International Symposium CRYSTALLIZATION IN GLASSES AND LIQUIDS Liechtemstein, 2000
The Role of Network Rigidity on Crystallization Behaviour of Glasses Isak Avramov(1), Ralf Keding (2) , Christian Rüssel(2) (1) Institute of Physical Chemistry, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria E-mail:
[email protected] (2) Otto-Schott-Institut, Universität Jena, Fraunhoferstrasse 6, D-07743, Germany ABSTRACT It is well known that oxide glasses build up continuous networks. Therefore the rigidity of the latter plays a key role in understanding glass properties. The network switches from floppy to rigid if the number of covalent bridges between the network formers exceeds a certain threshold value. One and the same structure could behave as rigid network in respect to some properties and could be floppy networks in respect to some other properties. Kinetics of phase transition (and most the nucleation processes) is very sensitive to the rigidity of the network. Some tiny floppy regions still exist inside the rigid network close above the threshold concentration. They can serve as active centers for the beginning of nucleation process. Much later, nucleation can appear in the rigid part of the network. Therefore, we predict a certain interval of concentration of rigid bounds in which a bimodal size distribution function of the formed phase will be observed. This prediction is confirmed by experimental data we find in literature.
INTRODUCTION One of approaches to treat glass formation is based on strength considerations. The stronger the bonds in the melt are, the more sluggish the rearrangement process will be. Oxide glasses are well described by the continuous-random-network model of Zachariasen [1]. The aim of the present article is to link this idea to the rigidity percolation theory. The latter shows [2-6] that networks change from floppy to rigid at a critical point where the number of constraints exceeds the number of degrees of freedom. CONSTRAINTS COUNTING MODEL Two sources of constraints are involved. The first one is generated by the fixed length of the bridges connecting nodes of the network (bridge stretching constraints BC). There is one BC constraint associated with each bridge. Therefore, the average number nl of BC per node is nl =
, 2
(1)
2 where the number of bounds is determined by the mean coordination . The fixed angle between bridges of one node give rise to the second sort of constraints (bridge bending constraints BB). For r coordinated node there are nb = 2r-3
(2)
BB constraints [5]. The overall constraints number is the sum n=n l + nb. If there are sr network formers NF with coordination r, the mean coordination is determined as: 4
< r >=
∑r s r =2 4
r
(3)
∑ sr r= 2
In real systems, network modifiers NF change the nature of some bounds in a way the latter impose no (or less) constraints. If there are sq NM , each of them modifying q connections the average number of constraining bounds becomes: < m >=< r >−
∑ qs ∑s
q
(4)
r
The properties of network is a rigidity percolation problem as we already discussed in Ref. [7]. The average concentration of rigid links is p = becoming rigid for p > p cr ≡
. Thus the network is
< mcr > .
The degrees of freedom per node is equal to the dimension d of the space. Every constraint disables one degree of freedom. Therefore, the fraction f of enabled degrees of freedom is given [5] as: f = d + 3 − 2.5 < m >
(5)
One can easily formulate a critical coordination of constraints at which the number of constraints is becoming equal the number of degree of freedom (f=0). For a three dimensional network (see [5] ) it becomes =2.4, (resp. pcr=0.6 for =4). The mean coordination can play an important role for the glass properties. In Fig.1 we illustrate this by presenting the dependence of glass transition temperature Tg on according to the data [8,9] for GexTe1-x and GexS1-x.
3 Boolchand at all [8,9] reasoned that glass forming tendency is maximal just at critical point. Crystallization is easier when networks are highly overconstrained ( >3) or highly underconstrained ( < 1.5). Up to this point, we derive rcr under the assumption that both BB and BC constraints are taking place In some cases the BC or BB constraints could be considered as broken. This will happen when the underlying bonding interaction is weaker than the testing force. If the investigated macroscopic property reflects vibrations of atoms, or when it requires only local rearrangement of the building units, the testing energy is about kBT. On the other hand, long distance motions (viscous flow, self diffusion etc.) require much higher activation energy. It is seen that different number of constraints will be observed for one and the same system depending on the investigated property. Therefore, two additional, extreme, cases should be considered briefly: i- No length constraints existing: Instead of Eq.(1) one has nl=0. In this case the critical point is at rcr(l) =
d +3 , i.e. for a three dimensional network the 2
critical number of angular constraints is 3 per node provided the length of the bonds is not fixed. ii- No bending constraints existing: Instead of Eq.(2) one has nb=0. This leads to a critical point at rcr( b ) = 2d . For a three dimensional network, the critical number of bonds is 6 provided the angle between them is not fixed. Fig.1 is obtained under the assumption that S and Te are two-fold coordinated. A sharp kink appears at a critical point of rcr≈ 2.7. One of the simplest explanations, why the observed value of rcr is about 12% above the expected one, is that some of the constraints are broken. Covalent bounds implies strong restrictions on both length and angle between bounds. On the other hand, ionic bonds impose much weaker constraints for the length and no constraints for theangle. Since Pauling [11] it is well known that there are no pure covalent or pure ionic bonds. Therefore, the declination of rcr from the predicted value can be
4
x 0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
800
800
750
Ge x S 1 - x
700
700
Tg [K]
650
600
600
550
500
500
Ge x T e 1 - x
450 400
400
350 2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
Figure 1:
Dependence of glass transition temperature Tg on mean coordination . - GexS1-x ; g - GexTe1-x
explained with the expected 90% covalent nature of bonds.
