Lateral instability controlled by constant electric field in an

acid-catalyzed reaction

Zsanett Viranyi, Antal Szommer, Agota Toth and Dezs}oo Horvath*

Department of Physical Chemistry, University of Szeged, P.O. Box 105, Szeged H-6701,Hungary. E-mail: horvathd@chem.u-szeged.hu

Received17th February 2004, Accepted 22ndMarch 2004F|rst published as an AdvanceArticle on theweb 4thMay 2004

We have investigated the effect of externally applied constant electric field on the pattern formation in theacid-catalyzed chlorite–tetrathionate reaction. Using electric field separating the reactant anions and theautocatalyst cation, lateral instability can be induced while planar fronts can be stabilized with the electric fieldin the opposite orientation. A three-variable model describes the behavior of the system and the results arein good agreement with the experiments. Linear stability analysis has also been carried out to determinethe onset of instability, and the simulations predict that the average wavelength does not vary significantlywith increasing the separation of ions.

I. Introduction

One of the simplest pattern formation occurs in reaction-diffu-sion fronts of an autocatalytic reaction1,2 when planar frontslose stability and cellular patterns evolve. The driving forcein the lateral instability of planar fronts is the slower diffusionrate of the autocatalyst at the front with respect to the reac-tant.3 If the diffusion coefficient of the autocatalyst is suffi-ciently smaller than that of the reactant, planar fronts losestability giving rise to cellular fronts in cubic or higher orderautocatalysis. Since diffusion coefficients are hard to tune inaqueous solutions, the first experimental realizations of thephenomenon have used immobile binding of the autocata-lyst,4,5 a technique that has been successful in slowing downthe activator in experimental studies of Turing patterns.6,7 Acareful look at the governing equations has revealed that theflux of the autocatalyst across the front has decreased as aresult of the decrease in its concentration behind the front.Therefore any effect in the system that lowers the build-up ofautocatalyst in the wake of the front leads to the destabiliza-tion of the planar symmetry and hence gives rise to theformation of cellular structures.In a series of theoretical works we have shown that the

concentration of the autocatalyst produced may be lowered bythe introduction of reversible removal into an immobilecompound,8 a slow irreversible decay,9 and—in case of ionicspecies—a differential migration parallel to the direction offront propagation.10,11 Unlike other studies of the effect ofexternal electric field on pattern formation,12–15 here a specificfield orientation only leads to the decrease in the overlap ofreacting species at the front without drastically changingthe front characteristics10,16 which then not only changes thevelocity of propagation but also results in the loss of stabilityfor planar fronts.The chlorite oxidation of tetrathionate is an acid-catalyzed

reaction and lateral instability is observed experimentally whenit is run with partial immobilization of the autocatalyst. Thereaction in slight chlorite-excess runs according to17

7ClO2� þ 2S4O6

2� þ 6H2O ¼ 7Cl� þ 8SO42� þ 12Hþ ð1Þ

with reaction rate r ¼ k[ClO2�][S4O6

2�][Hþ]2. Reversible bind-ing of the autocatalyst is achieved via binding the hydrogen

ions to carboxylate groups

M COOH Ð M COO� þHþ; ð2Þ

where M–COO� corresponds to sodium methacrylate in theimmobile matrix of an acrylamide/bisacrylamide/metha-crylate hydrogel and Kd is the dissociation constant of theimmobilizing agent.5

In this work, we use the chlorite–tetrathionate reaction toshow experimentally how electric field affects the lateral stabi-lity of planar fronts. Since hydrogen ions diffuse significantlyfaster than small ions in aqueous solutions, binding accordingto eqn. (2) is applied in the experiments as well. A linearstability analysis is also carried out on the three-variable modelbased on eqns. (1) and (2) to determine the phase diagramof the system. The dispersion curves characterizing the initiallyobserved patterns are then calculated for various parameters.

