Magnetic resonance imaging of the manipulation of a chemical wave using an inhomogeneous magnetic field

Article abstract of DOI:10.1021/ja0608287

The effects of applied magnetic fields on the traveling wave formed by the reaction of (ethylenediaminetetraacetato)cobalt(II) (Co(II)EDTA^2-) and hydrogen peroxide have been studied using magnetic resonance imaging (MRI). It was found that the wave could be manipulated by applying pulsed magnetic field gradients to a sample contained in a vertical cylindrical tube in the 7.0 T magnetic field of the spectrometer. Transverse field gradients decelerated the propagation of the wave down the high-field side of the tube and accelerated it down the low-field side. This control of the wave propagation eventually promoted the formation of a finger on the low-field side of the tube and allowed the wave to be maneuvered within the sample tube. The origin of these effects is rationalized by considering the Maxwell stress arising from the combined homogeneous and inhomogeneous magnetic fields and the magnetic susceptibility gradient across the wave front.

Full text of DOI:10.1021/ja0608287

Published on Web 05/16/2006
Magnetic Resonance Imaging of the Manipulation of a
Chemical Wave Using an Inhomogeneous Magnetic Field
Robert Evans,†,‡ Christiane R. Timmel,†,‡ P. J. Hore,‡ and Melanie M. Britton*,§,
Contribution from the Department of Chemistry, UniVersity of Oxford, Inorganic Chemistry
Laboratory, South Parks Road, Oxford OX1 3QR, U.K., Department of Chemistry,
UniVersity of Oxford, Physical and Theoretical Chemistry Laboratory, South Parks Road,
Oxford OX1 3QZ, U.K., Magnetic Resonance Research Centre, Department of
Chemical Engineering, UniVersity of Cambridge, New Museums Site, Pembroke Street,
Cambridge CB3 2RA, U.K., and School of Chemistry, UniVersity of Birmingham, Edgbaston,
Birmingham B15 2TT, U.K.
Received February 3, 2006; Revised Manuscript Received April 6, 2006; E-mail: m.m.britton@bham.ac.uk
Abstract: The effects of applied magnetic fields on the traveling wave formed by the reaction of
(ethylenediaminetetraacetato)cobalt(II) (Co(II)EDTA2-) and hydrogen peroxide have been studied using
magnetic resonance imaging (MRI) . It was found that the wave could be manipulated by applying pulsed
magnetic field gradients to a sample contained in a vertical cylindrical tube in the 7.0 T magnetic field of
the spectrometer. Transverse field gradients decelerated the propagation of the wave down the high-field
side of the tube and accelerated it down the low-field side. This control of the wave propagation eventually
promoted the formation of a finger on the low-field side of the tube and allowed the wave to be maneuvered
within the sample tube. The origin of these effects is rationalized by considering the Maxwell stress arising
from the combined homogeneous and inhomogeneous magnetic fields and the magnetic susceptibility
gradient across the wave front.
1. Introduction
that enables the wave to be observed using magnetic resonance
imaging (MRI).3 The NMR relaxation times of protons in water
molecules surrounding the paramagnetic Co(II) ions are sig-
nificantly shorter than for those surrounding diamagnetic Co-
(III) ions, and this difference in relaxation times produces the
image contrast necessary to visualize the traveling wave.
The transition from paramagnetic to diamagnetic cobalt
produces a magnetic susceptibility gradient across the wave
front. This, in conjunction with the associated concentration
gradient, is responsible for the magnetic field effects observed
for this wave in the presence of an inhomogeneous magnetic
field.3,4 In experiments conducted in a vertical tube, a descending
wave produced chemical fingers.3 Once a finger had formed,
the velocity of the wave could be manipulated by applying a
magnetic field gradient vertically along the length of the tube.
