Adam-Gibbs relaxation time-temperature

To proceed further, we make use of the Adam-Gibbs model for the temperature dependence of relaxation time x(T) of cooperative rearranging regions in glassforming liquids [41]... 

The Adam-Gibbs equation (4-10) can be tested directly by using the calorimetrically measured entropy difference AS to compute the temperature-dependence of the relaxation time, with B then being a fitting parameter. This has been done, for example, with the data for o-terphenyl shown in Fig. 4-11, and the predicted temperature-dependence of the viscosity is found to be in qualitative, but not quantitative, agreement with the measured viscosity (see, for example. Fig 4-12). The main reason for the failure in Fig. 4-12 is that the temperature Tj at which the entropy extrapolates to zero for o-terphenyl lies below the VFTH temperature Tq required to fit the viscosity data hence the predicted viscosity does not vary as rapidly with temperature as it should. 

For a small step in temperature, the fictive temperature Tf is never far from the actual temperature T hence r, as given by the Narayanaswamy or the Adam-Gibbs equations, doesn t vary much with time. Equation (4-27) then simplifies to the ordinary linear KWW equation, Eq. (4-1). For large AT, varies during the relaxation, and the asymmetry discussed earlier is predicted. Note, however, that in Eq. (4-27) is assumed to be a constant this is not strictly valid for large changes in temperature, but is usually acceptable even when AT is a few tens of degrees. 

The square tiling model has some attractive features reminiscent of real glasses, such as cooperativity, a relaxation spectrum that can be fit by the KWW equation, and a non-Arrhenius temperature-dependence of the longest relaxation time (Fredrickson 1988). However, the existence of an underlying first-order phase transition in real glasses is doubtful, and the characteristic relaxation time of the tiling model fails to satisfy the Adam-Gibbs equation. 

Actually, Adam and Gibbs claim to compute the average relaxation time T (T) which is then related to rj T) through tjoctG with G the shear rigidity assumed to have a negligible temperature dependence. 

Fig. 12. Adam-Gibbs plots of the dielectric relaxation time of 2-methyltetrahydrofuran (2-MTHF) and 3-bromopentane (3-BP) versus (Tsconi) . The lines are VTF fits, 7 fus is the fusion temperature, and Tb is the temperature below which the VTF equation applies. /I ag and Avf are prefactors in the Adam-Gibbs and VTF equations, respectively. Tk is the calorimetri-cally determined Kauzmann temperature, and To is the VTF singular temperature, which were set equal in the VTF (line) fits. (Reprinted with permission from R. Richer and C. A. Angell. Dynamics of glass-forming liquids. V. On the link between molecular dynamics and configurational entropy. J. Chem. Phys. (1998) 108 9016. Copyright 1998, American Institute of Physics.)...

The departure from Arrhenius behavior of the relaxation time arises, in the Adam-Gibbs theory, from the temperature dependence of the Sc term in equation (1), which itself is a consequence of the excess heat capacity ACp of equation (2). For constant Ap in equation (2), the degree of non-Arrhenius character, now called the fragility, is determined by the magnitude of ACp. A general but incomplete accord seems to exist between the fragility and ACp and exceptions, like the alcohols, can be rationalized by the presence of unusual Ap terms. 

Fig. 12.23 Relaxation time versus temperature for the blend system PS/PoClS open circles, pure PS (molecular weight 700 g/mol) triangles, 25 % PoClS squares, 50 % PoClS stars, 75 % PoClS open squares, pure PoClS. The solid lines are fits of the combined Adam/Gibbs - selfconcentration approach as described in the text. The dashed lines are fits of the Adam and Gibbs model to the data of the pure components. For details see reference (Data were taken fi om reference Cangialosi et al. (2005))...

Various phenomenological equations have heen used to describe the dependence of the characteristic relaxation time on temperature and structiu-e and sometimes pressure, including the TNM equation (120), equations derived hy Hodge (123) and Scherer (124), both based on the approach of Adam and Gibbs (125), the KAHR and similar equations (119,126), equations based on free volume, and several others (127,128). The essential idea in all of these equations is that the characteristic relaxation time depends on the instantaneous state of the material (ie, temperature, pressure, and some measure of structure— volume, 5, Tf, and/or Pf). The most widely used form is the TNM equation for isobaric structural recovery ... 

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