Derivation of the Poisson-Boltzmann Equation by Minimization

Derivation of the Poisson-Boltzmann Equation by Minimization of the Helmholtz Free Energy Using a Variational Principle

W. B. KLEIJN AND L. J. BRUNER

Department of Physics, University of California, Riverside, California 92521

Received August 9, 1982; accepted April 6, 1983

From the classical derivation of the Poisson-Boltzmann (P.B.) equation one does not obtain a clear insight into the approximations involved. As an alternative to sophisticated statistical theories we present a straightforward (mainly thermodynamic) derivation of the P.B. equation which does provide insight into the approximations made. In this derivation a free energy density is defined within the diffuse double layer. The free energy of the entire diffuse double layer can then be minimized, using a variational principle. The introduction of certain approximations results in the P.B. equation. The present method allows some corrections to the P.B. equation to be included.

I. INTRODUCTION

The Poisson-Boltzmann (P.B.) equation was developed independently by Gouy (1) and Chapman (2) to describe the ion distribution functions and the mean electrostatic potential near a charged interface. The P.B. equation has been found to give good pre- dictions for observable parameters over a wide range of experimental conditions (3-5).

Many authors have tried to evaluate the accuracy of the P.B. equation by comparing it to revised theories and computer experi- ments. Almost all of these studies make use of the "primitive model" (finite size ions in a continuous solvent). Recently, it has been shown that structural properties of the solvent become important close to the interface (6). Therefore results accurate for the primitive model may not be realistic close to the interface.

Monte Carlo simulations of the diffuse double layer using the primitive model have shown that the P.B. equation predicts the ion distribution functions for a 1:1 electrolyte accurately up to about 0.1 M (7-9). The mean potential function was found to be overestimated by the P.B. equation close to

20 0021-9797/83 $3.00 Copyright 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.

the interface at high surface charge densities (~0.05 C/m2). Thus, while uncertainty about the immediate vicinity of the interface re- mains, the P.B. equation gives good predic- tions at lower electrolyte concentrations.

Using certain approximations the P.B. equation can be derived from statistical mechanical theories. Thus, the P.B. equation has been shown to be the result of a zero-th order closure of a hierarchy of electrostatic equations (10). Similarly the P.B. equation can be obtained from the Boguliubov-Born- Green-Yvon integral equations (7, 11, 12) and also from the Ornstein-Zernike integral equations in the hypernetted chain approximation (13). The assumptions implicit in these statistical mechanical theories and the approximations made in its derivation from these theories provide insight into the assumptions implicit in the P.B. equation.

The assumptions implicit in the P.B. equation are not obvious from the equation itself. Our goal is to provide a straightforward derivation of the P.B. equation from well-known principles which demonstrates clearly the assumptions made. We derive the P.B. equation by minimizing the free energy within the diffuse double layer. Using a variational prin-

Journal of Colloid and Interface Science, Vol. 96, No, 1, November 1983

DERIVATION OF THE POISSON-BOLTZMANN EQUATION 21

ciple we optimize the independent functionals (potential and concentrations) under fixed boundary conditions.

II. THEORY

The free energy of an electrostatic system at constant temperature is (14)

Fe = [1]

where E is the dielectric constant, 4o is the permittivity of free space, E is the electric field, and the angular brackets indicate time averaging. Incorporated into Eq. [1] is the configurational entropy of the dipolar mop ecules of the dielectric medium. The diffuse double layer can be considered as a solvent medium of dielectric constant ~s containing hydrated ions of dielectric constant Ek. If C~ is the concentration (ions/unit volume) of ions of species k, Eq. [1] becomes

1 /We ---- ~ Es~0((E~) -- E fk(E~)ck) [21

k

where fk is a coefficient which depends on the geometry of the hydrated ion and on the dielectric constants es and ~k- In Eq. [2] E~ is the electric field that would exist if the dielectric constant of the hydrated ions, ek, were equal to that of the solvent, es. The second term on the right-hand side is of course just the polarization energy of the hydrated ions.

The electric field Eo~ can be attributed to all of the individual charges (both mobile ions and adsorbed surface charges) present in the system

E~ = Z Z EiEj i j

= Z E~ + E Z EiEj i i i

j i . [3]

In the preceding equation each subscript ranges over all the ions and adsorbed charges present. Since we will be concerned with the boundary layer adjacent to an infinite plane wall, we need be concerned only with scalar

field components directed perpendicular to the plane of the wall. Parallel feld components will vanish on time averaging and by symmetry. The time average of E2~ can be calculated if one assumes that there is no correlation in the locations of the ions:

= E + E E

22 KLEIJN AND BRUNER

cations and a fraction c-l) e is "occupied" by the anions, where ve is the exclusion volume of an ion. The contribution of this volume element to Eq. [3] (sums replaced by integrals) must therefore be diminished by -E~ (c+)2v,dr (cation-cation exclusion) and -E~(c-)2v,dz (anion-anion exclusion), and increased by 2E~c+c-vedr (anion-cation exclusion) where dr is the volume of the volume element. The total contribution due to the hard-sphere character is thus -E~(c -c - )2dr . Incorporating the hard-sphere character of the ions into Eq. [5] results now in

(EL) = f (e + + c-)d

(E , ) 2 - (E~) f~ (c + - c-)2v,dr. [61 +

The added term in Eq. [6] gives rise to the so-called cavity potential, which has been shown to be an important correction to the P.B. equation (15). Note that the cavity-potential contribution vanishes in the bulk solution where c + = c-.

Combination of Eqs. [2] and [6] followed by integration over space formally gives the free energy of the diffuse double layer. How- ever, the double integrals arising in the evaluation of the polarization energy of the ions do not allow use of the variational method. To prevent this, we will omit the first and last term of the right-hand side of Eq. [4] in the evaluation of the polarization energy of the ions. Thus we obtain

f Fo&= IgAc+ + c )

~0~s + -y {(Eoo) 2 - &(c + - c-)2ve

- fj(c + + c-)(e~)2}ld~ where Ve is the volume of an ion and

gj = ~-

[7]

[8]

Here it is assumed that & is a constant, i.e., that variations in the dielectric constant (and in particular its discontinuity at the interface) can be neglected. & is the hydration energy of the hydrated ions.

As mentioned before, the electric field E~o is a hypothetical field which would appear if the dielectric constant of the hydrated ions, ~h, reduces to that of the solvent. The matter of the dielectric constant of hydrated ions is considered further in Part B of the Appendix. From knowledge of the relation between the dielectric constant of the pure solvent and the actual mixture of solvent and ions, the Maxwell-Lewin formula (16), a coefficient relating the actual mean electric field, (E), and (E~) is obtained:

\2es + eh/)

x (E~) =- h(E~) [91

where vt is the total volume fraction of all ions present.

Thus the calculation of the electrostatic part of the free energy has been completed. The major assumptions in this calculation are (i) no spatial correlation between the ions exists, except for hard-sphere exclusion, (ii) the solvent medium in which the hydrated ions are submersed is a continuum with a well-defined dielectric constant (which may vary in space).

To complete the computation of the free energy of the diffuse double layer, another contribution must be introduced which can be obtained from either of two physical in- terpretations of the diffuse double layer system. One can consider the diffuse double layer system as a gas of ions in a background "ether" (the solvent) which screens the electrostatic interactions, or alternatively, as a mixture of ions and water molecules. In the first case (which we will not discuss in detail), the expression for the Helmholtz free energy of an ideal gas (again we neglect spatial correlations of the ions) must be added to the electrostatic contribution to obtain an

Journal of Colloid and Interface Science, Vol. 96, No. 1, November 1983

DERIVAT ION OF THE POISSON-BOLTZMANN EQUATION 23

expression for the free energy of the diffuse double layer. Using similar methods, such as those described below for the mixture interpretation, the P.B. equation can also be obtained.

Before we begin our discussion of the derivation of the P.B. equation from the mixture interpretation, we digress briefly to consider an important approximation made in the derivation of the P.B. equation (using either interpretation). In both cases, expressions developed for bulk phases are applied to an inhomogeneous region. This makes it pos- sible to define local free energy densities in the diffuse double layer. It has been shown (17) that the implicit assumption of such an approach is that the spatial correlation of the ions is short as compared to the inhomoge- neity of the system. For the diffuse double layer, this means that the (long-range) Cou- lombic correlations are neglected, which is consistent with our earlier assumptions. The finite size of the ions will invalidate the usage of expressions developed for bulk phases only at very high concentrations.

The free energy of mixing of the ions and water molecules can be separated into energy and entropy contributions. The energy of mixing (dissolution) has already been included in the electrostatic contribution to the free energy (Eq. [8]). As will be shown later, the energy of mixing does not contribute to the free energy of the diffuse double layer when one considers the reference state, and therefore, any nonelectrostatic parts of the mixing energy are not considered. Therefore only the entropy contribution due to mixing has to be included. If we assume that mixture to be ideal (18) we get

TSm = kT{-(1 - x + - x-) In (1 - x + - x-)

-x +lnx +-x - lnx -} [10]

where x + and x- are the mole fractions of cations and anions in the mixture.

