Magnetic resonance imaging of chemical waves in porous mediaAnnette F. Taylor and Melanie M. Britton Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 16, 037103 (2006); doi: 10.1063/1.2228129 View online: http://dx.doi.org/10.1063/1.2228129 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/16/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetic resonance imaging study on near miscible supercritical CO2 flooding in porous media Phys. Fluids 25, 053301 (2013); 10.1063/1.4803663 Models of water imbibition in untreated and treated porous media validated by quantitative magneticresonance imaging J. Appl. Phys. 103, 094913 (2008); 10.1063/1.2913503 Measuring flow resistivity of porous material via acoustic reflected waves J. Appl. Phys. 98, 084901 (2005); 10.1063/1.2099510 A magnetic resonance study of pore filling processes during spontaneous imbibition in Berea sandstone J. Chem. Phys. 119, 9609 (2003); 10.1063/1.1615757 Correlations between dispersion and structure in porous media probed by nuclear magnetic resonance Phys. Fluids 11, 259 (1999); 10.1063/1.869876

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Magnetic resonance imaging of chemical waves in porous mediaAnnette F. Taylora�

Department of Chemistry, University of Leeds, Leeds, LS2 9JT, United Kingdom

Melanie M. Brittonb�

Department of Chemistry, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom

�Received 10 April 2006; accepted 26 June 2006; published online 27 September 2006�

Magnetic resonance imaging �MRI� provides a powerful tool for the investigation of chemicalstructures in optically opaque porous media, in which chemical concentration gradients can bevisualized, and diffusion and flow properties are simultaneously determined. In this paper we givean overview of the MRI technique and review theory and experiments on the formation of chemicalwaves in a tubular packed bed reactor upon the addition of a nonlinear chemical reaction. MRimages are presented of reaction-diffusion waves propagating in the three-dimensional �3D� net-work of channels in the reactor, and the 3D structure of stationary concentration patterns formed viathe flow-distributed oscillation mechanism is demonstrated to reflect the local hydrodynamics in thepacked bed. Possible future directions regarding the influence of heterogeneities on transport andreaction are discussed. © 2006 American Institute of Physics. �DOI: 10.1063/1.2228129�

The coupling of nonlinear chemical reaction withdiffusion/dispersion and flow in porous media is of fun-damental importance in industrial processes and mayalso provide mechanisms for biological pattern forma-tion. Magnetic resonance imaging (MRI) can be used as anoninvasive method to probe the three-dimensional, opti-cally opaque medium. In this paper we review some ofthe mechanisms that give rise to chemical pattern forma-tion in a packed bed reactor and demonstrate how MRIcan be used to obtain information on the effect of diffu-sion and flow on these chemical structures. We discussthe propagation of reaction-diffusion waves in the net-work of channels in the packed bed in the absence offlow, and how stationary chemical waves are affected bydispersion and the nonuniform fluid velocity field in thepresence of flow. It is hoped that such studies may shedlight on certain biological processes that occur in hetero-geneous environments.

I. INTRODUCTION

Chemical/biological reaction in porous media is an im-portant area of research in a wide variety of disciplines fromengineering to environmental science.1 Applications includecatalysis in packed bed reactors,2 the use of micro-organismsto remove pollutants from soil �bioremediation�,3 and ana-lytical techniques involving chromatographic processes.4

Understanding how the heterogeneous medium influencestransport processes and how the mutual interactions betweenchemistry and transport may affect the outcome of the reac-tion is therefore of fundamental interest. The nature of thetransport may affect the mixing of reactants leading to varia-tions in product selectivity and yield.5 Fluid flow may serve

to accelerate or extinguish the reaction,6 and under certaincircumstances lead to the formation of chemical patterns.7

A tubular reactor packed with glass beads has provided ameans for the investigation of chemical pattern formation.8

In the absence of flow, nonlinear chemical reaction cancouple with diffusion leading to the propagation of reaction-diffusion waves in the network of channels created by thepacking material. Much theoretical research effort has goneinto understanding wave breakup and spiral formation instructurally heterogeneous media, largely as a result of insta-bilities observed in electrochemical wave propagation inheart and nerve tissue.9 Some of the main results are brieflydiscussed in Sec. II of this review. Flow-driven instabilities,such as the formation of traveling chemical waves via thedifferential flow-induced chemical instability �DIFICI� �Ref.10� and stationary concentration patterns via the flow-distributed oscillation �FDO� �Ref. 11� mechanism are alsodiscussed in that section. The latter have been proposed as ameans for biological pattern formation in, for example, geneexpression during segmentation.12 Reaction-inducedconvection13 may lead to chemical fingering in porous me-dia. The reader is referred elsewhere for a discussion on thisparticular type of instability.14

