Ionic Gradients at an Electrode above the Equilibrium Limit Current. 1

The charge and ionic concentration gradients next to an electrode where one member of a simple binary electrolyte is consumed is analyzed. It is shown...
0 downloads 0 Views 275KB Size
Download PDF
J. Phys. Chem. C 2007, 111, 3341-3348

3341

Ionic Gradients at an Electrode above the Equilibrium Limit Current. 1. Concentration and Charge Gradients in an Immobile Electrolyte Jonathan J. Van Tassel* and Clive A. Randall† Materials Research Institute, The PennsylVania State UniVersity, UniVersity Park, PennsylVania 16802 ReceiVed: July 27, 2006; In Final Form: NoVember 20, 2006

The charge and ionic concentration gradients next to an electrode where one member of a simple binary electrolyte is consumed is analyzed. It is shown that the thickness, profile, and formation time for a concentration gradient at the electrode can be expressed as a function of bulk voltage gradient and diffusivity of the consumed ion. When the concentration at the electrode approaches zero, this concentration gradient will move away from the electrode, becoming an equilibrium profile moving gradient layer, which moves away from the electrode at the speed of the counter ions in the bulk solution. The moving gradient layer will be followed by a thin, constant thickness transition layer and a growing unbalanced charge conduction layer. Conduction through this unbalanced charge layer will be by migration of a very low concentration of the consumed ion in the almost complete absence of the counterion. This layer is characterized by very high voltage gradients, high power dissipation (i.e., heating), and high stress gradients in the solvent.

1. Introduction In an ionic conductor current is carried by the motion of ions from one electrode to another. Without convection, this motion can only occur by ionic migration and diffusion. If one polarity of ion is consumed at an electrode and no ions of opposite charge are produced, such as by electrolysis of the solvent or electrode, the total concentration of ions adjacent to that electrode will be reduced. When the ionic concentration at an electrode approaches zero, there is a large, rapid, nonlinear rise in resistance. This limits the equilibrium current to a small multiple of what could be carried in the electrolyte by diffusion alone. The physics that lead to this limit current effect have been known now for one hundred years, and this effect is a fundamental concept in quantitative electrochemistry. However, what happens when an ionic conductor is pushed beyond the limit current has received relatively little attention. The limit current has existed mainly as something to be avoided through forced convection, addition of excess electrolyte, low overpotentials, microelectrodes, etc.1 A further reason for the neglect of conduction beyond the limit current is the difficulty of generating clear limit current behavior in a liquid electrolyte, due to the rapid onset of convection in the cell. As Levich2 pointed out: “... a motionless solution can be achieved in practice only in exceptional cases, for example, where the solution is immobilized by the addition of gelatin or agar-agar.” For practical purposes, a limit current only exists in a solid, a gelled liquid, or a liquid confined to a nanoporous solid. The history of this topic can be traced through a series of significant publications. The link between an electrochemical current and ionic diffusion in a charge balanced solution goes back to Nernst and Brunner (1904).3,4,5 How this leads to an equilibrium limit current for a cell was best explained by Levich (1959).2 Unbalanced charge conduction in a fluid was discussed by Felici (1971).6 More recent work has almost all been generated by interest in explaining the growth of ramified * Corresponding author. E-mail: [email protected]. † E-mail: [email protected].

structures during the electrodeposition of copper.7,8 Chazalviel (1981)9 was the first to consider that ion depletion at an electrode can lead to two layerssone characterized by the standard quasineutral assumption of a charge balanced electrolyte and a second characterized by unbalanced charge conduction. An analysis demonstrating the mathematical consistency of a moving gradient layer in the case of ramified growth of an electrodeposited layer was given by Bazant (1995).10 A simpler derivation of the formula for concentration across the moving gradient layer was given by Le´ger et al. (1998)11 along with very nice experimental measurements confirming the validity of the formula. More recently, steady-state conduction in a thin layer at a current beyond the limit current was analyzed by Chu and Bazant (2005).12 This paper is further development of analysis presented by Van Tassel (2005).13 This paper presents an analysis of the development of these layers at constant current using the example of an ethanol solvent with a HCl electrolyte. It is shown that the two most critical factors governing the development of the depletion layers in the quasi-neutral regime are the initial voltage gradient and mobility of the consumed ion. These two factors are incorporated in the equilibrium limit current thickness, which then provides a convenient factor for non-dimensionalizing these problems. Surprisingly, these layers are relatively unaffected by the total ionic concentration. When ionic concentration reaches a very low level at the electrode, the assumption of quasi-neutrality is violated and an unbalanced charge conduction layer is formed. This layer is characterized by very high voltage gradients, which can rise in a matter of seconds to the breakdown strength of the solvent. The thickness of the transition layer between quasi-neutral and unbalanced charge layers and the voltage rise rate are then functions of the mobility of both positive and negative ions, the initial voltage gradient in solution, and the total current flux through the system. The analysis performed here serves as an introduction to two subsequent papers. Here it is predicted that at constant current, and in the absence of convection, there will be an exponential

10.1021/jp064804y CCC: $37.00 © 2007 American Chemical Society Published on Web 02/02/2007

