3 Perturbation Theory, Ionic Fluids, and the Electric Double Layer DOUGLAS HENDERSON IBM Research Laboratory, San Jose, CA 95193
The properties of many fluids can be regarded as those of a simpler fluid (usually a hard-sphere fluid) plus some corrections. Perturbation theory, which is based on this idea, is reviewed briefly. Many earlier approaches, such as the virial series and the van der Waals theory, can be regarded as special cases of perturbation theory. Perturbation theory is applied to ionic fluids and is found to be useful provided that the coulomb potential is resummed. It is useful to restructure the perturbation expansion so that the mean spherical approximation is the leading term in the series. Finally, perturbation theory is applied to electrified interfaces, where results similar to those of the mean spherical approximation are obtained using simple arguments.
P
ERTURBATION THEORY HAS B E E N , at the very least, one of the most
significant developments in the theory of liquids during the past two decades. Perturbation theory combines accurate results for the thermodynamic properties with a pleasing physical picture and relatively straightforward numerical calculations. In particular, perturbation theory avoids the often frustrating convergence problems characteristic of the iterative procedures used in the numerical solution of integral equations arising from, for example, the hypernetted chain equation. Although it is only recently that the power of perturbation theory has been fully appreciated, perturbation theories have a venerable history. The van der Waals theory of dense gases and liquids is an early form of perturbation theory. The van der Waals theory is not surveyed here because the connection between it and perturbation has been pointed out previously (I). The virial expansion of a dense gas is another early form of perturbation theory. In this chapter, perturbation theory is briefly reviewed with an emphasis on pointing out its generality. Its application to electrolytes is considered. The chapter concludes with an application of perturbation theory to an interfacial problem, the electric double layer. 7.25/0 Society U55 16th St. N. W. Washington. 0. C. 20036
48
MOLECULAR-BASED STUDY OF FLUIDS
Perturbation
Theory
Our starting point is the free energy, A = -kT In j exp { - ( 3 0 } ^ . . . dr
N
+ terms independent of the density
(1)
where (3 = 1/fcT, T is the temperature, and 0(r . . . r ) is the potential energy of the N molecules whose centers of mass are at r . . . r . For simplicity, assume that the potential energy is pairwise additive x
N
x
N
(2) where r = \r - r j . Hence, the free energy becomes (j
t
A = -kT \ n eir^dr, . . . dr + J io [ r 2
f [ f
= 2
JO
*
+
6TT Jr„>o r 1 3 r 2 3
12
J dr_2 2 r
+ ". . .' j
1 2
6TT Jn/>o r r23
12
( ^ _ ^ f
Jri2>0 [ r
J
13
12
...]
+
4TT Jrij>0
12
r _
f^f^
J
Il rf '
d
K
2
(42)
f"
12
Thus, defining ^(r) as %r )
=
l2
— r
-£I
dr*
(43)
4TT Jr >o r r23
12
y
13
and taking the Fourier transform cg(fc) =
(44)
f %(r) sin fcr r
K Jo we have 4TTK
fc
2
2
+ K
(45) 2
Hence o\ This suggests that improved results might be obtained by taking the integrals for the region 0 < r < cr and combining them with the K term. Thus tJ
(j
{j
3
A
1 - - K(T + - K a 4 4
0
4
12irp
NkT
-
£
h
°
4
—7 K V + 8
2
• • •
+
i r ) d r
2
( 4 9 )
At first sight, K (1 - 3KO-/4 + 3K CT /4 - 7K CT /8 + . . .) seems to be an unpromising combination; it is, in fact, (2T) (1 + 3Ta/2) where K and T are related by 3
2
2
3
3
3
K
= 2T(1 + IV)
We note that 2T ^ K. Expanding
2 Thus,
4
(50)
3.
