Simha-Somcynsky free volume

To describe the kinetics of the structure relaxation process following sudden cooling or heating, the relaxation time x is first of all empirically related to the computed Simha-Somcynsky free volume f for the equilibrium liquid at a constant pressure, i.e.,... 

The quantity q has the physical interpretation of the free volume. It is the state variable used in the Simha-Somcynski equilibrium theory of polymeric fluids (Simha and Somcynski, 1969). The new variable p that we adopt has the meaning of the velocity (or momentum) associated with q. 

Lhx = Lnx = 0. The kinematics of q and p is chosen to be the same as if q is the position coordinate and p the momentum associated with it. In other words, whatever is the physical interpretation of q (e.g., free volume in the Simha-Somcynski theory), p is the momentum associated with it. 

The Simha and Somcynsky (S-S) [1969] cell-hole theory is based on the lattice-hole model. The molecular segments of an -mer occupy ay-fraction of the lattice sites, while the remaining randomly distributed sites, /i = 7 — y, are left as empty holes. The fraction /i is a measure of the free-volume content. The goal was to provide improved description of fluids, ranging from low-molecular-weight spherical molecules (such as argon) to macromolecular chains. The S-S configurational partition function is... 

Much of the work stems from Simha-Somcynsky (S-S) [1969] hole theory, developed originally to describe polymers in the liquid state. They introduced the free volume by using the formalism of vacant cells or holes in a lattice and developed an equation of state that could be used to calculate the fraction of sites occupied and hence the fractional free volume. As discussed in Chapter 6, the concept has been developed further by Simha and his co-workers. 

Yu et al. [1994] carried out PALS measurements on four PS fractions (4, 9, 25, and 400 kDa, respectively) versus temperature (Figure 10.4). They evaluated the free-volume fractions on the basis of the proportionality between the free-volume fraction as probed by o-Ps and the product of the o-Ps intensity I3 and the mean cavity volume assumed spherical, as sketched previously [Eq. (10.16)]. On this basis they observed agreement with the free-volume fraction predicted as given by the lattice-hole model [Simha and Somcynsky, 1969] over a range of temperatures above Tg, the proportionality constant C being a molar mass-dependent fitting parameter. 

Dlubek, G., Pionteck, J., and Kilburn, D., The structure of the free volume in poly (styrene-co-acrylonitrile) from positron lifetime and pressure-volume-temperature PVT) experiments I. Free volume from the Simha-Somcynsky analysis of PVT experiments, Macromol. Chem. Phys.,205, 500-511 (2004). 

Experimental data from our laboratories will be shown for an extensive series of amorphous polymers with glass transitions between Tg = 200 and 500 K. We discuss the temperature dependence of the hole-size distribution characterized by its mean and width and compare these dependencies with the hole fraction calculated from the equation of state of the Simha-Somcynsky lattice-hole theory from pressure-volume-temperature PVT) experiments [Simha and Somcynsky, 1969 Simha and Wilson, 1973 Robertson, 1992 Utracki and Simha, 2001]. The same is done for the pressure dependence of the hole free-volume. The free-volume recovery in densified, and gas-exposed polymers are discussed briefly. It is shown that the holes detected by the o-Ps probe can be considered as multivacancies of the S-S lattice. This gives us a chance to estimate reasonable values for the o-Ps hole density. Reasons for its... 

The free-volume concept dates back to the Clausius [1880] equation of state. The need for postulating the presence of occupied and free space in a material has been imposed by the fluid behavior. Only recently has positron annihilation lifetime spectroscopy (PALS see Chapters 10 to 12) provided direct evidence of free-volume presence. Chapter 6 traces the evolution of equations of state up to derivation of the configurational hole-cell theory [Simha and Somcynsky, 1969 Somcynsky and Simha, 1971], in which the lattice hole fraction, h, a measure of the free-volume content, is given explicitly. Extracted from the pressure-volume-temperature PVT) data, the dependence, h = h T, P), has been used successfully for the interpretation of a plethora of physical phenomena under thermodynamic equilibria as well as in nonequilibrium dynamic systems. 

Since Chapter 6 presents detailed discussion of Simha-Somcynsky lattice-hole theory, only an outline is provided here. The theory was derived for spherical and chain molecule fluids [Simha and Somcynsky, 1969 Somcynsky and Simha, 1971]. The model lattice contains a volume fraction y of occupied sites and h= —y of nonoc-cupied sites, or holes. From the Helmholtz free energy, F, the S-S equation of state was obtained in the form of coupled equations ... 