3342 J. Phys. Chem. C, Vol. 111, No. 8, 2007

Van Tassel and Randall

TABLE 1: Ionic Properties of Solution conductivity (µS/cm) molar conductivity (Λ) (cm2/Ω‚Mol) bulk HCl (Mol/m3) chloride ion mobility (νCl) (m2/V‚s) hydrogen ion mobility (νH) (m2/V‚s) codiffusion coeff of Electrolyte (DE) (m2/s)

case 1

case 2

case 3

2.31 51.3 0.045 2.19 × 10-8 3.13 × 10-8 6.61 × 10-10

9.58 49.9 0.192 2.13 × 10-8 3.04 × 10-8 6.40 × 10-10

35.4 47.1 0.751 2.01 × 10-8 2.87 × 10-8 6.07 × 10-10

rise in voltage in the system within the first few seconds of conduction. To explain why this behavior is not seen in a fluid medium, the second paper in this series demonstrates why the solution will transition to electroconvective motion in virtually the same instant that an unbalanced charge layer begins to form. While the existence of an ion depleted conduction layer seems to be a reasonable hypothesis, without clear, quantitative experimental evidence of its existence, the question remains whether this concept is valid or whether there exists some other mechanism which is being ignored. The third paper in this series then demonstrates how an unbalanced charge conduction layer can be stabilized and presents experimental evidence both for the validity of the equations governing this layer as well as the ultimate instability of this layer due to convection. 2. Experimental Section The electrolytic solution is a 99.5/0.5 wt % ethanol/water solvent with HCl as added electrolyte. Conductivity in this solvent has been well characterized by DiLisi et al.14 and confirmed by these authors in ref 15. With an association constant of 20 dm3 mol-1, the HCl can be assumed to be completely dissociated at the concentrations used here. The experimental apparatus is a deposition cell with polished platinum electrodes, a cell constant of 0.7 cm-1, and an average electrode spacing of 3.5 cm. The current flux in each of these trials was chosen to be proportional to the conductivity of the solvent so that the electric field at the cathode is similar in all three trials. 2.1. Cathode Reactions. The cathode reaction in this system is fairly simple. Hydrogen ions arriving at the surface in the form of hydronium or ethoxonium ions are neutralized to form hydrogen gas and water or ethanol. In the absence of other positive ions which could form a soluble salt with hydroxide or ethoxide ions there is no electrolysis of the solvent. Cathode Reaction

2H3O+ + 2e- w H2 + 2H2O 2.2. Anode Reactions. The chemistry at the anode is slightly more complex, with several reactions possible. The primary reaction is the electrolysis of water to produce hydronium or ethoxonium ions and oxygen gas. Reactions which consume chloride ions, either producing chlorine gas or ethylene chloride, will result in a net decrease in ionic concentration in solution. In an extended conduction test there was a drop in ionic concentration equivalent to ≈20% of the total electron flux through the solution. This means that 80% of the current at the anode results in the formation of H3O+/H2EtO+. Over the time span considered here, the molar current flux is not a significant fraction of the total quantity of ions in the suspension, and this reduction in ionic concentration is negligible. Primary Anode Reactions

6H2O w O2 + 4H3O+ + 4e2Cl- w Cl2(gas) + 2e-

2.3. Molar Conductivity, Ionic Mobility and Diffusion. The molar limit conductivity for HCl in this ethanol/water mixture is 53.4 S‚cm2/mol.14 This conductivity is the sum of the conductivities due to the mobility of the chloride and hydrogen ions in solution.

ΛOHCl ) λOCl- + λOH+

(1)

(The symbols in the above equation and all subsequent equations are defined in the section entitled Symbols.) Grahm et al.16 measured the molar limit conductance for the chloride ion in anhydrous ethanol as 21.9 S‚cm2/mol. The mobility of the chloride ion will be relatively unaffected by water content therefore this can be taken as an estimate for the limit conductance of chloride in this solvent as well. This gives a molar limit conductance for the proton of 31.5 S‚cm2/mol. Since ionic strength is assumed to affect both positive and negative ions equally, this ratio is used to divide the molar conductivity at each concentration between chloride and proton components. This molar conductivity is converted to an ionic mobility by dividing by the Faraday constant. Because of the relatively low concentrations used here, the Einstein relation can be used to estimate the diffusion coefficients, Di, from the ionic mobilities:

Di )

or RT kT νi ) ν |zi|e |zi|F i

(2)

These values are listed in Table 1 for the three ionic concentration conditions to be analyzed here. 3. Analysis 3.1. Overview. Before beginning on the quantitative analysis, it will be helpful to offer a qualitative description of our approach to the problem of ion depletion at an electrode. Electrode Boundary Layers. At each electrode in an electrochemical cell, there are two boundary layers. The first is the diffuse electrostatic boundary layer. This is the portion of the electrostatic double layer that extends into the solvent where ionic activities are governed by the Poisson-Boltzmann equation. The thickness for this layer is characterized by the Debye length. When current begins to flow, an electrochemical diffusion boundary layer is formed where the solution chemistry is changed by the production and consumption of chemical species at the electrodes. The thickness and composition of this layer is governed by the laws of mass transport: diffusion, migration, and convection. Equilibrium Conduction. A useful first step is to establish a scale on which the problem should be approached, which then allows some simplifying assumptions. The feature of interest here is an electrochemical boundary layer, specifically the ion depletion gradient at an electrode. A scale factor for this boundary layer can be generated by inverting the concept of the quasi-neutral limit current, which was best explained by Levich.2

Gradients in an Immobile Electrolyte

J. Phys. Chem. C, Vol. 111, No. 8, 2007 3343

Figure 1. Primary chemical reactions at the electrodes and ionic migration through the solvent.

