The Sanchez-Lacombe Lattice Fluid Theory

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 resulting equation of state for a pure component is given by Eq. (71). 

The reduced volume v, the reduced pressure p, and the reduced temperature f are defined by Eqs. (72a)-(72c). 

For mixtures the same equation of state is used, but the characteristic parameters r,e, and v are composition-dependent. Neau [65] gives an overview of different mixing rules proposed in the literature. Often-used mixing rules are given in Eqs. (73)-(75). In Eq. (75) k,j- is an adjustable binary interaction parameter which equals zero for i = j and which can fitted to binary experimental data. 

Neau [65] also showed that the expressions for the chemical potential used in earlier literature to calculate phase equilibria are thermodynamically inconsistent. According to Neau, the correct expression for the fugacity coefficient for the SL model is Eq. (77). 

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JI1 Jiang, S., An, L., Jiang, B., and Wolf, B. A., Pressure effects on the thermodynamics of fra 5-decahydronaphthalene/polystyrene polymer solutions apphcation of the Sanchez-Lacombe lattice fluid theory (experimeirtal data by S. Jiang), Macromol. Chem. Phys., 204, 692, 2003. 

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). 

Estimate the Hildebrandt solubility parameter using the Sanchez-Lacombe lattice fluid theory described in Chapter 2 and given as Equation... 

In this study we carried out doud-point measurements for binary system trans-decahydronaphthalene(TD)/ pwlystyrene (PS) in a pressure range up to 800bar to determine coexistence curves and critical lines. The purpose of this paper is to test whether the Sanchez-Lacombe lattice fluid theory (SLLFT) can describe the equilibrium behavior and pressure effects of polymer containing systems. The spinodals, the binodals, the FH interaction parameter, the enthalpy of the mixing, and volume change of the mixing for TD/PS system were calculated as a function of pressure, temperature and composition on the basis of the SLLFT. 

Sun et al. 1999, Phase-separation behavior of the system PES/phenoxy An application of the Sanchez-Lacombe lattice fluid theory, Macromol. ScL-Phys., Vol. 38, No. 1-2, PP. 67-74... 

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

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. 

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... 

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... 

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 and Lacombe supposed that in a binary polymer blend, free volume occupied Nq lattice sites, and the bulk polymer density p = NKN + Nq), where N = ILNiri and r, was the chain length of /th fraction, then they developed the lattice fluid theory to calculate Helmholtz free energy (Sanchez and Lacombe 1974 Sanchez 1978), as given by... 

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]. 

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]. 

A thermodynamic model meeting all the above requirements is presented in the next section. It is based on the Lattice-Fluid theory of Sanchez and Lacombe(7) as modified recently by the author (8-12).So far the model has been applied to solvent-homopolymer and homopolymer-homopolymer(both monodisperse) mixtures (1 0), to the gas solubility in polymeric liquids... 

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 ... 

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 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... 

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... 

In the lattice fluid theory, as formulated by Sanchez and Lacombe(Lacombe Sanchez, 1976 Sanchez Lacombe, 1976), the energy of mixing for binary polymer containing systems is related to the Gibbs energy per mer (indicated by the double bar) of the mixture (index M) and that of the pnire components (indexlor2) by... 

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. 

The decisive advantage of the original Elory-Huggins theory [1] lies in its simplicity and in its ability to reproduce some central features of polymer-containing mixtures qualitatively, in spite of several unrealistic assumptions. The main drawbacks are in the incapacity of this approach to model reality in a quantitative manner and in the lack of theoretical explanations for some well-established experimental observations. Numerous attempts have therefore been made to extend and to modify the Elory-Huggins theory. Some of the more widely used approaches are the different varieties of the lattice fluid and hole theories [2], the mean field lattice gas model [3], the Sanchez-Lacombe theory [4], the cell theory [5], different perturbation theories [6], the statistical-associating-fluid-theory [7] (SAET), the perturbed-hard-sphere chain theory [8], the UNIEAC model [9], and the UNIQUAC [10] model. More comprehensive reviews of the past achievements in this area and of the applicability of the different approaches are presented in the literature [11, 12]. 

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. 

Equation 19 is written for the mixture a polymer and Np low molecular weight components. In the lattice fluid theory of Sanchez and Lacombe, each species i is characterised by three parameters which refer to the given component in the pure phase the numter of cells occupied by a molecules of the species, r the characteristic volume of a lattice cell, Vj, and the characteristic energy, 6i. In equation 19 the subscript Np +i has been used to label polymeric properties so that and Mnp+i refer to the number of lattice cells occupied by a polymer molecule, and to the polymer molecular weight respectively. 

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