Structure of SiO2 Glass Zachariasen [1] and Warren & Biscoe [12] assume that structure of two- and multi- component oxide glasses and in particular silicates is a continuos random network. Finney and Bernal [13] describe the glass structure by means of Voronoi polyhedra. Gupta and Cooper [14] used distorted polytypes to describe the structure of glassforming melts. Following Mott [15], the coordination number of covalently bonded atoms is given by the 8-N rule, N being the number of outer shell electrons. The pure SiO2 glass has a structure [12,13] quite similar to that of cristobalite. It consists of SiO4 tetrahedra with shearing corners. The latter are absolutely rigid because the four bonds per Si atoms, predominantly covalent, impose seven constraints. Therefore the structure can be described as consisting of solid SiO4 tetrahedra. Each tetrahedra is four-fold coordinated; i.e. nl=2 (see Eq.1). A two-dimensional illustration of this
5 structure is presented on Fig.2. The oxygen atoms play role of ball and socket joints of this structure.
2+
Figure 2:
Structure of a silicate glass in a two-dimensional representation. g - Si; -O; -Me2+ion; - Me+ ion . Triangles stand for SiO4 tetrahedra.
The problem is whether to take into account bending constraints nb. The mechanical analog of the problem can be formulated as a question: “is there lubricant into socket”. Spectra analysis [16,17] show that in silicates the angle of the Si-O-Si bridge varies in a wide range around α=149o. The activation energy for rotation of the Si-O-Si bridge is relatively low (1300 cal/mol , i.e. it is about kBT). On the other hand, analysis of X-ray data [17] indicate that the angle of the Si-O-Si bridge changes with ±10o at this condition. Larger declination angles are conditioned by topological reasons require much higher energy. In other words one can assume that there are constraints for large angle deviations and there are no constraints for small ones. Conclusions: i- For small perturbations (sound propagation, IR spectra, diffusion of small ions like H, etc.) the network of pure SiO2 is floppy (n=n l=2 > 3).
6 Structure of silicate glasses Silicate glasses have structure similar to pure SiO2, only some of the Si-O-Si bridges are broken or modified. There could be two kinds of species in the system. First, there are Network formers (NF). They occupy the nodes of the network (e.g. in oxide systems these could be Si, Ge, Ti, As, etc.). Only NF belong to the network. Additionally there can exist some Network modifiers (NM) like Ba, Na, K etc. Oxygen makes bridges between NF. Some of them are broken or modified by NM as shown in Fig.2. We proposed earlier in ref. [7] to determine as the number of rigid links per NF. This is the doubled number of bridging oxygen per NF (because every oxygen links two NF). The number of bridging oxygen is determined by the total number of oxygen (in the chemical formula) minus the number of oxygen related to each NM. (this is the number of NM times the charge of each of them.). For example: in composition Na26Ca11Si76O176 is determined as: < n >= p=
2(176 − 26 − 2 * 11) = 3.37 , 76
3.37 = 0.84 4 . Rigidity and nucleation kinetics
In this section we give experimental evidence of the influence of network rigidity on nucleation and crystallization kinetics of glasses. We focused on nucleation process because it requires motion to interstitial distances. Data are from ref. [7,18]. Samples with varying were cooled from above the melting point at a rate of -10 K/min [7]. The crystalline phase formed was stoichiometric fresnoite Ba2TiSi2O8. The composition of the ambient glass varies about that of pure fresnoite 3+ by adding SiO2 (to increase the amount of NF) or by changing the state of Ti to Ti . (while four fold coordinated Ti is NF , the three fold coordinated is a NM). The undercooling temperature ∆T is the difference of the melting temperature and the maximum of the crystallization peak. The dependence of ∆T for on is shown in Fig.3. Although the network is becoming rigid when p exceeds pcr some tiny floppy regions still exists inside. One can assume that the nucleation starts at some points
7
500
∆T
[K]
400
300
200
100
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Figure 3:
Dependence [7] of critical undercooling ∆ T on mean number of constraining bounds.
