r

Indian Journal of ChemIstryVol. 17A, January 1979, pp, 11-:13

Effect of Ionic Migration on Oscillations & Pattern Formation inOscillatory Reactions

ASHWINI KUMARChemistry Department, University of Gorakhpur, Gorakhpur 273001

Received 15 December 1977; revised 13 April 1978; accepted 12 July 1978

The stability analysis is broadened in order to include ionic rnlgratton effects in Lotka andBrussalators models. Analysis shows that in the absence of electric field or ionic mlgratlon,oscillatory behaviour of dtssfpatlve structures is itnpossible although chernlcal oscillations maybe obtained, and it also shows that the periodicity of the space oscillations will strongly dependon the electric field.

OSCILLATORY phenomena in chemical systemshave aroused considerable interest in recentyearsl'2. Amongst the chemical reactions

Belousov-Zhabotiniskii reaction has been intensivelyinvestigatede". Experimental results of Rastogiet al.5-7 have shown that the concentration ofthe oscillating species changes periodically withtime. Normal mode analysis has been employed insolving reaction diffusion equation by Prigogine andGlansdroff", Rastogi and Kumar". In all previousanalysis diffusion was considered as the only trans-port process for the study of space oscillations.However, stability analysis Was broadened by JacobJorne10 in order to include ionic migration affects.Stability analysis of the Lotka+ and Brusslators"models has been studied in the present paper takingchemical reaction, diffusion and ionic migrationinto consideration and the results show that thereare certain conditions under which spontaneousoscillations and dissipative structures might develop.

Lotka ModelLotka model+ for auto-catalytic and auto-

inhibition is represented as shown below:k,

A+X --+2Xk,

X+Y --+2Yk,

Y --+ELet us consider that it is an homogeneous system

consisting of X and Y chemically active species.Both the components may diffuse and migrate underthe influence of an applied electric field. Consider-ing only one dimensional case, ignoring convection,at every location chemical species obey the con-servation equations of X and y (Eqs 1 and 2)ax 02X axat = k1AX-k2XY +Dx3r2 +U1Ear .;.(1)

ev 02y avTit = k2XY -k3Y +Dy or2 +U2E Tr ... (2)

In Eqs (1) and (2) klo k2 and s, are the arbitraryconstants, Ir; and Dy are the diffusion coefficients,r is the space and U,= ZiFui where u; is the ionicmobility, Zi its charge and F is the Faraday constant.

I

E is the applied field or the field caused by the ionsthemselves. At the steady state:Xo = k3/k2 and Yo= klA/k2 ... (3)

Let x and yare the perturbations from thesteady state.

x=X-Xo and y=Y-Yowhere x <{ x, and y <{ Yo'

Putting the values of X and Y in Eqs (1) and(2), we haveoX 02X oXot = klAx-k2Xoy-k2YoX+Dxor2 +U1Eor ... (4)

oy _ 02y oyot - k2Xoy-k2YOx-k3y+DY8r2+U2Ear ... (5)

We now consider the behaviour of such a systemin the container of finite dimension. Equations(4) and (5) are, in general, full matrix equationsand can be solved by Fourier analysist-. Followingstandard methods, we consider one set of Fouriercomponents for the perturbations.

coX = ~ [x(thJ exp (ikr)

k=O... (6)

coy = ~ [y(thJ exp (ikr) ... (7)

k=Owhere x (t)k and y(th are the time dependent, andexp (ikr) is space dependent (Where k is the wavenumber and r is the space). The characteristicmatrix Eqs (4) and (5) become

o[xo~hJ = (ikEU1-Dxk2)[X(thJ-k3[y(thJ (8)

o[yo~hJ = k1A[x(thJ +(ikEU 2-k2Dx)[y(t)kJ (9)

Let the solution of the equations be:

co[y(t)kJ = ~ [Yoh exp (Ak"t)

n=Owhere 'Ak,. gives the eigen values of the matrix.Hence characteristic matrix equation takes theform of Eq. (10)

21

r

INDIAN J. CHEM., VOL. 17A, JANUARY 1979

((ikEUl-k2Dx-),kn) -ks ) - 0

det klA (ikEU2-k2Dy-Akn) -... (10)

On solving determinant of Eq. (10), we get,A~"-Akn[ikE(U 1+TJ 2)-k2(D.+Dy)] +klksA+k4DxDy

_k2E2UP2-ikSE(D.V2+DyU1) = 0 ... (11)where

A = ikE(Ul+U2)-k2(Dx+Dyl+ ..1·[k4(D -D )2kl! 2' - 2 x y

_k2E2(UI-U2)2-4klkaA-2iEkS(U 1- U2)(Dx-Dy)]!... (12)

Equation (12) can be written in the form ofEq. (13).

