Ionic Gradients at an Electrode above the Equilibrium Limit Current. 3

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J. Phys. Chem. C 2007, 111, 3358-3365

Ionic Gradients at an Electrode above the Equilibrium Limit Current. 3. Stabilization of Ion Depleted Conduction by a Nanoporous Alumina Layer during Electrophoretic Deposition Jonathan J. Van Tassel* and Clive A. Randall† Materials Research Lab, The PennsylVania State UniVersity, UniVersity Park, PennsylVania 16802 ReceiVed: July 27, 2006; In Final Form: NoVember 20, 2006

The constant current electrophoretic deposition (EPD) of alumina powder from an acidic suspension is accompanied by an anomalous voltage rise across the deposited particulate layer. This voltage rise is much greater than can be accounted for by the simple blocking of the electrode by nonconducting particles. It is shown here that this voltage rise can be accounted for by the formation of a high resistance ion depleted conduction layer in the solvent at the cathode. This layer, which is highly unstable and therefore not seen in the pure solvent, is stabilized by the adsorption/desorption equilibria of ions on the alumina powder surface. This also explains instabilities seen in the EPD coating process. The stabilization of this type of layer is of interest beyond EPD for its ability to produce large pressure gradients in a fluid medium on the micrometer scale and a stable region of unbalanced charge and extreme voltage gradients in a nanoporous alumina layer.

1. Introduction It has been noted that during the electrophoretic deposition (EPD) of alumina from an acidic suspension there can be a large linear voltage rise across the deposited layer.1,2 This rise in resistance is much larger than can be accounted for by the blocking effect of the nonconductive particles alone. Even at maximum random particle packing density the deposited layer will retain ≈ 40 vol % open porosity filled by the electrolyte solution. If this electrolyte solution retained the same conductivity as the bulk solution, the resistance of this layer would be expected to rise by less than an order of magnitude. Yet, in practice, the resistance across this layer can rise by 3 orders of magnitude or more. Analysis of the thickness of the densely deposited layers and the voltage rise across them shows that this high resistance can be explained by the stabilization of an ion depleted conduction (IDC) layer within the particle deposition. The consumption of positive ions at the cathode and migration of negative ions away from the cathode can result in an ion depleted fluid layer next to the electrode, which only carries current by migration of positive ions without a balancing charge of negative ions. In a fluid medium this situation is extremely unstable and will transition to microscale convection before any significant quantity of unbalanced charge develops. To discover a fluid system where this type of layer is stabilized in spite of total potential drops of tens to hundreds of volts is quite remarkable and unexpected. However, to explain the nature of this conduction layer and how it is stabilized, it was first necessary to adequately explain why the normal case is so unstable. Thus, this series of articles begins with a review of ion depletion at an electrode in an immobile electrolyte solution, and its expected behavior.3 The second article then discusses why these layers are unstable to convection for a fluid medium.4 This article will discuss the role of alumina surfaces in suppressing or stabilizing the exceptionally high voltage and pressure gradients found in this * Corresponding author. E-mail: [email protected]. † E-mail: [email protected].

unique conduction layer. It should be noted that although in this experiment the IDC layer was stabilized by a particulate alumina, the surface chemistry and pore size of a nanoporous anodized alumina layer will be very similar, and it is likely that this type of environment can be created in this type of ordered nanoporous layer as well. The primary focus of this article is the stabilization of an ion depleted conduction layer. A more complete analysis of the EPD results of this experiment can be found in ref 6. 2. Experimental Section 2.1. Suspension Preparation. The Al2O3 powder is AKP50 from Sumitomo Chemical Co., Osaka, having an average particle diameter of 270 nm and a surface area of 10.0 m2/g. The powder was washed and hydrated as detailed in ref 5. Prior to mixing it was placed in a 135 °C drying oven for at least 1 h to remove excess condensed moisture. First, 7.99 g of this powder was added to 155.4 g of 99.5/0.5 wt % ethanol/water in a HDPE bottle. This yields a 1.01 vol % suspension of alumina particles. This was dispersed using alumina milling media on a vibratory mill. Conductivity of the suspension was increased by the addition of HCl from a 1 wt % solution in ethanol between the deposition trials. 2.2. Deposition Device. The deposition trials were performed using a deposition device designed to be immersed in suspension in a 250 mL, 6.2 cm inside diameter Pyrex beaker. This is diagrammed in cross section in Figure 1. The deposition electrode is a 25.4 × 25.4 × 0.5 mm alumina circuit substrate with a sintered and polished platinum coating on one side. The deposition substrate is clipped to a PTFE holder block by two spring loaded stainless steel hooks which also serve to provide electrical connection to the platinum surface. The holder block is placed onto a 5 mm thick PTFE masking disk with a square cut out which exposes a 5.2 cm2 area of the deposition electrode. The mask disk is mounted horizontally on three support posts 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.

