On the Chemical Free Energy of the Electrical Double Layer

Transcription

1 1114 Langmuir 23, 19, On the Chemical Free Energy of the Electrical Double Layer Marian Manciu and Eli Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 1426 Received September 26, 22. In Final Form: December 16, 22 The free energy of interaction between colloidal particles due to the overlap of their double layers was traditionally calculated either for systems of arbitrary geometry interacting at constant surface potential or at constant surface charge or for parallel plates interacting under arbitrary surface conditions. An expression is obtained for the change in the chemical contribution to the free energy of the double layers during a general interaction, which allows the calculation (within the Poisson-Boltzmann formalism of the interaction free energy for systems of arbitrary geometry and surface conditions. The change in chemical free energy depends not only on the values of the surface charge and potential at the final state but also on their values at each distance between infinity and the final state. A simple approximate expression for the change in the chemical free energy contribution is also proposed, which involves only the states at infinite and final distances. Its accuracy is tested for planar and parallel surfaces, with charges generated via the dissociation of surface groups. I. Introduction Gouy 1 and Chapman 2 were the first to predict the distributions of electrolyte ions in the vicinity of a charged surface, by assuming that the ions obey Boltzmann distributions and interact with a mean potential, which satisfies the Poisson equation. The distribution of ions between two planar charged surfaces (the difference in ions concentrations between middle distance and infinity was later related to the interaction force between the two surfaces by Langmuir. 3 The interaction free energy between parallel planar plates, separated by a distance l, could then be obtained by integrating the force from infinity to the distance l. The Langmuir procedure cannot be applied directly to nonplanar surfaces. However, an approximate method, the Derjaguin approximation, 4 in which the interaction between curved surfaces is calculated in terms of the interactions between planar surfaces, could be employed. The method is accurate only when the radii of curvature of the surfaces are sufficiently large compared to the Debye-Hückel length and to the distance of closest approach. Major progress was achieved by Verwey and Overbeek, 5 who proposed a direct method to calculate the interaction free energy between particles of arbitrary shapes, when either the surface potentials or the surface charges remain constant during the interaction. Their approach involved an imaginary charging process, in which the relation between the surface potential and the surface charge density was obtained by integrating the Poisson-Boltzmann equation for one of the above constraints. Another approach consisted of writing the free energy as the sum between the energy of the electrostatic field, an entropic term due the free ions and a chemical term due to the charge transfer to the surface. 6 However, expressions for the chemical free energy contribution have been derived only for interactions that occur either at constant surface charge density or at constant surface potential. For these cases, the latter method was shown to be equivalent to the imaginary charging approach. 6 Whereas the charging approach could be applied to any geometry but only at constant surface charge or potential, the Langmuir expression could be employed for any surface conditions but only for parallel planar plates. The addition of electrostatic, entropic, and chemical contributions would allow the calculation of the free energy of interaction for systems of any shape and any surface conditions, if one could derive a general expression for the chemical free energy contribution. There are colloidal systems for which the interactions can be well described by either constant surface charge density or potential. For example, when the surface charge is generated by the dissociation of surface groupsssuch as those of surfactantssand the dissociation constant is sufficiently large, the constraint of constant charge is a suitable one. In addition, in many cases the radii of curvature of the surfaces are large enough for the interactions to be accurately calculated by combining the Langmuir equation with the Derjaguin approximation. There are, however, colloidal systems, for which the above requirements are not fulfilled. For example, the surfaces of the proteins can have small radii, and their acidic and basic groups are not completely dissociated. In this case, the Verwey-Overbeek approach cannot be employed and the Langmuir-Derjaguin approximation is not accurate. The