Relaxation Adam-Gibbs equation

Assuming the validity of Adam-Gibbs equation for relaxation dynamics and the hyperbolic temperature dependence of heat capacity, the strength parameter is found to be inversely proportional to the change in heat capacity [see Eq. (2.10)] at the glass transition temperature [48,105]. 

Several well-known equations are available for interpreting the temperature dependence of viscosity, diffusion coefficient, and other relaxation rates for T > Tg. The Doolittle equation [18], the WLF equation [19], the Vogel-Fulcher equation [20], and the Adam-Gibbs equation [21] can be expressed in the same form. They are known to fit well with the relaxation data of liquids in equilibrium. The universal functional form is [20]... 

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. 

Although use of the Narayanaswamy equation is successful in the predictions of the data in Figs. 4-18 and 4-19, for polymeric liquids it fails badly and should be replaced by the Adam-Gibbs equation (Matsuoka 1992). For additional discussion of phenomenological theories of nonlinear relaxation, see Scherer (1992) and McKenna (1989, 1994),... 

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. 

The Adam-Gibbs equation for viscous liquid relaxation asserts that the time scale for re-equilibration after some perturbation is related to the excess entropy of liquid over crystal, according to... 

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. 

A number of equations exist to calculate the molecular mobility or its reciprocal, the relaxation time, x (Andronis and Zografi 1997 Di Martino et al. 2000 Mao et al. 2007 Yoshioka et al. 1994). The two most commonly used equations that give an indication on the mobility within an amorphous sample are the Kohlrausch-Williams-Watts (KWW) equation and the Adam-Gibbs (AG) equation. 

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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