J. Phys. Chem. C 2007, 111, 3349-3357
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Ionic Gradients at an Electrode above the Equilibrium Limit Current. 2. Transition to Convection 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
When one member of a simple binary electrolyte is consumed at an electrode, an ion depleted layer will form at that electrode. Under constant current conditions this layer will have a net electrostatic charge opposite that of the electrode. This leads to an attractive electrostatic body force between the solution in the depletion layer and the electrode which increases with proximity to the electrode. There are three primary processes that can occur in this solution: electric field relaxation, convective motion, and ionic diffusion/migration. By ranking these processes according to the speed with which their effects propagate through the solution, we show that this depletion layer is highly unstable and will transition to convective motion on the micrometer scale, generally before any voltage artifact is generated. A Rayleigh number for electrically forced convection is calculated and shown to rise by 3 orders of magnitude at precisely the time convection must begin. In this type of system, electrically driven convection will begin at the depletion electrode and grow out into the bulk of the solution following a moving ion depletion gradient layer. This is demonstrated by DC conduction in an ethanol solution containing added HCl.
1. Introduction The mechanisms of ion depletion at an electrode have been known for over 100 years now.1 The concept of an equilibrium limit current has been well established for almost 50 years.2 What occurs in an immobile electrolyte solution at a current slightly above the limit current is also yielding to analysis.3 However, why this limit current behavior is so difficult to observe in a fluid electrolyte solution has still not been well explained. In the previous paper in this series,4 we have explained exactly what we would expect of an electrolyte solution in the absence of convective motion. Between approximately 1/2 and 6/10 s after the current is turned on, we would expect to see a minor voltage rise of 100-200 mV due to the formation of an ionic depletion gradient at the electrode. After approximately 6/10 s, we would expect an exponential voltage rise on the order of tens to hundreds of volts per second. However, in experiments in a medium which is free to convect we do not see even a slight rise in voltage over this time frame. The simple physics of this system dictate that convective transport of ions to the electrode must either begin within this time frame or must have already begun before it. However, the authors are not aware of any adequate explanation of the transition to convection in the case of ion depletion at an electrode. In 1959, Levich2 pointed out the difficulty of obtaining a truly immobile solution, but offers only gravitation as a means for driving “natural convection” . In 1969, Felici5 published an interesting analysis of electrohydrodynamic (EHD) convection in a dielectric fluid charged by unipolar charge injection. Unfortunately, this work has only been followed up in the fields where unipolar charge injection is expected, and it has not been extended to analyzing the effects of unbalanced electrostatic charges due to concentration gradients in balanced * Corresponding author. E-mail:
[email protected]. † E-mail:
[email protected].
electrolytes during conduction (with the notable exception of Orlik6). Bruinsma and Alexander (1989)7 offered a somewhat problematic analysis of electroconvective instability, which ignored portions of the diffusion boundary layer. In an analysis of equilibrium conduction beyond the limit current Chazalviel (1990)8 proposed an unbalanced charge conduction layer, however, later in the same article he states that such a layer does not actually exist, calling it “unphysical” . The only mechanism that he offers for avoiding this unphysical state is that the initiation of ramified (branched-treelike) growth of an electrodeposit may trigger convection. Elezgaray et al. (1998)9 proposed a mechanism by which ramified growth could initiate based on a diffusion/reaction instability at the electrode. Several authors have looked at EHD convection at the existing tips of a growing ramified electrodeposit of copper.10-12 However, this will not apply to a system where there is no ramified growth of the electrode. While the transition to convection in the case of ion depletion at an electrode has not yet been adequately explained, in the opposite case of ion enrichment at an electrode, Orlik et al.7,14 have in a series of papers clearly demonstrated the forces initiating EHD convection originating at the edge of the electrochemical boundary layer in a electroluminescent solution. One mechanism of convection that is frequently mentioned in this context is gravitational convection due to density gradients in the solution, and in many typical electrochemical systems this may well provide adequate ionic transport. In most electrochemical systems, the objective is the efficiency of the electrochemical reaction itself. This leads to the use of high concentrations of reactants and low overpotentials. In these cases, the electrochemical boundary layers can grow slowly and have a significantly different density from the bulk solution. This density gradient is frequently sufficient to initiate gravitational convection of the bulk solution, which is then presumed to provide sufficient transport to prevent growth of the diffusion boundary layers to larger than the limit current thickness.
