Structural recovery nonlinearity

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 decrease in volume during physical aging is specifically known as volume recovery or volume relaxation. Volume recovery experiments include down-jump, up-jump, and memory experiments. The results of these experiments, which are shown and discussed later, demonstrate that structural recovery is both nonlinear, nonexponential and path-dependent. 

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 nonexponentiality of the structural recovery process is described by the nonexponentiality parameter in equation (15) the history dependence is accounted for by the integral of the reduced time (dt/r) and the nonlinearity is incorporated into the model by allowing the characteristic relaxation time r... 

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]. 

If ACp(T) were independent of temperature, the equilibrium configurational entropy SdT) from Eq. (2.21) would be given by AC >(T)ln(T/r2). This expression is the origin of Eq. (2.6), which, together with Eq. (2.5), introduces nonlinearity into structural recovery. If the temperature dependence of ACp(T) is well approximated by the hyperbolic expression, ACp(T) = A/T, which is the case for some glass-formers [75], then AS(T) = A(T — T2)/ TT2), which, after substitution into Eq. (2.20), leads to the equation... 

Multispecies toxicity tests are complex structures. Complex structures are nonequilibrium, historical, and nonlinear. To measure the recovery of such a structure is to measure a property that does not exist for a complex structure. 

Melt Viscosity. The study of the viscosity of poljmier melts (47-62) is important for the manufacturer who must supply suitable materials and for the fabrication engineer who must select polymers and fabrication methods. Thus, melt viscosity as a fimction of temperature, pressure, rate of flow, and poljmier molecular weight and structure is of considerable practical importance. Polymer melts exhibit elastic as well as viscous properties. This is evident in the swell of the polymer melt upon emergence from an extrusion die, a behavior that results from the recovery of stored elastic energy plus normal stress effects. Theoretical developments include a constitutive equation that correctly captures nonlinear behavior in both elongation and shear (63,64). 

This asymmetry in the recovery means that the structural-relaxation time depends not only on the temperature but also on the instantaneous structure. In particular, when the approach toward equilibrium is from above, the initial structure of the material will have a higher molecular mobility than that of the equilibrium structure, although the difference decreases as the departure from equiUbrium decreases. On the other hand, the opposite is true when the approach toward equilibrium is from below. This feature of isothermal structural relaxation is sometimes referred to in the literature as the nonlinearity of the process, in the sense that the structural relaxation cannot be described by the linear differential equation... 

Our focus has been the elucidation of specific features of the model which give rise to the most important aspects of glass transition behavior. Thus time dependence arises naturally from a consideration of the molecular aspects of the overall phenomena involved. The pronounced nonlinearity and asymmetry of behavior result from the structure dependency of the individual retardation times introduced in the model. In fact, our analysis shows that, on a time scale appropriately compensated to take account of structural dependence, non-linearity and asymmetry vanish. The existence of a multiplicity of recovery times in the model leads to memory, which is observed in real systems. 

The asymmetry of approaching equilibrium demonstrates a relaxation process subject to more complex thermal histories. That is, the material is initially equilibrated at temperature To that departs AT from the test temperature Ti, i.e.. To = Ti + AT or To = Ti - AT, in cases of down-jump or up-jump, respectively. Subsequently, the temperature is controlled to jump to Ti, where the volumetric evolution is then measured in the relaxation induced from the temperature jump. Although the magnitude of the temperature jump AT in both cases of the up-jump and down-jump is the same, the volume recovery towards equilibrium is found to be much faster in the down-jump experiment (Ti < To) than in the up-jump experiment (Ti > To). This result indicates the inherent nonlinearity of the structural relaxation process and has been widely used to determine that the aging response depends upon the instantaneous state of the material. 

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