Figure 3. Electrochemical boundary region near the cathode without convection is divided into four zones; (a) neutral region or bulk solution, (b) quasi-neutral moving gradient region, (c) charged layer or unbalanced charge region, and (d) diffuse electrostatic boundary layer (not shown).

Figure 2. Formation of a gradient as ions are depleted at the electrode.

The standard analysis begins with a cell with a given electrode spacing and ionic concentration, and with a charge carrying ion generated at one electrode and consumed at the other. It then shows that at equilibrium conduction the ionic concentration will decline linearly from the generating electrode to the consuming electrode. The slope of the concentration gradient is a function of current, and the slope which brings the ionic concentration to a theoretical zero point at the consuming electrode corresponds to the maximum, or limit, current of the cell. Additional current can be carried, but only by the initiation of convection or the violation of quasi-neutrality. For the solutions considered here, the ionic mobilities, concentrations and current density are known, and the question can be framed to ask what is the maximum cell thickness which can conduct this current without convection. Assuming a simple parallel plate cell, the maximum thickness for a given current density will be given by eq 3, and the dlim for the three cases considered here are all ≈20 µm (Table 2).

dlim ) 2

FDH+ 2DH+co co ) I J

(3)

The above treatment shows that the electrochemical boundary layer that is of interest here is on the order of tens of micrometers. Since we are interested here in a cathode electrochemical boundary layer, and given that the electrodes in the test cell are separated by more than two centimeters, it is reasonable to ignore both of the anode boundary layers and to treat this as a semiinfinite cell. At the cathode the Debye lengths for the three cases considered here begin at 6-26 nm. Although these layers will expand by a factor of 10-20 as ionic concentration declines next to the electrode surface, they will remain well below 0.5 µm, which can reasonably be neglected for this analysis. This is confirmed by the analyses of Smyrl and Newmann17 and Chu and Bazant.12 Moreover, the focus of this analysis is the potential gradients within the electrolyte affecting ionic migration. Therefore, the potential equations here will not include a concentration potential term. Analysis of the Problem. When the current is switched on the ionic concentration will go down at the cathode as H+ is consumed and Cl- migrates away in the electric field. This is shown schematically in Figure 2. The question is then, what happens when the concentration at the electrode approaches zero? The condition of constant current flux through the system means the concentration gradient

Figure 4. Ionic concentration next to the cathode (x ) 0) for case 2 at 0.2, 0.4, and 0.6 s after current is applied, calculated from eq 11.

cannot become less steep, nor is there an electrode 20 µm away from the cathode to block the migration of chloride ions away from the cathode. If convection is precluded, the only way for current to continue to flow through the system is for there to be a violation of the assumption of quasi-neutrality. The assumption of quasi-neutrality is simply that the difference between the number of positive and negative ions at any point in the solution is very small compared to the total number of ions in the solution at that point. This means that although there may be a sufficient electrostatic charge to change the gradient of the electric field, the current calculated by diffusion and migration in that electric field can be made on the assumption that the total charge of positive and negative ions are equal. This assumption is only violated in cases of very low ionic strength and large changes in the voltage gradient, such as occur at the edge of an ion depletion gradient. This leads to the schematic diagram shown in Figure 3. An ionic depletion gradient forms at the cathode. When the concentration at the cathode approaches zero, this gradient layer will begin to move away from the cathode. The volume between the gradient layer and the cathode is then filled by a layer of unbalanced charge conduction. There is, unfortunately, no analytic solution to the problem of combined diffusion and migration of ions which does not rely on the assumption of quasi-neutrality. Without this there is no single solution for the entire electrochemical boundary layer. However, a solution can still be generated by dividing the layer in two parts and applying the appropriate simplifying assumptions to each layer. The formation of the ion depletion layer and the shape of the subsequent moving gradient layer can be treated using the standard assumption of quasi-neutrality. In the unbalanced charge layer, the total ionic concentration is small while voltage gradients and the change in the voltage gradients are both large. Because of the high voltage gradient, it can be assumed that negative ions are effectively excluded from this layer and all conduction occurs by migration of positive ions. This leaves an intermediate layer where neither

