Magnetic Field Effects on Copper Electrolysis - The Journal of Physical

J. Phys. Chem. B 2001, 105, 9487-9502

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Magnetic Field Effects on Copper Electrolysis G. Hinds,*,† F. E. Spada,‡ J. M. D. Coey,† T. R. Nı´ Mhıócha´ in,† and M. E. G. Lyons§ Physics and Chemistry Departments, Trinity College, Dublin 2, Ireland, and Center for Magnetic Recording Research, UniVersity of California, San Diego, La Jolla, California 92093-0401 ReceiVed: February 14, 2001; In Final Form: June 25, 2001

The effect of a static magnetic field, B, on the electrolysis of copper in aqueous solution is investigated using linear sweep voltammetry, impedance spectroscopy, chronoamperometry, rotating disk voltammetry, and analysis of fractal growth patterns. Data are obtained in fields of up to 6 T. There is a large enhancement of the electrodeposition rate (up to 300%) from concentrated CuSO4 solution (c ∼1 M) when pH e 1. The effect of the magnetic field is equivalent to that achieved by rotating the electrode. From the pH, viscosity, field direction and concentration dependence of the field effect, the influence of field on the complex impedance, and the equivalence of field and electrode rotation, it is established that the magnetic field influences mass transport by forced convection. Convective flow is modified on a microscopic scale in the boundary layer close to the working electrode. There is no influence on the electrode kinetics. Turbulence sets in for our cell geometry when the product of field and current density exceeds a critical value of about 1000 N/m3. The competition between gravitational and magnetic forces is dramatically exhibited by the morphology and fractal dimensionality of planar electrodeposits in a flat circular cell. Quantitative comparison is made of the magnitude of various magnetic body forces inducing convection in typical experimental conditions. The results are discussed both in terms of Aogaki’s model of a streamline boundary layer, which predicts that the excess limiting current varies as B1/3c4/3, as observed experimentally, and in terms of the electrokinetic effect.

1. Introduction Magnetoelectrolysis is the term used for the effect of an applied magnetic field on heterogeneous electrochemical processes at an electrode/solution interface. Research into the influence of a magnetic field on electrochemical reactions has been ongoing for most of the past century. However, many of the published data on the subject are characterized by apparent contradictions and lack of reproducibility; no mechanism which can quantitatively account for all the observed effects has been established. In recent years, various effects have been reported in the literature under a wide range of conditions. Notable reviews of magnetoelectrolysis are those by Fahidy1 and Tacken and Janssen.2 In 1972, Mohanta and Fahidy3 reported that the electrodeposition of copper from acidifed copper sulfate solution was enhanced by the application of a magnetic field. The cathodic limiting current in a 0.05 M solution increased by 30% in a field of 0.7 T. They proposed that the magnetic field modified the hydrodynamic flow via the Lorentz force B FL ) B× j B B (N/m3), where Bj is the current density, such that the width of the boundary layer was reduced. This is sometimes called the magnetohydrodynamic (MHD) effect in the literature. The idea was developed by Aogaki et al.,4 who designed a hollow electrode cell where the field induced a well-defined flow pattern. They were able to obtain an approximate solution to the Navier-Stokes equation which showed that the diffusionlimited current should vary as c4/3 and B1/3, where c is the bulk electrolyte concentration. * Corresponding author. Telephone: +353-1-6081858. Fax: +353-16711759. E-mail: [email protected]. † Physics Department, Trinity College. ‡ University of California, San Diego. § Chemistry Department, Trinity College.

A different interpretation of magnetoelectrolysis came from Kelly,5 who showed in 1977 that the rate of corrosion of titanium electrodes in TiSO4 solution was increased by a factor of 2.5 by the application of a 2 T field. He suggested that the interfacial potential difference at the electrode/solution interface was altered by the field, thereby influencing the electrode kinetics. The papers which have been published on the subject since then may be divided into three main categories: those dealing with magnetic field effects on mass transport processes in solution,6-16 those concerned with the effects on the kinetics of the electrode reaction,17-21 and those relating to the effects on the morphology of the electrodeposit.22-29 The effect of the field on mass transport has been the most extensively studied. Many authors accept that the magnetohydrodynamic flow induced by the Lorentz force on the moving ions must be responsible for the observed effects, although no accurate quantitative model has been established due to the nonlinear nature of the hydrodynamic equations which govern the system. Other authors,11,30 however, believe that the force due to a B c/2µ0, concentration gradient, ∇c, of magnetic ions B FP ) χmB2∇ where χm is the molar susceptibility and µ0 is the permeability of free space (4π × 10-7 T mA-1), is the driving force for the field effects on mass transport. The effect of magnetic fields on both heterogeneous electron-transfer kinetics21 and electrochemical equilibria30 have proved more controversial, partly due to the difficulty of eliminating indirect mass transport effects. Hydrodynamics is the study of the effect of fluid motion on both physical and chemical processes in solution. To obtain an expression for the material flux near an electrode surface, it is necessary to solve the convective diffusion equation:

∂c ) -V b‚∇ B c + D∇2c ∂t

10.1021/jp010581u CCC: $20.00 © 2001 American Chemical Society Published on Web 09/12/2001

(1)

9488 J. Phys. Chem. B, Vol. 105, No. 39, 2001

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where D is the diffusion coefficient and b V is the fluid velocity. A limiting, static solution for uniform mass transport is:

jL )

nFDc δ

(2)

where jL is the limiting current density, n is the number of electrons transferred, F is Faraday’s constant, and δ is the thickness of the diffusion layer, but the solution to eq 1 requires a knowledge of the velocity profile, b V(r b) in the fluid, which is obtained by solving the Navier-Stokes and continuity equations:

∇ BP dV b +b V ∇b V)+ ν∇ 2 b V+B Fi dt F

(3)

∇ B ‚V b)0

(4)

The Navier-Stokes equation (eq 3) expresses Newton’s second law for an incompressible fluid element. The left-hand side of the equation is the acceleration of the fluid element and the right-hand side is the sum of the external volume forces on it. The first term on the right-hand side is the force due to the pressure gradient, ∇ B P, through the system. The second term accounts for the effect of viscous forces and only applies when one part of the fluid moves relative to another. Here ν is the kinematic viscosity of the fluid (m2/s). Momentum is transferred from faster moving layers of fluid to slower moving layers. The third term, B Fi, refers to any other external volume forces, gravitational, electrical or magnetic, acting on the fluid element, such as B FL or B FP. In general, the differential equations derived from the Navier-Stokes equation are nonlinear and complete analytical solutions do not exist. Numerical methods are usually employed. Dimensional analysis and similarity theory have been widely applied to hydrodynamic systems for the same reasons. The most important dimensionless variable characterizing the flow of a viscous fluid is the Reynolds number, Re:

Re )

UL ν

(5)

where U is a characteristic fluid velocity and L is a characteristic length. For the steady flow of an incompressible fluid in a particular geometry, the flow regime is entirely determined by the value of Re. Below a certain threshold value, Recrit, the flow is laminar; above Recrit the flow becomes turbulent. For a flat plate, Recrit ∼ 1500.31 The flow velocity must become zero at the surface of a solid body immersed in a fluid. This is known as the “no slip” condition. Hence, there is a region immediately adjacent to the solid surface where the flow velocity changes rapidly from zero to its value in the bulk stream. This region is known as the hydrodynamic boundary layer, and its thickness is denoted δ0. The retardation of fluid motion in the boundary layer is due to viscous forces alone. The concept of viscosity in convective flow is analogous to that of the diffusion coefficient for diffusive motion. Whereas the diffusion coefficient, D, is the proportionality factor between the flux of material and the concentration gradient, the kinematic viscosity, ν, is the proportionality factor between the flux of momentum and the velocity gradient. Although the boundary layer occupies a small volume (a typical thickness in a liquid of viscosity ν ) 10-6 m2/s moving with velocity U ) 0.1 m/s is 1 mm), its properties dominate the hydrodynamic behavior of the system. The copper/copper sulfate system is well suited to the study of magnetoelectrolysis due to the wealth of published data on both kinetic and transport phenomena in this system. It has been