10.1021/jp064807a CCC: $37.00 © 2007 American Chemical Society Published on Web 02/09/2007

Stabilization of an Ion Depleted Conduction Layer

Figure 1. Deposition cell crossection schematic (supports eliminated for clarity): (a) deposition direction; (b) particle depleted solvent at anode surface rising away from deposition zone.

During a deposition trial particles will move away from the counter electrodes. This creates an area of lower density fluid at the surface of the electrodes. The cell is designed so that this lower density, particle depleted fluid can rise to the surface of the suspension well above the electrophoresis zone. Undepleted fluid from below the surface of the suspension can then flow back down into the electrophoresis zone in the two quadrants not covered by the counter electrodes, replacing the depleted fluid. This flow pattern prevents gravitational convection or particle depleted solvent from the anode from affecting the deposition behavior at the cathode. In the center of the cylindrical volume the electrophoretic motion of the particles will become vertical, moving into the square mask cut-out toward the deposition electrode. Constant voltage/current is provided by a Keithly 2410 power supply which also provided voltage/current measurements. 2.3. Deposition Procedure. Deposition was carried out at constant current for a period of 2 min. The current was set according to conductivity of the suspension to yield the same starting voltage and, therefore, the same bulk electric potential gradient. After the deposition trial the deposition substrate was removed from the deposition device, exposing it to the ambient air for 10-15 s. The substrate was then placed into clear, as-received ethanol to rinse off an overlayer of deposited, but low density, alumina powder. This overlayer rinses off easily even with little or no agitation leaving a uniform, clearly defined layer which cannot be removed, even in part, except by sonication or mechanical wiping. The remaining deposited layers did not show evidence of any convection patterns, dried rapidly without cracking, and remained strongly adherent to the substrate. The resulting deposition was weighed. 2.4. Data. The deposition trials analyzed here fall into three sets. A single deposition trial from each of these three groups was chosen for detailed analysis. Deposition Set 1. Deposition trials in this set, conducted with progressively higher additions of HCl, yielded no significant deposition. The first five trials in this set covered a conductivity range from 0.6 to 1.2 µS/cm and showed voltage rises from 0.2 to 0.7 V. The final two deposition trials were conducted at a conductivity of 2.3 µS/cm and showed voltage rises of 1.0 and 1.1 V. A very small deposition of particles was observed as a slight fogging of the reflective platinum surface. The deposition weight of 0.6 g/m2 corresponds to a uniform monolayer of particles. That is, particles have deposited to the electrode, but there is no subsequent particle-particle deposition. The final deposition trial of this set from was chosen for detailed

J. Phys. Chem. C, Vol. 111, No. 8, 2007 3359

Figure 2. Relationship between conductivity and deposition weight. The three deposition trials that are selected for detailed analysis as cases 1,2, and 3 are indicated by arrows.

Figure 3. Voltage rise during deposition trials for cases 1, 2, and 3.

TABLE 1: Deposition Data conductivity (µS/cm) current density (I) (A/m2) initial voltage gradient at deposition electrode (V/cm) total current (C/m2) voltage rise (V) deposition weight (g/m2) deposit specific current (C/g) deposition thickness at 60 vol % density (µm)

case 1

case 2

case 3

2.31 0.32 13.9

9.58 1.25 13.0

35.4 5.0 14.1

38.3 1.0 0.6 66 0.24

150 3.3 8.1 18 3.4

600 18.9 26.1 23 11.0

analysis. Data for this trial are given in Table 1 and the voltage rise is shown in Figure 3. Deposition Set 2. This set of trials was marked by significant voltage rises during deposition, an overlayer of particles which rinsed off the substrate, and a uniform, dense deposition layer which did not rinse off. Conductivities range from 4.4 to 23 µS/cm. Voltage rises increased with conductivity from 2.0 to 14.0 V. Deposition weights also increased with conductivity from 5.8 to 16.9 g/m2 (Figure 2). Deposition Set 3. This set is comprised of two depositions conducted at a conductivity of 35.4 µS/cm. These depositions were distinguished from the depositions of set 2 by a nonlinear voltage rise during deposition. One deposition was chosen for detailed analysis and is referred to as case 3. Preliminary Data. Although this paper focuses on a particular deposition series, many experiments on this type of deposition of alumina have been conducted. Two primary results of these previous experiments are used here. The first regards deposition density. These types of depositions, from several micrometers to several tens of micrometers thick, marked by a rise of several volts during deposition and which cannot be removed even by

