© Copyright 2001 by the American Chemical Society
VOLUME 105, NUMBER 38, SEPTEMBER 27, 2001
LETTERS A Chemical Approach To Understand Fragilities of Glass-Forming Liquids K. J. Rao,*,†,‡ Sundeep Kumar,†,‡ and M. H. Bhat‡ Groupe Ionique du Solide, Ecole Nationale Superieure de Chimie et de Physique de Bordeaux, UniVersity of Bordeaux I, AVenue Pey Berland, BP 108 33402 TALENCE Cedex, France, and Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore, 560012, India ReceiVed: May 10, 2001; In Final Form: July 13, 2001
Fragility is viewed as a measure of the loss of rigidity of a glass structure above its glass transition temperature. It is attributed to the weakness of directional bonding and to the presence of a high density of low-energy configurational states. An a priori fragility function of electronegativities and bond distances is proposed which quite remarkably reproduces the entire range of reported fragilities and demonstrates that the fragility of a melt is indeed encrypted in the chemistry of the parent material. It has also been shown that the use of fragility-modified activation barriers in the Arrhenius function account for the whole gamut of viscosity behavior of liquids. It is shown that fragility can be a universal scaling parameter to collapse all viscosity curves on to a master plot.
The concept of fragility introduced by Angell1 in the context of glass-forming melts has brought about a major change of perspective in liquid-state research. It is based on an analysis of plots of logarithmic viscosity (ln(η)) as a function of the inverse (scaled) temperature (Tg/T). Liquids exhibiting linear Arrhenius behavior, are described as “strong” and those which depart from linearity (with Vogel-Tamman-Fulcher (VTF) behavior), as “fragile”. There is a steep drop in viscosity over a narrow range of temperature in “very fragile” liquids just above Tg, which causes a rapid loss of structural rigidity. Hence the use of the adjective “fragile”, which echoes the sense of the more familiar mechanical fragility. Several attempts have been made to quantify fragility in the literature using experimentally observed or extrapolated parameters.2-8 More recently the F1/2 fragility has been defined and related to ∆Tg/Tg by Ito, Moynihan, and Angell (IMA),9 where Tg and ∆Tg are the glass transition temperature and the * Author to whom correspondence should be addressed. E-mail: kjrao@ sscu.iisc.ernet.in. † University of Bordeaux I. ‡ Indian Institute of Science.
temperature width of the glass transition respectively.9 The ∆Tg/ Tg relation of IMA can be inverted to an F1/2 function, which provides a method of determining F1/2 fragilities from experimental ∆Tg/Tg values. These fragilities are found to be in good agreement with fragilities from viscosity data. While it is informative to cross correlate F1/2 values with other similar experimentally measured properties, it is desirable to seek an understanding of this seemingly fundamental property of liquids in terms of the physicochemical parameters, which determine the structure and bonding in the condensed state of matter. Fragilities represent a measure of the susceptance of molecular architecture to thermally induced breakdown. This breakdown causes the evolution of liquidlike configurational rearrangements of the constituent entities. Such rearrangements become facile when the cohesive interaction among constituent entities is nondirectional. The cohesive energy of a fragile liquid, therefore, is required to have a significant nondirectional component in it. An implication of this nature, albeit indirect, can be recognized as built into the IMA expression, because high fragilities result from low ∆Tg values and the latter are associated with high ionicity (nondirectional bonding) of glasses10 as in
10.1021/jp011802z CCC: $20.00 © 2001 American Chemical Society Published on Web 08/25/2001
9024 J. Phys. Chem. B, Vol. 105, No. 38, 2001
Letters
the case of CKN. Structural rearrangements and related molecular motions occur as a consequence of thermal excitation of bonds. Presence of a high density of low-energy excited states (high entropies associated with such excited states) accessible to thermal excitation is another factor, which contributes to facile configurational rearrangements. These states can be identified as the shallow wells in the energy landscape with similar depths as the potential well in which the glass is found (a conceptual (3N + 1)-dimensional energy hypersurface of an N particle glass-forming system which develops minima of varying depths on the surface as the system is cooled toward Tg. Glass is conceived as a state in one of the many nonglobal minima).11-13 Low energy configurational states may be expected to result from large bond distances, d, between constituent atoms/ions and high densities of such states can result when the atoms/ ions have high coordination numbers (many more ways of realizing equienergetic states). The latter is again high for large d as evident from crystallographic data. We, therefore, identify two factors as determinants of fragility. The first is the nondirectional component of cohesive energy. This is readily seen as the ionic part of the energy in polar covalent liquids but it is entirely cohesive energy in van der Waals bonded molecular liquids. The second is the distance d, the bond length of the weakest bond in the material. We can represent the ionic part of energy in polar covalent materials through the ionicity of bonding in them. Therefore, it scales as ∆χ2, the square of the difference in electronegativities of the bonded atoms. Combining the two factors, we propose a heuristic expression for fragility, which has the same attractive and simple form as the inverted IMA relation, but with different constants and contents.
