Recent Fluid Mixture Theory

Recent Fluid Mixture Theory.—The system CF4 + CH4 is at present unique among binary mixtures because (i) it is composed of simple non-polar spherically symmetric molecules, (ii) the system exhibits considerable deviations from ideality, and (iii) and x known explicitly and with high accuracy. G and F have been measured with high precision and (and hence TS ) can be estimated with somewhat lower precision. This binary system has therefore been used to assess the merits of various rival statistical theories of fluid mixtures. Some of these theoretical predictions are shown in Table 3. It is seen 

The next three theories, listed in Table 3, all make use of the so-called van der Waals approximation to relate the interaction parameters for the mixture to the parameters associated with the individual like and unlike interactions. They differ mainly in their choice of equation of state. The theory of Leland, Rowlinson, and Sather used the experimental equation of state of methane whereas the two other theories use explicit analytic equations of state, those of van der Waals and Guggenheim respectively. The LRS theory appears to 

The Flory theory bears certain similarities to the van der Waals and Guggenheim theories in that it makes use of an explicit equation of state. It can be seen from the results given in Table 3 that, although this theory is quite successful in predicting G , the theoretical values of F and are too low by a factor of more than two. 

Although certain of the above-mentioned theories are moderately successful in representing the experimental data of CF4 -t- CH and other fluorocarbon + hydrocarbon mixtures, experimental values of and x are required. At present there is no satisfactory method of obtaining these parameters a priori. Scott, in his 1958 review, considered the various possible factors that could lead to a weakening of the unlike interactions in such mixtures. He concluded that the three most significant were the presence of non-central forces, differences in ionization potential, and differences in size of the two component molecules. The use of the Kihara potential together with the Hudson and McCoubrey rule takes account of all these effects and thus the undoubted success of the Knobler treatment is not surprising. Criticisms could be levelled at his use of a spherically symmetric potential for substances such as n-hexane but the use of a more realistic potential such as the Kihara line-core potential is hardly justified until reliable experimental values for the ionization potentials of the fluorocarbons become available. 

Scott s statement made 16 years ago that a large amount of information has been obtained but we are yet far from a satisfactory explanation for all of it , is still pertinent. Our understanding of these systems has increased considerably over the intervening years but a wholly convincing quantitative theory is still lacking. 

---

Several colloidal systems, that are of practical importance, contain spherically symmetric particles the size of which changes continuously. Polydisperse fluid mixtures can be described by a continuous probability density of one or more particle attributes, such as particle size. Thus, they may be viewed as containing an infinite number of components. It has been several decades since the introduction of polydispersity as a model for molecular mixtures [73], but only recently has it received widespread attention [74-82]. Initially, work was concentrated on nearly monodisperse mixtures and the polydispersity was accounted for by the construction of perturbation expansions with a pure, monodispersive, component as the reference fluid [77,80]. Subsequently, Kofke and Glandt [79] have obtained the equation of state using a theory based on the distinction of particular species in a polydispersive mixture, not by their intermolecular potentials but by a specific form of the distribution of their chemical potentials. Quite recently, Lado [81,82] has generalized the usual OZ equation to the case of a polydispersive mixture. Recently, the latter theory has been also extended to the case of polydisperse quenched-annealed mixtures [83,84]. As this approach has not been reviewed previously, we shall consider it in some detail. 

All three areas will be addressed here. The application of classical density functional theory has led to some of the most important recent theoretical advances in SFE and these have been the subject of several authoritative review articles [10-16]. On the other hand, we know of no recent comprehensive review addressing theoretical approaches other than density functional theories (DFT) and the other two subject areas, particularly the last one, and it was this that motivated us to write this chapter. We hope that the somewhat broader coverage of molecular modeling research in SFE given in this chapter will be of benefit to researchers new to the field. We should mention that this Chapter is written from a perspective that is more strongly influenced by liquid-state statistical mechanics than by solid-state theory. The interests of the authors in the problem at hand are an outgrowth of their previous work on phase equilibrium in fluids and fluid mixtures. 

