Sanchez-Lacombe equation

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.

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. 

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 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... 

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. 

NAG Nagy, I., Loos, Th.W.de, Krenz, R.A., and Heidemami, R.A., High pressure phase equilibria in the systems linear low density polyethylene -1- n-hexane and linear low density polyethylene + n-hexane + ethylene Experimental results and modelling with the Sanchez-Lacombe equation of state, J. Supercrit Fluids, 37,115,2006. 

ANG Angelis, M.G.de, Merkel, T.C., Bondar, V.I., Freeman, B.D., Doghieri, F., and Sartim G.C., Hydrocarbon and fluorocarbon solubility and dilation in poly(dimethylsiloxane) comparison of experimental data with predictions of the Sanchez-Lacombe equation of state,/. Polym. Sci. PartB Polym. Phys., 37, 3011, 1999. 

Low-molecular-weight species (solvent, initiator, and monomer) undergo very fast transport between the phases, and they are assumed to be at interphase thermodynamic equilibrium at all times. The Sanchez-Lacombe equation of state [43, 44] was used for monomer and solvent, while an oversimplified partition coefficient was assumed for the initiator. 

Finally, the pure-component parameters in the Sanchez-Lacombe equation of state can be evaluated to reproduce puredensity versus temperature data), whereas the mixture parameters are evaluated through selected mixing rales. These in turn involve one or two binary interaction parameters per component pair, which are usually obtained by fitting binary data (for example sorption data for the pair C02-polymer). 

As mentioned above, the model was adapted to continuous reactors (CSTR) in order to further vahdate its prediction ability. In particular, the effect of changing the inlet monomer concentration on the MWD was investigated. Simulations were carried out using the identical set of model parameter values already considered for the batch simulations above. However, because of the higher operating temperature in the CSTR reactions (see Table 6.5) with respect to the batch experiments, the binary interaction parameters used in the frame of the Sanchez-Lacombe equation of state have to be updated. This has been done... 

Figure 2. a.) CO2 sorption in polystyrene and poly(n-butyl methacrylate). The solid lines are calculated results using the Sanchez-Lacombe equation of state with binary interaction parameters, 6y of 0.05 and -0.004 for PS/CO2 and PnBMA/C02 respectively, b.) CO2 sorption in PS, PVME and a PS/PVME blend at 20.6 "C. Solid lines are S-L fits using a 6y of -0.04 for PVME/CO2 and 0.018 for the pseudo-binary blend/C02 system (see section 4 for details). 

Figure 8. Compressibility of PVME/COi, PS-PVME 50-50 blend/COi, and PS/PVME mixtures at 20 C calculated using the Sanchez-Lacombe equation of state.

Borstar is an industrial olefin polymerization plant/technology, which combines different polymerization processes and reactor units, utilizing an advanced catalytic system. In the present work, a detailed model for the dynamic and steady-state simulation of this industrial plant has been developed. A comprehensive kinetic model for the ethylene-1-butene copolymerization over a two-site catalyst was employed to predict the MWD and CCD in the Borstar process. The Sanchez-Lacombe equation of state (S-L EoS) was employed for the thermodynamic properties of the polymerization system and the phase equilibrium calculations in the process units. 

Phoenix and Heidemann developed a new computer algorithm, based on the work of Michelsen and Koak, to calculate the cloud-point and the shadow curve of polydisperse polymer solutions in the framework of continuous thermodynamics using the Sanchez-Lacombe equation of state. The method was tested with (hexane + poly ethene) and (ethene+ polyethene). To describe... 

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