Chemical wave-electrical field interaction phenomena

Physica 26D (1987) 67-84 North-Holland, Amsterdam

CHEMICAL WAVE-ELECTRICAL FIELD INTERACTION PHENOMENA* Peter ORTOLEVA Department of Chemistry, Indiana University, Bloomington, IN 47405, USA Received 21 October 1985 Revised manuscript received 6 October 1986

Chemical waves in ionic media are shown to display a great variety of nonlinear phenomena. An overview of the application of bifurcation and scaling theory to the electrical field-chemical wave interaction problem is presented. It is discovered that electrical fields can induce effective negative phase diffusion coefficients in limit cycle systems and negative mode diffusion coefficients for systems near a bifurcation point. Waves are shown to exist in the system subject to an applied field when such waves do not exist in the field-free system. Entities like periodic static structures are shown to have an effective mobility and charge while fronts in some systems have an effective charge that can change sign with concentration. A host of new nonlinear equations describing these phenomena are posed. It is found that the variety of and potential to express nonlinear phenomena in reaction-transport systems is greatly enhanced by imposed or self-generated electrical fields.

1. Autonomous and imposed field effects The coupling of electrical fields and chemical waves and patterns in ionic systems is expected to have interesting consequences. Indeed, the most extensively studied chemical wave medium, the BZ system, involves ionic species as do many flames and bioelectrical phenomena. Propagation of chemical signals is due to the interrelationships among the concentration profiles of the participating species (and temperature profile in the case of a flame). Since these relationships can be dramatically affected by electrical fields, it is quite reasonable to expect that qualitatively new types of wave behavior can arise due to wave-electrical field interactions. The purpose of the present paper is to pose and solve new problems to demonstrate the existence of a great variety of electrical field-chemical wave phenomena. Wave-field interactions are of two distinct origins. Ionic concentration gradients lead to departures from charge neutrality. Strong electrical *Research supported in part by a grant from the Chemistry Division of the U.S. National Science Foundation.

forces act to minimize this charge density; hence ionic migration is strongly correlated by the resulting, so-called, Planck potentials set up by the composition gradients. Such autonomous potentials can tend to force reactants together or keep them apart depending on ionic charges and hence can alter the nature of wave propagation. Background electrolytes decrease these potentials by increasing the conductivity of the medium. Hence it is quite conceivable that as ionic strength is decreased, qualitatively new phenomena could arise in nonequilibrium, reaction-transport systems. The response of a reacting system to applied electrical fields is also expected to be quite interesting since such fields can force reactants (in a cross-gradient configuration) into or away from each other. Applied oscillatory fields can cause mixing of species in nonuniform, nonlinear systems and hence new types of coupling and nonlocal behavior should arise (see chap. XI of ref. 1). In the present article we illustrate a number of nonlinear phenomena that can occur as a result of chemical wave-electrical field interactions. Bifurcation, limit cycle perturbation and other scaling

0167-2789/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

P. Ortoleva / Chemical wave-electrical field interaction phenomena

68

techniques are used to demonstrate the nonlinear phenomena. Emphasis is placed on obtaining results that apply to a wide range of wave supporting media. Before presenting the new results that are the focus of this article, we briefly review the literature on chemical wave-electrical field interactions.

2. Earlier results 2.1. Formulation The dynamics of reacting ionic media can be described by the coupling of reaction-transport equations and Maxwelrs equations. For processes taking place on time scales greater than a millisecond, the latter reduce under most situations to Poisson's equation for the voltage V, i.e. for an N-species system V2 V _

4wF --

E

z • c,

N

z • c = ~

ziC.

(2.a)

of motion in the wave-fixed coordinate system ep = r - vt become

d [DdC

dff,[ d99 + ( o - E M ) C dE dcp

] +R=0,

(2.4)

4~rF - - 7 - z * C,

(2.5)

where E = - d V / d ~ p is the electrical field. The vector notations C, M and D ( Mij = MiSij where 6ij is the unit m a t r i x - a n d similarly for Dis) are introduced for convenience. Letting E o be E ( % ) for a reference point %, we can obtain a closed equation for C in the form

d--~L

dqg-

E°+----~J~odq~z*C

MC

dC +v~- +R=0.

(2.6)

This is an integro-differential equation for steady electro-chemical wave propagation [2].

i=l

The dielectric coefficient, e, is assumed to be constant; F is Faraday's constant and z i and C i are the valence and concentration of species i, i = 1, 2 . . . . . N. The concentrations are taken to evolve according to

2.3. Charge neutrality coupling

Here D i is the/-diffusion coefficient and M i is z i F times the mobility of species i.

Electrical forces are strong and hence even small local deviations of charge neutrality lead to appreciable fields. These Planck fields are always such as to import ions which would neutralize the local charge. In most experimental situations involving the BZ medium, charge neutrality is well maintained and hence we now investigate the form of the chemical wave equation in the charge neutrality limit. Rather than simply imposing the condition of charge neutrality, it is instructive to see how it naturally arises as a well-defined approximation [2]. First we introduce a set of dimensionless (primed quantities) by defining characteristic values (indicated with a bar)

2.2. Plane waves

C = CC',

0C

3t = - V ' J ~ +

Ri(C)'

(2.2)

where R i(C ) is the net rate of reaction due to all processes affecting i; R i may depend on all concentrations C = { C1, C2 . . . . . Cw }. The flux Ji for dilute solutions is of the form J~ = - D ,

VC i - MiC i VV.

(2.3)

m

For steady plane waves of velocity o in a onedimensional system along the r-axis, the equations

cp = Upep', v = Upv'/i,

_

_

_

D = DD', t = tt',

_

M= MM',

R = CR' f i ,

M= FD/RgT.

~2= £)i,

(2.7)

P. Ortoleva/ Chemical wave-electricalfield interactionphenomena The last equation c o n n e c t s / ~ to D via Einstein's relation (Rg and T being the gas constant and absolute temperature respectively). The parameter is a characteristic chemical reaction time. Inserting these definitions into the wave equation (2.6) we obtain, upon dropping the primes for convenience,

dC +v~---~ + R = 0.

(2.8)

a 2 = eRgT/4rrF2~p2C.

(2.9)

The parameter a is the ratio of the Debye length ( e R g T / 4 ~ F 2 C ) 1/2 to the characteristic reactiondiffusion length ~ associated with the wave profile. For solutions of high ionic strength (as in the usual BZ and most biological media) the Debye length is on the order of angstroms; hence a is usually very small. Our task then is to examine the behavior of the ionic wave system in the formal limit a ~ 0 . The formal "charge neutrality" development is carried out by assuming that all quantities may be expanded in a 2, namely oo" C = E c(n)°12n, ,=o

oo n=O

(2.10)

O ~ E u(n)ol2n,

and inserting (2.10) in (2.8). The term of order a - 2 must vanish and hence

f~S

dcp z * C (°) = 0.

