Changes in zeta potential caused by a dc electric current for thin double layers

Colloids and Surfaces A: Physicochem. Eng. Aspects 250 (2004) 67–77

Changes in zeta potential caused by a dc electric current for thin double layers Dennis C. Prieve Center for Complex Fluids Engineering, Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA Received 1 February 2004; accepted 13 July 2004 Available online 25 September 2004

Abstract An asymptotic solution was obtained to describe one-dimensional, steady-state transport of a symmetric binary electrolyte normal to two large parallel electrodes, in the limit in which the Debye length is infinitesimal compared to the distance separating the two electrodes. Despite the nonzero ion flux, Boltzmann’s equation continues to describe the relationship between either ion concentration and the electrostatic potential inside the diffuse part of the double layer, while local electroneutrality applies outside, even for current densities approaching the limiting value. In the absence of ion adsorption or dissociation reactions at the electrodes, the magnitude of any charge or zeta potential arising on the electrodes at zero current is determined by the equilibrium constant for the redox reactions which would exchange ionic charge carriers for electric charge carriers at the electrode surface. Nonzero current causes the ionic strength of the bulk to vary with position. This perturbs the Debye length of the diffuse cloud on either electrode: it is the local ionic strength just outside the cloud which determines the Debye length for that cloud. Nonzero current also changes the zeta potential. The dimensionless rate of change dζ/dJ was as large as 30. © 2004 Elsevier B.V. All rights reserved. Keywords: Zeta potential; Electrode; Double layer; Electric current

1. Introduction In electrophoretic deposition (EPD), an imposed electric field drives charged colloidal particles toward an electrode and deposits them on it. EPD is used in a wide variety of applications from the deposition of the primer coat of paint on automobile bodies, to the deposition of phosphorus coatings on cathode ray tubes, to deposition of composite coatings. When the electric field is not too large, double layer repulsion (resulting from the overlap between the counterion clouds on the electrode and the particle) can prevent the particle from coming into intimate contact with the electrode despite being attracted to the electrode by the applied electric field. Under these conditions, a single isolated particle remains nearly stationary (except for Brownian motion) near the electrode. The externally applied electric field, which attracts the particle toward the electrode also repels the oppositely charged counterion cloud surrounding the particle, causing electroosmotic flow around the particle in a direction E-mail address: [email protected]. 0927-7757/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2004.07.024

generally away from the electrode. Fluid thus forced away from the electrode must be replaced by fluid drawn toward the particle along the surface of the electrode. Even if electrically neutral, other colloidal particles also near the electrode can become entrained in this flow along the surface of the electrode and be drawn towards the particle undergoing electroosmotic streaming, resulting in apparent self-assembly of particles on the electrode. This is an important mechanism for two-dimensional selfassembly of like-charged particles near an electrode surface [1–3]. Solomentsev et al. [4] used video microscopy to observe the center-to-center distance between pairs of 10 ␮m polystyrene (PS) latex particles near an electrode surface as a function of time and electric field. Despite scatter in the results owing to lateral Brownian motion, the results quantitatively confirm electroosmotic flow as the mechanism of aggregation under these conditions [4]. Self-assembly of particles on electrodes surfaces has also been observed using alternating currents, although much larger electric fields were required than for direct current [5–7]. Moreover, particles near the electrode surface were

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observed to aggregate for frequencies less than 500 Hz but disaggregated for larger frequencies. A simple extension of the model which works for dc fields would suggest that the particles would aggregate for half of the ac cycle and disaggregate during the other half, leading to no net aggregation during a whole cycle. Moreover, the requirement of much larger electric fields for aggregation to occur in ac suggests that a more subtle mechanism is involved. Several mechanisms have been suggested for aggregation in ac fields. In the electrohydrodynamic model (EH) [1,6,8], the electric field acts on the diffusion layer which accumulates next to an electrode to drive electroosmotic flow. Although the charge density in this diffusion layer is typically very small or negligible compared to that inside the diffuse part of a double layer, the sign of charge alternates with the electric field during a cycle, such that the product of charge and electric field is proportional to E2 and of constant sign during any cycle, leading to net aggregation or net disaggregation. Solomentsev et al. [3,4] suggested that the electrokinetic mechanism (EK) which explains dc aggregation can also lead to net aggregation during each cycle in an ac field if the particle resides at different elevations above the electrode during the two halves of any cycle. Elevation dependence of the fluid flow and electric fields could break the antisymmetry which otherwise exists during the two halves of the cycle, leading to net aggregation or net disaggregation. In the EK mechanism, the counterion cloud on each particle experiences an electrical force which drives the electroosmotic flow. A counterion cloud also exists next to the electrode but, in modeling the electric field, a constantpotential boundary condition is often imposed on the electrode surface. This leads to no tangential component in the electric field and no electroosmotic flow driven by the electrode’s counterion cloud. However, a constant-potential boundary condition on the electrode represents only one limiting case. More generally, both the potential and the current density will vary with position on the electrode surface under the particle as a result of the insulating effect of the particle. Any variation in surface potential creates a tangential electric field on the electrode surface which can act on the charges inside the electrode’s counterion cloud to drive electroosmotic flow [9]. We will refer below to this mechanism as the electroosmotic model (EO). While all of the above studies have focused on the relative motion of pairs of particles tangential to the electrode, Fagan et al. [10] have observed the normal motion of single isolated particles near an electrode in ac fields using total internal reflection microscopy (TIRM). For some frequencies, amplitudes and electrolytes, the particle is observed to be drawn closer to the electrode by the ac field while in other cases the particle is pushed away from the electrode. This motion of the particle normal to the electrode correlates well with the tangential motion of pairs of particles: under conditions in which the particle is pushed away from the electrode by an ac field, pairs of particles are observed to aggregate. This correlation is expected if the same electroosmotic flow which

