Lattice theories hole theory

C. Lattice Theories (Hole, Cell and Free Volume Theories).. 238... 

The model of Marchetti et al. is based on the compressible lattice theory which Sanchez and Lacombe developed to apply to polymer-solvent systems which have variable levels of free volume [138-141], This theory is a ternary version of classic Flory-Huggins theory, with the third component in the polymer-solvent system being vacant lattice sites or holes . The key parameters in this theory which affect the polymer-solvent phase diagram are ... 

This principle serves as the basis for a number of models of fused salt systems. Perhaps the best known of these is the Temkin model, which uses the properties of an ordered lattice to predict thermodynamic quantities for the liquid state [79]. However, certain other models that have been less successful in making quantitative predictions for fused salts may be of interest for their conceptual value in understanding room temperature ionic liquids. The interested reader can find a discussion of the early application of these models in a review by Bloom and Bockris [71], though we caution that with the exception of hole theory (discussed in Section II.C) these models are not currently in widespread use. The development of a general theoretical model accurately describing the full range of phenomena associated with molten salts remains a challenge for the field. 

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 GC-EOS based on a series expansion of the nonrandom lattice-hole theory is written by[l,6],... 

Sanchez and Lacombe (1976) developed an equation of state for pure fluids that was later extended to mixtures (Lacombe and Sanchez, 1976). The Sanchez-Lacombe equation of state is based on hole theory and uses a random mixing expression for the attractive energy term. Random mixing means that the composition everywhere in the solution is equal to the overall composition, i.e., there are no local composition effects. Hole theory differs from the lattice model used in the Flory-Huggins theory because here the density of the mixture is allowed to vary by increasing the fraction of holes in the lattice. In the Flory-Huggins treatment every site is occupied by a solvent molecule or polymer segment. The Sanchez-Lacombe equation of state takes the form... 

Figure 2.17. Surface tension of linear alkanes as a function of (a) temperature and (b) redueed temperature. Cp Cg, etc. refer to alkanes with 1, 2,. .. CH2 groups per chain. Lattice hole theory, redrawn after Schlangen et al. (J. Phys. Chem. 100 (1996) 3607.)...

Given the diversity of relevant applications, it is not surprising that the characterization of voids in disordered systems has an appreciable history, which can be traced back to primitive hole theories of the liquid state (Frenkel, 1955 Ono and Kondo, 1960). While the early theories offer an admittedly rudimentary lattice description of voids, recent computational advances permit an exact (and highly efficient) characterization of the continuum void geometry present in particle packings in two (Rintoul and Torquato, 1995) and three dimensions (Sastry et al., 1997a). 

We want to pursue the subject by starting with a brief review for the present purposes of the essentials of lattice-hole theory, then follow with a consideration of free-volume mobility connections, and continue with some comparisons of experiment versus theory. Finally, we propose and sketch modifications of the theory. These may open the way to generalizations and more insightful relations to empirical formulations, such as the KAHR model [Kovacs et al., 1977, 1979] for volume relaxation. 

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

The hole theories have been developed as improvements over the cell models. Similar to the Ising fluid model, holes were placed on the lattice. Their presence addressed... 

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... 

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

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. 

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... 

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. 

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 ... 

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