Perturbation theory hard-sphere

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. 

Another important application of perturbation theory is to molecules with anisotropic interactions. Examples are dipolar hard spheres, in which the anisotropy is due to the polarity of tlie molecule, and liquid crystals in which the anisotropy is due also to the shape of the molecules. The use of an anisotropic reference system is more natural in accounting for molecular shape, but presents difficulties. Hence, we will consider only... 

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. 

Chandler-Andersen Perturbation Theory Directly in Terms of a Hard-Sphere Reference... 

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. 

To address the hmitations of ancestral polymer solution theories, recent work has studied specific molecular models - the tangent hard-sphere chain model of a polymer molecule - in high detail, and has developed a generalized Rory theory (Dickman and Hall (1986) Yethiraj and Hall, 1991). The justification for this simplification is the van der Waals model of solution thermodynamics, see Section 4.1, p. 61 attractive interactions that stabilize the liquid at low pressure are considered to have weak structural effects, and are included finally at the level of first-order perturbation theory. The packing problems remaining are attacked on the basis of a hard-core model reference system. 

The approach of Klupsch differs in several ways from the work of Haymet and Oxtoby. Klupsch uses a density-dependent direct correlation function which he approximates using hard-sphere perturbation theory. His ansatz for the crystal density is... 

The effect of ion pair formation to the hard-sphere contributions can be evaluated with the help of thermodynamic perturbation theory [25] and is included into the MAL terms,... 

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... 

Jog, P.K. and Chapman, W.G., Application of Wertheim s thermodynamic perturbation theory to dipolar hard sphere chains, Mol. Phys., 97(3), 307-319, 1999. 

Equations of state developed from perturbation theory are composed of a reference fluid equation to which are added perturbation terms. The reference fluid expresses the high repulsive energy that determines the molecular correlation and hence the structure of the fluid. The attractive energies that are relatively weak are treated as perturbations. Perturbation theory opens the door to the separate development of statistical theories for various intermolecular energies hard-sphere-fluid... 

In the perturbed hard-chain theory (PHCT) of Beret and Prausnitz [14] and Donohue and Prausnitz [15], the reference fluid is modeled as chains of tangential hard spheres. Since the fluid is stiU composed of hard spheres, albeit bonded, the CS eos is applied with modification to account for the bonding. The hard-sphere-chain equation is... 

The statistical-associated fluid theory (SAFT) of Chapman et al. [25, 26] is based on the perturbation theory of Wertheim [27]. The model molecule is a chain of hard spheres that is perturbed with a dispersion attractive potential and association potential. The residual Helmholtz energy of the fluid is given by the sum of the Helmholtz energies of the initially free hard spheres bonding the hard spheres to form a chain the dispersion attractive potential and the association potential,... 

Up to this point we have focused on the use of a single theory to describe both phases in a calculation of SFE. A somewhat less ambitious approach is to use a theory appropriate to each phase in the calculations. In particular, the cell theory can be used for the solid phase and a liquid state theory (e.g., a hard-sphere equation of state or thermodynamic perturbation theory) used for the fluid phase. This approach turns out to be at least as accurate as either the two-phase cell theory or DFT approaches described above and is often more accurate. Moreover, it has been more successful in the treatment of systems more complex than hard spheres. 

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]. 

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