pushes the particle away from the electrode also causes the particles to aggregate (as in the dc EK mechanism). Finite element simulations [11] of the flow fields developed in an ac experiment reveal that the EK mechanism can only push particles away from the electrode, whereas the experiments reveal that particles can also be drawn closer to the electrode by the ac field. While the EH mechanism simply accounts for the electrolyte dependence observed, the finite element simulations reveal that it is at least two orders of magnitude too weak to account for the motions observed [9]. This leaves the EO mechanism as the most likely mechanism to account for the behavior observed. Central to the EO mechanism is how the surface (or zeta) potential and current density vary with position on the electrode surface under the particle. In this manuscript, we explore how variations in the current density can induce variations in the zeta potential which in turn can cause electroosmotic flow along the electrode surface. Even in the limiting case of constant surface potential, variations in current density (and therefore variations in the zeta potential) are expected under the particle (known as the “primary current distribution”). To simplify the analysis sufficiently to allow an analytic solution, we restrict our attention to one-dimensional, steadystate transport of a 1-1 electrolyte. Here we obtain the onedimensional asymptotic solution for the profiles of ion concentrations and electrostatic potential in the double layer with current in the limit of very thin Debye lengths. The differential equations and model for the electrode kinetics used here are based on the more general unsteady analysis of Bonnefont et al. [12]. While our steady-state solution is also given there, the contribution of the present manuscript is a more thorough exposition of the steady-state solution. In particular, we are interested in how the zeta potential is perturbed by current density. While this manuscript was undergoing review, we learned of similar analysis by Bazant et al. [13] who have also extended their analysis to currents above the limiting value [14].

2. Theory 2.1. Diffuse layer with current The experiments cited above are performed in a cell consisting of a thin film of water held between two large parallel electrodes (see Fig. 1). When the dimensions of the electrodes are sufficiently large compared to their spacing 2l, we can treat the electrodiffusion of ions as one-dimensional. At steady-state, the 1-D flux Nix of ion species i is spatially uniform: dNix /dx = 0 for −l ≤ x ≤ +l. When fluid convection is negligible, the ion fluxes are related to the ion concentration gradient and local electric field by the Nernst-Planck equation:
 dci zi e dψ Nix = −Di + ci = const w.r.t. x (1) dx kT dx

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in half the cell the ion concentrations must be higher than c0 and in the other half of the cell the ion concentrations must be lower than c0 . To fix the total number of ions per unit area of electrode surface at 2lc0 , the concentration at the midplane must remain at c0 : at x = 0 :

Fig. 1. Schematic of fluid cell which consists of a thin film of water held between two large parallel electrodes.

where ci , zi and Di are number density, charge number (including sign) and diffusion coefficient of ions of species i, ψ is the local electrostatic potential, e is the charge on one proton, k is Boltzmann’s constant and T is the absolute temperature. Changes in the electrostatic potential are related to the local charge density by Poisson’s equation: d2 ψ 4πe  =− zi ci 2 dx ε

(2)

i

(1) for i = 1,2,. . .,n and (2) constitute n + 1 equations in n + 1 unknowns (ci for i = 1,2,. . .,n and ψ), where n is the number of ion species. The values of the ion fluxes depend on the voltage applied between the electrodes located at x = ±l and on the chemistry of the exchange reactions occurring there. To obtain an analytical solution, we will restrict our attention to symmetric binary electrolytes (i.e. n = 2 and z+ = −z− = z). Then (1) for i = + or − can be written as: −

N+ dc+ dy + c+ = D+ dx dx

(3)

−

N− dc− dy = − c− D− dx dx

(4)

c+ = c− = c0

y=0

and

we also choose this location as the reference state for electrostatic potential. In the absence of current, the second boundary condition on potential would usually be provided by specifying the value of the surface charge or the surface potential. Part of the motivation for this analysis is to determine the effect of current on these quantities, so we should not assume their values. In the related analysis by Bonnefont et al. [12], those authors assume continuity of the normal component of the displacement at the Stern layer which, in turn, relates the electric field at the inner edge of the diffuse layer to the potential drop across the Stern layer, the latter quantity is related to the kinetics of the electrode reactions. This is sufficiently complicated that we will present the details in a later section. For the current purpose of determining the functional form for the diffuse layer structure, we leave the value of the surface potential as an arbitrary parameter. 2.1.1. Dimensionless variables These equations will now be re-written in dimensionless form. Adding and subtracting (3) and (4), we have: −J+ − J− =

dC dy +ρ dX dX

(6)

−J+ + J− =

dρ dy +C dX dX

(7)

C = (c+ + c− )/2c0 , where J± = N± l/(2D± c0 ), ρ = (c+ − c− )/2c0 , and X = x/ l. (5) becomes: ε2

d2 y =ρ dX2

(8)

where y = ze␺/kT is the dimensionless electrostatic potential, which must satisfy Poisson’s Eq. (2):

where ε = (κl)−1 . The boundary conditions become at:

κ2 d2 y = − (c+ − c− ) dx2 2c0

X=0:

(5)

where κ2 = 8πz2 e2 c0 /εb kT is the Debye parameter, c0 is number concentration of z−z electrolyte present in the bulk solution in the absence of current and εb is the electrical permittivity of the water. We will further assume that the ion fluxes N+ and N− are prescribed. Since these fluxes must be uniform across the entire cell, any rate of consumption of ions at one electrode is matched exactly by a rate of production of the same ions at the opposite electrode. The total number of ions of either type within the cell must remain constant. In the next section we will see that, outside the boundary layer, the ion concentration profiles must be linear. This means that

C = 1, ρ = 0

and

y=0

(9)

Substituting (8) into (6) and integrating: C(X) = C0 − (J+ + J− ) X +

ε2 2




dy dX

2 (10)

where C0 is an integration constant. Substituting (8) and (10) into (7) yields a single equation for the electrostatic potential: 
  3 1 dy 3 dy 2 d y ε + [(J+ + J− )X − C0 ] − 3 dX 2 dX dX = J+ − J−

(11)

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With the term having the highest order derivative also being highest order in ε, in the limit that ε → 0 we anticipate the solution will exhibit a boundary layer. This is the asymptotic solution we seek. 2.1.2. Outer solution The leading term of the outer solution in this limit can be obtained by setting ε = 0 in (10): C0 (X) = 1 − (J+ + J− )X

(12)

To satisfy boundary condition (9), we chose C0 = 1. Making the same substitution in (11), then integrating: y0 (X) =

J+ − J− ln [1 − (J+ + J− )X] J+ + J −

(13)

Later we will need the inner limit (X → −1) of the outer solution, which is: i

(C0 ) = 1 + J+ + J− i

(y0 ) =

J+ − J − ln (1 + J+ + J− ) J+ + J −

(14) (15)

2.1.3. Inner solution To obtain the differential equation for the inner solution, we transform the independent variable by stretching the distance from the left boundary of the continuum region (i.e. X = −1): ξ =

X+1 ε

or

X = −1 + εξ 

(16)

In terms of this new independent variable, (11) becomes:
 d3 y 1 dy 3 dy − + [(J+ + J− )(εξ  − 1) − C0 ]  3   2 dξ dξ dξ = (J+ − J− )ε The leading term of the inner solution in this limit can be obtained by setting ε = 0 and C0 = 1:
 d3 y 1 dy 3 dy − − (1 + J+ + J− )  = 0 (17) 3   2 dξ dξ dξ The remaining coefficient can be absorbed by a second transformation of the independent variable:  ξ = ξ  1 + J + + J− (18) leaving:
 d3 y 1 dy 3 dy − − =0 dξ 3 2 dξ dξ

(19)

Notice that the current density no longer appears: the same differential equation results either with or without current. Armed with the Gouy-Chapman solution for the equilibrium

potential profile, we already know one particular solution for this differential equation: yi (ξ) = A + 4 tanh−1 (B e−ξ )

(20)

where the superscript i was added to remind us that this is the inner solution. Transformation (18) renormalizes the distance from the electrode by the Debye length evaluated at the ionic strength at inner edge of the outer region, which is given by (14), rather than the ionic strength at the midplane. By i inspection, we can see that A = yi (∞) = (y0 ) is the outer limit of the inner solution, which must be matched with inner limit of the outer solution given by (15), and that: B = tanh

ζ 4

(21)

is related to the potential drop across the diffuse part of the double layer: ζ = yi (0) − yi (∞). At equilibrium (no current), this potential drop is the usual “zeta potential”. In the discussion below we refer to this potential drop as the “zeta potential” whether current exists or not. Once the potential profile is known, we can determine the leading term of the total ion concentration and charge density by substituting (20) and C0 = 1 into (10) and (8): Ci (ξ) 1 + 6B2 e−2ξ + B4 e−4ξ = 2 1 + J + + J− (1 − B2 e−2ξ ) ρi (ξ) 1 + B2 e−2ξ = 4Be−ξ  2 1 + J + + J− 1 − B2 e−2ξ Finally, if desired, the concentration of the two ions can also be calculated:

2 1 ∓ Be−ξ i i (22) C± = C ± ρ = (1 + J+ + J− ) Be−ξ ± 1 In the absence of current, the ion concentrations would be related to the electrostatic potential according to Boltzmann’s equation. The general form of this relationship turns out to be:

2 i (ξ) C± 1 ∓ Be−ξ ∓[yi (ξ)−yi (∞)] =e (23) = 1 + J + + J− Be−ξ ± 1 where the second equation was obtained after substituting (20). With proper normalization, Boltzmann’s equation continues to hold even for current densities approaching the limiting current density. 2.1.4. Inner solution at right electrode The stretch transformation (16) and the equations that follow were developed for the left electrode at X = −1. In the next section, we will also need a description of the diffuse layer at the right electrode or X = +1. At the right electrode, (16) must be replaced by ξ  = 1−X or X = 1 − εξ  . ε

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The analog of (17) at the right electrode is
 1 dy 3 d y dy − − (1 − J+ − J− )  = 0 3   2 dξ dξ dξ 3