(2.11)

The only smooth function whose integral vanishes over an arbitrary interval is zero. Hence to lowest order charge neutrality is attained, Z * C (°) = O.

The terms to order a ° in (2.8) yield

d~[ dC(0) -

(2.12)

f~dfp

C(1)t

dC (0) + v(°) d----~ + R(C(°)) = 0.

(2.13)

The first order charge density z * C 0) is obtained by using charge neutrality, (2.12) and the fact that reactions do not create net charge (z * R = 0); applying z * (as defined in (2.1)) to both sides of (2.13) then implies d

Here/~o = F E o / R g T and

69

0,

I= -z*D--

dC (0) dcp

+ z * MC(°)[ ffT° + f ~ d ~ z * C(X)]

(2.14)

The quantity I is the electrical current. From (2.14) it is clear that the current is a constant of steady wave motion for the charge neutrality constrained (a << 1) system. Denoting this current I o we obtain, dropping the superscript (0),

d [DdC-ff~MC d~[ ~

]

+v

= I o + z • D dC/dcp z*MC

+R(C)=O,

(2.15) (2.16)

The quantity/~ contains a contribution due to the Planck electric field, i.e. that field due to the first order charge density z * C (1), manifest here in the concentration gradient terms. It is operative only for nonscalar diffusion (all D i not equal). /~ also has an ohmic contribution I o / z * MC, z * M C being the electrical conductivity. Eq. (2.15) has several interesting and rather general implications: a) A Planck potential propagates along with an electro-chemical wave; this potential increases with ionic gradients; b) the characteristics of wave propagation depend on the current I 0 applied along the w a v e - thus the wave speed v can depend on I0;

P. Ortoleva / Chemical wave-electrical field interaction phenomena

70

c) the Planck field and through current effects are strongest in systems with small conductivity z * MC; and d) the current 10 is a constant of steady wave motion. 2.4. Known wave-field interaction phenomena The response of steady waves to a constant current I 0 depends strongly on the reaction mechanism. Model [2-6] and experimental [6, 7] studies have shown that an interesting set of phenomena are possible: a) Multiple types of waves can exist in a medium subjected to a current 10 while the current-free system has only one type of wave (see fig. 1); Wave Field

Phenomena

~ "5

~

ultiplicity

b) the effect of an applied current may saturate as I o becomes large indicating that one or more species have been driven out of the wave-driving reaction zone, changing the character of the wave (see fig. 2); c) a cutoff current Io, c may exist beyond which waves do not exist; the effect is asymmetric in that waves started in the field-free system ( I 0 = 0) will be annihilated when a super-critical current is turned on only if the current and wave velocity are parallel or antiparallel depending on the reaction mechanism (see fig. 3); d) a sufficiently large applied current in an excitable, two-dimensional medium supporting a circular outward going pulse can cause a crescent shaped wave as in fig. 4; note the two stabilized free ends which exist only in the presence of the applied field; if the current is turned off the crescent degenerates into a pair of counter-rotating spirals (as in fig. 4); e) in an excitable one-dimensional medium, a pair of pulses propagating in opposite directions may be induced once an appropriate supra-

Saturation

o

point

Io Fig. 1. Schematic plot of steady wave velocity v as a function of applied current I o showing the existence of a range of I o where more than one type of wave m a y propagate even though only one wave exists in the field-free ( I 0 = 0) medium. Fig. 2. Same as fig. i except that for this type of system the velocity response is seen to saturate for large I o. Fig. 3. Same as fig. 1 except that for this type of system no waves in a given direction (relative to the field) m a y p r o p a g a t e - i . e , these waves are annihilated beyond a critical through current Io, c.

Fig. 4. (a) Crescent wave created when a circular pulse is subjected to a supra-annihilation field so that the wave propagating in the "unfavored direction" (towards the positive electrode in the case shown) are annihilated. This creates a pair of free wave ends. (b) The crescent degenerates into a pair of counter rotating spirals when the applied field is removed.

P. Ortoleva / Chemical wave-electrical fieM interaction phenomena

threshold perturbation is applied; if a current I 0 less than the annihilation current Io, c but greater than a lower current I0,m is applied, only one pulse in a "favored" direction is emitted; and f) otherwise static patterns may be made to propagate under an applied static electrical field. In the sections to follow we demonstrate some of these and other phenomena using a range of nonlinear perturbation techniques.

3. L i m i t c y c l e s y s t e m s

3.1. Phase mobility in high conductivity media For limit cycle systems the local phase theory [11, 12] may be extended to include applied field effects. Let 'P(t) be the trajectory in C-space denoting the limit cycle-i.e, d~I'/dt = R ( q ) . Then in the local phase picture C -- '/'(t + cp) where the local phase cp varies slowly in space and time. Limit cycle perturbation theory is a multiple time scale analysis that yields an equation for the spatio-temporal behavior of cp. Let us develop it here to include electrical effects. For a medium with high background conductivity z * M C is large and approximately constant. In that case the main effect is the response to an ohmic field (equal to the current density divided by the conductivity). Consider spatial variations on a length scale 8 -1/2 for a system in a weak static applied ohmic field 81/2E (8 << 1), the resuits of field-free limit cycle perturbation theory generalize the local limit cycle phase (¢p) dynamics to 3cp Ot1 = Dp V'2rp + A IWepl2 - g E - WcP, (3.1)

71

the limit cycle; the latter corresponds to the mode of marginal stability of the limit cycle to changes of phase (see ref. 12 for more details on limit cycle perturbation theory). Also t I = S t is the appropriate time variable for the slow evolution of the phase. The parameter tt acts like an electrical mobility for the phase. It can be positive or negative (depending on kinetics and species valences and mobilities) and hence may be conceived of as containing a factor reflecting the "valence" of the phase function. The magnitude and sign of tt depend on the detail of the reaction mechanism. For periodic plane waves along the r-direction, parallel to a constant applied field E we have (3.2)

¢p = Vt I "4- k r ,

v = Ak 2 - trEk.

Thus if 0~c is the frequency of the homogeneous cycle then the frequency t~(k) of waves of wavelength 2 ~ r / k is given by %(1 + v(k)); hence o~(k) k~0o~ (1 - i s E k + Ak2).

