Correlation Between Thermodynamic and Kinetic Properties of Glass-Forming Liquids
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Correlation Between Thermodynamic and Kinetic Properties of Glass-Forming Liquids Oleg N. Senkov1, and Daniel B. Miracle2 1 Materials and Processes Division, UES, Inc., 4401 Dayton-Xenia Rd., Dayton, OH, 45432 2 Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson AFB, OH, 45433 ABSTRACT Correlations between three characteristic temperatures: glass transition, Tg, Kauzmann, Tk, and Vogel-Fulcher-Tammann, To, were identified from the analysis of more than 60 metallic and non-metallic glass-forming materials. It was found that Tg ≥ Tk ≥ To and Tk is the geometric mean of Tg and To (Tk2 = Tg To). The relation Tk ≥ To indicates that the excess total entropy of a super-cooled liquid ∆S approaches zero at a higher temperature than the configurational entropy ∆S conf , and such behavior was explained by the stronger temperature dependence of the excess g −c vibrational entropy of the liquid, ∆S vib , than that of the corresponding glass, ∆S vib .A relationship between the fragility index m, reduced excess heat capacity ∆Cp(Tg)/Sm, and reduced glass transition temperature, Trg, was identified using the found correlation between the characteristic temperatures.
INTRODUCTION Three key temperatures are often used to characterize relaxation behavior and glass transition of super-cooled liquid.1,2,3 These are the glass transition (Tg), Kauzmann (Tk) and Vogel-Fulcher-Tammann (VFT, To) temperatures. By definition, Tg is the temperature, below which the physical properties of an amorphous material vary in a manner similar to those of a solid (glassy state), and above which the amorphous material behaves like a viscous liquid (supercooled liquid state). The glass transition can be identified by differential scanning calorimetry (DSC) as the second order endothermic transition or by thermo-mechanical analysis (TMA) as the transition associated with the change in the coefficient of thermal expansion (CTE).4 In the glass transition range, the Maxwell relaxation time, τ, of the supercooled liquid increases sharply and becomes comparable to the experimental time scale. Therefore, Tg is often defined as a temperature at which the relaxation time approaches the value of τg ≈ 100 s.1,5 The VFT temperature To is a kinetic parameter of the VFT equation, which is widely used to describe the non-Arrhenius dependence of τ on temperature T of supercooled liquids: 1-5 DTo ⎝ T − To ⎛
τ = το exp⎜⎜
⎞ ⎟ ⎟ ⎠
(1)
According to this equation, τ becomes infinite at To; however, this prediction cannot be verified because of the glass transition and because the system is bound to fall out of equilibrium as To is approached. Therefore, To is determined by extrapolation of the experimental data collected above Tg to the temperature range much below Tg. Other parameters in Equation (1) are the minimum relaxation time at infinitely large temperature, τo ≈ 10-14 s, which almost does not depend on liquid constitution;1,5 and the strength parameter, D, which characterizes liquid
fragility (i.e. the level of d
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