3360 J. Phys. Chem. C, Vol. 111, No. 8, 2007

Van Tassel and Randall

TABLE 2: Ionic Properties of Solution case 1 conductivity (µS/cm) molar conductivity (Λ) (cm2/Ω‚Mol) bulk HCl concentration (Mol/m3) hydrogen ion mobility (νH+) (m2/V‚s)

case 2

case 3

2.31 51.3

9.58 49.9

35.4 47.1

0.045

0.192

0.751

3.13 × 10-8

3.04 × 10-8

2.87 × 10-8

TABLE 3: Surface and Colloid Properties case 1 electrophoretic mobility (µm‚cm/V‚s) surface potential (mV) ave. particle electrostatic charge (C) surface adsorbed HCl (µmol/g)

0.97

case 2 0.85

case 3 0.47

79 51 26 2.52 × 10-16 2.64 × 10-16 2.40 × 10-16 21.4

27.5

31.6

vigorous agitation during rinsing, have a density of approximately 60 vol %. A slightly higher or lower actual density will not significantly affect the conclusions reached here. The second fact used here is the location of the excess voltage that arises during deposition. Experiments using a platinum wire as a voltage probe have shown that the voltage rise during deposition is almost completely accounted for by a potential difference across the deposited layer. For example, a constant current deposition with an initial potential of 20 V shows a rise in voltage to 40 V over the course of the deposition. The final voltage breakdown will be approximately as follows: 20 V due to resistance of the solution; 1 V to drive convection in solution due to concentration gradients; 19 V across the deposited layer. The wire probe can be pressed against the deposited layer, and as long as the deposited particles are not scraped away, a 19 V potential difference will be measured between the deposition electrode and the solvent side of the deposition. 2.5. Suspension Description. The properties of the suspending solvent and electrolyte for the three deposition trials chosen for detailed analysis are given in Table 2. The properties of the particles in suspension are given in Table 3. Electrophoretic mobility values are calculated from the data presented in ref 6. The net electrostatic charge is calculated using the estimation formula of Loeb et al.7 3. Discussion and Analysis 3.1. Voltage Rise without Particles. During constant current conduction in an ethanol/water solvent with an HCl electrolyte, the ionic content of the solvent will be rapidly depleted at the cathode.3 In the absence of particles, this ion depleted layer at the cathode transitions to convection when the ionic concentration reaches a theoretical zero point at the cathode surface.4 This convective layer grows outward from the cathode at the migration speed of the counter ions in solution. This layer has a lower average ionic concentration than the starting bulk solution and therefore a lower conductivity. To drive a constant current across this growing layer the total voltage across the cell must rise. This additional voltage drop also provides the power to drive convection. A key attribute of this voltage rise is that it an be eliminated by stirring the solution and will grow again when the stirring is stopped. Figure 4 shows the voltage rise across the deposition cell for a HCl solution without particles with a conductivity of 10 µS/ cm given a starting voltage of 20 V. In this case the chloride

Figure 4. Voltage rise during constant current conduction across the deposition cell in the absence of particles.

ions move away from the cathode at a speed of 30 µm/s and the convective layer will grow to an average thickness of 3.6 mm with a 5 V rise over the 120 s of the experiment. The curvature of the voltage rise line and the ultimate value of 24.5 V are functions of the cell geometry. 3.2. Ionic Content in the Bulk. When alumina particles are introduced into the system the particle surfaces will adsorb a large quantity of HCl from solution. The powder then acts as a buffer against ionic concentration changes in solution, releasing ions as the solution concentration goes down and adsorbing them as concentration goes up in a rapid, reversible equilibrium between the two states. Dissolved ion concentration can be determined by conductumetry, with the adsorbed ion concentration determined from the adsorption isotherm in ref 6. For the three cases considered here, the concentration of adsorbed HCl ions are 0.865, 1.11, and 1.28 mol/m3. This gives a ratio of adsorbed HCl on the particle surfaces to HCl in solution of 20×, 6×, and 2×. 3.3. Conduction and Ionic Migration in the Bulk. Ions adsorbed to the particle surfaces have an equally dramatic impact on ionic migration in the bulk solution. While negative chloride ions in solution migrate away from the cathode, chloride ions on the surface of the positively charged alumina particles are carried by electrophoretic migration toward the cathode. The migration of Cl- ions in solution toward the anode is 1.36, 5.32, and 21.3 µmol/s‚m2. The flux of Cl- ions on the particle surfaces toward the cathode is 11.7, 12.3, and 8.47 µmol/s‚m2. Because almost all of the ions adsorbed on the particles are present as balanced charge, the contribution of particle electrophoresis to conduction is negligible. The increase in conductivity due to the net electrostatic charge on the moving particles is only 1.9, 0.4, and 0.06% of the total current for the three suspensions respectively. 3.4. Accumulated Particle Layer at the Cathode. The particles’ contribution to ionic transport comes to an end when electrophoretic motion is stopped by the electrode surface. At this point the role of the particles is to buffer ionic concentration changes in solution. The extent of this buffering depends on the density of the accumulated particle layer at the electrode. Here this layer will only be considered as an accumulation of particles. The formation of an adherent deposition and colloidal stability calculations can be found in ref 6. The particles in case 1 have a very high electrostatic stability with a Debye length of 26 nm (κa ) 5). On the basis of this and the applied electric field, the density of the accumulated particle layer at the electrode is estimated at 25 vol %. The particles in case 2 have a lower stability and a debye length of 12.5 nm (κa ) 11). On the basis of drying observations of similar depositions, this low density accumulation is estimated at 35 vol % density. In case 3, the Debye length is again shorter, 6.3 nm (κa ) 21), with an energy barrier to floccing just high enough to give moderate stabilization in the bulk suspension.


