KAHR model,

We want to pursue the subject by starting with a brief review for the present purposes of the essentials of lattice-hole theory, then follow with a consideration of free-volume mobility connections, and continue with some comparisons of experiment versus theory. Finally, we propose and sketch modifications of the theory. These may open the way to generalizations and more insightful relations to empirical formulations, such as the KAHR model [Kovacs et al., 1977, 1979] for volume relaxation. 

Modeling of relaxation behaviour in the glass can be done using either phenomenological models or models that attempt to describe bulk behaviour using thermodynamic or molecular arguments. An example of the former is the transparent mulitparameter model of Kovacs, Aklonis, Hutchinson, and Ramos (29), now commonly referred to as the KAHR model. They used a sum of exponentials and a normalized departure from equilibrium, 8 = (v-v ,)/vco, where v is the volume at time t and Voo is the volume at equilibrium. The distribution of relaxation or retardation times is a function of 6 and can be written ... 

Fhenmisnidogical Descripttm of Bw Vdonie Reooveiy. We have employed a hybrid (29 of the Tool-Narayanaswamy-Mt nihan-KAHR models of the kinetics of structural recovery (16-20) to obtain relevant phenomoiological fits to peiature-jump eiqieriments. The volume departure from equilibrium, 5, is related to the thermal history and material prc rties as ... 

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 ... 

The two essential features of structural relaxation and structural recovery have been brought out by experiments (Figs. 2.6 and 2.7). They are (1) the structural-relaxation time depends not only on temperature T, but also on the instantaneous structure (nonlinearity) and (2) the time dependence of the structural-relaxation process is not a simple exponential function (non-exponentiality). Naturally, a viable model must incorporate these two features. There are two such models. The one formulated by Moynihan and co-workers [31] is based on the constructs of Tool [27] and Narayanaswamy [30] and is known as the TNM model. The other is the KAHR model developed by Kovacs and co-workers [32]. Both models account for nonlinearity and non-exponentiality and they are essentially equivalent. We shall describe only one of them, the TNM model and its variations. A review of the KAHR model can be found in [8]. 

There is no doubt that, by capturing the two important features of structural relaxation, nonlinearity and non-exponentiality, the TNM and the KAHR models can explain qualitatively the structural relaxation even for complicated thermal histories. Quantitatively, good agreement with experimental data can also be reached by... 

The Kovacs-Aklonis-Hutchinson-Ramos (KAHR) model... 

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... 

The KAHR model assumes that each depends on the total departure from equilibrium i,e. on S not on i) which couples the set of equations (72). Furthermore, the total departure is the sum of the individual departures, Le,... 

The solution to equation (72) now depends upon the specific temperature (pressure) and structure dependence of the i,-. These dependencies are put into the KAHR model in a manner equivalent to the time-temperature superposition principles of viscoelasticity theory. Then KAHR assume that, by a change in temperature or S, each retardation time is shifted by the same amount and that... 

Equation (79) is a general equation in which the recovery function R(z) and the shift factors a and a are unknown. Aa is unknown but easily measured. In the KAHR model, R(z) is a function which can be expressed by a spectrum of retardation times (equation 80). A more easily visualized function is the so-called Kolrausch -Williams-Watts (KWW) function introduced to describe glassy kinetics by Moynihan et... 

Glass Formation and Glassy Behavior The forms of a and a used in the KAHR model are... 

Figure 36 Calculations of log vs. departure from equilibrium <5. Calculations are based on the KAHR model. This figure is...

According to the coupling model, relaxation parameters e.g. ordering parameters or mechanisms in the KAHR model) do not exhibit simple exponential decay. Rather, the physics of complex systems requires cooperativity among the relaxing species which acts to slow the simple constant relaxation rate into a time-dependent rate. Then one finds that the relevant variable initially decays at a constant rate, PFq = Tq S but at times larger than a characteristic time the rate becomes... 

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... 

Concurrently with experimental investigations, structural relaxation has also received a great deal of interest from theoretical studies. Several kinetic or thermodynamic models were developed to capture the three essential ingredients of structural relaxation. Among these models, the most famous two are the Tool-Narayanaswamy-Moynihan (TNM) model [96-98] and the Kovacs-Aklonis-Hutchinson-Ramos (KAHR) model [99]. 

In the KAHR model, the volume recovery behavior can be expressed by the normalized departure from equilibrium J (= (V - Voo)/Voo), where Voo is the equilibrium volume. The volume response is determined as [99]... 

The shift factors a-r and as follow the exponential forms suggested by the KAHR model [99]... 

The KAHR model can be extended to predict mechanical response shift factors ate if relationship between ate (or v) could be ascertained. Based on the findings of Struik [111] and McKenna [89], the logarithmic ate data appear to be well represented by a straight line versus 5 (or v) for materials that do not reach... 

Solid lines in Fig. 3.13 are fits to the TNM model. To model the structural relaxation behavior of polymer nanoparticles, the we utilized the relationship Tf — Ta = hl Cp in the TNM model, where zlCp is the change in heat capacity at Tg, 5 is the departure from equilibrium of enthalpy, analogous to <5 in the KAHR model, as presented in the earlier section in this chapter. <3 for enthalpy relaxation is given by ... 

Ramos, A.R., Hutchinson, J.M., Kovacs, A.J. Isobaric thermal behavior of glasses during uniform eooling and heating. 3. Predictions from the multiparameter kahr model. J. Polym. Sci. Polym. Phys. Ed. 22, 1655 (1984)... 

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