suggested that the increase in the rate of transport of Cu2+ ions in an applied field may arise due to the MHD effect,3,4,7 the generation of a nonuniform magnetic field in the diffusion layer,11,16 or concentration changes in the bulk solution.12 Research into the effect of the field on the electron-transfer kinetics has revealed that while there appears to be no change in the Tafel slopes,17,18 surface processes may be indirectly influenced by mass transport phenomena.21 Interesting field effects have also been reported on both the anodic dissolution32 and nucleation and growth33 of copper. Work by Mogi et al.23,24 has shown an influence of a magnetic field on the pattern formation of fractal electrodeposits grown in a horizontal planar cell. The authors suggest that the Lorentz force on the motion of cations in solution is responsible for the altered morphology. These results have been modeled by a simplified simulation of two-dimensional diffusion-limited aggregation.34,35 In a preliminary study,36 we found that the effect of a static magnetic field on the rate of mass transport during the electrodeposition of copper is most pronounced at low pH (pH < 1). We also established the importance of competition between gravitational and magnetic forces during the growth of fractal electrodeposits.37 Here, we present an extensive study of the effect of magnetic fields in the range 0-6 T on the electrochemical behavior of the redox couple Cu2+/Cu in aqueous solution. Linear sweep voltammetry, impedance spectroscopy, chronoamperometry, rotating disk voltammetry, scanning electron microscopy, X-ray diffraction, and analysis of fractal growth patterns obtained with different orientations of the flat cell and the magnetic field are all used to build up a coherent picture of these fascinating and often unexpected effects. Besides the magnetic field, the variables are electrolyte concentration, pH, viscosity, and ionic strength. Our data establish that the primary influence of the magnetic field on the electrodeposition and electrodissolution of copper is to induce convection in the bulk solution. Quantitative comparison is made of the magnitude of the various magnetic body forces acting in typical experimental conditions. 2. Experimental Methods Copper sulfate solutions of various concentrations were made up using anhydrous CuSO4 in triply distilled water. The solution pH was varied in the range from 0.4 to 3.5 by the addition of 2 M H2SO4. The electrochemical experiments were carried out using a standard three-electrode cell of diameter 25 mm and depth 50 mm which could be placed centrally in the bore of a cylindrical magnet. The electrodes used for the various experiments were as follows: working (copper, platinum, glassy carbon), counter (copper, graphite, platinum) and reference electrodes (silver/silver chloride). The copper working electrodes were prepared using fine-grain sandpaper and rinsed with sulfuric acid and triply distilled water. The platinum and glassy carbon working electrodes were prepared by polishing with 0.3 µm alumina before being rinsed with triply distilled water. Four different magnets were used to supply a static magnetic field to the cell during the various electrochemical experiments. A Halbach cylinder with a fixed static field of 0.5 T in a 54 mm bore or a 48 mm bore Multimag38 permanent magnet system capable of delivering variable fields in the range 0-1.0 T in any direction transverse to the bore was used for the lower field experiments. Some data were also obtained in a 2 T laboratory electromagnet. For the high-field experiments, a superconducting magnet was used to apply axial fields of up to 6 T to the cell, mounted in its 60 mm bore. The copper fractal electrodeposits were grown in a flat circular cell (22 mm diam, 0.2 mm thick)

Magnetic Field Effects on Copper Electrolysis

J. Phys. Chem. B, Vol. 105, No. 39, 2001 9489

Figure 1. Effect of magnetic field on the voltammogram for a Cu working electrode in 0.75 M CuSO4 at pH ) 0.5.

with a graphite cathode at the center and a copper ring anode. The cell could be mounted either horizontally or vertically. A constant voltage power supply was used to apply a voltage of 6 V across the cell. The resulting fractal patterns were photographed using a digital camera and analyzed using the box-counting method.39

Figure 2. Effect of magnetic field on the cathodic (b) and anodic (O) limiting currents at pH 0.5. Cathodic limiting currents are taken at -0.5 V and anodic limiting currents at +1.5 V with respect to Ag/AgCl (Figure 1). The data are fitted to eq 6.

3. Results 3.1. Linear Sweep Voltammetry. The effect of a magnetic field on the voltammogram for a copper working electrode in 0.75 M CuSO4 solution at pH ) 0.5 is shown in Figure 1. In each case, the potential was swept from +2 to -2 V at a sweep rate of 10 mV/s. Two distinct regimes may be identified on each branch: the activation regime close to the rest potential and the mass transport regime, characterized by the current plateau. A third regime is hydrogen evolution, which is identifiable on the cathodic branch below -800 mV. No effect of the field is immediately obvious in the activation regime. However, this in itself is not sufficient evidence to eliminate the possibility of an influence of the field on the electrode kinetics since mass transport effects may obscure kinetic effects even at relatively low overpotentials. The crossing of the curves in the hydrogen evolution regime indicates that in higher fields the hydrogen evolution reaction is somehow suppressed. The effect of the field is most pronounced in the mass transport regime. On the cathodic branch, the current density at -500 mV increases by a factor of 4, from 1300 A/m2 in zero field to 5400 A/m2 in a field of 6 T. In fact, such is the enhancement of mass transport in the field that a limiting current is not reached on the cathodic branch before hydrogen evolution sets in. The currents flowing in the anodic branch are smaller than those on the cathodic branch by a factor of 2-3. In contrast to the cathodic branch, a limiting current is observed on the anodic branch in all fields. Similar magnetic field effects are observed here: the current density at +1.5 V is increased from 600 A/m2 in zero field to 1900 A/m2 in 6 T. No significant effect of field direction on the magnitude of the field effects was observed in our cell. The effect of field magnitude on the enhancement of limiting current is shown in Figure 2. The effect tends to saturate in the larger fields of 5-6 T. The data are fitted to

jL ) jL0 + aBm

(6)

where for the cathodic branch jL0 ) 1200 ( 100 A/m2, a ) 2300 ( 200 A/m2, and m ) 0.35 ( 0.03 and for the anodic

Figure 3. Effect of pH on (a) limiting current and (b) current enhancement for a Cu working electrode in 0.75 M CuSO4 in an applied magnetic field of 0.5 T.

branch jL0 ) 600 ( 50 A/m2, a ) 680 ( 70 A/m2, and m ) 0.36 ( 0.04. The pH was adjusted by varying the concentration of the supporting electrolyte, H2SO4. Figure 3a shows how the field effect on the deposition current at -500 mV varies with pH for a copper working electrode in 0.75 M CuSO4. The enhancement in the deposition current is plotted as a function


Figure 4. Effect of magnetic field on the voltammogram for a Cu working electrode in 0.75 M CuSO4 at pH ) 3.5.

Figure 5. Effect of magnetic field on the cathodic (b) and anodic (O) limiting currents at pH 3.5. Cathodic limiting currents are taken at -0.5 V and anodic limiting currents at +1.0 V with respect to Ag/AgCl (Figure 4).

of pH in Figure 3b. The extent of current enhancement is typically 10 to 15% over the pH range 1.5-2.5, but as the solution becomes more acidic, a marked increase in current enhancement is observed. At pH 0.7, the current enhancement is 117% in a field of 0.5 T. No such dependence of the field effects on solution pH was observed on the anodic branch. The effect of a magnetic field on the voltammogram for a copper working electrode in an unacidified 0.75 M CuSO4 solution at pH ) 3.5 is shown in Figure 4. The effect of the field on the limiting current is clearly more pronounced for the anodic branch than for the cathodic branch at the higher pH value (Figure 5). On the anodic branch, the limiting current density at +1.5 V increases from 990 A/m2 in zero field to 1800 A/m2 in a field of 6 T. On the cathodic branch, the current density at -500 mV increases from 1610 A/m2 in zero field to 1870 A/m2 in a field of 3 T. Note that in higher fields the current enhancement falls off with increasing field. Again, the crossing of the curves in the hydrogen evolution regime of Figure 4 is noted, together with the lack of any obvious effect in the activation regime. The effect of using Na2SO4 instead of H2SO4 as the supporting electrolyte is demonstrated in Figure 6. A similar enhancement in deposition current is obtained for a glassy carbon working electrode in 0.1 M CuSO4 with 1 M concentra-

Hinds et al.

