Sanchez-Lacombe lattice-fluid model

Optimizing solvents and solvent mixtures can be done empirically or through modeling. An example of the latter involves a single Sanchez-Lacombe lattice fluid equation of state, used to model both phases for a polymer-supercritical fluid-cosolvent system. This method works well over a wide pressure range both volumetric and phase equilibrium properties for a cross-linked poly(dimethyl siloxane) phase in contact with CO2 modified by a number of cosolvents (West et al., 1998). 

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]. 

For components solubilities in polymer particles, pressure in the process units and flash calculations in the Flash unit the Sanchez-Lacombe EoS is used. It is appropriate for polymer mixtures and derives from a lattice-fluid model (Kirby McHugh, 1999) ... 

Heidemann et al also presented a discontinuous method to calculate spinodal curves and critical points using two different versions of the Sanchez-Lacombe equation of state and PC-SAFT. Moreover, Krenz and Heidemann applied the modified Sanchez-Lacombe equation of state to calculate the phase behaviour of polydisperse polymer blends in hydrocarbons. In this analysis the polymer samples were represented by 100 pseudo-components. Taimoori and Panayiotou developed a lattice-fluid model incorporating the classical quasi-chemical approach and applied the model in the framework of continuous thermodynamics to polydisperse polymer solutions and mixtures. The polydispersity of the polymers was expressed by the Wesslau distribution. 

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... 

Lattice Fluid Mode/ The Lattice Fluid model developed by Sanchez and Lacombe (15,16) introduces vacancies into the classical incompressible Flory-Huggins model. The lattice vacancy is treated as a pseudo particle in the system. The free energy of an incompressible binary polymer solution is then converted to that of the bulk polymer. The equation of state for a polymer is given below ... 

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... 

Sanchez-Lacombe [1976,1978] The Sanchez and Lacombe [1976,1978] equation of state (S-L) is based on the Ising fluid model. The authors followed the Guggenheim [ 1966] approach, placing A -mers and No holes in an A -lattice. Hard-core volumes of the s-mer, as well as its flexibility, were assumed independent of Tand P. Furthermore, only the nearest neighbors of nonbonded mers contributed to the lattice energy ... 

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 Theoiy (SAFT) models to obtain a better representation. 

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