With the appropriate changes to the second transformation (18)  ξ = ξ  1 − J + − J− the final differential Eq. (19) remains unchanged along with the form of the particular solution (20). Thus, the description of the inner solution obtained in the previous section for the left electrode also holds at the right electrode, except that J± at the left electrode is everywhere replaced by –J± at the right electrode. 2.2. Effect of current on zeta In the previous section we saw that, with proper renormalization, the equilibrium description of the diffuse layer continues to hold, even for current densities approaching the limiting value. Instead, the main effect of current is to change the concentration of ions outside the diffuse region, to change the Debye length and possibly to change the potential drop across the diffuse layer. These conclusions are independent of which redox reactions occur at the electrode. We will now focus on this latter change for a particular reaction. In this section we follow the approach presented by Bonnefont et al.[12] The reaction at cathode involves the reduction of the cation C to form a neutral species N; the inverse reaction is assumed to occur at the anode: Cz+ + ze−
N

(24)

The kinetics of the electrode reaction are described by the Butler-Volmer equation:  ze j = kO cN exp −αO (ψr − ψe ) − kR c+ kT  ze × exp +αR (ψr − ψe ) kT where the first term gives the rate of oxidation of neutral species (causing current to flow in the +x direction) and the second term is the rate of reduction of cations, j is the net electrical current density flowing in the +x direction, c+ and cN are the concentrations of cations and neutral species at the reaction plane x = −l, αO and αR are the transfer coefficients (taken to equal 1/2) and kO and kR are the rate constants for the two reactions, which are assumed to be independent of concentration, potential or current. The electrode reactions are assumed to occur at the edge of the continuum region of the fluid, denoted as x = −l in Fig. 2. The kinetics of the reaction depend on the electrostatic potential drop ψr − ψe , where ψr is the potential at the reaction plane and ψe the potential inside the nearby metal electrode at x ≤ − (l + ␭S ). A similar Butler-Volmer equation could be written for the second electrode at x = +l. Besides replacing the concentrations and potentials with their counterparts for this second

Fig. 2. Schematic showing the reaction plane on either electrode.

electrode, the sign of the current density j must be reversed (since the same positive charges flowing away from the left electrode will be flowing toward the right electrode). The region −(l + ␭S ) < x < −l is called the Stern layer, which is assumed to possess no free charges and to have an electrical permittivity εS which is smaller than that of the bulk solvent ␧b . In the absence of free charges, Poisson’s equation requires the electric field Ex to be constant inside the Stern layer and equal to −(ψr −ψe )/λS at the left electrode. Although ions might adsorb at the reaction plane or ionizable groups on the electrodes might dissociate to generate a nonzero surface charge density, we will neglect such effects here in order to minimize the number of parameters. Duval et al. [15–18] have considered such effects as well as more realistic electrode reactions and favorably compared their predictions with experiments in a number of situations. Here we overlook such effects in order to focus on our main conclusion: even in the most simple of situations, current perturbs the zeta potential. Without adsorption of ions or dissociation of surface groups, there can be no surface charge density at the reaction plane. In the absence of surface charges in the reaction plane, the normal component of the electric displacement must be continuous at the reaction plane: at x = −l :

εb

dψ ψr − ψ e = εs dx λs

(25)

The ratio εs /λs is the capacitance of the Stern layer, to which Bonnefont et al.[12] assign a value of 80 ␮F/cm2 . Taking εs to be 1/8 the bulk value, they estimate that λs is about 0.1 nm. This equation plus its counterpart at the other electrode provides the two boundary conditions on the potential in terms of the two yet unknown potential drops across each Stern layer. These two potential drops are in turn related to the current j = ze (N+ − N− ) by the Butler-Volmer equations. Besides the transport of ions we also need to model the transport of neutral species. The flux of neutral species is related to the flux of ions through the stoichiometry of the electrode reactions: NN = −DN

dcN = −N+ dx

As with the ions, an appropriate boundary condition is: at x = 0 :

cN = γc0

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D.C. Prieve / Colloids and Surfaces A: Physicochem. Eng. Aspects 250 (2004) 67–77

where 2lγc0 is the total number of neutral molecules present per unit area of electrode. In terms of dimensionless variables, these conditions become DN dCN = 2J+ D+ dX at X = 0 :

lim B = ∓1

C→±∞

where CN = cN /c0 . Integrating D+ J+ X DN

(26)

We will also need to express the Butler-Volmer equation in terms of dimensionless variables J+ = KO CN e−αO V − KR C+ e+αR V

εs /εb V C= √ 4 1 + J+ δ

(32)

The negative square root was chosen in (31) to insure that −1 < B < + 1. With this choice of sign for the root, the asymptotic behavior is

CN = γ

CN (X) = γ + 2

where

(27)

and

lim B = −C

C→0

Before applying (27), we will need to evaluate the concentrations C+ and CN next to the electrode. Evaluating (22) at ξ = 0:
 1−B 2 C+ = (1 + J+ ) (33) = (1 + J+ )e−ζ 1+B where the second equation was obtained with the help of (21). Evaluating (26) at X = −1:

where V = (ψr − ψe )ze/kT is the dimensionless potential drop between the reaction plane and the metallic interior of the electrode while Ki = ki l/2zeD+ c0 are the dimensionless rate constants (i = R or O). In the expressions just above, we also have recognized that for reaction (24) the flux of anions must vanish, or J− = 0, leaving the total dimensionless current J equal to the dimensionless cation flux J+ . To make the form of (27) more recognizable to electrochemists, we partition the potential drop V into an equilibrium value V0 (causing the net current density to vanish) and an “overpotential” ␩:[19]

For a given value of the current J+ , V is determined as the root of the transcendental algebraic equation obtained by substituting (31)–(34) into (27). Evaluating (33) and (34) for J+ = 0 and substituting the result into (29) yields 
KO γ V0 = ln +ζ (35) KR

V = V0 + η

where ζ is related to B by (21).

