Physrca 2D (1981) 536-544 North-Holland Pubhshmg Company

EXPERIMENTS ON ELECTRIC FIELD - BZ CHEMICAL WAVE INTERACTIONS: ANNIHILATION AND THE CRESCENT WAVE

R. FEENEY, S. L. SCHMIDT and P. ORTOLEVA Department of Chemrstry, Jndrana Untuersrty, Bloomrngton, Indrana 47405, USA

Recerved 28 January 1980

Expertmental evidence of the mfluence of an electrtc field on chemtcal waves is found m the Belousov-Zhabotmsku (BZ) medium. Effects found mclude (1) the vartatton of wave veloctty wtth field, (2) the exrstence of a crttrcal field beyond whtch waves propagatmg toward the negative electrode are anmhtlated, and (3) a new crescent-shaped wave phenomenon The experrmental findmgs are m qualitattve agreement wtth earher theorettcal predrcttons based on a new reductton of the FKN kmettcs A theory for the crescent wave IS also presented

1. Introduction

Chemical waves are propagating composition disturbances. They may have a great variety of profiles and have been extensively studied theoretically and experimentally (see ref. 1 for reviews). In the most well known wave sup- porting medium [2], the Belousov-Zhabotinskii (BZ) system, most of the species relevant to propagation are ionic. This suggests that ap- plication of electric fields to these BZ waves might have profound effects. In a recent series of theoretical studies [3-S] we have addressed this question and demonstrated that many in- teresting effects could be observed. It is the purpose of this study to demonstrate that many of these interesting new phenomena can, in fact, be observed in a slight modification of the BZ system.

Wave propagation in the BZ system can be understood on the basis of the Field-Koros- Noyes (FKN) mechanism 161 by focusing on the three fundamental species, Br-, HBrOz and Fe(phen):+. The basic process driving the wave apparently is autocatalytic HBr02 production; a model based on this autocatalytic process leads to a Fisher type equation [7] which predicts a velocity greatly in excess of the observed value.

The main factor that interferes with the HBr02

autocatalytic front is the Br- ahead of the wave. This effect was shown to bring the theoretical- ly predicted velocity more into line with the observed velocity both by numerical simulations [8] and by a new Stefan moving boundary method [9]. The effect of the Fe(phen)3+ in the electric field free medium is to restore the sys- tem back to the “rest state” that it had in advance of the oncoming HBr02 autocatalytic front (see fig. 1).

The three central BZ wave species all are affected differently by an electric field applied across a wave. If the field is in the direction of propagation of a field-free pulse then Br- is driven opposite to the oncoming wave, Fe(phen)3+ is dragged in the direction of the field-free motion and HBr02 is not directly affected. Thus, it is not surprising that applied electric fields could in principle severely alter the properties of BZ wave propagation. This was, in fact, the conclusion of our study of ref. 5 which analyzed BZ field effects based on various reductions of the FKN mechanism. We found that the effect of Br- was to further slow the wave down when the field was in the direc- tion of field free propagation and to speed it up when polarity was reversed-both effects being

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R. Feeney et al./BZ chemical waoe interactions. annihdation and the crescent waoe 531

readily understood in terms of the retarding effects of Br- on autocatalytic HBr02 produc- tion. More precisely, the electrical effect on the wave due to Br- alone is such that the velocity attains a plateau value, the Fisher limit, for E < EF and is decreasing to - ~0 for E > Er. For the known BZ mixture EF is of order 500 V/cm and the plateau region is not observable. The effect of Fe(phen)3f is much more profound. When it is pushed into the wave by a field in the direction of field free propagation it tends to terminate the HBrOz autocatalysis. Thus, in ref. 5, we predicted the existence of an annihilation field E, beyond which the BZ waves are des- troyed. In particular we obtained

E, = v,(0)/.4Fe3+ (1.1)

where up(O) is the velocity of the field-free wave and &3+ is the mobility of Fe3+(phen). This is a very intuitive result - annihilation is obtained when the species Fe(phen)3+ that shuts off the HBrOz autocatalytic wave can just keep up with this wave because the Fe(phen)‘+ is driven by the field at a velocity EaAFe3+ equal to that of the HBr02 wave.