Nuclear magnetic resonance �NMR�, or magnetic reso-nance imaging �MRI�, can be used to monitor chemical re-action in porous media.15,16 It has advantages over othertypes of imaging such as x-ray tomography in that it is si-multaneously able to determine the pore network of the me-dium and the chemical composition,17,18 through the mea-surement of NMR relaxation times and chemical shift ofnuclei such as 1H. Transport properties including fluid ve-locities and diffusion coefficients of water molecules canalso be determined from local differences in relaxation timein the porous medium. Structures such as chemical wavescan be visualized by exploiting the change in relaxation timeof water molecules surrounding the metal catalyst as it oscil-

a�Electronic mail: A.F.Taylor@leeds.ac.ukb�Electronic mail: M.M.Britton@bham.ac.uk

CHAOS 16, 037103 �2006�

1054-1500/2006/16�3�/037103/8/$23.00 © 2006 American Institute of Physics16, 037103-1

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lates between oxidative states.19,20 An overview of the MRItechnique is given in Sec. III of this paper and in Sec. IV, wepresent MR images of stationary and propagating chemicalwaves in a packed bed reactor in the presence and absence offluid flow. Finally, in Sec. V, we discuss future directions.21

II. THEORY: NONLINEAR CHEMICAL REACTIONIN A TUBULAR PACKED BED REACTOR

A tubular packed bed reactor provides an idealized labo-ratory environment for studying the influence of a porousmedium on chemical reaction and transport. The porosity orvoid fraction of the medium, �, is defined by Vp /VT where Vp

is the volume of the fluid and VT is the total volume of themedium calculated from the sum of Vp and the volume of thesolid phase, Vs.

22 The packing material might consist of cata-lytic particles that take part in the reaction or glass particlessimply present to provide plug-flow. The idealized plug-flowmodel assumes a solute is carried by the flow with no radialconcentration gradient and no axial dispersion �Fig. 1�a��.However, dispersion plays a significant role in the distribu-tion of chemicals in the packed bed �Fig. 1�b��.17 The tem-poral evolution of a chemical species undergoing reaction ina packed bed reactor can be described by equations of theform

da

dt= f�a� + D

d2a

dz2 − Vda

dz, �1�

where a represents the concentration of the species of inter-est, f�a� represents chemical reaction �kinetic� terms, and z isthe axial coordinate. The diffusive term in Eq. �1� incorpo-rates a coefficient D generally referred to as the diffusioncoefficient in the absence of flow and the dispersion coeffi-cient in the presence of flow. The dependence of D on bothrandom molecular motion and the additional dispersive mix-ing arising with flow is accounted for in the following equa-tion:

D = Deff +dV

PeL, �2�

where Deff is the effective diffusion coefficient in the packedbed, d is particle diameter of the packing, PeL is the Pecletnumber, and V is the interstitial �average� fluid velocity.23

The effective diffusion coefficient may be lower than themolecular diffusion coefficient of the solute, Dm, as a resultof geometrical hindrance due to the tortuosity, �, of thepacked bed �Deff=Dm /��.24 The relative contributions of dif-fusion and dispersion to D depends on the flow rate, withdiffusion dominating at low flow �D=Deff� and hydrody-namic dispersion dominating at high flow �D=dV /PeL�.Axial dispersion �in the direction of flow� is generally greaterthan the radial dispersion. The final term in Eq. �1� representsadvection with interstitial fluid velocity V=V0 /�, where V0 isthe flow velocity in the absence of packing and � is theporosity. In reality, both the dispersion coefficient and thefluid velocity are a function of axial and radial position.There is a distribution of flow velocities amongst the chan-nels, and the mean flow velocity through a channel is depen-dent on the geometry and tortuosity of that channel and thetopology of the pore space �Fig. 1�c��.21,25 However there aremore high-velocity channels close to the wall than in thecenter of the tube. This variation in flow is due to structuraleffects imposed by the wall of the tube.