The first and second order perturbation contributions could be evaluated by using rigorous expressions for and gjj as obtained from an integral equation theory. Such an approach has been recently undertaken with good results [301, 303], Unfortunately, the expressions are quite lengthy and somewhat inconvenient for further differentiation. For this reason, we will evaluate the perturbative contributions of the free energy by using a Van der Waals like one fluid approximation. In this approximation, one considers that the radial distribution function of the Lennard-Jones fluid mixture may be expressed in terms of the radial distribution function of a pure effective Lennard-Jones fluid with composition dependent parameters, try and ey, yet to be determined. More specifically, one assumes that gij (r) may be expressed in terms of the radial distribution function of a pure fluid as follows. 

In Sec. 3 our presentation is focused on the most important results obtained by different authors in the framework of the rephca Ornstein-Zernike (ROZ) integral equations and by simulations of simple fluids in microporous matrices. For illustrative purposes, we discuss some original results obtained recently in our laboratory. Those allow us to show the application of the ROZ equations to the structure and thermodynamics of fluids adsorbed in disordered porous media. In particular, we present a solution of the ROZ equations for a hard sphere mixture that is highly asymmetric by size, adsorbed in a matrix of hard spheres. This example is relevant in describing the structure of colloidal dispersions in a disordered microporous medium. On the other hand, we present some of the results for the adsorption of a hard sphere fluid in a disordered medium of spherical permeable membranes. The theory developed for the description of this model agrees well with computer simulation data. Finally, in this section we demonstrate the applications of the ROZ theory and present simulation data for adsorption of a hard sphere fluid in a matrix of short chain molecules. This example serves to show the relevance of the theory of Wertheim to chemical association for a set of problems focused on adsorption of fluids and mixtures in disordered microporous matrices prepared by polymerization of species. 

Panagiotopoulos et al. [16] studied only a few ideal LJ mixtures, since their main objective was only to demonstrate the accuracy of the method. Murad et al. [17] have recently studied a wide range of ideal and nonideal LJ mixtures, and compared results obtained for osmotic pressure with the van t Hoff [17a] and other equations. Results for a wide range of other properties such as solvent exchange, chemical potentials and activity coefficients [18] were compared with the van der Waals 1 (vdWl) fluid approximation [19]. The vdWl theory replaces the mixture by one fictitious pure liquid with judiciously chosen potential parameters. It is defined for potentials with only two parameters, see Ref. 19. A summary of their most important conclusions include ... 

Efforts have been made to develop EOS for detonation products based on direct Monte Carlo simulations instead of on analytical approaches.35-37 This approach is promising given recent increases in computational capabilities. One of the greatest advantages of direct simulation is the ability to go beyond van der Waals 1-fluid theory, which approximately maps the equation of state of a mixture onto that of a single component fluid.38... 

The equations for conservation of mass, momentum, and energy for a one-component continuum are well known and are derived in standard treatises on fluid mechanics [l]-[3]. On the other hand, the conservation equations for reacting, multicomponent gas mixtures are generally obtained as the equations of change for the summational invariants arising in the solution of the Boltzmann equation (see Appendix D and [4] and [5]), One of several exceptions to the last statement is the analysis of von Karman [6], whose results are quoted in [7] and are extended in a more recent publication [8] to a point where the equivalence of the continuum-theory and kinetic-theory results becomes apparent [9]. This appendix is based on material in [8]. 

Equations of state offer a number of advantages over activity coefficient models for example, they can be applied to both low and high pressures, for properties other than phase equilibria, and the density is not required as an input parameter. However, often they are more difficult to develop for complex fluids and mixtures than are activity coefficient models. Very many equations of state have been proposed for polymers Section 16.7 discusses the reason. Recent reviews have been presented. " " We will not attempt to cover all the various approaches, but essentially discuss in detail only two of them, which seem promising for polymer solutions and blends the cubic equations of state and the SAFT (Statistical Associating Fluid Theory) method. 