Figure 6. Effect of supporting electrolyte (1 M) on the voltammogram for a glassy carbon working electrode in 0.1 M CuSO4 at pH 0.5 with and without an applied magnetic field of 0.5 T.

Figure 7. Effect of CuSO4 concentration on cathodic limiting current in zero field (b) and in an applied magnetic field of 0.5 T (O) at pH 0.4. The data in the field are fitted to eq 7.

tions of each species. Note that for Na2SO4, the hydrogen evolution reaction does not occur in this potential range. These data establish that the current enhancement shown, for example, in Figure 3b must be attributed to the ionic strength of the electrolyte, rather than to its acidity. The concentration dependence of the magnetic field effect on electrodeposition at constant pH was also investigated using linear sweep voltammetry. The working electrode was glassy carbon, and the sweep rate was 20 mV/s. The variation in the current density at -800 mV as a function of electrolyte concentration is displayed in Figure 7. As expected, the concentration dependence was found to be linear in zero field, but in an applied field of 0.5 T, the deposition current was found to vary nonlinearly. The enhancement in limiting current is greater at higher concentrations. The data in the field are fitted as

j ) Rcm

(7)

where R ) 6000 ( 200 A/m2 and m ) 1.40 ( 0.06. The effect of increasing the electrolyte viscosity was investigated by the addition of glycerol in varying concentrations. Figure 8 shows the effect of the field on the deposition current at -400 mV for a glassy carbon electrode in 0.1 M CuSO4 at pH 0.5. In zero field, the current in the mass transport limited



Figure 8. Effect of glycerol concentration on the limiting current at a glassy carbon working electrode in 0.1 M CuSO4 at pH 0.5. Data were obtained with and without an applied field of 0.5 T.

Figure 10. (a) Effect of magnetic field on the electrodeposition current at a glassy carbon rotating disk electrode in 0.01 M CuSO4 at pH 1.0. (b) Effect of a field of 0.5 T (equivalent to a rotation rate of 10 rpm). Figure 9. Effect of a 0.5 T field on the Tafel plot for a Cu-modified electrode in 0.1 M CuSO4 at pH 0 (rotation rate is 10 000 rpm).

regime shows a slight decrease in magnitude with increasing glycerol concentration. However, in a field of 0.5 T, the current enhancement is drastically reduced as the glycerol concentration is increased. The current enhancement is reduced from 131% in the absence of glycerol to just 2% when the concentration of glycerol is 500 mL/L, corresponding to a kinematic viscosity ν ) 7 × 10-6 m2/s. The effect of a magnetic field on the heterogeneous electrontransfer kinetics of the Cu2+/Cu oxidation and reduction reactions was also investigated using linear sweep voltammetry. The work was carried out using a glassy carbon rotating disk electrode with a rotation rate of 10,000 rpm in order to eliminate mass transport effects. Before each experiment, a thin layer of copper was deposited on the electrode. In each case the charge passed during this electrodeposition was 0.15 C, giving an estimated layer thickness of 1 µm. Tafel plots for this electrode in 0.1 M CuSO4 at pH 0 are presented in Figure 9. Good Tafel behavior was observed on both cathodic and anodic branches. The exchange current density was found to be 10 A/m2 for each branch. From the Tafel slopes, the cathodic and anodic chargetransfer coefficients were calculated to be 0.45 and 0.90, respectively. These kinetic parameters are unaffected by the magnetic field. The data in 0 and 0.5 T superpose perfectly. The effect of a static magnetic field on electrodeposition of copper under conditions of enhanced mass transport was

examined using a rotating disk electrode, where there is a high rate of convection, and with a microelectrode, where there is a high rate of diffusion. Results in conditions of forced convection using a glassy carbon rotating disk electrode in Figure 10 show the effect of an applied field of 0.5 T on the voltammograms obtained at several different rotation rates in 0.01 M CuSO4 (pH ) 1.0). The limiting current density with a stationary electrode in the absence of an applied field is 4.5 A/m2. When a magnetic field of 0.5 T is applied to the cell, the limiting current density increases to 9.4 A/m2. The effect of the field on the quiescent solution is quantitatively equivalent to a rotation rate of 10 rpm (Figure 10b). When the field was applied to a rapidly rotating electrode, enhancements of just 2 to 3% were observed on the limiting currents at each rotation rate. Voltammograms for Pt microelectrodes of various dimensions in 0.05 M CuSO4 at pH 0.5 are shown in Figure 11. Enhancements of only 1 to 2% in the deposition current at -500 mV were observed with the application of a 0.5 T field. 3.2. Chronoamperometry. The effect of a magnetic field on the diffusion of Cu2+ was investigated using chronoamperometry. A potential step of magnitude 500 mV was applied to take the working electrode from the rest potential, where no faradaic reaction occurs, to a final value where all electroactive species that reach the electrode are instantaneously reduced. This corresponds to a cathodic potential in the diffusion-limited current plateau of Figure 1, where the electrode kinetics are significantly faster than the rate of mass transport. In quiescent


Hinds et al.

Figure 11. Effect of a 0.5 T field on the voltammograms for Pt microelectrodes of various dimensions in 0.05 M CuSO4 at pH 0.5.

solution, the rate of reaction, and hence the measured current response, is solely determined by the rate of diffusion. For a reduction reaction, such as the electrodeposition of copper, the current, I, is given by the Cottrell equation40

I(t) )

nFAD1/2c (πt)1/2

(8)

where A is the electrode area (j ) I/A). Equation 8 shows that the diffusion-limited current decreases as t-1/2. This is due to the decrease in concentration gradient with time as the diffusion layer thickness grows. However, natural convection, arising from density differences in the solution, eventually perturbs the concentration gradient and prevents further growth of the diffusion layer. This results in a steady-state current at long times. At short times, a capacitative current also flows due to reordering of the charge distribution in the double layer. This contribution is only important on a time scale of 100 ms and can usually be neglected thereafter. A plot of I against t-1/2 is known as a Cottrell plot. Under conditions of semi-infinite linear diffusion, such a plot will be linear, enabling the determination of the combination nAD1/2 from eq 8. Deviations from linearity occur at short times due to the charging of the double layer and at long times due to natural convection. The effect of a static magnetic field on the chronoamperometric response of a copper electrode in 0.05 M CuSO4 at pH 0.5 is shown in Figure 12a. The current density at long times is increased by 85% in the field, from 43 A/m2 in zero field to 79 A/m2 in a field of 0.5 T. The Cottrell plot for this experiment is shown in Figure 12b. Linear behavior is observed at intermediate times both in the absence and presence of the field. This part of the graph may be extrapolated to the origin, thereby demonstrating the expected behavior, and the slope yields a diffusion coefficient of 7 × 10-10 m2/s, in good agreement with the literature.41 The value of D is unchanged by the magnetic field. Deviations from linearity at short times are due to charge ordering in the double layer. Given that these deviations occur for t-1/2 > 2, we can conclude that the capacitative currents which interfere with the faradaic response are negligible after 250 ms. At long times, the current density in the Cottrell plot in a field of 0.5 T levels off before that in zero field, to a value consistent with the linear sweep voltammogram. These deviations from linearity at long times are due to convective effects, which prevent further growth of the diffusion layer, so that eq 8 no longer holds.

Figure 12. Effect of a 0.5 T field on (a) the current response at a Cu working electrode in 0.05 M CuSO4 at pH 0.5 following a cathodic potential step of 500 mV and (b) the Cottrell plot for the same data.

