J. Phys. Chem. 1988, 92, 501-507 TABLE III: Energetic Contributions and the Born Energy of Solvation as a Function of q / u i for an Ionic Charge of Q* = 14.152" add/ai
10.0
3.33 t
-AUlD -AUSOL -AAB
-AU~D
-AUSOL
-AAB
-AUSOL
-AAB
1.11
0.53
(0.910) 0.606 (0.749) 0.522 (0.467) 0.342
(0.770) 0.569 (0.634) 0.487 (0.395) 0.323
(0.667) 0.534 (0.550) 0.453 (0.342) 0.296
(0.528) 0.466 (0.434) 0.391 (0.269) 0.250
(0.347) 0.337 (0.286) 0.279 (0.177) 0.173
(1.351) 0.910 (0.975) 0.707 (0.691) 0.527
t = 4.18 (1.144) (0.993) 0.856 0.803 (0.825) (0.716) 0.658 0.610 (0.584) (0.505) 0.443 0.483
(0.785) 0.703 (0.566) 0.523 (0.398) 0.373
(0.516) 0.504 (0.371) 0.366 (0.260) 0.257
(1.642) 1.143 (1.044) 0.806 (0.822) 0.650
(1.390) 1.073 (0.883) 0.743 (0.694) 0.593
(0.952) 0.872 (0.605) 0.578 (0.473) 0.452
(0.624) 0.616 (0.397) 0.395 (0.308) 0.306
e
-AUlD
2.0
= 2.05
= 10.95 (1.205) 1.004 (0.766) 0.684 (0.601) 0.541
"The quantity in parentheses is the corresponding energy for Q* = to convert to units of kBT multiply by
1. In order
tribution from the C R A M potential to AUDD* is not negligible. For instance, a t e = 10.95, we found it to generally cause an increase in AUDD* greater than 10%. 4. Conclusion In this work the IPA model has been applied to an ionic fluid at infinite dilution. In particular, we have looked at contributions to the Born energy and compared our results with those from an MSA calculation. Our analysis was carried out at a simple level and did not include an accurate description of the solvent particle structure close to the ion arising due to the nonlocal entropy (excluded volume) effect. Thus, compared with the MSA, our model significantly overestimates solvation energies. On the other hand, the IPA functional contains mechanisms neglected by the MSA and it is their role in determining equilibrium properties which we wish to emphasize in this report. So far the LHNC9 approximation, which has also been applied to ionic solvation, has not been discussed in any detail, but it too is a linear theory and
501
is therefore subject to the same criticisms directed toward the MSA. Application of the recently solved fully nonlinear H N C theoryZZwill improve the accuracy but, so far as we can see, thermodynamic consistency will not be guaranteed. Moreover, these theories are based upon approximations of a more mathematical than physical nature and therefore do not necessarily provide physical insight commensurate with their numerical accuracy. We have found that attractive angular correlations and other field-dependent contributions to the electrostriction effect play a large part in determining the excess particle adsorption, which can contribute significantly to solvation energies. These mechanisms are neglected in the MSA theory. In addition, the nonlinear functional we have employed is able to give a qualitatively correct description of the saturation effect upon solvation energies which will be important when considering small or highly charged ions. The interplay of these mechanisms is important. For instance, the extra particles attracted from the bulk by highly charged ions act to push thermodynamic properties in the opposite direction as compared to the saturation effect. This may go some may toward mitigating the error made in the MSA approximation. However, the saturation effects we did find were significant, with a depression of greater than 20% in the reduced Born energies not uncommon. Nevertheless, we expect in view of our neglect of detailed hard-sphere structure that the results in Table I1 still underestimate the saturation effect. A more accurate description will be afforded by a more sophisticated nonlocal free volume functional of the type discussed in ref 20. However, the present model provides an important first step in an IPA model description of ionic solvation. It describes in a simple but plausible way all the major electrostatic mechanisms for moderately polar solvents. While the present applications have been chosen to allow comparisons with the MSA theory, which is at its best for small weakly charged ions, our S-GvdW theory should become more accurate as the size of the ion becomes much greater than that of the dipolar fluid. We believe our theory will be found useful in understanding the solvation of large molecular ions or colloidal particles. Subsequent work will then have to show the added effects of entropic nonlocalities and, for strongly polar solvents, cooperative correlation effects which go beyond the pair approximation used here for dipoledipole angular correlations. Taken together, these mechanism pose a formidable challenge and it is well worth while to prepare for it by studying smaller subsets of mechansims in isolation. This has been the aim of the present work. (22) Fries, P. H.; Patey, G. N. J . Chem. Phys. 1985, 82, 429.
