Narayanaswamy equation

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),... 

During annealing, the temperature dependence of relaxation time of macroscopic quantities is often described by the Narayanaswamy-Moynihan equation... 

A comparison of this equation with Eq. (2.22), yields the desired relationship between fragility index and Narayanaswamy-Moynihan nonlinear parameter x [115]... 

We must now specify an appropriate expression for the dependence on 7 and T of the relaxation time Ty. Garden and Narayanaswamy (1970 Narayanaswamy 1971) proposeda simple generalization of the Arrhenius equation ... 

The Narayanaswamy expression fits volumetric relaxation data well over a range of temperatures for some glass formers (Rekhson et al. 1971 Mazurin 1977 Scherer 1992). But the equation has been criticized for its lack of a physical basis, as well as for its prediction of an Arrhenius temperature-dependence of the relaxation time at equilibrium. Furthermore, near the glass transition, the best-fit value of A // is much larger than the activation energy of the relevant molecular conformational transitions. 

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. 

Moynihan s formulation [5] of the Tool-Narayanaswamy [7] model is used in tins woilc. In Moynihan s equations, the Active temperature, Tf, originally d ned by Tod [78], is used as a measure of the structure of the glass. The evdution of Active temperature is represented by the generalized stretched exponential Kohlrausch-William-Watts(KWW) function [76,77] ... 

The results suggest that one of the shortcomings of the Tod-Narayanaswamy-Moynihan model of structural recoveiy, the dei ndence of modd parameters on thermal history is not due to thermal gradients in the DSC sample itself. Other explanations [72] need to be examined, including the validity of the equations used for the relaxation time, the assumption of thermoifaeological and theimostructural simplicity, and the way in which tte nonlinearity is incorporated into the model. 

Equation 13.6 is referred to as the Tool-Narayanaswamy-Moynihan (TNM) equation to acknowledge the contributions of different authors to the theoretical model. Both Ah and x have no clear physical meaning. However, in their treatment of the glassy state, Gibbs and DiMarzio (1958) (GD) postulated that a second thermodynamic transition T2 existed below Tg at which the configurational entropy Sc is zero. The concept of a temperature at which Sc = 0 was introduced by... 

Models of structural recovery include the Kovacs-Aklonis-Hutchinson-Ramos (KAHR) model (119), Moynihan s model (120), and Ngai s coupling model (121). These models are based on work done originally by Narayanaswamy (122), incorporating the ideas of Tool (13). The models of stnictiual recovery reflect the nonlinear and nonexponential effects observed experimentally. The historical development of these equations has been detailed (7,8) only a brief description follows. The KAHR formulation (119), which is written in terms of a departure from equilibrium S rather than in terms of Tf, is conceptually easier to use when the full three-dimensional PVT surface is considered ... 

Nonlinearity is accounted for in the Tool-Narayanaswamy-Moynihan (TNM) model by replacing r in the linear equations (2.1) and (2.2) by r given by the so-called Tool-Narayanaswamy (TN) equation ... 

In addition to controlling the magnitude of structural relaxation, fragility also affects the kinetics of the relaxation process. This is easily seen by inspection of the Tool-Narayanaswamy-Moynihan (TNM) equation [53] ... 

One of the most convenient tools for practical determination of fictitious temperatures is thermomechanometry [396] see Fig. 54, where the time dependence of fictitious temperature can be obtained on the basis of the Tool-Narayanaswami relation [391,396,400] by the optimization of viscosity measurements (logr] T,Tf versus temperatures) using the Vogel-Fulcher equation again. 

Several developments based upon these ideas were made by Narayanaswamy, Kovacs and co-workers and Moynihan and co-workers. The result has been a tremendous improvement in our ability to understand and describe the kinetic behavior described above. In fact, once the model is developed, a single non-linear equation describes all of the major features of the phenomenology. In the next section, we will develop this equation following the approach of Kovacs and co-workers, the so-called KAHR model, because of this author s familiarity with the model. We note that the resulting equations are formally identical with those of Narayanaswamy and Moynihan and co-workers, the differences being not in the underlying assumptions of the... 

Upon bringing a glass from equilibrium at a temperature To = Ti ATto the new temperature the volume (enthalpy) recovery towards equilibrium exhibits either autocatalytic or autoretarded behavior depending upon whether the approach towards equilibrium is from below or above. We previously referred to this behavior as the asymmetry of approach in T-jump experiments (Figure 30). As described above, the phenomenological equations of the Narayanaswamy-... 

There are then three important facets to equation (92). First, at equilibrium, the decay function is non-exponential (equation 90). Second, the observed retardation time depends upon S and T. Third, both n and t appear inside the integral, which implies that the history-dependent change of both these parameters is inherent in the model and so they affect the volume-recovery response differently than in the Narayanaswamy-KAHR-Moynihan-type models. For the same isothermal recovery after a T-jump, we can compare equation (92) with the response for the KAHR model... 

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