purpose of this article is to obtain a general expression for the chemical free energy contribution, when neither the surface potential nor the surface charge density are constant on the surfaces during the interaction. It will be shown that the change in the chemical free energy contribution depends on the trajectory ψ S ψ S (σ, where ψ S is the surface potential and σ the surface charge density, during the interactions from an infinite to the final separation distance. Using this expression for the change * To whom correspondence may be addressed: address, feaeliru@acsu.buffalo.edu; phone, ( ext 2214; fax, ( (1 Gouy, G. J. Phys. Radium 191, 9, 457. (2 Chapman, D. L. Philos. Mag. 1913, 25, 475. (3 Langmuir, I. J. Chem. Phys. 1938, 6, 893. (4 Derjaguin, B. V. Kolloid-Z. 1934, 69, 155. (5 Verwey, E. J.; Overbeek, J. T. G. Theory of the stability of lyophobic colloids; Elsevier: Amsterdam, (6 Overbeek, J. T. G. Colloids Surf. 199, 51, /la CCC: $ American Chemical Society Published on Web 1/28/23

2 Chemical Free Energy of the Electrical Double Layer Langmuir, Vol. 19, No. 4, in the chemical free energy, one can recover the results obtained via the imaginary charging process, for constant surface charge or potential conditions, or via the Langmuir procedure for parallel plates and arbitrary surface conditions. However, it also can be used without any of the restrictions involved in the traditional theories. Furthermore, a simple approximate expression for the chemical free energy will be suggested, which is more efficient computationally. In that approximation, the free energy of the system does not depend on the trajectory ψ S ψ S (σ, but only on the values of σ and ψ S at infinite and at the final separation distance. The approximate expression is exact in the two limiting cases, namely, interactions at constant surface potential and at constant surface charge density. Finally, the approximate free energy is compared with the exact one (obtained both from the present theory and the Langmuir procedure for two planar surfaces, when the charge is generated via the dissociation of surface groups. II. The Double Layer Free Energy The total free energy of a double layer is the sum of an electrostatic free energy F el, an entropic term due to the mobile ions F ent and a chemical free energy. 6 The first one is given by F el 1 2 V ɛɛ ( ψ2 dv (1 the integral being carried out over the entire volume V of the system. In the above expression, ψ is the electric potential, ɛ the vacuum permittivity, and ɛ the relative dielectric constant. Note that in eq 1 the charges are assumed fixed (hence their entropy is not included, but the entropy of the dipoles of the medium is accounted for through ɛ. The entropic term due to the mobile ions of the electrolyte takes into account the differences between the ions distributions in the double layer and their distribution in the bulk 6 F ent -T S V ( c i ( c ln i i c i - c i i + c dv (2 where k is the Boltzmann constant, T the absolute temperature, c i the concentration of ions of species i in the double layer, and c i its concentration in the bulk and the subscript i runs over all ionic species. For an 1-1 electrolyte of concentration c E, whose ions obey Boltzmann distributions, the above expression becomes -T S 2c E V [( sinh ( cosh ( ] dv (3 where e is the protonic charge. The above two contributions to the free energy of the double layer are both positive. The third term, the chemical free energy contribution, which is responsible for the spontaneous formation of the double layer, should be negative and larger than the sum of the previous two terms. During interactions at constant surface potential ψ S, the surface charge decreases when two surfaces approach each other and ion adsorption or molecular group dissociation is involved in the surface charging. The system (liquid and surfaces is in thermodynamic equilibrium, and a transfer of charges from the liquid to the surfaces occurs to ensure the equality of the electrochemical potentials of the ions in the bulk and near the surface. Denoting by µ the change in the chemical part of the electrochemical potential of an ion of charge q brought from the bulk to the surface and by n the number of ions adsorbed per unit surface area, the chemical free energy (due to the adsorption of charges per unit area, is given by 5 F ch ψs const n µ (4 The thermodynamic equilibrium implies that µ bulk µ surface + qψ S (5 Since µ µ surface - µ bulk -qψ S, eq 4 becomes F ch ψs const n µ -nqψ S -σψ S where σ is the surface charge density and ψ S is the surface potential. For an interaction at constant surface charge, the charge is fixed on the surface and it is not at thermodynamic equilibrium, and the change in the chemical free energy during the interaction is zero. Hence, the chemical free energy is a constant which