HENDERSON
59
Perturbation Theory
Retaining only the first term gives the mean spherical approximation (MSA) (6, 7). As is seen in Figure 1, the first (MSA) term gives fairly good agreement with computer simulations (8-10). In principle, Equation 52 gives a series of corrections to the MSA. At low concentrations, where K is small, the corrections are negligible. At higher concentrations the convergence is fairly poor and a Pade summation is required. Henderson and Blum (IJ) have suggested changing the expansion parameter from K to 2T. They obtain A - A
0
(2T) (1 + 3Ta/2)
(2T)
12irp
16TTP
3
NkT
4
(53)
h (r)dr + 0
The higher order terms in this expansion are given by Henderson and Blum. Since 2T ^ K , this series of corrections to the MSA is better behaved. In fact, the corrections are negligible at normal ionic concen-
0.25
0.50
0.75
1.00
C o n c e n t r a t i o n (m/l)
Figure 1. Internal energy of a 2:2 model ionic solution. Conditions, cr = 4.2 A; e = 78.358; and T = 298.16 K. Key: o, computer simulation values of van Megen and Snook (10); computer simulation values ofValleau et al. (8, 9); and —, MSA results.
60
MOLECULAR-BASED STUDY OF FLUIDS
trations. At the densities characteristic of fused salts, the correction terms would make an appreciable contribution. At low concentrations, the correction terms, given in Equations 52 and 53, are not the most important corrections to the MSA result. This is because there is, in the fourth-order term, the contribution AA NkT
= - }
PP 2
/ a
(58)
where £l and i l are variables specifying the orientation of molecules 1 and 2, (x is the dipole moment 2
l
D(12) = 3(Ai • r ) (fe • f ) - iii • U 12
12
(59)
and (Li and f are unit vectors. Before considering this system of hard spheres with embedded point dipoles, a few general comments about the application of perturbation theory to molecules with nonspherical pair potentials are in order. Nonspherical molecules are a more complex system than the simple spherical systems discussed so far. The dipolar hard spheres considered here are an especially simple system because the hard core is spherical. The u-expansion is appropriate and everything proceeds in a reasonably straightforward manner. For more complex systems, where the core is nonspherical, the situation can be more complex. If a spherical reference system is used then some of the perturbation energy may be large and positive. If so, a w-expansion is inappropriate and an/-expansion is preferable, at least for the regions where the perturbation is large and positive. However, the penalty we pay is that in the region where Ui(r Cl) is negative, /(r,ft) can be very large, which might result in convergence problems. Ideally one would like a nonspherical reference fluid. Should such a reference fluid be available, perturbation theory might be more widely useful for nonspherical potentials. Expressions for the free energy of many fluids consisting of nonspherical molecules are available (2). However, there are no expressions presently available for g(r,fl) for such systems. Fortunately, such problems need not concern us when considering dipolar hard spheres. Using the u-expansion 1 2
9
A=A + 0
2 ((V)"A
(60)
n=l
where A is the free energy of hard spheres of diameter a 0
A, NkT
(61)
62
MOLECULAR-BASED STUDY O F FLUIDS
and uf(12) is the pair potential divided by |x. The angular integral in Equation 61 is zero. Hence A = 0. After performing the angular integrations 2
Y
A /NkT 2
= - | p | r-%(r)dr
(62)
and A /NkT 3
= ^ P / " go(123) dr dr
(63)
1 + 3 cos 0i cos 0 cos 0 ; r
(64)
2
123
2
3
where 2
"123
=
3
3
Vri2rl3 23/ r
and r and 0 are the sides and interior angles of a triangle formed by molecules 1, 2, and 3. We see that these expressions for dipolar hard spheres are quite similar to the virial expansion and to the expressions for charged hard spheres. Only the ring diagram survives in the third-order term. In fact, if we took only the diagrams that contribute to the virial expansion and then expanded in powers of (3 we would obtain Equations 38 and 60-63. The integral in Equation 62 is easily evaluated. Barker et al. (13) have calculated the integral in Equation 63. As is seen in Figure 2, the free energy series obtained from Equation 60 converges very slowly. The perturbation terms seem to alternate in sign. Rushbrooke et al. (14) have employed the Pade sum {j
{
and found it to be in good agreement with computer simulations (15). This is shown in Figure 2. There is some indirect evidence (15) from computer simulations that the terms in the perturbation sum, Equation 60, are negligible for n > 4. Hecht et al. (16) have obtained formal expressions for A and A , and Tani et al. (17) have made some progress towards calculating A . It will be interesting to see if a truncated series agrees with the simulation results. In the case of the charged hard spheres, it was found helpful to 4
5
4
3.
