Sanchez-Lacombe

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). 

Figure 11.8. Viscoelastic scaling factor for the PDMS-C02 system predicted by eq. 11.1, with the Sanchez-Lacombe equation of state employed to evaluate specific volume, as calculated by Gerhardt et al. (1998). The solid line is the prediction at 50 °C, and the broken line is the prediction for 80 °C. The predicted curves are compared to the data of Gerhardt et al. (1997) A data for 50 °C data for 80 °C.

PEG (2) + ethanol (3) system. The ethanol to PEG6000 weight ratio is 95 5. Symbols are experimental cloud point compositions. Solid lines are determined using the Sanchez-Lacombe EOS. (Reprinted with permission from Ref. [9], with permission from Elsevier)... 

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... 

Costas et al. (1981) and Costas and Sanctuary (1981) reformulated the Sanchez-Lacombe equation of state so that the parameter r is not a regression parameter, but is actually the number of segments in the polymer molecule. In the original Sanchez-Lacombe treatment, r was regressed for several n-alkanes, and it was found that the r did not correspond to the carbon number of the alkanes. In addition, the Sanchez-Lacombe equation of state assumes an infinite coordination number. Costas et al. (1981) replaced the segment length r as an adjustable parameter with z. This modification involves the same number of adjustable parameters, but allows r to be physically significant. Thus, the model is more physically realistic, but there have been no definitive tests to determine whether this improves the correlative results from the model. 

The second major difference between the Panayiotou-Vera and the Sanchez-Lacombe theories is that Sanchez and Lacombe assumed that a random mixing combinatorial was sufficient to describe the fluid. Panayiotou and Vera developed equations for both pure components and mixtures that correct for nonrandom mixing arising from the interaction energies between molecules. The Panayiotou-Vera equation of state in reduced variables is... 

Xiong, Y, and Kiran, E., 1995. Comparison of Sanchez-Lacombe and SAFT model in predicting solubility of polyethylene in high-pressure fluids. J. Appl. Polym. Set, 55 1805-1818. 

Orbey, H., Bokis, C.P., and Chen, C.-C., Equation of state modeling of phase equilibrium in the low-density polyethylene process the Sanchez-Lacombe, Statistical Associating Fluid Theory, and the polymer-SRK equation of state, Ind. Eng. Chem. Res., 37, 4481, 1998. 

Binary interaction parameter in Sanchez-Lacombe equation-of-state theory. 

The Sanchez-Lacombe equation-of-state provides a good example to help clarify the rather abstract discussion given above. It will now be discussed further. It is given by Equation 3.26 for a pure molecular liquid or gas. The variable r is defined by Equation 3.27, where M is the molecular weight and R is the gas constant. If T, p and p are known, Equation 3.26 can be solved iteratively to estimate the density as a function of temperature and pressure. Since the reduced density p depends on M through the variable r defined by Equation 3.27, it is not equal for all molecules at the same combination of T and p values. Consequently, for ordinary molecules, the Sanchez-Lacombe equation-of-state is not a corresponding states theory. 

Figure 3.7. Reduced density as a function of reduced temperature and reduced pressure for polymers, calculated by using the Sanchez-Lacombe equation-of-state in the limit of infinite molecular weight where it becomes a corresponding states theory. Each curve is labeled by the value of the reduced pressure that was used in its calculation.

Now consider the mixture of a supercritical fluid or a gas with a polymer. The Sanchez-Lacombe equation-of-state treatment is described by the equations below, where the subscripts 1 and 2 refer to the gas and the polymer, respectively. The weight and the volume fractions of the gas (wj and Oj) and the polymer (W2 and O2) in the mixture add up to 1. The definition of the... 

The fit of P-x data for the acetone-propane binary system (Gomez-Nieto and Thodos, 1978) is shown in figure 5.9. The best fit of the Sanchez-Lacombe... 

Figure 5.9 Phase behavior of the acetone-propane system. The symbols represent experimental data (Gomez-Nieto and Thodos, 1978). The solid lines represent calculations with the Sanchez-Lacombe EOS with...

Figure 5.10 Comparison of calculated (lines) and experimental cloud point data (symbols) of the poly(ethylene-co-methyl acrylate) (69 mol%/ 31 mol%)-acetone-propane system (Hasch et al., 1993). The polymer concentration is fixed at 5wt%. The calculations are performed with the Sanchez-Lacombe EOS with kij and 17,y set equal to zero for the EMAt9/3i-acetone pair, kij = 0.030 and rj/y = 0.000 for the propane-acetone pair, and kij = 0.023 and 77,/ = -0.002 for the EMA soi-propane pair. The weight average and number average molecular weights of EMA69/31 are 58,900 and 31,000, respectively.

The Sanchez-Lacombe EOS is a mean-field equation that does not directly account for hydrogen bonding and polar interactions. But Sanchez and Balazs (1989) show that it is possible to mimic the trends in the experimental data if the mixture parameters are allowed to vary with temperature. Therefore, when dealing with polar polymers or polar solvents, it may be necessary to force the mixture parameters to vary with temperature to obtain a representative fit of experimental data. In some cases, improved fits of the Sanchez-Lacombe equation to experimental data can also be obtained if the characteristic parameters of the solvent and the solute are obtained by fitting P-V-T data in the region where the mixture data were obtained. The improved fit of mixture data with characteristic parameters of the light component obtained in this manner is usually at the expense of a poor fit of the vapor pressure curve. 

Calculating Binary, Vapor-Liquid Equilibria Using the Sanchez-Lacombe Equation of State... 

Subroutine SLCHEMPOT is used to calculate the chemical potentials of the components with the Sanchez-Lacombe EOS. The line numbers start from 1 in this subroutine. 

Kiszka, M. B., M. A. Meilchen, and M. A. McHugh. 1988. Modeling high-pressure gas-polymer mixtures using the Sanchez-Lacombe equation of state. J. Appl. Polym. Sci. 36 583. 

Pottinger, M. T., and R. L. Laurence. 1984. The PVT behavior of polymeric liquids represented by the Sanchez-Lacombe equation of state. J. Polym. Sci. Polym. Phys. Ed. 22 903-907. 

Flory-Orwoll-Vrij, 1964] (FOV), [Sanchez-Lacombe, 1976-8] (S-L), and [Simha-Somcynsky, 1969] (S-S). Large deviations (< 0.01 mL/g) were observed for S-G, the two following relationships were useful only at low P and over small P-ranges, whereas S-S consistently provided the best representation of data over extended ranges of T and P, with deviations < 0.003 mL/g, comparable to the experimental uncertainties. The FOV model can be expressed as ... 

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