3.3. Impedance Spectroscopy. The effect of a 0.5 T field on the Bode plot for a copper working electrode in a quiescent solution of 0.2 M CuSO4 at pH 0 is shown in Figure 13a. In general, the magnitude of the impedance is slightly greater in the field. The effect of the field on the phase angle, φ, is not significant. At low frequencies, a constant phase angle of about 16° is observed in both field-on and -off cases. This is lower than the value expected for a system under pure diffusion control (45˚), which suggests that the reaction is not completely reversible. The same data are plotted using the Nyquist representation in Figure 13b. At high frequencies, a depressed semicircle characteristic of a quasi-reversible electrode reaction under kinetic control is observed. The high-frequency intercept of this semicircle gives the solution resistance, Rs ) 0.6 Ω, independent of field. The charge-transfer resistance, Rct, is given by the difference between the two intercepts. The low-frequency intercept is shifted from around 15 Ω in zero field to around 25 Ω in a field of 0.5 T, indicating that the charge-transfer resistance is increased by the field. The charge-transfer resistance is related to the exchange current density, j0 ) I0/A, by42

Rct )

RT nFI0

(9)

According to eq 9, the effect of the field is to decrease the exchange current density from 10 A/m2 in zero field (in agreement with the Tafel plots in Figure 9) to 6 A/m2 in a field of 0.5 T. The double layer capacitance may be estimated from



Figure 13. (a) Bode plot, (b) Nyquist plot, and (c) plot of Z′ against ω-1/2 for a Cu working electrode in 0.2 M CuSO4 at pH 0.5 with (O) and without (b) an applied field of 0.5 T.

the frequency, fmax, corresponding to the peak in the semicircle of Figure 13b using the equation42

Cd )

1 2π fmax Rct

(10)

From Figure 13b, fmax changes from 26 Hz in zero field to 15 Hz in a field of 0.5 T. According to eq 10, the double layer capacitance is around 350 µF in both field-on and -off cases. The double layer thickness, xd, may then be estimated by using the parallel plate approximation:

Cd )

0A xd

(11)

Figure 14. SEM images of copper films deposited with (a) zero applied field, (b) an applied field of 0.5 T, and (c) a rotation rate of 10 rpm.

where is the relative permittivity of the medium and 0 is the permittivity of free space. Taking ∼ 80 for water and with A ) 8.5 × 10-5 m2, we obtain xd ∼ 0.2 nm, which is a reasonable value. The thickness of the double layer is unaffected by magnetic field. At low frequencies, a straight line of constant phase angle is observed, consistent with a quasi-reversible electrochemical reaction under diffusion control. The effect of the magnetic field


Figure 15. X-ray spectra of copper films deposited in (a) zero applied field and (b) an applied field of 0.5 T.

on the diffusion coefficient of Cu2+ may be determined from a plot of Z′ against ω-1/2 in this limit,42 as shown in Figure 13c. The slope of this plot is unchanged when a field of 0.5 T is applied to the cell, which again shows that the diffusion coefficient, D, is unaffected by the field. 3.4. Analysis of Electrodeposited Copper. The surface morphology of the electrodeposited copper was examined using scanning electron microscopy. Copper films of micron thickness were deposited on a glassy carbon working electrode (3 mm diam) from unstirred solutions of 0.75 M CuSO4 at pH 0.5 both in the absence and presence of the field. A standard threeelectrode cell was used, with the deposition potential held at -500 mV until a fixed amount of charge had passed. The films

Hinds et al. were removed using a razor blade and coated in gold for imaging by scanning electron microscopy. The surface of a representative film deposited in zero field (Figure 14a) is compared to that deposited in a field of 0.5 T (Figure 14b). The surface of the film deposited in the field is considerably rougher. The surface of a copper film deposited at the same rate on a glassy carbon rotating disk electrode at 10 rpm is shown in Figure 14c for comparison. The effect of the field on the crystal structure of the electrodeposited copper was investigated using X-ray diffraction. A larger glassy carbon electrode, of diameter 7 mm, was used in order to increase the amount of copper deposited. The copper was deposited from unstirred solutions of 0.2 M CuSO4 at pH 0.5 at a fixed potential of -500 mV. In each case the charge passed during this electrodeposition was 4 C, giving an estimated layer thickness of 3.8 µm. Growth of the 3.8 µm layer of copper in zero field took around 240 s, whereas in the presence of a field of 0.5 T, the layer had reached this thickness after 180 s. The films were removed using double-sided tape and attached to a microscope slide for X-ray analysis. The X-ray diffraction patterns in Figure 15 show only peaks characteristic of fcc copper with a lattice parameter of 0.3615 nm. No significant effect of the field was observed on either the crystalline structure or the texture of the electrodeposited copper. 3.5. Fractal Electrodeposits. Fractal electrodeposits grown in the 22 mm flat circular cell in a horizontal orientation from 0.2 M CuSO4 at an applied voltage of 6 V are shown in Figure 16. In the absence of an applied magnetic field, dense radial growth is observed (Figure 16a). When a magnetic field is applied perpendicular to the plane of the cell, a branched spiral pattern forms instead (Figure 16b). These growth patterns are

Figure 16. Copper electrodeposits grown around a central cathode in a horizontal flat circular cell (22 mm diameter, 0.2 mm thick): (a) in zero applied field, (b) 0.4 T vertically upward (c) 0.4 T vertically downward, and (d) 1 T horizontally.



Figure 17. Copper electrodeposits grown around a central cathode in a vertical flat circular cell (22 mm diam, 0.2 mm thick): (a) in zero applied field, (b) 0.4 T horizontally perpendicular to the plane of the cell, and (c) 1 T vertically parallel to the plane of the cell.

TABLE 1: Form of Copper Electrodeposits in Zero Field or 1T cell

B)0

B)1T (vertical)

B)1T (horizontal)

horizontal

dense radial dense unidirectional

branched chiral stringy unidirectional

stringy unidirectional branched chiral

vertical

chiral, and if the field direction is reversed, the fractal spirals in the opposite sense (Figure 16c). When the magnetic field is applied parallel to the plane of the cell, a stringy deposit is formed (Figure 16d). A notable feature of these stringy deposits is their asymmetry. The growth was observed to predominate in the direction where the Lorentz force and the gravitational force both act downward. Fractal electrodeposits grown with the flat circular cell in a vertical orientation are presented in Figure 17. Without an applied field, a dense vertical growth is observed above the cathode (Figure 17a). When the field is applied perpendicular to the plane of the cell, a branched spiral pattern is formed (Figure 17b). When the field is applied parallel to the plane of the cell, a unidirectional stringy deposit forms (Figure 17c). The influence of cell orientation and field direction on the growth patterns of the copper electrodeposits is summarized in Table 1. The effect of the magnetic field on the fractal dimensionality, df, of the electrodeposits grown in a horizontal cell is shown in Figure 18a. There is a broad increase in fractal dimensionality in a perpendicular field, but a clear decrease,

from 1.8 to 1.6, occurs when the field is applied in-plane, corresponding to the increasingly stringy nature of the deposit. The radius of curvature of the main branches of the fractal electrodeposits formed when a field is applied perpendicular to the plane of a horizontal cell decreases with increasing field. The influence of magnetic field on the chirality of these deposits is shown in Figure 18b. Chirality is defined here as the inverse radius of curvature and its sign depends on the direction of the spiral. 3.6. Turbulence. The effect of the magnitudes of both the magnetic field and the current density on the flow regime in the CuSO4/H2SO4 solution was investigated quantitatively. A transparent rectangular cell, of dimensions 4 × 4 × 1 cm3, was used with two parallel copper plate electrodes of working area 6 × 4 mm2 glued to opposite walls to act as anode and cathode during electrolysis. The solution used was 0.75 M CuSO4 in 2 M H2SO4. The current through the cell was controlled by a galvanostat. The cell was placed in the bore of the Multimag and illuminated from below. In zero applied field, the solution remained stagnant during electrolysis. In the presence of a magnetic field, however, a rotational flow of the electrolyte was observed throughout the cell. The flow velocity was on the order of 0.1 m/s. The velocity of this rotational flow increased if either the magnetic field or the current density was increased. Furthermore, the rotational flow was observed even when the magnetic field was applied perpendicular to the electrode surface. At a given magnetic field, the rotational flow showed a transition from laminar to turbulent flow as the current density was increased. The threshold current density, jt, required to induce the onset of turbulence in the solution was identified


Hinds et al. 4. Discussion A summary of the magnetic field effects on copper electrodeposition and electrodissolution observed in this work is provided in Table 2. Upon examination of the data, the most striking effects are the impressive increase in the limiting current and the dramatic change in morphology of the fractal electrodeposits. There is no significant influence on the copper reaction energy or kinetics. Since both field effects are manifested in mass transport-controlled regimes, it is reasonable to conclude that the primary effect of the magnetic field is to increase the rate of transport of electroactive species to or from the electrode. The enhancement of mass transfer in the field must be due to some force of magnetic origin. There are four possible forces which could be responsible for the observed effects. Two of these are due to the interaction of the field with the magnetic properties of the electrolyte. One is the paramagnetic gradient force, B FP:

B FP )

χmB2∇ Bc 2µ0

(12)

which arises from the variation in the paramagnetic susceptibility of the diffusion layer due to the concentration gradient of Cu2+ ions. Here, χm is the molar susceptibility of Cu2+. The second force is the field gradient force, B FB:

B FB )

Figure 18. (a) Fractal dimensionality, df, of electrodeposits grown with the cell horizontal as a function of applied field. The fractal dimensionality is shown for both vertical (b) and horizontal (O) fields. (b) Chirality of electrodeposits grown in a horizontal cell with a vertical applied field. Chirality is defined as the inverse radius of curvature of the main branches of electrodeposits such as those in Figure 16b,c.

