Sanchez-Lacombe EOS

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)... 

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. 

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. 

For polymer systems, a lot of EOS have been developed (e.g., Flory-Orwoll-Vrij EOS [10], Sanchez-Lacombe EOS (11), Panayiotou-Vera EOS [12], lattice gas EOS [13], group-contribution lattice fluid EOS [14]). In all these equations, the pressure is introduced via empty lattice sites allowing the compressibility of the lattice. 

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) ... 

The pure component lattice fluid parameters for PMMA, CO2 and C2H4 have been obtained using the P-V-T data reported in (77) for the polymer and in (18) for the penetrants. The parameters values obtained in this work for the Sanchez-Lacombe EOS are recorded in Table I. They are in good agreement with the values reported in the literature for the same components (79). 

LIU Liu, D., Li, H., Noon, M.S., and Tomasko, D.L., C02-induced PMMA swelling and multiple thermodynamic property analysis using Sanchez-Lacombe EOS,... 

The characteristic pressure for a binary mixture for the Sanchez-Lacombe EOS is expressed as ... 

The Sanchez-Lacombe EOS has been applied to PMMA/SAN [29], polycarbonate (PC), tetramethyl polycarbonate (TMPC) andpoly(e-caprolactone) binary and ternary blends [30],... 

Flory et al. (1964) (FOV), Sanchez-Lacombe (1976,1977,1978) (S-L), Simha and Somcynsky (1969) (S-S), Prigogine et al. (1953, 1957) (P), Dee and Walsh (1988) (D-W), Hartmann and Haque (1985) (H-H), and Sanchez and Cho (1995) (S-C), and tabulated the respective corresponding state values P, V, and T ) for most common polymers. These comparisons span across the different types of EoS models, from cell models (FOV, P, D-W), to lattice-fluid (S-L) and hole (S-S) models, to semiempirical approaches (H-H, S-C), comparing the validity of distinctly different EoS approaches across large numbers of different homopolymers and copolymers. All reviews seem to build a consensus on the comparative accuracy of the various EoS Zoller (1989) reported large deviations (<0.01 mL/g)... 

The expression to define the spinodal curves in the phase diagram of binary blends was obtained from the criteria for equilibrium and stability and from the Gibbs free energy expression. EOS theory such as the lattice fluid theory of Sanchez-Lacombe... 

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... 

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. 

Within the framework of the Sanchez-Lacombe theory, we set up the following combining rules for the molecular weight and the scaling parameters for P(EO-b-DMS) block copolymer ... 

To estimate the change in volume (swelling), AVpoi, the Sanchez-Lacombe equation of state SL-EOS [17,33-35] (10,11) using the equation of DeAngelis [36] (12)... 

According to the classical nucleation theory, the free energy barrier for bubble nucleation and thereby the nucleation rate are functions of the bubble pressiu e, PbubUe-In computer simulations of polymeric foaming processes, almost all previous research has approximated the value of Pbubbie by the saturation pressure, P at- In this paper, the thermodynamic equilibrium condition and the Sanchez-Lacombe (SL) equations of state (EOS) are enployed to determine the value of PbubUe- It is shown that the Pbubbk approximation using Psat will lead to significant overestimations of the nucleation rate and the final cell density. 

In this work, the thermodynamic eqmhbrium condition and the Sanchez-Lacombe (SL) equation of state (EOS) are employed to determine the value of Pbubbie.o- By comparing the computer-simulated cell density using the thermodynamieally determined PbubUe,o, to that according to the approximated Pbubble, o with Psat, tiie impact of using... 

Since the size of a critical bubble is in the sub-micron level, statistical thermodynamic theories shoidd be employed to determine the chemical potential of the gas in the critical bubble. In this study, following the approach suggested by Li et. al. [16], the values of tiG.bubUe and Ra,soiution at specified values of T, P ys, and C are determined based on the Sanchez-Lacombe (SL) equation of state (EOS), which is expressed as [17-18] ... 

The Sanchez and Lacombe LF EoS considers a compressible lattice for the representation of microstates of pure fluids and fluid mixtures. Such a lattice is made of cells, whose volume depends on mixture composition, which can be either empty or occupied by molecular segments of the components considered. The statistical analysis of the possible combinations of molecules in the lattice and the evaluation of the energetic... 

Simha and Somcynsky 1969 Patterson 1969, 1982 Patterson and Robard 1978 Sanchez and Lacombe 1976, 1977 Sanchez 1983, 1984). The equation of state (EoS) theories of mixtures are based on the principles discussed in Sect. 2.4.1. Formally, the computation of the partition function for a single component or for a mixture of components is similar, yielding the Helmholtz free energy of mixing. 

A pure fluid is completely characterized by the three molecular parameters, v r, and , and the scale factors T P, and p. P v /PP = 1 and Mir = v p = PP p /P. In principle, any thermodynamic property can be utilized to determine these parameters. Saturated vapor pressure data is useful as they are readily available for a variety of fluids. A compendium of such data is available for organic liquids. The lattice fluid theory of Sanchez and Lacombe as described is similar to the van der Waals EOS as discussed in the earlier section for small molecules. The virial form of the EOS of Sanchez and Lacombe can be written as follows ... 

The characteristic parameters for Sanchez and Lacombe free volume EOS for selected polymers are given in Table 2.4. 

The EOS predictions of Sanchez and Lacombe are shown in Figure 2.5 for PMMA at 25°C at infinite molecular weight. The Sanchez and Lacombe (SL) EOS allows for the effect of polymer molecular weight in the EOS. The characteristic temperature values in Table 2.4 for the same set of polymers as indicated before for SL, EOS, are a lot lower compared to other EOS values. The proximity of the characteristic temperatures of polymers to operating temperatures in the molding and other polymer processing operations may indicate that this model may be a better representation of polymer PVT behavior compared with other models. 

Sanchez and Lacombe developed the lattice fluid EOS theory using statistical mechanics. Gibbs free energy can be expressed in terms of configurational partition function Z in the pressure ensemble. In the lattice fluid theory development the problem is to determine the number of configurations available for a system of N molecules each of which occupies r sites and vacant sites or holes. Mean field approximation was used to evaluate the partition function. The SL EOS has the capability to account for molecular weight effects, unlike other EOS theories. Characteristic lattice fluid EOS parameters were tabulated for 16 commonly used polymers. 

The EOS developed by Sanchez and Lacombe using lattice fluid theory... 

The EOS of Sanchez and Lacombe in terms of reduced pressure, temperature, and density is a transcendental equation. In combining this equation with the free energy of mixing equation from the binary interaction model, is a numerical solution needed or can an analytical solution be obtained ... 

The EOS developed by Sanchez and Lacombe using the lattice fluid theory can be used to obtain estimates of Hildebrandt solubility parameter, 8 . Equation (2.61) for isothermal compressibility, p, and volume expansivity, k, from the lattice fluid EOS for polymers is used to obtain an expression for 8 as... 

The technique of Axisymmetric Drop Shape Analysis-Profile ADSA-P) [10,11] was used for image analysis and experimental parameters. Surface or interfacial tensions were obtained by fitting the Laplace equation of capillarity from the shape and dimensions of the acquired axisymmetric menisci [12]. The value of surface tension was generated as a fitting parameter after a least square algorithm was employed to minimize the difference between experimental drop profiles and theoretical ones [13]. During this procedure, the density difference between polystyrene and carbon dioxide was an input parameter [14, 15, 16], which was determined by the Sanchez and Lacombe (S-L) equation of state (EOS). 

---