For negative gradients, corresponding to a magnetic field larger
at the top of the tube than at the bottom, an acceleration of the
propagating finger was measured. Conversely, a slight decrease
in the velocity of the wave was measured for positive gradients.3
The origin of these observations was discussed in terms of the
action of the Maxwell force, FM ) (V/µ0)∆øV(B‚∇)B on a drop
of one solution surrounded by another where ∆øV is the
difference in volume magnetic susceptibilities of the two
solutions, µ0 is the vacuum permeability, V is the volume of
the drop, and B is the applied magnetic field vector. Therefore,
Traveling waves have been observed in a variety of chemical
reactions and result from a coupling between autocatalysis and
diffusion.1 Waves formed in these chemical systems provide
an excellent opportunity to model biological wave formation,
such as in chemotaxis and calcium waves. One such wave-
forming reaction occurs between (ethylenediaminetetraacetato)-
cobalt(II) (Co(II)EDTA2-) and hydrogen peroxide, at pH 4.2
Hydroxide ions autocatalyze this reaction, in which cobalt(II)
is oxidized to cobalt(III) according to:
2Co(II)EDTA2- + H2O2 f 2Co(III)EDTA- + 2OH- (1)
A chemical wave is produced by introducing hydroxide ions in
a localized region of the reactive Co(II)EDTA2-/H2O2 solution.
Propagation of the wave occurs as the autocatalyst diffuses to
neighboring regions. During the reaction, paramagnetic Co(II)
ions are oxidized to diamagnetic Co(III) ions, and it is this
difference, in the magnetic properties of reactants and products,
‡ Department of Chemistry, University of Oxford, Physical and Theoreti-
cal Chemistry Laboratory.
§ Magnetic Resonance Research Centre, Department of Chemical
Engineering, University of Cambridge.
School of Chemistry, University of Birmingham.
† Department of Chemistry, University of Oxford, Inorganic Chemistry
Laboratory.
(1) Epstein, I. R.; Pojman, J. A. An Introduction to Nonlinear Chemical
Dynamics; Oxford University Press: Oxford, 1998.
(2) Yalman, R. G. J. Phys. Chem. 1961, 65, 556.
(3) Evans, R.; Timmel, C. R.; Hore, P. J.; Britton, M. M. Chem. Phys. Lett.
2004, 397, 67.
(4) Boga, E.; Ka´da´r, S.; Peintler, G.; Nagypa´l, I. Nature 1990, 347, 749.
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Evans et al.
the Maxwell force per unit volume for a drop of Co(III)EDTA-
solution surrounded by Co(II)EDTA2- solution is 2.1 N m-3
,
for which ∆øV ) 0.184 × 10-5 3
.
In our earlier experiments,3 the sample was subjected to a
longitudinal magnetic field gradient superimposed on the
homogeneous magnetic field B0 produced by the vertical
superconducting magnet of the MRI spectrometer oriented along
the axis (z) of the magnet bore, i.e., Bz ) B0 + zGz, where
B0 ) 7.0 T and Gz ) ∂Bz/∂z ) (0.2 T m-1. In the work
presented here, transVerse gradients are applied to create a field
Bz ) B0 + yGy, where Gy ) ∂Bz/∂y ) (0.2 T m-1 and
-2 mm < y < 2 mm. Thus, the magnetic field in the z-direction
varies in a linear fashion across the cylindrical sample tube
rather than along its length.
Figure 1. Schematic diagrams indicating image orientation and fields-of-
view for multiple xy slices (a) and a zy slice (b). In both diagrams the gray
area represents the field-of-view, which is the region of the tube held within
the radio frequency coil.
In this paper, MRI has been used to follow the formation
and propagation of the traveling wave formed in the reaction
between Co(II)EDTA2- and hydrogen peroxide. Descending
traveling waves were initiated in vertical tubes, inside the MRI
spectrometer. By applying linear transverse field gradients, the
wave could be directed as it propagated downward, and the
formation of a chemical finger could be forced. By switching
the orientation of the transverse gradients the tip of the finger
could be maneuvred.
Figure 2. Schematic representation of the timings of imaging sequences
and gradient trains, which are looped n times. The time value τ varies from
11 to 30 s, depending on the experiment. ∆ is the time between imaging
experiments.