χmcB∇ BB µ0

(13)

which is the force due to the field gradient, ∇ B B, in the solution, i.e., when the field is nonuniform. B FP and B FB are thus driving forces due to a magnetic energy gradient. The other two forces, the Lorentz and the electrokinetic forces, are due to the interaction of the field with the electric current. The Lorentz force, B FL,

j B B FL ) B×B

(14)

arises from the motion of charge across lines of magnetic flux, whereas the electrokinetic force, B F E,

B FE )

Figure 19. Effect of magnetic field on the threshold current density, jt, required to induce the onset of turbulence in a rectangular cell consisting of two parallel Cu electrodes in 0.75 M CuSO4 and 2 M H2SO4.

visually at several different values of magnetic field. The results are shown in Figure 19. The threshold current density decreases significantly with increasing magnetic field, from 4700 A/m2 in a field of 0.2 T to 950 A/m2 in 1 T. It was noted that the product jtB remains practically constant at around 950 N/m3 as the field is increased.

E| σd B δ0

(15)

results from the stress on the charge carriers in the diffuse double layer under the influence of a nonelectrostatic field, B E|, parallel to the electrode surface.43 This nonelectrostatic field is created by the motion of charge across the double layer in the presence of a magnetic field parallel to the electrode surface. All four forces are body forces with units of newtons per cubic meter. 4.1. Diffusion. It is difficult to see how a field on the order of 1 T could have any direct effect on the diffusion of Cu2+. The instantaneous velocity of Cu2+ ions in aqueous solution is on the order of 400 m/s, giving a Lorentz force per ion of qVB ) 1.3 × 10-16 N. Estimating the time between collisions, τ ∼ 1 ps, and using ∆V⊥ ) FLτ/m, where m is the mass of the Cu2+ ion (1.06 × 10-25 kg), we obtain the change in transverse velocity ∆V⊥ ∼ 10-3 m/s, which is 5 orders of magnitude lower than V. The direct influence of the Lorentz force on the diffusive movement of ions in solution is therefore expected to be completely negligible. A similar order of magnitude argument applies to the electrokinetic force and the field gradient force. (The field gradient in our experiments was typically ∇B ) 1 T/m). The effect of the paramagnetic force here is also



TABLE 2: Summary of Magnetic Field Effects Observed in This Worka property

measured parameter

technique

field effect

free energy of reaction electron-transfer kinetics

rest potential, Veq exchange current density, j0 charge-transfer coefficients, Ra,c double layer capacitance, Cd limiting current density, jL diffusion coefficient, D fractal dimensionality, df chirality, ξ surface morphology lattice parameter, a0 current density, j

open circuit LSV LSV IS LSV IS, CA box counting ruler SEM XRD LSV

no no no no yes no yes yes yes no yes

mass transport morphology crystal structure hydrogen evolution

a LSV ) linear sweep voltammetry, IS ) impedance spectroscopy, CA ) chronoamperometry, SEM ) scanning electron microscopy, and XRD ) X-ray diffraction.

Figure 21. Schematic diagram of the mechanism of the mass transport enhancement.

Figure 20. Effect of magnetic field on the diffusion layer thickness for both anodic and cathodic limiting currents.

negligible, but for a different reason. The driving force for diffusion is

B FD ) -RT∇ Bc

(16)

Both B FD and B FP act along the same line and depend linearly on the concentration gradient. For a field of 1 T, the ratio FP/FD is 10-6 at room temperature. In any case, the enhancement in mass transport is also observed for diamagnetic cations such as Zn2+, for which B FP has the opposite sign and is at least an order of magnitude smaller.44 The negligible effect of the field on the diffusion coefficient is confirmed by the chronoamperometric data in Figure 12b and by the impedance data in Figure 13c. The slope of the Cottrell plot is unchanged by the application of a 0.5 T field and yields D ) 7 × 10-10 m2/s, in good agreement with the literature.41 However, we have seen that the field does have a dramatic effect on the diffusion-limited current density, given by

Bc jL ) nFD∇

(17)

If the diffusion coefficient remains constant, then the concentration gradient, ∇ Bc, must increase in the field. By use of the linear approximation, ∇ B c ) c/δ, where δ is the thickness of the diffusion layer, it is possible to quantify the effect of the field on δ by using the data in Figure 2. Results are shown in Figure 20. The diffusion layer thickness decreases with increasing field, and the most rapid variation occurs in the low-field region. This narrowing of the region where the concentration gradient occurs causes the increase in the diffusional flux toward the cathode or away from the anode.

4.2. Convection. Convection is usually responsible for such a reduction in the diffusion layer thickness. At a given potential, δ grows as t1/2 (eq 8) until natural convection, arising from density differences in the solution, eventually prevents the diffusion layer from extending any further into the bulk solution and a steady-state concentration gradient results. Forced convection, such as that provided by stirring the solution, may be used to control the thickness of the diffusion layer. The equivalence of the effect of a 0.5 T field with that of rotating the electrode at 10 rpm (Figure 10b) supports the idea that the field acts to induce convection in the solution. Further evidence is provided by the steady-state behavior observed in the Cottrell plot of Figure 12b. In zero field, the current density at long times (t > 20 s) begins to deviate from semi-infinite linear diffusional behavior as natural convection perturbs the concentration gradients. In a field of 0.5 T, this deviation sets in at much earlier times (t ∼ 2 s) and leads to an 85% higher steady-state current density (Figure 12a), which indicates that a stronger pattern of convection is induced by the field. The direct effect of the field on the convective flow must take place in the solution outside the diffusion layer, because within the diffusion layer, transport by convection is negligible since D∇c . ν∇V. This is supported by observations in the turbulence experiment that rotative convection occurs throughout the cell. Classical hydrodynamic theory shows that a tangential velocity, U, of electroactive species can bring about a significant increase in the concentration gradient in the diffusion layer.31 It is clear that this velocity has its origins in the entrainment of Cu2+ in the convective flow of the solvent. Since the flow of current is normal to the electrode surface, a tangential flow could be induced by the Lorentz force. A schematic view of this process for the electrodeposition of copper is shown in Figure 21. In the absence of an applied field, the diffusion layer thickness is a function of concentration, potential and time. Holding these constant, we observe a steady diffusion layer


Hinds et al.

