3D diagrams of equations of viscous flow of silicate glass-forming melts Ana F. Kozmidis-Petrović ⁎ University of Novi Sad, Faculty of Technical Sciences, Trg D. Obradovića 6, 21000 Novi Sad, Serbia abstract article info Article history: Received 2 February 2012 Received in revised form 14 February 2012 Available online 8 March 2012 Keywords: Silicates; Viscosity; 3D diagrams Vogel–Fulcher–Tammann (VFT) equation and Mauro, Yue, Ellison, Gupta and Allan (MYEGA) equation for the viscous flow are here expressed using the parameters characteristic for Avramov and Milchev model. This en- abled us to represent these functions through 3D diagrams and compare them directly. The 3D diagrams of these functions were compared as functions of the two variables for different temperature (or ratio T/T r ) and in a wide range of x. T r is referential temperature at which η Tr =10 13 d Pa s. The 3D diagrams were also used to present the differences of these functions. It followed from the 3D dia- grams that for x ~ 0.5 and x > 0.8 the most significant differences occur in the values of logη. For that reason equations are additionally tested on concrete silicate systems with such composition of x. The VFT and the MYEGA functions are then modified so that through T r they include lubricant coefficient k. We show that the 3D diagrams of the so modified VFT and MYEGA functions can predict the behavior of viscosity of silicate systems, both when a component with large k and when a component with small k is present. With these some more frequent anomalies at certain parts of the 3D diagram were observed with the VFT function. The MYEGA function proved to be just as successful as the AM function in predicting the lubricant effect and showed good agreement with the experimental results. © 2012 Elsevier B.V. All rights reserved. 1. Introduction Different models have been suggested for describing the tempera- ture dependency of viscosity and it is therefore extremely important to determine which of them is the most successful. Three is the usual number of free parameters for most of the con- temporary viscosity equations. The most popular viscosity model with three parameters is the Vogel–Fulcher–Tammann (VFT) equation [1–3] log η VFT T ; x ð Þ¼ log η ∞V x ð Þþ Bx ðÞ T -T o x ðÞ ð1Þ where T is temperature and x is composition. The three parameters are: constant B(x), pre-exponential factor η ∞V and T o (x)a finite temperature at which the viscous flow ceases. These parameters can be determined by fitting. The VFT equation is empirical, although there are papers where attempts are made to achieve this starting from the mean value of activation energy. Avramov and Milchev model (AM) [4–7] determines the tempera- ture dependence of average jump frequency. According to this model viscosity depends on the entropy S of the system and through it on the temperature and on composition [8,9]. In the AM model viscosity is the function on the entropy S of the system according to equation [9] log η AM ¼ log η ∞A þ log η g η ∞A   exp - 2 S-S g   ZR 2 4 3 5 ð2Þ Here reference state is denoted with subscript g with entropy S g . Z is the number of escape channels and logη ∞A is pre-exponential parame- ter. Papers [10–12] in the application of the AM model choose as a reference state the glass transition temperature T g when η =10 12 [Pa·s]. As Y-Z Yue states in his paper [13] it is well-known that even for the same glass, the measured T g values are quite diversified, depending on the experimental methods used and the conditions under which they are performed. According to the analysis presented in this paper, there is an excellent agreement between T g determined using DSC method at the rate of 10 K/min and T g at viscosity of 10 12 [Pa·s]. Avramov in [14] uses referential temperature T r instead of T g . Referen- tial temperature T r is the temperature at which the viscosity is 10 13 [dPa·s]. This temperature is just the one assumed by Yue (10 13 [dPa·s] = 10 12 [Pa·s]). The number of free parameters of viscosity equa- tion can be reduced to two by assuming that at T r the viscosity is 10 13 [dPa·s]. Journal of Non-Crystalline Solids 358 (2012) 1202–1209 ⁎ Tel.: + 381 214852287; fax: + 381 216350770. E-mail address: analeto@yahoo.com. 0022-3093/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2012.02.022 Contents lists available at SciVerse ScienceDirect Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/ locate/ jnoncrysol