Several factors exist which mediated against these effects being observed previously: (1) the high conductivity of the typical wave mixture; (2) the relatively high wave velocity compared to a typical ionic drift velocity induced by fields of reasonable strength; (3) the presence of bubbles to destroy wave front geometry; and (4) the disruption of the kinetics due to the ohmic heating caused by the high currents needed to produce appreciable electric fields. In the present study, we modified the typical trigger wave medium - denoted solution I - and used filter paper, as suggested by Winfree [lo], to surmount these difficulties. In the course of our studies we developed two new media-denoted solutions II and III. The recipes for the three solutions are given in the appendix. Solutions II and III replace the usual counter ions Na+ and SO:- by H+ and BrO;. This allows a reduction

of ionic strength at given [H+l and [BrO;]. By reducing the ionic strength (and the conduc- tivity) higher fields can be induced at a given current density. Thus the ameliorization of problem (1) automatically alleviates problem (2). Furthermore, the ohmic heating, uEz, is reduced as o is reduced, where u is the conductivity and E is the electric field. Two additional benefits resulted from the introduction of the new solu- tions: 1) the zero-field wave velocity was found to be lower in solutions II and III than in I; this may be due to the salt effect on the kinetics of the reactions; 2) gassing was considerably decreased, possibly for the same reason. The use of filter material completely eliminated problem (5) although the theoretical results of ref. 5 indicate the need for higher fields on such material relative to the higher transport rates in the free liquid (problem 2). The coincidence of all these factors due to our modifications of the BZ medium allowed us to observe the phenomena reported in this work. We believe that these observations open up a new direction in the study of chemical waves.

2. Comparison of solution properties

Let us now compare some properties of the three solutions. The typical mixture used by earlier researchers [lo], denoted solution I, has a pH of around 0.05. These figures (and those for other solutions) change during the course of the reaction. Solution II was developed to reduce the conductivity of the system by eliminating the irrelevant K+ and SOi ions by directly synthesizing HBrO3 via a procedure outlined in ref. [ll]. The pH of this mixture is approximately 0.38 initially with an ionic strength of 0.49. Note this is a decrease by a factor of 3.4 below the ionic strength of solution I and hence (roughly) a similar factor for the decrease of the conduc- tivity.

Decreasing [H+] and the ionic strength has

538 R Feeney et al /BZ chemical waue mteractrons annrhM~on and the crescent waue

interesting effects on the trigger waves. Unlike solution I (and III) where extensive gassing is observed (in a dish or on filter paper) after about 5 min, little or no gassing is found for about 30 min. Another important effect is the lowering of the velocity by lowering the [H+] [BrO;] product since

up(O) a WJ’lWOJI”*, (2.1)

where q(0) is the velocity of planar waves in the field free medium [12].

Solution III was designed to imitate as closely as possible the [H+] and [BrO;] of solution I while decreasing the conductivity. The pH of solution III is 0.05 as in solution I while the ionic strength is only 0.901, down by a factor of about 1.8 from solution I. The field free wave velocity in solution III is essentially the same as for solution I although some subtle differences in wave profile and color were observed.

3. Experimental method

The experiment was carried out on Metricell GA-4 grid membrane, pore size 0.8 pm, in strips on a glass plate between Al foil electrodes using solution II as the reaction medium. Electrodes were placed 4.0 cm apart with facing edges parallel, held rigidly by being sandwiched under glass at the edge of the test membrane. The electrodes were connected to a dc voltage sup- ply, with a polarity reversing toggle switch in series for fast voltage reversal studies. Results were recorded photographically in 35 mm for- mat on Kodak Plus-X film at l/30 s and F4. The paper was illuminated from the side with blue light and photographed through an 80B (blue) filter at 10 s intervals.

The reagents were mixed to form solution II and several drops placed on the membrane, which was allowed to saturate. Excess solution was removed from the membrane, and an 18 mm cover slip placed in the middle to slow

evaporation. A given potential was applied and the results recorded photographically as a func- tion of time (see fig. 2).

4. Results

A velocity response curve for solution II is shown m fig. 3. All runs are taken during the first several minutes of reaction to minimize variations due to an overall change of con- ditions as reactants are used up. Three measurements were simultaneously taken from each photograph of a circular wave that formed spontaneously under the cover slip. We measured the velocity of a wave propagating toward the positive and negative electrodes and perpendicular to the field. Waves propagating perpendicular to the field should not be effected (as shown theoretically in the next section). Thus, the variations of the velocity of per- pendicular waves shown in fig. 1 are caused by ohmic heating or evaporation and serve as a control to “normalize” the results.

