Somcynsky and Simha

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 Simha-Somcynsky (S-S) [Simha and Somcynsky, 1969 Somcynsky and Simha, 1971] equation of state incorporates the hole fraction, h, a direct measure of/(for details, see Chapter 6). Thus, it was natural to modify empirical equation (16.35) by replacing/calculated from the density by h computed from the equation of states, as well as replacing h]q by its more general, constant stress homolog, )] [Utracki, 1974,1983a,b, 1985,1986 Utracki and Simha, 1981,1982,2001b Utracki andGhijsels, 1987] ... 

Flory-Orwoll-Vrij, 1964] (FOV), [Sanchez-Lacombe, 1976-8] (S-L), and [Simha-Somcynsky, 1969] (S-S). Large deviations (< 0.01 mL/g) were observed for S-G, the two following relationships were useful only at low P and over small P-ranges, whereas S-S consistently provided the best representation of data over extended ranges of T and P, with deviations < 0.003 mL/g, comparable to the experimental uncertainties. The FOV model can be expressed as ... 

Several attempts have been made to derive predictive relationships for the Newtonian viscosity of semidilute and concentrated polymer solutions. Simha and co-workers [Simha, 1952 Utracki and Simha, 1963 Simha and Somcynsky, 1965 Simha and Chan, 1971 Utracki and Simha, 1981] explored the possibility of developing a principle of corresponding states based on the c,M) scaling equivalent to the packing of hard spheres ... 

Somcynsky, T., and Simha, R., Hole theory of liquids and glass transition, J. Appl. Phys., 42, 4545 548 (1971). 

Equations of state derived from statisticai thermodynamics arise from proper con-figurationai partition functions formuiated in the spirit of moiecuiar modeis. A comprehensive review of equations of state, inciuding the historicai aspects, is provided in Chapter 6. Therefore, we touch briefly in oniy a few points. Lennard-Jones and Devonshire [1937] developed the cell model of simple liquids, Prigogine et al. [1957] generalized it to polymer fluids, and Simha and Somcynsky [1969] modified Pri-gogine s cell model, allowing for more disorder in the system by lattice imperfections or holes. Their equations of state have been compared successfully with PVT data on polymers [Rodgers, 1993]. 

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 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] ... 

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]. 

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. 

The main purpose of this example is to provide a very simple but still physically meaningful illustration of the Legendre time evolution introduced above. The physical system that we have in mind is a polymeric fluid. We regard it as Simha and Somcynski (1969) do in their equilibrium theory but extend their analysis to the time evolution. As the state variables we choose... 

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. 

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. 

It is not surprising that attempts have been made to derive equations of state along purely theoretical lines. This was done by Flory, Orwoll and Vrij (1964) using a lattice model, Simha and Somcynsky (1969) (hole model) and Sanchez and Lacombe (1976) (Ising fluid lattice model). These theories have a statistical-mechanical nature they all express the state parameters in a reduced dimensionless form. The reducing parameters contain the molecular characteristics of the system, but these have to be partly adapted in order to be in agreement with the experimental data. The final equations of state are accurate, but their usefulness is limited because of their mathematical complexity. 

A very interesting semi-empirical equation of state was derived by Hartmann and Haque (1985), who combined the zero-pressure isobar of Simha and Somcynsky (1969) with the theoretically derived dependence of the thermal pressure (Pastime and Warfield, 1981). This led to an equation of state of a very simple form ... 

Simha, R. Somcynsky, T., "On the Statistical Thermodynamics of Spherical and Chain Molecule Fluids," Macromolecules, 2, 342 (1969). 

Returning now to our more general discussion of cquations-of-state, note that a concise review of cquations-of-state for the PVT behavior of polymers was provided by Zollcr [39], who compared the ability of several equations-of-state (some developed empirically and some developed based on fundamental theoretical considerations) to represent PVT data for homopolymers, copolymers and polymer blends. He concluded [39] that, among theoretically-based equations-of-state, the Simha-Somcynsky equation [40,41] represents the available data best over extended ranges of temperature and pressure. 

Since this chapter is not intended to be a review of polymer thermodynamics, but to provide information on diverse thermodynamic aspects pertinent to polymer blends, only EoS derived by Simha and Somcynsky [1969], will be discussed in some detail. 

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