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. 

Figure 4.12 The points give the measured viscosity-temperature relationship for e-terphenyl, while the shaded regions are the viscosities predicted by the Adam-Gibbs equation (4-10) using A5(7 ) measured for o-terphenyl. The two shaded regions represent alternative fits of the Adam-Gibbs parameters, one fit to the high-temperature, and the other to the low-temperature, data. (From Greet and Turnbull, reprinted with permission, from J. Chem. Phys. 47 2185, Copyright 1967, American Institute of Physics.)...

Adam-Gibbs equation. Most flexible polymers have energetically favored trans states and disfavored gauche states, with one trans state for every two gauche states. If U is the energy difference per mole between the trans and gauche states, and all such states contribute independently to the energy, then the partition function for these conformations is... 

However, the low entropy obtained at low temperatures in the Miller theory is assumed to slow down the kinetics via the Adam-Gibbs equation. The configurational entropy, plotted in Fig. 4-13, has a linear portion extending from U/RT = 0.75 to 3.0 this can be fitted by... 

Other expressions for r (r, Tf) have been proposed (Scherer 1992). The most satisfying of these, both conceptually and quantitatively, is a nonequilibrium version of the Adam-Gibbs equation ... 

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 temperature dependence of iq in physical aging may be expressed in terms of the Adam-Gibbs equation (27,28)... 

The VTE and the Adam-Gibbs equations are useful because they permit the use of thermodynamic data, obtained by calorimetric methods, to describe molecular motions in undercooled liquids and glasses. Most investigations of pharmaceutical solids rely on calorimetric measurements, and for that reason their discussion is deferred until the fundamentals of the methods have been described in the following section. 

Resnlts plotted according to the WLF eqnation could be predicted also from the molecular kinetic equation and show that the two approaches are compatible. The Adam-Gibbs equations also lead to a value of (Fg - Fj) = 55 K, so the theory appears to resolve most of the differences between the kinetic and thermodynamic interpretations of the glass transition. 

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... 

Figure 11. Self-diffusion coefficient plotted vs. temperature, for several lattice sizes. The solid and the dotted line are fits to the Vogel-Fulcher and Adam-Gibbs equation, respectively. Prom (Binder, K. Baschnagel, J. Bdhmer, S. Paul, W. Phil Mag. 5, in press.).

Figure 11 also reproduces the Vogel-Fulcher fit of Figure 10 and additionally shows a fit to the Adam-Gibbs equation (48),... 

The reversible step may be related to the dynamic crossover in protein hydration water at To 345 5K. NMR self-diffusion results [19] indicate that at this temperature a sudden change in hydration water dynamics occurs and the inverse diffusion constant switches from low-temperature super-Arrhenius behavior to high-temperature Arrhenius behavior. Neutron techniques (QENS) have also been used to study protein hydration water at this high-r crossover. Figure 21 shows the atomic MSD of protein hydration water at the low-r crossover measured using MD simulation. These crossovers can also be shown theoretically. Whenever the slope of an Arrhenius plot of the D T) changes, the specific heat has a peak. The well-known Adam-Gibbs equation (AGE) shows this as... 

Figure 22. (a) Experimental Cp of a water-lysozyme solution [14] Inset S (r) versus T calculated from integration of the experimental Cp. (b) Arrhenius plot of Dq/D versus 1000/r obtained according to the Adam-Gibbs equation [74]. 

In more recent work, Cheng et al. (2002) reported measurements of the low-shear viscosity for dispersions of colloidal hard spheres up to (f) = 0.56. Nonequilibrium theories based on solutions to the two-particle Smoluchowski equation or ideal mode coupling approximations did not capture observed viscosity divergence (Cheng et al. 2002), although the Doolittle and Adam-Gibbs equations still appeared to hold. 

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