10.1021/jp064805q CCC: $37.00 © 2007 American Chemical Society Published on Web 02/09/2007
3350 J. Phys. Chem. C, Vol. 111, No. 8, 2007 Because of this predominance of gravitational convective motion visible in the bulk solution, both Levich2 and Newman13 use “natural convection” as a synonym for gravitational convection. The experiments of Huth et al.14 and Marshall et al.15 have shown a bulk vertical convection pattern consistent with gravitational convection, which coexists with horizontal electroconvective vortices at the tips of growing branches of a ramified copper electrodeposit. These papers illustrate what is likely a very common occurrence. Gravitational forces will set the direction of convection after which both gravitation and EHD effects work to drive convection in the same direction. This can make it difficult to disentangle the relative contributions of gravitational and EHD forces. While this picture of gravity dominated convection is correct for typical electrochemical systems, there are many systems at the other end of the spectrum which feature high overpotentials. These occur in phenomena such as dielectrophoresis,16 dielectrophoretic alignment in polymers,17 electrorheological fluids,18 anodization,19 and electropolishing.20 In these cases, the overvoltage can range from tens to thousands of volts and the voltage gradients from tens to tens of thousands of volts per centimeter. In these systems, the diffusion boundary layers form too rapidly and are too thin for gravitational convection to play any role in ionic transport, but currents continue to flow far beyond the limit current that is possible in an immobile electrolyte. In one specific example, understanding the nature of this convection is vital for understanding electrophoretic deposition (EPD) at an electrode where ion depletion occurs. In EPD particles which are stably suspended in a solvent must move to a deposition electrode by electrophoresis through the solvent. At the deposition electrode, the electrochemical environment must be sufficiently different from the bulk suspension that the repulsion that keeps the particles from floccing in the bulk is overcome or eliminated, allowing the particles to deposit.21 Convection at the deposition electrode can disrupt both of these processes. Convective motion can be much faster than electrophoretic motion of the particles, bringing in excess particles in spots while washing them away in others. The mixing of convection can also eliminate the electrochemical diffusion layer at the surface, which in many cases is necessary to allow deposition of the particles. This paper will use a simple hydraulic model to illustrate how an ion depletion diffusion layer becomes extremely unstable as the ionic concentration at the electrode approaches zero under the assumption of quasi-neutrality. If this instability translates rapidly into convection, then this quite adequately explains the complete lack of a voltage jump at a time when a exponential rise would be expected in an immobile electrolyte. This article represents further development of concepts first developed in ref 22. 2. Experiment and Results The electrolytic solution is a 99.5/0.5 wt % ethanol/water solvent with HCl as added electrolyte. The voltage trace in Figure 1 is typical for all three of the solution concentrations listed in ref 4. The electrochemical cell was designed to control gravitational convection at the cathode with the horizontal cathode at the top of the cell and vertical anodes at the sides, Figure 1. The cell constant is 0.7 cm-1 with an average electrode spacing of 3.5 cm. The cathode is a 25.4 × 25.4 × 0.5 mm alumina circuit substrate with a sintered platinum coating on one side, which was polished to a mirror finish. This is placed onto a 5 mm thick PTFE masking disk with a square cut out which exposes
Van Tassel and Randall
Figure 1. Cell crossection schematic (supports eliminated for clarity); (a) horizontal, polished platinum cathode; (b) platinum foil anode. The ion depletion gradient moves downward from the cathode surface.
Figure 2. Total potential between cathode and anode for the first 4 s after a constant current is applied.