2. Experimental Section
Sodium hydroxide, EDTA, cobalt(II) chloride, and hydrogen
peroxide (35% solution by volume), all of A.C.S. grade, were
obtained from Aldrich and were used without further purifica-
tion. Solutions of 0.02 M Co(II)EDTA2- were prepared by
dissolving equimolar quantities of EDTA and CoCl2 in deionized
water and adjusting the pH to 4.0. The reacting solution was
made from 0.02 M Co(II)EDTA2- and hydrogen peroxide
solutions, in a 9:1 ratio. The concentration of sodium hydroxide
solution used to initiate the traveling wave was 0.016 M.
The reaction was studied in a 5 mm NMR tube and the wave
initiated inside the spectrometer magnet, using a delivery device
to introduce a drop of the sodium hydroxide solution on top of
the Co(II)EDTA2-/H2O2 solution. MRI experiments were
performed on a Bruker DMX-300 spectrometer equipped with
a 7.0 T superconducting magnet, operating at a proton resonance
frequency of 300 MHz. All MRI experiments were carried out
at a temperature of 22 ( 0.2 °C. The tube was imaged using a
25 mm radio frequency coil, which had a maximum vertical
observational region of 30 mm.
Characterization of the reacting solution and experimental
details can be found elsewhere.3 Images were obtained using
the fast imaging, multiple spin-echo sequence, RARE.5 The
orientations of images are shown in Figure 1. The zy images
had a slice thickness of 1 mm and were positioned in the center
of the tube. The vertical and horizontal fields of view were 50
and 12.5 mm, respectively. The corresponding pixel size was
195 µm × 195 µm. Multiple-slice experiments, which collected
either six or ten xy images, were also acquired. Each slice had
a thickness of 1 mm and separation of 1.2 mm and was
composed of a 64 pixel × 64 pixel array with a field-of-view
of 5 mm in both directions. The T2 relaxation time of the
Co(III)EDTA- solution was sufficiently long3 that an image
could be obtained from a single signal acquisition, so that the
imaging time was 1 s for the zy images and 3 s for the xy
multiple-slice images.
To follow the effect of magnetic field gradients on the
traveling wave, trains of gradient pulses were applied between
image acquisitions. Gradient trains were generated using the
imaging gradients of the spectrometer and comprised pulses
which were switched on for 2 ms and off for 1 ms, cycled 2000
times, with amplitude of +0.2 T m-1 or -0.2 T m-1. This
produced an average gradient of (0.133 T m-1 over a period
of 6 s. Constant gradients were not applied, as they could have
damaged the gradient coils. Three gradient trains were applied,
at 5 s intervals, between imaging experiments (Figure 2).
Relatively long time intervals between images were chosen to
minimize the influence on the wave from the magnetic field
gradients involved in the imaging sequence. For negative
gradients, the field (Bz) decreased from right to left across the
sample/image and increased for positive gradients. Heating of
the sample due to the imaging sequence and gradient pulses
was negligible.
3. Results
In Figure 3 a series of typical zy images are shown over a
period of approximately 7 min following wave initiation. Once
the reaction was initiated at the top of the Co(II)EDTA2-/H2O2
solution by adding a drop of NaOH solution, a traveling wave
formed, which propagated downward, initially with a flat,
horizontal interface. After a time, the wave front distorted and
a finger formed. The propagation of this finger was faster than
that of the flat wave front, and it is believed that this
enhancement of the wave velocity is associated with convection.
In these experiments, where no magnetic field gradients were
applied except for those required for the imaging, the horizontal
position at which the finger developed was found to be random,
(5) Hennig, J. J. Magn. Reson. 1988, 78, 397.
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Figure 5. Time series of seven MRI zy images; parameters are the same
as in Figure 3. Image (a) is taken 165 s after the wave was initiated by the
addition of a drop of NaOH solution. Following image (a) magnetic field
gradient trains were applied between imaging experiments, with a time delay,
τ, of 11 s between gradient trains and imaging experiments. Applied
gradients were in the +y direction with the magnetic field increasing from
right to left, as indicated by the arrow. Images were collected every 51 s
and shown at 267 s (b), 369 s (c), 471 s (d), 522 s (e), 573 s (f), and 624
s (g) after wave initiation.
