Sanchez-Lacombe theory lattice fluid model

Redlich-Kwong equation of state and Soave modification Peng-Robinson equation of state Tait equation for polymer liquids Flory, Orwoll, and Vrij models Prigogine square-well cell model Sanchez-Lacombe lattice fluid theory... 

Blends of various homopolymers with AMS-AN copolymers were systanatically examined for miscibility and phase separation temperatures in cases where LCST behavior was detected. The experimental data was used to calculate the interaction energy using the Sanchez-Lacombe lattice fluid equation of state theory. The analysis assumes that the experimental phase separation temperatures are represented using the spinodal curve and that the bare interaction energy density AP was found to be independent of temperature. Any dependence on the B interaction parameter with temperature stems frtrni compressibility effects. AP was determined as a function of cqxtlymer composition. AP,y values obtained for blends of the various homopolymers with AMS-AN copolymers woe then compared with corresponding ones obtained from SAN copolymers. Hwy-Huggins values were calculated from the experimental miscibility limits using the binary interaction model for comparison with AP,y values. 

Like the Flory-Huggins model, the Sanchez-Lacombe lattice fluid theory is based on the assumption that segments of solvent molecules and polymer molecules occupy the lattice sites of a rigid lattice, but vacant lattice sites are also allowed. The number of vacant lattice sites, and as a consequence the total number of lattice sites, are pressure-dependent, and in this way compressibility is introduced. 

The other equation of state model widely noted is the Sanchez-Lacombe lattice fluid theory [26-28]. The Sanchez-Lacombe equation of state is ... 

The phase behavior of polymer/SCF mixtures can be described using versions of the lattice fluid (LF) model such as that developed by Sanchez and Lacombe [17]. The LF equation of state is relatively simple, and has been successfully used to describe either polymers dissolved in SCFs, or SCFs dissolved in polymers [18,19], including phenomena such as retrograde vitrification. The statistical associating fluid theory (SAFT) [20] can also describe the phase behavior of polymers dissolved in SCFs. The SAFT model, while somewhat more cumbersome to implement than the LF model, is especially well-suited for polymers with varying backbone architecture, such as branched polymers or copolymers. Both the Sanchez-Lacombe and SAFT models have been incorporated into commercially available modeling software [21]. 

Nine different equations-of-state, EOS theories are described including Flory Orwoll Vrij (FOV) Prigogine Square Well cell model, and the Sanchez Lacombe free volume theory. When the mathematical complexity of the EOS theories increases it is prudent to watch for spurious results such as negative pressure and negative volume expansivity. Although mathematically correct these have little physical meaning in polymer science. The large molecule effects are explicitly accounted for by the lattice fluid EOS theories. The current textbooks on thermodynamics discuss... 

Recent advances in statistical thermodynamics and better understanding of intra- and intermo-lecular interactions, thanks to accurate experimental measurements and molecular simulations using realistic force fields, have contributed significantly to this end. Many of the recent thermodynamic models based on statistical mechanics are rooted in the pioneering work of Guggenheim [1] and Flory [2] on lattice models for complex fluids, including polymers. The lattice fluid (LF) theory of Sanchez and Lacombe [3,4] is probably one of the most widely used and successful lattice models. 

The Sanchez-Lacombe model [48-50] is a lattice fluid model in which each component is divided into parts that are placed in a lattice. The different parts are allowed to interact with a mean-field intermolecular potential. By introducing an appropriate number of vacant sites (holes) in the lattice, the correct solution density can be obtained. SAFT [51-53] is based on the perturbation theory. The principle of perturbation theory is that first a model is derived for some idealized fluid with accurately known properties, called the reference fluid . Subsequently, the properties of this model are related to those of a real dense fluid. By expanding this reference fluid into power series over a specified parameter, the power terms can be regarded as corrections or "perturbations for the reference fluid as compared to reality. Obviously, the more the reference model approaches reality, the smaller the corrections are. Therefore, the key issue for applying perturbation theory is deriving the most suitable reference fluid. 

Equation 27 represents the basic equation for the NELF model based on the Sanchez and Lacombe lattice fluid theory it provides the explicit dependence of the chemical potential of each penetrant species of a multicomponent mixture on temperature, volume and composition. In view of equation 12 and equation 14 at given temperature, volume and composition this equation is valid for any pressure... 

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... 

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 polymer solutions warrant use of a special class of lattice models such as Florry-Huggins. For correlation purposes Sanchez-Lacombe method is sufficient but one may also use Statistical Association Fluid Theory (SAFT) models to obtain a better representation. 