(3.3)

The shift of the frequency of these small k wave trains is proportional to k and to the strength of the field. The dependence of o: on odd powers of k is a reflection of the fact that the symmetry of space is broken by the applied field-the frequency depends on the direction of wave propagation relative to the applied field. Phase fronts may be obtained as with the field-free case [6]. Using the nonlinear transformation [11] ¢p = (Dp/A)ln A , r = ( D p / A ) r ' , we get (dropping the prime) OA 0 t 1 = X72A - - J k E "

WA,

(3.4)

I~ =- fo1 d t f M 0'/' Ot '

where the phase diffusion coefficient Dp and frequency renormalization factor A are the same constants as in the field-free theory; # ( t ) is assumed to have period one; f is the adjoint function to d ' t ' / d t for the linearized dynamics a b o u t

Traveling wave solutions A = A ( ~ = r - Ctl) to this equation are obtained in the form A C= e

-(c-xe)t:+a(~)l,

(3.5)

for arbitrary c and function a ( c ) . Nonlinear superposition holds as for the field-free case [6]. For

72

P. Ortoleva/ Chemical wave-electricalfield interactionphenomena

example a linear combination of A c and Axe corresponds to a plane periodic wave under the influence of an electrical field colliding with a region of homogeneous oscillation. 3.2. Nonlinear field coupling and induction of negative phase diffusion When the phase diffusion coefficient Dp, the frequency renormalization coefficient A and the phase mobility # are small, interesting new field-related terms enter the equation of phase dynamics. In particular one finds that effective phase diffusion and frequency renormalization parameters arise that depend in sign and magnitude on the imposed field strength. We start with the full reaction-transport equations for a system with space-and time-independent field and high ionic strength and conductivity; in that case the field E is related to the applied current I o and the (essentially constant) conductivity z * MC via E = lo/z * MC and

aC at = r D V 2 C - S E ' M V C + R ( C ) '

(3.6)

where the symbols have their usual meaning; space is scaled as 6-1/z and the field strength as 81/2 for a scaling parameter 6 << 1. Assuming that Dp, A and /~ scale as Vp=Opa,

A=A'6,

#=~'6,

(3.7)

the local phase dynamics theory [1, 12] yields the following equation: acp at 2

D; V% + a'l v' l 2 -

v' 0

+Al(v2)% + A21 v l' + A3I v 012

2

D p = O ; . - F n l E2 ,

-

~=At+n2

g2.

re-if

+ ....

(3.9)

In the presence of the field, the phase diffusion coefficient and the frequency renormalization factor are replaced by modified quantities/)p and z~ that depend on the field strength. Thus for a system with B 1 < 0 the effective phase diffusion coefficient could become negative as the field is increased beyond a critical value (-Dp/B1) 1/2 even though the field-free system had a positive phase diffusion coefficient. Note that /)p can even go negative for equal diagonal diffusion systems (for which Dp is positive) and hence there is no phase instability for most uncharged systems since the diffusion coefficients of all aqueous species are essentially the same. Clearly a host of phenomena are embedded in the above fourth order phasefield equation and more detailed analysis would be of interest. 3.3. Coupled phase-field dynamics

+A. Vn0- Vl V l: + A, V 0" V(V%) + a 6 ( V 2 q o ) 2 + A 7 vZl vcpl 2

- B I ( E " W)2~o+ B2(E" W~) = + B 3 W2cp(E • Wcp) + B,(E" W) wz~P q- B5E* V] V¢P[ 2 -t- g 6 Vq0 ° V ( E *A¢~) + B 7 w2(E" Vcp ) + 0)2.

The time t 2 = 62t is that appropriate for the small Dp, A, # systems of interest here. The 092, A and B constants are expressible as cycle average quantities analogous to those defining Dp, A and/~; the specific expressions may be obtained by carrying out the local phase multiple time scale procedure as in chap. V of ref. 1 or in ref. 12. Although all the terms in the above fourth order local phase equation are not easily interpreted, the B 1 and B 2 terms have interesting implications. For a one-dimensional system along the r direction we have

(3.8)

The condition of local charge neutrality may be violated on length scales of interest to wave propagation if the ionic strength and conductivity are not too large. Let us examine the problem anew within the local phase theory without making the high ionic strength expansion. Coupled equations for the local phase ¢p and the electrical field E are obtained as follows.

73

P. Ortoleva / Chernical wave-electrical field interaction phenornena

The starting point of the development is the conservation of mass equation for the column vector of concentrations C and Poisson's equation. The development is much like the applied field theory of the previous section but with interesting differences. One proceeds by introducing a space scaling factor 8 such that r = 8 - 1 / 2 r t and a pair of times t and tl/& The electric field is a gradient of a potential so that a scaling E = 81/2E ' is conjectured. With these scaled variables the starting point for the development is (dropping primes)

[ ~---[+ 8 ~-~i-~ll c = 8[ D v 2 C - M V . ( CE)] + R( C), 4~rF

8V" E =

e

z * C.

(3.10) (3.11)

We expand the field and concentration in powers of 8 (with coefficients C,, E,). To leading order we get

OC° = R( Co), Ot

z * C O= 0,

(3.12) (3.13)

dropping the subscript 0 on E, %CO= Dp ~r2CO-I-AI~7¢P12- (f ,M--~"-E°~rlVt~ ) 011

-(f, Mq'V'E).

(3.17)

The inner product (A, B) for any two column vector functions of t is defined via

N ( A , B ) = L l d / 1 ~ Ai(t)Bi(t )

0

(3.18)

i=1

for the N species system. We assume that f is normalized such that (f, ag'/Ot)= 1. Note that the last two terms in (3.15) depend on the form of the limit cycle since E depends on the short time t as well as r and t 1. This is a dramatic deviation from the constant field case where these terms take a universal form. A second equation is obtained by multiplying (3.15) by z *, using (3.16) and z * ~2 = 0 (since reactions do not produce charge) to find e

0

4¢rF 0t

(v.E)

= z

[ O'~t 2

• D ---~v CO+

- V "(z * Mg'E),

02~

~

Ivcol 2

]

(3.19)

showing that CO is locally on the limit cycle '/', CO= q'(t + c0(r, q ) ) ,

(3.14)

and that the evolution is charge neutral. The latter is a direct consequence of the assumption about long length scale spatial variations. Unlike for the high conductivity medium, however, charge neutrality does not hold to all orders in & The lowest order field E 0 and the phase q0 are determined next. Collecting terms of order 8 yields

]c1 = - -~1 + D VECo- V'(MCoEo), (3.15) ~'E°=

4~rF e z * C 1,

where the unperturbed cycle g' is evaluated at t + CO(r, q). Eqs. (3.17), (3.19) constitute a coupled set of equations for ~ and V, E = - VV. Note for the case of high constant background conductivity z * M ' / ' is approximately constant in space so that (3.19) reduces to V2V = O. Thus for V fixed at the boundaries E is independent of the short time t and hence the second to last term in (3.17) reduces to - / ~ E . VCO as obtained in the constant E case. The full E, COdynamics is more complex than the decoupled case. In fact it has yet to be demonstrated that the coupled problem (3.17,19) can sustain steady plane waves.