was taken as zero 5 F ch σconst (6a (6b As a result of this choice, the double layer free energy is in this case always positive. Actually this constant is negative, equal to the free energy required to charge the surfaces at infinite separation distance, and the double layer free energy is negative. Let us now derive an expression for the chemical free energy for a general interaction (when both the surface charge density and surface potential are changing during the interaction. When two particles (of arbitrary geometry approach each other from an infinite separation, the surface charge density and the surface potential, for an infinitesimal area δa on the surface of a particle, depend on the distance and follow a trajectory ψ S ψ S (σ (see Figure 1. This trajectory can be thought of as composed of successive infinitesimal transformations at constant surface potential (σ i, ψ S,i f σ i+1, ψ S,i and constant surface charge density (σ i+1, ψ S,i f σ i+1, ψ S,i+1. The change of the chemical free energy during the transformations at constant surface potential is given by -ψ S,i (σ i+1 - σ i δa, and it is zero for the transformations at constant surface charge. Adding all the changes when two particles approach from infinity to a final separation distance l, one obtains for the change F ch (l of the chemical contribution to the free energy F ch (lδa (- σ( ψs (σ dσδa (7 In the above integral, the Poisson-Boltzmann equation should be solved for each separation distance l and the appropriate surface boundary conditions (constant surface charge density, constant surface potential, dissociation of surface groups, or adsorption of ions. Consequently, ψ S and σ are functions of l; eliminating l one obtains ψ S ψ S (σ. The function ψ S (σ describes the unique trajectory of the system during the interaction, and the integral is performed along this trajectory (the path 1 f 2 in Figure 1. The total free energy of the double layer is obtained by adding eqs 1, 3, and 7

3 1116 Langmuir, Vol. 19, No. 4, 23 Manciu and Ruckenstein Figure 1. The surface potential ψ S as a function of the surface charge density σ when two interacting particles approach from infinite separation to the final separation distance l. F DL (l 1 2 V ɛɛ ( ψ2 dv + 2c E V [( sinh ( cosh ( ] dv + (C - ψs σ( (σ dσ da (8 the last integral being taken over the entire surface of the system, and the constant C (independent of l representing the chemical free energy at infinite separation. As shown by Overbeek, the first two terms of eq 8 can be transformed into surface integrals using Green s theorem V ɛɛ ( ψ2 dv + 2c E V [( sinh ( 1 - cosh ( + ] dv ( φs (ξ,l dξ da (9 To perform the integral in the right-hand side of the above expression, the Poisson-Boltzmann equation should be solved for a fixed value of l and all the imaginary surface charges ξ between and to obtain the imaginary surface potentials φ S. The surface potential φ S depends in this case on the values of ξ and l. From eqs 8 and 9 one obtains F DL (l ( φs (ξ,l dξ da + (C - ψs σ( (σ dσ da (1 Expression 1 is valid for a system of arbitrary shape and arbitrary surface conditions. It will be shown in what follows that one can recover the results obtained via the traditional procedures, within their domains of validity. For interactions at constant surface charge, the integral over σ in the second term vanishes and the well-known expression due to Verwey and Overbeek, which assumed C, is recovered 5 F DL (l σconst ( σ φs (ξ,l dξ da (11a For interactions at constant surface potential, ψ S (l ψ S ( ψ S and using the corresponding Verwey-Overbeek choice C -σ( ψ S (, one obtains, integrating by parts the first right-hand side term of eq 1, the expression F DL (l ψconst ( φs (ξ,l dξ da + (-σ( ψ S ( - ψ S ( σ( dσ da ξ,φ (φ S ξ S ψ S (l ψ ξ,φs - S (l ξ(φs,l dφ S da - (ψ S ( da - ( ψ S ξ(φs,l dφ S da (11b which is the Verwey-Overbeek expression for the free energy for interactions at constant surface potential. The free energy of the system is always defined up to an arbitrary constant; the choice C in the Verwey- Overbeek approach leads to positive values for the free energy of the system at constant surface charge and negative at constant surface potential. However, the interaction free energy (the difference between the free energy of the system at the final separation distance and the free energy of the system at infinite separation is not affected by the choice of the constant C. Let us now show that eq 1 leads to the same result as the Langmuir equation for the force between two identical planar surfaces and arbitrary surface conditions. Using eq 1, one obtains the following expression for the force per unit area, between two identical planar surfaces Π DL (l - 1 df DL (l - d A dl dl 2 φs (ξ,l dξ + d dl 2( ψs σ( (σ dσ (12 where the factor 2 accounts for the two identical surfaces.

