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Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B, February 2007, 48 (1), 61–63 Viscosity activation energy Isak Avramov Institute of Physical Chemistry, 1113 Sofia, Bulgaria Accepted 6 October 2006 The standard procedure of determining the activation energy for shear viscosity from the slope L of the Arrhenius plot, lgη against 1/T leads frequently to overestimated values. The article discusses the significance of L and shows that an important information can be retrieved from it value as the fragility parameter can be formulated as α=0·007Lg/Tg. Fragility parameter α=1 is an indication for a strong glass with constant activation energy. It is demonstrated that the average jump frequency model describes fairly the dependence of viscosity on chemical composition. On the contrary, models based on mean activation energy considerations fail completely. 1. Introduction Most of the models of shear viscosity assume that it is a thermally activated process. To move, the molecules have to overcome activation energy E(T) barriers(1) created by the resistance of the surrounding building units. Therefore, the jump frequency, is related to the vibration frequency ν∞ according to equation Ê E (T ) ˆ n = n • exp Á Ë RT ˜¯ (1) where R is the ideal gas constant. On the other hand, viscosity η is related to the jump frequency ν of the building units according to the Frenkel equation h= RT 1 3pVm n (2) where Vm is the molar volume. Maxwell gives an alternative expression in the form h= G• n (3) G∞ being the shear modulus. Therefore, viscosity is also thermally activated Ê E (T ) ˆ h = h• exp Á Ë RT ˜¯ (4) There is a fundamental question: how to determine experimentally the activation energy and how to interpret it. Although the answer seems trivial, the situation is quite tricky and misleading. The aim of the present article is elucidate this problem. 2. Activation energies problem The conventional procedure of determining the activation energy is to plot experimental data in Arrhenius Email [email protected] coordinates, logη against 1/T. In the high viscosity region, experimental data give a straight line. Therefore, a widespread fallacy is that the slope determines directly the activation energy according to L0 = 1 E 2 ◊3 R The provisional slope Lo, given by Equation (5), should be equal to the experimentally determined slope L if the activation energy was constant. Since the experimental data resemble a straight line, it seems that this assumption is right. However, every curve resemble to a straight line if the investigated interval is sufficiently short (even the Equator looks like a straight line to the horizon limits). When viscosity is studied in wider interval, the declination from a straight line is becoming evident. The situation is more complicated with relaxation experiments because, in this case, it is much more difficult to widen the experimental interval. Hopefully, it was proven(2–6) that relaxation process and shear viscosity are controlled by the jump frequency of the same building units, i.e. the relaxation time (reciprocal of the jump frequency) is proportional to viscosity η. If the activation energy depends on temperature the slope L is much larger the value of the provisional slope L0 Ê ˆ 1 Á ∂ ˜ L = L0 + E (T )˜ 2 ◊ 3RT Á ∂ 1 ÁË ˜¯ T (6) Unreasonably high values of E(T) are obtained if the second term, in brackets, of Equation (6) is neglected. At any temperature T, the absolute value of the activation energy can be determined from Equation (4) provided the preexponential constant η∞ is estimated properly. Taking into account typical values of the shear modulus of glasses, the expected value is about Physics and Chemistry of Glasses: European Journal of Glass Science and Technology Part B Volume 48 Number 1 February 2007 PC48_061-063 Avramov.indd 61 (5) 61 04/04/2007 12:29:59 I. Avramov: Viscosity activation energy logη∞≈−1. Earlier,(7) on the basis of experimental data of more then 100 substances, we have demonstrated that logη∞ varies in the limits −2≤logη∞≤1·5 [dPa s]. Let us determine the activation energy at temperature Tg at which ηg=1013·5 [dPa s]. Equation (4) permits the determination of the activation energy at Tg in the following way E(Tg)=2·3R(13·5−lgη∞)Tg=εRTg (7) where ε is a dimensionless constant having a value e = 2 ◊ 3 log hg h• = E (Tg ) RTg ª 32 ± 10% (8) Earlier, Bartenev has shown(8,9) that, for relaxation of polymers ε≈31, i.e. this parameter is about the average value reported here. With these considerations, the activation energy for viscous flow is given by E(Tg)=εRTg =265Tg±10% [J/mol] (9) In the literature, there is no unified definition of the glass transition temperature. Here we adopt ηg=1013·5 [dPa s] as a suitable definition of Tg. In this way Tg is always in the glass transition region and will coincide, within 2% of accuracy, with the value determined according to other methods. Additional advantage of this definition is that the relaxation time at this temperature is estimated to be logτg≈2. The comparison of Equations (5) and (8) determines the provisional slope Lo as L0 = e Tg 2 ◊3 (10) The dimensionless ratio α a= ( ) = 2 ◊3 L eT L (T ) L Tg 0 g g g ª 0 ◊ 07 Lg Tg ± 10% (11) is a measure of the “fragility” of the system. In other words, it determines how fast the activation energy is changing with temperature. In Equation (11) Lg stands for slope at the glass transition temperature, Lg≡L(Tg). 3. Influence of chemical composition on viscosity Data on fragility α on more then 100 substances are summarised in Ref. 7. It was shown experimentally (for alkaline and for most of the alkaline earth silicates), that α depends on composition x in the following way α(x)=1+6x (12) The chemical composition is expressed by the molar fraction x of the modifying oxides, respectively, 1−x is the molar fraction of SiO2. The same result, Equation (12), is derived theoretically(7) within the framework of the “jump frequency” model. A brief comparison between the “jump frequency” model and the “average activation energy” models is given in the Discussion part. Equation (11) is based only on the assumption that viscous flow is a thermally activated process. Therefore, the definition of the dimensionless fragility parameter α is universal. It accounts for the rate at which the activation energy changes with temperature. By means of Equations (5) and (6) α is expressed as follows a = 1+ ˘ ∂ E (T )˙ ∂ (1/T ) ˚T =T ( ) g Tg E Tg (13) It is seen there is relationship between composition x and sensitivity of the activation energy to temperature changes in the corresponding glass transition interval ˘ ∂ E (T )˙ ∂ (1 /T ) ˚T =T g ( ) Tg E Tg = 6x (14) 4. Discussion There are several models(7,10–12) capable of describing the temperature dependence of viscosity with sufficient accuracy. To distinguish which of them is better one has to test their possibility to describe with no new assumptions, or new adjustable parameters, the dependence of viscosity on other chemical composition. Here we discuss viscosity of SiO2, when low concentration of modifying oxides is added. In this case, a tiny fraction Qi(x) of the SiO4 tetrahedra is expected to have i broken Si–O–Si bridges (i varies between 1 and 4). Detailed investigation on Qi(x) is given in Ref. 13 Qi = 4! (1 - p)4- i pi i ! ( 4 - i )! (15) The mobility of SiO4 tetrahedra is connected with much lower activation energy in the presence of broken bridges. If E0 is the activation energy of SiO4 tetrahedra with no broken bonds and Ei(T) is the corresponding value of SiO4 tetrahedra with i broken bonds, the average activation energy is given approximately as follows E = 4 ÂQ (x) E (T ) i i (16) i=0 It is seen that at low concentrations of the modifying oxides the average value of the activation energy remains almost unchanged. However, experimental 62 Physics and Chemistry of Glasses: European Journal of Glass Science and Technology Part B Volume 48 Number 1 February 2007 PC48_061-063 Avramov.indd 62 04/04/2007 12:30:01 I. Avramov: Viscosity activation energy H solid line in Figure 1 is drawn according to Equation (17), for Ei≈Ei−1/2. The