A number of equation of state theories have been used to model phase behavior of polymers in supercritical fluids. For example the lattice-fluid theory of Sanchez and Lacombe[4U 42] includes holes on the lattice in order to model compressibility. The lattice-fluid theory has been applied to model phase behavior of both homopolymers and copolymers in supercritical fluids[32, 38, 43, 44]. The statistical associating fluid theory (SAFT)[43,45-48] and corresponding state models[49] have also been employed to model compressible polymer-solvent mixtures. Figure 1 gives the pressure-concentration phase diagram for poly(dimethyI siloxane) in CO2 modeled with the lattice-fluid equation of state[50]. 

A thermodynamic model meeting all the above requirements is presented in the next section. It is based on the Lattice-Fluid theory of Sanchez and Lacombe(7) as modified recently by the author (8-12).So far the model has been applied to solvent-homopolymer and homopolymer-homopolymer(both monodisperse) mixtures (1 0), to the gas solubility in polymeric liquids... 

Sanchez and Lacombe [15] developed a molecular theory of classical fluids based on a well-defined statistical mechanical model. The model fluid reduces to the classical lattice gas in one special case. It can be characterized as an Ising or lattice fluid. The model fluid undergoes a liquid-vapor transition. Only three parameters are required to describe a fluid. These parameters have been determined and tabulated for several fluids. 

There is growing interest in what have been called lattice gas (fluid) models. These envisage a fluid to be a mixture of molecules and holes. In essence they are lattice-graph models in which some of the lattice sites are occupied while others remain empty (holes). Originally introduced by Sanchez and Lacombe they have been more recently developed by them - in terms of an equation-of-state approach (see p. 305). Such models offer an attractive and combinatorially transparent alternative to free volume (holes) extensions of corresponding states theories (see next section), which have been much described by Dayantis. Thornley and Shepherd comment that preliminary results using this model indicate that it might be the most accurate so far . 

The decisive advantage of the original Elory-Huggins theory [1] lies in its simplicity and in its ability to reproduce some central features of polymer-containing mixtures qualitatively, in spite of several unrealistic assumptions. The main drawbacks are in the incapacity of this approach to model reality in a quantitative manner and in the lack of theoretical explanations for some well-established experimental observations. Numerous attempts have therefore been made to extend and to modify the Elory-Huggins theory. Some of the more widely used approaches are the different varieties of the lattice fluid and hole theories [2], the mean field lattice gas model [3], the Sanchez-Lacombe theory [4], the cell theory [5], different perturbation theories [6], the statistical-associating-fluid-theory [7] (SAET), the perturbed-hard-sphere chain theory [8], the UNIEAC model [9], and the UNIQUAC [10] model. More comprehensive reviews of the past achievements in this area and of the applicability of the different approaches are presented in the literature [11, 12]. 

Several other equation-of-state models have been proposed The lattice-fluid theory of Sanchez and Lacombe (1978), the gas-lattice model proposed by Koningsveld (1987), the strong interaction model proposed by Walker and Vause (1982), and the group contribution theory proposed by Holten-Anderson (1992), etc. These theories are reviewed by Miles and Rostami (1992) and Boyd and Phillips (1993). The lattice-fluid theory of Sanchez and Lacombe has similarities with the Flory-Huggins theory. It deals with a lattice, but with the difference from the Flory—Huggins model in that it allows vacancies in the lattice. The lattice is compressible. This theory is capable of describing both UCST and LCST behaviour. 

One early considered approach was to extend Flory Huggins-like lattice models by introducing empty lattice sites (holes) so that the number of holes in the lattice is a measure of the density of the system. Density changes in the system are realized via a variation of the hole number. Equations of state based on this idea are, for example, the Lattice-Fluid Theory from Sanchez and Lacombe [4] and the Mean-Field Lattice-Gas theory from Kleintjens and Koningsveld [5]. 

Contemporary Approaches. Numerous advanced theories have been formulated in the last decades to reproduce or even predict experimental findings for polymer containing mixtures. Most of them are particularly suitable for the description of some phenomena and special kinds of systems, but all have in common that they have lost the straightforwardness characterizing the Flory-Huggins theory. The following, incomplete collocation states some of the wider used approaches These are the different forms of the lattice fluid and hole theories (38), the mean field lattice gas model (39), the Sanchez-Lacombe theory(40), the cell theory (41), various perturbation theories (42), the statistical-associating-fluid-theory (43) (SAFT), the perturbed-hard-sphere chain theory (44), the... 

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