4 Chemical Free Energy of the Electrical Double Layer Langmuir, Vol. 19, No. 4, The first term is the Verwey-Overbeek expression for the interactions at constant surface charge density and the second accounts for the variation of the surface charge with the separation distance. The derivative of the first term of eq 12 is given by d dl ( φ φs (ξ,l dξ S (ξ,l dξ + ψ S (σ d dl and the derivative of the second term by Consequently, from eqs 12-14, one obtains It remains only to prove that eq 15 is equivalent to the Langmuir equation where ψ m (l is the potential at the middle distance between plates. This task was performed earlier. 7 For the completeness of the presentation, the details of the proof that eqs 15 and 16 are equivalent are reproduced in the Appendix. Therefore, the present treatment of the double layer interaction leads to the same results for the interaction free energy as the imaginary charging approach for systems of arbitrary shapes and constant surface potential or constant charge density and to the same results as the Langmuir equation for parallel plates and arbitrary surface conditions. It can be, however, used for systems of any shape and any surface conditions, since it does not imply any of the above restrictions. III. Approximate Expressions for the Change in Chemical Free Energy A notable consequence of eq 7 is that the values of surface charge density and surface potential at a given separation distance l are not sufficient to calculate the free energy of interaction between surfaces. To calculate that free energy, one has to solve the Poisson-Boltzmann equation at each separation distance along a trajectory from infinity to the separation distance l. Here we will propose a simple approximation for the change of the chemical free energy, which simplifies significantly the problem. As shown in Figure 1, the transformation from 1( f 2(l is bounded by two extremal processes, 1 f 3 f 2 and 1 f 4 f 2. For the first, the change in the chemical free energy, per unit area, calculated using eqs 6a and 6b, is while for the second d dl ( ψs σ( (σ dσ ψ S (σ d dl Π DL (l -2 φ S (ξ,l (13 (14 dξ (15 Π DL,Langmuir (l 2c E (cosh( m (l - 1 (16 F 132 F 13 + F 32 -ψ S ( ( - σ( + F 142 F 14 + F 42 - ψ S (l( - σ( (17a (17b Therefore, the change in the chemical free energy is bounded by F 132 and F 142. A simple approximation for the change in the chemical free energy, per unit area, is the arithmetic average of those two bounds This expression is exact for both constant surface charge and constant surface potential interactions. One can also construct approximate expressions for the change in the chemical free energy by extending those at constant surface charge (6b or at constant surface potential (6a to general processes. In the first case which will be named in what follows constant surface charge approximation. In the second case, since the chemical free energy is defined in terms of σ and ψ S at each distance l, a natural approximation is which, as our calculations have shown, is a very poor approximation when the surface potential is not constant. Since eq 6a implies that ψ S (l ψ S (, other two legitimate choices for the approximate change in the chemical free energy would be or F ch (l (19 F ch (l -(ψ S (l - σ( ψ S ( F ch (l -(ψ S ( - σ( ψ S ( F ch (l -(ψ S (l - σ( ψ S (l (2a (2b (2c where the constant surface potential is assumed to be either at infinite separation distance or at the distance l. Expressions 2b and 2c are the two bounds proposed earlier (eqs 17a and 17b, respectively. A simple modality to improve the approximation is to use as constant surface potential in eq 6a an intermediate value between ψ S ( and ψ S (l. By employing their average, one obtains F ch (l -( - σ( (ψ S (l + ψ S ( 2 (2d which coincides with eq 18. This expression will be named in what follows the constant surface potential approximation, although it is exact when either the surface charge or the surface potential is constant. The accuracy of these approximations, when neither the surface potential nor the surface charge density are constants, will be investigated below, for two identical, parallel planar surfaces. It will be assumed that the surface charge is due to the dissociation of surface groups (for instance dissociation of surfactant molecules. 8,9 In general, the electrolyte counterions are the most abundant ions in the vicinity of the surface and therefore they can control the surface charge via reassociation. Denoting by c E the concentration (in the reservoir of an 1:1 electrolyte, by N the number of sites per unit area, and by x the fraction of dissociated sites, the dissociation equilibrium provides the expression K D (xnc + E (1 - xn xc + E (1 - x x 1 - x c E exp (- S (21 where K D is the dissociation constant, c E+ denotes the cation concentration in the liquid in the vicinity of the surface, e is the protonic charge, and the Boltzmann distribution was assumed for the counterions (note that ψ S is negative. The surface charge density is therefore given by F ch (l - (ψ S (l + ψ S ( ( - σ( 2 (18 (7 Ruckenstein, E. J. Colloid Interface Sci. 1981, 82, 49. (8 Ninham, B. W.; Parsegian, V. A. J. Theor. Biol. 1971, 31, 45. (9 Prieve, D. C.; Ruckenstein, E. J. Theor. Biol. 1976, 56, 25.

