Confinement of paramagnetic ions under magnetic field influence:

Lorentz- versus concentration gradient force based explanations

Tom Weier1, Kerstin Eckert2, Sascha Muhlenhoff2, Christian Cierpka1,

Andreas Bund3, Margitta Uhlemann4

November 19, 2007

1 Forschungszentrum Dresden–Rossendorf, P.O. Box 51 01 19, 01314 Dresden, Germany2 Institute of Aerospace Engineering, Technische Universitat Dresden, 01062 Dresden, Germany3 Institute of Physical Chemistry and Electrochemistry, Technische Universitat Dresden, 01062 Dresden, Germany4 IFW Dresden, P.O.Box 27 01 16, 01171 Dresden, Germany

Abstract

Concentration variations observed at circular electrodes with their axis parallel to a magnetic and normal tothe gravitational field have previously been attributed elsewhere to the concentration gradient force only. Thepresent paper aims to show that Lorentz force driven convection is a more likely explanation.

preprint, finally published in Electrochemistry Communications 9 (2007), 2479–2483

In magnetoelectrochemistry, i.e. in electrochemistryinside a magnetic field, different forces of magnetic originand their relative importance are actively debated. Anoverview of the forces under discussion can be found in[1]. Recently, the so called “concentration gradient force”or “paramagnetic gradient force” [1]

~F∇c =χmB2

2µ0

~∇c (1)

has attracted a great deal of attention, see, e.g. [2–10]. InEq. (1) χm denotes the molar susceptibility, B the magni-

tude of the magnetic induction ~B, µ0 the vacuum perme-ability and c the concentration of the electroactive species,respectively. ~F∇c is considered in the literature as a truebody force. Main arguments in favor of the existence ofthis force result from unexpected low deposition rates, e.g.of cobalt ions, in presence of ~B [5].

A more established and generally accepted force is theLorentz force

~FL = ~j × ~B (2)

where ~j is the current density and ~B the magnetic induc-tion. Changes in mass transfer, i.e. the limiting currentdensity, are usually attributed to convection generated orinfluenced by the Lorentz force. Sometimes, this is calledthe “MHD effect”, where MHD is an acronym for magne-tohydrodynamics. Especially in cases, where the magneticinduction is normal to the working electrode (WE) sur-face, Lorentz forces are often assumed to be absent. Evenif this might be true in the direct vicinity of the electrode,Lorentz forces can originate anywhere in the cell, whenelectric and magnetic fields are not strictly parallel. Con-vection arising from those Lorentz forces will influence the

mass transfer at the electrode since it is generally not con-fined to its origin only. This is even more so in small cellswhich have to be used in the narrow gaps of electromag-nets.

Unfortunately, in the vast majority of papers on mag-netic field effects on electrochemical reactions, convectionis either assumed or neglected a priori but rarely directlyobserved. Flow visualizations are a valuable step towardsflow measurement, but might be misleading if not inter-preted with care, as shown by e.g. [11]. However, withthe advent of Particle Image Velocimetry (PIV) an easilyapplicable measurement technique for flow fields is readilyavailable. Interferometry has been used by O’Brien andcoworkers, e.g. [2, 12], to measure concentration fieldsin electrochemical cells under magnetic field influence. Ifone tries to dodge the considerable experimental effortrelated with interferometry, synthetic schlieren [13], i.e.background oriented schlieren (BOS) [14], is one of sev-eral alternative solutions and especially attractive if a PIVsystem is already available since it uses the same compo-nents and algorithms.

White and coworkers [15–18] have shown that Lorentzforces are generated at disc microelectrodes (diameter6 ≤ d ≤ 250 µm) whose axes are parallel to a magneticfield. These azimuthal Lorentz forces are due to radialcurrent densities at the perimeter of the electrodes anddrive an azimuthal primary flow which in turn generatessecondary flows. Somewhat in contrast to these findings,Leventis et al. [7, 9] claim absence of convection at mil-lielectrodes (diameter 0.5 ≤ d ≤ 3mm). Based on thisassumption, Leventis et al. [7, 9] attribute the indeed sur-prising pinning of buoyant boundary layers in presence of~B to the sole action of ~F∇c.

1

The present communication aims to identify whethera possible ~F∇c or ~FL are responsible for the confinementof paramagnetic ions in presence of ~B. For this purposewe demonstrate the Lorentz force generated convection atmillielectrodes via PIV while the corresponding concentra-tion variations are visualized by BOS and interferometry.On analyzing key situations such as deposition, dissolu-tion and the behavior after switching back to open circuitpotential, we could not find any need to include ~F∇c toconsistently explain our observations.