Figure 5. Adsorption isotherm for HCl on alumina.5 Circle shows the total adsorption for case 1. Gray line shows the linear adsorption assumption.

In this case the particles are estimated to be compacted to approximately 50 vol % density. There are two possible states for these layers, both of which would explain the observed viscosity, density and rinsing behaviors. The first is that the particles form a very low-density deposition with a very low average number of particle contacts caused by EHD effects between the particles. This structure is then easily broken by small shear forces and the layer rinses easily. A second possibility is the formation of a dispersed but ordered colloidal state with a finite shear strength. This seems especially likely for cases 1 and 2. An overview of disorderorder transition for charged particles and the effect on viscosity is given in ref 8 sections 10.6 and 14.5. The density estimates above would be reasonable for either case. A systematic study of particle density prior to drying has not yet been done. As a result the above estimates are approximate, however, a very wide range of variation in these densities, i.e., (15%, would not affect the conclusions of this paper. 3.5. Conduction in the Accumulated Particle Layer. As particles accumulate at the electrode at a higher density than in the bulk suspension, volumetric conductivity will be reduced by the blocking effect of the nonconductive particles. However, conduction will be enhanced by electroconvective motion in the pores between particles. The extent to which these effects counterbalance each other is beyond the scope of this paper, however, most of the arguments here are based on ionic fluxes in a constant current environment rather than potentialconductivity analysis. In addition, voltage changes due to changing particle density will be small compared to changes due to the gradient and ion depleted conduction layers. Therefore, these conductivity changes will not affect the basic conclusions of this paper. 3.6. Case 1. Suppression of Voltage Rise. The first set of depositions are marked by a suppression of the voltage rise that occurs in the solution without particles. This is due to the buffering effect of the particles which prevents the ionic concentration from approaching the zero concentration point at which either convection or a significant voltage rise would start. Case 1 is chosen as representative example of this type of behavior for detailed analysis. Figure 5 shows the adsorption isotherm for HCl on the particle surfaces. To simplify the calculations here, for the conditions of case 1 the adsorption isotherm can reasonably be replaced by a linear adsorption between zero and the bulk ionic concentration. This is shown by the solid gray line in Figure 5. As mentioned above, with electrophoresis of the particles


Figure 6. Ion depletion gradient at end of 120 s experiment in case 1. Gray area indicates thickness of accumulated particle layer at this time.

there is a net flux of chloride ions toward the cathode. This means that HCl will be carried to the accumulated particle layer faster than it can be depleted at the electrode. Therefore, there can only be an ionic depletion within the accumulated particle layer where electrophoretic motion is stopped. Returning to the assumption that the particles will accumulate at a 25% volume density (an interparticle separation of 80-90 µm), this means that in the accumulated particle layer there will be 480 times more HCl adsorbed to the particles per unit volume than HCl dissolved in solution. The current conducted through this layer will still depend on the diffusion/migration of ions in solution, but the depletion of ions will proceed at 1/480th of the speed in the solvent alone. Placed in terms of the diffusion equation this is

∂c ∂ 2c ∂ 2c 1 ) DE 2 ) RDE 2 ∂t 480 ∂x ∂x

(1)

The boundary conditions are the same as in ref 3 and the solution for the depletion gradient is then

c(x,t) ) co -

[(

) ( )

4RDEt J π 2DH+

1/2

exp

-x2 4RDEt x erfc

(x )] x

(2)

4RDEt

where R is the ratio of bulk dissolved ions to adsorbed ions in the accumulated layer. Plotting this equation for the depletion at the electrode shows a depletion time of 270 s, 150 s longer than the duration of this experiment. Figure 6 shows the concentration gradient at the end of the 120 s. Using the linear desorption assumption, the profile of this gradient is exactly the same as the for depletion gradient without the particles. The only difference is the depletion time. The accumulated particle layer will grow at a rate of 0.375 µm/s, reaching a thickness of 45 µm at the end of the experiment. This is shown as the gray area in Figure 6. As can be seen the depletion gradient is completely contained within the accumulated particle layer. The ionic concentration does not approach zero at the cathode surface. 3.7. Case 2. Stabilization of an Ion Depleted Conduction Layer. The second set of deposition trials are marked by dense deposited layers and linear voltage rises that range from 2 to 14 V. A deposition from the middle of this range with a voltage rise of 3.3 V was chosen for analysis and designated as case 2.