5 1118 Langmuir, Vol. 19, No. 4, 23 Manciu and Ruckenstein Figure 2. The interaction free energy (per unit area as a function of the separation distance l for two identical plates, planar and parallel, at T 3 K, ɛ 8, c E.1 M and: (a K D 1. M, N sites/m 2 ; (b K D 1. M, N sites/m 2 ; (c K D.1 M, N sites/m 2 ; (d K D.1 M, N sites/m 2. The continuous thick line represents the exact result; the up triangles represent the upper bound, the down triangles represent the lower bound, the circles represent the constant surface charge approximation, and the crosses represent the constant surface potential approximation. σ -exn - en 1 + c E exp(- S K D (22 When the dissociation constant K D is large, (c E /K D exp- (- S /, 1 and σ =-en. Then the interaction can be well approximated by that at constant surface charge density. However, for low values of K D, the interaction can no longer be approximated by assuming constant surface charge or constant surface potential. Another relation between the surface charge and surface potential is provided by the equation σ -ɛɛ ( dψ(z dz S (23 where the z direction is normal to the surface. Equation 23, combined with eq 22 provides one of the boundary conditions. The other boundary condition is provided by the symmetry of the system, which implies that the derivative of the potential at the middle distance vanishes dψ(z dz z1/2 (24 Consequently, one has to solve the Poisson-Boltzmann equation, which for a uni-univalent electrolyte has the form d 2 ψ dz 2ec E sinh 2 ɛɛ ( (25 for the above boundary conditions. The free energy of interaction between two identical plates with charges generated through dissociation will be calculated both exactly and approximately, and the results will be compared for various values of the dissociation constant. For a given separation distance l, the solution of the Poisson-Boltzmann equation was obtained via numerical integration. From the value of the potential at the middle distance between the plates ψ m (l ψ(l/2, the disjoining pressure was calculated using the Langmuir equation. The interaction free energy was subsequently obtained by integrating the disjoining pressure from infinite to the final separation distance l. In a second method, the free energy was calculated by adding the electric (eq 1, entropic (eq 3, and chemical contributions, and the interaction free energy was obtained by subtracting the free energy for infinite separation. The integrals were calculated using the numerical solution of the Poisson-Boltzmann equation for ψ(z,l and. The change in the chemical free energy was calculated using several expressions, namely, eq 7 (the exact expression, eq 19 (the constant surface charge approximation, eq 2b (the lower bound, eq 2c (the upper bound, and eq 2d (the constant surface potential approximation.

6 Chemical Free Energy of the Electrical Double Layer Langmuir, Vol. 19, No. 4, Figure 3. The ratio between the approximate interaction free energy, based on the change in chemical free energy contribution provided by eq 18 and the rigorous interaction free energy. The values of the parameters are the same as those given in Figure 2. When the exact expression was employed, the results obtained always coincided (within numerical errors, as expected, with those provided by the Langmuir procedure. For large dissociation constants (implying small c E /K D ratio, the interaction is well described, at low surface potentials, by the constant surface charge density approximation. In Figure 2a, the free energy is calculated using c E.1 M, T 3 K, ɛ 8, K D 1. M, and N sites/m 2. In this case, the calculations based on the expressions for the change in the chemical free energy (eq 19 and eqs 2b-d are all good approximations to the rigorous result obtained either using eq 7 for the change in the chemical energy or via the Langmuir procedure. The increase of N to sites/m 2 (Figure 2b displaces the σ-ψ S equilibrium toward larger surface potentials. The constant surface potential approximation (based on eq 2d provides in this case a much better agreement with the exact result than the constant surface charge approximation (based on eq 19. For the low dissociation constant K D.1 M (c E.1 M, T 3 K, ɛ 8, N sites/m 2, Figure 2c, the constant surface charge approximation is inaccurate even at low potentials (ψ S <.1 V. When the site density was increased to N sites/ m 2 (c E.1 M, T 3 K, ɛ 8, K D.1 M, Figure 2d, the constant surface charge approximation predicted attraction between the plates. In Figure 3 the ratio between the interaction free energy calculated on the basis of the approximate (eq 2d and the rigorous (eq 7 expressions for the change in the chemical free energy is plotted against the distance. In all the cases investigated, the approximate expression for the chemical free energy was accurate. In summary, when the surface charge was generated by the dissociation equilibrium of surface groups, the surface charge density was almost constant for large dissociation constants and low surface potentials. In this case, the neglecting of the change in the chemical free energy (eq 19 was a good approximation. However, in the other cases (small dissociation constants or large surface potentials it was, in general, inaccurate and sometimes even predicted double layer attraction. In all the cases investigated, the surface potential has changed, sometimes drastically, during interactions. However, a suitable choice of the surface potential (the arithmetic average of the surface potentials at infinity and at distance l in the constant surface potential approximation for the change in the chemical free energy provided always a good approximation. This approximation is exact when either the surface potential or the surface charge are constant. IV. Conclusions One can calculate the free energy of a double layer by adding to the free energy of the electric field, an entropic contribution due to mobile ions and a chemical free energy contribution due to the charge transfer between bulk and interface. However, expressions for the latter contribution are provided in the literature only for interactions at constant surface charge density or constant surface potential. In this paper, an expression for the change in the chemical free energy which is valid for interactions at any surface conditions was derived; it depends not only on the values of the surface charge and potential at the final separation distance but also on their values at all distances between infinite and the final one. The present approach reduces to the traditional ones within their range of application (imaginary charging processes for double layer interactions between systems of arbitrary shape and interactions either at constant surface potential or at constant surface charge density, and the procedure based on Langmuir equation for interactions between planar, parallel plates and arbitrary surface conditions. It can be, however, employed to calculate the interaction free energy between systems of arbitrary shape and any surface conditions, for which the traditional approaches cannot be used.