a 5.2 cm2 area of the deposition electrode. The mask disk is mounted horizontally above a cylindrical volume 1.5 cm high and 6 cm in diameter. The counter electrodes are two platinum foils which each cover one quadrant of the sides of this cylindrical volume. Figure 2 shows the total voltage across the cell when a constant current is first applied. Prior to application of the current the cell was allow sit with cathode and anode shorted for 45 min to allow adsorption to equilibrate and any convection to dissipate. The vertical voltage rise is due to the ohmic resistance of the solution. The additional curved voltage rise over the first 100 ms is due to electrode adsorption and double layer capacitance. After this initial rise there is a gradual increase of voltage across the cell of approximately 5 V over 2 min. This additional gradual voltage rise can be eliminated by simple stirring of the solution. 3. Analysis and Discussion This analysis begins where the analysis in ref 4 leaves off. Positive ions, hydronium and ethoxonium, are consumed at the cathode and chloride counterions migrate away from the cathode at 28 µm/s. Under the quasi-neutral assumption, at 0.65 s after the current is switched on, the ionic concentration should go to zero. The ionic concentration does not actually go to zero, but the solution does become unbalanced with a higher concentration of positive than negative ions in a thin transition layer. If there is no convective motion of the electrolyte solution, the gradient layer will move away from the cathode at the 28 µm/s speed of the counterions in the bulk solution. This would open a very high voltage unbalanced charge conduction layer. One tenth of a second later, 0.75 s after the current is turned on, voltage would rise by 2.5 V and continue to rise exponentially after that. 3.1. Electrode Boundary Layers. At an electrochemically conducting electrode, there are two boundary layers. The first is the diffuse portion of the electrostatic boundary layer. This
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is most commonly referred to as the “double layer” , where “double layer” is very frequently used to mean only the diffuse half of the Gouy-Chapman two-layer model.23 The primary characteristic of this layer is a balance between diffusion of ions in one direction and electrophoretic migration in the opposite direction. This migration/diffusion balance is expressed in the familiar Poisson-Boltzmann equation. This layer is normally very thin, measured in nanometers, and diffusion through this layer is very rapid. Thus, the concentration and electric field gradients in this layer are relatively unaffected by a current flux through the layer.24 Although not demonstrated here, it can be shown that this boundary layer is very stable against convective motion. Any external disturbance will be almost instantly damped. The voltage drop across this layer is set by the total voltage across the system and the ionic concentration at the outer edge of the layer. In the constant current case this will be regulated to yield the same average cation activity at the surface in order to maintain the same reaction rate. As ionic concentration drops next to the electrode this voltage will go up. This will undoubtedly be a portion of the voltage rise during the experiment, however, since this voltage rise will be uniform across the electrode surface, this layer will be considered here as part of the electrode and the solution side of this layer as an equipotential plane. The second layer at a conducting electrode is the electrochemical boundary layer, electrochemical diffusion layer or simply the diffusion layer. This layer is marked by an electrochemical environment that is different from the bulk solution due to diffusion of reactants to the electrode and/or diffusion of soluble products away from the electrode. This layer can be marked by either ionic depletion as ions from the bulk are consumed at the electrode, or by ionic enrichment as ions are generated. This layer begins at a thickness scale measured in micrometers and, if undisturbed, grows at the migration/ diffusion speed of ions in solution until the anode and cathode layers meet. Only an ion depletion electrochemical boundary layer will be considered here, and the only concern will be the stability and convection of this layer. 3.2. Force on the Solvent without Convection. We begin here with the hypothetical case where an ion depletion layer has formed and has begun to move away from the electrode opening an unbalanced charge conduction layer. Figure 2 shows the body force on the fluid next to the cathode for the conductivity and current flux conditions of case 2 from ref 4, without convection. The body force on the fluid at any point is simply the product of the net electrostatic charge density per unit of solvent times the electric field at that point.