(28)

Substituting this into (27), setting J+ = η = 0, αO = 1 − β and αR = ␤, then solving for V0 yields: 
KO CN (29) V0 = ln KR C + which represents Nernst’s equation for this reaction. With the help of (28) and (29), (27) becomes: J+ = J0 (e−αO η − eαR η )

(30)

CN = γ − 2

D+ J+ DN

(34)

2.3. Total potential drop Of particular interest to experimentalists is how the current depends on the total potential drop, measured between the metal electrode and some convenient counter-electrode. As a computationally convenient reference electrode, we choose the mid-plane located at x = 0 in Fig. 2. We can then calculate the potential of the left electrode relative to the mid-plane by summing the following contributions (see Fig. 2):

where J0 = (KO CN )αR (KR C+ )αO is the dimensionless exchange current density. Thus (30) is another form of the Butler-Volmer equation which is equivalent to (27). In terms of dimensionless variables, boundary condition (25) becomes:

ψe − ψmp = (ψe − ψr ) + (ψr − ψio ) + (ψio − ψmp )

 4B εs V 1 + J+ 2 = B −1 εb δ

The three contributions being summed represent the potential drop across the Stern layer, across the diffuse layer (zeta) and the ohmic drop across the electrically neutral bulk solution. The two equations above have been written for the left electrode; the total potential drop for the right electrode can be obtained from that for the left electrode just by reversing the sign of the current. Because −V represents the dimensionless potential drop across the Stern layer, we will refer to it as the “Stern potential”. However, in the colloids literature “Stern potential” usually means the potential of the Stern plane ψr relative to

after substituting particular solution (20) and transformation (18), where δ = ␬λs is the thickness of the Stern layer normalized by the Debye length. Solving the quadratic for B, this boundary condition requires the following choice for the integration constant: √ 1 − 1 + 4C2 B= (31) 2C

or in dimensionless terms: ye − ymp = −V + ζ + yo (−1)

(36)

D.C. Prieve / Colloids and Surfaces A: Physicochem. Eng. Aspects 250 (2004) 67–77

the bulk solution just outside the double layer or relative to ψio : in the overly simplified treatment described above, that usual definition of Stern potential is equivalent to the zeta potential. Instead, we use the term “Stern potential” here to denote one of the three contributions to the total potential drop given by (36). At equilibrium (no current), the ohmic drop [last term in (36)] vanishes according to (15), leaving −V0 + ␨ as the total potential drop. Rearranging (35) yields:
 KR ζ − V0 = ln (37) KO γ which relates the total potential drop at equilibrium to the relative rate constants (KR /KO ) and the neutral species concentration (γ).

3. Results and discussion 3.1. Structure of diffuse layer with current In the classic Gouy-Chapman model for the diffuse part of the double layer, the local ion concentrations are related to local electrostatic potential by Boltzmann’s equation which requires the ions to be at equilibrium: the electrochemical potential of any ion species is uniform across the diffuse cloud. Boltzmann’s equation can also be obtained from the more general Nernst-Planck Eq. (1) if the ion flux Nix everywhere vanishes. Of course, for a steady-state electric current to occur, at least one of the ions must have a nonzero flux. The concentration of this ion would not then be expected to satisfy Boltzmann’s equation. Yet, the analysis performed here reveals that, with the proper choice of reference state for electrostatic potential and the proper renormalization of ion concentration, the two quantities are still related by Boltzmann’s equation, even for electrical currents approaching the limiting current: see (23). The reference state for both quantities is the location just outside the diffuse cloud, which is assumed to be infinitesimally thin. Boltzmann’s equation does not however apply outside the diffuse cloud in the presence of current. Outside the diffuse cloud, the ion concentrations and potential are given by (12) and (13). Combining these two equations, we obtain:

 J+ + J − o Co (X) = exp y (X) (38) J+ − J − The factor involving the ion currents J+ and J− causes this relationship to deviate from Boltzmann’s equation. If one or the other of these ion fluxes vanishes, this factor becomes +1 or −1, reducing this relationship to Boltzmann’s equation for the species having zero ion flux. However, the species with the nonzero flux does not satisfy Boltzmann’s equation. Instead the two ion species have the same concentration: electroneutrality occurs. Newman [20] also showed that electroneutrality can be assumed in the region outside the double layer in his asymptotic

73

analysis of convective diffusion of symmetric binary electrolytes during 1-D transport to a rotating disk with currents below the limit current. Indeed Newman [20] also obtained an inner solution for the electric field which has the same distance dependence as in the equilibrium Gouy-Chapman model. Newman’s problem differs slightly from this one in that the boundary condition on electrolyte concentration requires it to approach a specified constant infinitely far from the rotating disk. The present problem involves a bounded outer region with a nonzero concentration gradient. Nonetheless, the assumption of electroneutrality outside the diffuse layer is well entrenched in electrochemical analysis [19]. A well-informed reviewer pointed out that Levich [21] showed that a Boltzmann-like relationship applied between the ion concentration and potential inside the diffuse layer for currents below the limiting current. However, Levich’s conclusion appears to require that the current conducting ion have a sign opposite to that of the surface change on the electrode. No such requirement is needed in the present analysis. A simple physical explanation can be provided for the persistence of Boltzmann’s equation. When the electrode spacing 2l is made large enough compared to the Debye length κ−1 , the macroscopic externally applied electric field will always become infinitesimal compared to the electric field inside the diffuse part of the double layer in the absence of current: |∆ψ|