thickness, δ, as shown in part A of Figure 21. If we then apply a magnetic field, the Lorentz force induces a tangential velocity component, U, in the bulk solution. This leads to the formation of a hydrodynamic boundary layer of thickness δ0, as shown in part B of Figure 21. The velocity gradient in the boundary layer creates mixing in the diffusion layer, thus reducing its thickness. The net result is an increase in the rate of diffusion toward the electrode, which increases the current, I, in the bulk solution, leading to an increase in U and a further decrease in δ. This positive feedback process is limited by viscous forces in the boundary layer, which stabilize the flow rate at a steady-state value. The enhancement of the electrodeposition rate of copper is more pronounced at higher concentrations of supporting electrolyte (low pH), and this enhancement is also observed if Na2SO4 is used as the supporting electrolyte (Figure 6). In quiescent solution, transport in the bulk solution occurs primarily through migration, under the influence of the weak electric field. In acidified copper sulfate solution, this migrational transport is mainly due to the supporting electrolyte. The relative contributions of each ionic species in solution to the overall conductivity may be estimated from the transport number, ti, defined by

ti )

|zi|ci µi Σ|zj|cj µj

(18)

where zi is the charge on the ion and µi is the ionic mobility. The ionic mobility can be obtained by dividing the equivalent conductivity of an ion by the Faraday constant. The transport number of an ion in a given solution is thus the ratio of its own conductivity to the total conductivity of the solution. The mobilities of H+, HSO4-, SO42- and Cu2+ in aqueous solution are 36.2 × 10-8, 5.2 × 10-8, 8.3 × 10-8, and 5.6 × 10-8 m2 s-1 V-1, respectively.41 In a 0.75 M CuSO4 solution at pH 0.5 the dominant anion is HSO4 (1.02 M) with SO42- being present in much lower concentration (∼0.042 M). The transport number for the ions in this solution calculated from equation 18 are 0.44 (H+), 0.21 (HSO42-) and 0.32 (Cu2+), respectively, which means that only 32% of the current is carried by Cu2+ in the bulk solution. The interaction of the magnetic field with the migration of supporting electrolyte is thus the main cause of convection in the cell. The Lorentz forces add for the positive and negative species migrating in opposite directions. In the absence of supporting electrolyte, the effect of the field is reduced due to the lower conductivity of the bulk solution (Figure 3). Of course, within the diffusion layer, the entire current is due to diffusion of Cu2+, except when the hydrogen evolution reaction is taking place. It has been noted by Lee et al.10 that the rate of transport of electrically neutral species, such as nitrobenzene, can also be significantly enhanced by an external static magnetic field. Since the Lorentz force acting on a neutral species is zero, the enhancement in mass transport must occur through the convective effect of the field on the supporting electrolyte. The Lorentz force may thus be represented as a magnetic body force acting on a fluid element through which a distributed and continuous current passes, rather than in terms of the direct influence of the field on the random motion of individual ionic species. The tangential velocity may be related to this body force through the Navier-Stokes equation to obtain4

1 2 FU ) jBL 2

(19)

when the field is applied at right angles to the flow of current.

Here, L is the length of the electrode. By use of a typical value for j (103 A/m2), it follows from eq 19 that the tangential velocity induced by the field at an electrode with dimensions of 1 cm should be on the order of 0.1 m/s. This is in agreement with the experimentally observed flow rates. The relationship between the diffusion layer thickness and the tangential velocity derived by Levich31 is

δ ≈ D1/3ν1/6(x/U)1/2

(20)

Taking x ∼ 1 cm, ν ∼ 10-6 m2/s, D ) 7 × 10-10 m2/s, and U ∼ 0.1 m/s, we obtain δ ∼ 28 µm in a field of 1 T, in excellent agreement with the experimental data in Figure 20. The reduction in the magnitude of the magnetic field effects under various conditions adds weight to the idea that fieldinduced convection in the solution is the dominant effect. The current enhancement at a rotating disk electrode, where forced convection due to the rotation of the electrode is far stronger than the field-induced convection, is reduced to only 2 or 3% (Figure 10a). A similar reduction in current enhancement is observed at microelectrodes, under conditions of enhanced diffusion (Figure 11). Most convincing though, is the effect of adding glycerol to the solution. A gradual reduction in the effect of the field on the limiting current as a function of glycerol concentration is observed (Figure 8). This is because as the glycerol concentration is increased, the viscosity of the solution increases, damping out the field-induced convection. Experiments in both high and low fields show that the limiting current at low pH during both electrodeposition and electrodissolution varies approximately as B1/3, in agreement with other reports in the literature.4,7,14 The scale of the magnetic effect is comparable in both processes. The fact that the exponent m < 1 means that a large part of the effect of the field on mass transport is achieved in relatively low fields, ∼1 T (Figure 2). The data are in particularly good agreement with experiments carried out by Aogaki et al.,4 who derived the B1/3 dependence by incorporating the effect of the Lorentz force on the current flow into the Navier-Stokes equation. However, their derivation is applicable only to the orthogonal geometry of their channel electrode cell and does not explain the independence of the magnetic field effects observed here on field direction. We found the limiting current to vary linearly with concentration in zero field and approximately as c4/3 in a field of 0.5 T at constant pH. Similar power-law dependencies have been observed by Aogaki et al.4 with copper sulfate and by Leventis et al.14 with various electroactive organic compounds. The transition from laminar to turbulent flow as the current density is increased at constant magnetic field provides further evidence that the field effects scale with the Lorentz force. When the bulk flow velocity, U, given by rearrangement of eq 19, is substituted into the equation for the Reynolds number (eq 5), we obtain a magnetic Reynolds number, Rem:

Rem )

( ) 2jB F

1/2 3/2

L ν

(21)

The observation that the transition to turbulence occurs at a constant value of the product jtB provides quantitative support for this form of Rem. The use of jtB ∼ 950 N/m3, L ∼ 1 cm, F ∼ 1000 kg/m3, and ν ∼ 10-6 m2/s yields Rem ∼ 1400, which compares favorably to the Reynolds number for a flat plate, Recrit ∼ 1500. Further work is required to determine whether the transition to turbulence depends on L and ν as predicted by eq 21.

Magnetic Field Effects on Copper Electrolysis 4.3. Electrode Kinetics. The effect of a magnetic field on the kinetics of the electrode reaction has been the subject of much debate in the literature. Our Tafel plots were taken in a regime of strong forced convection, to eliminate any mass transport field effects on the electrode process. The exchange current density for Cu2+/Cu was determined to be 10 A/m2 from both the impedance data (Figure 13b) and from the Tafel plots (Figure 9). This is an intermediate value, which suggests that the copper reaction may be considered quasi-reversible.40 The well-defined Tafel behavior and the depressed semicircle in the Nyquist plot strongly support this analysis. According to the Tafel plots, no change in any of the kinetic parameters was observed in a field of 0.5 T. However, the impedance data indicated that the exchange current density decreases to 6 A/m2 in 0.5 T. Furthermore, work by Aogaki et al.18 has also shown a suppression of the exchange current density by a magnetic field in quiescent solution. The discrepancy between the two experiments may be related to the fact that the Tafel plots were carried out in a regime of forced convection, whereas the impedance data were collected in stagnant conditions. The kinetic data from the Tafel plots should therefore be more reliable. The decrease in exchange current density in the impedance experiments and in quiescent solution may be an indirect mass transport effect, rather than a direct effect on the electron-transfer kinetics. The conclusion that the field has no effect on the kinetics is supported by Tafel data of Chopart et al.17 and the magneto-impedance experiment of Devos et al.21 From the Tafel plots, the cathodic transfer coefficient was found to be 0.45, which is consistent with a rate-determining step involving a one-electron transfer. The values for j0 and Rc are in good agreement with the values established in the classic paper by Mattsson and Bockris.45 However, the anodic Tafel slope, which is inversely proportional to the transfer coefficient, was found to be exactly half that of the cathodic branch, rather than one-third as observed by Mattsson and Bockris, yielding an anodic transfer coefficient, Ra ) 0.9, compared to 1.5 reported previously.45 The reason for this difference is not clear. In any case, the behavior is unchanged in the presence of the field. From the value of the cathodic transfer coefficient, we can conclude that during electrodeposition the electron-transfer proceeds in two steps: slow

+ Cu2+ aq + e 98 Cuaq fast

Cu+ aq + e 98 Cu

The first step is rate-determining. It is possible in principle that a magnetic field could affect the rate of such a reaction. There have been many studies of magnetic field effects on the rates of chemical reactions, but to date the only well-established effects have been those on the recombination of radical pairs.46,47 Applied magnetic fields either enhance or impede the coherent oscillation of the radical pair between its singlet and triplet states, thereby modifying the rate of recombination. However, it is unlikely that a magnetic field could influence the rate of a heterogeneous reaction, since the magnetic energy is negligible compared to both the thermal energy and the electrostatic potential energy. 4.4. Electrochemical Equilibria. The fact that no field effect is observed on the rest potential is unsurprising. The rest potential is determined by the chemical equilibrium between aqueous Cu2+ and metallic Cu and is therefore entirely governed by thermodynamic parameters. Even in a field of 6 T, the magnetic energy in a paramagnetic system is orders of magni-

J. Phys. Chem. B, Vol. 105, No. 39, 2001 9499 tude lower than the activation energy or the thermal energy. The ratio of the magnetic energy, EM, to the thermal energy, ET, is given by