Figure 3. Time series of six MRI zy images of a traveling wave formed
in the reaction of Co(II)EDTA2- with H2O2. The slice thickness of each
image is 1 mm, and a region of 45 mm (vertical) × 16 mm (horizontal) is
displayed. Image (a) is taken 127 s after the wave was initiated by the
addition of a drop of NaOH solution. Subsequent images were taken 186
s (b), 264 s (c), 318 s (d), 342 s (e), and 415 s (f) after initiation. Signal
intensity is high (bright) where Co(III)EDTA- ions predominate and low
(dark) where Co(II)EDTA2- ions predominate. The vertical white lines in
each image indicate the sides of the NMR tube.
depicted here are acquired over a period of 3 s and show more
fully the three-dimensional (3D) structure of the finger.
By applying magnetic fields which vary horizontally across
the sample according to Bz ) B0 + yGy prior to finger formation,
it was discovered that the propagating wave could be directed
and the position of the finger controlled. In Figure 5, a series
of zy images are shown where, after the first image, trains of
positive y gradients are applied between imaging experiments,
during which the magnetic field is greater on the left side than
on the right side. The images are taken at 51 s intervals, with
the first image taken 165 s following wave initiation.
The progress of the wave can be more clearly followed using
the graph shown in Figure 6. The displacements of the wave
front along the two dashed vertical lines shown in Figure 6a
are plotted against time. Initially no gradients are applied, and
the displacement of the wave is matched on both sides of the
tube. From a point 178 s after wave initiation, a series of gradient
trains are applied between images (as shown by the schematic
in Figure 6b). A finger is formed on the right-hand side of the
tube (the region of low field) 470 s after wave initiation (Figure
5d), and its propagation is significantly accelerated in contrast
to the wave propagation on the left-hand side, which continues
to proceed at the original velocity. Once the finger has formed,
its velocity remains constant, in agreement with previous
studies.3
To demonstrate that the finger develops in a fixed position
and that no other fingers are formed outside of the central yz
slice, six xy images were taken following trains of negative y
gradient pulses, Figure 7. The horizontal slices clearly demon-
strate that the finger forms in a specific, and fixed, position.
The finger lies hard against the side of the tube, down its length,
except for the tip or leading-edge which is located toward the
center of the tube. The early part of the autocatalytic reaction
takes part in this region of the finger.
We have observed that the position of finger formation is
perfectly reproducible and dependent only on the direction of
the applied y gradient. When positive y gradients are applied,
and the magnetic field is at its greatest on the left side, the finger
forms on the right side of the tube. In cases where a negative
gradient is applied, the finger forms on the left side. With
repeated application of these gradient trains additional fingers
did not develop, in contrast to the situation in the absence of
Figure 4. Multiple xy slice images taken through the length of a chemical
finger formed 333 s after wave initiation. Each image has a slice thickness
of 1 mm, with a separation of 1.2 mm between images, and a field of view
of 5 mm in both x and y directions. Slice position 1 is the highest and is
positioned close to the wave initiation position. The multiple-slice experi-
ment for these images was performed between the zy images shown in d
and e of Figure 3. The rectangle indicates the corresponding position of
the zy images, and the circle represents the edge of the tube.
and frequently a second finger was formed. Figure 4 shows a
series of six xy images taken between the two zy images shown
in d and e of Figure 3. The rectangular box in Figure 4 indicates
the slice position for the zy images (Figure 3d,e). The xy images
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Evans et al.
Figure 6. (a) MRI zy image of the chemical finger formed 573 s after wave initiation following the application of magnetic field gradient trains. Applied
gradients were in the +y direction with the magnetic field increasing from right to left. (b) Plot of chemical wave displacement against time along the two
lines highlighted in the image in (a). Above the graph the timings for the gradient pulses are indicated, with respect to the data points.
the gradient pulses (Figure 3). However once a finger was
formed, its position could be controlled by switching the
direction of the magnetic field gradient. A typical example of
this is shown in Figure 8. The first image is taken 20 s after the
wave was initiated, prior to the application of the gradient pulse
trains. The following two images are then taken following the
application of a set of negative y gradient trains. The resulting
finger is clearly seen in Figure 8c. Following this image the
direction of the gradient train is switched and a train of positive
y gradient pulses were applied. The position of the finger then
clearly switches to the right side of the tube and continues to
propagate down this side. It is important to note that only the
tip of the wave is manipulated and the part of the finger
containing the fully reacted Co(III)EDTA- solution remains
fixed on the left side of the tube. Furthermore, a second finger
starts to form on the right side of the tube from the interface at
the top (Figure 8d).