κ |ζ| l In particular, the ratio of these two electric fields is ε → 0 and Boltzmann’s equation is preserved as (23). On the other hand, without current no electric field exists outside the diffuse part of the double layer so that any externally applied electric field causes Boltzmann’s equation to be replaced by (38). Although current does not alter the functional relationship between ion concentration and potential inside the diffuse cloud, it does alter this relationship outside the diffuse cloud. These latter changes also affect the ion concentration profiles inside the cloud. In particular, the ion concentrations just outside the diffuse cloud depend on current: see (14). It is these ion concentrations rather than those at the mid-plane of the cell which determine the Debye length inside the diffuse cloud: recall that ξ  is the distance from the electrode normalized by the Debye length evaluated for the ionic strength at the mid-plane. On the other hand, ξ is normalized by the ion concentrations given by (14): see (18). Besides changing the Debye length, current might change the potential drop across the diffuse layer. These conclusions are independent of which redox reactions occur at the electrode. We will now focus on this latter change in one particular case. 3.2. Equilibrium zeta When cations undergo reaction (24) and no other reaction or ion adsorption occurs, then the electrostatic potential drops

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renormalized by the current-dependent factor appearing in (32). A word of caution about the values of Stern potentials: V = 4 represents a potential drop of approximately 0.1 volt across the Stern layer. If the Stern layer has a thickness of 0.1 nm, this corresponds to an electric field of about 109 volt/m in the Stern layer. Such enormous electric fields might induce other phenomena or other electrode reactions than those considered here. 3.3. Effect of current on profiles Fig. 3. . Effect of relative rate constants (KR /KO ) and neutral species concentration (γ) in the absence of current (i.e. at equilibrium) on the potential drop across the Stern layer, across the diffuse layer (ζ) and on the total potential drop (ζ − V0 ). The linear relationship involving the total is given by (37).

Fig. 5 shows the electrostatic potential and concentration profiles for one particular current (J+ = 0.49) and one particular set of rate constants (KR = KO = 100). Other parameters whose values must be specified are DN = D+ , z = 1, γ = 1, δ = 0.01, αO = αR = 1/2 and ε = 0.05. This particular choice of constants corresponds to KR /KO γ = 1, for which Fig. 3 reports that the equilibrium (i.e. J+ = 0) zeta potential vanishes. Nonetheless, the application of current causes a substantial potential drop across the diffuse clouds next to both electrodes. As J+ → 0.5, the current is limited by diffusion of the neutral species to the left electrode: notice that CN appears to vanish at X = −1. Actually, it does not quite vanish for the slightly smaller current used for the calculation in Fig. 5; if instead, we had used J+ = 0.5, then CN would vanish at X = −1 but the potential drop across the left diffuse cloud would diverge. Changing the sign of the current reflects all the profiles about X = 0. The left and right halves of the

Fig. 4. Correlation between the potential drop across the diffuse layer (zeta) and the potential drop across the Stern layer (−V).

across the diffuse cloud and the Stern layer is determined, in the absence of electric current, solely by the relative values of the two rate constants KO and KR and the bulk concentrations. Fig. 3 shows how the potential drops in the absence of current depends on the relative rate constants. Other parameters which must be specified for this calculation but were not systematically varied include αR = αO = 1/2, εs /εb = 1/8 and δ = 0.01. Increasing the reduction rate KR or decreasing the oxidation rate KO γ tends to favor reduction of cations which, at equilibrium, is offset by a more positive value of the Stern potential −V0 [see (27)].1 This more positive Stern potential in turn induces a more positive value of the zeta potential [see (31) and (32)]. Also shown in Fig. 3 is the total potential drop ζ-V0 between the metal electrode and the bulk solution outside the double layer. The linear relationship between this total and the abscissa is given by (37). Fig. 4 shows the cross-correlation between the values of the Stern and zeta potentials of Fig. 3 (where J = 0 was assumed). This relationship between the potentials also holds in the presence of current, provided the Stern potential is 1

This is not the usual Stern potential: see second paragraph before Section 3.

Fig. 5. Profiles of concentrations (b) and potential (a) in the presence of current J+ = 0.49. Black curves represent the composite asymptotic solutions whereas the grey curves represent just the outer solution.