(3.16) 4. Interactions near criticality

where the matrix f~ is OR/aC evaluated at the cycle g'(t + CO). Multiplying (3.15) by f, the null vector of the adjoint of O/%t - 12, and integrating over a period of the period-one cycle we get,

Again consider a high background electrolyte medium in a constant field E. The critical region scaling (bifurcation) approach [11, 15, 16] may be

74

P. Ortoleoa / Chemical waoe-electrical field interaction phenomena

used to obtain quite general results on wave-electrical field interaction phenomena. The various critical behaviors (onset of multiple, uniform steady states, limit cycles, periodic wave trains or static patterns) each have their special features. 4.1. Cubic and quintic fronts Consider the case of the bifurcation of multiple uniform steady states. Let C be the column vector of concentrations in a reference steady state and let (~/'1, ~2 . . . . . ~,) = '/" be the column vector of amplitudes of the modes of the homogeneous stability problem linearized about C. Then following ref. 13 we introduce the matrix of mode diffusion coefficients D *, a diagonal matrix of mode relaxation rates/~ and a column vector ~/that is quadratic or higher order in the ~/"s that arises from the nonlinear part of the reaction rate terms. First we write C = C + T ~ where T is a matrix that transforms the linearization matrix to a diagonal matrix F (i.e. / ' = T - I ( ~ R / a C ) T ). With this, we have

For the above cubic mode dynamics one may obtain propagating fronts connecting the two possible stable steady states (i.e. zeros of 3'+ + B~k2 + A~b3). The velocity of propagation along the field is simply the field-free velocity plus ~11E. Different scaling yields a similar result for the quintic mode dynamics that must be considered when A > 0. The field only affects the wave speed; the profile for the single mode problem with DI~ > 0 is unaffected. 4.2. Field-induced negatioe critical mode diffusion Here the situation is more interesting. Scaling the field, D ~ and ~11 to bring higher order derivative and field terms into balance with quintic terms in the mode amplitude we obtain a fourth order equation for the critical mode amplitude ~p in the form

a~

at - DI~ ~72~ + ~ ~74%pq- alv~12+ ~k

+V~b + B+ 2 + a~b3. a~

--

at

v2+ (4.3)

= D * V 2 ~ - / x E " V@ + F ~ + n(~/'), # = T 1MT.

(4.1)

D* = T - 1 D T . One may then proceed as for the

field-free case as in refs. 6, 13 by using a scaling of the field similar to that for space. Let r = 8-Xr ' for 8 << 1 and x a scaling exponent for space. Then E = 8xE ' defines the scaled field E'. (In what follows we drop the prime on E.) Letting ~b now represent the amplitude of the (assumed) single unstable mode (taken to be mode 1) the scaling procedure yields the following cubic mode dynamical equation:

All quantities here except '/" are constants given by the scaling procedure. The electrical interaction brings in a number of interesting terms nonlinear in ko and also a term quadratic in the field. Solutions of this equation corresponding to propagating fronts or spatially periodic patterns are sure to demonstrate a richness of phenomena. A most interesting aspect of (4.3) is that it demonstrates the possibility of electrical field induced, effective negative mode diffusion coefficients. Consider a one-dimensional system along the coordinate r. The D~ and X terms in (4.3) combine to give a diffusion-like term with effective diffusion coefficient b n given by

a4~

at = Dx~ V2q~ - / ~ n E ° V~ + Y~ + B~b2 + A~P3

/ ~ n ( E ) = D~I + X E 2.

(4.4)

(4.2) for constants y, A, B given by the scaling procedure.

This implies that even for positive mode diffusion coefficient systems, D~ > 0, the spatial instabilities associated with a negative diffusion coefficient

P. Ortoleva/ Chemical wave-electricalfield interactionphenomena

may be obtained if X < 0 and E 2 > - D * / X . Thus in such a system patterns would be attained when the field exceeds its critical value while such patterns do not exist in the field-free medium. Indeed the effective mode diffusion coefficient can even be negative for an equal diagonal diffusion coefficient system. 4.3. Periodic

wave extinction

and subcritical

75

Amplitude

E=O

E
bifurcation Near the bifurcation point for a one-parameter family of periodic waves, one may demonstrate extinction and modification of the range of existence of waves. We consider here the imposition of a weak field E = E2A 2 along a traveling periodic wave of amplitude measured by a small parameter A. Let k be the wave vector and w be the wave frequency; then it is most convenient to calculate the waves as 2~r-periodic functions of the moving coordinate p = k r - wt for a wave train moving along the r coordinate. With this C(p, k, E, A) satisfies

k2D

__d2C dp 2

dC + ( w - k E 2 M A 2 ) - ~ + R ( C , X) = 0, (4.5)

where X is a bifurcation parameter. The parameter A measures the amplitude of the deviation of C from a homogeneous reference state C(~), i.e. R(C, x) = 0. The bifurcation theory proceeds by expanding F( = C - C) and )~ about critical values (0, he) in powers of A as for the field free theory. In fact by construction the theory is identical to the latter up to second order in A. As for the field free case, we assume that there is a critical wavevector k c such that at he the wave of wavelength 27r/k¢ has infinitesimal amplitude and no other waves exist. As ), exceeds hc by an amount of order A 2 a band of wavevectors around k¢ exists in which there are finite amplitude waves. We scale this band of allowed wavevectors by introducing a parameter x such that k 2 = k~ + xA. It is important to preserve the sign of the product

E)E c

K Fig. 5. Dependence of periodic wave train amplitude on wave vector k as given in eq. (4.27) obtained from a bifurcation analysis of a system with high background electrolyte. Beyond a critical field E c no waves traveling in the unfavored direction exist. The range of existence and amplitude of the favored waves increases with increasing field strength.

Ek since we expect that propagation parallel and anti-parallel to the field is different. Hence we let e2A 2 = E k / I k I and hence E k = e 2 [ k 2 c + x A ] 1/2

(4.6)

Thus e a is positive or negative when k and E are parallel or anti-parallel but e a contains no A dependence. With this the formalism of the field-free periodic wave train theory carries through and we find the amplitude relation

dA2=a(X_Xc)_b(kV_k2)2

- -Ek ~kcm

(4.7)

for constants a, b and m given by the bifurcation theory. The above result has interesting implications as suggested in fig. 5. For concreteness let us assume that the "wave mobility" rn is positive. The parameters a and b are both positive when the

P. Ortoleva/ Chemical wave-electricalfield interaction phenomena

76

Amplitude

E--0 ~ ~ . ~ Rez+_

K

0

Fig. 7. Wavevector dependence of periodic wave amplitude in cases where the range of existence of waves includes k = 0. Resolving the apparent discontinuity in amplitude at k = 0 (as given in (4.30)), requires a more careful scaling as done for the Hopf bifurcation in fig. 8.