II. Modeling study

A. Governing equations

The system with eqns. (1)–(2) can be described in a constantelectric field by

@ S4O2�6

� �@t

¼ DS4O2�6H2 S4O

2�6

� �� 2r� ES4O

2�6

@ S4O2�6

� �@x

;

ð3Þ

@ ClO�2

� �@t

¼ DClO�2H2 ClO�

2

� �� 7r� EClO�

2

@ ClO�2

� �@x

; ð4Þ

@ Hþ½ �t@t

¼ S@ Hþ½ �@t

¼ DHþH2 Hþ½ � þ 12r� EHþ@ Hþ½ �@x

; ð5Þ

where Ei ¼ ziDiFEx/(RT ) for every component is based onthe Einstein relation,10,11 and [Hþ]t ¼ [Hþ]þ [M–COOH] isthe total concentration of hydrogen ion. The factorS� 1þKd[M–COO�]t/(Kdþ [Hþ])2 arises from the presenceof an immobile binding agent with the overall concentrationof [M–COO�]t ¼ [M–COO�]þ [M–COOH]. Simplification ofeqns. (3)–(5) by introducing dimensionless variables and theassumption that small aqueous ions—with the exception of

PCCP

www.rsc.o

rg/pccp

R E S E A R C H P A P E R

3396 P h y s . C h e m . C h e m . P h y s . , 2 0 0 4 , 6 , 3 3 9 6 – 3 4 0 1 T h i s j o u r n a l i s Q T h e O w n e r S o c i e t i e s 2 0 0 4

DOI:10.1039/b402382j

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hydrogen ion—have the same diffusion coefficients yield

@a@t

¼ ~HH2a� 2abg2 � zAe

@a@x

; ð6Þ

@b@t

¼ ~HH2b� 7abg2 � zBe

@b@x

; ð7Þ

s@g@t

¼ d~HH2gþ 12abg2 � zCde

@g@x

; ð8Þ

where a, b and g are the dimensionless concentrations ofS4O6

2�, ClO2�, and Hþ, respectively, relative to [S4O6

2�]0 , andt ¼ k[S4O6

2�]30t is the dimensionless time scale. The dimen-

sionless spatial coordinates are given as x ¼ x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik S4O

2�6

� �30=D

qin the direction of front propagation and Z—analogously tox—perpendicular to that, defining ~HH

2 ¼ @2/@x2þ @2/@Z2.The relative diffusivities are defined by d ¼ DHþ/D and

the dimensionless electric field in the x-direction is

e ¼ ExFD1=2

�RT

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik S4O

2�6

� �30

q� �. The reversible binding is

expressed as s ¼ 1þKcm/(Kþ g)2 where K ¼ Kd/[S4O62�]0

and cm ¼ [M–COO�]t/[S4O62�]0 .

Ahead of the front (x!1), the boundary conditions corre-spond to the reactant solution as a ¼ 1, b ¼ (kþ 7)/2 andg ¼ 0, where k ¼ 2[ClO2

�]0/[S4O62�]0� 7 is the chlorite

excess. In the reaction the tetrathionate ion is the limitingreagent and hence a ¼ 0, b ¼ bs and g ¼ gs behind the front(x!�1), where the concentration of the hydrogen ionproduced gs and that of the remaining chlorite bs can bedetermined from the solution of planar fronts.

B. Planar fronts

For steady state solution we introduce z ¼ x� ct travelingcoordinate with c being the planar front velocity, and trans-form eqns. (6)–(8) into

0 ¼ d2a

dz2� 2abg2 þ c� zAeð Þda

dz; ð9Þ

0 ¼ d2b

dz2� 7abg2 þ c� zBeð Þdb

dz; ð10Þ

0 ¼ dd2g

dz2þ 12abg2 þ cs� zCdeð Þdg

dz: ð11Þ

The composition of the product solution can be determined byusing the component balances, i.e., integrating eqns. (9)–(11)for the entire space, and applying zero-flux boundary con-ditions. For the limiting reagent this yields a value for theintegral of the reaction rate from eqn. (9)

Zþ1

�1

abg2dz ¼ c� zAe2

:

Considering eqns. (10)–(11) in which

Zþ1

�1

cs� zCdeð Þdgdz

dz ¼ cS� zCdeð Þg½ �þ1�1;

where S ¼ 1þ cm/(Kþ g), we obtain—after rearrangement—that

bs ¼ � 7

2

c� zAec� zBe

þ kþ 7

2;

gs ¼ 6c� zAe

cS� zCde¼

�BþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2 þ 24K c� zAeð Þ c� zCdeð Þ

q2 c� zCdeð Þ ;

where B ¼ ccmþK(c� zCde)� 6(c� zAe).