Finally we observe that by rotating the direction of the
magnetic field gradient in the xy plane, the tip of the chemical
finger can be controllably positioned. To reduce the effect of
the imaging gradients a limited number of slices was taken and
so only a limited region of the finger could be observed by one
set of xy images. To follow the rotation of the finger, slices
were taken close to the finger tip as it propagated downward.
In Figure 9 a set of xy images are shown of the part of the
finger just above the tip, where it lies hard against the side of
the tube, using Bz fields of the form Bz ) B0 + (y cos θ Gy +
x sin θ Gx), with Gx ) Gy. The first image shows the orientation
of the finger formed after a train of positive y gradient pulses.
Subsequent images have trains of gradient pulses applied
between imaging experiments which have been rotated by 45°.
It can be seen that, as the gradient is rotated, so too is the
Figure 7. Multiple xy slice images taken through the length of a chemical
finger formed 246 s after wave initiation following the application of
position of the tip of the finger. As was seen in Figure 8, only
the tip can be directed, and the fully reacted part of the finger
remains fixed in position. So, as the finger propagates down-
ward, the tip is rotated, thus creating a spiral down the length
of the tube.
magnetic field gradient trains. Applied gradients were in the -y direction
with the magnetic field increasing from left to right, as indicated by arrow.
Each image has a slice thickness of 1 mm, with a separation of 1.2 mm
between images, and a field of view of 12.5 mm in both x and y directions.
Slice position 1 is the highest and is positioned near where the wave was
initiated.
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Figure 8. Series of MRI zy images; parameters are the same as in Figure 3. Image (a) was taken 20 s after wave initiation. Following image (a), magnetic
field gradient trains were applied between imaging experiments and subsequent images were collected 71 s (b), 123 s (c), and 175 s (d) after wave initiation.
For the images shown in (b) and (c) the applied gradients were in the -y direction. Following image (c) the direction of the gradients were switched to +y.
The arrows indicate the direction in which the magnetic field increases during the gradient trains, which were applied prior to acquiring the images.
Figure 9. Time series of MRI xy images following the tip of the finger as it propagates downward. Gradient trains are applied between imaging experiments
where the direction of the gradient is rotated in the xy plane. The arrows indicate the direction in which the magnetic field increases during the gradient
trains, which were applied prior to acquiring the images. The position of each xy slice is moved to follow the part of the finger just above the tip, where it
touches the wall of the tube. The positions of these slices, relative to the position of the chemical wave just after initiation, are 2.0 mm (a), 6.3 mm (b), 10.9
mm (c), 16.9 mm (d), 18.1 mm (e), 18.4 mm (f).
4. Discussion
have no effect on the wave propagation. For a field of this form,
the only nonzero gradient is ∂Bz/∂y so that, from eq 2, Fx )
Fy ) 0, and Fz ) kByGy ) 0, because there is no applied field
in the y-direction.
However, in the absence of electric currents in the sample or
time-dependent electric fields, Maxwell’s equation, ∇ × B )
0, means that the following relation must hold:
In general terms, the Maxwell force is given by
FM ) (V/µ0)∆øV(B‚∇)B )
Fx
Fy
Bx∂Bx/∂x + By∂Bx/∂y + Bz∂Bx/∂z
Bx∂By/∂x + By∂By/∂y + Bz∂By/∂z
Bx∂Bz/∂x + By∂Bz/∂y + Bz∂Bz/∂z
) k
(2)
( )
(
)
Fz
∂Bp ∂B
)
q, p,q ) x,y,z
(4)
where k is the collection of constants (V/µ0)∆øV. Hence, for a
force Fq to act on the wave in direction q, at least one term of
the form Bp∂Bq/∂p needs to be nonzero.