3344 J. Phys. Chem. C, Vol. 111, No. 8, 2007

Van Tassel and Randall

TABLE 2: Conduction in Quasi-Neutral Region current flux (I) (A/m2) equilibrium limit current thickness dlim ) [J/2coDH+]-1 (µm) bulk voltage gradient (V/cm) bulk Cl- ion velocity (µm/s) depletion time to (s)

case 1

case 2

case 3

0.32 21.8

1.25 23.1

5.0 21.4

13.9 30.3 0.56

13.0 27.7 0.63

14.1 28.3 0.59

assumption is valid, however, this intermediate layer is shown to be small, and the solutions for the moving gradient and unbalanced charge conduction layers can be joined by matching potentials and potential gradients. Note. As ionic concentration declines in the gradient layer, ionic mobility will rise toward the limit value at zero concentration. To simplify the analysis, the mobilities and diffusion constants are assumed to be constant at their bulk values in the gradient layer and are taken at their zero concentration limit values in the unbalanced charge layer. 3.2. Region A. Bulk Conduction and Mobility. Table 2 gives the quantities necessary for analyzing conduction and ion depletion in the quasi-neutral regime. To calculate the electric field in the bulk solution near the cathode the specific current is divided by the conductivity to get the electric field gradient. The velocity of the chloride ions in the bulk is then the mobility times the field gradient. Once a stable gradient layer is formed, it will move away from the cathode at this velocity. 3.3. Region B. Formation of the Depletion Gradient. The first step in this process is the formation of a depletion gradient layer at the electrode. During the formation of this gradient layer the quasi-neutral assumption will apply:

cCl- ≈ cH+ ) c

(4)

Both ionic species will be exposed to the same electric field, which allows an equation to be written for the diffusion and migration of each ion, eqs 5 and 6. 2

∂c ∂ ∂φ ∂c c ) D H+ 2 + ν H+ ∂t ∂x ∂x ∂x

( )

(5)

∂c ∂2c ∂ ∂φ c ) DCl- 2 - νCl∂t ∂x ∂x ∂x

(6)

( )

Using eq 2 to convert the ionic mobilities to functions of the diffusion constants, the potential gradient can be eliminated converting this to a simple diffusion equation:

DH+DCl∂c ∂2c ) DE 2; DE ) 2 ∂t D ∂x H+ + DCl-

(7)

Since no chloride ions are generated at the cathode boundary of the system, x ) 0, the net flux of chloride ions will be zero:

0 ) DCl-

∂φ ∂c | - νCl-c |x)0 ∂x x)0 ∂x

(8)

The constant current condition then means that there must be a constant molar flux, J, of protonated ions at the electrode surface, x ) 0, equal to the total current flux across the system, I;

∂c I ∂φ ) J ) -DH+ |x)0 - νH+c |x)0 F ∂x ∂x

(9)

These equations can then be combined to give the concentration and voltage gradients at the electrode. The result is, of course, that the gradient at the electrode is the same as the gradient in the equilibrium limit current case:

∂c J | )∂x x)0 2DH+

(10)

Using eq 11 as one boundary condition, while at the other boundary, c f co as x f -∞, the solution of eq 7 for the depletion gradient is then

c(x,t) co

)1-

J 2DH+co

[(

) ( )

4DEt π

1/2

exp

-x2 - x erfc 4DEt

(x )] x

4DEt (11)

The voltage gradient over the entire depletion layer can then be obtained for by substituting eq 11 into eq 16:

[

|

∂φ ∂φ 1 ) c ∂x c o ∂x

x)∞

-

]

RT DH+ - DCl- ∂c F D + + D - ∂x H

Cl

(12)

According to eq 11 the concentration at the cathode (x ) 0) will go to zero after a time to:

to )

[

]

π 2DH+co 2 π ) d 2 4DE J 4DE lim

(13)

This shows that in all three of the example cases the concentration will approach zero six tenths of a second after the current is switched on. If we substitute the solution for to from eq 13 back into eq 11, we get an equation for the profile of the depleted layer after a theoretical zero concentration has been reached:

c(x,t) co

[ ( [ ])

) 1 - exp -

1 x π dlim

2

-

(

x 1 x erfc dlim xπ dlim

)]

(14)

The shape of the depletion gradient is shown to be a function only of xJ/2DH+co or more simply x/dlim. 3.4. Region B. Quasi-Neutral Moving Gradient Layer. As will be shown below, the quasi-neutral assumption is only broken at the very edge of this concentration depletion gradient. Because of the electric field within this gradient, Cl- ions will still migrate toward the anode, and the gradient layer must move with them. The question is then how the profile of this moving gradient layer will change as it moves away from the electrode. There are three ways the gradient profile could evolve. It could (1) become continually steeper, (2) achieve an equilibrium profile, or (3) become continually less steep, with the same total concentration drop spreading over a wider distance, yielding a continuously declining gradient. The first case is self-limiting. The maximum possible gradient would be a discontinuous step in concentration. Diffusion, however, provides a limit on how steep the gradient can become. The gradient is also limited in how flat it can become. Similar to an equilibrium limit current, the current flux will set a minimum possible gradient through the layer. If the gradient profile is then limited both by how steep and how flat it can become, it must achieve some profile between the two extremes. Since there is no input that would cause it to oscillate, it must approach a steady, equilibrium profile. If there is a moving reference frame where the concentration profile is constant, it follows immediately from conservation

