Fluid dipolar hard sphere

The high population of ion pairs near criticality motivated Shelley and Patey [250] to compare the RPM coexistence curve with that of a dipolar fluid. It is now known that a critical point does not develop in a system of dipolar hard spheres [251]. However, ion pairs resemble dumbbell molecules comprising two hard spheres at contact with opposite charges at their centers. Shelley and Patey found that the coexistence curves of these charged dumbbells are indeed very similar in shape and location to the RPM coexistence curve, but very different from the coexistence curve of dipolar dumbbells with a point dipole at the tangency of the hard-sphere contact. 

Mixtures of equisized charged spheres were also treated by the MSA. Such a system is then uniquely characterized by the ratio of the critical temperatures of the pure components. Harvey [235] found that a continuous critical curve from the dipolar solvent to the molten salt is maintained until the critical temperature of the ionic component exceeds that of the dipolar component by a factor of about 3.6. This ratio is much higher than theoretically predicted for nonionic model fluids. We recall that for NaCl the critical line is still continuous at a critical temperature ratio of about 5. Thus, the MSA of the charged-hard-sphere-dipolar-hard-sphere system captures, at least in part, some unusual features of real salt-water systems with regard to their critical curves. 

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. 

Finally, C tld /,.[ = cdd = c Jm is the Fourier transform of the replica-replica direct correlation function (blocking function), and the connected function is defined as usual by cc = cdd — cdd, and similarly for hc. Let us recall that the replicated particles are the dipolar hard spheres, i.e. the annealed fluid in the partly quenched mixture. 

In what follows we discuss the phase behavior of the Stockmayer fluid in the presence of disordered matrices of increasing complexity. All results are based on a variant of the HNC equation [see Eq. (7.49)], which yields very good results for bulk dipolar fluids [268, 322]. Moreover, subsequent studies of dipolar hard-sphere (DHS) fluids [defined by Eq. (7.59) with ulj = 0] in disordered matrices by Fernaud et al. [323, 324] have revealed a very good performance of the HNC closure compared with parallel computer simular tion results. The integral equations are solved numerically with an iteration procedure. To handle the multiple angle-dependence of the correlations... 

The analytic solution of the SSOZ-MSA equation for polar hard dumbbells came before any serious consideration was given to calculating the dielectric constants of such systems by computer simulation. At the time, there was considerable controversy about the simulation methods used to calculate the dielectric constant, and for the model systems then in vogue (dipolar hard spheres and the Stockmayer fluid) there was also debate about the correct value of the dielectric constant. Today, this problem is becoming better understood in particular, the quality of the simulation work has improved greatly, and this has allowed meaningful conclusions to be drawn about the relative merits of simulation methods. 

Agrofonov, Martinov, and Sarkisov have recently proposed another theory for dipolar hard spheres based on the HNC approximation. Their approach is similar in spirit to that followed in the usual thermodynamic perturbation theory (TPT) of dipolar fluids. It is assumed that h 2) can be expanded in the power series... 

Ewald boundary conditions were first applied to dipolar fluids by Jansoone, and Adams and McDonald have studied dipolar lattices. Recently Ewald methods have been applied in MD calculations for Stoclcmayer fluids by Pollock and Alder and in MC calculations for dipolar hard spheres by Adams and by De Leeuw et al. De Leeuw et al. also report results for the effective potential <>(1 12), which is Just tpbc(12). The details of these recent results are described in Section III.D, but there are two important points worth noting here. 

QHNC and Ewald results for Stockmayer fluids have been compared by Pollock and Alder (Figs. 11 and 12), and Adams has made similar comparisons for dipolar hard spheres (Figs. 9 and 10). We know from the work... 

Hard Spheres with Dipoles and Quadrupoles. LHNC and MSA results for fluids of hard spheres with dipoles and quadrupoles are shown in Fig. 20. In both approximations e decreases with increasing quadrupole moment. The LHNC results are particularly dramatic, since the dipolar hard-sphere or Q = 0 value is very large to begin with. As discussed in Section III.B.3, the QHNC approximation is not very satisfactory for dipole-quadrupole systems, since solutions are not found for some values of ju and Q. When solutions can be obtained, however, the QHNC e also decreases sharply with quadrupole moment. 

Fig. 22. Comparison with the dielectric constant of real liquids LHNC results for dipolar hard spheres at p 0.8 (solid curve), Stockmayer fluids at p — 0.8, 7 —1.35 (dashed curve) ONS Onsager. The dots are the experimental results for the following Uquids (1) CH3I, (2) CHjCl. (3) NHj, (4) CHj— C—CH3, (5) CH3F, (6) CH3— C—H, (7) QHjNOj, (8)...