EM χmB2 ) ET 2µ0RT

(22)

For Cu2+ at room temperature, χm ∼ 2 × 10-8 m3/mol, and this ratio is 3 × 10-6 in a field of 1 T at room temperature. For a ferromagnetic material, however, the magnetic energy is BMVm, where M is the magnetization of the material and Vm is the molar volume (∼10-5 m3), which does become only an order of magnitude smaller than the thermal energy in high fields (B ∼ 10 T). Application of a magnetic field to a ferromagnetic system causes the chemical equilibrium to shift to the side which has the greater magnetization, to minimize the magnetic energy. Such an effect was reported by Yamamoto et al.48 on the equilibrium between the ferromagnetic hydride LaCo5Hx and hydrogen, in fields in the range 2-14 T. A general thermodynamic theory of magnetic field effects on chemical equilibria has been published by Yamaguchi et al.49 Waskaas and Kharkats30 have reported shifts of the order of tens of millivolts in the rest potentials of Fe, Ni, and Co electrodes in paramagnetic solutions of their ions, in fields of up to 0.8 T. These effects are puzzlingly large, since the energy shifts due to the ferromagnetic magnetization in 0.8 T are 0.11, 0.03, and 0.08 mV, respectively. In any case, no such effect is seen with copper. 4.5. Fractals. Mass transport is the rate-limiting factor during the growth of the fractal electrodeposits. Pattern formation in these deposits is very sensitive to the growth conditions,50 so they offer a sensitive test of the influence of the magnetic field. The fractal dimensionality of zinc deposits has been shown to increase with both electrolyte concentration and voltage as their form changes from diffusion-limited aggregation (DLA) to dendritic.51,52 Most of the deposits we observe are characterized as intermediate between DLA and dendritic in form. The broad increase in fractal dimensionality, df, as a function of B, when the field is perpendicular to the plane of the horizontal cell, is related to the more compact spirals formed as the field is increased. The increasingly stringy nature of the deposits grown when the field is parallel to the plane of this cell is reflected in the sharp decrease in df with field for this orientation. The deposits grown in the horizontal cell with a vertical magnetic field are similar to those previously reported by Mogi et al.23,24 The results with the vertical cell establish unambiguously that convection of the solvent near the cathode is important for mass transport. The competition between the Lorentz force and natural convection has a direct bearing on the pattern formation, such as the change from unidirectional to branched when the cell is vertical and the field is applied perpendicular to the cell. (Figure 17b). Evidence that the magnetic body force is significant is also seen in the unidirectional nature of the stringy deposits in the horizontal cell whenever B is applied in-plane, and particularly the fact that these strings tend to grow in the sense where the gravitational force and the Lorentz force add (Figure 16d). The relevant quantity here is the dimensionless ratio:

Rg )

jB ∆Fg

(23)

where ∆F is the density difference between the copper-rich and copper-depleted electrolyte which normally drives convection. Rg will be 1 when the Lorentz and gravitational body forces are equal in magnitude. For the fractal electrodeposits, j ∼ 2 × 105 A/m2 at the beginning of the experiment and ∆F ∼ 100


Hinds et al.

TABLE 3: Typical Forces Acting in Aqueous Copper Sulfate Solutiona force

expression

typical value (N/m3)

driving force for diffusion (F BD) driving force for migration driving force for forced convection driving force for natural convection viscous drag paramagnetic force (F BP) field gradient force (F BB) Lorentz force (F BL) electrokinetic force (F BE) magnetic damping force

RT∇ Bc zFc∇ BV F(rω)2/2δ0 ∆Fg b Fν∇2V χmB2∇ B c/2µ0 χmcB∇ B B/µ0 B× j B B σdB E|/δ0 σV b× B B×B B

1010 1010 105 103 102 104 101 103 103 101

a T )298 K, c ) 103 mol/m3, δ ) 10-4 m, z ) 2, V ) 1 V, F ) 103 kg/m3, d ) 10-2 m, ω ) 102 rad/s, δ0 ) 10-3 m, ∆F ) 102 kg/m3, ν ) 10-6 m2/s, V ) 10-1 m/s, B ) 1 T, χm ) 10-8 m3/mol, ∇B ) 1 T/m, j ) 103 A/m2, σd ) 10-1 C/m2, E| ) 10 V/m, σ ) 102 (Ω m)-1.

kg/m3, so Rg ∼ 1 when B ) 5 mT. This argument indicates how relatively low fields can produce observable effects. As time progresses, the current flows to the growing ends of the branches, but it is difficult to determine j because the active area is unknown. Experimentally, the chiral behavior exhibited by the deposits shown in Figure 16b,c was observed in fields as low as 5 mT. The chirality can be modeled by superimposing a field-dependent probability of moving in a particular direction on random walkers on a lattice.34,35 Dimensional analysis may be used to estimate from the observed chirality the scale on which the forces responsible for the convective effects are acting. The driving force for convection is balanced, in equilibrium, by the viscous drag, B Fdrag ) 6πνrFV, where r is the size of the moving object. If the field is to produce observable chirality, the ratio

RB )

qB 6πνFr

(24)

of the transverse to the longitudinal forces acting on the moving objects is expected to be on the order of 1, which means that when B ) 1 T and q ) (4/3)π r3σ, where σ is the positive charge density in solution, r should be on the order of 10 µm. This may be interpreted as the size of an eddy or of a convection cell. This provides some support for an idea recently put forward by Aogaki53 that microscopic vortex motion on the scale of 10 µm drives the macroscopic convection in the solution. The most notable feature of the field effects on fractal growth is the marked dependence of the patterns formed by copper electrodeposits on the relative orientations of cell and magnetic field. This is in contrast to the independence of the magnetic field effects on field direction in the other electrochemical experiments. This difference must be attributed to the influence of the constrained geometry of the flat cell on the convection cells in the solution. Since the field effects scale with current density, they should be most pronounced at the growth points, such as those at the tips of fractal branches, or at rough electrode surfaces. 4.6. Summary. A summary of the main forces acting in copper sulfate solution is given in Table 3. The three dominant forces in the absence of an applied field are the driving forces for diffusion, migration, and forced convection (for the case of a rotating disk electrode). Their relative magnitudes require some qualification. First, the driving force quoted for diffusion applies only in the diffusion layer; in the bulk solution this force is negligible. Second, although the driving force for migration is

significantly larger than that for forced convection, its effect in terms of material flux is many orders of magnitude smaller than that suggested by the value in the table because the electric field is screened by the presence of the other charged species in solution, and the applied voltage is most effective only within the electrostatic double layer. In quiescent solution, the net rate of migration in the bulk solution is essentially controlled by the rate of diffusion in the diffusion layer. The migrational current flows through the bulk solution to compensate for the charge imbalance at each electrode, thus preserving electroneutrality. In stirred solutions, however, the flux of material occurs on a macroscopic scale, and convection is the dominant transport mechanism. The equivalence of the effect of the field with that produced by a rotating disk electrode was demonstrated experimentally (Figure 10b). The application of a field of 0.5 T produces the same enhancement in limiting current as that achieved with a rotation rate of 10 rpm. This experimental value may be compared with theory by equating the magnitudes of the two relevant driving forces in Table 3: the Lorentz force and the force driving convection at a rotating disk electrode. Using the hydrodynamic relation31

δ0 ) 10δ

(25)

combined with eq 2 yields the equivalence between the rotation rate and the applied field:

ω≈

(

)

