Hole equilibrium fraction

Fig. 2. Maxima of the curves defining the equilibrium glass transition temperature. It is treated as a thermodynamic anomaly at which the most stable hole configuration is reached under the close packing of holes and flex bonds. R is the gas constant, z is the lattice coordinate number, and /ir(=/r) is the hole fraction at T, [11]...

Yuh and Stolka (1988) described the temporal features of the photocurrent transients of TPD doped PC. The authors concluded that holes reach dynamic equilibrium within a small fraction of the transit time. A plateau in the photocurrent transient then occurs, followed by a dispersion of arrival or transit times. The dispersion was attributed to positional disorder. The results were in agreement with the simulations of Marshall (1978). A key result of the study was that W scaled with thickness as L l/2. Field-independent dispersion at very long times was attributed to fluctuations of hopping site energies. Over a wide range of fields, the mobilities were independent of thickness. 

The subscripts ale, e and w refer to alcohol, ethylene and water, respectively. The term in pressures is usually written Kp, and the partial pressures are derived from the total pressure multiplied by mole fraction. The expression in activity coefficients, because of its similarity in form to Kp, is often abbreviated Ky it is not of course an equilibrium constant. In short, we may write K=KpKr The activity coefficients are determined from the generalized chart shown as Fig. 7.7, from known or estimated pressures. However, to start with we know only K and the total pressure, and must estimate starting values of each pressure in order to find values for y in each case. We can then estimate Kp as K/ffi and calculate equilibrium pressures once more. If the new values for y are appreciably different, then another iteration (calculation) is called for. The approach is similar to that of a golfer approaching his hole he arrives by successive approximation. Let us see how this works out in practice. 

The hole fraction, h, computed from the PVT data, was found to correlate with the static (thermodynamic equilibrium) and dynamic properties (e.g., flow of polymers) [Utracki, 1986 2005], as well as their blends, foamed compositions [Utracki and Simha, 2001a,b], composites, and nanocomposites (see Chapter 14). 

The hole fraction h was then calculated from a numerical solution of the pressure equation P = -(dFldV)T. This equation is valid for the equilibrium state however, the specific assumption that the free energy is a minimum has not been made here. Therefore, it is usual to calculate the h values from the specific volume below Tg (P) from this equation using the scaling parameters P, V, and T determined for the liquid. These h (and y) values are considered to be sufficiently good approximations for conditions not too far from equilibrium [McKinney and Simha, 1976 Robertson, 1992]. 

Figure 11.15b also shows an interesting way of extrapolating the equilibrium hole fraction h T,P = 0) to lower temperatures. When considering the vacancies of the S-S lattice as Schottky point defects, their equilibrium concentration heq may be expressed by the Schottky equation. 

Robertson et al. [1984] developed a stochastic model for predicting the kinetics of physical aging of polymer glasses. The equilibrium volume at a given temperature, the hole fraction, and the fluctuations in free volume were derived from the S-S cell-hole theory. The rate of volume changes was assumed to be related to the local free volume content thus, it varied from one region to the next according to a probability function. The model predictions compared favorably with the results from Kovacs laboratory. Its evolution and recent advances are discussed by Simha and Robertson in Chapter 4. 

Starting in the melt, proceeding to the transition region and continuing into the glassy state, sets of equilibrium, and non-equilibrium processes are considered. We examine the consequences of a unified view derived from a lattice-hole model, involving a hole fraction h to account for the structural disor r. 

However, the equilibrium condition (6.37) can be written more conveniently in terms of the mole fractions of empty and occupied holes, which are defined, respectively, as... 

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