Figure 7. Adsorption isotherm for HCl on alumina. Circle shows the total adsorption for case 2. Gray line shows a linear adsorption assumption. Black line is a modified adsorption assumption.

Figure 9. Concentration gradient layers at the end of 120 s deposition time shown at two length scales. (a) The concentration at the surface of the accumulated particle layer (gray area) will rise to 1/2co as the layer grows. Inside the accumulated particle layer there is a slow moving gradient layer and an ion depleted conduction layer (dark gray). (b) Outside the accumulated layer a diffusion gradient will extend 1 mm into the bulk suspension.

Figure 8. Evolution of the concentration gradient layers under the modified desorption assumption of Figure 7. At 0.16 s after initiation of current concentration at the electrode drops to 1/2co. At 2.0 s the concentration will begin to approach zero at the cathode. Black line calculated using eq 2. Gray line calculated based on semiinfinite diffusion. Gray area next to cathode line is accumulated particle layer thickness at 2.0 s.

Here, as in the previous case, there is a net flux of ions toward the cathode due to electrophoretic migration of the alumina particles. This means that the ionic concentration in the system cannot drop to the point that the assumption of quasi-neutrality is violated, except where particle motion is blocked at the cathode. Therefore, an ion depleted conduction layer will be contained within the accumulated particle layer at the electrode. The adsorption isotherm for HCl on the particles is shown in Figure 7, with a circle indicating the conditions of case 2. Replacing this isotherm with a simple linear model, as shown by the gray line, is much less accurate than in case 1 above. An equally simple and slightly improved model is shown by the black line. The assumption for this is that there is no desorption until the ionic concentration in solution drops by 50%, and desorption is linear with solution concentration after that. This simplification allows a step by step description of ion depletion and the growth of the accumulated particle layer which captures all of the major aspects of this process. The results of this assumption are shown in Figure 8. At 0.16 s after the current is first switched on, the ionic concentration at the cathode will drop to 50% of the bulk value. Over this range, depletion is very rapid as the only ions available are those dissolved in solution. This time frame is also too short for there to be a significant accumulation of particles at the electrode. The next stage of depletion will be slowed by the desorption of ions from particles in solution. As mentioned above, at the bulk density of 1 vol % the particles hold 5.78 times more ions than are available in the bulk solution. The desorption of these ions significantly slows depletion next to the cathode, so that it will

take an additional 2 s for the concentration in solution to drop to less than 5% of the bulk concentration at the electrode. Over these 2 s, particles also begin accumulating at the electrode. At only the bulk velocity of the particles of 11 µm/s with an accumulated layer volume density of 35%, the accumulated layer would grow to 0.6 µm over these 2 s. However, particles will be accelerated toward the electrode in this layer due to both the higher electric field in the gradient layer and the higher ζ potential of the particles as the solution ionic concentration goes down. Thus, when the concentration approaches zero at the cathode and an ion depleted conduction layer begins to form, it will be within an accumulated layer of particles ≈1.0-1.5 µm thick. Since the net flux of ions on the surface of the particles toward the electrode is greater than the flux of chloride ions away, the ion depleted conduction layer will continue to be contained within the accumulated particle layer. As the experiment progresses both the accumulated particle layer and the ion depleted conduction layer will continue to grow. As the accumulated particle layer grows, the ionic concentration at the surface rises back to 1/2co. At the end of 120 s, the accumulated particle layer will encompass a slow moving gradient layer and an ion depleted conduction layer (Figure 9a). Outside of the accumulated layer a diffusion gradient extends from the edge of the accumulated layer nearly 1 mm into the solvent (Figure 9b). An estimate of the thickness of the ion depleted layer can be made by a simple accounting for the chloride ions in a control volume next to the cathode. Establishing a boundary 1 mm away from the electrode, over a 120 s period there will be a total flux of Cl- ions in solution out of the volume of 630 µmol/m2. Depletion in solution, region A in Figure 9, can account for 36 µmol/m2. The depletion gradient layer in the accumulated particle layer accounts for 350 µmol/m2, which leaves 240 µmol/ m2. This would be generated by complete depletion and desorption from a 35 vol % dense layer of particles 6.2 µm thick. If the very high voltage gradients in the ion depleted conduction layer compressed that layer of particles from 35%



Figure 10. (×) Calculated ion depletion layer thicknesses; (O) deposition layer thicknesses based on experiment.