For a straightforward demonstration of Lorentzforce generated convection at millielectrodes, we usea petri dish geometry of axial symmetry which al-lows for an undisturbed development of the primary az-imuthal flow. Fig. 1 shows the flow at the electrolyte

x /mm

y/m

m

0 10 20 30 40 50 60

10

20

30

40

50

60

|u| /mms-1

54.543.532.521.510.5

5/mms -1

r /mm

z/m

m

0 2 4 6 8

2

4

6

8FL /AU

0.2000.1300.0850.0550.0360.0240.0150.010

WE

CE

Magnet

CE

Ø 1.5 WE

5

10

Ø 20Ø 52

Figure 1: Lorentz force driven azimuthal flow on the sur-face of an electrolyte filled petri dish. Sketch of the assem-bly (bottom insert) and Lorentz force density distributionin the meridional plane (top insert).

surface. It results from the magnetic induction due toa permanent magnet, positioned with its south pole up-wards below the petri dish, and the current density distri-bution between the WE just below the electrolyte surfaceand the counter electrode (CE) on the bottom of Hereand in all of the experiments described in the following,the WE is completely isolated at its perimeter. the petridish. Of course, such a simple experiment may provokecriticism, especially regarding the role of the inhomoge-neous magnetic field. However, since the main cause ofintense Lorentz forces is the radial current density distri-bution at the WE edge, the details of the magnetic fielddistribution are not overly important. This can be seenfrom the calculated Lorentz force density shown in thetop insert of Fig. 1 and will be further substantiated inthe following.

To visualize concentration distributions andto measure the secondary flow, optical de-

mands enforce the use of a rectangular cell.Fig. 2 displays the Lorentz force distribution in

Fc

∆

CCu2+

Fc

∆

CCu2+

copper dissolution

copperdeposition

FL /AU

0.0290.0270.0250.0230.0210.0190.0170.0150.0130.0110.0090.0070.0050.0030.001

WE WEb)

B

WE

E

FL

CE

a)

Figure 2: Lorentz force distribution due to a homogeneousmagnetic field and radial currents near the WE (a). Di-

rection of a possible ~F∇c at the WE for copper dissolutionor deposition (b).

the midplane of a rectangular cell calculated from ahomogeneous magnetic field parallel to the WE axis andthe primary current density. Though the exact shape ofthe Lorentz force density distribution further away fromthe WE differs slightly from what is shown in Fig. 1, themain feature, i.e. the Lorentz force concentration aroundthe rim of the WE is preserved. It is mainly determinedby the radial current densities there. In this small volume,even the magnetic field of the permanent magnet used inthe petri dish experiment is comparatively homogeneous.

To discriminate between the impact of a possible ~F∇c,Eq. (1), and ~FL, Eq. (2), on the confinement of the buoy-ant diffusion boundary layer to the millielectrode we ex-amine several exemplary cases, including deposition anddissolution. We use aqueous solutions of CuSO4 andH2SO4 in connection with copper electrodes since theCu2+ ions display a noticeable paramagnetism (χm =1.33 · 10−9 m3/mol). For the copper deposition, the Cu2+

concentration increases with increasing distance from theWE while the opposite takes place for the dissolution.Since ~F∇c would be directed towards the area of highermagnetic susceptibility (see Fig. 2), oppositely oriented~F∇c should occur in both cases as detailed below:

1. Cu2+ deposition: ~F∇c would point away from theelectrode, ~FL present (leading to a clockwise rota-tion).

2. Anodic dissolution of copper: ~F∇c would be directedtowards the anode, ~FL present (causing a counter-clockwise rotation) and

3. Switching back to open circuit potential (in presence

of ~B): After the decay of the inertial forces, ~F∇c

would remain the only force of magnetic origin since| ~FL |= 0.

2

Without a magnetic field, natural convection sets inquickly and dominates the concentration field. For in-stance in the case of copper deposition, a plume formedby solution from which Cu2+ ions have been removed,rises up from the electrode due the density difference tothe surrounding fluid.

In sharp contrast to that is the behavior un-der a magnetic field applied parallel to the WE

shift / pixel

2mm

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

a) b)

Figure 3: BOS images of copper deposition at the WE ina cubic cell (20×20×20mm3). 44 s after the potential step(a) and 5 s after switching back to open circuit potential(b).

as shown by BOS measurements in Fig. 3(a). Thecontours are proportional to |∇c|. Cu2+ ion depleted so-lution accumulates now at the WE. Although this be-havior resembles that observed by [7], it can hardly be

explained by the action of ~F∇c. For the copper deposi-tion, the latter would point away from the electrode. ~F∇c

should therefore lead to an intensified removal of depletedsolution from the WE instead of an accumulation there.Fig. 3(b) shows the gradients of the concentration distri-bution shortly after switching back to open circuit poten-tial when the magnetic field is still present. The depletedsolution does not stick to the WE, but rises upwards fol-lowing a slightly curved path.

Velocity measurements in the midplane of the cubiccell, corresponding to Fig. 3(a), are depicted in Fig. 4.