F ) qE
(1)
The body force is a vector pointing toward the cathode; i.e., the solution is attracted to the cathode. In the gradient layer, the electrostatic charge is calculated from the gradient of the electric field:
∂ 2φ q ) 2 ∂x o
(2)
In the gradient region the electrostatic charge and electric field both go up exponentially as the total ionic concentration drops. This gives a very rapid rise in the body force over the last few micrometers of the gradient layer. In the unbalanced charge layer, continuity gives the result that charge and electric field are inversely related; therefore,
Figure 3. Electrostatic body force on ethanol next to the cathode for current and conductivity conditions of case 2. The force shown on a logarithmic scale is given in multiples of the gravitational force on ethanol. Position is in the reference frame of the moving gradient layer.4
their product, the body force, is constant, Figure 3. This is the straight portion of the curve. The sharp corner near x ) 0 is due to the change of assumptions from quasi-neutrality to unbalanced charge conduction. There is no analytic solution for this transition region, but numerical calculations show that a more realistic treatment of the transition layer will result in a slight rounding of this transition but not a change in its basic shape. With the attractive force on the solvent rising toward the cathode, it is not immediately obvious why this layer should be convectively unstable at all, much less severely unstable. If this were a gravity/density gradient, this would be the definition of stability. 3.3. Stimulus and Response Speeds in the Solution. The fact that this layer is severely unstable to convection comes from the orders of magnitude of difference in the transmission speed of changes in the solvent due to (1) the electric field, (2) convection, and (3) diffusion/migration, and in the response times of the corresponding changes of the solvent (1) electrostatic charge/ionic flux, (2) fluid momentum, and (3) ionic concentration differences. Changes in electric field are effectively transmitted instantaneously through the solvent. This results in a change in the direction and speed of ionic migration, and therefore flux in the solvent. For our purposes here, this change is also effectively instantaneous. Wherever there is an ionic concentration gradient there will also be a change in electrostatic charge. This only requires a small relative change in the position of positive and negative ions. This ionic relaxation process feeds back to modify the electric field in the solvent. This whole adaptation to a changed electric field occurs on the scale of milliseconds. Convection can move parcels of solvent with different ionic concentration or chemical composition through the solvent at speeds from millimeters to centimeters per second. The time necessary for convection to begin will depend on the accelerating force per unit volume in the fluid. Diffusion and migration are the slowest processes in the solution. The maximum diffusion speed in this problem is ≈3 mm/s in the highest concentration gradient regions. The maximum migration speed occurs for the positive ion in the charge depleted zone, reaching ≈5 cm/s. However, much more typical is the range of speeds in the charge balanced region of 30 µm/s up to 4 mm/s. This means that inhomogeneities in ionic concentration will be very slow to dissipate when compared to electric field changes and convective motion. 3.4. Convective instability. In the following discussion, we will begin with an extreme case and apply a large perturbation. The conclusions are exactly the same as would be generated by stability analysis using an infinitesimal perturbation, but our purpose here is primarily clarity of presentation.
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Figure 4. Two-dimensional circular vortex at the electrode (a) is modeled by two tubes hydraulically connected at the electrode surface (b).
Simple Model. Convection at the electrode will take the form of a vortex, with fluid at one side moving toward the electrode and at the other moving away. Since the primary force here is perpendicular to the electrode, the relevant question is what is the relative force between two columns of fluid with one moving toward and the other moving away from the electrode. This permits the creation of a very simple conceptual and analytical model of this flow; the two sides of the vortex are replaced by two tubes, as shown in Figure 4. One end of the tubes is open to the bulk solution while the other end terminates at the cathode. At the cathode end of the tubes there is a bridge which allows fluid motion from one tube to the other. This allows the two sides of the vortex to be treated as two columns of solution which can be analyzed in one dimension. To illustrate the forces driving convection, the case shown in Figure 5 is taken as a baseline case. In each of the tubes a gradient layer is assumed to have formed uniformly and moved away from the electrode by 21/2 µm. A finite perturbation is then added. A 1/2 µm layer of fluid from the depleted layer is moved instantaneously from tube B to tube A. Two cases at opposite extremes are then considered, constant current and constant voltage through each of the tubes. Constant Current. The constant current case is the simplest and most intuitive. In this case the electrostatic body force on the solution remains the same, but in tube A the depleted layer is thicker. The integral of the body force from the bulk to the electrode gives the hydrostatic pressure at the electrode. With a thicker depleted layer, an identical body force is integrated over a longer distance, and the pressure at the electrode in tube A is higher than at the electrode in tube B. This would force the displaced fluid back into tube B until the depleted layers are the same thickness again. Thus, in the case of a uniform current flux across the electrode surface, this layer would be stabilized against convection. However, while simple and intuitive, this case is also unusual. For there to be a constant current density across the electrode area, the reaction rate must be strictly limited by the electrode, regardless of solution side conditions. A much more normal case will be where the electrode is at a uniform potential and current density varies across its surface. Constant Voltage. When conducting an experiment at constant current, the power supply will adjust so that the same total current is conducted through the solution, but there is no requirement that the current flux be uniform across the electrode surface. Given a uniform potential at the electrode and the much higher conductivity of the bulk solution relative to the gradient and depleted layers, these layers are best described as separating two equipotential planes. Returning to the two tube example, when a portion of the depleted layer moves from tube B to tube A, the effective resistance of tube A increases dramatically while that of tube
Figure 5. Flow under constant current. If a constant current is maintained in each of the two tubes leading to the electrode, pressure forces will resist any flow between the tubes. In this case convection will be strongly damped.
