Influence of magnetic forces on electrochemical mass transport
G. Hinds*a, J.M.D. Coeya and M.E.G. Lyonsb
a
b
Physics Department, Trinity College, Dublin 2, Ireland.
Chemistry Department, Trinity College, Dublin 2, Ireland.
* Tel: +353-1-6081858; Fax: +353-1-6711759; email:
[email protected]
Abstract
Enhancements of the order of 100 % in the mass transport limited current for electrodeposition have been observed in magnetic fields of order 1 T. The effect of the field is to induce convection in the solution and it is equivalent to rotating the electrode or stirring the solution. In this communication, a quantitative comparison is made of the magnitude of various body forces which have been proposed to account for the experimentally observed effects, with a view to identifying the likely source of the field enhancement. When the magnetic field is uniform, the Lorentz force and the electrokinetic force both contribute significantly to the field enhancement.
Keywords
Magnetic field effects, electrodeposition, convection, body forces.
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1. Introduction
Magnetic field effects on electrochemical reactions may be divided into three categories - those relating to mass transport, electrode kinetics and deposit morphology. Notable reviews of the field are those by Fahidy [1] and Tacken and Janssen [2]. The effect of a magnetic field on mass transport is well established. The field acts to induce convection in the solution thus increasing the diffusion-limited current [3] [4]. The effect of the field on the heterogeneous electron transfer kinetics is more controversial. Some authors report that the field has no influence [5], whereas others explain modifications in the exchange current density in a field in terms of transitions between magnetic quantum states in the ions [6]. In recent years, clear field effects on the morphology and texture of electrodeposits have been reported under various conditions [7-10].
We focus here on the effect of the field on mass transport, which has been most extensively studied. Dramatic modifications in the rate of transport may be observed in the presence of the field. For example, the limiting current density at a copper electrode in 0.75 M CuSO4 (pH = 0.5) increases by a factor of four in a field of 6 T (T) (Fig. 1) [10]. The magnitude of the current enhancement depends on electrolyte concentration, solution viscosity and on the nature and concentration of the supporting electrolyte. The field somehow induces convection in the solution and is qualitatively equivalent to gentle stirring, such as that produced by a rotating disk electrode at about 100 rpm. Attempts have been made to model the field induced convection using the relevant hydrodynamic equations:
2
dv P v v 2 v Fmag dt
(1) c v c D 2 c t
(2)
In principle, the theoretical determination of the field dependence of the limiting current begins with the Navier-Stokes equation (Eq. (1)), which expresses the kinematics for an incompressible fluid element. The left hand side of Eq. (1) is the acceleration of the fluid element. The right hand side is the sum of the external body
forces on the element, with terms due to the pressure gradient, P , viscosity, , when
one part of the fluid moves relative to another and the magnetic body force, Fmag , responsible for the induced convection. Solution of Eq. (1) yields the velocity profile,
v r , in the fluid, which may then be substituted into the convective-diffusion
equation (Eq. (2)) to obtain the concentration gradient, c , close to the electrode and hence the limiting current. In general however, these differential equations are nonlinear, which means that complete analytical solutions do not exist and numerical methods are required. Here, a comparison is made of the order of magnitude of the
various magnetic body forces, Fmag , acting in typical experimental conditions in order to determine the most likely sources of magnetic field induced convection.
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2. Discussion
A summary of the typical body forces acting in aqueous electrolytes is shown in Table 1. The four dominant forces in the absence of an applied field are the driving forces for diffusion, electromigration and convection, both natural and forced (for the case of a rotating disk electrode). The relative magnitudes quoted for typical conditions need some qualification. Firstly, the quoted driving force for diffusion applies only within the diffusion layer; in the bulk solution this force is negligible. Secondly, although the driving force for electromigration greatly exceeds that for forced convection, its effect in terms of material flux is limited because the electric field is screened by the presence of the other charged species in solution. In quiescent solution, the net rate of electromigration 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.
There are five possible forces which could be responsible for the observed magnetic field effects. Two of these depend on the magnetic properties of the electrolyte and the others are related to the movement of charged particles. The energy, Emag, of the electrolyte in the magnetic field is -(1/2)MB (J/m3) where M (A/m) is the magnetization induced by the field B (T). Now M = mcB/0, where m is the molar susceptibility, hence:
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E mag
1 m cB 2 2 0
(3)
The related force Fmag E mag includes two terms:
m B 2 c m cBB Fmag 2 0 0
(4)
The first term is the paramagnetic gradient force, FP :
m B 2 c FP 2 0
(5)
which arises from the variation in the paramagnetic susceptibility of the diffusion layer due to the concentration gradient of the cations there. The second term is the
field gradient force, FB :
m cBB FB 0 (6)
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which is the force due to the field gradient, B , in the solution when the field is non uniform. FP and FB are thus driving forces due to a magnetic energy gradient. Two other forces - the Lorentz force and the electrokinetic force - are due to the interaction
of the magnetic field with the electric current. The Lorentz force, FL :
FL j B
(7)
arises from the motion of charge across lines of magnetic flux, while the electrokinetic shear stress, S E :
S E d E || (8)
is the corresponding force on the charge carriers in the diffuse double layer under the
influence of a nonelectrostatic field, E || , parallel to the electrode surface. This effective electric field represents the interaction of charge moving across the double layer with a magnetic field applied parallel to the electrode surface. Here d is the charge density (C/m2) in the diffuse layer. The last force in Table 1 is related to the conductivity, , of the electrolyte. 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 [11]:
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FM v B B
(9)
where v B is the nonelectrostatic field due to the applied magnetic field. All five forces are body forces with units N/m3.
