Dipolar hard spheres

The solution detennines c(r) inside the hard core from which g(r) outside this core is obtained via the Omstein-Zemike relation. For hard spheres, the approximation is identical to tire PY approximation. Analytic solutions have been obtained for hard spheres, charged hard spheres, dipolar hard spheres and for particles interacting witli the Yukawa potential. The MS approximation for point charges (charged hard spheres in the limit of zero size) yields the Debye-Fluckel limiting law distribution fiinction. 

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... 

Zhu J and Rasaiah J C 1989 Solvent effects in weak electrolytes II. Dipolar hard sphere solvent an the sticky electrolyte model with L = a J. Chem. Phys. 91 505... 

The extent of the agreement of the theoretical calculations with the experiments is somewhat unexpected since MSA is an approximate theory and the underlying model is rough. In particular, water is not a system of dipolar hard spheres.281 However, the good agreement is an indication of the utility of recent advances in the application of statistical mechanics to the study of the electric dipole layer at metal electrodes. 

Atom dynamics Group contribution and rigid bonds/angels Specific adsorption Dipolar hard sphere SPC, ST2, TIPS Polarizable H Bonds... 

More realistic treatment of the electrostatic interactions of the solvent can be made. The dipolar hard-sphere model is a simple representation of the polar nature of the solvent and has been adopted in studies of bulk electrolyte and electrolyte interfaces [35-39], Recently, it was found that this model gives rise to phase behavior that does not exist in experiments [40,41] and that the Stockmeyer potential [41,42] with soft cores should be better to avoid artifacts. Representation of higher-order multipoles are given in several popular models of water, namely, the simple point charge (SPC) model [43] and its extension (SPC/E) [44], the transferable interaction potential (T1PS)[45], and other central force models [46-48], Models have also been proposed to treat the polarizability of water [49],... 

Figure 2. Solvent-averaged potential for charged hard-sphere ions in a dipolar hard-sphere solvent. MC approximation by Patey and Valleau (16) and LHNC approximation by Levesque, Weis, and Patey (11). Also shown are the primitive model functions for solvent dielectric constants 9.6 and 6.

Figure 10. The compressibility factor for a charg and dipolar hard sphere mixture pr icted by perturbation thmiy is compared with the results of Monte Carlo simulation. Tl e elementaiy electronic charge is denoted e.

Figure 11. The ion cluster size distribution obtained from computer simulations of the charged and dipolar hard sphere mixture at several states half charge 1 Molar (A) fully charged, 1 Molar (B) fully charged, 0.4 Molar (C) and half charge, 0.4 Molar (D).

More refined continuum models—for example, the well-known Fumi-Tosi potential with a soft core and a term for attractive van der Waals interactions [172]—have received little attention in phase equilibrium calculations [51]. Refined potentials are, however, vital when specific ion-ion or ion-solvent interactions in electrolyte solutions affect the phase stability. One can retain the continuum picture in these cases by using modified solvent-averaged potentials—for example, the so-called Friedman-Gumey potentials [81, 168, 173]. Specific interactions are then represented by additional terms in (pap(r) that modify the ion distribution in the desired way. Finally, there are models that account for the discrete molecular nature of the solvent—for example, by modeling the solvent as dipolar hard spheres [174, 175]. 

With regard to real electrolytes, mixtures of charged hard spheres with dipolar hard spheres may be more appropriate. Again, the MSA provides an established formalism for treating such a system. The MSA has been solved analytically for mixtures of charged and dipolar hard spheres of equal [174, 175] and of different size [233,234]. Analytical means here that the system of integral equations is transformed to a system of nonlinear equations, which makes applications in phase equilibrium calculations fairly complex [235]. 

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. 

Again uq(R) is the hard sphere potential. This is necessary to keep the molecules from overlapping. The parameter is the dipole moment of molecule i. The factor D(i, j) is a term that depends on the orientation of the dipoles i and j and need not concern us here. We can call this potential the dipolar hard sphere potential. 

To move beyond the primitive model, we must include a molecular model of the solvent. A simple model of the solvent is the dipolar hard sphere model, Eq. (16). A mixture of dipolar and charged hard spheres has been called the civilized model of an electrolyte. This is, perhaps, an overstatement as dipolar hard spheres are only partially satisfactory as a model of most solvents, especially water still it is an improvement. 

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. 

Weis, J.J., and Levesque, D. Chain formation in low density dipolar hard spheres a Monte Carlo study. Phys. Rev. Left. 1993, 71, p. 2729-32. 

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. 

Figure 2. Minimum eigenvalue of the stability matrix (see Eq. (24) in the text) and isothermal compressibility for equilibrated mixtures of charged and dipolar hard spheres.

On the basis of Wertheim s solution of the MSA for dipolar hard spheres it is convenient to define a polarization parameter which is obtained directly from the relative static permittivity, that is, by solving the equation... 

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 same approach can be applied to investigate the explosivity conditions of the H20-NaCl system. We have selected the Anderko-Pitzer (AP) equation of state,which is based on realistic physical hypotheses. It describes H20-NaCl by means of statistical thermodynamic models developed for dipolar hard spheres. This assumption is reasonable at high temperatures, where NaCl is known to form dipolar ion pairs. However, for this reason, this equation of state is only applicable above 573 K, 300°C. 

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. 

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. 

Wertheim has shown that for dipolar hard spheres (3.14), (3.12b), and (3.12d) can be transformed to yield equations formally analogous to those occurring in the PY approximation for hard spheres. This allows us to obtain analytic expressions for h 2), c(12), the various thermodynamic properties, and the dielectric constant. The dielectric constant of dipolar hard spheres is given by the equations... 

Two further approximations that build on the MSA solution are derived in Section II. These are the so-called LIN and L3 approximations, which have been investigated by Stell and Weis for dipolar hard spheres. Both theories substantially improve upon the MSA. 

The LHNC and QHNC approximations have not been solved analytically, but numerical solutions can be obtained by iteration. This is also true of the MSA except for the previously discussed dipolar hard-sphere system solved by Wertheim. The details of the numerical solution are described in Refs. 30, 38, 58, and 59. Essentially, (3.11) and the appropriate closure relations are written in terms of c" " and -q "" and iterated until a solution is obtained. This means that all equations defining a particular approximation are simultaneously satisfied. The present problem is very similar to that... 

The LHNC or QHNC e can be obtained from either (3.7a) or (3.8), since both routes must give e consistently. Dipolar hard spheres have been studied extensively using both the LHNC and QHNC theories, and before discussing the results it is convenient to introduce the reduced variables p = pd and ji = which are sufficient to characterize... 

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... 

Fig. 1. Values of for dipolar hard spheres at p = 0.8 and ja = 2.75 long dashes,...

Unfortunately, the discussion above proves to be rather academic, since (3.36) has a serious practical problem. The relationship between e and g is such that for e> 10 (/x >l for dipolar hard spheres), small uncertainties in g lead to very large errors in e. Thus except for systems with relatively small dielectric constants, (3.36) is not very useful. 

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