A shooting method is used to solve the equivalent of eqns.(9–11)

dadz

¼ v1; ð12Þ

dbdz

¼ v2; ð13Þ

dgdz

¼ v3; ð14Þ

dv1dz

¼ 2abg2 � c� zAeð Þv1; ð15Þ

dv2dz

¼ 7abg2 � c� zBeð Þv2; ð16Þ

dv3dz

¼ � 12abg2

d� cs� zCdeð Þv3

d; ð17Þ

where eqns. (12)–(14) are the defining equations for the auxiliaryvariables. The sought front profile is represented by a trajectorythat leaves the unstable steady state corresponding to theboundary condition at z ¼ �1 and arrives at the stable steadystate corresponding to the original initial condition at z ¼ þ1.

C. Linear stability analysis

It is convenient to introduce m(g) ¼ 1/s(g) for expressing thebinding and hence the continuity equation for hydrogen ionin the moving coordinate system (eqn. (8)) becomes

@g@t

¼ dm@2g

@z2þ dm

@2g@Z2

þ c� zCdmeð Þ @g@z

þ 12abg2m: ð18Þ

The perturbations in the concentrations are introduced as

a ¼ a0 zð Þ þX1k¼1

a1;k zð ÞFk Z; tð Þ ¼ a0 zð Þ þX1k¼1

a1;k zð ÞeotþikZ;

b ¼ b0 zð Þ þX1k¼1

b1;k zð ÞFk Z; tð Þ ¼ b0 zð Þ þX1k¼1

b1;k zð ÞeotþikZ;

g ¼ g0 zð Þ þX1k¼1

g1;k zð ÞFk Z; tð Þ ¼ g0 zð Þ þX1k¼1

g1;k zð ÞeotþikZ;

ð19Þ

where a0 ,b0 ,g0 are the planar front solutions resulting in

m ¼ m0 zð Þ þX1k¼1

m1 zð Þg1;k zð ÞFk Z; tð Þ

¼ m0 zð Þ þX1k¼1

m1 zð Þg1;k zð ÞeotþikZ;

with m0 ¼ m(g0(z)) and m1 is dm0/dg evaluated at g0 .Substitution of the variables leads to

oþ k2 0 00 oþ k2 00 0 oþ dm0k

2

0@

1A a1;k

b1;kg1;k

0@

1A ¼ LL

a1;kb1;kg1;k

0@

1A ð20Þ

for the terms linear in the perturbation for each k, where

LL¼

d2

dz2þ c�zAeð Þ d

dz�2b0g

20 �2a0g20 �4a0b0g0

�7b0g20

d2

dz2þ c�zBeð Þ d

dz�7a0g20 �14a0b0g0

12m0b0g20 12m0a0g

20 L3;3

0BBBBB@

1CCCCCA

ð21Þwith

L3;3 ¼ dm0d2

dz2þ c� zCdm0eð Þ d

dzþ dm1

d2g0dz2

� zCdm1edg0dz

þ 12a0b0g0 2m0 þ m1g0ð Þ:

T h i s j o u r n a l i s Q T h e O w n e r S o c i e t i e s 2 0 0 4 P h y s . C h e m . C h e m . P h y s . , 2 0 0 4 , 6 , 3 3 9 6 – 3 4 0 1 3397

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The k ¼ 0 mode is the translation of the planar front in thedirection of front propagation

a zþ dzð Þ ¼ a0 zð Þ þ da0=zð Þdz ;b zþ dzð Þ ¼ b0 zð Þ þ db0=dzð Þdz ;g zþ dzð Þ ¼ g0 zð Þ þ dg0=dzð Þdz ; ð22Þ

which is an invariant solution, for which we have

0 ¼ LLda0=dzdb0=db0dzdg0=dz

0@

1A:

Comparison of eqn. (22) with eqn. (19) and the equation aboveyields that o ¼ 0 for mode k ¼ 0 since