B drops equally dramatically. Using the hierarchy of effects laid out above, the first change will be in electric field, current flux, and electrostatic charge. The change in electrostatic charge and electric field will translate into an immediate change in body
Transition to Convection force and therefore hydrostatic pressure. The concentration profile in the gradient layer will initially be assumed to be constant, and the forces that would drive convection will be assessed. In the base case when the gradient layer has moved away from the electrode by 2.5 µm, the total voltage drop over the 100 µm next to the electrode is 2.09 V. This is taken as the constant voltage for the following analysis. When 1/2 µm of fluid is displaced from tube B to tube A, the current in tube A will drop by 33% while the current in tube B will increase by 55%. Although the total voltage drops are equal through each of the tubes, the electric fields are higher in tube B. The electric field gradients are also higher in tube B, which indicates higher net electrostatic charge. This higher electrostatic charge acted on by higher electric fields leads to much higher body forces on the solvent in tube B. This is shown in the graphs in Figure 6. The pressure at the electrode in tube B rises to 140 Pa from 110 Pa while the pressure drops to 80 Pa in tube A. In this case an initial disturbance will lead to a pressure difference that will drive the system further in the direction of the disturbance. Once this instantaneous electrostatic/electrodynamic analysis is done it is now necessary to look at the slower responding phenomena, convection and diffusion. Convection is driven by pressure differences. In this case, the instability of the charge depleted layer drives convection while convection increases the instability of the charge depleted layer. On the other hand, the increased conduction in tube B will cause the gradient layer to become steeper and recede faster from the electrode, with the opposite occurring in tube A. Thus, the change in speed of the gradient layers acts to dampen the instability. In the two tube example of Figure 6, the gradient layer in tube B would accelerate to 45 µm/s away from the cathode while slowing to 20 µm/s in tube A. This gives a relative speed of 25 µm/s. To compare this to convection it is necessary to estimate possible convection speeds. This can be done by combining the pressure differential with an estimate of the mass to be accelerated. To get an order of magnitude estimate of the mass, a conservative estimate of an initial vortex size of 30 µm is taken. This would obviously include both the depleted layer as well as the entire gradient layer. This is then roughly approximated by making the two tubes in the above example 30 µm long. The pressure difference between the two tubes is 60 N/m2 with a mass to be accelerated of 0.048 kg/m2. This gives an acceleration of 1250 m/s2. At this acceleration the convection speed would exceed the relative speeds of the gradient layers in 20 ns. If this pressure were stable over a distance of 0.5 µm the solvent would accelerate to 3.5 cm/s, more than 1000 times the speed of the gradient layers. As stated in the beginning, the example used here is somewhat exaggerated. Not only are the forces quoted here far more than necessary to drive convection, the voltages hypothesized here clearly do not show on the voltage trace, Figure 2. The example here illustrates the inevitability of convection, but the real solution must transition much earlier. 3.5. Initiation of Convection. Analogy to Thermal Convection. If we accept now that the ion depleted layers at an electrode are unstable, the question is then at what point in the development of these layers is the instability large enough that convection begins? To get an estimate of this we must by necessity turn to the most similar system which has already been measured. This is the case of Rayleigh-Benard instabilitysthe transition to thermal convection in a fluid between two flat
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Figure 6. Disturbance under constant voltage. At a constant voltage, a small disturbance will lead to pressure forces which drive the system farther from equilibrium. This system is convectively unstable.
plates, where the bottom plate is at a higher temperature than the top plate. When the temperature difference is small, viscosity of the fluid will damp any motion that might begin while thermal conduction will dissipate any inhomogeneities that could drive convection. At higher temperatures, instabilities are too large
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to diffuse away and the power available to drive convection is greater than the power dissipated by viscous resistance. The thermal convection problem is somewhat unique in that it can be linearized and solved analytically. A solution was first proposed by Rayleigh.25 Chandrasekhar26 solved the system of equations for flow that was bounded by two rigid plates and for flow bounded by a rigid plate on one side with a free surface on the other. The layer was calculated to transition to convective motion at a Rayleigh number of 1100 for one free surface and 1708 for the case of two rigid surfaces, where the Rayleigh number is calculated according to eq 3.