Propagating or stationary chemical waves might be ob-served in a packed bed reactor if the reaction includes feed-back through, for example, autocatalysis. The prototypicalchemical oscillator is that of the Belousov-Zhabotinsky �BZ�reaction which can be simulated with a scaled 2-variableOregonator model26

�u

�t=

1

��u�1 − u� − fv

�u − q��u + q�� +

�2u

�z2 − ��u

�z, �3�

�v�t

= u − v + ��2v�z2 − �

�v�z

, �4�

where u and v are the dimensionless concentrations of theautocatalyst �HBrO2� and oxidized form of the metal catalyst�Mox� respectively; the parameters �, f , and q are related torate constants and initial concentrations; � is the ratio ofdiffusion or dispersion coefficients=Dv /Du and � is the di-mensionless flow rate. The BZ reaction has been widely ex-ploited as a means of testing theory concerning chemicalpattern formation.27 Oscillations are of the relaxation type inthis reaction, with a sharp increase in the oxidized form ofthe catalyst v, followed by a more gradual return to the re-duced steady state.

In the absence of flow, Eqs. �3� and �4� support reaction-diffusion waves. The wave speed, c0, of reaction-diffusionwaves depends on the �pseudo-first order� rate constant ofautocatalysis and the diffusion coefficient of theautocatalyst.28 When performed in a packed-bed reactor, thereaction medium consists of inexcitable obstacles that createa network of channels in which the waves can propagate.The distribution of channel widths in the packed bed as wellas connectivity of channels influences wave propagation.Waves emerging from a small channel to a channel of largerwidth will collapse if the radius of the emerging wave issmaller than that of a critical radius �rcrit=D /c0� predicted bythe Eikonal relation29

FIG. 1. Transport of solute with concentration a �black corresponds to thehighest concentration of a in the grey scale� in a tubular packed bed reactor.�a� Plug-flow; �b� axial �upper� and radial �lower� dispersion of solute;�c� tortuous fluid path in the packed bed.

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c = c0 − D� �5�

where the wave speed, c, is reduced compared to the planarwave speed, c0 by an amount that depends on the diffusioncoefficient D of the autocatalyst and the wave curvature �=1/r, with r equal to the radius of an expanding wave.

Simulations in 2D have demonstrated that increasing thenumber or size of randomly distributed obstacles beyond acritical value leads to wave breakup, spiral formation, andultimately wave collapse.30 Spatial clustering of obstaclesresults in larger channel widths and therefore favors wavepropagation. Simulations of 3D chemical waves in structuredexcitable media are computationally intensive but some as-pects of the problem have recently been considered.31 It issuggested that the increase in channel connectivity achievedin three dimensions allows the presence of a greater percent-age of inexcitable obstacles in the reaction domain beforewave collapse is observed.

An alternative method for the emergence of propagatingchemical waves has been proposed. The differential flow-induced chemical instability �DIFICI or DIFI� relies on dif-ferential transport of species.10 This is achieved in thepacked bed reactor by the selective binding of the catalyst ofthe BZ reaction to the packing material �an ion exchangeresin� while the other reactants are fed to the reactor from areservoir.32 In simulations, the diffusion and flow terms ofthe catalyst v in Eq. �5� are removed and conditions are suchthat the system is in a stable steady state in the absence offlow. Upon the addition of flow, a convective instability isestablished in which a perturbation from the steady state isamplified locally and subsequently advected from the reactor�Fig. 2�a��. The result is a train of chemical waves propagat-ing with the flow. Such waves have a significantly greatercharacteristic wave speed and wavelength than reaction-diffusion waves.33

Flow and diffusion-distributed structures �FDS� are amechanism for the formation of stationary concentration pat-terns from an oscillating chemical reaction in an open flow,provided the boundary is constantly-forced chemically.11,34

The latter is achieved by pumping the BZ reactants into atubular packed bed reactor from a continuous stirred tankreactor �CSTR� operated under conditions such that the re-

action is in a steady state in the CSTR but would oscillate ina batch reactor.35 With the ideal plug-flow model, each vol-ume element is assumed to leave the CSTR in the same stateand oscillate at the same position along the tube thus givingrise to a flow-distributed oscillation �FDO�. However, experi-ments and simulations of Eqs. �3� and �4� indicate that dis-persion plays a role in the formation and wavelength of thestationary patterns.36 Waves form via a “wave-splitting”mechanism whereby a wave front appears some distancealong the tube and splits into two parts, one traveling withthe flow �fast� and one traveling against the flow �slow�,eventually settling to give a stationary band of high autocata-lyst concentration �Fig. 2�b��. Intrinsic FDO periods calcu-lated from the product of the wavelength and inverse flowvelocity are thus reduced compared to the natural �batch�oscillatory period. There also exists a minimum flow rate forwhich the patterns can be observed; less than this criticalflow rate the reaction may display advective instabilities thatresult in bulk oscillations in the tube, or convective instabili-ties that give rise to traveling waves.