to 60% density, the layer thickness would be 3.6 µm, very close to the experimental value of 3.4 µm. Although this is good evidence that the dense deposited layer is formed in an ion depleted conduction layer, far more convincing evidence is provided by the voltage behavior of the depositions. All of the depositions in set 2 exhibited significant voltage rises as well as dense deposition layers. Assigning 1 V to compensate for the increased resistivities of the gradient layers, the question then becomes, what thickness of ion depletion conduction layer would account for the remaining voltage rise? These thicknesses are calculated using the potential equation from ref 9

φ ) φo -

[

]

2 2JF 3 νH+ro

1/2

x3/2

(3)

and are plotted in Figure 10 as “×” s. Superimposed on these calculated values are circular spots indicating the exprimental deposition thickness based on a 60 vol % density. The standard deviation is only 4% for all of those depositions where the net migration of Cl- ions in the bulk solution is toward the cathode, and therefore the ion depletion conduction layer is expected to be contained within the accumulated particle layer. A significant deviation of 30% appears for the highest conductivity cases, and this will be discussed in section 3.9. This provides very strong evidence that a stable ion depleted conduction layer is formed at the electrode and that it is this layer which accounts for the formation of a dense deposited layer of alumina particles. 3.8. Mechanism of Stabilization. In the previous paper it was shown that an ion depletion conduction layer is extremely unstable and is unlikely to be observed in a fluid medium. This is due to ionic concentration gradients which move with the fluid and forces which in turn move the fluid due to the presence ionic concentration gradients in an electric field. The presence of a particle layer breaks this linkage by effectively anchoring the concentration gradients to the particle layer regardless of fluid motion. A concentration gradient in solution between the particles is also a gradient of adsorption on the surface of the particles. With the density of ions on the surface of the particles at least 2 orders of magnitude higher than in solution, motion of the solution through the particles can only move the concentration gradient very slowly. Since the primary forces driving circulation are perpendicular to the electrode, a circulation cell can be modeled by two tubes of solution leading from the bulk concentration to the cathode with a hydraulic connection between the tubes at the cathode surface, Figure 11. The motion of solvent in one tube toward the cathode is balanced by an equal motion in the other tube away. It is assumed that, as in case 2, that they are filled with 35 vol % alumina particles and that the bulk solution ionic concentration is 0.192. A current is passed through these

Figure 11. Circulation model. Two tubes of fluid lead from the bulk solution to the cathode surface. After a gradient and ion depleted layer has formed uniformly in each tube, it is assumed that a disturbance completely fills tube B with solution at the bulk ionic concentration, while tube A is completely filled with ion depleted solvent.

columns so that an ionic concentration gradient develops and begins to move away from the cathode. For simplicity and clarity only one point in this concentration gradient layer will be singled out for consideration, that will be the point where the solution ionic concentration between the particles has dropped to 0.045 mol/m3 (the conditions of case 1). At this point in each tube the net ionic concentration in solution, excluding the 35 vol % occupied by particles is 0.030 mol/m3. The particles have adsorbed 2.14 mol/m2 of ions, and the surface area density is 1.49 × 107 m2/m3, which gives a molar concentration of adsorbed ions of 31.8 mol/m3. Looking at the small circle on the graph in Figure 0.5, the slope of the adsorption/desorption isotherm at this point is 10 µmol/m2 for each mol/m3 in solution concentration change. Now consider a very large hydrodynamic disturbance where the solution in column B is instantly, entirely replaced with solution at the bulk molar ionic concentration, and the solution in column A is entirely replaced with solution which is entirely ionically depleted. The solution is now out of equilibrium with the particle surfaces, and ions will be adsorbed from the solution in tube B and desorbed into solution into tube A. After the solutions have reequilibrated the solution concentration in column A will have gone down by 0.1% and risen in column B by 0.5%. If the concentration gradient at this point in the two columns is 8.35 × 103 mol/m4, then even with this very large disturbance the concentration gradient will have moved by a maximum of only 18 nm in tube B. This is less than the Debye layer thickness at this point and is on a scale that it can be eliminated very rapidly by diffusion. Since it is movements in the gradient layer that result in differential hydrostatic pressures which drive hydraulic motion, if these gradient layers are anchored to an immobile particle bed, small fluctuations cannot trigger forces sufficient to drive convective motion and an ion depleted conduction layer will be stabilized. 3.9. Case 3. Stabilization Co-Existing with Convection. The robustness of this stabilization is illustrated by the final deposition, case 3. In this case the nonlinear voltage rise is much higher than can be explained by ionic depletion in the bulk solution alone, and the deposition is thicker than an ion depleted conduction layer with the same voltage drop. What distinguishes this case of deposition is the net ionic flux in the system. In the cases at lower solution conductivity the migration of particles toward the cathode in the bulk solution moves more ions toward the cathode than are moved away by migration in solution. This means that an ion depleted layer will be contained within the accumulated particle layer and convection is not necessary in the bulk solution. For case 3, the flux of chloride ions in solution away from the cathode is 21/2 times greater than the flux of chloride ions carried on the