Wertheim s formulation of his SSC approximation, which we have already discussed in the context of nonpolarizable fluids in Sections II and III, applies to the more general case of polar-polarizable fluids. In describing this case we use his notation. For polarizable dipolar hard spheres, the approximation is defined by the integral equations ... 

High-temperature-approximation calculations have been performed primarily for the Lennard-Jones fluid and the square-well fluid. Higher order pei turbation theory results for the Lennard-Jones fluid, the square-well fluid,and the dipolar hard-sphere fluid have also been obtained. The Fade approximant method has also been used to extend the results. ... 

Fries, P. H. and G. N. Patey. 1985. The solution of the hypemetted-chain approximation for fluids of nonspherical particles—A general-method with application to dipolar hard-spheres. Journal of Chemical Physics. 82, 429. 

Ferroelectric fluid phases have also been observed in simulations of disk-shaped particles with embedded dipole moments along the symmetry axis [ 147,148] (see Sect. 4) and in a dipolar hard sphere model carrying two parallel dipole moments displaced equally from the sphere centre [ 149]. In the latter model antiferroelectric arrangement of the particles is, however, equally preferred at sufficiently large separations (0.3a) of the two dipole moments. 

P. Attard, D. R. Berard, C. P. Ursenbach, G. N. Patey. Interaction free energy between planar walls in dense fluids an Omstein-Zernike approach for hard-sphere, Lennard-Jones, and dipolar systems. Phys Rev A 44 8224-8234, 1991. 

The structure formation in an ER fluid was simulated [99]. The characteristic parameter is the ratio of the Brownian force to the dipolar force. Over a wide range of this ratio there is rapid chain formation followed by aggregation of chains into thick columns with a body-centered tetragonal structure observed. Above a threshold of the intensity of an external ahgn-ing field, condensation of the particles happens [100]. This effect has also been studied for MR fluids [101]. The rheological behavior of ER fluids [102] depends on the structure formed chainlike, shear-string, or liquid. Coexistence in dipolar fluids in a field [103], for a Stockmayer fluid in an applied field [104], and the structure of soft-sphere dipolar fluids were investigated [105], and ferroelectric phases were found [106]. An island of vapor-liquid coexistence was found for dipolar hard spherocylinders [107]. It exists between a phase where the particles form chains of dipoles in a nose-to-tail... 

The extension of the ROZ formalism to confined molecular fluids has recently been carried out for adsorbed diatomic molecules [6] and dipolar fluids confined in hard sphere matrices [18, 19], In the case of ionic matrix, new features of the system have to be taken into account. On one hand, we have now a two component matrix (with positive and negative ions). This case was already considered in [14, 15] for the primitive model electrolyte adsorbed in an electroneutral charged matrix. On the other hand, we have to deal with two different temperatures the matrix temperature, (h (at which the ionic fluid is equilibrated before quenched) and the fluid temperature fi, at which the fluid is adsorbed in the solid matrix. As usual when dealing with molecular fluids one starts with an expansion of the correlation functions in terms of spherical harmonics as follows,... 

Figure 3. Dielectric constant of the dipolar fluid embedded in a charged matrix quenched at low temperature 3r,e2/a = 1. ROZ vs GCMC results. HNC results for the corresponding equilibrated mixture and ROZ results for an equivalent system with a neutral (hard sphere) matrix are included for comparison.

We also compare in both figures the dielectric constant values of the dipolar fluid when adsorbed in the ionic and in the hard sphere matrix. A clear influence of the presence of charges in the matrix is then observed. The ionic matrix lowers the response of the dipoles to an external field, i.e. lowers the value of the dielectric constant of the fluid in the given state. This can easily be understood, since the local electric field that the matrix charges generate... 

In the case of polar fluids, a fluid of dipolar spheres provides the simplest model for discussion, in which the reference potential < o(l 2) is a hard-sphere term and the perturbing w(l 2) is an ideal dipole-dipole term ... 

The adsorption of dipolar fluids in porous solids induces modifications of their bulk properties. Quantitative estimates of these changes on the dielectric properties and spatial correlations of the DHS fluid adsorbed in uncharged or charged disordered or random matrices of immobile hard spheres have been obtained by grand canonical MC simulations [203,204] for different densities of the adsorbed fluids and porosities of the matrices. 

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