Sanchez-Lacombe lattice fluid equation state

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). 

The Sanchez-Lacombe lattice-fluid equation of state reads ... 

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. 

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... 

Find the isentropic volume expansivity for systems described using the Sanchez-Lacombe lattice fluid theory equation of state. The isentropic volume expansivity as defined by Equation (2.80). 

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

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]. 

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 ... 

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... 

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. 

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. 

The model appears to describe accurately sorption isotherms when the equation of state parameters of both polymer and penetrant are determined. Like the Flory-Huggins modef, the Sanchez-Lacombe model assumes that the different components mix randomly in a lattice. Unlike the Flory-Huggins model, the Sanchez-Lacombe model permits some lattice sites to be empty, which allows holes or free volmne in the fluid. The addition of free volume to the lattice permits volume changes upon mixing components. The amount of absorbed penetrant in the polymer is determined by equating the chemical potential of the penetrant and the chemical potential of the penetrant in the mixture and by satisfying the equation of state of the pure penetrant phase and of the polymer-penetrant mixture. At fixed temperature and pressure, these conditions are met by equations 5-7. 

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]. 

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 ... 

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. 

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 . 

Another variant of the lattice model for fluids and fluid mixtures has been developed by Lacombe and Sanchez.In their treatment the fluid lattice contains empty sites (holes), whose equilibrium concentration is temperature dependent. Exchange interaction energies are taken proportional to contact surfaces, as in Flory s theory, but hole-hole and hole-mer interactions are zero. Close-packed molecular volumes, independent of temperature and pressure, appear in place of hard-core volumes. The holes provide the free volume, and there is no need to introduce a separation of internal and external molecular degrees of freedom or a parameter c. These considerations lead to a reduced equation of state for an r-mer fluid in the form... 

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