Figure 12. Boundary infiltration. At an edge of a deposition, fluid from the bulk can infiltrate along the electrode-deposition boundary displacing ion depleted solution outward. If the deposition begins to break away from the electrode, solvent can rush in very rapidly, “popping” the deposition off the surface (a). The positively charged flake may sediment out or may be redeposited on the electrode (b).

particles toward the cathode. This means that when an ion depleted layer forms, it will grow faster than the accumulated particle layer. When the ion depleted layer extends past the accumulated particle layer, it will trigger convection in the bulk solution just as in a solution without particles. A stabilized ion depleted conduction layer is necessary to explain the large voltage rise, while convection in the bulk is mandatory to provide adequate ionic transport. This means the growth of a deposition containing an ion depleted conduction layer with a gradient/buffer layer at the surface which stabilizes that ion depleted layer. Outside of the accumulated particle layer convection transports both ions and particles to the surface of the accumulated particle layer. The buffer layer is able to stabilize an ion depleted conduction layer even in the face of varying ionic concentrations and convective flows outside of the growing deposition layer. 3.10. Breakdown of Stabilization. The powder stabilizes the formation and growth of an ion depleted conduction layer by delinking the motion of the gradient layer from the motion of the fluid. These layers, however, are still fundamentally unstable, as is illustrated by two of the mechanisms that can cause this stability to break down during deposition. Flaking. During this type of deposition the fluid at the electrode surface is at a significantly higher pressure than the bulk. This is stable as long as the gradient layer is a uniform distance from the electrode and is held in place by an immobile layer of particles. At an edge or defect in the coating, ion containing solvent can flow toward the electrode, allowing the ion depleted solvent to flow away. This can occur quite rapidly and this fluid motion can cause flakes of deposited particles to “pop” off of the deposition electrode. This is shown schematically in Figure 12. Figure 13 shows an example of this behavior in an actual deposit. An area bare of deposition has grown as scallop shaped flakes of deposited powder have popped off the electrode. The edge where a previous flake has come off provides an edge where solvent can flow under the deposition, rapidly displacing ion depleted solvent, and breaking off another flake. The flakes retain a positive surface charge and are frequently redeposited. (All depositions are carried out in an upward direction, against gravity, therefore sedimentation would normally carry the higher density flakes away from the deposition.) Lumping and Pitting. As explained above, the stabilization of an ion depleted conduction layer depends on the gradient layer being pinned to the particle bed. The particle bed then can also be a source of instability that leads to convective flow. Anyplace where the gradient layer is not a uniform distance from the electrode, and therefore the ion depleted conduction layer has a nonuniform thickness, there will be a driving force for convection. Solvent will be pulled toward the electrode where the gradient layer is closer and pushed away where it is farther from the electrode. Once this convection begins, it will

Figure 13. Deposition experiment. The deposition is progressively flaked away from right to left with some flakes redepositing on the surface.

Figure 14. Nonuniform consolidation can move the gradient layer closer to the electrode at a point. This will initiate an inflow of bulk solvent and allow an outflow of ion depleted solvent (a). Inflow of particle containing solvent will result in a lump while outflow of ion depleted solvent washes out a pit with some of the washed out material redepositing (b).

be reinforced by the convective flux of ions, and instead of being suppressed by the accumulated powder layer, it will be pinned in place. There are two ways the gradient layer can end up a nonuniform distance from the electrode. The first way would be for there to be a nonuniform density in the accumulated particle layer. An area of lower particle density will have a lower buffering capacity and the gradient layer will move away from the electrode faster. Where the gradient layer has moved farther away there will be a convective outflow of ion depleted solvent. This will move the gradient layer away faster, reinforcing the flow. This low-density region can come from the deposition of a large, low-density particle floc or the deposition of a foreign object such as a textile fiber. This situation can also occur if the particle layer does not consolidate uniformly. If the particle layer forms a rigid deposit at 50 vol % density, the forces in the ion depleted conduction layer may bring it to a critical stress where it suddenly collapses to a higher density. The gradient layer will move closer to the electrode along with the powder. This will both trigger and pin a convective cell, as shown in Figure 14a. Outflowing solvent will then wash away deposited material while inflowing solvent will deposit additional particles and redeposit material washed away by the outflow resulting in a large irregular lump. This is shown schematically in Figure 14b, and a example is shown in Figure 15. Once this convective process has begun at one point on the deposition it will spread. Each convective cell will destabilize the layer around it, generating new convective cells. If the deposition shown in Figure 15 had been ended sooner, it would have yielded a very smooth and uniform deposited layer. If it had been extended longer, the lumping and pitting would have spread across the entire surface. This can lead to very irregular depositions or sometimes the deposited layer being removed entirely.