<)'c Fig. 6. Wave vector (k) dependence of the real part of the complex conjugate pair of stability eigenvalues z _+ as a system parameter X passes through its critical value Xc. Such a field-free system behavior can lead to the periodic wave dependence on field shown in fig. 5.

real part of the pair of complex conjugate stability eigenvalues z+(k, ~) (that lead to wave propagation in the field-free problem) are as in fig. 6. The onset of waves does not occur at the field-free value X = X~ but when X = ~c where

h~ = ~ + MEk,

Ik[ near kc.

(4.8)

Waves in the "favored" direction Ek < 0 bifurcate earlier than do the unfavored ones, Ek > 0. Also, the range of existence of the favored waves expands and that of the unfavored ones contracts as E increases when X > ?%. Finally there exists a critical field E c such that no waves propagate in the unfavored direction. The situation is summarized in fig. 5. The direction selectivity of waves is very dramatic for cases where the field free lower cutoff value k 2 (see fig. 5) is negative. In this case there is a j u m p discontinuity in the amplitude A(k, E, h) as k passes through zero as indicated in fig. 7,

where 0 + and 0 - are positive and negative infinitesimals respectively. Some caution is needed in interpreting this result, however. The above develo p m e n t assumes that the range of allowed wave vectors is on the order of (X - Xc) 1/2. Hence if the range of allowed k extends from kc (k c > 0 and constant as A ~ 0) to zero, we must assume that k c is small for consistency. This suggests that a more careful analysis of small k behavior is needed wherein k~ scales with a power of A. This is done below for the H o p f bifurcation. It is known from the field free theory [6] that a family of standing wave solutions (parametrized by k) also emerges at a propagating wave bifurcation [11]. Such standing waves can be unfolded into a two-parameter family of propagating toroidal waves (see ref. 3). With these results it is expected that, under the influence of an applied electrical field, the standing waves can be made to migrate as multiply periodic (toroidal) disturbances.

4.4. Electro-Hopf bifurcation:

unfavored wave

tenacity and subcritical waves

A(k=O+,E)-A(k=O-,E) 1

- d,/2 { [ a ( ) t - ) t c ) - b k J - E k c m ] 1/2 -[a(h-hc)-bk•+Ekcm]l/2},

K

(4.9/

N e a r a H o p f bifurcation to a homogeneous limit cycle there exists a family of plane, periodic waves [17, 18]. For such a system the equation of motion of the local (complex) amplitude ff of the

P. Ortoleva/ Chemical wave-electricalfield interactionphenomena

oscillatory m o d e was obtained. It is straightforward to generalize the field-free theory to include an applied electrical field. If the applied, constant, ohmic field E is small, i.e. E = AEx, A being a measure of the average amplitude of the disturbance, then '/" satisfies O~ _ D+ V,2~ _ ]~E1 ° Vll~ nt- ot~ -/Iq, 3t2

I2q~

(4.10)

77

Amplitude E increasing

// 0

for complex constants D +, /~, a and f. Here rl/A and t2/A 2 denote space and time respectively. Consider now plane waves of wavevector k. The amplitude W k ( = Iq'klA) of steady plane waves is found from (4.10) to satisfy

Amplitude

E increasin O

W 2 Re f = a (h - h~) - k 2 R e D + + Ek I m / t , (4.11) where a ( = R e a ) and R e D + are taken to be positive constants; h is a system parameter such that as h passes through its critical value h e a homogeneous H o p f bifurcation occurs; the above results are valid for h - Xc of order A 2. For consistency the result is limited to small h - hc, k and E. In what follows we assume E Re/~ > 0 for concreteness. Interestingly the supercritical (h > he) and the subcritical (h < Xe) cases are rather different. For supercritical conditions there is always a domain for existence of waves for both positive and negative k. Waves exist in the interval k L < k < k v where 1 ( E im/~_+ [ ( E im/~)2 ktJ, L = 2 R E D + 1/2

+4a(h-h~)

RED+]

}.

(4.12)

Since a and Re D+ are assumed to be positive, it is clear that k L < 0 no matter how large E becomes. N o t e in fig. 8a, however, that as E increases the range of wave existence shifts dramatically to the right (in fact k u ---, oo as E --* ~ ) . The existence of waves in the (vanishingly small) range in the unfavored direction (here k < 0)

K

Fig. 8. Amplitude of periodic wave trains associated with a Hopf bifurcation. Note in the case (b) where there is no homogeneous cycle (i.e. a system parameter X is below its critical value he) it may still be possible to propagate waves in a "favored" direction (k > 0 here).

is simply that, no matter how large E is, there is always a small enough k so that the concentration gradients are so small that the field effect is not strong enough to cause annihilation. Note that as E --* ~ the amplitude has a discontinuity at k = 0. In the subcritical case we note that k L > 0. This means that only waves in the favored direction m a y propagate. Such waves only exist in fields for which (E Im/t) 2 > -4a(h-Xc)ReD+.

(4.13)

The field-supported subcritical waves constitute a truly electrical effect. Note in fig. 8b that their range of existence vanishes as E reaches the cutoff value E o = 2[a(X¢ - X) Re D+]l/2/Im #. The waves that propagate when E is near this cutoff value are of wave vector k ~ given by k 0 = E 0 I m / ~ / 2 Re D+.

(4.14)

P. Ortoleva/ Chemical wave-electrical field interaction phenomena

78

4.5. Bifurcation of electrical modes In previous sections we have considered the electrical field to be weak so that the bifurcating states had many of the same properties as the field-free patterns. Only the amplitude and frequency were affected by the applied field to lowest order. In this section we consider the consequences of having the field of order zero in the pattern amplitude A of periodic waves, i.e. as A --* 0 the magnitude of the applied field is held constant. The dynamics of concentration column vector deviation c from a homogeneous steady state C ( k ) is given, for a constant electric field E, by ac - D V'2c - M E . ~rc + 12(X)c + Bc 2 at

+ Tc 3 + . . . .

(4.15)

Here 12(h) is the matrix of linearization of the reaction rates about C ( k ) while B and T are analogous matrices for quadratic and trilinear contributions (the latter written in symbolic form Bc 2, Tc 3, respectively, to avoid lengthy summation expressions). The linear problem has harmonic solutions for wavevector k of the form exp ( i k . r + zt) with z a solution of

det[zI+k2D+iMk.g-I2(k)]=O.

