preprint, finally published in Electrochemistry Communications 9 (2007), 2479–2483
Confinement of paramagnetic ions under magnetic field influence:
Lorentz- versus concentration gradient force based explanations
Tom Weier1 , Kerstin Eckert2 , Sascha Mühlenhoff2 , Christian Cierpka1 ,
Andreas Bund3 , Margitta Uhlemann4
November 19, 2007
1
Forschungszentrum Dresden–Rossendorf, P.O. Box 51 01 19, 01314 Dresden, Germany
Institute of Aerospace Engineering, Technische Universität Dresden, 01062 Dresden, Germany
3
Institute of Physical Chemistry and Electrochemistry, Technische Universität Dresden, 01062 Dresden, Germany
4
IFW Dresden, P.O.Box 27 01 16, 01171 Dresden, Germany
2
Abstract
Concentration variations observed at circular electrodes with their axis parallel to a magnetic and normal to
the gravitational field have previously been attributed elsewhere to the concentration gradient force only. The
present paper aims to show that Lorentz force driven convection is a more likely explanation.
mass transfer at the electrode since it is generally not confined to its origin only. This is even more so in small cells
which have to be used in the narrow gaps of electromagnets.
Unfortunately, in the vast majority of papers on magnetic field effects on electrochemical reactions, convection
is either assumed or neglected a priori but rarely directly
observed. Flow visualizations are a valuable step towards
flow measurement, but might be misleading if not interpreted with care, as shown by e.g. [11]. However, with
the advent of Particle Image Velocimetry (PIV) an easily
applicable measurement technique for flow fields is readily
available. Interferometry has been used by O’Brien and
coworkers, e.g. [2, 12], to measure concentration fields
in electrochemical cells under magnetic field influence. If
one tries to dodge the considerable experimental effort
related with interferometry, synthetic schlieren [13], i.e.
background oriented schlieren (BOS) [14], is one of several alternative solutions and especially attractive if a PIV
system is already available since it uses the same components and algorithms.
White and coworkers [15–18] have shown that Lorentz
forces are generated at disc microelectrodes (diameter
6 ≤ d ≤ 250 µm) whose axes are parallel to a magnetic
field. These azimuthal Lorentz forces are due to radial
current densities at the perimeter of the electrodes and
drive an azimuthal primary flow which in turn generates
secondary flows. Somewhat in contrast to these findings,
Leventis et al. [7, 9] claim absence of convection at millielectrodes (diameter 0.5 ≤ d ≤ 3 mm). Based on this
assumption, Leventis et al. [7, 9] attribute the indeed surprising pinning of buoyant boundary layers in presence of
~ to the sole action of F~∇c .
B
In magnetoelectrochemistry, i.e. in electrochemistry
inside a magnetic field, different forces of magnetic origin
and their relative importance are actively debated. An
overview of the forces under discussion can be found in
[1]. Recently, the so called “concentration gradient force”
or “paramagnetic gradient force” [1]
χm B 2 ~
∇c
F~∇c =
2µ0
(1)
has attracted a great deal of attention, see, e.g. [2–10]. In
Eq. (1) χm denotes the molar susceptibility, B the magni~ µ0 the vacuum permetude of the magnetic induction B,
ability and c the concentration of the electroactive species,
respectively. F~∇c is considered in the literature as a true
body force. Main arguments in favor of the existence of
this force result from unexpected low deposition rates, e.g.
~ [5].
of cobalt ions, in presence of B
A more established and generally accepted force is the
Lorentz force
~
F~L = ~j × B
(2)
~ the magnetic inducwhere ~j is the current density and B
tion. Changes in mass transfer, i.e. the limiting current
density, are usually attributed to convection generated or
influenced by the Lorentz force. Sometimes, this is called
the “MHD effect”, where MHD is an acronym for magnetohydrodynamics. Especially in cases, where the magnetic
induction is normal to the working electrode (WE) surface, Lorentz forces are often assumed to be absent. Even
if this might be true in the direct vicinity of the electrode,
Lorentz forces can originate anywhere in the cell, when
electric and magnetic fields are not strictly parallel. Convection arising from those Lorentz forces will influence the
1
The present communication aims to identify whether mands enforce the use of a rectangular cell.
a possible F~∇c or F~L are responsible for the confinement Fig. 2 displays the Lorentz force distribution in
~ For this purpose
WE
of paramagnetic ions in presence of B.
