Narayanaswamy-Moynihan nonlinear

The ratio of Eqs. (2.38) to (2.39) evaluated at the glass transition temperature yields the following expression for the Narayanaswamy-Moynihan nonlinear parameter x [115] ... 

Here, Cv h(T) and Svlh(T) are the vibrational contributions to the heat capacity and the entropy, respectively. Note that the slope of the replica symmetry-breaking parameter with respect to temperature is not unity as predicted by one-step replica symmetry breaking. Rather, the slope is governed by three factors the Narayanaswamy-Moynihan nonlinearity parameter x, the Kauzmann temperature, and the ratio of the Kauzmann temperature to the glass transition temperature. 

One theory that describes the temperature dependence of relaxation time and structural recovery is the Tool-Narayanaswamy-Moynihan (TNM) model developed to describe the often nonlinear relationship between heating rate and Tg. In this model, the structural relaxation time, x, is referenced as a function of temperature (T), activation enthalpy (Ah ), universal gas constant (R), hctive temperature (7)), and nonlinearity factor (x) (Tool, 1946 Narayanaswamy, 1971 Moynihan et al., 1976) ... 

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. 

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 ... 

The concept was developed further by Narayanaswamy (1971) and Moynihan et al. (1976) who treated the kinetics of the aging processes by describing the relaxation toward an equilibrium state in terms of a nonequilibrium decay function. As this depended on the departure from equilibrium, the kinetics are also nonlinear. To allow for this, the average relaxation time, t, was made a function of both temperature and structure and expressed 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]. 

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