Fluids perturbation theory

M. Ross,/. Chem. Phys., 71, 1567 (1979). A High Density Fluid-perturbation Theory Based... 

The first step in quantitative description of pure polyamorphic fluid is a selection of the model that can qualitatively describe a possible multiplicity of critical points in wide range of temperatures and pressures. A great many of explanations of multicriticality in monocomponent fluids (perturbation theory models semiempirical models lattice models, two-state models, field theoretic models, two-order-parameter models, and parametric crossover model has been disseminated after the pioneering work by Hemmer and Stell Here we test more extensively the modified van der Waals equation of state (MVDW) proposed in work and refine this model by introducing instead of the classical van der Waals repulsive term a very accurate hard sphere equation of state over the entire stable and metastable regions... 

Ross proposed a soft-sphere perturbation theory for the equation of state of the exponential-6 fluid [14]. Ross, Ree and others successfully applied this equation of state to detonation and shock problems [15-18]. Kang et al. also derived a fluid perturbation theory designed to work at high-density [19]. Computational cost is a significant difficulty with equations of state based on fluid perturbation theory. W. Byers Brown... 

These equations provide a convenient and accurate representation of the themrodynamic properties of hard spheres, especially as a reference system in perturbation theories for fluids. 

Kirkwood derived an analogous equation that also relates two- and tlnee-particle correlation fiinctions but an approximation is necessary to uncouple them. The superposition approximation mentioned earlier is one such approximation, but unfortunately it is not very accurate. It is equivalent to the assumption that the potential of average force of tlnee or more particles is pairwise additive, which is not the case even if the total potential is pair decomposable. The YBG equation for n = 1, however, is a convenient starting point for perturbation theories of inliomogeneous fluids in an external field. 

In perturbation theories of fluids, the pair total potential is divided into a reference part and a perturbation... 

Barker J and Henderson D 1967 Perturbation theory and equation of state for a fluids II. A successful theory of liquids J. Chem. Phys. 47 4714... 

Larsen B, Rasaiah J C and Stell G 1977 Thermodynamic perturbation theory for multipolar and ionic fluids Mol. Phys. 33 987... 

Smith W R 1972 Perturbation theory in the classical statistical mechanics of fluids Specialist Periodical Report vol 1 (London Chemical Society)... 

The fugacity coefficient of thesolid solute dissolved in the fluid phase (0 ) has been obtained using cubic equations of state (52) and statistical mechanical perturbation theory (53). The enhancement factor, E, shown as the quantity ia brackets ia equation 2, is defined as the real solubiUty divided by the solubihty ia an ideal gas. The solubiUty ia an ideal gas is simply the vapor pressure of the sohd over the pressure. Enhancement factors of 10 are common for supercritical systems. Notable exceptions such as the squalane—carbon dioxide system may have enhancement factors greater than 10. Solubihty data can be reduced to a simple form by plotting the logarithm of the enhancement factor vs density, resulting ia a fairly linear relationship (52). 

We would like to recall that Xa p) is the fraction of molecules not bonded at an associative site now it is a function of the averaged density p(r). A generalization of the perturbational theory allows us to define Xa p) similar to the case of bulk associating fluids. Namely... 

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. 

As we have already pointed out, the theoretical basis of free energy calculations were laid a long time ago [1,4,5], but, quite understandably, had to wait for sufficient computational capabilities to be applied to molecular systems of interest to the chemist, the physicist, and the biologist. In the meantime, these calculations were the domain of analytical theories. The most useful in practice were perturbation theories of dense liquids. In the Barker-Henderson theory [13], the reference state was chosen to be a hard-sphere fluid. The subsequent Weeks-Chandler-Andersen theory [14] differed from the Barker-Henderson approach by dividing the intermolecular potential such that its unperturbed and perturbed parts were associated with repulsive and attractive forces, respectively. This division yields slower variation of the perturbation term with intermolecular separation and, consequently, faster convergence of the perturbation series than the division employed by Barker and Henderson. 

The exp-6 model is not well suited to molecules with large dipole moments. To account for this, Ree9 used a temperature-dependent well depth e(T) in the exp-6 potential to model polar fluids and fluid phase separations. Fried and Howard have developed an effective cluster model for HF.33 The effective cluster model is valid for temperatures lower than the variable well-depth model, but it employs two more adjustable parameters than does the latter. Jones et al.34 have applied thermodynamic perturbation theory to... 