III. THEORY: MRI

MRI has the great advantage of being able to examinesystems noninvasively. While the spatial resolution of MRIis modest compared to optical microscopy, it is able to probeoptically opaque systems and provide spatially resolved in-formation concerning chemical and physical properties, in-cluding precise velocity and diffusion measurements. Thetheory behind magnetic resonance imaging �MRI� and veloc-ity imaging can be found, in detail, elsewhere.37,38 An over-view of these techniques is given here.

Magnetic resonance �MR� probes the behavior of nucleithat possess nuclear spin. When these nuclei are placed intoa static magnetic field �B0� they precess at a frequency ���dependent on the magnetogyric ��� ratio of the nucleus andthe strength of B0. The nuclear spin energy levels also be-come nondegenerate, and at thermal equilibrium there willbe a Boltzmann distribution of nuclei amongst these energylevels. The most probed nucleus is 1H, as it has high naturalabundance and relatively high sensitivity. 1H nuclei have twoenergy levels: spin up �where the nuclear spin vector isaligned parallel to B0� and spin down �where it is alignedantiparallel to B0�, and there is a slight excess of nuclei withthe lower energy spin up orientation. Transitions betweenthese energy levels are produced by applying a radio-frequency �rf� pulse, at the resonance frequency � to thesystem. This causes a redistribution of nuclei amongst theenergy levels and is necessary for producing the MR signal.Immediately after a 90° rf excitation pulse, there will beequal populations of spin up and spin down nuclei and theirspin vectors will be aligned and so have phase coherence.Following the rf pulse the system relaxes to thermal equilib-rium. The rate at which this occurs is characterized by tworelaxation processes: spin-lattice �T1� and spin-spin �T2�. TheT1 relaxation time determines the rate at which the ratio ofspin up to spin down nuclei return to the Boltzmann distri-bution. The T2 relaxation time determines the rate at whichthe nuclear spin vectors lose phase coherence. These relax-

FIG. 2. Simulations of reaction-diffusion-advection waves in the dimension-less Oregonator model of the BZ reaction. �a� Space-time plot illustratingpropagating waves via the DIFICI mechanism with v profile at the tenthtime unit; �b� space-time plot illustrating formation of stationary waves viathe FDO mechanism with v profile at the 50th time unit. �See Refs. 33 and36 for more details.�

037103-3 MRI of chemical waves in porous media Chaos 16, 037103 �2006�

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ation times are frequently used as a means of producing con-trast in MR images.

MR images are produced by applying magnetic field gra-dients to the sample, so making the nuclear precessional fre-quency spatially dependent,

��r� = ��B0 + G · r� , �6�

where ��r� is the resonance frequency at position r in thesample and G is the local magnetic field gradient. By apply-ing magnetic field gradients in 1, 2, and 3 directions it ispossible to produce 1D, 2D, or 3D images. If the signal isacquired in the presence of a magnetic field gradient, spins atdifferent positions will have different resonance frequencies�frequency encoding�. If the magnetic field gradient is ap-plied prior to signal acquisition, spins at different positionswill have different phases, arising from the spatially depen-dent precessional frequency imposed during the gradientpulse �phase encoding�. A combination of frequency andphase encoding methods are generally used to produce 2Dand 3D images. Typical resolution for MR images is in therange of 10–200 �m.

An important part of imaging is the ability to distinguishbetween regions of different chemical composition or physi-cal environment, and this is achieved through image contrast.There are four main quantities that can be used to createcontrast within an image: spin density, T1 or T2 relaxationtimes, chemical shift or molecular motion �flow and diffu-sion�. Contrast is produced through variations in signal in-tensity between pixels and is dependent on the imaging se-quence used, the values of key imaging parameters and thephysical or chemical properties of nuclei within those pixels.MRI has been used to probe chemical waves by exploitingthe relaxation time difference of solvent �i.e., water� mol-ecules surrounding the transition metal catalysts. The relax-ation time �T1 and T2� of solvent molecules is shortened inthe presence of paramagnetic species.39 During the formationof waves the oxidative state of the transition metal ion cata-lyst periodically changes and hence the relaxation time ofmolecules surrounding these ions also oscillates and so be-haves as a “MRI indicator.” In MRI studies of chemicalwaves, manganese is frequently used as the catalyst/indicator, where there is an interconversion between Mn2+

and Mn3+. There are more unpaired electrons in the Mn2+

aquo ion than Mn3+ and as a result the relaxation time formolecules surrounding Mn2+ is shorter. This produces thenecessary contrast with which to visualize chemical waves.