where molecules move faster (in terms of Thorpe's ideas [5] these are regions controlled by ''floppy'' modes). In a sense this picture resembles very much a percolation problem. If the size R of the floppy region is smaller than the size r of the critical nucleus, the crystallization cannot start. When the system is completely floppy the critical supercooling ∆T for the crystallization to start is controlled by the nucleation kinetics. Otherwise the size R of the floppy region impose some limitations on the process. The crystallization will start when the critical undercooling ∆T is so high that the size of the critical cluster r(∆ ∆T) is becoming equal to R. The radius R and the concentration of the floppy regions diminish very rapidly with p [7].
R~
1
[ p − p cr ]ϑ
(6)
where for three dimensional systems the power is ϑ =0.85. The floppy regions play the role of active centers in the nucleation process. In this respect, they are of importance in the range (2.4 ≤ ≤ 3.1) [18]. If such a system is annealed then nucleation will take place in the active centers (floppy regions). Upon further annealing, nucleation in the
8 rigid part may also take place [18]. Therefore, a bimodal size distribution function of crystals formed should be observed. If the annealing temperature is lowered, the crystallization in the rigid network is impeded very much. Because crystallization is connected with a rearrangement of molecules, it leads to the formation of elastic stress which cannot relax that easy, especially in the rigid part at low temperatures. For this reason, the crystals in the rigid part might be too tiny to be observed (if they appear at all) with the respective method used. Hence, a unimodal distribution function is expected. By analogy, an unimodal distribution function is also expected if p3.1 the floppy regions are too tiny and diluted to play an important role, therefore again an unimodal distribution is expected. CONCLUSIONS Oxide glasses (silicates, borates, germanates etc.) have a continuos random network structure. The building units are rigid tetrahedra (SiO4) or rigid triangles (BO3). Flexibility of the oxygen bridges plays a key role determining whether the network is rigid or floppy. References 1. W. Zachariasen, J. Am. Chem. Soc 54, 3841 (1932). 2. J. Phillips, M. Thorpe, Solid State Comm. 53 84 (1986) 3. J. Phillips, J. Non Cryst. Sol. 73 153 (1979) 4. M. Thorpe, J. Non-Cryst. Sol. 57, 355 (1983). 5. M. Thorpe, Phys. Rev. B 40, 535 (1989) 6. Thorpe M., J. Non-Cryst. Sol. 182, 135 (1995) 7. I. Avramov, R. Keding, C. Rüssel, J. Non-Cryst. Sol., in press 8. J. Malek, J. Non-Cryst. Sol., 107, 323 (1989) 9. M. Mitkova, P. Boolchand, J. Non-Cryst. Sol. 240, 1, (1998) 10. P. Boolchand,, X. Feng, D. Selvanathan, W. Bresser in “Rigidity Theory and Applications” Ed. M. Thorpe, P. Duxbury, Kluwer Acad./Plen. Publ. 1999 p.279 11. L. Pauling “General Chemistry 12. B. Warren, J. Biscoe, J. Amer. Ceram. Soc. 21, 259 (1938) 13. G. Finney, J. Bernal, Nature 213 1079 (1967) 14. P. Gupta, A. Cooper, „Topologically Disordered Networks of Rigid Polytopes“ in Proc. XVI Intern. Congress on Glass, Madrid 1992, Bol. Sociedad Espagniola de Ceram. Vidio 31c (1992) 15 15. N. Mott, Philos. Mag. 19 835 (1969) 16. P. Flubacher, A. Leandbetter, J. Morrison, B. Stoicheff, J. Phys. Chem. Solids, 12, 53 (1959) 17. A. Lazarev “ Vibration Spectra and Structure of Silikates” (in russian) Nauka, Leningrad
9 18. I. Avramov, R. Keding , C. Rüssel, R. Kranold, to be published 19. J. Ziman in: Models of Disorder (Cambridge Univ. Press, London 1979) p.375