In our present discussion of the diffuse double layer, we have not yet considered the fact that the ions and water molecules are in

equilibrium with the bulk solution (the reference level of the free energy). When re- moving ions and water molecules from the bulk solution to create a diffuse double layer, the free energy of the bulk solution dimin- ishes by

Fb = g~(c + + C-) + kT{(1 - x + - x-)

In (1 - x~ - xS) + x + In x~-

+ x- In x8 [11]

where x~ and Xo are the mole fractions of cations and anions in the bulk solution. In Eq. [ 11 ] the first term describes the hydration energy and the second term describes the entropy. If the dielectric constant does not vary within the diffuse double layer the hydration term will cancel out in calculation of the free energy of the diffuse double layer (wall effects are neglected).

From combination of Eqs. [7], [9], [10], and [ 11] an expression for the free energy of the diffuse layer can be obtained:

f Fdr=f [2eOes{h2(E)2 -g j (c+-c )2/.) e

- h2f~(c + + c-)(E) 2) - kTl (1 - x - - x +)

ln(} -x - -X~)xo +x- In (x~)

+ x + In dr [12]

where it has been assumed that the dielectric constant of the solvent does not vary within the diffuse double layer.

The complete character of the Coulombic interactions has not yet been incorporated into our discussion. This is most conve- niently accomplished by use of the Poisson equation for the mean electrostatic potential, denoted here as g,:

dx h = -q (c + - c-)/~O~s [131

where a fiat diffuse double layer has been assumed. With Eq. [13] and from the rela-


24 KLE I JN AND BRUNER

tions x + = v~c + and x- = v~c- the free energy density of the diffuse double layer can be written as a function of c +, d~k/dx = (E>, and d/dx(h(d~/dx):

o ~ Fdx

fo ( de, d ( d~]]dx [14] = F C+,dx,dx h~x/!

(for a flat diffuse double layer). With the variational principle (19) the

functionals c + and ~ of Eq. [ 14] can be op- timized so that the free energy of the diffuse double layer is at a minimum. At this point we will assume that the coefficient h (Eq, [9]) can be set equal to unity. Including the coefficient h in our derivation is straightforward but makes the computations tedious. The variational derivative ofF(c +, ~', ~") in c + is

6c + -- e0e~{-f?p '2} - kTln = 0 [151

where the equalities x+-/x~ = c+-/c~ are assumed to hold. Equation [ 15] can also be written as

C + C- / - - ,oes f ,~b '2~

4Co = exp[ kT ]" [16]

The variational derivative in 1/' is

-~ = ,o,s -~x + g~ve ~x 2 (c + - c - ) 1 2 2 d2ff ,2 d E0~s

+ ~ ~ f, ~ + G,,of, dx ((c+ - c-)~b')

eseo kT~--;~ In c - = 0 . [171 q

In the concentration range for which the P.B. equation is used, (c + + c-)fj ~ 1 and therefore the fourth term can be neglected compared to the first. Then Eq. [17] can be in- tegrated with ~ = ~' = 0 in the bulk:

c -= Co exp[(q~+gjve(c+-c -)

l eoe.sfj~'2)/kT ] [18] 2

Combination of Eqs. [13], [16], and [18] gives a modified P.B. equation:

d2d/ I ( _qd /g jVe(C+C- ) ~x z = Co exp - -

1 exp[(q~k+g, ve

(C+--C - ) - leOeOf ,~/2) /kT ] . [19]

This modified P.B. equation includes a cavity potential correction and a correction for the polarization energy of the ions. Using the variational method, the effect of the electrolyte concentration and saturation (depending on the analytical expression used) can also be included in the modified P.B. equation. Omission of the correction terms in Eq. [ 19] gives the classical P.B. equation.

III. CONCLUSIONS

In the present derivation the P.B. equation is obtained by minimizing the free energy of the diffuse double layer. Such a derivation provides useful insight into the approximations involved without employing sophisticated statistical theories. In summary the approximations leading to the P.B. equation are ;

(1) The solvent medium is assumed to have a dielectric constant which is well defined on the molecular scale used; i.e., the molecular structure of the solvent is ignored. Saturation of the dielectric constant is assumed not to occur. Corrections due to saturation can be included by modifying Eq. [ 13] appropriately, but the use of such (macroscopic) corrections is doubtful on the molecular scale considered.

(2) No spatial correlations between the ions in the diffuse double layer are assumed to exist. Strictly speaking, this is not true for the derivation of the entropy contribution (Eq. [10]) from the equations for an ideal mixture. In this derivation ions and molecules are assumed to be of similar (finite) size


DERIVAT ION OF THE POISSON-BOLTZMANN EQUATION 25

(18). The hard-sphere character of the ions can be included in the electrostatic contribution to the free energy by introducing a cavity potential term (Eq. [6]). If the P.B. equation is derived from a gas model for the ions the hard-sphere character could be included by a van der Waals type correction. These corrections are inaccurate if the parameters of the diffuse double layer change significantly within a length equal to the ra- dius of the ion (17). Since correlations arising from Coulombic interaction are long range they cannot be included in a method using local free energy densities.

