log |P| (Nrri2)
82 C (r,s) =
5.2 Mean field approximation
In lowest order approximation correlations are ignored so that P^ay(r,s) = pa(r) py(s). Then with the ideal gas as a reference system (u^OjS) = 0) integration of Eq. (5.1.7) yields the mean field result
mf id 1 T"' f
y = 7 + T > P (r) P (s) U (r,s) dr ds (5.2.1)
2 J a y aT
a,y
with the equilibrium profile determined from Eq. (5.1.9) as
Ho(r) = ^ - <Da(r) = kßTln { X* pa(r)} + y(t ) (5.2.2) Va(r) = X J P (s) u (W) ds (5.2.3)
Y
In mean field theory the pair potential contributes to the profile only through the mean potential \|/a(r). Rearranging gives the Boltzmann equation
Pa(r) = zaexp- ß { y a(r) + C>a(r) } (5.2.4)
In the absence of any mean or external potentials, the fugacity za=A,a“3 exp{ß|ia} reduces to the density of species a in a uniform bulk fluid.
The Poisson-Boltzmann (PB) description of the double layer follows if all short range interactions and image potentials due to dielectric
discontinuities are ignored. Note that the self image interaction (part of the external field) should remain in any rigorous mean field analysis. However, since in reality this is screened, including it while ignoring correlations gives strange results at large separations. Hence image charge effects are not yet considered, and the pair potential is taken to be the Coulomb interaction in an infinite medium with uniform dielectric constant £j
u (r,s) = - Ü - (5.2.5)
“Y El lr-sl
where qa is the charge of the species a. Taking the total mean electrostatic potential as \|/(r) = ( \|/a(r) + Oa(r)) / qa the non-linear PB equation emerges as
V2\|/(r) = ~ — X V o ^ P t - ß V ^ r ) )
^ a
83
84
The PB result for the thermodynamic potential is now obtained. To do this, substitute Eq. (5.2.1) into Eq. (5.1.8), and take H0 to be the external, configuration independent, electrostatic free energy only. Then the
thermodynamic potential consists of an ideal term (less the chemical potential part) and mean electrostatic terms; i.e. with the ionic charge density Q(r) = Sq^ pa(r), and an external charge density a(r), we have
Qpb = y ld - X n a J p o(r) dr + 4-JQ(r)y(r) dr + ^ J a ( r ) y ( r ) dr (5.2.7)
a 2 2
Use of the optimised profile Eq. (5.2.2), gives
£2PB = -kBT X j p a(r> - 4 j Q(r) \|/(r) dr + y j a ( r ) y ( r ) dr
= ~kBT X [ p a(r) ^ - “^“ Je(r)(V \|/(r) ) 2 d1* + Ja(r)\jf(r) dr (5.2.8)
In obtaining the final line, the middle term has been identified with the mean electrostatic energy. The PB pressure is obtained by differentiation with respect to separation.
The term H0 we have used above does not include the Lifshitz
interaction. The reason is that the Coulomb potential Eq. (5.2.5) is that for a infinite uniform dielectric medium. Since the image interactions have been excluded in the PB analysis, it is not appropriate to add here the Lifshitz interaction which also arises from dielectric discontinuities. It is also
inconsistent to include dipolar fluctuations while ignoring those due to ions. In next approximation both image charge effects and correlations in the electrolyte will be included. It is then essential to include the Lifshitz contribution in H0. Otherwise, as will be made explicit, when dielectric discontinuities are present the primitive model gives a spurious long range repulsion which is cancelled identically by the zero frequency Lifshitz term (see parts I and II).
5.3 Images and correlations
The mean field approximation above includes no correlations. Both the direct and indirect correlation functions were taken to be zero. A better approximation follows from Eq. (5.1.13) if we take the second functional derivative of the non-ideal p art of the mean field free energy Eq. (5.2.1). This gives
c (r,s) = -ßu (r,s) (5.3.1)
ay ay
Note th a t this closure is a Debye-Hückel approximation which is equivalent to the m ean spherical model with no hard core radius. In fact, this is the correct asymptotic form for the direct correlation function, which is to be expected since the m ean field superposition approximation for the pair correlation function is also exact for large particle separations. Use of this closure provides an improved description of the double layer. For bulk electrolytes Debye-Hückel theory fails a t high concentrations because short range effects become important. However, the thermodynamics is given correctly in the low dilution limit, since there the system is dominated by the tail of the Coulomb potential. This provides some motivation for persisting w ith this approximation. However, the real justification will come a
posteriori, when the results are compared with more sophisticated calculations.