Armstrong and co-workers observed waves in the BZreaction19 using a GRASS experiment,40 where images wereT2 weighted. By using this small-angle imaging technique,the acquisition time of each image was reduced to 2–3 s.They found that waves could be visualized using eithermanganese19 or ruthenium.20 Ferroin, however, was found tobe a poor MRI indicator20 because ferric ions are produced inacidic conditions, which dominate the relaxation time of thesolution, eliminating any image contrast. Britton and co-workers used an alternative imaging sequence, RARE,41 tovisualize waves in the 1,4-cyclohexanedione-acid-bromate�CHD�,42 and the BZ reaction21 in packed beds. Again, man-ganese was used as the catalyst and MRI indicator. The

RARE imaging sequence is a multiple-echo sequence andalso produces T2 weighed images on the order of seconds.Using this sequence, it was also possible to spatially quantifycatalyst concentration.18 Other transition metal ions used toimage chemical fronts include cobalt43 and iron.44

In order to measure transport processes such as flow anddiffusion, a technique based on the pulsed gradient spin echo�PGSE� experiment37 is used. By applying two narrow mag-netic field gradient pulses of duration �, separation , and ofgradient strength g, phase shifts arise, not from spin position,but from spin displacement over the time scale of . In thecase of diffusion, where molecular motion is incoherent,there is a distribution of phase shifts, resulting in an attenu-ation of the MR signal. The attenuation of the signal is thendependent on �, �, , g, and the diffusion coefficient of themolecules, D. In practice the MR signal is acquired over arange of g values, keeping � and constant, and the diffu-sion coefficient is then calculated using the Stejskal-Tannerrelationship,45

I

I0= exp�− �2�2g2D� −

�

3 , �7�

where I is the measured signal intensity and I0 is the signalintensity when g=0. This type of experiment has been usedto probe convection in the CHD reaction.46 In the case offlow, where molecular motion is coherent, there is a netphase shift, , which is dependent on �, �, , g, and the flowvelocity v,37

= �vg� . �8�

Velocity and diffusion coefficients are measured in thedirection of the applied PGSE magnetic field gradient. Inorder to image flow and diffusion, it is necessary to incorpo-rate these motion encoding �PGSE� gradients into an imag-ing sequence. For the measurement of flow a minimum oftwo images is necessary, which, for example, are acquiredusing two PGSE gradient amplitudes �g�, so that a phaseshift, and hence velocity, can be accurately measured foreach pixel in the image. Details regarding the implementa-tion and analysis of velocity imaging in porous media havebeen recently reviewed by Mantle and Sederman.16

IV. EXPERIMENT: MRI OF CHEMICAL WAVESIN A PACKED BED REACTOR

MRI has already proved to be a useful tool for visualiz-ing reaction-diffusion waves in the liquid phase. It has alsobeen demonstrated using MRI that convective effects influ-ence the propagation of BZ �Ref. 10� and CHD �Ref. 46�chemical waves, in tubular reactors of internal diameters 4.3and 10 mm, respectively. In the presence of such reaction-induced convection, the dispersion coefficient �in the verticaldirection� was found to be three times greater than the diffu-sion coefficient. This results in increased wave velocities andthe formation of complex chemical patterns.

The ability of MRI to probe opaque systems enableschemical waves formed in porous media to be uniquely ob-served. When the BZ reaction was performed in tubes ofdiameter 5 mm and 10 mm packed with 0.5 and 3.2 mm

037103-4 A. F. Taylor and M. M. Britton Chaos 16, 037103 �2006�

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glass beads, respectively, spherical and planar reaction-diffusion waves were observed.47 The wave velocity wasfound to decrease by a factor of 2.5 as waves passed from thesolution above the packing material into the heterogeneousmedium. It was suggested that the reduction in wave velocityarises from a lower effective diffusion coefficient Deff due tothe tortuosity of the packed bed. The change in wave veloc-ity in the packed bed might also be explained from consid-eration of an enhanced wave velocity in the solution abovethe packing material due to reaction-induced convection.