F)
0.22 - x [0.22 + x]
(1)
where x ) 0.04[(10 - ∆χ2)/d]. (Compare with the IMA relation, F1/2 ) [(0.151 - x′)/(0.151 + x′)], with x′ ) ∆Tg/Tg). We now proceed to calculate F using x values and discuss its theoretical significance later. The numerical constants in eq 1 and in x have been determined on the following basis. Since the leading term in the expansion of Pauling ionicity expression14 is ∆χ2/4, and since F1/2 should tend to high values in highly ionic materials, the numerator in x is assumed to scale as 1 - ∆χ2/4. We have approximated this term to vary as (10 - ∆χ2) keeping in view the limits to ∆χ values. The constants 0.22 and 0.04 have been fixed on the basis of the fragility of SiO2, again keeping in view the lowest possible values of d. Since the cohesive energy is due to van der Waals interactions, the intermolecular bonding is nondirectional in glassforming molecular liquids such as salol and the use of ∆χ is not relevant in them. But retention of the constant 10 in x severely overestimates the total energy, it represents. Also, it leads to very low values of fragility. Therefore x has to be redefined to scale down the energy it represents, for the case of molecular liquids. VDW energy has the form 3R2I/4d6, where I is the ionization potential and R is the polarizability.15 Since ionization potential should again scale as χ and R2/d6 is only a small numerical constant (R ≈ 4πr3/3 and r ) d/2), we have fixed x as 0.08χ/d on the basis of the reported (IMA) fragility of toluene. d is now considered as an effective bond distance and is calculated as (Vm/N)1/3 where Vm is the molar volume and N, the Avogadro number.16 This will now enable use of same eq 1 to evaluate fragilities of both polar covalent and molecular liquids.
TABLE 1: Glass-Forming Materials with Their ∆χ, d, F, and F1/2 Values composition
∆χ (or χ) d (Å)
GeO2 SiO2 As2S3 As2Se3 Se PbO-Li2O-B2O3 (20:40:40) Pb2P2O7-Li4P2O7 (50:50) Ag2P2O7-Li4P2O7 (50:50) ZnCl2 Li2O-GeO2-P2O5 (40:20:40) B2O3
1.43 1.54 0.40 0.37 0 1.84 1.68 1.65 1.51 1.96 1.40
P2O5
1.25
Na2O-3SiO2 PbO-PbCl2 (50:50) BeF2 K2O-MoO3-P2O5 (20:40:40) K2O-WO3-P2O5 (20:40:40) Li2SO4-Li2O-B2O3 (20:40:40) PbO-PbF2 (50:50) SnO-NaPO3 (05:95) NaPO3 Li2SO4-Li2O-P2O5 (20:40:40) K2SO4-Na2SO4-ZnSO4 (22.5:07.5:70) KCl-2BiCl3 CKN
1.75 1.43 2.41 1.59 1.58 1.98 1.84 2.11 2.14 2.13 1.84
HDW H2O propylene glycol glycerol toluene salol butyronitrile sorbitol OTP
0.29 2.55 2.44 2.5 2.36 2.46 2.37 2.52 2.39
2.04 2.44
F
1.74 1.62 2.24 2.41 3.8 1.96 1.96 1.96 2.32 1.96 1.48 2.76 1.63 2.78 2.21 3.07 1.54 2.61 2.61 3.12 2.59 2.26 2.26 3.12 3.77
0.09 0.08 0.11 0.15 0.35 0.24 0.20 0.19 0.25 0.27 0.01 0.31 0.03 0.29 0.27 0.36 0.34 0.32 0.31 0.48 0.37 0.38 0.39 0.52 0.52
3.14 3.79 5.67 2.38 3.1 4.96 4.94 5.62 6.56 5.26 5.88 6.84
0.49 0.67 0.77 0.14 0.54 0.70 0.69 0.74 0.76 0.72 0.73 0.77
F1/2
ref.