One proceeds by substituting Eq. (5.3.1) into the Omstein-Zemike equation and obtain an expression for the indirect correlation function. The pair potential coupling constant integral can then be formally evaluated to give the next approximation for 2T.
To tu rn on the pair potential, invoke a coupling constant i.e.
u a7(r,s;X)=X u ^ fo s). Then if 2T = tFmf + Drcorr, with the mean field free energy given by Eq. (5.2.1), one has
l
y =
f
dA,f
p (r) p (s) h (r,s;X) u (r,s) dr ds J J a y ay ay a,y 0 85 (5.3.2)86
The required partially coupled indirect correlation function h (r,s;A.) follows'*» from the Omstein-Zemike equation ( with ca^(r,s;X) = -ßÄ.ua>y(r,s)) as
h (r,s;X) = -ßXu (r,s) - ßX X f u s(r,t) p ft) h^(t,s;>.) dt (5.3.3)
cry ay J ao 5 5y
5
Integration of this equation, substitution into Eq. (5.3.2), followed by the coupling constant integration, then gives for the correlation free energy
corr
y
Pa(r) p (s) U^(r3) u^(s,r) dr ds+ Pa(r) p^(s) p5(t) u^(r,s) §(s,t) uga(t,r) dr ds dt (5.3.4)
By discretising this expression (replace the integrals by Riemann sums), it becomes clear that it represents a sum of traces of powers of a matrix. The expression can be resummed if we determine the eigenvalues A of the integral equation
p (s) u (r,s) f (s) ds
y ay y A f (r)a (5.3.5)
Then the correlation free energy may be rewritten as sums of powers of the eigenvalues corr S -k T 2 oo oo kBT V - V - 2 r f InU-Aj) + A. L i=i (5.3.6)
Note that the term in Ai represents the self interaction which cancels the first term in the expansion of the logarithm. Since the self image interaction will later be added via the external field, it is actually only the (infinite) direct Coulombic self interaction which is excluded from the free energy.
Further progress requires determination of the eigenvalues. For notational convenience first write the pair potential as ua(y(r,s) = qyu(r,s) and define the local inverse Debye length by
K2(r) = I(r) = X , q2 p (r) (5.3.7)
£l el a “ “
87
eigenfunction fa(r) =qaflr) may be rewritten
- ß J I(s) u(r,s) f(s) ds = A f(r) (5.3.8)
Note that the profile remains unspecified in the present analysis.
Now particularise to planar geometry. The electrolyte is confined to a region of dielectric constant ex of width d = h + 2w separating two half spaces of dielectric constant (fig* 5.1) with the distance of closest approach of the ions to each surface being w (the zeroth order Stem layer). As the area A of the planar surfaces becomes infinite, the spectrum of the eigenvalues
becomes continuous in the (x,y) direction parallel to the interfaces. Then in Fourier space the sum over wave vectors k is replaced by a two-dimensional integral with a density of modes A/(2jc)2. The correlation free energy can then
be written (with dk=2xkdk)
coir kRT A r^ r 1
S = —--- V { ln [ 1 - A.(k) ] + A.(k) } dk (5.3.9)
8tc2 J m
and the eigenvalue equation becomes
- ß j u f k ^ z ^ 1 ^ ) ffz^ = A(k)f(zx) (5.3.10)
The circumflex denotes the two dimensional Fourier transform of the cylindrically symmetric potential, and the eigenfunction is an implicit function of k.
Because u(r,s) is the potential at r due to a charge in the electrolyte at s, it satisfies V2u = 0, I z I > h/2, and, V2u = -(47t/Ej) 5(r-s), I z I < h/2. In Fourier
space these equations can be written
ü"(k,z,z1) - k2u(k,z,Zj) = 5(z-z1) (5.3.11)
*1
and the boundary condition can be shown to be
k e 2 ü(k,d/2,z1) + ^ ü*(k,d/2,z1) = 0 (5.3.12)
Using these results the eigenvalue equation (5.3.10) can be rewritten as
F ig u re 5.1 In planar geometry, the width of the electrolyte is h and that of each Stem layer (from which the ions are excluded) is w. The separation of the surfaces is d=h+2w, which is also the width of the intervening uniform dielectric er This region separates two half-spaces with dielectric constant e2.
88