0 ¼ LL

a1;0b1;0g1;0

0@

1A : ð23Þ

The onset of instability can be determined by studying thenature of the dispersion relation (o(k)) at the origin. Small per-turbations are amplified in time resulting in a cellular structureonly if the slope of the curve at the origin is positive. The onsetof convection, therefore, occurs when do/dk2 changes its signat k ¼ 0.If we consider the small neighborhood of k ¼ 0 with k2 ¼ E

where E is an infinitesimally small positive number, the growthrate o ¼ O(E), and for a given k

a1;k ¼ a1;0 þ ~aa Eð Þ;b1;k ¼ b1;0 þ ~bb Eð Þ;g1;k ¼ g1;0 þ ~gg Eð Þ:

Following the substitution into eqns. (20)–(21), the terms withzeroth-order in E returns the planar solution (eqn. (23)) whilethose with first-order in E lead to

o Eð Þ þ k2 Eð Þ 0 0

0 o Eð Þ þ k2 Eð Þ 0

0 0 o Eð Þ þ dm0k2 Eð Þ

0B@

1CA

�a1;0b1;0g1;0

0B@

1CA ¼ LL

~aa Eð Þ~bb Eð Þ~gg Eð Þ

0B@

1CA : ð24Þ

The solvability condition for eqn. (24) requires that

Zþ1

�1

C1C2C3ð Þoþ k2 0 0

0 oþ k2 0

0 0 oþ dm0k2

0B@

1CA

a1;0b1;0g1;0

0B@

1CAdz¼ 0 ;

ð25Þ

where C1 , C2 , and C3 are the components of the eigenvectorcorresponding to the zero eigenvalue of the adjoint matrixoperator (L*).Since o/k2! do/dk2 as E! 0, rearranging eqn. (25)

provides the slope of the dispersion curve at the origin as

dodk2

¼ �

Rþ1

�1C1a1;0 þC2b1;0 þC3dm0g1;0� �

dz

Rþ1

�1C1a1;0 þC2b1;0 þC3g1;0� �

dz: ð26Þ

Components of the eigenvector can be determined by solving

LL� C1

C2

C3

0@

1A ¼ 0 ; ð27Þ

where

LL� ¼

d2

dz2� c�zAeð Þ d

dz�2b0g

20 �7b0g

20 12m0b0g

20

�2a0g20d2

dz2� c�zBeð Þ d

dz�7a0g20 12m0a0g

20

�4a0b0g0 �14a0b0g0 L�3;3

0BBBBB@

1CCCCCA

;

with

L�3;3 ¼ dm0

d2

dz2þ 2d

dm0dz

� c� zCdm0eð Þ

d

dzþ d

d2m0dz2

þ dm1d2g0dz2

þ 12a0b0g0 2m0 þ m1g0ð Þ:

III. Numerical study

A. Two-dimensional calculations

The temporal evolution of cellular structures was investigatedby solving eqns. (6)–(8) with an explicit Euler method andapplying zero-flux boundaries. The standard 9-point formulawas used to approximate the Laplacian on 401� 401 gridpoints with h ¼ 0.2 and a time step of Dt ¼ 0.001. For thecalculations d ¼ 3, K ¼ 0.001 and cm ¼ 3.6 or 4.2 were chosen.

B. Onset of instability

A shooting method based on eqns. (12)–(17) was used to deter-mine the velocity of propagation and the planar front profilewith a relative error of 10�14. A relaxation method was thenapplied on 100 001 grid points with h ¼ 5� 10�4 to solveeqn. (27) with a relative error of 10�8 for the components ofC. For boundary conditions, the concentration profiles andthe components of C were matched to exponential decays gov-erned by the eigenvalues obtained from the stability analysis ofthe fix points in the appropriate phase space, corresponding tothe limits at �1 in the physical space. In a Newton–Raphsonscheme the slope of the dispersion was evaluated accordingto eqn. (26) by varying cm until the onset of instabilitywas reached within a preset error of 10�6 for a given valuesof e. The CVODE package18 was used for the numericalintegration.