0.10
(9)
0.18 (17) 0.31 (18) 0.25 (19) 0.25 (19) 0.33 (20) 0.39 (9) 0.30 (9) 0.43 (21) 0.40 0.34 0.52 0.43 0.34 0.36 0.48 0.50
(22) (23) (18) (24) (25) (25) (18) (26)
0.05
(9)
0.56 0.62 0.74 0.74 0.54 0.75 0.75
(9) (9) (9) (9) (9) (9) (9)
Fragilities calculated on the basis of eq 1 alongwith F1/2 fragilities from IMA relation are presented in Table 1; the data used in the computations are also given. F and F1/2 are directly compared in Figure 1. Of the 34 data points, 10 are common with those in the IMA plots9 including 7 of molecular liquids. An additional 14 have been added to the F1/2 list from available ∆Tg/Tg data.17-26 The rest are only from eq 1 and for convenience put on the diagonal dotted line. Fragilities of B2O3, P2O5, CKN, and water have been calculated using two d values each and are discussed below. Most importantly, the agreement between F and F1/2 is rather remarkable whereVer we could compare. Small inaccuracies in bond distances arising from the use of listed ionic radii, etc., do not seriously affect the calculated values of F. Therefore, the fragilities in the molten state of materials are determined by the same physicochemical factors, namely electronegativities and radii of constituent particles, which determine their structures in the solid state. ∆χ for materials such as SiO2, is simply ∆χ ) χ(O) - χ(Si). For materials such as K2SO4, ∆χ ) χ(SO4)0 - χ(K). χ(SO4)0 has been computed as the geometric mean of electronegativities.27 d is either from the reported28,29 bond length (Si-O ) 1.62 Å) or the sum of ionic radii, (in K2SO4, d ) r(K+) + r(SO42-)). In polymeric materials such as NaPO3, d is assumed as r(Na+) + r(O-), (r(O-) ) 1.28 Å rather than 1.36 Å of O2-). For discrete anions such as SO42- and NO3-, their listed effective thermochemical radii15 have been used. In complex systems such as Na2SO4-K2SO4- ZnSO4 composition weighted 〈∆χ〉 is used.
Letters
Figure 1. Comparison of F and F1/2. O are from ref 9, ∆ are calculated from reported ∆Tg/Tg values, and 0 are for which ∆Tg/Tg values are not available, ( is HDW (high-density water) ,and 1 is water above 230 K. b is B2O3, * is P2O5, + and 3 with dot at center are for CKN. The inset shows the variation of fragility as a function of mole % Ge in the Ge-Se system.
The two F values of B2O3 result from two different d values, both based on the assumption that a significant number of tetrahedrally coordinated borons are present even in pure B2O3 glass itself 29,30 due to B- r O+ bonding (with 3 bonded oxygens). B--O or B- r O+ can both be involved in bond excitation, the corresponding d values are 1.48 Å and ∼(1.48 + 1.28) Å and the corresponding fragilities are 0.01 and ∼0.33, respectively. The former makes much sense to the emerging opinion (also see the cited Kauzmann plot in ref 9) that B2O3 may be one of the strongest liquids. The latter value is, however, quite close to the value in the IMA plot.9 For the case of P2O5 also, two (dP-O and dPdO plus r(O-)) d values have been used assuming that PdO may also be involved in interlayer bonding. In CKN, two distances are considered (i) distance between K+NO3- (the weakest bond) and (ii) the distance between Ca2+ and K+ in K2[Ca(NO3)4] complex, which it forms. Both fragilities are presented in Figure 1, but we are unable to make any comparison with literature values. The computed F of BeF2 is somewhat high and the origin of this exceptional behavior is unclear to us. Water is a perennially enigmatic liquid.31 We have computed F of HDW by ignoring hydrogen bonds and considering the relevant excitations of H3O+- -OH- bonds (d ) 2rOH-) and F of ordinary water above 230 K by considering H2O as a molecular liquid. HDW appears to be indeed in the class of SiO2 in terms of fragility (F ) 0.14),9 while the low-pressure water is quite fragile (F ) 0.54) as rightly remarked by IMA. We now return to a discussion of the theoretical basis of eq 1 which seems to predict F values very well. The values of fragilities in this model is determined by x ) x(χ, d). Therefore, x is of fundamental significance to the transport behavior of the material in its molten state. In polar covalent liquids, (10 ∆χ2) in x scales as residual covalent (directional) part of interaction energy. Hence, x ) 0.04 (10 - ∆χ2)/d has the dimensions of a residual directional force. This force hinders local molecular rearrangements in the liquid. Therefore, for a liquid on its cooling course and approaching Tg smaller values