(4.16)

F r o m this we see that z is not an even function of k . E l [E I because the symmetry of space is broken by the imposed field. A few general statements can be made about the eigenvalues z ( k 2, k . E , X). If z(0,0, k) is a real eigenvalue then simple matrix eigenvahie perturbation theory shows that z has the limiting behavior Rez -

z(O,O, h) + f l k 2,

k---~O

Im z(0,0, X) = 0,

for a constant y. This is, in fact, the first manifestation of the asymmetry of the problem due to the symmetry-breaking field E. Furthermore it is clear that as k --* oe z has the behavior

z -

-k2Dj,

(4.19)

k~oe

where Dj is one of the eigenvalues (assumed positive) of the diffusion matrix. With the above constraints one can conjecture a number of interesting scenarios for z( k 2, k . E, X) and then carry out the bifurcation calculations to determine their consequences for the nonlinear problem. For illustrative purposes consider a onedimensional system along the field of strength E and study the bifurcation of plane periodic waves. To be considered is the case shown in fig. 9. We assume Im z :~ 0 over the range of k shown. For E = 0 the system is, by assumption, of the type that can support a periodic wave train for k greater than a critical value k c. From fig. 9, we expect that as E increases from zero, an increasing interval in k of waves in an "unfavored" direction will be forbidden while the waves propagating in the favored direction will exist over a larger range of k than for the field-free case. Furthermore, in the subcritical case k < 2% it is also quite clear that there can exist a range of k where favored waves can propagate even though no waves can exist in the field free, subcritical system. Rez

E =0

K

(4.17) where fl is a constant (not necessarily the same for all z). For complex z(0,0, X) we find that Rez-

k~O

Rez(O,O, 2 ~ ) + y k . E ,

Imz(0,0, X)~0 (4.18)

Fig. 9. Dependence of the real part of the stability eigenvalue complex conjugate pair z+ on k with varying applied field as described in subsection 4.5.

P. Ortoleva/ Chemical wave-electricalfield interactionphenomena

The bifurcation theory of the periodic wave system of fig. 9 follows in direct analogy to that for the field-free or weak-field cases. The main difference is that one must treat waves in the favored and unfavored directions separately since they bifurcate at different (E-dependent) critical values of the system parameter )~. The final result will be a cubic equation for the h, k, E-dependent amplitude whose structure will be, in general, quite complex with respect to its E dependence. Interestingly this dependence should be sensitive to the details of the reaction mechanism and the charges of the species involved.

79

Thus we keep k fixed and examine the traveling solutions induced as E increases from 0. Insertion of these expansions into the wave equation and collecting terms to various orders in E we find D d2~b° + R ( ~ b o ) = 0 , drp 2

(5.4)

t~bx + (Wl - M ) ~-~P-~°= 0,

(5.5)

where d2

L = Dd--~2+ ~2(qo)

(5.6)

5. Field-induced waves from static structures

A static dissapative structure in an infinite medium may be induced to move when a constant electrical field is applied (see also ref. 5). Let ~bo(kr, k ) be a field-free, periodic static structure. In an infinite medium such structures can exist in some finite range of wavelength 2~rk -1. We now construct traveling wave solutions ~b(~, k, E ) for small fields. The coordinate {p is that moving with the wave, assumed to have velocity v (k, E), q0 = k ( r - vt). The basic equation of steady wave motion in one dimension under an applied field E is taken to be

and I2 is the linearization matrix for R about ~b0. The operator L has the translational invariance null vector O~b0/3q0:

L a~o/a~0 = o.

(5.7)

Corresponding to this null vector we assume the existence of the null vector of the adjoint of L. We introduce the bracket notation ( a I for the null vector of the adjoint of L, denoted L ÷. We take the normalization (alO~bo/3~> = 1,

(5.8)

where

D d2~ + (,o- a c e ) dq. dq: a7 +R(~')=°'

{a = k v , ( a l L = 0.

(5.9)

(5.1) In the above we have the notation where R is the reaction rate, and D and M have their usual m e a n i n g - e x c e p t that D includes a factor k 2 and M has a k factor included in their definition. Both ~0 and ~b are 2or-periodic functions of q0. Finally {a is kv. For small fields we invoke expansions in powers of E:

,~(k, e ) = ~ l ( k ) e + o~2(k)e 2 + ....

(5.2)

¢ ( ~ , k, e ) = ~o(~, k) + ~1(~, k ) e + . . . .

(5.3)

+

d2

L,, = D.g~T~ + a . ,

(5.10)

N (FIG) =

• f_"F~(qo)G,(qo)dq~

(5.11)

i=1 for the N-species system, for arbitrary row vector ( F I and column vector IG). The adjoint null vector (al may be used to analyze the various order perturbation equations. Taking the inner product of the order E equation

80

P. Ortoleva/ Chemical wave-electricalfieM interactionphenomena

(5.5) with (al yields

~o1 = m k -
6. Variable pseudo-valence (5.12)

With this value of w, we can invert (5.5) to find

~x = L - l ( M - ink) O~o/aq).

(5.13)

Note that an additional contribution to ~1 proportional to 0g,0/0q0 can always be absorbed by redefining the origin of the q0 coordinate. Although it apparently has not been proven formally, it appears that all static periodic structures in infinite, one-dimensional systems have a point within a period about which ~b0 is symmetric. Hence O+0/O¢p is odd about this point and therefore so is ~1. Hence inducing propagation in the static structure leads to associated asymmetry in the propagating pattern. The static periodic structure is induced to move with velocity v(k, E) that has the limiting behavior

v( k, E ) e~,om( k )E.

(5.14)

Thus m ( k ) plays the role of a pseudo-mobility for the structure. Its sign determines whether the structure moves parallel (m > 0) or anti-parallel (m < 0) to the field. Quite generally from (5.1) we can show that if tk(q0, k, E) is a solution with velocity v(k, E) then ~(-qo, k, E) is also a solution for - E and has velocity - v ( k , E). Hence v is odd in E and the correction to (5.14) is of order E 3. One-dimensional static patterns are just one example of symmetry broken states of otherwise translationally invariant systems. Since all these states constitute a one-parameter degeneracy in one spatial dimension due to translational invariance, it is clear that the present calculation opens up the more general question of the drift and distortion of these states (including higher dimensional static patterns or circular, spiral, and other center waves) due to an imposed electrical field.

A front in a multiple species ionic system can act as an entity whose valence is variable with concentration in a way that depends on reaction mechanisms and the charges of the species involved. To demonstrate this point consider a chemical analogue of a competitive predator-prey problem, 0x ~ [ = Dx W 2 X - MxE" WX + ( Z - 1 ) X + e f ( X , Y, Z ) , 3Y

(6.1)

= Dy I;72y -- M y E " V Y

+ ( Z - 1 ) Y + eg(X, Y, Z ) ,

(6.2)

OZ 8t - Dz ~72Z - MzE. tzZ + I-(X+

Y)Z+~h(X,Y,Z).