RaRB )
F1gβ(T1 - T2)d3 µR
(3)
These calculations have been shown to correspond very well to experimental results where the transition from quiescence to convection reliably occurs at a Rayleigh number of 1700 ( 50. Unfortunately, for the EHD case the forces cannot be linearized even on the first order, making this a very challenging problem for pure analytic methods. In comparing these two problems, it is important to remember here that the numerical value of unitless numbers such as the Rayleigh or Reynolds numbers have no physical meaning. They are strictly indexes made up of the important physical properties of a system and which correlate well to behavior in that system. When this index is applied to another system, not only is there no reason to expect specific values in one system to correspond to behaviors in a new system but also there is a reasonable chance that this index may not correlate to behavior at all. It is possible, however, to take encouragement from the analysis of Felici5 whose theoretical analysis of electroconvective instability in the case of unipolar charge injection showed that the critical value of the Rayleigh number for that case could be within an order of magnitude of the thermal case. To create an equivalent Rayleigh number for ion depleted convection it is necessary to begin with as explicit a definition of the Rayleigh number for the thermal case as possible. Put in words, this is as follows: “Take the difference between the weight of two cubes of a fluid of dimension d, one at temperature T1 and the second at T2 and divide this force by the product of the viscosity and thermal diffusivity of the fluid.” The force in this formulation can be rewritten as a pressure acting on an area d2, allowing the index to be re-formulated as
Pgd2 RaRB ) µR
(4)
where Pg is now the pressure difference at the lower surface due to gravitational force between two columns of fluid of height d with one at T1 and another at T2. We can now directly substitute the equivalent properties available to characterize our system to generate a Rayleigh number for ion depleted convection:
PEdlim2 RaIDC ) µDE
(5)
In this case PE is the pressure difference at the electrode between a column of fluid which features an ion depletion gradient and one which is filled with ion depleted solvent (Figure 7), and where the diffusion coefficient for the electrolyte has replaced the thermal diffusivity. There are several points where this new number can be significantly different from the thermal convec-
tive case: force, in the thermal case the force is uniformly distributed over the fluid unit while in the depletion case the body force occurs as almost a step function at the edge of the depletion layer; length, in the thermal case the separation between the top and bottom plates is defined exactly and sets the scale of the initial convective vortices while in the depletion case the concentration gradient is an exponential function with a clear boundary at the edge of the electrode electrostatic boundary layer, but no clear outer boundary; diffusion, in the case of thermal diffusion there is a linear relationship between the body force gradient and thermal gradient while in the ion depletion case ionic diffusion has a very nonlinear effect on the force gradient, declining almost as a step function away from the electrode. Keeping in mind that any of these could affect the value by 1 or 2 orders of magnitude, we will proceed with the calculation using the following values: viscosity at 25 °C, µ ) 1.07 × 10-3 N‚s/m2; codiffusion coefficient for HCl DE) 6.61 × 10-10 m2/s; length scale, the equilibrium limit current distance dlim ) 23 × 10-6 m. If we then guess that the Rayleigh number for transition to convection calculated in this manner is between 17 and 17000, we come up with an electrostatically induced pressure at the electrode of between 0.2 and 20 Pa. This corresponds to an excess voltage drop over the gradient layer ranging from 30 to 200 mV, a voltage gradient range from 90 to 3000 V/cm, and a time ranging from 1/10 s before to to almost exactly to. For this specific case the RaIDC will exceed 100 000 at 1/10 s after to with an excess voltage of 2.5 V. While not a proof, this analogy to the thermal convection system shows that it is not unreasonable to expect this type of convection to begin at, or very close to, the depletion time for ions at an electrode. Moreover, if this type of convection does begin at the low end of this range of RaIDC numbers, it completely explains the absence of a voltage spike at to shown in Figure 1. It is also reasonable to expect this transition to come sooner than later. In the thermal convection problem surface roughness on the heated plate is negligible on the scale of the convective flow. In our case even though the electrode is polished, there are still pores and scratches that are of significant size on the 20 µm length scale of this system. Convection that initiates early on these irregularities would spread rapidly across the surface. A very good demonstration of the role of surface roughness in initiating electroconvection can be found in ref 27. 3.6. Stable Convection. Two Tube Model. Returning to the two tube example, we will assume convection has started and the fluid in tube B is moving toward the electrode and fluid in tube A moving away. This motion will advance very quickly until the gradient layer comes up against the electrode in tube B. After that solvent flowing into tube B will exactly match the motion of the gradient layer relative to the solvent so that the gradient layer is stationary relative to the electrode. Virtually all of the current will be carried through this tube, with tube A filing up with ion depleted solvent, Figure 7. The gradient layer will stabilize at a distance just far enough from the cathode that the electrostatic force on the solvent creates enough pressure to drive the convection. If the two tubes in our example are the same size, the conductivity of tube A will be almost zero while the conductivity of the solution in tube B will be essentially the same as the bulk. If we impose the condition that the whole system conduct the same current then the voltage must double to drive the same current through half the area. This doubling of voltage means a doubling of the voltage gradient and a doubling of the migration speed of the gradient layer. To keep the gradient layer
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Figure 7. Stable convection in two tube case. Almost all current will be carried by solvent in tube B while tube A fills with ion depleted solvent. The fluid will flow at a speed just sufficient to keep the gradient layer fixed relative to the electrode.