C. Dispersion relation

For the calculations, we introduced the perturbations in thefollowing form: a1,kexp(ikZ), b1,kexp(ikZ), and g1,kexp(ikZ),where a1,k , b1,k , and g1,k now contain the time dependenceas well. Following previous work,11 we obtain for the first-order terms that

@a1;k=@t@b1;k=@t@g1;k=@t

0@

1A ¼ LL

a1;kb1;kg1;k

0@

1A�

k2 0 00 k2 00 0 dm0k

2

0@

1A a1;k

b1;kg1;k

0@

1A :

ð28Þ

The comparison of eqns. (20) and (28) reveals that with areasonable trial function (y1 ,y2 ,y3) for the perturbation, aftera transient period we have

@y1@t

=y1 ! o;@y2@t

=y2 ! o;@y3@t

=y3 ! o ; ð29Þ

and at the same time

y1 ! a1;k; y2 ! b1;k; y3 ! g1;k;

since there exists a distinct value for o; and all other modesdecay rapidly. A similar relaxation method was used to solveeqn. (28), as described in the previous section for the onsetof instability. The growth rate (o) was calculated with a

3398 P h y s . C h e m . C h e m . P h y s . , 2 0 0 4 , 6 , 3 3 9 6 – 3 4 0 1 T h i s j o u r n a l i s Q T h e O w n e r S o c i e t i e s 2 0 0 4

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relative error of 10�6 by incrementing k and taking the pertur-bation calculated at each step as the trial function for the next.Throughout the theoretical study, the selected values for the

parameters corresponded to the experimental system: cm ¼ 0–4.8, d ¼ 3. The scaled constant for the binding (K ¼ 0.1) wassomewhat greater than the experimental value to achieve rea-sonable speed during the integrations because its value hadonly minor effect on the stability in reversible binding as longas it was less than one.8 The unit scale of the dimensionlesselectric field strength corresponded to the order of 1 V cm�1.

IV. Modeling results

In the chlorite–tetrathionate system the reactants are nega-tively charged, while the autocatalyst is positively charged,therefore the positive electric field tends to enhance the overlapof the key species leading to an increase in the rate of reactionat the front. Hence planar fronts exhibit an increasing velocityof propagation with e at a constant binding. A slowing down isobserved in the opposite orientation as shown in Fig. 1, wherethe negative field tends to separate the ions, leading eventuallyto the quenching of the autocatalytic reaction for variousimmobilization of the autocatalyst. Beyond the critical fieldstrength, only independent electrophoretic fronts exist foreach ion with a charge-dependent drift velocity and with anincreasing width due to diffusion.At sufficiently strong binding (cm > 3.360), planar fronts

lose stability and cellular fronts develop with the criticalcm—defining the onset of lateral instability—strongly depen-dent on e (see Fig. 2). At a given composition, the initially pla-nar structure can therefore be destabilized on increasing thenegative electric field strength and planar structures can be sta-bilized from an initially cellular pattern on increasing the posi-tive electric field strength, as shown in an example in Fig. 3.The evolving patterns can be characterized by the dispersion

curve for a given condition. Fig. 4 clearly demonstrates that onincreasing positive field strength the region of instabilityshrinks while at e ¼ 0.00727 it disappears. Further increaseonly leads to stabilization at shorter timescales. The oppositelyoriented electric field, on the other hand, results in the separa-tion of important ions which destabilizes the initially planarstructure according to Fig. 5. On increasing the negative fieldstrength the region of instability expands at first, however,close to the appearance of electrophoretic fronts, only thegrowth rate of the most unstable mode keeps increasing, thewave number becomes almost constant. This suggests thattowards the critical field strength, the initially evolving pat-terns have approximately equal wavelength independent of e,while their amplitude gradually increases and the cellularstructures appear at shorter time scales.

V. Experimental

Throughout the experiments crosslinked polyacrylamide gelwas used to ensure a convection-free environment. The gel

Fig. 1 Velocity as a function of the electric field for various cm withd ¼ 3.

Fig. 2 Phase diagram showing the regions of stable planar reactionfronts (SP); lateral instability (LI); electrophoretic fronts (EF) in the(cm–e)-plane. Curves calculated from the linear stability analysisrepresent the onset of instability with dashed line and the extinction ofreaction-diffusion fronts with solid line.

Fig. 3 Images of fronts with e ¼ 0.02 (top) and e ¼ �0.02 (bottom)at cm ¼ 4.2 ([M–COO�]0 ¼ 21 mM). Dark regions correspond to thereactant, light regions to the product solution.

