Simha-Somcynsky theory

The physical insight involved in the Simha-Somcynski theory and an additional insight that we need to extend it to the time evolution will now be expressed in the building blocks of (55). We shall construct a particular realization of (55). We begin with the state variables. They have already been specified in (58). 

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. 

Nies, E., Stroeks, A., Simha, R., and Jain, R. K., LCST [lower critical solution temperature] phase behavior according to the Simha-Somcynsky theory apphcation to the n-hexane/polyethylene system. Colloid Polym. Sci., 268, 731-743 (1990). 

Curro, J. G., Lagasse, R. R., Simha, R., Application of the Simha-Somcynsky Theory to polymer glasses. Bulletin of the American Physical Society, 26, pp. 368-368 (1981). 

Rudolf et al. (1998) used the modified cell model of Dee and Walsh and the Simha-Somcynsky theory to investigate the phase behavior, excess volumes, the influence of pressure on miscibility, and the causes of miscibility. It was found that the theory of Dee and Walsh yields results similar to the previously investigated theories, whereas the Simha-Somcynsky theory does not. A modification of the latter theory for mixtures again resulted in predictions similar to that of Dee and Walsh and the earlier investigated theories. 

For the description and prediction of thermodynamic data, e.g. volumetric and compositional derivatives of thermodynamic functions of state, many theoretical models are available. The Simha-Somcynsky theory can be considered to be very succesful if one is interested in the quantitative description of thermodynamic properties. Especially, for the equation of state properties this has been shown on many occasions. For the phase behavior of polymer systems, the theory hasn t been evaluated yet in great detail. In this contribution the influence of composition, temperature and molar mass distribution of the polymer is studied for the system polystyrene/cyclohexane. 

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. 

Theoretical treatments of the equation of state are based on the lattice theory. The Simha-Somcynsky theory suggests a hole theory of polymeric liquids by determination of the reduced parameters p, v, and T (78,79). This statistical... 

FIG. 10 LCM binary binodals (heavy curves), spinodals (light curves) and critical point (O) in terms of (T — Tcrit) VS. (x2 — X2, ) for 100, 50, and 1 bar (top to bottom). Calculated for w-hexane/polyethylene with the Simha-Somcynski theory [64]. Mole fraction of pol5mier X2-... 

Additional examples of equation of state models include the lattice gas model (Kleintjens et al, [33,34], Simha-Somcynsky hole theory [35], Patterson [36], the cell-hole theory (Jain and Simha [37-39], the perturbed hard-sphere-chain equation of state [40,41] and the modified cell model (Dee and Walsh) [42]. A comparison of various models showed similar predictions of the phase behavior of polymer blends for the Patterson equation of state, the Dee and Walsh modified cell model and the Sanchez-Lacombe equation of state, but differences with the Simha-Somcynsky theory [43]. The measurement and tabulation of PVT data for polymers can be found in [44]. 

Fig. 6. Experimental (circles) and theoretical (curves) isotherms for polypropylene. The curves are based on the Simha-Somcynsky theory and show good agreement with the data (o = 1.1954 cm3/g,p = 530 kPa, T = 11,155 K) (78,79).

Figure 24 Reduced volume vs. reduced temperature for poly(vinyl acetate) at two different reduced pressures. Solid lines are calculated from Simha-Somcynsky theory assuming that the hole fraction is frozen at Tg. Dashed hnes are for the equilibrium liquids below Tg (after ref. 173, with permission)...

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. 

Wang, M., Takishima, S., Sato, Y, and Masuoka, H., Modification of Simha-Somcynsky equation of state for small and large molecules. Fluid Phase Equilibria, 242. 10-18 (2006). Wang, W, Liu, X., Zhong, C., Twu, C. H., and Coon, J. E., Simplified hole theory equation of state for liquid polymers and solvents and their solutions, Ind. Eng. Chem. Res., 36, 2390-2398 (1997). 

Equation (8.9) is obtained from the first relation in Eq. (8.8) after separating external variables, P and T, and m int al variable, the cell volume. This equation with appropriate definition of A and B (see below) is identical to the first equation of the coupled Simha-Somcynsky equation of state. Its form allows for comparison of cell and hole theories. In terms of the S-S theory,Jhe reduced van der Waals quantities A and B depend only on reduced cell volume, W = >/ , and are given by... 

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. 

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... 

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 ... 

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. 

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)... 

In the past several theoretical studies have been concerned with the mutual solubility of linear polyethylene and M-alkanes. In the course of such investigations phase behavior, or pVT relations, of pure n-alkanes has to be dealt with. In the following, three of such models will be discussed briefly Flory s Equation of State theory (EoS), the Mean-Field Lattice Gas (MFLG) model, and the Simha-Somcynsky (SS) theory. 

The Simha-Somcynski Hole Theory In the MFLG theory the effects of compressibility are related to the presence of vacancies on the lattice. On the other hand, in the EoS theory of Flory and coworkers a completely filled lattice is assumed and the pVT contributions are due to changes in the volume of the lattice sites or cells. Finally hole theories, which for polymer systems were initiated by Simha and... 

The first applications of the Simha-Somcynski (SS) hole theory concentrated on the equation of state of low and high molar mass components and mixtures thereof [59-63]. The results proved that the SS theory is quite successful in this respect. In a subsequent application the miscibility behavior of solutions has been considered [64-66] including the system -hexane/PE. Several refinements have been introduced into the SS theory which resulted in a more accurate evaluation of thermodynamic properties without the introduction of additional adjustable parameters [66-69]. 

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