Tool-Narayanaswamy-Moynihan model

Tool-Narayanaswamy-Moynihan (model of kinetics of the glass transition) [Tool, 1946a,b Narayanaswamy, 1971 Moynihan et al., 1976] two-parameter theory... 

The temp, modulated DSC (TMDSC) technique could be used for heat capacity spectroscopy in the low frequency range. The measured property was the complex heat capacity. The frequency-dependent relaxation transition measured by TMDSC occurred in the temp, range of the thermal Tg. The non-equilibrium of the glassy state thus influenced the TMDSC curves. Experimental evidence was the dependence of the shape of the heat capacity curve on the thermal history. A theoretical description of the influence of the non-equilibrium state on the spectroscopic curves, based on the Tool-Narayanaswamy-Moynihan model, is presented. 23 refs. 

Weyer S, Merzlyakov M, Schick C. Application of an extended Tool-Narayanaswamy-Moynihan model. Part 1. Description of vitrification and complex heat capacity measured by temperature-modulated DSC. Thermochim Acta 200l i77(l-2) 85-96. 

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) ... 

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 Narayanaswamy-Moynihan model (NMM), discussed in detail by Donth and Hempel [49], is an improvement on this approach. Instead of postulating a simple exponential relaxation mechanism governed by a single relaxation time [47], the nonexponential stractural relaxation behavior and the spectmm effect were studied. It was assumed that the whole thermal history Tit) started from a thermodynamic equilibrium state where 7 (to) = 7 (to)- Tool s fictive temperature is defined by... 

In addition to the free volume [36,37] and coupling [43] models, the Gibbs-Adams-DiMarzo [39-42], (GAD), entropy model and the Tool-Narayanaswamy-Moynihan [44—47], (TNM), model are used to analyze the history and time-dependent phenomena displayed by glassy supercooled liquids. Havlicek, Ilavsky, and Hrouz have successfully applied the GAD model to fit the concentration dependence of the viscoelastic response of amorphous polymers and the normal depression of Tg by dilution [100]. They have also used the model to describe the compositional variation of the viscoelastic shift factors and Tg of random Copolymers [101]. With Vojta they have calculated the model molecular parameters for 15 different polymers [102]. They furthermore fitted the effect of pressure on kinetic processes with this thermodynamic model [103]. Scherer has also applied the GAD model to the kinetics of structural relaxation of glasses [104], The GAD model is based on the decrease of the crHiformational entropy of polymeric chains with a decrease in temperature. How or why it applies to nonpolymeric systems remains a question. 

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... 

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 ... 

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]. 

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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