The frequency shift factor, ar, has been related to the free-volume fraction,/ [Ferry, 1980]. There is a direct correlation between/and the Simha-Somcynsky (S-S) hole fraction, h [Utracki and Simha, 2001b]. Under ambient pressure, h depends on the reduced temperature [Utracki and Simha, 2001a] ... 

Simha et al. have used the hole theory of Simha-Somcynsky (S-S) (Simha and Somcynsky 1969) as a starting point to develop further the idea of fi ee volume. In their vacant cells or holes in a polymer lattice constitute the fi ee volume arising from inefficient chain packing. An equation of state was developed to calculate the Iraction of occupied lattice sites and hence the fractional free volume. 

The hole theory offers an excellent basis to evaluate the phase behavior of polymer systems. The description of the spinodal conditions are almost quantitative without the introduction of empirical parameters. The cell free volume is very important for this quantitative success. The influence of polydispersity on the spinodal conditions in the Simha-Somcynsky theory is not restricted to the mass average molar mass. 

Free-volume theory is an improved cell or lattice model for the liquid state by introduction of vacancies in the lattice. In the free-volume theory, the Simha-Somcynsky equation of state of a polymer system is written as (10)... 

The most successful statistical theory of liquids is that derived by Simha and Somcynsky. The model considers liquids to be mixtures of voids dispersed in solid matter, i.e., a lattice of unoccupied and occupied sites. The occupied volume fraction, y (or its counterpart the free volume fraction f = 1 - y), is the principal variable y = P, T). From die configurational partition function the configurational contribution to the Helmholtz molar free energy of liquid i was expressed as [3] ... 

Other free volume theories such as those of Sanchez and Lacombe and Simha and Somcynski are based on a lattice model and all or part of the free volume arises from vacancies on the lattice, unlike the Flory theory where free volume arises from an overall increase in molecular separations. Such theories are discussed in the chapter on polymer solutions (Volume 2, Chapter 3) and have not been much used in relation to polymer mixtures. Their use may -well prove to be valuable since, especially using the theory of Simha and Somcynski, they much better describe the properties of the pure components. 

Because of the success of the empirical free-volume relations in describing the behavior of glassforming liquids, there have been many attempts since the Cohen and Turnbull free-volume model to quantify the concept and make the free-volume physics more than a convenient way to correlate data. The reader is referred to the literature for a general look at the various models and also for some specific developments. " However, due to space limitations, we limit our discussion to the cell model of Simha and Somcynsky " and the extensive developments of this model which have been carried out over the years by Simha and co-workers. 

Somcynsky model differs markedly from that used by Ferry and co-workers (WLF-type), or that which can be derived from the Gibbs-DiMarzio lattice model. The Simha-Somcynsky model results in values of h=f 0.07, while the WLF empiricism and the Gibbs-DiMarzio lattice model give values of the free-volume fraction of about 0.025, as mentioned previously. The discrepancy with the WLF approach may come from the arbitrary choice of a value of unity for parameter B in the Doolittle equation. 

The Simha-Somcynsky model is successful in describing much of the thermodynamics (PVT) of glass-forming substances. Its one major drawback is that it does not develop the kinetics, and therefore the transition phenomenon, a priori. However, the model is useful as a quantitative free-volume description of glassy behavior. As we will see, the cell-model limitations on prediction of the entropy (or energy) equation of state still hold for the Simha-Somcynsky description of the glass. 

Equation (52) o (55) describes the experimental data quite well except at low molecular weights (large values of M % and has been used extensively to describe the T -M relationship for polymers. We note, however, that Gibbs and DiMarzio have pointed out that the glass is not a true iso-free-volume state, and the free-volume description of Simha and Somcynsky also precludes such a description of the glass transition. 

There has been little use of the free-volume models to describe or predict thermodynamic functions other than the PVT equation of state for glasses, because these models are primarily concerned with molecular mobility. However, the cell model of Simha and Somcynsky has been used to predict the heat capacities of poly (vinyl acetate) in the liquid and glassy states. Table 3 shows a comparison between theory and experiment for Cp and in the liquid and glassy states, as well as for ACp, AC and AS at the glass transition. Although the values of and C obtained from the model are less than 20% of the observed ones, values for C -C are reasonably close to those actually observed. The values for ACp, AC, and AS obtained from the model are less... 

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