Fig. 4 Dispersion curves for fronts at various electric field with d ¼ 3and cm ¼ 4.5 ([M–COO�]0 ¼ 22.5 mM).

T h i s j o u r n a l i s Q T h e O w n e r S o c i e t i e s 2 0 0 4 P h y s . C h e m . C h e m . P h y s . , 2 0 0 4 , 6 , 3 3 9 6 – 3 4 0 1 3399

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was polymerized a day prior to carrying out the experiments,as described previously.5 A rectangular piece of gel was placedin the reactant solution for 30 min with the composition givenin Table 1. The continuous stirring of the solution above thegel ensured the homogeneous loading. Excess water was thenwiped off and the gel was layed between two Plexiglas plateswith a pair of platinum wires pressed onto opposite ends ofthe gel. A front was initiated electrochemically and its propa-gation was monitored using an imaging system. The electricfield was introduced by a galvanostat (Electroflex EF1307)parallel to the direction of front propagation, as shown inFig. 6. The constant electric field was achieved by adding 2M sodium nitrate as an inert conducting salt to the reactantsolution which suppressed the change in conductance in thecourse of the reaction to less than 1%. The applied electric fieldstrength was determined from the iR-drop arising across thegel according to

E ¼ I

Gl;

where I is the applied constant current, G is the conductance ofthe gel, and l is the distance between the platinum electrodes.The conductance was measured with a conductivity meter(Radelkis, OK-112) before and after each experiment withoutmoving the electrodes. The reaction fronts were examined suf-ficiently far from the electrodes, therefore any effects that couldarise from the product of electrolysis may be neglected.

VI. Experimental results

Upon introduction of the conducting salt at high concentra-tion, the mobility of free hydrogen ion increases resulting ina higher diffusion coefficient for the autocatalyst. The frontvelocity in the absence of electric field is therefore five timeshigher than in previous experiments, with significantly smallerionic strength.5 Fig. 7 illustrates that the more positive theapplied electric field, the greater the velocity of propagationis, as predicted by considering the charges. The sharp endsof the velocity curves near quenching (cf. Fig. 1.) are hard toobserve experimentally, since instead of the actual concentra-

tion profiles, the color change in pH indicator present in thesolution is monitored.The faster diffusing autocatalyst requires a higher concentra-

tion of sodium methacrylate as binding agent in the gel matrixfor the appearance of cellular structures. Under our experi-mental conditions in the absence of external electric field, theonset of lateral instability for planar fronts increases from33%5 binding to 65–70%, corresponding to cm ¼ 3.9–4.2.Immobilization above the onset of instability yields a cellu-

lar structure, as shown in Fig. 8(a) in the absence of electricfield. In gels of constant binding, the positive electric field—

Fig. 5 Dispersion curves for fronts at various electric field with d ¼ 3and cm ¼ 4.2 ([M–COO�]0 ¼ 21 mM).

Fig. 6 Scheme of the experimental setup. The position of electrodesfor negative electric field applied.

Fig. 7 Velocity as a function of the electric field for [M–COO�]0 ¼ 15 mM with respect to the velocity (c0) observed in theabsence of electric field.

Fig. 8 Cellular fronts at E ¼ 0 V cm�1, t ¼ 3.5 h (a) and stabilizedplanar fronts at E ¼ 0.080 V cm�1, t ¼ 1.5 h (b) with [M–COO�]0 ¼ 21 mM. Dark regions correspond to the reactants and lightregions to the products. The white bar corresponds to 1 cm.

Table 1 Composition of reactant solution

[K2S4O6]/mM 5.0

[NaClO2]/mM 20.0

[NaOH]/mM 1.0

[Congo red]/mM 0.574

[NaNO3]/M 2.0

3400 P h y s . C h e m . C h e m . P h y s . , 2 0 0 4 , 6 , 3 3 9 6 – 3 4 0 1 T h i s j o u r n a l i s Q T h e O w n e r S o c i e t i e s 2 0 0 4

