Hard fluid model application

The model of diffusion of hard spheres is applicable to interpret self-diffusion in liquids which behave according to the van der Waals physical interaction model (56). This might be the case for simple dense fluids at high temperature, T Tg, but it is an oversimplified model for the real diffusion of small organic penetrants in polymers. The functional relationships derived in the model of hard-spheres have been reinterpreted over course of the time, leading to a series of more sophisticated free-volume diffusion models. Some of these models are presented briefly below. 

The potential U(r ) is a sum over all intra- and intermolecular interactions in the fluid, and is assumed known. In most applications it is approximated as a sum of binary interactions, 17(r ) = IZ > w(rzj) where ry is the vector distance from particle i to particle j. Some generic models are often used. For atomic fluids the simplest of these is the hard sphere model, in which z/(r) = 0 for r > a and M(r) = c for r < a, where a is the hard sphere radius. A. more sophisticated model is the Lennard Jones potential... 

Finally, we mention an interesting recent study by Chandler that extended the Gaussian field-theoretic model of Li and Kardar to treat atomic and polymeric fluids. Remarkably, the atomic PY and MSA theories were derived from a Gaussian field-theoretic formalism without explicit use of the Ornstein-Zernike relation or direct correlation function concept. In addition, based on an additional preaveraging approximation, analytic PRISM theory was recovered for hard-core thread chain model fluids. Nonperturbative applications of this field-theoretic approach to polymer liquids where the chains have nonzero thickness and/or attractive forces requires numerical work that, to the best of our knowledge, has not yet been pursued. 

In view of the success of the methods based on hard-sphere theories for the accurate correlation and prediction of transport properties of single-component dense fluids, it is worthwhile to consider the application of the hard-sphere model to dense fluid mixtures. The methods of Enskog were extended to mixtures by Thome (see Chapman Cowling 1952). The binary diffusion coefficient >12 for a smooth hard-sphere system is given by... 

In the following section, expressions are given for the transport coefficients of dense as semblies of smooth hard-spheres and applied to the rare gases. The only adjustable parameter here is the core size. For molecular fluids, there is the possibility of translational-rotational coupling. The effects of this are discussed in Section 10.3 in terms of the rough hard-sphere model, which leads to the introduction of one additional parameter - the translational-rotational coupling factor - for each property, and the results are applied to methane. For dense nonspherical molecular fluids, it is assumed that the transport properties can also be related directly to the smooth hard-sphere values with proportionality factors ( roughness factors ) which account for effects of nonspherical shape. The application of this method is described in 10.4 with reference to alkanes, aromatic hydrocarbons, alkanols and certain other compounds. 

Repulsive interactions are important when molecules are close to each other. They result from the overlap of electrons when atoms approach one another. As molecules move very close to each other the potential energy rises steeply, due partly to repulsive interactions between electrons, but also due to forces with a quantum mechanical origin in the Pauli exclusion principle. Repulsive interactions effectively correspond to steric or excluded volume interactions. Because a molecule cannot come into contact with other molecules, it effectively excludes volume to these other molecules. The simplest model for an excluded volume interaction is the hard sphere model. The hard sphere model has direct application to one class of soft materials, namely sterically stabilized colloidal dispersions. These are described in Section 3.6. It is also used as a reference system for modelling the behaviour of simple fluids. The hard sphere potential, V(r), has a particularly simple form ... 

The present chapter is organized as follows. We focus first on a simple model of a nonuniform associating fluid with spherically symmetric associative forces between species. This model serves us to demonstrate the application of so-called first-order (singlet) and second-order (pair) integral equations for the density profile. Some examples of the solution of these equations for associating fluids in contact with structureless and crystalline solid surfaces are presented. Then we discuss one version of the density functional theory for a model of associating hard spheres. All aforementioned issues are discussed in Sec. II. 

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. 

Our main focus in this chapter has been on the applications of the replica Ornstein-Zernike equations designed by Given and Stell [17-19] for quenched-annealed systems. This theory has been shown to yield interesting results for adsorption of a hard sphere fluid mimicking colloidal suspension, for a system of multiple permeable membranes and for a hard sphere fluid in a matrix of chain molecules. Much room remains to explore even simple quenched-annealed models either in the framework of theoretical approaches or by computer simulation. 

The Emerman model described in the previous section is hardly applicable to the carbon black-filled CCM as the black particles have sizes of hundreds angstrom and such a composite, compared with the molding channel size, may be considered as a homogeneous viscous fluid. Therefore, the polymer structure, crystallinity and orientation play an important role for such small particles. The above-given example of manufacture of the CCM demonstrates the importance of these factors being considered during processing of a composite material to and article with the desired electrical properties. 

One possibility for this was demonstrated in Chapter 3. If impact theory is still valid in a moderately dense fluid where non-model stochastic perturbation theory has been already found applicable, then evidently the continuation of the theory to liquid densities is justified. This simplest opportunity of unified description of nitrogen isotropic Q-branch from rarefied gas to liquid is validated due to the small enough frequency scale of rotation-vibration interaction. The frequency scales corresponding to IR and anisotropic Raman spectra are much larger. So the common applicability region for perturbation and impact theories hardly exists. The analysis of numerous experimental data proves that in simple (non-associated) systems there are three different scenarios of linear rotator spectral transformation. The IR spectrum in rarefied gas is a P-R doublet with either resolved or unresolved rotational structure. In the process of condensation the following may happen. 

Three main properties render clay suitable for making ceramic materials its plasticity when wet, its hardness when dry, and the toughness, increased hardness, and stability that it acquires when fired. The addition of water to dry clay produces a clay-water mixture that, within a narrow range of water content, has plastic properties it is deformed, without breaking or cracking, by the application of an external stress, and it retains the acquired shape when the deforming stress is removed. Wet clay mixtures can, therefore, be modeled, molded, or otherwise made to acquire a shape that will be retained after the forming operations. Water-poor mixtures are not plastic, however, and excess water results in mixtures, known as slips, that are too fluid to retain a shape, as shown in Table 56. 

As a first application of the model of coke deposition we have described the impact of coke on the cracking reaction. As the cracking reaction is hardly affected by the catalyst but dictated by temperature and residence times, the impact of coke is via the latter parameter (reduction of the hold up of the fluid phases). The results in Figure 9 show that a very good performance description is thus obtained. 

In recent years, a number of investigators have studied the phase equilibria of simple fluids in pores by the application of density functional theory. Semina] studies were carried out by Evans and his co-workers (1985,1986). Their approach was considered to be the simplest realistic model for an inhomogeneous three-dimensional fluid . The starting point was a model intrinsic Helmholtz free energy functional F(p), with a mean-field approximation for the attractive forces and hard-sphere repulsion. As explained in Section 7.6, the equilibrium density profile of the fluid in a pore was obtained by minimizing the grand potential functional. 

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