J. Phys. Chem. B, Vol. 105, No. 38, 2001 9025 of x (as in high fragility liquids) facilitate the exploration of low-energy configurational states deep into the supercooled region. As x tends toward zero, the supercooled liquid would appear to continue its journey toward the (lowest energy) ground state, which in highly ionic glasses is known to be stopped just short of Kauzmann thermodynamic limit. In the viscosityreduced temperature plots low values of x pushes the viscosity curve farthest from the Arrhenius line and the value of (Tg TK) tends toward very low values. The interpretation of 0.08χ/d in molecular liquids would be similar because of the sense in which the term has been used. It is tempting to associate the ideal F ) 1 liquid with the hard sphere fluid in which the interaction is entirely nondirectional and there is no attractive component. The formulation of fragility in terms of x has the advantage that it provides a unified approach to the rheology of all classes of liquids, ionic, covalent, molecular, or even metallic (not discussed here) and quantifies the viscosity behaviors through F. Fragilities of even complex melts such as GexSe1-x may be evaluated when the excitable bonds are thoughtfully identified. In Se, we have seen that interchain Se-Se VDW bonds are excited (∆χ ) 0 and d is twice the van der Waals radii). In the composition Ge20Se80 (tGeSe4), which is Ge(Se2)2, more likely that Se-Se bond in Ge-Se-Se-Ge links are excited (again ∆χ ) 0 and d ) Se-Se covalent bond distance). Ge33.3Se66.7 (tGeSe2) is analogous to SiO2 and the relevant bond for excitation is Ge-Se (∆χ ) 0.54 and d ) rGe-Se). In the intermediate compositions, one can use bond population weighted fragility contributions since fragilities (susceptances) can be treated as additive. In fact, the variation of fragilities as a function of composition shown as inset to Figure 1, quite remarkably reflects the fragility dip in Ge20Se80 composition. Since the liquids explore large numbers of configurations immediately above Tg in more fragile liquids, the commonly observed sudden rise in configurational heat capacity is well anticipated.6 It is interesting to note that presence of such a high density of low energy states was introduced by Angell and Rao, on a heuristic basis by associating a large ∆S with low-energy excited states in their bond-lattice model.32 A sharp rise of both CP and ∆CP values were thus achieved. Also, in liquids with high F values we may expect the presence of frozen low-energy configurational states below Tg in the resulting glasses. In such glasses, dielectric relaxation may be expected to be governed by stretched exponential behavior with low β values as indeed reported by Ngai.33 Although the most dramatic manifestation of fragilities occurs in the metastable supercooled region close to Tg in ln(η) vs Tg/T plots, they are already expressed in the equilibrium liquid at Tm (Tg/T ≈ 0.6). At temperatures not much higher than Tm (Tg/T ∼ 0.5), most liquids are Arrhenius and the reduced activation barriers (E/2.303RTg) for viscosity are in the reverse order of steepness indices or fragilities. Since the ln(η) trajectories do not generally intersect, the ordering remains unaffected in any viscosity related measurements of fragility in the region close to Tg. In view of its (experimental) inverse relation with reduced activation barriers above Tm, fragility can be considered as an equilibrium property. Therefore, our formulation of fragility using fundamental liquid constants seems justified. During supercooling, the system is driven by a thermodynamic compulsion to seek lower energy configurational states. It occurs easily in the more fragile liquids and, therefore, viscosities remain low. However, as the liquids plummet into sufficiently low energy states, they realize that further down in temperature scale the density of states has hopelessly decreased (low configurational