The application of this approach to the hard-sphere system was presented by Ree and Hoover in a footnote to their paper on the hard-sphere phase diagram. They made a calculation where they used Eq. (2.27) for the solid phase and an accurate equation of state for the fluid phase to obtain results that are in very close agreement with their results from MC simulations. The LJD theory in combination with perturbation theory for the liquid state free energy has been applied to the calculation of solid-fluid equilibrium for the Lennard-Jones 12-6 potential by Henderson and Barker [138] and by Mansoori and Canfield [139]. Ross has applied a similar approch to the exp-6 potential. A similar approach was used for square well potentials by Young [140]. More recent applications have been made to nonspherical molecules [100,141] and mixtures [101,108,109,142]. 

For the calculations, different EoS have been used the lattice fluid (LF) model developed by Sanchez and Lacombet , as well as two recently developed equations of state - the statistical-associating-fluid theory (SAFT)f l and the perturbed-hard-spheres-chain (PHSC) theoryt ° . Such models have been considered due to their solid physical background and to their ability to represent the equilibrium properties of pure substances and fluid mixfures. As will be shown, fhey are also able to describe, if not to predict completely, the solubility isotherms of gases and vapors in polymeric phases, by using their original equilibrium version for rubbery mixtures, and their respective extensions to non-equilibrium phases (NELF, NE-SAFT, NE-PHSC) for glassy polymers. 

Finally, we mention that very recently three other integral equation approaches to treating polymer systems have been proposed. Chiew [104] has used the particle-particle perspective to develop theories of the intermolecular structure and thermodynamics of short chain fluids and mixtures. Lipson [105] has employed the Born-Green-Yvon (BGY) integral equation approach with the Kirkwood superposition approximation to treat compressible fluids and blends. Initial work with the BGY-based theory has considered lattice models and only thermodynamics, but in principle this approach can be applied to compute structural properties and treat continuum fluid models. Most recently, Gan and Eu employed a Kirkwood hierarchy approximation to construct a self-consistent integral equation theory of intramolecular and intermolecular correlations [106]. There are many differences between these integral equation approaches and PRISM theory which will be discussed in a future review [107]. 

Some thermodynamic aspects of polymer-polymer compatibility are discussed in this work.The analysis is made with the Lattice-Fluid theory of polymer solutions as modified recently by the author.The theory has been extended to multicomponent and multigroup systems and is,thus,applicable besides others to mixtures of polydisperse polymers and mixtures of random copolymers. The effect of pure component properties,of pressure and of polydispersity on the critical behavior of polymer mixtures are examined.Theoretical estimations are compared with experimental data whenever available.A satisfactory agreement is observed between theory and experiment. 

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... 

The ability to model Selexol-based unit operations in Aspen Plus or Aspen HYSYS was recently made possible by the inclusion of the Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT) physical property model. As in Aspen HYSYS (see Section 6.1.1), a single chemical DEPG be used as a proxy for the mixture. Simple example files for using PC-SAFT with Selexol for one- or two-unit operations are included with the Aspen Plus distributiOTi, and an example for using PC-SAFT in Aspen HYSYS is available for download to subscribers of the Aspen Technology support website. 

There is growing interest in what have been called lattice gas (fluid) models. These envisage a fluid to be a mixture of molecules and holes. In essence they are lattice-graph models in which some of the lattice sites are occupied while others remain empty (holes). Originally introduced by Sanchez and Lacombe they have been more recently developed by them - in terms of an equation-of-state approach (see p. 305). Such models offer an attractive and combinatorially transparent alternative to free volume (holes) extensions of corresponding states theories (see next section), which have been much described by Dayantis. Thornley and Shepherd comment that preliminary results using this model indicate that it might be the most accurate so far . 

---