F /AU
0.029
we demonstrate the Lorentz force generated convection at
0.027
0.025
millielectrodes via PIV while the corresponding concentra0.023
F
0.021
tion variations are visualized by BOS and interferometry.
0.019
On analyzing key situations such as deposition, dissolu0.017
0.015
tion and the behavior after switching back to open circuit
0.013
B
0.011
potential, we could not find any need to include F~∇c to
0.009
E
CE
0.007
consistently explain our observations.
0.005
0.003
For a straightforward demonstration of Lorentz a)
0.001
force generated convection at millielectrodes, we use
CCu
CCu
a petri dish geometry of axial symmetry which alcopper
copper
Fc
deposition
lows for an undisturbed development of the primary azdissolution
Fc
WE
WE
imuthal flow. Fig. 1 shows the flow at the electrolyte b)
L
L
2+
2+
∆
∆
WE
60
FL /AU
8
5/mms-1
4
50
z /mm
2
0 r /mm 2
40
CE
4
30
Figure 2: Lorentz force distribution due to a homogeneous
magnetic field and radial currents near the WE (a). Direction of a possible F~∇c at the WE for copper dissolution
or deposition (b).
the midplane of a rectangular cell calculated from a
homogeneous magnetic field parallel to the WE axis and
the primary current density. Though the exact shape of
the Lorentz force density distribution further away from
the WE differs slightly from what is shown in Fig. 1, the
main feature, i.e. the Lorentz force concentration around
the rim of the WE is preserved. It is mainly determined
by the radial current densities there. In this small volume,
even the magnetic field of the permanent magnet used in
the petri dish experiment is comparatively homogeneous.
To discriminate between the impact of a possible F~∇c ,
Eq. (1), and F~L , Eq. (2), on the confinement of the buoyant diffusion boundary layer to the millielectrode we examine several exemplary cases, including deposition and
dissolution. We use aqueous solutions of CuSO4 and
H2 SO4 in connection with copper electrodes since the
Cu2+ ions display a noticeable paramagnetism (χm =
1.33 · 10−9 m3 /mol). For the copper deposition, the Cu2+
concentration increases with increasing distance from the
WE while the opposite takes place for the dissolution.
Since F~∇c would be directed towards the area of higher
magnetic susceptibility (see Fig. 2), oppositely oriented
F~∇c should occur in both cases as detailed below:
0.200
0.130
0.085
0.055
0.036
0.024
0.015
0.010
6
6
8
|u| /mms-1
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
20
10
Ø 1.5
WE
CE
y /mm
5
10
0 x /mm 10
Magnet
Ø 20
Ø 52
20
30
40
50
60
Figure 1: Lorentz force driven azimuthal flow on the surface of an electrolyte filled petri dish. Sketch of the assembly (bottom insert) and Lorentz force density distribution
in the meridional plane (top insert).
surface. It results from the magnetic induction due to
a permanent magnet, positioned with its south pole upwards below the petri dish, and the current density distribution between the WE just below the electrolyte surface
and the counter electrode (CE) on the bottom of Here
and in all of the experiments described in the following,
the WE is completely isolated at its perimeter. the petri
dish. Of course, such a simple experiment may provoke
criticism, especially regarding the role of the inhomogeneous magnetic field. However, since the main cause of
intense Lorentz forces is the radial current density distribution at the WE edge, the details of the magnetic field
distribution are not overly important. This can be seen
from the calculated Lorentz force density shown in the
top insert of Fig. 1 and will be further substantiated in
the following.
To visualize concentration distributions and
to measure the secondary flow,
optical de-
1. Cu2+ deposition: F~∇c would point away from the
electrode, F~L present (leading to a clockwise rotation).
2. Anodic dissolution of copper: F~∇c would be directed
towards the anode, F~L present (causing a counterclockwise 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
| F~L |= 0.