9026 J. Phys. Chem. B, Vol. 105, No. 38, 2001
Figure 2. Variation of log(ηr) as a function of Tg/T, for different F values. The range of log(ηr) is 17 decades. The O are experimental points of o-terphenyl.34
entropy, Sc). Accessing the sparcely populated states through cooperative rearrangements (increasing barriers) results in a viscosity increase, more as a thermodynamic consequence. On the other hand, in strong liquids the low energy states are not explored efficiently because of a high viscosity due to high barriers to rearrangement (breaking of directional covalent bonds). A large part of the entropy of the strong liquid, therefore, remains frozen mainly because of the prohibitive kinetics of structural rearrangement in the liquid. We may now examine whether the present fragilities reproduce the log(η/η0) versus Tg/T plots. In these plots the range of log(η/η0) is 17 decades. For F ) 0 (Arrhenius liquid), therefore, log(η/η0) ) log(ηr) ) 17. The reduced activation barrier, which is d log(ηr)/d(Tr-1) (Tr ) T/Tg) is equal to Eη/2.303RTg ) Er )17.0. We introduce the function log(ηr) ) Er/Tr1/(1-F) to generate the reduced viscosity plots, shown in Figure 2 for different F values. The plots are astonishingly similar to the experimental viscosity-reduced temperature plots. As an example, the behavior of the experimental viscosities of oterphenyl for which F and F1/2 values are, respectively, 0.77 and 0.75 are reproduced well by F ) 0.80 line in Figure 2. Neither the chosen functional form of log(ηr) nor the observed agreement with experimental results are fortuitous for the following reason. The function log(ηr) ) Er/Tr1/(1-F) can be written as [Er/TrF/(1-F)]/Tr ) Er(eff)/Tr, which is the Arrhenius form with a Tr-dependent activation barrier. In fact, [Er /TrF/(1-F)] is to be compared with (B/2.303ScTg) of the celebrated AdamGibbs (AG) equation.35 The Tr-dependent, effective reduced activation barriers are obtained by differentiation as [Er/(1 F)]/TrF/(1-F). The variation of these barriers for various F values are shown in Figure 3. It is clear in high F liquids Er(eff) is low to start with and increases as Tg is approached. This variation is the essence of fragile behavior. Since Er(eff) is largely independent of Tr in strong liquids, (B/TgSc) is expected to be a constant and in highly fragile liquids it increases, as we noted earlier. B itself is unlikely to be a constant (see ref 4 for comments on B ) DT0) in this approach although it represents a barrier to configurational changes. Since B/TSc () (B/TgSc). (1/Tr)) varies as (Er/TrF/(1-F))(1/Tr), constancy of B would require
Letters
Figure 3. Variation of Er(eff) ) [Er/(1 - F)]/TrF/(1-F) (Er ) Eη/ 2.303RTg) as a function of Tg/T for different F values. The dotted vertical line at Tg/T ) 0.6 indicates the equilibrium melt temperature for most liquids. The inset shows variation of the number of particles in CRR as the high-temperature liquid is cooled toward Tg for various values of F.
that Sc ∼ TrF/1-F. However, for liquids of intermediate F values, Sc ∼ ∫(∆CP/T) dT ∼ T-1 (for constant ∆CP), and this enables a direct derivation of VTF relation from AG equation. Such hyperbolic temperature dependence of S concomitantly requires B to vary as Tr-1/(1-F). Since (B/TgSc) varies as Tr-F/(1-F), it simply follows that F/(1 - F) ) (d ln(B/TgSc))/(d ln(Tr-1)) ) c. A log-log plot of (B/TgSc) vs Tr-1 can be used to obtain F as c/(1 + c). (B/TgSc) itself is evaluated from classical Arrhenius plots. For glass-forming liquids data in the range of 0.8Tm to 1.2Tm should yield F. Most obviously, F can now be used as a viscosity scaling parameter and in log(ηr) vs Tr-1/(1-F) plots all the viscosity lines collapse onto a universal viscosity line. The temperature-dependent activation energy barriers in Figure 3 can be interpreted as n‚∆µ of the AG equation, where ∆µ is the activation barrier per rearranging