HENDERSON
Perturbation Theory
63
)3ju /a 2
3
Figure 2. Free energy of a dipolar hard-spherefluid(pa = 0.8344) as a function of reduced dipole moment. Key: •, computer simulation values ofValleau and Patey (15); , 2 and 2 + 3, results of Equation 60 when truncated after 2 and 3 terms, respectively; —, MSA results; solid curve, results of Equation 65. 3
remove the MSA results from the perturbation series and write the series as a correction to the MSA. This could also be done for the dipolar hard spheres. If this is done
A/NkT = A /NkT MSA
-
where y = 4TTPP|A /9. The similarity to Equation 52 is striking. Terms important at low densities can be constructed in a manner analogous to Equation 57. We should not expect this series of correction terms to the MSA result to converge quickly. A Pade summation may still be required. It may or may not be preferable to the original series, Equation 60. 2
64
M O L E C U L A R - B A S E D STUDY O F FLUIDS
However, in view of the fact that A is not too bad an approximation to A, as is seen in Figure 2, using A as a starting point, rather than A , does not seem too bad an idea. At low densities, terms analogous to those in Equation 57 would have to be included in the perturbation expansion. M S A
M S A
0
Perturbation theory can also be applied to the calculation of the dielectric constant. The result is (e -
l)(2e + 1) 9e 3 cos e - 1 2
3
g (123)dr dr + . . 0
( l3 23) r
r
2
3
(67)
3
Including only the first term in the above series gives the Onsager result for e (18). This series for e can be rewritten by expressing it as a correction to the MSA for e rather than as a correction to the Onsager result. Hence (e - l)(2e + 1)
(e - l)(2e + 1)
9e
9e
^=1
MSA
The integrals in Equations 67 and 68 can be calculated by the same techniques as those used in the calculation of the integral in Equation 63. However, the numerical problems are more difficult because the integral is long ranged. Despite this, Tani et al. (17) have been able to calculate this integral and thereby obtain e. Their results are promising but still preliminary and so are not presented here. The perturbation theory presented here is based upon the u-expansion. Dipolar hard spheres have been treated by /-expansion techniques also (19). The method has some advantages but is more complex since the angle averages cannot be performed analytically. In the past two sections, perturbation theory has been applied to the ions and the solvent separately. What is needed is a treatment of a mixture of ions and dipoles. This has not yet been done. The formulation of perturbation theory for this system would be more difficult as dipoles as well as ionic terms will have to be resummed to avoid divergences.
3.
65
Perturbation Theory
HENDERSON
Electric Double
Layer
Let us consider first ions near a charged electrode. As before, we consider the ions to be charged hard spheres of diameter a. The electrode is approximated as a uniform, hard, charged wall. First let us consider the case where the solvent is a uniform dielectric medium. If the electrode is charged, there will be an accumulation near the electrode of ions whose charge is opposite to that of the electrode. We can then speak of a double layer of charge. To apply perturbation theory to the system, we consider the electrode to be a large ion whose diameter is R » a and whose charge is Q. Eventually, we will take the limit R - ^ 00. Since the concentration of this large ion is l/N
A
= o" A
2w* / 2
2
d r
where the sums are over the bulk charged hard spheres and go (r) is the radial distribution between the large hard sphere and the bulk hard spheres. Hence, the excess free energy due to the presence of this large ion is A = 28
J(R
2e
Jo
= _
r
+ a)/2
d
gg(r) dr 6
0 V
1
_ ^!2! r 2e Jo
r
m
d
r
+...
7o)
(
Replacing the divergent integral in Equation 70 by the ring sum of which it is the first member gives
AA = -
2e
Jo
KQ f 2
2
~ dr
e
Kr
+ —^2e Jo
00
-
hS(r)dr + . . . 2e
J(R+