2
Since the primary flow is driven by the azimuthal
Lorentz force around the WE rim, Fig. 4 displays the
secondary flow only. Deposition and dissolution differ in
direction and magnitude of the primary azimuthal rotation but display the same topology of the secondary flow.
This sort of secondary flow is consistent with the flow visualizations given by Grant et al. [18]. One vortex torus,
represented in the two-dimensional section as a double
vortex, dominates the flow field. In the center of the vortex torus, situated between the WE and the CE, a jet
is formed which impinges on the WE. This impinging jet
generates a stagnation region in which the velocity is almost zero. Thus, the boundary layer can grow further
there in diffusive manner which is equivalent to an accumulation of depleted or enriched solution in case of deposition or dissolution, respectively.
Without a magnetic field, natural convection sets in
quickly and dominates the concentration field. For instance in the case of copper deposition, a plume formed
by solution from which Cu2+ ions have been removed,
rises up from the electrode due the density difference to
the surrounding fluid.
In sharp contrast to that is the behavior under a magnetic field applied parallel to the WE
shift / pixel
0
a)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
b)
1
The stagnation region is also clearly visible in the interferograms for both, the deposition, Fig. 5(a)–(b), and
the dissolution, Fig. 5(c)–(d). It is marked in Figs. 5(a)
and (c) by an arrow and characterized by a minimum
fringe deflection with respect to the imposed horizontal
position. Since the interferometer employed is of lateral
shearing type [19], the fringe deflections are proportional
to ∂c/∂y. They are more pronounced for the dissolution
because the current density is approx. 1.5 times higher,
but they are still visible for the deposition, too. Most
important, the spatial occurrence of the fringe deflection
has the shape of a truncated cone. Experiments with
several electrode types showed that the envelope of the
cone along which the fringe deflections occur is exactly
correlated with the position of the jet-like outflow of the
toroidal vortex in Fig. 4. Thus, a certain part of the Cu2+ ion deficit [excess] produced by deposition [dissolution] is
carried away with the meridional flow and becomes visible
along the cone.
2mm
Figure 3: BOS images of copper deposition at the WE in
a cubic cell (20×20×20 mm3 ). 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). The
contours are proportional to |∇c|. Cu2+ ion depleted solution accumulates now at the WE. Although this behavior resembles that observed by [7], it can hardly be
explained by the action of F~∇c . For the copper deposition, the latter would point away from the electrode. F~∇c
should therefore lead to an intensified removal of depleted
solution from the WE instead of an accumulation there.
Fig. 3(b) shows the gradients of the concentration distribution shortly after switching back to open circuit potential when the magnetic field is still present. The depleted
solution does not stick to the WE, but rises upwards folThe interferograms are captured inside a homogeneous
lowing a slightly curved path.
magnetic field of 0.6 T. The electrode geometry (cf. inset
Velocity measurements in the midplane of the cubic
in Fig. 5(c) ) differs by intention from that in Fig. 3. It is
cell, corresponding to Fig. 3(a), are depicted in Fig. 4.
a 1.0 mm Cu wire (thin white lines in Fig. 5) glued into
CE
a conically protruding PVC housing. This construction
serves two purposes: (i) It demonstrates that the slope
of the truncated cone is a function of the geometry behind the WE. Since this electrode geometry is more constrained, the cone becomes more shallow as compared to
WE
Fig. 4. (ii) An asymmetry in embedding the Cu wire in
the PVC–chassis causes an asymmetry of the cone.
With this fluid flow picture in mind we look once
again on the behavior after switching back to open circuit potential. In the interferograms for both dissolution
5/mms
2 mm
and deposition, Fig. 5, the boundary layer start immediately to fall down or to rise, respectively, in a plume
Figure 4: In–plane velocity components in the midplane like manner, see also the BOS visualization, Fig. 3(b).
of the cell (secondary flow).
If the current is switched off, the Lorentz force vanishes.
-1
3
a)
b)
c)
concentration boundary layer can grow in diffusive manner. A minor influence of F~∇c onto this effect cannot be
excluded up to now and is studied currently in a work in
progress.
d)
y
Acknowledgements
We are indebted to Adrian Lange, Gerd Mutschke, and
Gunter Gerbeth for many fruitful discussions. Financial
support from Deutsche Forschungsgemeinschaft (DFG) in
frame of the Collaborative Research Centre (SFB) 609 is
gratefully acknowledged.