(3) The polarization energy of the ions is neglected. If one assumes a fixed dielectric constant for the hydrated ion a correction for this effect is rather straightforward (Eq. [2]).

(4) Changes in the hydration energy of the ions are neglected. The hydration energy of the ions is dependent upon the dielectric constant of its environment (Eq. [8]). Changes in the hydration energy due to the discontinuity of the dielectric constant at the interface are called "image effects" and are often significant. Dielectric saturation also leads to sharp modifications of the P.B. equation (20), but the use of macroscopic theories gives information of little quantitative value.

(5) The compressibility of the solvent medium is ignored. This leads to a small er- ror (15).

Bell and Levine (21) have also considered the approximations cited above in the con- text of a "local thermodynamic balance" derivation of the P.B. equation. These authors also calculated the free energy of the diffuse layer, using the result to determine the electrochemical potentials of the ion species in the diffuse layer. Imposition of the require- ment of constancy of the electrochemical potentials, with summation over species, led them to a first integral for the P.B. equation with correction terms. The novelty of our approach resides in the use of the variational method, rather than in the discovery of any hitherto unsuspected correction terms.

From the Monte Carlo studies it has been shown that recent statistical-mechanical theories give a reasonable description of the primitive model of the diffuse double layer (9). The accuracy of the P.B. equation for the primitive model can be assessed from these theories, or directly from the Monte Carlo studies. As mentioned in the introduction the P.B. equation was found to give a satisfactory description of the primitive model in a large range of the parameters involved. Estimates for the polarization energy correction are probably fairly accurate (approximation 4).

However the magnitude of approximations (1) and (4) are most difficult to evaluate because of the molecular structure of the solvent. Only recently a first approximate theory which treats the solvent molecules and the ions on equal footing has been presented. No Monte Carlo studies of such systems are presently available. The approximate theory predicts that effects due to solvent structure are important close to the interface. Thus, conclusions reached for the primitive model are probably not accurate for a real system.

In conclusion it can be said that the structure of the diffuse double layer away from the interface is well understood. The P.B. equation gives a good description of this region in most cases. Properties of the diffuse double layer close to the interface are presently not well understood and, therefore, the accuracy of the P.B. equation in this region is not certain. However it appears justified to say that the P.B. equation is a good approximation also for this region at low electrolyte concentrations, and surface charge densities.

APPENDIX

A. Macroscopic and Microscopic Electric Fields

It is useful to rewrite Eq. [4] in the form

((E~ - (E~)) 2) -- ~ ((Ei- (El))2). [AI] i

This result states that, in the absence of spa-


26 KLEIJN AND BRUNER

tial correlation of charges, the mean squared fluctuation of the macroscopic field about its mean value at any point in space, (Eo~), is obtained by summation over all charges (both mobile and adsorbed) of the corre- sponding quantity for each charge. Restrict- ing the summation to a 1:1 electrolyte of mobile anions and cations identical except for the sign of their charge, Eq. [A1] becomes

correction for the polarization energy of the ions, can be carded out using the variational method with the right-hand side of Eq. [A 1 ] set equal to zero. Retaining the term, N(E2i), introduces the ion hydration energy, which does not enter the P.B. equation, in our approximation, and the cavity potential correction which has been included in our derivation.

((E~ - (E~}) 2} = N((E~ - (E~}) 2} [A2]

where N is the total number of such charges in the system. Removal of adsorbed charges from the summation at this point is justified by the argument presented following Eq. [5] above. If at this point we set (El} = 0, and N = fv (c+ + c-)dr, Eq. [A2] reduces to Eq. [5] of the text. We can, on the other hand, permit (Ei} 4: 0, and estimate an upper limit for (E~} 2 as follows, setting

(E~) ~ N(Ei). [A3I

Combining Eqs. [A2] and [A3] yields

((E~ - (E~}) 2} I ((gi - (Ei}) 2 [A41 (Eo~} 2 N (Ei} 2

which states that the fractional mean squared fluctuation in the macroscopic field varies in inverse proportion to the total number of particles contributing to that field. In this form the result clearly resembles familiar results of the calculation of energy and density fluctuations in many-particle systems (Ref. (18), pp. 34-38). If we expand Eq. [A2] and substitute from Eq. [A3], we get

(E2} - (1 -1 ) (E ,~}2 ~ N(E~}. [A5]

Since expansion of Eq. [A2] with (Ei} = 0 yields

(E2} - (Eo~} 2 = N

DERIVATION OF THE POISSON-BOLTZMANN EQUATION 27

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Derivation of the Poisson-Boltzmann Equation by Minimization