fit is obtained for a mixed random network of boroxol groups and B O 3 triangles [19]. Taking into account the relative insensitivity of these methods to local variation of order a paracrystalline arrangement cannot be excluded. The present structural methods are not capable of fully discriminating between these hypotheses which crudely express our ignorance of the middlerange order actually present in glasses. Even the modelling techniques are of little help as the number of atoms on which they generally operate is insufficient to correctly simulate a domain of adequate extension.
References [1] [2] [3] [4] [5]
B.E. Warren, H. Krutter and O. Morningstar, J. Am. Ceram. Soc. 19 (1936) 202. B.E. Warren, X-ray Diffraction (Addison Wesley, New York, 1969). R.L. Mozzi and B.E. Warren, J. Appl. Cryst. 2 (1969) 164. T. Masumoto, T. Fukunaga and K. Suzuki, Bull. Am. Phys. Soc. 23 (1978) 467. P.A.V. Johnson, A.C. Wright, C.A. Yarker and R.N. Sinclair, cited by C.N.J. Wagner, J. Non-Crystalline Solids 42 (1980) 3. [6] M. Misawa, D.L. Price and K. Suzuki, J. Non-Crystalline Solids 37 (1980) 85.
J. Zarzycki / Glass structure
43
[7] J.F. Sadoc and J. Dixmier, in: The Structure of Non-Crystalline Materials, ed. P,H. Gaskell (Taylor and Francis, London, 1977) p. 85. [8] P.H. Fuoss, W.K. Warburton and A. Bienenstock, J. Non-Crystalline Solids 35-36 (1980) 1233. [9] P.J. Bray and J.G. O'Keefe, Phys. Chem. Glasses 4 (1963) 37. [10] P.J. Bray, S.A. Feller, G.E. Jellinson ans Y.H. Yun, J. Non-Crystalline Solids 38-39 (1980) 93. [11] D.E. Sayers, F.W. Lytle and E.A. Stern, J. Non-Crystalline Solids 8 (1972) 401. [12] R.F. Bell and P. Dean, Phil. Mag. 25 (1972) 1381. [13] J.D. Bernal, Proc. Roy. Soc. 280A, (1964) 299. [14] C.A. Angell, P.A. Cheseman, J.H.R. Clarke and L.V. Woodcock, in ref. 7, p. 194. [15] S. Brawer and M.J. Weber, J. Non-Crystalline Solids, 38-39 (1980) 9. [16] S. Brawer and M.J. Weber, Energy and Technology Review (Lawrence Livermore Laboratory, April 1980) p. 9. [17] J. Zarzycki, Proc. X Int. Congr. Glass, Kyoto, Japan, No. 12 (1974) p. 28. [18] J. Zarzycki, Les Verres et l'Etat Vitreux (Masson, Paris, 1982) p. 172. [19] R.N. Sinclair, J.A. Desa, G. Etherington and A.C. Wright, J. Non-Crystalline Solids 42 (1980) 107.