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by enhancing the mixing of ions—stabilizes the planar struc-ture presented in Fig. 8b. The difference in behavior is morestriking in the temporal evolution of front amplitudes, definedas the distance in the direction of propagation between theforemost and hindmost segment of the front (see Fig. 9a).At weaker immobilization, planar fronts are stable in the

absence of electric field and they may be destabilized uponapplication of negative electric field as a result of the increasedseparation of the reacting ions, as shown in Fig. 9b. The regionof lateral instability is however relatively narrow, therefore nosignificant variation in the average wavelength can be deter-mined before reaching the domain of electrophoretic fronts.In conclusion, we have shown experimentally that ionic

migration—parallel to the direction of propagation—inducedby external electric field will affect the stability of planar auto-catalytic fronts. In positive electric field where the mixing of

the reactants and the autocatalyst is enhanced, the flux ofthe autocatalyst is greater with respect to that of the reactantsacross the front. Under these conditions, planar fronts may bestabilized. In the oppositely oriented field, a slight separationof the key ions leads to a decrease in the flux of the autocata-lyst, which then results in the appearance of cellular fronts atlower binding. Greater negative field strength completelyquenches the reaction giving rise to separately drifting planarelectrophoretic fronts. In accordance with the experiments,the theoretical study of the three-variable model based onthe empirical rate law reveals that the region of lateral instabil-ity wedged between those of the stable planar reaction frontsand the electrophoretic fronts is narrow, which is a directconsequence of the increased mobility of hydrogen ions becauseof the large concentration of the applied conducting salt.

Acknowledgements

This work was supported by the Hungarian Scientific ResearchFund (OTKA F031728).

References

1 I. R. Epstein, J. A. Pojman, An Introduction to NonlinearDynamics: Oscillations, Waves, Patterns, and Chaos, OxfordUniversity Press, 1998.

2 W. van Saarloos, Phys. Rep., 2003, 386, 29.3 D. Horvath, V. Petrov, S. K. Scott and K. Showalter, J. Chem.

Phys., 1993, 98, 6332.4 D. Horvath and K. Showalter, J. Chem. Phys., 1995, 102, 2471.5 D. Horvath and A. Toth, J. Chem. Phys., 1998, 108, 1447.6 V. Castets, E. Dulos, J. Boissonade and P. De Kepper, Phys. Rev.

Lett., 1990, 64, 2953.7 Q. Ouyang and H. L. Swinney, Nature (London), 1991, 352, 610.8 E. Jakab, D. Horvath, A. Toth, J. H. Merkin and S. K. Scott,

Chem. Phys. Lett., 2001, 342, 317.9 A. Toth, D. Horvath, E. Jakab, J. H. Merkin and S. K. Scott,

J. Chem. Phys., 2001, 114, 9947.10 D. Horvath, A. Toth and K. Yoshikawa, J. Chem. Phys., 1999,

111, 10.11 A. Toth, D. Horvath and W. van Saarloos, J. Chem. Phys., 1999,

111, 10 964.12 H. Sevcıkova, M. Marek and S. C. Muller, Science (Washington,

D. C.), 1992, 257, 951.13 O. Steinbock, J. Schutze and S. C. Muller, Phys. Rev. Lett., 1992,

68, 248.14 H. Sevcıkova, I. Schreiber and M. Marek, J. Phys. Chem., 1996,

100, 19 153.15 B. Schmidt, P. De Kepper and S. C. Muller, Phys. Rev. Lett.,

2003, 90, 118 302.16 H. Sevcıkova and M. Marek, Physica D, 1984, 13, 379.17 I. Nagypal and I. R. Epstein, J. Phys. Chem., 1986, 90, 6285.18 S. D. Cohen and A. C. Hindmarsh, Comput. Phys., 1996, 10, 138.

Fig. 9 The amplitude as a function of time (a) at E ¼ 0 V cm�1 (solidline) and at E ¼ 0.080 V cm�1 (dashed line) with [M–COO�]0 ¼ 21mM. The amplitude of a destabilized planar front (b) at E ¼ �0.028V cm�1 with [M–COO�]0 ¼ 18 mM.

T h i s j o u r n a l i s Q T h e O w n e r S o c i e t i e s 2 0 0 4 P h y s . C h e m . C h e m . P h y s . , 2 0 0 4 , 6 , 3 3 9 6 – 3 4 0 1 3401

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