Gradients in an Immobile Electrolyte

J. Phys. Chem. C, Vol. 111, No. 8, 2007 3345

of mass that that reference frame must be moving away from the cathode at the speed of the Cl- ions in the bulk solution. Furthermore, to maintain this constant profile, all of the Clions in this gradient layer must move at the same speed as the reference frame. Since there can be no charge accumulation in solution, current flux must be constant at every point across the moving gradient layer, and consequently the current flux carried by the Cl- ions at any point in the moving gradient layer is proportional to their concentration. This can be expressed as

νClcCl∂cCl∂φ - νClc J ) DClνCl- + νH+ cCl-o ∂x ∂x Cl-

(15)

The right-hand side of this equation is the flux of chloride ions. On the left [νCl-/(νCl- + νH+)]J is the proportion of net ionic flux due to Cl- ions in the bulk, and cCl-/cCl-o is the ratio of Clions at a point in the gradient layer to the concentration of Clions in the bulk. On the basis of the assumption of quasi neutrality, as current flows through the solution there will be no significant net accumulations of either positive or negative charge. To obey this restriction, the net ionic flux must be constant at all points through the gradient layer. This criterion can be expressed by

J ) (DCl- - DH+)

∂c ∂φ - (νCl- + νH+)c ∂x ∂x

(16)

Using eq 4, cCl- is replaced by c in eq 16, which can then be multiplied by (νCl- + νH+)/νCl- to give:

(νCl- + νH+) ∂c c ∂φ J) DCl- - (νCl- + νH+)c co νCl∂x ∂x This can be subtracted from eq 16 to give

( ) [ 1-

(17)

]

DCl- (νCl- + νH+) ∂c c J ) (DCl- - DH+) co νCl∂x (18)

The term in square brackets becomes simply:

(DCl- - DH+) -

DCl-(νCl- + νH+) ) -2DH+ νCl-

(19)

Equation 18 can then be rearranged:

-J 1 dx ) dc 2coDH+ co - c

(20)

Integrating we obtain:

-J x + k1 ) -ln(co - c) 2coDH+ B

(21)

where k1 is a constant of integration. With this integration x changes from a relative to an absolute position index, so the subscript B is added to indicate position in region B (Figure 3). This can be solved to give an expression for concentration with the value for the integration constant k1 determined by boundary conditions. Since c will only approach co asymptotically, there is no point x where c ) co. On the other end, there is a theoretical point where c ) 0. This point is at the electrode surface at to and moves away from the surface at the speed of the chloride ions in the bulk solution. This moving reference frame can be defined by setting xB to be zero at the point where the moving

Figure 5. Equilibrium moving concentration gradient in quasi-neutral region for three cases: (a) relative concentration, eq 22; (b) potential gradient, eq 24.

gradient reaches a theoretical zero concentration point, with xB defined in terms of x as xB ) x - tνCl- ∂φ/∂x; t > to. The resulting concentration profile is

[

]

[ ]

xB c J ) 1 - exp xB ) 1 - exp co 2coDH+ dlim

(22)

In this equation J, DH+, and co will be positive; therefore, for increasing negative values of x, c will approach co. As with the depletion gradient, the thickness and profile of the equilibrium moving gradient layer is a function of xJ/2DH+co or x/dlim. Once again because the current flux is proportional the concentration, the thickness of the moving gradient layers in all three cases is similar. When the relative concentration is plotted as a function of absolute distance, in Figure 5a), we see that the three cases overlay each other almost exactly. Equation 22 has been derived three times before. Each time in the context of the electrodeposition of a ramified structure, where the tips of the branched structure were observed to grow at the speed of the migration of the counter ions in bulk solution. Barkey and LaPorte18 and Leger et al.11 both derived this profile beginning with an a priori assumption of the speed of the layer based on experimental observation. Chu and Bazant12 make no a priori assumption about the speed of the gradient layer but then show through the boundary conditions on the system that the speed must be the same as the average velocity of the counter ions in bulk solution. Here we began only with the assumption that a gradient must exist and that it cannot become either infinitely flat nor steep. This moving gradient layer is an intrinsic feature of any motionless solution where ion depletion occurs at an electrode and is not only a feature of systems showing ramified growth of a conductive electrodeposited material. From this approximation, it is possible also to solve for the potential, field and field gradient in the moving gradient layer, and the results are listed below. (Details of the derivation of these formulas can be found in the Supporting Information.)

φ)

(

[ ]) ( [ ]) [ ]( [ ])

xB RT -J x+ ln 1 - exp F dlim co(νCl- + νH+)

+ φo

xB ∂φ -J RT ) dlim-1 1 - exp ∂x co(νCl- + νH+) F dlim

xB xB ∂ 2φ RT -2 (d ) ) exp 1 exp lim F dlim dlim ∂x2

(23)

-1

(24) -2

(25)

3346 J. Phys. Chem. C, Vol. 111, No. 8, 2007

Van Tassel and Randall

Figure 6. Ion depletion gradient at to and equilibrium moving gradient layer plotted as a function of reduced gradient layer thickness x/dlim.