fixed relative to the electrode the convective flow will have to move at two times the migration speed of the anions in the bulk solution, in this case 55 µm/s. Stated more generally, the inflow velocity will be the ratio of the outflow crossectional area to the inflow crossectional area times the migration speed of the counterions in the bulk solution. In the case of equilibrium growth of a ramified electrodeposition, there can be a stable gradient layer which moves ahead of the convecting layer.9 In this case the ionic migration in the inflowing portion of the vortex will increase in inverse proportion to the decreased ionic concentration. Thus, if our counterion moving away from the electrode is Cl- with an net migration velocity in the bulk of VCl bulk, the vortex inflow velocity can be estimated by eq 6. This model generates flow velocities which are consistent with the vortex flow velocities measured by Huth et al.14
vin )
Ain co ClV Aout cin bulk
(6)
An important point here is that the convection velocity will be set by the speed of the counterions moving away from the electrode. The gradient layer will move just enough relative to the electrode that the power available is sufficient to drive convection at exactly this speed. The essentially unlimited power that is available to drive convection that comes from the electrostatic charge in the transition and ion depleted layers explains why convection can occur even with the addition of a gelling agent to the solution.28 Stable ConVection in Real Systems. The two tube example above is a case where geometric constraints on the system allow a stable convection pattern to develop. Having a fixed geometric constraint on the system on the same scale as the vortices is a
condition for the formation of a stable vortex or system of vortices. These geometric constraints can be broken into electrode distance and pinning. If the electrodes are close enough together that a single vortex between the electrodes can provide sufficient transport to suppress the formation of additional vortices, a stable system of vortices can form. This requires either the current be very low, leading to very slow vortex motion, or the electrodes be very close together. Examples of stable vortices between closely spaced electrodes are given in refs 8 and 9. Vortices can also be pinned by asperities on the surface. Any part of the conductive electrode that sticks out above the surface will concentrate the electric field at its tip.9 Likewise a nonuniformity in the conductivity of the electrode will focus the electric field on areas with the highest conductivity gradient. 3.7. Free Convection. In many electrochemical cells for electrodeposition or EPD, the electrode is a uniform conductive surface, there is a large distance between the anode and cathode, and a current is used which is several orders of magnitude higher than the limit current for the cell. In this case, we expect torroidal vortices will form at the surface, with the diameter and spacing on the scale of the ion depletion gradient layer, Figure 8a. These vortices will increase ionic transport next to the electrode, but beyond these vortices, counterions in the bulk solution will still be migrating away from the electrode and ion depletion will continue. Therefore, the outer edge of these vortices will grow out into the bulk solution at the speed of the equilibrium moving gradient layer, i.e., the speed of the counterions in the bulk, Figure 8b. The resulting oval crossection of the vortices will not be stable and these vortices will break apart, merge and grow, Figure 8c. The larger vortices will then provide more ionic transport to the surface. However, at some point the vortices will become large enough that they do not provide sufficient transport in the layer immediately adjacent to the electrode. Once again a steep ionic concentration gradient layer will form generating a new set of small vortices, Figure 8d. In some cases, a stable system can form where a large vortex transports ions in from the bulk and a set of small tubular vortices aligned with the larger vortex flow provide enhanced transport at the electrode surface. This is particularly common where the larger vortex is pinned by some inhomogeneity in the system. In other cases, particularly where the cell and electrode are uniform on the scale of the gradient layers, there will be a continuous, chaotic, “boiling” layer of vortices being generated, breaking, merging, growing, and regenerating.