(6.3)

for arbitrary functions f, g, h; E is a constant ohmic electric field in the medium with high background electrolyte content. Note that as the "degeneracy" parameter e ~ 0 the reaction properties of the two "predators" X and Y become identical. In the degenerate limit ~ ~ 0 the system has the uniform steady states X, Y, Z given by X=g, = (1 - g),

(6.4)

Z=I, for arbitrary g in 0 < g _< 1. The question arises as to the behavior of smooth spatial variations which are everywhere like (6.4) but with a slow dynamics for #. The latter scaling picture is accomplished by seeking solutions on the spatial scale r ' = el/2r and on a time scale t' = et for a weak field el/2E'. Using scaling techniques [1], (dropping the primes for convenience) we obtain art %t - D ( g ) Vzlz - M(II)E • Wg + F ( g ) ,

(6.5)

D(tt) = ( 1 - g)Dx + gDy,

(6.6)

M(bt) = (1 - g ) M x + gMy,

(6.7)

r ( g ) = (1 - g ) f ( g , 1 - #, 1) - gg(/~, 1 - g, 1).

(6.8)

P. Ortoleva/ Chemical wave-electricalfield interactionphenomena

The variable /z (0
/z(rp)

1 1 + e ~°

f l ± = ½ [ E _ ( E 2 + 2)1/z],

81

Fig. 10. Velocity response to an appfied field E for competitive predator dominance front described in section 6.

We expect that a host of interesting phenomena can be sustained with systems of more degenerate species and possible other types of behavior than multiple steady states (i.e. dynamical asymptotic states as discussed in chap. VII of ref. 1). A scaling approach to competitive species in ecological systems has also been investigated by Fife.

(6.10)

1 -2a v + = 2fl+

7. Charge degeneracy 7.1. Formulation

The velocity response to the field o ( E ) is shown in fig. 10 where the velocity is seen to saturate as E---, + ~ for the (+)-wave that moves to the right. (The wave associated with v_ that moves to the left has similar velocity response to that shown in fig. 10 except the latter is inverted through the v = E = 0 origin.) These results are in sharp contrast to the case M x = M y = 1 for which v = + (1 -2a)/21/2+ E, a simple linear response. The velocity response for the system of variable pseudo-valence has a solution which is static (v 0) as E ~ oo. Apparently this is due to the fact that the pseudo-charge of/~ is of opposite sign in the two homogeneous states # = 0,1 and hence the competing predators are kept away from each other's territory. Indeed the field-free fronts correspond to a front of dominance of one species over the other (the one that wins depending on the sign of e) (recall t' = et) and the form of the functions f and g.

Chemical reactions do not produce any net charge and hence it is sometimes possible to find a one-parameter family of homogeneous steady states with varying charge density. We consider here the dynamics of the charge density using a scaling approach based on this charge density degeneracy. Consider a system with retarded transport (e << 1) evolving according to aC

Ot = eD v 2 C -

eV'(MCE)

V.E=fp,

+ R(C),

(7.1) (7.2)

N

p = ~., z , C i - z * C.

(7.3 /

i=1

Examples of such a system would be a porous medium or a biological cell mass. Here p is the

82

P. Ortoleva/ Chemical wave-electricalfieM interaction phenomena

charge density, f is 4~r times Faraday's constant divided by the dielectric constant and the other symbols have their usual meaning. Assume that the homogeneous system has a steady state CO

R(Co) = 0 .

(7.4)

In some cases such a steady state problem can admit a family of solutions Co that depend on the charge density P0. We have introduced the parameter e to limit our treatment to a system of low conductivity and diffusivity. Hence the appropriate time scale for the evolution of nonuniformities should be of the order of e-1 as long as we restrict the initial data so that C is near Co(Oo) everywhere. Thus we introduce a time t ' = et so that O0 and E o are functions of r and t'. Note that z • CO= P0; hence the field E 0 associated with Po satisfies

V" Eo =fPo.

be obtained. If a net charge is injected at a point in the medium, then the resulting coulomb repulsion should induce an outwardly propagating spherical shell of charge. This driving force coupled to reaction induced state multiplicity could lead to phenomena involving charge induced switching between steady states. Wave phenomena in the equations of charge density dynamics (7.4)-(7.6) have yet to be examined.

8. Strong-field phenomena When electrical fields are strong, chemical waves can take on an entirely new character. For illustrative purposes, consider the case of periodic plane waves. Waves along the r coordinate with frequency 60 and wavevector k are described by

k2D dEc + (60 - EkM) dC dopE ~ + R ( C ) = O,

(8.1)

(7.5)

What remains is to obtain another equation coupling Oo and E 0.

where q9 = k r - 60t. For strong fields it is instructive to examine the limit E ~ o¢, k ~ 0 with e( = Ek) held constant. The perturbation procedure starts by introducing the expansions

7.2. Charge density dynamics Since reactions do not produce charge, z * R = 0. Thus applying z * to (7.1) yields as e-~ 0 (dropping the prime on t henceforth)

60 = 600(e) + 601(e)k 2 q- . . . .

(8.2)

C = Co(¢p, e) + C1(~, e)k 2 + . . . .

(8.3)

To lowest order we find OPo ~t = z * D V2Co(Po) + z * M V ' [ C o ( P o ) E o ] . (7.6) Thus (7.4-6) yield the coupled charge-field dynamics that arises due to charge degeneracy in systems of slow transport. The dependence of CO on 0o can be quite nonlinear if the reaction rates in (7.4) are nonlinear. Hence we conclude that the charge density dy° namics in these systems can be highly nonlinear in a way that depends critically on the chemical kinetics. When (7.4) yields multiple steady states, interesting charge density wave phenomena may

[600 - eM]

+

R ( C o ) = O.

(8.4)

We seek 2or-periodic solutions. Clearly in some systems this problem could have solutions for e beyond a critical value whereas no solution may exist for e - - 0 . Alternatively if solutions exist for e = 0 (i.e. a homogeneous limit cycle) then there can exist an upper bound on e beyond which no solution exists. The theory may be carried out to higher order. In the usual way 60a is determined via a solubility condition involving D dEC0//dep2 projected onto

P. Ortoleva / Chemical wave-electrical field interaction phenomena

the null space of the dynamics (8.4) linearized about a given 2~r-periodic solution. For the high-field problem, interesting results follow from the analysis of (8.4). A Hopf bifurcation approach using e as a parameter suggests itself. Furthermore all the richness of period doubling and chaos onset as e changes is a possibility. A great number of interesting results should flow from the analysis of (8.4). A simple model serves to illustrate the realm of the possible. Consider the case dX

( °~° -- mx)-d--~ + 1 B X - A Y= dY (°~°- my)-d--~ + 1 B Y +

1

A0

COS 2 0 = "~ + m my ~--

(8.10.)

x"

Clearly if the field is strong enough and M x ~ My then the field can cause the system to have discontinuous behavior as it attempts to circulate about the field-free orbit X 2 + y2 = R 2. Since cos 2 0 lies between 0 and 1, a necessary condition for the existence of a solution to (8.10) is 2A 0

(my_rex) < 1 .