The stability of reaction-diffusion waves in a packed bedreactor is illustrated with MR images acquired by putting amanganese-catalyzed BZ reaction in a mixed sulfuric-phosphoric acid medium into a tube of internal diameter16 mm, containing 1 mm glass beads �Fig. 3�.21 Verticalslices �zx plane� were obtained from the center of the tube atsuccessive time intervals of 16 s. Signal intensity is shown ingrey scale and is dependent on the relaxation time of watermolecules. Regions which are bright correspond to a pre-dominance of Mn3+ ions and dark regions correspond to apredominance of Mn2+. Pixels containing beads have no sig-nal and thus are black. The bead-free BZ reaction solution atthe top of the tube displays the expected disordered patternof Mn3+ waves due to convective effects. In the region con-taining beads, the reaction self-organizes into simple propa-gating waves, with a source at the bead/solution interfaceand a second source at the base of the tube. Planar wavespropagate down from the interface and annihilate withupward-propagating spherical waves. There was no evidenceof any spiral wave formation under these conditions, indica-tive of the strong connectivity of channels as a result of thelarge porosity.

The ability of MRI to probe transport properties simul-taneously with the observation of wave formation provides avaluable opportunity for studying flow-distributed oscilla-tions in packed bed reactors. Experiments21 were performedin which the manganese-catalyzed BZ reaction was pumpedfrom a CSTR through a tubular reactor of internal diameter16 mm, packed with glass beads of 1 mm diameter. Twostock solutions were fed into the CSTR, which was operatedunder steady state conditions. The interstitial fluid velocity iscalculated from the experimentally-determined flow rate andthe estimated porosity of 0.41.48 Propagating waves were

observed initially, followed by the appearance of stationaryconcentration waves.

The structure of the stationary waves was determined byobtaining MRI images of the tubular packed bed reactor atvarious locations. In a vertical �zx� slice taken from the cen-ter of the reactor, light regions correspond to the oxidizedform of the catalyst, Mn3+, and glass beads appear as blackspheres �Fig. 4�. The image shows the wave front �excitedreaction solution� with sharp leading edge upstream and dif-fuse tail downstream. With the interstitial fluid velocity of1.2 mm s−1, the wave front extends over approximately20 mm in the axial direction, for a particular x coordinate.There is a distinct radial concentration gradient resulting in aV-shaped wave front in the axial plane and a spreading of thewave from the center to the edges of the reactor in successivehorizontal images �Fig. 5�. The 2D axial images can be usedto reconstruct the wave front in three dimensions. The result-ant 3D front is conical in shape with a hollow center and atotal approximate length of 45 mm �Fig. 6�. The wavelengthof the pattern was obtained for two different fluid velocities:with a flow velocity of 0.7 mm s−1 the wavelength was98 mm giving an intrinsic FDO period of 140 s, and withflow velocity of 1.2 mm s−1 the wavelength was 134 mmgiving an FDO period of 112 s, compared to the natural os-cillatory period of 160 s determined from well-stirred batchBZ reactions with the same initial concentrations.

The V-shape of the front can be explained by the radialvariation in fluid velocity.25 Data concerning local fluid ve-locity was extracted by MRI.21 A 2D radial slice was ob-tained from a 3D velocity map of a packed bed �Fig. 7�a��.The beads appear as dark spheres and the grey scale corre-sponds to the z component of the fluid velocity. The radialvelocity oscillates as function of the bed porosity, increasingat the walls of the reactor to almost three times the velocityin the center of the reactor �Fig. 7�b��. The wave front stillonly extends over 45 mm in total, despite the large variationin flow velocity, indicating the presence of strong radial cou-pling through dispersion. The reduction in intrinsic FDO pe-riod with increasing flow rate can be explained by the in-creased axial dispersion in accordance with Eq. �2�.

