Mean field lattice gas model

A number of theoretical models have been proposed to describe the phase behavior of polymer—supercritical fluid systems, eg, the SAET and LEHB equations of state, and mean-field lattice gas models (67—69). Many examples of polymer—supercritical fluid systems are discussed ia the Hterature (1,3). 

To illustrate the complexity of the phase behavior in a more compact way it is instructive to employ a mean-field lattice-gas model. The relative simplicity of the grand potential... 

KEN Kennis, H.A.J., Loos, Th.W. de, DeSwaan Arons, J., Van der Haegen, R., and Kleintjens, L.A., The influence of nitrogen on the liquid-hquid phase behavioiu of the system n-hexane-polyethylene experimental results and predictions with the mean-field lattice-gas model, (experimental data by Th.W. de Loos), Chem. Eng. Sci., 45, 1875, 1990. 

KLE Kleintjens, L.A., van der Haegen, R., van Opstal, L., and Koningsveld, R., Mean-field lattice-gas modelling of supercritical phase behavior, J. Supercrit. Fluids, 1, 23, 1988. 

The mean-field lattice gas model is a molecular model for small-molecular and macromolecular systems that can reliable predict thermodynamic equilibrium properties. The model contains several adaptable empirical parameters for pure substances and for the mixtures, whereas no mixing rules are involved. The use of critical conditions in the adaption of parameters for binary mixtures to experimental data is discussed here, taking as an example the vapour-1iquid critical behaviour of the system ethylene-naphtalene. 

The introduction of a relevant expression for the critical determinant in the mean-field lattice gas model for binary systems is discussed here. It leads to an alternative and thermodynamic consistent method of adjusting two-particle interaction functions to experimental critical binary 1iquid-vapour densities. The present approach might lead to new developments in the determination of MFLG parameters for the mixture in small-molecule mixtures and in polymer solutions and polymer mixtures (blends). These relevant critical conditions appear because of the extra constraint, which is the equation of state, put on the hole model, and are... 

For polymer systems it is common to use rigid lattice models (29-32). However, since in SFE the phases differ in density such treatments are unsuitable. Lattice gas description (32-36) can circumvent this problem and combine the advantage of lattice statistics with the flexibility of equation of state models. Fig. 2 shows an example of the phase behaviour for the system linear polyethylene/n-hexane at nearcritical conditions, calculated with the mean-field lattice-gas model (35). 

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]. 

The mean-field lattice gas model (MFLG) represents a pure component by a lattice, the sites of which are randomly occupied by i moles of molecules. The molecules are allowed to occupy rrii lattice sites each the volume per lattice site, vq, is kept constant. The total number of sites equals q- - iWi(= where 0 is the amount of vacant sites in moles. Pressure and temperature changes affect the density of the system and can be dealt with by appropriate variations of Hq. In one of the more elaborate versions of the model (see, e.g., ref [53]), the Helmholtz free energy, AA, of mixing Hq vacant sites with i moles of molecules is... 

Contemporary Approaches. Numerous advanced theories have been formulated in the last decades to reproduce or even predict experimental findings for polymer containing mixtures. Most of them are particularly suitable for the description of some phenomena and special kinds of systems, but all have in common that they have lost the straightforwardness characterizing the Flory-Huggins theory. The following, incomplete collocation states some of the wider used approaches These are the different forms of the lattice fluid and hole theories (38), the mean field lattice gas model (39), the Sanchez-Lacombe theory(40), the cell theory (41), various perturbation theories (42), the statistical-associating-fluid-theory (43) (SAFT), the perturbed-hard-sphere chain theory (44), the... 

In many production routes, and also during processing, polymer systems have to undergo pressure. Changes in the volume of a system by compression or expansion, however, cannot be dealt with in rigid-lattice-type models. Thus, non-combinatorial free volume ( equation of state ) contributions to AG have been advanced [23 - 29]. Detailed interaction functions have been suggested (but all of them are based on adjustable parameters, for blends, e.g., Mean-field lattice gas [30], SAFT [31], specific interactions [32]), and have been succesfully applied, for example, by Kennis et al. [33]. 

Lacombe model) [53] and the Mean-Field Lattice-Gas theory [54]. These two approaches were also successfully appHed to polymer/carbon dioxide systems (see, e.g., [24, 28, 45, 52, 55, 56]). However, to achieve a quantitative description, a large set of parameters, most of them temperature dependent, has to be determined. 

Modelling Separators. In many polymerization processes pressure has been applied for several decades to control the thermodynamic state in the reactor as well as to effect downstream separations of products and reactants. As is shown above, recent developments in thermodynamic modelling, such as the Statistical Associating Fluid Theory (SAFT) and the Mean Field Lattice Gas (MFLG) model, make it possible to draw up adequate correlations of the influence of p,T, overall composition and copolymer composition on the phase behaviour, even in systems involving many components. 

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-Huid Theory firom Sanchez and Lacombe [4] and the Mean-Field Lattice-Gas theory firom Kleintjens and Koningsveld [5]. 

In the past several theoretical studies have been concerned with the mutual solubility of linear polyethylene and M-alkanes. In the course of such investigations phase behavior, or pVT relations, of pure n-alkanes has to be dealt with. In the following, three of such models will be discussed briefly Flory s Equation of State theory (EoS), the Mean-Field Lattice Gas (MFLG) model, and the Simha-Somcynsky (SS) theory. 

Coleman ST, McKinnon WR, Dahn JR (1984) Lihtium intercalation in LixMogSeg a model mean-field lattice gas. Phys Rev B 29 4147 149... 

The simplest treatment of the lattice-gas model is through the mean-field or randommixing approximation, which is treated in a number of textbooks (see, e.g.. Refs. 1 and 4). We give a short summary of its application to liquid-liquid interfaces, since it nicely illustrates under what conditions the phases separate. 

The lattice gas model of Bell et al. [33] neither gave any detailed mechanism of the orientational ordering nor separated the contributions of the headgroup and the acyl chain. Lavis et al. [34] discussed Ref. 33 critically and concluded that the sharp kink point in the isotherm at transition was an artifact of the mean field approximation used. An improved correspondence to experimental data was claimed by the use of the real-space renormalization group method [35]. The same authors returned to the problem [35] and concluded that in addition to the orientation of the molecules, chain melting had to be included in a model which could interpret the phase transitions. 

Although the simple mean-field expression (Eqn (7.10)) for a lattice-gas model has been used to understand intercalation systems qualitatively... 

The results shown in Figures 2 and 4 are intuitively obvious, and reflect the well known fact that the critical temperature in the system depends primarily on the strength of molecular interactions. In particular, in the lattice gas models the maximum of Tc is reached for the system of particles characterized by a" corresponding to the highest interaction between adsorbed particles. This can be readily demonstrated by considering the prediction of a very simple mean-field theory in the Bragg-Williams approximation. 

Figure 9.9 Adsorption isotherms for a lattice gas model consisting of shell and axial sites at the indicated temperatures (reduced by the pair interaction well depth) and various values of the reduced chemical potential. While mean field (MF) results exhibit a discontinuous shell-filling transition at T = 1, essentially exact Monte Carlo (MC) results show a near discontinuity there. (Adapted from Ref. [31, 32])...

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