13 Equation of State of Ionic Fluids 1
2
3
Douglas Henderson , Lesser Blum , and Alessandro Tani 1
IBM Research Laboratory, San Jose, CA 95193 Department of Physics, University of Puerto Rico, Rio Piedras, PR 00931 Institute di Chimica Fisica, Université di Pisa, via Risorgimento 35, 56100 Pisa, Italy
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2
3
An ionic fluid is modelled as a mixture of dipolar hard spheres (the solvent) and charged hard spheres (the ions). The free energy is expanded in a power series in the inverse temperature. Some of the terms in this expansion are infinite. However, these terms can be resummed to give a finite result. This expansion reduces to previously known results for pure dipolar hard spheres and pure charged hard spheres. As is the case for the pure dipolar and charged hard sphere systems, the convergence of the expansion can be enhanced by means of a Padé approximant. Simple expressions for the integrals appearing in the expansion are given. Most theoretical studies of the equation of state of ionic fluids have been based upon the primitive model in which the solvent is modelled as a dielectric continuum and the ions are modelled as charged hard spheres. It is clearly desirable to model the solvent as a collection of molecules. A simple model of the solvent would be the dipolar hard sphere system. Although deficient as a model of water, the dipolar hard sphere system may be a fairly reasonable model for many organic solvents. Even for water, the dipolar hard sphere model is a first step and is clearly an improvement over a dielectric continuum. Using this model of the solvent, an ionic fluid can be modelled as a mixture of dipolar hard spheres and charged hard spheres. Such a model would include the pure solvent and the molten salt as limiting cases and could, in principle, be applied to a continuously miscible system. This model was first discussed by Blum and colleagues and Adelman and Deutch who solved analytically the mean spherical approximation (MSA) for the restricted model in which all the spheres have the same diameter. Extensions and generalizations have also been g i v e n . Patey and colleagues - have considered simplified versions of the hypernetted chain ( H N C ) approximation for this fluid. There is also the modified Poisson-Boltzmann (MPB) approximation considered by Outhwaite and the study of Adelman and C h e n . Unfortunately, there are few exact results or computer simulations which can be used to determine the accuracy of the theory. There is the simulation of Patey and Valleau for very dilute solutions. Very recently, the one-dimensional mixture of charged and dipolar hard spheres was solved exactly by Vericat and B l u m . In the high temperature limit, the one-dimensional M S A is recovered. 1
2
3-5
7
5
6
8
9
1 0
0097-6156/ 86/ 0300-0281 $06.00/ 0 © 1986 American Chemical Society
Chao and Robinson; Equations of State ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
282
EQUATIONS OF STATE: THEORIES AND APPLICATIONS
The solution of the M S A is analytic but implicit. Explicit results can be obtained only by some expansion. The M P B results are resonably simple but are restricted to small dielectric constants or low densities. The H N C results are numerical. Clearly, there is a need for a simple explicit theory. Perturbation theory is a very promising candidate for such a simple explicit theory. In perturbation theory, the free energy is expanded in powers of the inverse temperature using a hard sphere fluid as a reference system. This theory, which we have reviewed in previous A C S symposia, is a very satisfactory theory of simple fluids, such as the inert gases and the pure dipolar and pure charged hard sphere fluids. Prior to this work, it has not been applied to a mixture of dipolar and charged hard spheres. However, it should be a good approximation for this system also. 11
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12
In this publication, we apply perturbation theory to this mixture. For simplicity, we restrict ourselves to the case where the dipolar and charged hard spheres have an equal diameter, σ. The general case of differing diameters would be rather complex. However, we give some plausible expressions for the case where the dipolar diameter differs from the ion diameter σ{. Perturbation Expansion 11
13
Using perturbation t h e o r y , - the Helmboltz free energy can be expanded in powers of 0 = l / k T , where k is Boltzmann's constant and Τ is the temperature. To third order, the result is Ρ
(
Λ
~
Λ
θ
)
=
- \
Ρβ Σ^{ 2
8° (12) ( 1
Γ
2
ij -
}
Wk/
1
g° ( 23)dr dr k
i j
j k
i k
2
3
,
(1)
ijk where An is the free energy of the reference hard sphere system and includes the entropy of mixing term, Σ x^nxj, p = N / V , the number of hard spheres per unit volume, X . S I N J / N , the fraction bf hard spheres of species i, gjj( 12) and g (123) are the reference pair and triplet correlation functions for a pair or triplet of species i and j or i, j, and k, respectively, and the terms in angular brackets are orientationally averaged pair interactions, i.e., ijk
= J u^{\2)aQ dn / v
J άΩ^Ω
2
(2)
2
= J u ( 1 2 ) u ( 2 3 ) u ( 1 3 ) d f i d « d f i / J dti^^d^ ij
jk
ik
i j
j k
i k
1
2
3
.
(3)
In principle, other terms are present in E q . (1). However, for the system considered here, they vanish either because for the dipolar hard spheres 3
< ( 1 2 ) > = = 0 υυ
(4)
j
or because of the charge neutrality of the charged hard spheres, x
2
Σ i i = j
0
5
where zj is the valence (including sign) of the ion of species i. For simplicity, we assume that the salt is binary and symmetric and that all the hard spheres have the same diameter, a. Thus, x = x = x / 2 , x = 1-x and υ ^ ( Γ ) = ο ο for ra, 1
2
3
Chao and Robinson; Equations of State ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
13.
283
Equation of State of Ionic Fluids
HENDERSON ET A L .
2 2
u ( r ) = u (r) = n
= - u (r)
22
u (r)=
(6)
12
- i ^ t f - ^ ) r
1 3
u (r) = ϊ ψ r
(7)
(? · £ )
23
(8)
2
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and 2
r where species 1 and 2 are the charged hard spheres and species 3 is a dipolar hard sphere. The quantities e and μ are the electronic charge and the magnitude of the dipole moment of the dipolar hard spheres, respectively, and z= | z\ \. The caret above a vector indicates that it is a unit vector. Finally, D(12) = 3(M! ' r ) ( £ 1 2
10
' 1 2 ) - 0*1 * M )
( >
f
2
2
Thus, 2
= 2/3
(11)
2
(12)
and = I
The triplet orientation averages have been worked out by Rasaiah and Stell cos# >ccd = - T ^
14
and are
13
< >
>cdd = I {cos(0 - 0 ) + cos0 cos0 }
(14)
1 + 3cos0,cos0