We observe that waves propagating toward the positive electrode increase their velocity

,” ------ ___ . . ____-____-_-__-_--__

‘\ **

‘\ . \ \ -7 ‘;; \ I

I I \

-7 \ I 05

_ t I- I

\

x * : \ 7 f

\ , / x 1 0. -__

_~WO Itexpanded sc~lel

I I I I c I I I 1 1 1 -8 -6 -4 -2 0 J/ 0 2 4 6 8 10

qJ(ZO2S)

Fig 1 Wave profile as predIcted m the theory of ref 1 based on a new “IU-ator” reduction of the FKN kmetlcs Reduced concentrations x, y and z are proportIona to [HBrO& [Br-] and [Fe(phen):‘] respectively (see ref 1 for more precise defimtions) The spatml coordinate 4 IS that moving with the wave whereas the expanded scale c(- 54) IS Included to show the narrow pulse structure near the

origin

R Feeney et al.lBZ chemtcal wave interactions* annlhllatlon and the crescent wave

with increasing applied field strength. Waves propagating toward the negative electrode have their velocity decreased as the field strength is increased in absolute value.

Let us take the convention that the voltage is positive if the field-free wave propagates towards the positive electrode. Then between -IO- and - 15 V/cm we find that the wave is annihilated. This effect is shown in the temporal sequence in the photographs shown in fig. 2 where an electric field stronger than the anni- hilation level is applied across an initially cir- cular wave. Note that the part of the wave propagating essentially toward the positive electrode is accentuated. As one goes around the circle the component of the field perpendicular to the wave front decreases and hence the velocity decreases. Most dramatic is that there is a region of the circle that is com- pletely eliminated on the side of the circle whose direction of propagation is towards the negative electrode. This is a very important observation. It completely eliminates the pos- sibility that the effects observed are caused by some hydrodynamic flow superimposed on the wave. This annihilation phenomenon was in fact predicted using a new “IU-ator” reduction of the FKN kinetics [5]. In fact this theory predicts an annihilation field of about + 9 V/cm. The agreement of theory and experiment increases our confidence that we are seeing a true electric field effect.

Reversal of the field (with strength less than the annihilation field) leads to a reversal of the various tendencies in an expected manner. If the field strength is beyond the annihilation value, complex patterns of partial arc waves are formed. Reversing the field periodically at a large field strength (- 50 V/cm) produced a “disco” effect with waves dancing in rapid res- ponse to the voltage reversals. This disco effect is more than humorous; by oscillating the field we are sure that’no electrode reaction products (such as H+ and OH-) are transported into the observation area, causing a trivial concentration

0 -

539

Fig. 2 Breakdown of a cucular tngger wave into a crescent upon subjection to a supra-annihdation field ( f ) mdlcates the posItIon of the (?) electrode and t is the time (t = 0 when the field is turned on). (Note the apparent overall concentration gradlent is not real but an artifact of the

dlumination configuration.)

540 R Feeney et al IBZ chemml wave interactions annhdation and the crescent waoe

gradient and not a true electrical field effect. Another argument in favor of a true electrical effect stems from the observation that if the electric field is turned on, annihilation occurs at any point in the medium (for waves propagating in the negative direction) in a supra-anm- hilation field. Also the free ends of the partial circular wave continue to move toward the negative electrode. If OH- was moving in a front from the negative electrode and causing the annihilation, then the free ends would move away from the negative electrode, in contradic- tion to the observed effect.

Our experiments with circular waves in supra- annihilation fields serve to demonstrate the existence of a new wave phenomenon, the “crescent” wave. This is evident in the last photograph in fig. 2. The crescent wave is a purely electrical field phenomenon. If the field is shut off it degenerates into a pair of Winfree’s counter rotating spiral waves [lo]. The presence of the applied field stabilizes the free ends of the crescent.

Our velocity response u(E) is shown in fig. 3 in normalized form so as to compare it with the theory of ref. 5 based on a new reduction of the FKN mechanism. The theory and experiment are found to be in good agreement on this

annlhllatlon

I I I I I 1 -10 -05 00 05 10

E/E,

Fig. 3 Velocrty response to an applied electnc field for the BZ waves of solution II The velocity IS normalized to its zero field value whde the field IS normahzed to the anm- hdatlon field E.. Experimentally we observed - IO a E. a -15 V/cm whde the theory of ref 1, the sohd hne m this figure, predicts E. = - 9 V/cm. The existence of an anm- hilatlon field is strong evidence for a true electrical effect on

BZ trigger waves

normalized graph although absolute numbers, see the caption of fig. 3, are not quite so good.

