Simha lattice-hole theory

Utracki, L. A., and Simha, R., Analytical representation of solutions to lattice-hole theory, Macrvmol. Theory Simul., 10, 17-24 (2001). 

Simha, R., and Xie, H., Applying lattice-hole theory to gas solubility in polymers, Polym. Bull., 40, 329-335 (1998). 

Consolati, G., Quasso, R, Simha, R., and Olson, G. B., On the relation betwen positron annihilation lifetime spectroscopy and lattice-hole-theory free volume, J. Polym. Sci. B, 43, 2225-2229 (2005). 

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 ... 

Jain, R. K., Simha, R., Lattice-hole theory Bulk properties and surface tension of oligomers and polymers, Journal of Colloid and Interface Science, 216(2), pp. 424 28 (1999). 

Simha, R., Polymer and oligomer melts, thermodynamics, correlations, and lattice-hole theory. Polymer Engineering and Science, 36(12), pp. 1567-1573 (1996). 

To improve on the cell model, two other classes of models were developed, namely, lattice-fluid and lattice-hole theories. In these theories, vacant cells or holes are introduced into the lattice to describe the extra entropy change in the system as a function of volume and temperature. The lattice size, or cell volume, is fixed so that the changes in volume can only occur by the appearance of new holes, or vacant sites, on the lattice. The most popular theories of such kind were developed by Simha and Somcynsky or Sanchez and Lacombe. ... 

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... 

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]. 

Simha, Robertson, and their coworkers have pursued this line of attack with commendable results, using the Simha-Somcynsky hole theory [21] (S-S) as a starting point. The theory uses the formalism of a polymer lattice in which vacant cells or holes constitute the free volume arising from inefficient chain packing. An equation of state was developed to calculate the fraction of occupied lattice sites and hence the fractional free volume. 

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. 

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. 

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... 

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. 

The S-S theory describes the structure of a liquid by a lattice model with cells of the same size and a coordination number of z = 12. The disordered structure of the liquid is modeled by allowing an occupied lattice-site fraction y=y(V,73 of less then 1. The configurational or Helmholtz free energy, F, is expressed in terms of the volume V, temperature 7] and occupied lattice-site fraction y = y(K7), F=F(V,T,y). The value of y is obtained through the pressure equation P = —(9F/9V)r and the minimization condition (dFldy)v,T = 0. The hole fraction is given by the fraction of unoccupied lattice sites (holes or vacancies), which is denoted by h, h(P,T) = —y(P,T). This theory provides an excellent tool for analyzing the volumetric behavior of linear macromolecules but was also applied successfully to nonlinear polymers, copolymers, and blends. Several universal relationships where found which allow an approximate estimation of the fraction of the hole (or excess) free volume h and the total or van der Waals free volume/ [Simha and Carri, 1994 Dlubek and Pionteck, 2008d]. For more details, see Chapters 4, 6, and 14. 

The invariant hole size and energy of hole formation in the Lacombe-Sanchez model implies a dependance of the internal energy on the density that is strictly of the van der Waals form, as in Flory s theory. Another, somewhat more complicated, model developed by Simha and his collaborators is similar to that of Sanchez and Lacombe in its use of a liquid lattice with vacant sites, but it also retains features of Prigogine s earlier cell model in the c parameter for external degrees of freedom and a lattice energy with a density dependence based on an effective (6-12) pair potential. Like the other theories, this one has been successful in correlating equation of state data both for neat polymer liquids and for nondilute solutions. 

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