∂q
∂p
Consequently, when a magnetic field Bz ) B0 + yGy is applied
to the sample, there must also be a “concomitant” field gradient
in the y-direction: By ) zGy. Thus:
In the experiments reported by Evans et al.,3 it was clear that
the applied field, with its longitudinal gradient, Bz ) B0 + zGz,
produced a force in the z-direction:
Fy Bz∂By/∂z ) (B0 + yGy)Gy ≈ B0Gy
(5)
Fz Bz∂Bz/∂z ) (B0 + zGz)Gz ≈ B0Gz
(3)
(note that with Gy ) 0.2 T m-1 and |y| < 2 mm, we have
2
(using B0 . zGz). At first sight, however, it would appear that
a transverse gradient superimposed on the spectrometer field,
i.e., Bx ) By ) 0, Bz ) B0 + yGy, as employed here, could
yGy , B0 and Fz ) kByGy ) kzGy , Fy). These so-called
concomitant magnetic fields are known to create a number of
artifacts in MRI at low fields, particularly in experiments that
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Evans et al.
are sensitive to phase evolution.6 In the work presented here
the concomitant gradients generate no detectable artifacts of this
type but are responsible for the ability to manipulate the wave
using magnetic forces that are transverse to the direction of
propagation.
be driven “upwards” by an applied magnetic field gradient. In
relation to the last point, Evans et al.3 have shown that positive
field gradients lead to a significant increase in the downward
wave front velocity but negative field gradients retard the
velocity to a smaller extent, presumably as a result of convec-
tion. Viscosity and concentration control of the reaction mixture
and suppression of convection might allow even more dramatic
magnetic manipulation of such wave reactions.
Equation 5 clearly demonstrates that Fy is the dominant force
component for this system and that it changes direction when
the applied gradient is reversed. As shown in Figures 5-9, the
sign of Gy determines the initial point of formation for the
chemical finger as well as its consequent motion within the tube.
As expected, the finger (of diamagnetic reaction product) is
exclusively formed in the region of lowest field. Consequent
application of magnetic field gradients allows the finger to be
“pushed” around within the tube as the (diamagnetic) finger
consistently moves toward the lowest field region of the tube.
Finally it is interesting to note that only the leading-edge of
the finger is affected by the gradient trains. As seen in Figure
8, when the gradient direction is reversed during an experiment,
only the most recently reacted part of the finger can be
manipulated by the applied field. In contrast, regions of the
finger that formed prior to switching the gradients remain
unaffected. Possible explanations for this phenomenon might
be based on the increased significance of hydrodynamic
instabilities at the leading edge of the finger or on the curvature
of the wave front. Further work is underway to investigate the
origin of this effect.
5. Conclusion
It has been shown that the spatial evolution of a descending
wave front, for the autocatalytic oxidation of Co(II)EDTA2-
,
can be controlled by applying magnetic field gradients of
suitable orientation. Both the position of finger formation and
its consequent propagation direction can be controlled repro-
ducibly by appropriately chosen magnetic field orientations.
Maxwell stress can account for this observed behavior.
The experimental methods described in this paper are
expected to be suitable for the manipulation of other traveling
chemical waves. The prerequisite for the success of magnetically
controlled spatial wave evolution is the existence of magnetic
susceptibility gradients across the reaction boundary. The
exciting possibility of controling polymer synthesis based on
radical chain carriers is just one potential application of this
technique at present under investigation.
By applying gradient pulse trains of variable direction spatial
patterns such as spirals (Figure 9) can be set up within the tube.
The design of such spatial patterns is presently limited by several
factors: (i) it is only possible to manipulate the tip of the finger
at its point of formation; (ii) the chemical reaction proceeds in
all directions, and hence, the finger becomes blurred as the
reaction proceeds (Figure 9d,e,f); (iii) the reaction cannot easily
Acknowledgment. M.M.B. thanks EPSRC for an Advanced
Research Fellowship and Prof. L. F. Gladden and the MRRC,
Cambridge University, for support. C.R.T. thanks the Royal
Society for a University Research Fellowship and the EPSRC
for financial support. We are grateful to Prof. R. J. Ordidge
and Prof. G. A. Brooker for advice on concomitant field
gradients.
(6) Norris, D. G.; Hutchison, J. M. S. Magn. Reson. Imaging 1990, 8, 33.
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