An analogy may be drawn between the phase behavior of weakly attractive monodisperse dispersions and that of conventional molecular systems provided coalescence and Ostwald ripening do not occur. The similarity arises from the common form of the pair potential, whose dominant feature in both cases is the presence of a shallow minimum. The equilibrium statistical mechanics of such systems have been extensively explored. As previously explained, the primary difficulty in predicting equilibrium phase behavior lies in the many-body interactions intrinsic to any condensed phase. Fortunately, the synthesis of several methods (integral equation approaches, perturbation theories, virial expansions, and computer simulations) now provides accurate predictions of thermodynamic properties and phase behavior of dense molecular fluids or colloidal fluids [1]. 

Fickett in "Detonation Properties of Condensed Explosives Calculated with an Equation of State Based on Intermolecular Potentials , Los Alamos Scientific Lab Rept LA-2712 (1962), pp 34-38, discusses perturbation theories as applied to a system of deton products consisting of two phases one, solid carbon in some form, and the other, a fluid mixt of the remaining product species. He divides these theories into two classes conformal solution theory, and what he chooses to call n-fluid theory. Both theories stem from a common approach, namely, perturbation from a pure fluid whose props are assumed known. They differ mainly in the choice of expansion variables. The conformal solution method begins with the assumption that all of the intermolecular interaction potentials have the same functional form. To obtain the equation of state of the mixt, some reference fluid obeying a common reduced equation of state is chosen, and the mixt partition function is expanded about that of the reference fluid... 

W. Fickett in "Detonation Properties of Condensed Explosives Calculated with an Equation of State Based on Intermole-cular Potentials , LosAlamosScientific-LabRept LA-2712(1962), pp 38-42, reports that pseudopotential theories are obtd by an approach completely different from perturbation theories. The problem of defining a system of detonation products consisting of both solid carbon in some form and a fluid mixt of the remaining product species has been formally rearranged to a single fictitious substance with an extremely complicated compn- temp-dependent potential function , called the pseudopotential. The fictitious substance corresponding to this potential is clearly non-conformal with the components of the mixt... 

The analysis of ordinary differential equation (ODE) systems with small parameters e (with 0 < generally referred to as perturbation analysis or perturbation theory. Perturbation theory has been the subject of many fundamental research contributions (Fenichel 1979, Ladde and Siljak 1983), finding applications in many areas, including linear and nonlinear control systems, fluid mechanics, and reaction engineering (see, e.g., Kokotovic et al. 1986, Kevorkian and Cole 1996, Verhulst 2005). The main concepts of perturbation theory are presented below, following closely the developments in (Kokotovic et al. 1986). 

In order to determine the thermodynamic properties by means of the perturbation theory, the thermodynamic properties of the reference system are needed. Here, the expressions for the equation of state and the radial distribution function of a system of hard spheres are included for both the fluid and solid reference states. A face-centred-eubic arrangement of the particles at closest packing is assumed for the solid phase. 

The perturbation theories [2, 3] go a step beyond corresponding states the properties (e.g., Ac) of some substance with potential U are related to those for a simpler reference substance with potential Uq by a perturbation expansion (Ac = Aq + A + Aj + ). The properties of the simple reference fluid can be obtained from experimental data (or from simulation data for model fluids such as hard spheres) or corresponding states correlations, while the perturbation corrections are calculated from the statistical mechanical expressions, which involve only reference fluid properties and the perturbing potential. Cluster expansions involve a series in molecular clusters and are closely related to the perturbation theories they have proved particularly useful for moderately dense gases, dilute solutions, hydrogen-bonded liquids, and ionic solutions. 

Integral equation methods provide another approach, but their use is limited to potential models that are usually too simple for engineering use and are moreover numerically difficult to solve. They are useful in providing equations of state for certain simple reference fluids (e.g., hard spheres, dipolar hard spheres, charged hard spheres) that can then be used in the perturbation theories or density functional theories. 

Dimitrelis, D and Prausnitz, J.M. Comparison of two Hard-Sphere Reference Systems for Perturbation Theories for Mixtures, Fluid Phase Equilibria. Vol. 31. 1986, pp. 1-21. 

In considering fluids, a variety of approaches have been used to compute entropy and free energy a free-volume method,111-112 thermodynamic perturbation theory,113-116 thermodynamic integration,117-121 umbrella sampling,122-124 and a Monte Carlo recursion method.125-126 The entropy of association of two protein molecules in water has also been computed.127... 

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