When discussing the five magnetic forces listed in Table 1, it is necessary to consider not only their magnitudes but the direction in which they act. It is well established that the Lorentz force (Eq. (7)) has a significant effect on the diffusion limited current during an electrode process. More recently, it has been claimed [12,13] that the paramagnetic gradient force (Eq. (5)) is responsible for field effects in paramagnetic solutions. However, it is unlikely that such a force could have any effect on the global rate of transport of electroactive species. The paramagnetic force only becomes significant in the diffusion layer. It arises from the gradient in magnetic susceptibility due to the concentration gradient of paramagnetic ions such as Cu2+ in this region.
Furthermore, since it acts in the direction of c , parallel to the thermodynamic driving force for diffusion, it would have to be comparable in magnitude to this force to produce any observable effect. The ratio of these two forces is of order 10-6 at room temperature. Hence we expect the effect of this force on mass transport to be negligible.
The field gradient force may become significant in non-uniform magnetic fields, whether on the scale of the cell or on a microscopic scale at the surface of ferromagnetic electrodes. Under typical experimental conditions, the field gradient
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force (Eq. (6)) is of order 10 N/m3 (1% of the Lorentz force). This force can become dominant if a field gradient is imposed deliberately (B >> 1 T/m) [14,15]. This conclusion is borne out by data from Mohanta and Fahidy [16], who reported that a comparable enhancement in limiting current is observed in non-uniform fields whose average value is one tenth of the uniform field strength required.
Recent work by Olivier et al [17] 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 non-equipotential working electrode or by the application of a magnetic field parallel to the electrode surface. The electrokinetic stress, S E , on the charge density, d, 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 1 by dividing by a characteristic length, x, and then scaling with the dimensions of the hydrodynamic boundary layer, 0. Taking x ~ 1 nm gives FE ~ 109 N/m3, which when scaled by a factor x/0 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.
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The magnetic damping force listed in Table 1 is negligible in aqueous solution, where the conductivity is relatively low ( ~ 102 (m)-1). In conducting melts such as those used in metal and semiconductor processing, ~ 106 (m)-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 melts [18] depends on this damping force. In such conditions, the magnetic damping force is of order 105 N/m3 in a field of 1 T.
3. Conclusion
Three of the magnetic body forces ( FL , FE , FB ) may be comparable in magnitude, depending on the experimental conditions. FB is absent in a uniform magnetic field, but both FL and FE always play some role in inducing convection. Despite the fact that it is larger than any of the other magnetic forces, the paramagnetic gradient force
( FP ) cannot exert any significant influence on mass transport since it is negligible compared to the driving force for diffusion ( FD ), and acts in the same direction. The
magnetic damping force ( FM ) is negligible due to the relatively low conductivity of aqueous solutions.
Acknowledgements
This work was supported by Enterprise Ireland under contract ST/1999/181. We are grateful to Professor Alain Olivier for some helpful discussions.
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Table 1 Typical forces acting in aqueous electrolytes
Expression RTc zFcV
Typical value (N/m3)
(r)2/20 g
105
2v m B2 c / 2 0 m cBB / 0 jB d E || / 0 v B B
102
Force Driving force for diffusion ( FD )
Driving force for electromigration Driving force for forced convection Driving force for natural convection Viscous drag
Paramagnetic force ( FP ) Field gradient force ( FB ) Lorentz force ( FL ) Electrokinetic force ( FE ) Magnetic damping force ( FM )
1010 1010
103
104 101 103 103 101
(T =298 K, c = 103 mol/m3, = 10-4 m, z = 2, V = 1 V, = 103 kg/m3, d = 10-2 m, = 102 rad/s, 0 = 10-3 m, = 102 kg/m3, , = 10-3 Ns/m2, v = 10-1 m/s, B = 1 T, m = 10-8 m3/mol, B = 1 T/m, j = 103 A/m2, d = 10-1 C/m2, E|| = 10 V/m, = 102 (m)-1)
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Figure 1
Effect of magnetic field on the cathodic () and anodic (o) limiting current for a copper electrode in 0.75 M CuSO4 (pH = 0.5) [10].
6000
5000
2
| j | (A/m )
4000
3000
2000
1000
0 0
1
2
3
B (T)
13
4
5
6