Figure 8. Unstable convective cells. Without physical constraints “boiling” convection can occur where convection cells form, grow, break, merge, and reform at the surface, as the moving gradient layer (MGL) moves farther from the electrode.
3356 J. Phys. Chem. C, Vol. 111, No. 8, 2007 3.8. Moving Electrode. A special case of stable intermediate scale convection occurs during the stable ramified (branched) growth of metal electrodeposits.29 Prior to ion depletion, the metal is deposited evenly across the electrode. After the transition to convective flow, deposition only occurs where there is an inflow of solvent bringing ions from the bulk solution. This causes the growth of small protrusions from the surface. However, the small initial vortices do not provide enough material transport for these small branches to grow at the speed that the gradient layer moves away from the electrode. Vortices will combine and grow until a single vortex can transport enough ions to a single growing branch that that branch can grow at the speed that the gradient layer recedes. At this point the vortices and the number of growing branches will stabilize, and growth proceeds prismatically across the cell. Convection in the frame of the moving gradient layer will be stable, with the growing tips of the deposit effectively creating a moving electrode surface. Each branch will grow in a fractal like pattern due to depletion instability at the tip of the electrodeposited branch and the inward flow of the main vortex, but this oscillation will occur around a central point as vortices stabilize each other. The relationship between stable vortex size and the thickness of the moving gradient layer can be seen in the branch density vs voltage gradient in a ramified copper electrodeposit.29
Van Tassel and Randall layer in a rigid electrolyte, will be filled by an electroconvective layer in a mobile solution. Symbols co c d dlim DE E F g P q RaRB RaIDC T R β o Fl µ φ
ionic concentration in bulk solution (mol/m3 or mM) ionic concentration (mol/m3 or mM) distance between parallel surfaces (m) equilibrium limit current distance (m) codiffusion coefficient of electrolyte (m2/s) electric field (V/m) force (N) gravitational acceleration (m/s 2) hydrostatic pressure (N/m2) solution net specific electrostatic charge (C/m3) Rayleigh number for Rayleigh-Benard convection Rayleigh number for ion depletion convection temperature (K) thermal diffusivity of fluid (m2/s) thermal expansion coefficient of fluid (K-1) relative dielectic constant permittivity constant (N/V2) fluid density solvent viscosity (cP or mPa‚s) electrostatic potential (V)
4. Conclusions The fact that an ion depletion gradient forms at an electrode where one polarity of ion is consumed has been well documented, as has the existence of an equilibrium moving gradient layer in the case of a moving electrode.10 What has not been well explained is why an unbalanced charge layer does not form when the ionic concentration at the electrode approaches zero. With an attractive electrostatic force between the solvent and electrode this layer would appear to be inherently stable. However, by considering the relative propagation speeds of electric field, convective transport, and ionic migration/diffusion, it becomes obvious why this layer is severely unstable and can transition to convection without a trace of voltage rise due to ionic depletion. Once an unbalanced charge layer begins to form at an electrode, the increased pressure at the electrode can accelerate a flow of ion depleted solvent away from the electrode to a speed orders of magnitude higher than diffusive motion can suppress the instability. Once an instability begins, the attraction on the solvent moving toward the electrode goes up while the attraction on the fluid moving away goes down. This pressure difference moves ion depleted solvent laterally along the electrode from the inflowing to the outflowing fluid column. The energy available to initiate convection can be indexed by a Rayleigh number for ion depletion convection, RaIDC. In the example case given here, from 0.1 s before the ion depletion time to 0.1 s after, the RaIDC rises by 4 orders of magnitude. Thus, at exactly the time convection is needed to suppress the formation of an ion depleted conduction layer, more than enough energy becomes available to drive a rapid transition to convection at the edge of the diffusion boundary layer. This convection will not change the physics of the moving gradient layer. There is no driving force for electroconvection in the bulk solution and the gradient layer will continue to move away from the electrode at the speed of the counterions. However, the space between the moving gradient layer and the electrode that would be filled by an ion depleted conduction
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