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potential profile are not antisymmetric: the outer solution at larger current densities displays a slight downward curvature in electrostatic potential. In order to see the curvature in the central region, we have severely truncated the range of potentials displayed. Reducing the magnitude of the current density causes the profiles to tend toward φ = 0 and Ci = 1 for all X. In order to make the diffuse cloud clearly visible on these graphs, we have chosen a fairly large value for ε (0.05). For such thick counterion clouds, the integral of the concentrations might differ from 2lc0 which was implicitly assumed in our boundary conditions. Values of ε in our experiments are much smaller and any changes in the integral then are negligible. Finally, a word about the concentration gradients in the bulk outside the diffuse clouds: the reader might be surprised 1) that dCN /dX = 1 and dC+ /dX = −0.5 although the stoichiometry of the reaction requires equal but opposite fluxes (assuming z = 1) for the cations and the neutral species and 2) that dC− /dX = −0.5 although the anion flux must vanish (since anions are not involved in the surface reactions). In addition to diffusion, electromigration also contributes to the flux; indeed the contribution from electromigration neutralizes the diffusive flux of anions and it also makes the flux of cations equal but opposite to that of the neutral species. This can be confirmed analytically by examining (6) and (7) in the bulk where ρ = 0. 3.4. Effect of current on zeta Fig. 6a shows the effect of current density on the potential drop across the diffuse cloud [calculated as yi (0) − yi (∞) from the inner solution] at the left electrode. We will refer to this potential drop as the “zeta potential” whether current exists or not. Results for four sets of rates constants are displayed with all sets corresponding to KR /KO γ = 1, for which Fig. 3 reports that the equilibrium (i.e. J+ = 0) zeta potential vanishes; indeed, all four curves in Fig. 6a pass through the origin. Positive currents represent the net production of cations by oxidation at the electrode while negative currents represent the net consumption of cations by reduction. Transporting cations away from (toward) the electrode requires a more positive (negative) zeta potential than exists in the absence of current. For large but equal values of both rate constants, the slope in the vicinity of zero current is dζ/dJ = 2.92, but for the smallest but equal value of both rate constants, the slope becomes an order of magnitude larger. To understand why the slope is strongly dependent on the rate constants, we need to recall the kinetics at the left electrode which are prescribed by (27). To change the current from zero to some positive value requires that V become more negative than its equilibrium value. Neglecting the corresponding changes in the surface concentrations CN and C+ , larger changes in V are required to achieve the same current when the rate constants are smaller. These larger, negative changes in V in turn cause

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Fig. 6. Effect of current density on the potential drop across the diffuse cloud (a) and the ion concentrations (b) evaluated at the left electrode when the rate constants are varied keeping their ratio constant. In (b), the cation concentration is denoted by the black curves while the anion concentration is denoted by the grey curves.

larger, but positive changes in ζ: the relationship between V and ζ is determined from (31) and (32) and is plotted in Fig. 4. Fig. 6b summarizes the corresponding changes induced by current in the ion concentrations at the surface of the electrode. Not shown are the corresponding changes induced by current in the concentration of neutral species. According to (26), the neutral species concentration varies linearly with current from CN = 3 for J = −1 to CN = 0 for J = 0.5, regardless of the values of the rate constants. Note in Fig. 6b the logarithmic scale for ion concentrations, which is used because ion concentrations undergo changes by several decades as the current changes. For positive currents, the curves for cation and anion concentration appear to be mirror images reflected about C = 1. This is expected if the ion concentrations obey Boltzmann’s equation: the same Boltzmann factor by which the counterion concentration is increased also acts to decrease the co-ion concentration. However, this symmetry does not seem to apply for negative currents, suggesting non-Boltzmann-like behavior. In (23) we showed that the ion concentrations do indeed satisfy Boltmann’s equation, provided the ion concentrations are properly renormalized. In the case of Fig. 6b, this renormalization factor is 1 + J+ . If both cation and anion concentrations were divided by 1 + J+ , the renormalized curves for cation and anion would be exact mirror images about C = 1. Moreover, if we cross-correlated the renormalized cation (anion) concentration with the corresponding zeta potential (Fig. 6a), the curves on semi-log coordinates would collapse to a single straight line having negative (positive) slope.

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Fig. 7 is similar to Fig. 6 except the two rates constants are not equal. As a consequence, the equilibrium (i.e. J+ = 0) zeta potential differs from zero for all but one of the curves. The equilibrium zeta potentials are consistent with Fig. 3. The left half of the curves are all similar in shape but just shifted vertically by the shift in the equilibrium zeta potentials. 3.5. Current versus voltage

Fig. 7. Effect of current density on the potential drop across the diffuse cloud (a) and the ion concentrations (b) evaluated at the left electrode when the rate constants are varied keeping their product constant. In (b), the cation (anion) concentration is denoted by the black (grey) curves and −1 the upper to lower (lower to upper) curves correspond to KR = KO = 0.01, 0.1, 1, 10 and 100.

Fig. 8. Effect of current density on the potential drop across the Stern layer (a) and the total potential drop [black curves in (b)] evaluated at the left electrode when the rate constants are varied keeping their ratio constant. In (b), the grey curve denote the ohmic potential drop between the outer edge of the diffuse layer and the midplane.

The total potential drop was defined by (36). The three contributions being summed represent the potential drop across the Stern layer, across the diffuse layer (zeta) and the ohmic drop across the electrically neutral bulk solution. The equations above have been written for the left electrode; the total potential drop for the right electrode can be obtained from that for the left electrode just by reversing the sign of the current. Fig. 8 shows how the total potential drop depends on current for the five sets of rate constants used by Fig. 6. Also shown in Fig. 8 are two of the three contributions [−V and y0 (−1)] to the total; the third contribution (ζ) can be found in Fig. 6. All three contributions display similar trends with current and with the value of rate constants: all three display a monotonic increase with current, with a greater rate of increase displayed for positive currents than for negative cur-

Fig. 9. Effect of current density on the potential drop across the Stern layer (a) and the total potential drop [black curves in (b)] evaluated at the left electrode when the rate constants are varied keeping their product constant. In (b), the grey curve denote the ohmic potential drop between the outer edge of the diffuse layer and the midplane.