same values are applied in calculating the dotted line according to <E> model. The superior accuracy of the jump frequency approach is readily seen. 5. Conclusions Figure 1. Dependence of the viscosity on composition at 1473 K. Solid line is according to the jump frequency model while the dotted line is according to the mean activation energy approach. The parameters used are as follows: Eo=380 kJ/mol; Ei=Ei−1/2, i=1 to 4. The solid points are experimental results from Refs 14–18 It is fallacious to determine the activation energy E(T) from the slope L of the Arrhenius plot of viscosity against reciprocal temperature. In the glass transition interval the slope Lg plays an important role as the ratio α=0·07(Lg/Tg)±10% determines the “fragility” parameter, indicating the sensibility of the activation energy on temperature. In other words, determining how “short” is the glass. There is a strong experimental indication that the average jump frequency model is capable to describe with superior accuracy viscosity as compared to the mean activation energy approaches. Acknowledgment data(14–18) show that viscosity drops fast as illustrated by the solid points in Figure 1. It gives viscosity at 1473 K in dependence of the molar fraction x of the modifying oxides. Most of the approaches are based on sophisticated models determining the average activation energy <E>. Afterwards, the viscosity is determined by introducing the value of <E> into Equation (4). The result is illustrated by a dotted line in Figure 1. The complete failure of the “average energy” models is evident. Once the effective activation energy E0 for motion of SiO4 tetrahedra with no broken bond is fixed, it is not possible to find a set of Ei values suitable to describe the experimental data by means of <E> models. The alternative approach (see Refs 7,13,19) is first to determine the mean jump frequency <ν> according to n = 4 ÂQ (x)n (T ) i i=0 i (17) where the jump frequencies νi(T) are determined by the corresponding activation energies Ei according to Equation (1). Than viscosity is determined by introducing the average jump frequency <ν> in Equation (2) (or alternatively in Equation (3)). It is seen that the mean jump frequency is not determined by the mean the activation energy, i.e. <ν>≠ν∞ exp(−<E>/RT). The The author appreciates support of the Project INTERCONY. References 1. Glasstone, S., Laider, H. & Eyring, H. The Theory of Rate Processes. Princeton Univ. Press , New York. London, 1941. 2. Moynihan, C. T., Easteal, A. J. & DeBolt, M. A. J. Am. Ceram. Soc., 1976, 59, (1976) 12. 3. Moynihan, C. T., Easteal, A. J. & DeBolt, M. A. J. Am. Ceram. Soc., 1971, 54, 491. 4. Avramov, I., Grantscharova, E. & Gutzow, I. J. Non-Cryst. Solids, 1987, 91, 386. 5. Scherer, G. in "Glass 89" Proc. XV Int. Congr. on Glass, Leningrad, 1989, p.254. 6. Mazurin, O. V. J. Non-Cryst. Sol., 1977, 25, 130. 7. Avramov, I. J. Non-Cryst. Solids., 2005, 351,3163. 8. Bartenev, G. Structure and Mechanical Properties of Inorganic Glasses (in Russian). Moskow: Lit. po Stroitelstvu, 1966. 9. Bartenev, D. Sanditov, Dokl. Acad. Nauk SSSR, 1989, 304, 1378. 10. Cohen, M. & Turnbull, D. J. Chem. Phys., 1970, 52, 3038. 11. Macedo, P. & Litovitz, T. J. Chem. Phys., 1963, 42, 245. 12. Adam, G. & Gibbs, J. J. Chem. Phys., 1965, 43, 139. 13. Avramov, I., Rüssel, C. & Keding, R. J. Non-Cryst. Solids, 2003, 324, 29. 14. Buckermann, W.-A., Müller-Warmuth, W. & Frischat, G. H. Glastech. Ber., 1992, 65, 18. 15. Stebbins, J. F. J. Non-Cryst. Solids, 1988, 106, 359. 16. Emerson, J. F., Stallworth, P. E. & Bray, P. J. J. Non-Cryst. Solids, 1989, 113, 253. 17. Mazurin, O., Streltsina, M. & Shvaiko-Shvaikovskaya, T. Handbook of Glass Data, Elsevier, Amsterdam, 1983. 18. Partington, J. An advanced treatise of physical chemistry, Vol. 3, Longmans, London, New York, Toronto, 1957, p. 367. 19. Avramov, I. J. Non-Cryst. Solids, 2000, 262, 258. Physics and Chemistry of Glasses: European Journal of Glass Science and Technology Part B Volume 48 Number 1 February 2007 PC48_061-063 Avramov.indd 63 63 04/04/2007 12:30:01