(8.111

(8.5) (8"6/

A0 (o~0 - mx) sin20 + (o~0 - my) cos 2 0 ' (8.7)

where Ao=A(Ro). This equation can be integrated; taking 0(cp = 0) = 0, we get 0-

This occurs when

-1<

where B and A are functions of R 2 = X 2 + y2 and m x = eMx, m y = eMy. For small e the orbits are attracted to the stable zeros of B(R). Let R o be such a stable radius. Then for smooth solutions of the problem as e--,0, X = R 0 s i n 0 , Y= R o cos 0, where 0 satisfies dO dqo

83

- m y )O + l ( tH y - r e x ) s i n 2 0 = 2 A o q o .

Clearly if M x 4: My and kE is sufficiently large this condition can be satisfied. A more detailed examination of this model is required to determine whether the non-existence of smooth solutions to (8.7) implies an annihilation phenomenon or the emergence of shock structures. In either case it is clear that the large field limit of wave-field interactions retains much of the interesting physics of the full problem.

9. Electrical interactions with propagating discontinuities Consider a species Y that is affected by several particularly fast reactions. Then for Y (2.15) reads d [D dY dg~ t Yd--~~

(8.8)

I 0 + z * D dC°/d~o - yDy dY/dep

Since 0(~ + 2~r) = 0(cp) + 2~r we obtain

--

~ o = A o + (mx + my)~2.

+vdY 1 -d~ + ~Ry (Y' C°) = 0 ,

(8.9)

Thus the wave frequency is a function of the field as one might expect. F r o m (8.7) we see that an interesting effect of the field is that, unlike the field-free case, rotation around the cycle is not at constant rate. Hence X and Y are not simple harmonic functions of qo as they are in the field-free problem. Indeed a striking possibility presents itself-dO/dqo can diverge.

z , MCO + yMy Y

] MyY

1 (9.1)

where y is the valence of Y and C° is the column vector col {0, C2,..., C u } of all species concentrations except that of Y (taken to be species 1) has been removed. Consider the limiting behavior of this system as $ ~ 0, $ being a time scale ratio. When 8 ---, 0 and the net rate Ry as a function of Y has multiple zeros then it is known [8-10] that such a system can support pulses, fronts, wave

84

P. Ortoleva/ Chemical wave-electricalfieM interaction phenomena

trains and a host of other chemical wave phenomena in uncharged systems. These phenomena involve the propagation of very narrow transition layers across which Y jumps between stable zeros of Ry. Here we investigate electrical aspects of these propagating discontinuity phenomena. Scale space via cp = 81/2ep' to examine a Y transition layer near ~o= 0 where we assume Y to j u m p between a pair of stable zeros of Re(Y, C °) for the given value of C O. Then, keeping only leading order terms, assuming large currents, I 0 = I~8-1/2 and dropping the primes on Id and qo' for convenience, we get

d [ z*MC ° dY dep [ z , MCO + yMyyDy dep

(

+ vo - z , M C O + y M y Y ]

Y

1

+ R y ( Y , C °) = 0 ,

References (9.2)

where v ---, VoS-l/2 as 8 ~ 0 for nonzero constant v0. This equation can yield quite interesting deviations from the nonionic case if the mobility of species other than Y is small. In that case as Y increases its effective diffusion coefficient decreases and hence profiles can become very sharp. This could greatly enhance autocatalytic Y production at a front because the Y produced at a front would tend to contain itself since the effective Y diffusion coefficient would be lowered due to increased Y at the front. Let X represent a slow variable-i.e, one for which there is no 8 -1 term in its reaction rate. Then in the wake of the propagating Y discontinuity, we have X variations on a length scale 8-1/2 [8]. Letting ~ = (p8 -1/2 define a long length scale spatial variable, the wake structure is determined by

v o - z , M C O + y M y Y -d--~-+ R / = 0 .

greater than a critical value. That this is possible depends on the relative sign of v0 and the product of M x and I 0. Note also that for a given I 0 this annihilation effect (or other qualitative change of behavior) could set in as the background electrolyte concentration is decreased; in that case the z * M C ° term in (4.3) would not cancel the effect of the M x I o term. If the width of the transition zone across which Y jumps is of the order of the Debye length (as could be obtained at low ionic strength for fast reactions) then large charge densities (and hence fields) could be generated within the advancing Y transition layer. Thus autonomous fields can be very important in multiple time scale systems.

(9.3)

Note that if the bracketed term changes sign as Y varies, it is clear that if solutions exist for zero applied current ( I 0 = 0) they may not exist for 10

[1] P. Ortoleva, The Variety and Structure of Chemical Waves, in the Synergetics Series, H. Haken, ed. (Springer, Berlin, 1987). [2] S.L. Schmidt and P. Ortoleva, J. Chem. Phys. 67 (1977), 3771; 71 (1979), 1010. [3] S.L. Schmidt and P. Ortoleva, J. Chem. Phys. 74 (1981) 4488. [4] R. Feeney, S.L. Schmidt and P. Ortoleva, Physica 2D, (1981) 536. [5] D. Kondapudi and A. Nazarea, Biophys. Chem. 8 (1978) 71. [6] S.L. Schmidt and P. Ortoleva, in: Chemical Oscillations and Traveling Waves, R.J. Field and M. Berger, eds. (Wiley, New York), pp. 333-418. [7] H. Sevcikova and M. Mareck, Physica 13D (1984) 379. [8] P. Ortoleva and J. Ross, J. Chem. Phys. 63 (1975) 3398. [9] G.A. Carpenter, J. Diff. Eqns. 23 (1977) 335. [10] P.C. Fife J. Chem. Phys. 69 (1976) 554. [11] P. Ortoleva and J. Ross, J. Chem. Phys. 60 (1974) 5090. [12] P. Ortoleva, in: Theoretical Chemistry, vol. 4, H. Eyring and D. Henderson, eds. (Academic Press, New York, 1978), p. 235. [13] P. Ortoleva, Toroidal Phenomena in Reaction-Diffusion Systems, J. Chem. Phys. submitted for publication. [14] A. Nitzan and P. Ortoleva, Phys. Rev. A 21 (1980) 1725. [15] G.F.G. Auchmuty and G. Nicolis, Bull. Math. Biol. 37 (1975) 323. [16] D.H. Sattinger, Topics in Stability and Bifurcation Theory (Springer, New York, 1973). [17] F. Hoppenstead, R.M. Miura and D. Cohen, SIAM J. Appl. Math. 33 (1977) 217. [18] M. Delle Donne and P. Ortoleva, J. Stat. Phys. 20 (1979) 473. [19] P.C. Fife, private communication (1984).