References
[1] G. Hinds, J. Coey, M. Lyons: Electrochemistry Communications 3 (2001) 215.
[2] R. O’Brien, K. Santhanam: J. Appl. Electrochem. 27
(1997) 573.
[3] Y. Yang, K. M. Grant, H. S. White, S. Chen: Langmuir
19 (2003) 9446.
[4] K. Rabah, J.-P. Chopart, H. Schloerb, S. Saulnier,
O. Aaboubi, M. Uhlemann, D. Elmi, J. . Amblard:
J. Electroanal. Chem. 571 (2004) 85.
[5] M. Uhlemann, A. Krause, J. Chopart, A. Gebert: J. Electrochem. Soc. 152 (2005) C817.
[6] A. Bund, H. H. Kuehnlein: J. Phys. Chem. B 109 (2005)
19845.
[7] N. Leventis, A. Dass: J. Am. Chem. Soc. 127 (2005) 4988.
[8] J. Coey, F. Rhen, P. Dunne, S. McMurry: J. Solid State
Electrochem. 11 (2007) 711.
[9] N. Leventis, A. Dass, N. Chandrasekaran: J. Solid State
Electrochem. 11 (2007) 727.
[10] A. Krause, J. Koza, A. Ispas, M. Uhlemann, A. Gebert,
A. Bund: Electrochimica Acta (2007). In press.
[11] J. M. Cimbala, H. M. Nagib, A. Roshko: J. Fluid Mech.
190 (1988) 265.
[12] R. O’Brien, K. Santhanam: J. Electrochem. Soc. 129
(1982) 1266.
[13] G. H. S.B. Dalziel, B. Sutherland: G. Carlomagno,
I. Grant (eds.) 8th Int. Symp. on Flow Visualization. Sorrento, Italy (1998), 62.1–62.6.
[14] H. Richard, M. Raffel: Meas. Sci. Technol. 12 (2001) 1576.
[15] S. R. Ragsdale, J. Lee, X. Gao, H. S. White:
J. Phys. Chem. 100 (1996) 5913.
[16] S. R. Ragsdale, H. S. White: Anal. Chem. 71 (1999) 1923.
[17] K. M. Grant, J. W. Hemmert, H. S. White: J. Electroanal. Chem. 500 (2001) 95.
[18] K. M. Grant, J. W. Hemmert, H. S. White:
J. Am. Chem. Soc. 124 (2002) 462.
[19] W. Merzkirch: Flow Visualization. Academic Press, Orlando (1987).
Figure 5: Interferograms of the copper dissolution (a) and
the deposition (c) after 35 s (b) and (d) show the respective situation immediately after switching back to open
~ is homogeneous, of a magnitude of
circuit potential. B
0.6 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 rotational
motion for a while. This is the reason why the rising
and falling plumes take a curved path. Now one may
indeed wonder in what the action of F~∇c could consist
in. F~∇c , described by Eq. (1), is independent of any current. Therefore absence of the latter should not influence
the distribution of confined paramagnetic ions at the electrode if F~∇c would be the main reason for the accumulation. Consider once more the case of the copper dissolution. F~∇c is directed here towards the electrode. Thus it
should promote the confinement even more. However, no
effect going beyond that of the copper deposition with an
oppositely directed F~∇c is visible.
These findings are not restricted to the sort of mag~ or electrolyte. They occur for both very
netic field, B,
~ where already small | B
~ |∼
and moderate homogeneous B
0.3 T are sufficient. Furthermore, they are not specific
for the copper sulphate electrolyte selected. Moreover, if
repeating the experiments with the chemistry of [7] qualitatively the same phenomena are found.
To conclude, the experiments brought strong evidence
that the confinement of paramagnetic ions at circular electrodes is caused to a large extent by the Lorentz force
driven convection and not by the action of a possible concentration gradient force. Specific for this convection here
is the appearance of a stagnation point area into which the
4