Nonlinear Electric Field Effects on the Thermodynamic Properties of Dipolar Fluids C. E. Woodward*+ and S. Nordholm Department of Theoretical Chemistry, University of Sydney, N.S. W. 2006, Australia (Received: June 27, 1986; In Final Form: July 1 , 1987) The free energy functional, used in the preceding article to study solvation energies, is used here to investigate the thermodynamics of a uniform dipolar fluid in a constant electric field. Apart from the free energy, we consider the polarization, the chemical potential and electrostriction, and gas-liquid coexistence. Most of the results obtained are for dipolar hard spheres, but the extension to Stockmayer systems is trivial. 1. Introduction
The investigation of the influence of electric fields on the bulk thermodynamic properties ofa dipolar fluid has until recently been largely confined to systems which can respond linearly to the 'Present address Physical Chemistry 2, Chemical Centre, University of Lund, Box 124, S-221 00 Lund, Sweden.
0022-3654/88/2092-0501$01 S O / O
applied field. This can no doubt be partly attributable to the difficulties involved in placing the nonlinear systems within a modern statistical mechanical framework. Much of the pioneering work in this area has been carried out by Nienhuis and Deutch' and by Haye and StelL2 Despite these formal treatments, (1) Nienhuis, G.; Deutch, J. M. J . Chem. Phys. 1971, 56, 1819.
0 1988 American Chemical Society
Woodward and Nordholm
502 The Journal of Physical Chemistry, Vol. 92, No. 2, 1988
however, approximate quantitative theories have been slow in coming forward. Those that have been proposed usually involve mean-field assumptions3 or else remain at the linear response or small field limit. In the preceding paper we used a recently developed free energy functional t h e ~ r y to ~ -study ~ the solvation energy of an ion in a dipolar solvent. In this article we use that functional to consider in detail the electrostatic and thermodynamic properties of the uniform dipolar fluid in a constant electric field. In particular we look at the behavior of the polarization and the chemical potential as a function of increasing field strength, studying the nonlinear effects of polarization saturation and electrostriction. Both the mean spherical approximation (MSA) and the linearized hypernetted chain (LHNC)* theories have been used previously to study dipolar fluids in electric fields. However, phenomena involving nonlinear response to the field could not be studied due to the linear approximations inherent in these theories. Higher order expansions of the hypernetted chain (HNC) approximation have also been applied to study electrostriction; however, Rasaiah et al.9 found it necessary to go beyond the H N C closure itself to achieve thermodynamic consistency, even to second order in the field strength. We show that the free energy functional we employ gives a thermodynamically consistent description of these phenomena to all orders in the field strength. As in the preceding paper, however, we restrict ourselves to fluids with small dielectric constant where our approximate functional is most accurate. We also investigate the effect of the electric field on gas-liquid coexistence, studying in particular a Stockmayer model for hydrogen chloride. Recently Quint et a1.I0 have studied the saturation pressure of a dipolar liquid in equilibrium with its gas in the presence of an electric field, using a compressible fluid model; however, this work only considered second-order effects. 2. Theory Consider a large, uniform dipolar hard-sphere sample, with density p and temperature T , in the presence of some constant electric field of magnitude Eo.The particles have a diameter u with an imbedded point dipole of strength I.L. The independent pair approximation (IPA) free energy functional for this system has the following form
J
constant. AGvdWfp] is a generalized van der Waals fun~tional,~l-'~ which estimates the hard-sphere contribution to the free energy. There are several choices one can make for this functional, and we have chosen to use the most accurate Carnahan and Starling f ~ r m . ' ~ The J ~ constrained reference state averaged Mayer function (CRAM)potential, V C w M(r,a) introduces orientational correlations to what would otherwise be a mean-field functional. It has been developed and applied by us in a recent series of article^^-^ and constitutes an independent pair approximation to orientational correlations between dipoles. In this way the functional goes beyond mean-field theories as used, for example, by Herye and Stel13 and Tarazona et al.I5 It is useful to write
where fb is the free energy density. The thermodynamics for this system are easily extracted from the free energy. However, it must be remembered that the interactions acting between ordered dipoles are long-ranged and, hence, thermodynamic properties will, in general, exhibit a dependence upon the shape of the fluid sample, subject to the applied field. The long-ranged interactions are contained within the mean-field estimate of the dipole-dipole interaction energy, Le., the third term in eq 1. All other terms are local in the sense that they involve short-ranged correlations. Fortunately, the long-ranged terms, as they occur in particular derivations, can be identified with quantities that arise in macroscopic electrostatics. Thus they are easily handled and, in addition, the resulting shape dependence of our microscopic equations are automatically consistent with macroscopic electrostatics. Furthermore, as we start with the free energy, our results for derivative quantities will also be thermodynamically consistent. This latter result may seem almost trivial, but it is worth noting, given that similar self-consistency is not shared by some integral equation appro ache^.^ For example, consider the derivative of the free energy density with respect to the field strength at constant temperature and density
= -I.LLPL(a)
J
j/2P2SdrI Jd"2
e(r12-u)~v~RAM(r12;a)
- Jdr PL(a)PI.LEO (1)
where the notation is consistent with that used in the previous paper. In particular, a (which is here constant) is the orientational order parameter and
L ( x ) = cotanh (x) - l / x
Eo)= ZPb'%)Ed j=l
(23)