J. Phys. Chem. C, Vol. 111, No. 8, 2007 3365 deposition, specifically: flaking, lumping, and pitting. Understanding and using the ion depletion effect allows the EPD of densely packed particle coatings many micrometers thick with an automatic leveling effect due to the high conductivity contrast between the ion depleted layer and the bulk solution, similar to what is achieved in electrocoating. Beyond EPD, this effect is interesting for its ability to produce high-pressure gradients on a micrometer scale with no moving parts. The buffering effect of the particle bed allows a large electrostatic force to be applied to a fluid without inducing convection. This can enable much higher pressure electrostatic pumping than is possible by simple capillary boundary layer mechanisms. Finally, this type of layer recreates the voltage gradient and ionic imbalance conditions that are usually only found in the diffuse electrostatic boundary layer of a surface. However, the electrostatic boundary layer is usually only measured in nanometers. The micrometer scale of the IDC layers represents an increase in reaction volume of 3 orders of magnitude and offers a practical route to the study of chemical reactions in an extremely high potential gradient environment. Symbol

Figure 15. Pictures of a defect in an otherwise uniform powder deposition, ≈ 90 µm thick. Top picture shows pits where the solvent outflow has washed away deposited powder (black areas). The bottom picture, taken at a 20° angle to the deposition surface, shows how inflowing solution has built up a much thicker, ≈200 µm, very irregular lumped surface.

3.11. Pressure Gradients. One of the more interesting effects of this type of conduction layer is the very large pressure gradients that can be generated. The pressure gradient is linear as shown in ref 3:

dP JF ) dx νH+

(4)

For case 3 this results in a pressure gradient of 166 Pa/µm. This is several orders of magnitude higher than would be expected from simple capillary conduction.10-12 4. Conclusions In a fluid medium where ionic gradients move along with the fluid, an ion depleted conduction layer is extremely unstable and is therefore unlikely to be observed except under perfect conditions. However, if the ionic gradients can be held in place by an ionic buffer which does not move with the fluid, this layer can be stabilized. During the electrophoretic deposition of alumina powder, the powder particle surfaces provide the ionic buffering effect necessary to stabilize an ion depleted conduction layer. The existence of a low conductivity ion depleted layer can then explain: the large voltage rise seen across the deposited powder layer, the existence of two density modes in the accumulated/ deposited layer, and the thickness of the densely deposited layer. Furthermore, the breakdown of stabilization of this layer explains some of the defects seen in this type of electrophoretic

c DE F J t x R r o κ-1 κa νH+ φ

ionic concentration (mol/m3) diffusion coefficient of electrolyte (m2/s) Faraday constant ionic flux (mol/s‚m2) time (s) distance from cathode (m) ratio of solution ionic concentration to adsorbed ionic concentration in the accumulated layer relative dielectric constant permittivity constant Debye length (m) diffuse layer thickness index hydrogen ion mobility (m2/V‚s) electrostatic potential (V)

Acknowledgment. This work was funded in part by the the Penn State NSF-IUCRC Centers for Dielectric Studies and Particulate Materials Center and by a grant from the Intel Corporation. References and Notes (1) Sarkar, P.; Huang, X.; Nicholson, P.S. Ceram. Eng. Sci. Proc. 1993 14, 707. (2) Sarkar, P.; Nicholson, P. S. Ceram. Trans. 1996, 62, 271. (3) Van Tassel, J. J.; Randall, C. A. J. Phys. Chem. C 2007, 111, 3341. (4) Van Tassel, J. J.; Randall, C. A. J. Phys. Chem. C 2007, 111, 3349. (5) Van Tassel, J.; Randall, C.A. J. Colloid Interface Sci., 2001, 241, 302. (6) Van Tassel, J.; Randall, C.A. J. Mater. Sci. 2006, 41, 8031. (7) Loeb, A. L.; Overbeek, J. Th. G.; Wiersma, P. H. The Electrical Double Layer Around a Spherical Colloidal Particle; MIT Press: Cambridge, MA, 1961. (8) Russel, W.B.; Saville, D.A.; Schowalter, W.R. Colloidal Dispersions, Cambridge Univ. Press: Cambridge, U.K., 1989. (9) Chazalviel, J.-N. Phys. ReV. A 1990, 42, 7355. (10) Rice, C. L.; Whitehead, R. J. Phys. Chem. 1965, 69, 4017. (11) Tavares, M. F. M.; McGuffin, V.L. Anal. Chem. 1995, 67, 3687. (12) Wan, Q-H. Anal. Chem. 1997, 69, 361.

Ionic Gradients at an Electrode above the Equilibrium Limit Current. 3

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