5/mms -12 mm

WE

CE

Figure 4: In–plane velocity components in the midplaneof the cell (secondary flow).

Since the primary flow is driven by the azimuthalLorentz force around the WE rim, Fig. 4 displays thesecondary flow only. Deposition and dissolution differ indirection and magnitude of the primary azimuthal rota-tion but display the same topology of the secondary flow.This sort of secondary flow is consistent with the flow vi-sualizations given by Grant et al. [18]. One vortex torus,represented in the two-dimensional section as a doublevortex, dominates the flow field. In the center of the vor-tex torus, situated between the WE and the CE, a jetis formed which impinges on the WE. This impinging jetgenerates a stagnation region in which the velocity is al-most zero. Thus, the boundary layer can grow furtherthere in diffusive manner which is equivalent to an accu-mulation of depleted or enriched solution in case of depo-sition or dissolution, respectively.

The stagnation region is also clearly visible in the in-terferograms for both, the deposition, Fig. 5(a)–(b), andthe dissolution, Fig. 5(c)–(d). It is marked in Figs. 5(a)and (c) by an arrow and characterized by a minimumfringe deflection with respect to the imposed horizontalposition. Since the interferometer employed is of lateralshearing type [19], the fringe deflections are proportionalto ∂c/∂y. They are more pronounced for the dissolutionbecause the current density is approx. 1.5 times higher,but they are still visible for the deposition, too. Mostimportant, the spatial occurrence of the fringe deflectionhas the shape of a truncated cone. Experiments withseveral electrode types showed that the envelope of thecone along which the fringe deflections occur is exactlycorrelated with the position of the jet-like outflow of thetoroidal vortex in Fig. 4. Thus, a certain part of the Cu2+-ion deficit [excess] produced by deposition [dissolution] iscarried away with the meridional flow and becomes visiblealong the cone.

The interferograms are captured inside a homogeneousmagnetic field of 0.6 T. The electrode geometry (cf. insetin Fig. 5(c) ) differs by intention from that in Fig. 3. It isa 1.0 mm Cu wire (thin white lines in Fig. 5) glued intoa conically protruding PVC housing. This constructionserves two purposes: (i) It demonstrates that the slopeof the truncated cone is a function of the geometry be-hind the WE. Since this electrode geometry is more con-strained, the cone becomes more shallow as compared toFig. 4. (ii) An asymmetry in embedding the Cu wire inthe PVC–chassis causes an asymmetry of the cone.

With this fluid flow picture in mind we look onceagain on the behavior after switching back to open cir-cuit potential. In the interferograms for both dissolutionand deposition, Fig. 5, the boundary layer start imme-diately to fall down or to rise, respectively, in a plumelike manner, see also the BOS visualization, Fig. 3(b).If the current is switched off, the Lorentz force vanishes.

3

a) d)c)b)

y

Figure 5: Interferograms of the copper dissolution (a) andthe deposition (c) after 35 s (b) and (d) show the respec-tive situation immediately after switching back to opencircuit potential. ~B is homogeneous, of a magnitude of0.6 T and aligned in normal direction to the electrodes.The inset in (c) shows the entire electrode geometry.

Owing to its inertia, the fluid remains in rotationalmotion for a while. This is the reason why the risingand falling plumes take a curved path. Now one mayindeed wonder in what the action of ~F∇c could consistin. ~F∇c, described by Eq. (1), is independent of any cur-rent. Therefore absence of the latter should not influencethe distribution of confined paramagnetic ions at the elec-trode if ~F∇c would be the main reason for the accumula-tion. Consider once more the case of the copper dissolu-tion. ~F∇c is directed here towards the electrode. Thus itshould promote the confinement even more. However, noeffect going beyond that of the copper deposition with anoppositely directed ~F∇c is visible.

These findings are not restricted to the sort of mag-netic field, ~B, or electrolyte. They occur for both veryand moderate homogeneous ~B where already small | ~B |∼0.3T are sufficient. Furthermore, they are not specificfor the copper sulphate electrolyte selected. Moreover, ifrepeating the experiments with the chemistry of [7] qual-itatively the same phenomena are found.

To conclude, the experiments brought strong evidencethat the confinement of paramagnetic ions at circular elec-trodes is caused to a large extent by the Lorentz forcedriven convection and not by the action of a possible con-centration gradient force. Specific for this convection hereis the appearance of a stagnation point area into which the

concentration boundary layer can grow in diffusive man-ner. A minor influence of ~F∇c onto this effect cannot beexcluded up to now and is studied currently in a work inprogress.

Acknowledgements

We are indebted to Adrian Lange, Gerd Mutschke, andGunter Gerbeth for many fruitful discussions. Financialsupport from Deutsche Forschungsgemeinschaft (DFG) inframe of the Collaborative Research Centre (SFB) 609 isgratefully acknowledged.

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