Figure 8. Matching of regions B and C. Position indicated in the moving gradient layer, region B, coordinates. (a) Positive ion concentration, eqs 22 and 28. (b) Potential vs position, lines indicate position of match points.

TABLE 3: Violation of Quasi-Neutrality 10% ion imbalance (µm) Cl- concn (µmol) relative concn c/co Figure 7. (a) Concentration gradients, eq 28, and (b) potential gradients, eq 29, in an unbalanced charge conduction layer.

When the potential for the three conduction cases is plotted in Figure 5b, the interesting result is that even though the conductivities of the three solutions considered here span an order of magnitude, the voltage gradients are identical across the gradient layer. It also should be noted that over most of the gradient the electric field is similar to the bulk electric field and only begins to diverge significantly over the last 2-3 µm of the concentration gradient. 3.5. Transition from Depletion Gradient to Moving Gradient. A diffusion dominated depletion gradient will grow at the electrode until the quasi-neutral assumption is violated. Once this happens the gradient will start to move away from the electrode and evolve to an equilibrium profile. If this moving profile were significantly different from the depletion profile, we would expect a significant transition period from depletion to equilibrium motion that would have to be accounted for. Since both the depletion gradient at to and the equilibrium moving gradient are functions of x/dlim they can be plotted together, as shown in Figure 6. Here it can be seen that the difference is relatively small and match what would be expected from the change from a transient to an equilibrium profile. More importantly, where the potential and concentration gradients are highest, the two functions are identical. 3.6. Violation of Quasi-Neutrality. There is no clear point on this gradient where the quasi-neutral assumption is suddenly violated. It simply gradually loses validity as concentration drops. To get an index of this, Table 3 gives the location where the concentration of unbalanced positive charge necessary to produce the electric field gradient of eq 25 reaches 10% of the balanced ionic concentration calculated by eq 22.

case 1

case 2

case 3

-0.65 1.3 0.029

-0.41 3.3 0.018

-0.25 8.7 0.012

3.7. Region C. Unbalanced Charge Conduction Layer. The equations for the quasi-neutral region rely on a tight link between diffusion in a concentration gradient and migration in the local electric field, where slight charge imbalances adjust the electric field to maintain quasi-neutrality. In the thin region at the edge of the gradient where the excess of unbalanced charge becomes a significant fraction of the total concentration, this linkage is broken and the current begins to be dominated by the migration of positive ions. In this region, the concentration of positive ions will drop less rapidly than predicted by eq 22 while the concentration of negative ions will continue to drop at close to the same rate. (The concept and equations for the unbalanced charge conduction layer presented here are derived from Chazalviel.9) The ionic concentration never actually becomes zero, but at some point it becomes so low and the voltage gradient so high that the assumption of quasi-neutrality will no longer be valid. Given the exponential decline in concentration and rise in voltage through the quasi-neutral gradient region, the edge of this region can be considered as a point at which conduction changes to conduction purely by cations with a zero concentration of anions. With the high voltage gradient in this region, the migration current will be so much larger that diffusion can be ignored. This allows the current flux to be written as a function only of electric field and cation concentration:

J ) -νH+cH+

∂φ ∂x

(26)

The electric field gradient will then be determined by the cation concentration:

FcH+ ∂2φ )2 o ∂x

(27)

Gradients in an Immobile Electrolyte

J. Phys. Chem. C, Vol. 111, No. 8, 2007 3347

TABLE 4: Matching Regions B and C region B position (xB) (µm) region C position (xC) (µm) voltage gradient (V/cm)

case 1

case 2

case 3

-0.24 0.12 -1080

-0.15 0.08 -1690

-0.01 0.05 -2680

Combining these equations, separating variables and integrating to determine concentration as a function of position gives

c H+ )

[ ] Jo 2FνH+

1/2

xC-1/2

(28)

The constant of integration is set to zero by taking a boundary condition of c ) ∞ at x ) 0. This expression can be substituted into eq 26 to give the potential

[ ]

2JF ∂φ )∂x νH+o

1/2

xC1/2

(29)

Substituting in the parameter values for the three cases used here, at an xC position of 1 µm, the voltage gradient is already 0.31, 0.61, and 1.21 MV/m. 3.8. Matching Regions B and C. The assumptions for region B lead to a zero ionic concentration at xB ) 0; in region C, the concentration of positive ions goes to infinity at xC ) 0. Clearly both of these cases are nonphysical, and a transition from one region to the other at reasonable values of ionic concentration is needed. To do this, the two potential equations are matched where the electric field and electric field gradient are equal, with the absolute potential set to the value for region B. This matches the first three terms of the Taylor series expansion of each potential equation. The result is a solution which is continuous in electric field but which is necessarily discontinuous in balanced ionic concentration due to the different simplifying assumption in each region. The position of the match points in the coordinates of each region and the voltage gradients at the match points are given in Table 4. The most significant point that arises from this analysis is that the higher the bulk ionic concentration and current flux, the thinner the transition layer. In practical terms, the higher the ionic concentration, the faster the transition from quasi-neutral to unbalanced charge behavior. The careful reader will note that the equations for the unbalanced charge conduction region are based on a static, not moving, gradient. However, given the minimum voltage gradient at our matching point of 1 kV/cm and a positive ion mobility of 3.15 × 10-8 m2/(V s), the lowest ion migration speed in this layer is 3 mm/s. This is 2 orders of magnitude larger than the 30 µm/s migration speed of the layer, allowing the use of the static solution. An analytic solution for the transition layer would have to solve for electric field and ionic concentration in a region where both the concentration of unpaired positive ions and the concentration of negative chloride ions are both too large to ignore. An exact solution for this transition region is beyond the scope of this paper, however, this region is relatively thin, less than 1/2 µm, with a voltage gradient on the order of 1000 V/cm. An uncertainty of 20% in the voltage drop across this transition would only yield an error of (100 mV, which will not affect the basic conclusions here. 3.9. Voltage, Heat, and Stress. The above analysis predicts three primary effects in an immobile electrolyte solution: a sharp voltage rise after the depletion time, high power dissipation in

