Structural relaxation enthalpy

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) ... 

Privalko, V. P, Demchenko, S. S., and Lipatov, Y. S. (1986). Structure-dependent enthalpy relaxation at the glass transition of polystyrenes. Macromolecules 19(3), 901-904. 

The structural relaxation time defines the activation enthalpy Ah for small deviation from equilibrium... 

Structural relaxation times determined from enthalpy relaxation studies with sucrose and trehalose [37,50] are given in Figure 8. The structural relaxation times observed are qualitatively similar to those estimated for the fragile glass... 

Figure 8 Structural relaxation times for quench-cooled glassy disaccharides as determined from enthalpy relaxation data. Structural relaxation times were obtained by a fit of the data to the stretched exponential function (see [37,50]). ( ) Data for sucrose obtained by differential scanning calorimetry on annealed samples [37], (O) Data for sucrose obtained by isothermal microcalorimetry [50]. (A) Data for trehalose obtained by isothermal microcalorimetry [50].

The problems associated with freeze drying of peptides and proteins for therapeutic use have also received calorimetric attention recently - particularly, attempts to understand and interpret the dynamics of amorphous solids. Structural relaxation time is a measure of molecular mobility involved in enthalpy relaxation and thus is a measure of the dynamics of amorphous (glassy) solids. These dynamics are important in interpretation of the physicochemical properties and reactivities of drugs in amorphous formulations. The authors conclude that microcalorimetry may provide data useful for rational development of stable peptide and protein formulations and for control of their processing . 

In the previous section, at temperatures above 7g, the Johari-Goldstein relaxation time has been shown to correspond well to the primitive relaxation time, and both are related to the structural a-relaxation time by Eq. (10). This equation should continue to hold at temperatures below T,. However, testing this relation in the glassy state is difficult because of either the scarcity or the unspecified thermal history of the data on the a-relaxation time xa. In fact, a reliable characterization of the structural relaxation can be acquired only at equilibrium, and such condition is rarely satisfied below Tg. Glassy systems are nonergodic, and their properties can depend on aging time and thermal history. Anyway, for glasses in isostructural state with a constant Active temperature Tf, both ra and To as well as Xjg should have Arrhenius dependences with activation enthalpies Ea, Eq, and Ejg respectively. Eq. (10) leads us to the relation... 

Moynihan et al., irrespective of the direction of the temperature change, the direction of the structural relaxation process is always toward equilibrium. That is, when the glass is heated or cooled, the enthalpy H changes in such a fashion as to move toward the equilibrium value. Consequently, on reheating, H follows a path different from that found on cooling, as indicated in Fig. 3. 

The program is then simply to start at a high temperature, where p T) — Peq( ) lower the temperature at a fixed q<0. The result forp T) can then be used in (8.2) to (8.4) for to obtain results that can be compared directly with experiment. The only quantity that we must specify in addition to those in the equilibrium theory is the relaxation time r T). Since t(T ) is to describe the relaxation by diffusion of structural modes represented by the variation of p, it should have the same temperature dependence as the shear viscosity rj. That is, we suppose that the same microscopic movement processes underlie self-diffusion, viscosity, and structural relaxation. This supposition is consistent with existing theories and with a number of experimental results indicating that the activation enthalpy A A for volume or enthalpy relaxation is generally the same as the activation enthalpy for the viscosity tj. We therefore assume that r(T) can be expressed by the Doolittle equation. 

Figure 3 Changes in enthalpy or volume with time during an isothermal structural relaxation following a temperature jump. Reproduced with permission from Moyniham ...

For a temperature or pressure jump, the rate of the structural relaxation is expressed in terms of a characteristic relaxation time t. As a working approximation, the rate of approach of the density and configuration to their equilibrium values can be treated as first-order kinetic processes. Thus, the change in enthalpy accompanying the relaxation is expressed by... 

The phenomenon described here is termed physical aging or structural relaxation. It can be detected through the time evolution of not only thermodynamic properties such as specihc volume or enthalpy but also mechanical or dielectric properties. 

We have criticized the use of enthalpy at constant total volume, Ey, in the enthalpy ratio since it is the free volume and not the total volume that controls the structural relaxation, and the free volume Vf(T,P) is not the same for a given total volume VCT,P) but depends on T,P pairs [Dlubek et al., 2005b]. Considering these arguments, we suggest employment of the activation enthalpy at constant specific (hole) free volume, Eyf=R[(d In T/dJ )]yf, which gives [Dlubek et al., 2007a, b]... 

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