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rents and a much greater rate of increase (at least for positive currents) for smaller values of the rate constants. We have already presented the close cross-correlation between two of the contributions in Fig. 4. While it does not depend on the rate constants, the remaining contribution y0 (−1) increases monotonically with current when J− = 0 [see (15)]. So, the total potential drop also displays these characteristics. Fig. 9 shows how the total potential drop depends on current for the five sets of rate constants used by Fig. 7. Once again, the trends in the total potential with current and with the rate constants mimic the corresponding trends displayed in Fig. 7 by the zeta potential.

4. Conclusions Applying an electric current between two large parallel electrodes requires a nonzero ion flux and generates an ohmic drop in the electrolyte solution. Using matched asymptotic expansions, we obtain the asymptotic solution for 1-D, steady transport of a symmetric binary electrolyte in the limit in which the Debye length is infinestimal compared to the electrode spacing. To leading order, the bulk of the solution can be treated as electrically neutral whereas Boltzmann’s exponential relationship between ion concentrations and the electrostatic potential persists inside the diffuse part of the double layer, even for current densities up to the limiting value, despite the nonzero ion flux. The application of current causes the ionic strength of the bulk solution to vary linearly with position. The Debye lengths for the thin double layers next to either electrode will be different and correspond to those calculated from the usual formula using the local value of the ionic strength just outside each double layer. Any current perturbs the potential drop across the diffuse layers (i.e. the zeta potential). The amount of the change depends on the kinetics of the electrode reactions. We explored one simple case in which the Faradaic reaction at the cathode is the reduction of cations to form neutral species and no other charging mechanism exists (i.e. we ignore any adsorption of ions and ignore any dissociation of ionizable surface sites). Then the equilibrium (i.e. zero current) zeta potential is determined by the relative rate constants and the bulk concentration of neutral species. Even when this equilibrium zeta potential vanishes, application of a current perturbs the zeta potential. Dimensionless perturbations dζ/dJ were as large as 30.

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While the electrode reactions, steady-state conditions, geometry and electrolyte considered here are overly simple, the main conclusions (electroneutrality of the bulk, ion concentrations in diffuse layer satisfy Boltzmann’s equation) ought to be applicable in more complex situations as long as the Debye length remains very thin. Acknowledgments The simple explanation for the presistence of an equilibrium (Boltzmann) structure for the thin double layer in the presence of current was suggested by Prof. Hermann van Leeuwen of Wageningen Agricultural University. Prof. Paul J. Sides of Carnegie Mellon Univ pointed out Newman’s [19] derivation of the more familiar form of the Bulter-Volmer Eqs. (27) and (30). Financial support for this work was partially provided by the National Science Foundation under Grant CTS-89875.

References [1] M. Giersig, P. Mulvaney, J. Phys. Chem. 97 (1993) 6334. [2] M. B¨ohmer, Langmuir 12 (1996) 5747. [3] Y. Solomentsev, M. B¨ohmer, J.L. Anderson, Langmuir 13 (1997) 6058. [4] Y. Solomentsev, S.A. Guelcher, M. Bevan, J.L. Anderson, Langmuir 16 (2000) 9208. [5] M. Trau, D.A. Saville, I.A. Aksay, Science 272 (1996) 706. [6] M. Trau, D.A. Saville, I.A. Aksay, Langmuir 13 (1997) 6375. [7] J. Kim, S.A. Guelcher, S. Garoff, J.L. Anderson, Adv. Colloid Interf. Sci. 96 (2002) 131. [8] P.J. Sides, Langmuir 17 (2001) 5791. [9] J.A. Fagan, P.J. Sides, D.C. Prieve, Langmuir 20 (2004). [10] J.A. Fagan, P.J. Sides, D.C. Prieve, Langmuir 18 (2002) 7810. [11] J.A. Fagan, P.J. Sides, D.C. Prieve, Langmuir 19 (2003) 6627. [12] A. Bonnefont, F. Argoul, M.Z. Bazant, J. Electroanal. Chem. 500 (2001) 52. [13] M.Z. Bazant, K.T. Chu, B.J. Bayly, SIAM Journal, in press. [14] K.T. Chu, M.Z. Bazant, SIAM Journal, in press. [15] J. Duval, J. Lyklema, J.M. Kleijn, H.P. van Leeuwen, Langmuir 17 (2001) 7573. [16] J. Duval, J.M. Kleijn, J. Lyklema, H.P.v. Leeuwen, J. Electroanal. Chem. 532 (2002) 337. [17] J.F.L. Duval, G.K. Huijs, W.F. Threels, J. Lyklema, H.P. van Leeuwen, J. Colloid Interf. Sci. 260 (2003) 95. [18] J.F.L. Duval, J. Colloid Interf. Sci. 269 (2004) 211. [19] J.S. Newman, Electrochemical Systems, Prentice-Hall, Englewood Cliffs, NJ, 1973. [20] J. Newman, Trans. Faraday Soc. 61 (1965) 2229. [21] V.G. Levich, Dokl. Akad. Nauk SSSR 67 (1949) 309.