the unbalanced charge layer, and a rapid buildup of compressive stress on the unbalanced charge layer. Voltage. During most of the time of the formation of the depletion gradient, the voltage rise will be very small. Only at the very last stages will voltage rise by 100 to 200 mV. Once the gradient layer is fully formed it will move away from the cathode at at the speed of the counter ions in the bulk solution, opening an unbalanced charge conduction region and beginning a very rapid voltage rise. In the example cases, the cell voltage would be essentially flat for the 0.6 s depletion time. Given the 30 µm/s migration speed of the chloride ions, after an additional 1/ s, a 15 µm thick ion depleted layer will open between the 2 gradient layer and the electrode. To maintain a constant current, the voltage in these three cases will have to rise by 12, 24, and 47 V. In the highest current case, the voltage would reach 1000 V within 5 s. More significantly, in this time the voltage gradient will have reached the 15 MV/m breakdown strength of even a very pure ethanol solvent.19 In a gelled electrolyte, this breakdown strength is likely to be much lower. Heating. Driving a constant current against the high voltage gradients of the unbalanced charge layer will result in high power dissipation (i.e., heating). In the highest current case considered here the heat flux will rise to 1 kW/m2 at 1.3 s after to. CompressiVe Stress. Because of the unbalanced charge there will be an electrostatic attraction between the solvent in the unbalanced charge layer and the electrode. The product of eq 28, the electrostatic charge concentration, and eq 29, the electric field, gives the body force on the solvent. Because of the inverse proportionality between electric field and ionic concentration in this layer, the spatial term xC drops out and the force on this fluid layer is constant.

JF ∂P ) ∂xC νH+

(30)

For the conditions of case 2 the attraction between the solvent and the electrode is 5000 times the gravitational force on an equivalent volume of ethanol, generating a pressure gradient of 40 Pa/µm. 4. Conclusions When one polarity of ion from an electrolyte is consumed at an electrode and no ions of opposite polarity are produced, the total ionic concentration will go down at that electrode. If a current higher than the equilibrium limit current for the cell is applied and convective motion is prevented, the ion depletion gradient at the electrode will separate and move away from the electrode at the speed of the counter ions in solution. This opens a region of unbalanced charge conduction marked by extremely high voltage gradients. Although first observed in a system involving ramified growth of a cathode electrodeposition, this moving gradient layer is a feature of any electrochemical system where the ionic concentration is depleted at an electrode. Quasi-Neutral Gradient Layers. A significant point that has been brought out here is that the thickness and formation speed of these layers can be written as functions of the equilibrium limit current thickness dlim . Although in the mathematic treatment here we have been using the ionic flux and bulk ionic concentration, these are not usually the most convenient quantities for practical electrochemical systems. Noting that the ionic flux can be written as the product of the molar conductivity, bulk molar concentration, and electric field, J ) Λ/F co ∂φ/∂x, the reduced boundary layer distance, x/dlim, can be