7 112 Langmuir, Vol. 19, No. 4, 23 Manciu and Ruckenstein ψ(z,l z -( 2c E ɛɛ 1/2( ( 2 cosh (z,l - 2 cosh( m (l 1/2 (A.4 Since the change in the chemical energy depends on all the values of the surface potential and surface charge density, when the particles approach from infinity to the final separation distance, the calculation of the interaction free energy involves the solution of the Poisson-Boltzmann equation at all the points along this trajectory. For systems with complicated shapes, this procedure is inconvenient. For this reason, an approximate expression for the change in the chemical free energy was also proposed, which depends only on the values of the surface potential and charge density at the infinite and final separation distances. It was shown that this approximation is accurate. which leads to the relation ψ(z,l -ɛɛ z z (2c E ɛɛ 1/2 (2 cosh( S (l - 2 cosh( m (l 1/2 (A.5 Appendix Integrating eq 15 by parts, one obtains for the disjoining pressure between two identical, parallel and planar surfaces separated by the distance l φ Π DL (l -2 S (ξ,l dξ -2 ( φs (ξ,l dξ + 2ψ S (σ -2 (φ S ξ ξ,φ S ψ S (l ψ ξ,φ S - S (l ξ(φs,l dφ S + 2ψ S (σ 2 ( ψ S (l ξ(φs,l dφ S - 2σ(ψ S ψ S (l (A.1 For the integral in the last right-hand side of the equality, one can use an expression derived by Verwey and Overbeek (eq 37b in ref 5 5 ψ - S ξ(φs,l dφ S -c E l( ( cosh m (l ( 2ɛɛ c E (3 m (l/ 1/2 e 2 S (l/ ( 2 cosh(y - 2 cosh( m (l the derivative of which with respect to the distance is given by - ( ψ S ξ(φs,l dφ S -c E ( cosh ( m (l c E l sinh( m ψ m + ( 2ɛɛ c E (3 ( 2 cosh(y - 2 cosh( m (l y 1/2 ( 2ɛɛ c E (3 1/2 sinh( m e 2 ψ m 1 Integrating once the Poisson-Boltzmann equation, one obtains 1/2 1/2 e 2 ( m (l/ ( S (l/ + m (l/ S (l/ (2 cosh(y - 2 cosh( m (l 1/2 dy (A.2 dy (A.3 and by direct integration, from the surface to the middle distance to the expression 5 m (l/ S (l/ By employing eq A.6, the last right-hand-side term in expression A.3 becomes ( 2ɛɛ c E (3 1/2 e 2 and cancels the second term of the right-hand side of eq A.3. Using expression A.5, eq A.3 becomes Finally, from eqs A.1 and A.8, one obtains which is the Langmuir equation, eq 16. LA dy (2 cosh(y - 2 cosh( m (l 1/2 sinh( m ψ m m (l/ S (l/ 1 (2 cosh(y - 2 cosh( m (l 1/2 ( 2ɛɛ c E 1/2( ( (3 e 2 cosh S (l 2 e ψ S (l - 2( l 2c E e2 1/2 ɛɛ dy c E l sinh( m ψ m - ( ψ S ξ(φs,l dφ S -c E ( cosh ( m (l -c E ( cosh ( m (l (A.6 (A.7-2 cosh( m (l σ(ψ S ψ S (l Π DL (l 2 ( ψ S (l σ(φs,l dφ S - 2σ(ψ S ψ S (l 2c E (cosh( m (l (A.8-1 (A.9