5. Theory of the crescent wave

A simple theory of the crescent wave will now be presented. We proceed by examining the effect of an electric field applied at an arbi- trary angle to a plane wave and then assume that the curvature of the wave front is sufficiently small that a wave of arbitrary geometry behaves locally like a plane wave.

5.1. Orientational behavior of field-plane wave

interactions

Let c represent a column vector of concen- trations of chemical species taking place in the wave mechanics. Then, as in ref. 3, we take c to evolve according to

2 = DV% + ME . Vc + F(c), (5.1)

where E is the ohmic electric field, D and M are matrices of diffusion and mobility coefficients and F are the rates of reaction. Plane wave solutions for waves propagating along a con- stant electric field of strength E in the y-direc- tion have profile xp(&, E) where x,, satisfies

Ox;: + ( vp + ME)x; + F(xP) = 0 (5.2)

and Q(E) is the field dependent wave velocity, &, = y - v,t is a spatial coordinate moving with the wave and ’ = d/d+,. This plane, parallel case has been studied in detail in ref. 5 for several reductions of the FKN mechanism.

For the purpose of constructing a simple theory of crescent waves we need to know the response of a plane wave propagating in a direction of a (unit length) vector li to an ohmic field E at arbitrary angle with respect to ri. As one might expect, the result is the same as if

R Feeney et al./BZ chemacal waoe otteractions* anmhilatron and the crescent waue 541

the wave was propagating parallel to a field of magnitude fi l E, i.e., the component of the field perpendicular to the wave front. Indeed, the solution

xa(fi - r - ut, E) = x,,(A . r - ut, fi - E), (5.3)

u(E, ri) = u,(fi * E), (5.4)

is easily shown to satisfy (5.1) and corresponds to a plane wave propagating along the direction A in the presence of an arbitrarily oriented constant field E.

5.2. An approximate wave front equation

As observed experimentally, we assume that the plane wave xP is a pulse. In creating the crescent we allow a circular wave to grow to a radius R. and then turn on an ohmic field Ej taken to be in the y-direction (9 being a unit vector along the y-direction). If the field is supra-annihilation, i.e. jEI> -E,, then a crescent is formed. We shall try to approximate the phenomenon assuming that the wave looks essentially like a plane wave bent in an ap- propriate shape’. This appears to be consistent with the observations and must be correct if the curvature is sufficiently small. From the result of subsection 5.1 above we expect that to a good approximation

c = ~~(4, Eii ‘9) + 6c, (5.4)

where SC is “small” and 4(r, t) is an appropriate coordinate telling what phase of the plane wave one is close to at r, t. Also, ri = V4//lV41. To see in what sense this is reasonable and to get an equation for $J, we substitute (5.4) into (5.1) and compare with (5.2). One finds that in order that 6c be small we must have IV41 = 1, V24 small, &$/at = u,(Eii l 9). We find that assuming fi is slowly varying we may take

4 = ri - r - u,(Eri . f)t + a,

where (Y is also slowly varying.

(5.5)

5.3. The exploding crescent

Let us follow the wave via the profile of one species. We may always choose conventions such that this species reaches an extremum as the pulse passes at 4 = 0, i.e. this species (say number 1) reaches the maximum value xP,l(0) as the pulse passes by. Let us represent the crescent in polar coordinates r, 8. At t = 0 the wave front is a circle of radius denoted Ro. For a circle ri = 3 and hence we have

ri(r, t = 0) = ?, (5.6)

Ly(r, t =0)=-R,-,. (5.7)

where we have chosen the initial circle to denote the location of the extremum in xP,l. But since fi and (Y are slowly varying we assume, very crudely, that they maintain these initial values. Hence 4 takes the form (noting ri l 9 =

cos 6) C$ = r - u,,(E cos tl)t - Ro. Denoting the wave front to be at radial distance R(8, t) we thus obtain, since d(r = R, 8, t) = 0,

R(f3, t) = Ro + u,(E cos 0)t. (5.8)