20nFDcB Fr2

1/2

(26)

where r is the radius of the electrode. In the above experiment, r ) 3 mm, B ) 0.5 T, and c ) 10 mol/m3, yielding ω ∼ 1 rad/s (10 rpm). Equation 26 predicts that the equivalent rotation rate increases with increasing electrolyte concentration and decreasing electrode radius. It also depends on the diffusion coefficient of the ionic species and the number of electrons transferred in the electrode reaction. This explains the apparent discrepancy between the equivalent rotation rates reported in the literature. The data obtained at high concentrations by Aaboubi et al.7 are in good agreement with eq 26. Of the five magnetic forces listed in Table 3, we have already seen that the Lorentz force has a significant effect on the diffusion-limited current during an electrode process. In recent years, however, several papers have been published11,30 claiming that the paramagnetic gradient force (eq 12) is responsible for experimentally observed field effects. However, it is not evident how such a force could have any effect on the rate of transport of electroactive species. The paramagnetic force only becomes significant in the diffusion layer, where it arises from the gradient in paramagnetic susceptibility due to the concentration gradient of paramagnetic Cu2+ ions in this region. Since it acts in the same direction as the thermodynamic driving force for diffusion, it would have to be comparable in magnitude to this force to have any measurable effect. The ratio of these two forces is equal to the ratio of magnetic energy to thermal energy, given by eq 22, which we have shown is on the order of 10-6 at room temperature. Therefore, we expect this force to have a negligible effect on mass transport. As mentioned previously, the only circumstances in which it is envisaged that the magnetic energy might play a role is in the determination of chemical equilibria for ferromagnetic systems in high fields. The magnetic energy gradient does become significant in nonuniform fields. In the permanent magnet flux sources used in this work, the field gradient force is on the order of 10 N/m2



(1% of the Lorentz force). The interference of this force with the effect of the Lorentz force may explain the independence of the field effects on the orientation of the field. This force becomes dominant if a field gradient is deliberately applied (∇B . 1 T/m).16,54 This conclusion is supported by data from Mohanta and Fahidy,55 who reported that a comparable enhancement in limiting current is observed in nonuniform fields whose average value is one tenth of the uniform field strength required. Recent work by Olivier et al.43 has demonstrated that the effect of a magnetic field on the limiting current is equivalent to that produced by a tangential electric field close to the electrode surface. Such an electric field may be created by the use of a nonequipotential working electrode or by the application of a magnetic field parallel to the electrode surface. The electrokinetic stress, B SE, on the charge density, σd, in the diffuse double layer induces a tangential flow which is transmitted to the bulk solution via viscous forces. This force per unit area acts on the scale of a few nanometers from the electrode surface and may be compared to the other body forces in Table 3 by dividing by a characteristic length, x, and then scaling with the dimensions of the hydrodynamic boundary layer, δ0. Taking x ∼ 1 nm gives FE ∼ 109 N/m3, which when scaled by a factor x/δ0 yields FE ∼ 103 N/m3, a magnitude comparable to that of the Lorentz force. Both forces are similar in origin; the main difference between them is the scale on which they operate. The question remains as to whether the flow on a microscopic scale drives the macroscopic flow or vice versa. Since both forces are comparable in magnitude, it is likely that both play some role in the interplay between the two scales of flow, and that both must be considered in the quantitative analysis of field induced mass transport enhancement. The magnetic damping force listed in Table 3 is negligible in aqueous solutions, where the conductivity is relatively low, σ ∼ 102 (Ωm)-1. In conducting melts such as those used in metal and semiconductor processing, σ ∼ 106 (Ωm)-1, and this force becomes significant. The extensive use of static magnetic fields to control convection during growth of crystals such as silicon from conducting melts56 depends on this damping force. The argument is that longitudinal flow in the direction of B is unimpeded, but transverse flow with velocity V is damped because it is opposed by a force:57

B F ) σV b×B B×B B

(27)

where b V × B B is the nonelectrostatic field due to the applied magnetic field. In such conducting melts, the magnetic damping force is on the order of 105 N/m3 in a field of 1 T. Given the asymmetric nature of both the Lorentz force and the electrokinetic force, our observation that the diffusion-limited current is practically independent of the orientation of the applied magnetic field is somewhat surprising. Significant effects of field orientation have been reported at microelectrodes,10 in systems where both reactant and product species are in solution,6 and in chronoamperometric measurements on copper at low current density.58 We found no marked effect of field orientation for copper using conventional electrodes. The rotational flow observed in the cell when the field is applied perpendicular to the electrode is evidence that convection is induced in that configuration. It appears that there is always a component of current flowing perpendicular to the field, regardless of its orientation. This can be explained if the electric field is not everywhere perpendicular to the electrode surface, either because of microscopic roughness or because of the electrode geometry in the cell. Another possibility is that a current may be set up

in the direction of any magnetic field gradient that may be present, as a result of FB (eq 13). Referring to Table 3, at low fields and at low concentrations, there is a delicate balance between the magnetic body forces (FL and FE) and the gravitational forces driving natural convection. Some detailed hydrodynamic analysis based on real macroscopic and microscopic flow patterns would be needed to account numerically for the enhancement of mass transport in a magnetic field in a cell with a particular electrode geometry. 5. Conclusions The magnetic field effects on copper electrodeposition and electrodissolution are explained by field-induced convection. The effects depend on cell and electrode dimensions. The tangential fluid velocity induced by the field results in the formation of a hydrodynamic boundary layer at the electrode surface, which decreases the diffusion layer thickness and thus increases the rate of mass transport. This is manifested experimentally in the higher currents observed in the field when the electrode process is under diffusion control. The effect of the field on the morphology of the deposited copper may then be directly related to the modified mass transport rate. It is still unclear, however, where the induced convection originates. Equivalent effects would be observed whether convection was driven by the action of the Lorentz force in the bulk solution or on a nanometer scale in the diffuse layer due to the electrokinetic force. The magnitudes of the magnetic body forces involved (FL, FE, FB) may be comparable, depending on the experimental conditions. It is therefore likely that all three play some role in inducing convection. The paramagnetic gradient force is not expected to exert any significant influence on mass transport as it is negligible compared to the driving force for diffusion. No effects on electrode kinetics or electrochemical equilibria are observed. Acknowledgment. This work was supported by Enterprise Ireland under Contracts ST/1996/771 and ST/1999/181. J.M.D.C. is grateful for a Fullbright Scholarship. References and Notes (1) Fahidy, T. Z. J. Appl. Electrochem. 1983, 13, 553. (2) Tacken, R. A.; Janssen, L. J. J. J. Appl. Electrochem. 1995, 25, 1. (3) Mohanta, S.; Fahidy, T. Z. Can. J. Chem. Eng. 1972, 50, 248. (4) Aogaki, R.; Fueki, K.; Mukaibo, T. Denki Kagaku 1975, 43, 504; 509. (5) Kelly, E. J. J. Electrochem. Soc. 1977, 124, 987. (6) Iwakura, C.; Edamoto, T.; Tamura, H. Denki Kagaku 1984, 52, 596; 654. (7) Aaboubi, O.; Chopart, J.-P.; Douglade, J.; Olivier, A.; Gabrielli, C.; Tribollet, B. J. Electrochem. Soc. 1990, 137, 1796. (8) Mori, S.; Satoh, K.; Tanimoto, A. Electrochim. Acta 1994, 39, 2789. (9) Kim, K.; Fahidy, T. Z. J. Electrochem. Soc. 1995, 142, 4196. (10) Lee, J.; Ragsdale, S. R.; Gao, X.; White, H. S. J. Electroanal. Chem. 1997, 422, 169. (11) O’Brien, R. N.; Santhanam, K. S. V. J. Appl. Electrochem. 1997, 27, 573. (12) Noninski, V. Electrochim. Acta 1997, 42, 251. (13) Ragsdale, S. R.; Grant, K. M.; White, H. S. J. Am. Chem. Soc. 1998, 120, 13461. (14) Leventis, N.; Chen, M.; Gao, X.; Canalas, M.; Zhang, P. J. Phys. Chem. B 1998, 102, 3512. (15) Shinohara, K.; Aogaki, R. Chem. Lett. 1998, 1223. (16) Grant, K. M.; Hemmert, J. W.; White, H. S. Electrochem. Commun. 1999, 1, 319. (17) Chopart, J.; Douglade, J.; Fricoteaux, P.; Olivier, A. Electrochim. Acta 1991, 36, 459. (18) Aogaki, R.; Negishi, T.; Yamato, M.; Ito, E.; Mogi, I. Physica B 1994, 201, 611. (19) Lee, C.-C.; Chou, T.-C. Electrochim. Acta 1995, 40, 965. (20) Yonemura, H.; Ohishi, K.; Matsuo, T. Chem. Lett. 1996, 661.

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Magnetic Field Effects on Copper Electrolysis - The Journal of Physical

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