3348 J. Phys. Chem. C, Vol. 111, No. 8, 2007 rewritten as Λ/2DH+F ∂φ/∂x x. Given that molar conductivity only changes gradually with changes in concentration and that this change is likely to be proportional to the change in the positive ion diffusivity, the first term in this quantity will be close to constant for any given ion pair. This means that the thickness of the gradient layer is close to a linear function of the voltage gradient while being relatively independent of ionic concentration. This makes it clear why the profiles for our three example solutions are so similar. Each was run at close to the same voltage gradient with the same ion pair. In the region where the quasi-neutral assumption applies, the behavior of these three solutions will be almost identical. Ion Depleted Conduction Layer. In a rigid electrolyte medium where convection is not possible, an ion depleted conduction layer will form and grow as the ion depletion gradient moves away from the electrode. The transition from quasi-neutral to unbalanced charge behavior will occur on a submicrometer scale, and in this case on a millisecond time scale. Although the largest effect on the voltage rise due to this layer comes from the total current flux, mobility of both ions and the voltage gradient in the bulk will also affect how this layer develops over time. Although the possibility of this type of layer forming has been known for several years now, obtaining unambiguous quantitative observations of an ion depleted conduction layer is difficult simply because of the extreme nature of this layer. Some of the problems that can occur in this layer are as follows: melting of gelsfor electrochemical systems immobilized by the addition of agar or similar gel material, rapid heating at the electrode can melt a thin layer of gel, allowing electroconvection to commence; solvent breakdownsat constant current the voltage gradients in this layer can very quickly reach the dielectric breakdown strength of the medium; gel dryings for an electrolytic solution which consists of a solvent kept from moving by a polymer gel, the pressure gradient in the ion depleted conduction layer can drive out the liquid solvent, drying the polymer at the interface. This may be a significant effect especially in a system run continuously at just slightly above the equilibrium limit current. ConVection in Electrodeposition Systems. One of the most important implications of this analysis relates to fluid, rather than rigid, electrolyte solutions. As will be shown in the following article, in a fluid electrolyte these depletion layers will transition to convection as soon as the depletion gradient is fully formed, with the scale of that convection depending on the scale of the gradient layer. This means that the time to the initiation of convection at the electrode and the scale of that convection will be functions of the voltage gradient and cation mobility. This offers a new route for understanding the formation of surface texture in electrodeposition. The action of additives for brightening, leveling and grain refining in electrodeposited metal coatings has mainly been considered in terms of adsorption to the metal surface or participation in surface reactions. However, many of these additives also chelate the cation in solution, significantly changing its electrophoretic mobility, and thereby changing the formation rate and thickness of the depletion gradient. This is also significant for the interpretation of Hull cell results,20 where variation of the voltage gradient across and electrode is related to changes in surface finish. Acknowledgment. This work was funded in part by the Penn State NSF-IUCRC Centers for Dielectric Studies and Particulate Materials Center and by a grant from the Intel Corporation.

Van Tassel and Randall Supporting Information Available: Text giving the derivation of the equations for potential in the moving gradient region, eqs 23-25. This material is available free of charge via the Internet at http://pubs.acs.org. Symbols co c d dlim DCl DH DE E F I J k P q T to t x  o λOi Λ νCl νH φ

ionic concentration in bulk solution (mol/m3 or mM) ionic concentration (mol/m3 or mM) distance between parallel electrodes (m) equilibrium limit current distance (m) chloride ion diffusion coefficient (m2/s) hydrogen ion diffusion coefficient (m2/s) codiffusion coefficient of a 1-1 ion pair (m2/s) electric field (V/m) Faraday constant current density (A/m2) ionic flux (mol/s‚m2) Boltzmann constant pressure solution net specific electrostatic charge (C/m3) temperature (K) time to attain a mathematical zero ionic concentration at the electrode time from to (s) distance parallel to electric field (m) relative dielectic constant permittivity constant molar conductivity of species i (cm2/Ω‚mol) molar conductivity (cm2/Ω‚mol) chloride ion mobility (m2/V‚s) hydrogen ion mobility (m2/V‚s) electrostatic potential (V)

References and Notes (1) Newman, J. S. Electrochemical Systems, 2nd ed.; Prentice Hall: Englewood Cliffs, NJ, 1991. (2) Levich, Veniamin, G. Physicochemical Hydrodynamics, Prentice Hall: Englewood Cliffs, NJ, 1962. (3) Nernst, W. Z. Phys. Chem. 1904, 47, 52. (4) Brunner, E. Z. Phys. Chem. 1904, 47, 56. (5) Brunner, E. Z. Phys. Chem. 1907, 58, 1. (6) Felici, N. ReV. Ge´ n. EÄ lectr. 1969, 78, 717. (7) Brady, R. M.; Ball, R. C. Nature 1984, 309, 225. (8) Cork, R. H.; Pritchard, D. C.; Tam, W. Y. Phys. ReV. A 1991, 44, 6940. (9) Chazalviel, J.-N. Phys. ReV. A 1990, 42, 7355. (10) Bazant, M. Z. Phys. ReV. E 1995, 52, 1903. (11) Le´ger, C.; Elezgaray, J.; Argoul, F. Phys. ReV. E 1998, 58, 7700. (12) Chu, K. T.; Bazant, M. Z. SIAM J. Appl. Math. 2005, 65, 1485. (13) Van Tassel, J. J. Ph.D. Thesis, The Pennsylvania State University, 2004. (14) De Lisi, R.; Goffredi, M.; Liveri, V. T. J. Chem. Soc., Faraday Trans. I 1976, 72, 436. (15) Van Tassel, J.; Randall, C. A. J. Colloid Interface Sci. 2001, 241, 302. (16) Graham, J. R.; Kell, G. S.; Gordon, A. R. J. Am. Chem. Soc. 1957, 79, 2352. (17) Smyrl, W. H.; Newman, J. Trans. Faraday Soc. 1967, 63, 207. (18) Barkey, D. P.; LaPorte, P. D. J. Electrochem. Soc. 1990, 137, 1655. (19) Brie`re, G. B. Brit. J. Appl. Phys. 1964, 15, 413. (20) Dini, J. W. Electrodeposition: the Materials Science of Coatings and Substrates; Noyes Pubs.: Park Ridge NJ, 1993.