To proceed we need the field dependence of up. From the experiments presented here and the

theory of ref. 5 it is reasonable to assume that there is (approximately) a linear relation be- tween up and the field. Letting b be a constant, we thus assume

up(%) = up(O) + b8, (5.9)

where 8 is a field strength. Combining this with (5.8) we get the cardioid

R(B, t) = Ro+ [up(O) + bE cos @It, E cos 8 > E,. (5.10)

The restriction in (5.10) insures that annihilation does not occur. However, at the back of the crescent this does occur in a region around

542 R Feeney et al ISZ chemcal waue mferactmns annrhht~on and the crescent wave

0 = 7~ defined by

cos 8 < E,IE. (5.11)

These results may now be compared to the observations of fig. 2. The crest of the wave is at a distance R(8, t) from the center of the initial circle. The development is shown in fig. 4 for a sequence of increasing times. The domain of annihilation (5.11) is indicated by a dotted segment of the wavefront. Note that as time advances the free ends of the comet move towards the negative electrode. If the destruc- tion of the back of the circular wave was due to a front of OH- coming from a negative elec- trode reaction then the free end would be mov- ing towards the positive electrode. In fact since there are actual free ends (as evidence by the

0

0 Fig 4 Exploding crescent wave predlcted on the basis of the theory of sectlon 4 Solid lmes mdlcatmg posltion of wave crest Dotted segments indicate regions where the component of the field m the dIrectIon of propagation is

sufficient to cause anmhdatlon

fact that they curl around into Winfree spirals if the field is suddenly turned off) argues against a pH gradient model which would predict a varia- tion of velocity via formula (2.1) but no stable free end. Thus, again, we find detailed evidence of the fact that the observed effects are true electrical field-chemical wave phenomena

A word of caution is in order here. The above theory surely must break down at the angles 8, such that cos 8, = EJE at which annihilation of the front causes a free end. Mathematically describing these dangling end effects is beyond the scope of the present study. We observe, however, that an end cannot just curl around and form a Winfree spiral (as they do if the field is suddenly turned off). As they attempt to do so, the wave normal becomes more antiparallel to the field and the curl is always annihilated. Thus, the dangling ends have a clear measure of stability although their detailed nature is not given by our present theory.

In conclusion, the crescent wave IS a second

example of a stable free ended reaction diffusion wave - Winfree’s spiral being the first such wave.

6. Remarks

It is our purpose here to demonstrate the existence of a multiplicity of phenomena which occur when chemical waves are subjected to electrical fields. The theory of ref. 5 predicts a velocity response function as indicated by the solid line in fig. 3, m qualitative agreement with the present experiments. Perhaps the most stringent test of the theory is the prediction of an annihilation field as found in the present experimental study. A consequence of this annihilation phenomenon is the creation of crescent waves.

All these effects were made accessible by using solution II which was designed to mini- mize adverse factors as mentioned in section 1. In work m progress we are fine-tuning these

R. Feeney et al./BZ chemrcal wave rnteractrons* anmluletron and the crescent wave 543

experiments to establish better controls and obtain more accurate measurements. As a byproduct of these experiments and the theoretical considerations we hope to gain new insight into the wave propagation kinetics of the BZ waves as described by FKN and more recently postulated mechanisms [13]. Indeed, the theoretical analyses of refs. 3-5 show that the velocity response depends sensitively on chemical kinetics. Thus it is hoped that our

future, more carefully controlled experiments will help to determine which mechanism is most accurate.

Acknowledgements

The authors gratefully acknowledge very useful exchanges and encouragements from R. J. Field, R. M. Noyes and A. T. Winfree.

Appendix

Reagent Contamer # Stock cone Rxn cone Partral vol

H2S04 KBrO, NaBr

CHdCOOHh

Ferroin

HBrO3 NaBr CHr(COOH)z Ferroin

HBr03 HBr CHz(COOH)z Ferroin

Solutron I

0514M 0.473 M 0.972 M 1.389 M 0.025 M

Solution II

0.363 M 0.972 M 1389M 0.025 M

0 473 M 666M 1389M 0025M

0 363 M 0334M 0.057 M 0 163 M 0003M

0256M 60 0.057 M 05 0 163 M 10 0003M 1.0

0334M 60 0392M 05 0 163 M 10 0003M 10

60

05 10 10

In all three cases, mtx reagents m contamers 1, 2 and 3, and allow the Brl produced to dissipate When solutron is clear, strr m the Ferrom

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