particle. Assuming that at temperatures sufficiently above Tg (Tg/T e 0.4), most of the configurational entropy arises from single particle (n ) 1) movement, the variation of the number of particles present in cooperatively rearranging region (CRR) have been evaluated and shown in the inset to Figure 3. It is seen that in very fragile liquids, the number of particles involved in cooperative rearrangements can be quite high (F ) 0.80: there are 40 particles in CRR at Tg). Fragilities of liquids are, therefore, determined by the nature of the chemical bonding in them. It is thus encrypted in the chemistry of the parent material. In eq 1, x representing a directional force can possibly be formulated more rigorously. But the importance of such directional force, which controls the viscosity behavior, is quite well reflected by the present formulation. It is based on the use of electronegativity and bond distances, which are uniquely meaningful chemical parameters. In glass-forming liquids fragility plays an important role, because the supercooled liquid, which has missed the crystallization opportunity, is in search of lower wells in the energy landscape11-13 in the remaining less-than-a-third of its temper-
Letters ature life (Tg/Tm ≈ 2/3). The extent to which it succeeds is determined by the magnitude of the fragility. Acknowledgment. We thank Professors J. Etorneau (ICMCB) and A. Levasseur (ENSCPB) for their kind encouragement. References and Notes (1) Angell, C. A. Relaxation in Complex Systems; Ngai, K., Wright, G. B., Eds.; National Technical Information Service, U.S. Department of Commerce, Springefield, VA, 1985; 22161, p 1. (2) Angell, C. A.; MacFarlane, D. R.; Oguni, M. Ann. N.Y. Acad. Sci. 1986, 484, 241. (3) Donth, E. J. Non-Cryst. Solids 1982, 53, 325-330. (4) Hodge, I. M. J. Non-Cryst. Solids 1996, 202, 164-172. (5) Zhu, D. M. Phys ReV. B 1996, 54, 6287-6291. (6) Xia, X.; Wolynes, P. G. Cond. Matt. /9912442. (7) Richert, R.; Angell, C. A. J. Chem. Phys. 1998, 108, 9016-9026. (8) Rault, J. J. Non-Cryst. Solids 2000, 271, 177-217. (9) Ito, K.; Moynihan, C. T.; Angell, C. A. Nature 1999, 398, 492495. (10) Angell, C. A.; Sichina, W. Ann. N.Y. Acad. Sci.1976, 279, 53. (11) Goldstein, M. J. Chem. Phys. 1969, 51, 3728. (12) Stillinger, F. H. Science 1995, 267, 1935. (13) Debenedetti, P. G.; Stillinger, F. H. Nature 2001, 410, 259-267. (14) Pauling, L. Nature of the Chemical Bond; Cornell University Press: Ithaca, NY, 1960. (15) Huheey, J. E. Inorganic Chemistry; Harper & Row Publishers: New York, 1983; pp 78 and 266.
J. Phys. Chem. B, Vol. 105, No. 38, 2001 9027 (16) CRC Handbook of Chemistry and Physics; CRC Press: Boca Raton, FL, 1994. (17) Rao, K. J.; Mohan, R. J. Phys. Chem. 1980, 84, 1917. (18) Rao, K. J. Unpublished data. (19) Ananthraj, S.; Rao, K. J. Ph.D. Thesis, University of Bangalore, 1991. (20) Kumar, S.; Murugavel, S.; Rao, K. J. J. Phys. Chem. B 2001, 105, 5862-5873. (21) Rao, K. J.; Rao, B. G.; Elliott, S. R. J. Mater. Sci. 1985, 20, 1678. (22) Selvaraj, U.; Rao, K. J. J. Non-Cryst. Solids 1985, 72, 315. (23) Selvaraj, U.; Sundar, H. G. K.; Rao, K. J. J. Chem. Soc., Faraday Trans I 1989, 85, 251-267. (24) Rao, B. G.; Sundar, H. G. K.; Rao, K. J. J. Chem. Soc., Faraday I 1984, 80, 3491. (25) Bhat, M. H.; Berry, F. J.; Jiang, J. Z.; Rao, K. J. J. Non-Cryst. Solids 2001, 291 (1,2), 000-000. (26) Sundar, H. G. K.; Rao, K. J. J. Chem. Soc., Faraday Trans. I 1980, 76, 1617. (27) Sanderson, R. T. Polar CoValence; Academic Press: New York, 1983. (28) Wong, J.; Angell, C. A. Glass Structure by Spectroscopy; Marcel Dekker: New York, 1976; Chapter 2. (29) Balta, P.; Balta, E. Introduction to the Physical Chemistry of the Vitreous State; Abagus Press: Kent, England, 1976; Chapter 4. (30) Rao, K. J. Proc. Indian Natl. Sci. Acad. 1986, 52A, 176. (31) Angell, C. A. Larecherche 1982, 5, 584-593. (32) Angell, C. A.; Rao, K. J. J. Chem. Phys. 1972, 57, 470. (33) Ngai, K. J. Non-Cryst. Solids 1996, 203, 232. (34) Angell, C. A. J. Phys. Chem. Solids 1988, 49 (8), 863-871. (35) Adam, G.; Gibbs, J. H. J. Chem. Phys. 1965, 43, 139-146.