FIG. 3. �a� Schematic diagram indicating the orienta-tion of vertical MR images. �b�–�d� MR images of trav-eling �reaction-diffusion� waves in the manganese cata-lyzed BZ reaction, in a packed bed of 1 mm glassbeads. Images are taken from the center of the bed at aninterval of 16 s. Where the signal intensity is high�bright regions� Mn3+ predominates and where it is low�dark regions� Mn2+ predominates. �See Ref. 21.�

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V. DISCUSSION AND FUTURE DIRECTIONS

The mutual interaction of transport and chemical reac-tion is particularly important in porous media such as packedbed reactors. One goal of our work is to begin to understandthe influence of microscopic heterogeneity on macroscopicchemical patterns. MRI is a powerful tool able to resolvechemical concentration gradients in an optically opaque, po-rous medium and determine diffusion, dispersion, and flowproperties of the medium. We have observed the self-organization of nonlinear chemical reaction in a porous me-dium and found that heterogeneities in the 3D structure ofthe bed stabilize wave propagation in the absence of flow,while heterogeneities in the flow velocity alter the 3D struc-ture of stationary concentration patterns in the presence offlow.21

Planar and spherical reaction-diffusion waves were ob-served propagating in the 3D network of channels in theporous matrix of the packed bed reactor. Neither wavebreakup nor spiral waves were observed under these condi-tions. The porosity of the medium �0.41� was much lowerthan that predicted for wave-break up in theoretical studies�0.65 in 2D and 0.55 in 3D�,31 possibly because larger ob-stacles were used in the packed bed which created largeraverage channel widths. A systematic experimental investi-gation of the wave activity as a function of the void fraction

FIG. 4. MR �zx� image of a stationary wave taken vertically through thecenter of a packed bed. The image was taken in a 16 mm diameter tubefilled with 1 mm glass beads at a flow rate of 100 mm3 s−1. �See Ref. 21.�

FIG. 5. �a� Schematic diagram indicating the orientation of the horizontalMR images shown in �b�. �b� MR �xy� images taken horizontally through thelength of a stationary wave, at the end �i�, middle �ii�, and tip �iii� of thewave. Image �iii� is at a position further upstream than the image �i�. Con-ditions for this image are the same as Fig. 4. �See Ref. 21.�

FIG. 6. Three-dimensional renderings of a stationary wave, at two orienta-tions, produced from a series of 2D xy images. In �a� the grey scale is usedto highlight the structure and surface of the wave. In �b� the grey scaleindicates the height in the bed of fluid in the wave, with dark regions beingupstream and light regions being downstream. �See Ref. 21.�

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would provide information on the degree of heterogeneityrequired for wave breakup in three dimensions. Characteriza-tion of the average channel width as well as the connectivityof channels is important.

Further experiments are required to establish how thevelocity of waves depends on the structure/composition ofthe medium. Previous work suggests that the packing actssimply to reduce convective effects. However, the packedbed reactor presents possibilities for the investigation of theeffects of anisotropic diffusion on wave propagation49

through periodic variations in the structure of the packingthat restricts diffusion of the solvent in certain directions.Additionally, the transport of the positively charged catalystmight be influenced by interactions with the negative surfacecharges on the packing material in the reactor.50 Such inter-actions would depend on the nature of the catalyst and theparticles used, and may lead to the formation of new 3Dpatterns.

The flow-distributed oscillation mechanism provides amechanism for the formation of stationary concentration pat-terns in a packed bed reactor with a boundary feed of chemi-cals. The wavelength of the pattern depends on the dispersivecharacteristics of the packed bed. There is a reduction in theintrinsic FDO period calculated from the product of the in-verse flow velocity and wavelength �compared to the batchFDO period� by an amount that increases with increasingflow rate, in agreement with Eq. �2�. The nonuniform flowvelocity profile in the bed is reflected in the 3D structure ofthe wave front. Radial variations in flow velocity result in thecharacteristic V-shape while radial dispersion prevents thewave front from being extended over a much larger region.Additional experiments might be performed to obtain valuesof the dispersivity as a function of the packing materialand/or chemistry.

Numerical simulations predict that FDO patterns formvia a “wave-splitting” mechanism, and this was verified inexperiments exploiting the ferroin-catalyzed BZ reaction.36

The transient patterns observed in preliminary MRI investi-gations of the manganese-catalyzed BZ reaction in a packedbed reactor do not appear to correspond to wave-splitting.21

It was suggested that interactions of the ferroin catalyst withnegatively-charged surface groups on the silicon glass beads

may destabilize wave formation at low flow velocities.51

Thus, while the appearance of stationary concentration pat-terns does not theoretically depend on the nature of the os-cillating chemical reaction or the catalyst used, interactionsof the chemical constituents with the packing material mayaffect the stability and formation of patterns. This will be thesubject of further investigation.

ACKNOWLEDGMENTS

We thank our collaborators A. J. Sederman, L. F.Gladden, and S. K. Scott.

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