2Akn = a+ib ± (p+iq)t ... (13)

wherea~ -k2(Dx+Dy); .b=kE(U1+U2)

p = k4(Dx-Dy)2-k2E2( U1- U2).2-~-Ak1ksA. q = -2k3E(U1-U2)(Dx-Dy)

Equation (13) can be transformed into Eq. (14)(see Appendix)'

Akn = (X+i~ ... (14)where

(X = ia ± i[t(P+VP2+q2)]!

~= ib+t[{2(P~Jp+q2)}!]

Instability would occur only if the real part ofAkn is positive. In the absence of electrical field,i.e. E =0(X = ia± i(P)E=O

= _ k2(Dx

2+Dy) ± i[k4(Dx-Dy)2-4k1kaA]! ... (15)

Instability would occur if

[k4(Dx-Dy)2_4klksAJ! > k2(Dx+Dy)and oscillations would occur if

4k1kSA > k4(Dx-Dy)2If D; = Dy, then one will get stable damped oscil-

lations when, (X = -k2Dx± i(k1kaA)2

At k = 0, one will get sustained oscillations aslong as A is present in the system. The results Iare similar to those obtained by Rastogi andKumar",

An inspection of the result in the presence ofan electric field shows that di's~ipative structurewill occur if

Gt =ia±tEHP+VP2+q2)JL>O ... (16)for a particular value of k. The electric field Eappears in P and q. Increasing p2 and .q2will widenthe range of dissipative structures.

Brusslators ModelThe chemical scheme proposed by Prigogine and

coworkers" is given below:

22

I

•. _ •.••••••••.••. 4

A----+XB +X----+Y +D2X+Y----+3X

X----+EThe net reaction is A+B----+D±E where only

two species, i.e. X and Yare independent variablesand forward rate constants are kept unity. I Takingdiffusion and migration into consideration, thekinetic equation for the reaction model can beexpressed by Eqs (17) and (18)

0; = A+X2Y -bX-X+D};~ +U1Eo: ... (17)

oy _ 2 02y oyot -BX-X Y+DYor2+U2Eor ... (18)

At steady state, we HaveXo=A and Yo=B/A ... (19)where, the subscript 0 denotes the steady, state,Let us suppose that in the neighbourhood of steadystate X and Yare: X = Xo+x and Y = Yo+y,where x~Xo and y~Yo and are the small pertur-bations. Putting the values of X and Y in. Eqs(17) and (18), we have

"'.,.1,' IC)

OX _ 2 02X oXat -Xoy+2XoYox-Bx-x+Dxor2+UlEor .... (2.0)

~ 2 o~ ~at = Bx-Xoy-2XoYox+DYor2+U2Eor ... (21)

Putting the values Xo and Yo in Eqs (20) and(21) we haveox 2 02X oXot = Ay+Bx-x+Dx or2+U1E or ... (22)

~ o~ ~ot = -A2y- Bx+Dy or2+U2E or ... (23)

; .Equations (22) arid (23) are treated in the manner

discussed above for Lctka model.We have the characteristic matrix equation (24)

d ((B-1-.k2DX+ikEU1-Akn) A2 ) = 0

et -B (ikEU2-k2Dy-Akn)... (24)

On solving characteristic matrix Eq. (24), we getB-(1A2)-k2(Dx+Dy)+ikE(U1+U2)A~= 2' ±

t[{B-(1+A2)-k2(Dx+Dy)}2-k2£2(U1-U2)2-4(1 +k2Dx) (A2+k2Dy) +4Bk2Dy+2ikE{(Ul +U2)(B-1-k2Dx-k2Dy-A2)-2U2(B-1-k2Dx)+2U 1(A2+k2Dy)}]1/2 ... (25)Putting Eq. (25) in the form of Eq. (26)

2Akn =:a+ib± (p+iq)1/2 ... (26)wherea = B-(1+A2)-k2(Dx+Dy); b = kE(U1+U2)P = [{B-(1+A2)-k2(Dx-Dy)}2-k2£2(U1-U2)2-

4(1 +k2Dx) (A2+k2Dy) +4Bk2Dyq = 2Ek[(U1 +U2HB-(1+B2)-k2(Dx+Dy)-2U2

(B-l-k2Dx)+2U1(A2+k2DylJThis can be ,transformed into Eq. (27) (see

Appendix)'

,

rKUMAR: EFFECT OF IONIC MIGRATION ON OSCILLATIONS

Ak"= ex.+i~ ... (27)where ex. and ~ have the some significance asin Lotka's model.

Instability would occur only when the real partof Ak" is positive. In absence of electric field

ex. = ia ± i(P)E~O. .,

ex. = HB-(1+A2)-k2(Dx+D;y)] ± H{B-(1+A2)-k2(Dx+Dy)}2+4BBk2Dy-4~1.+k2Dx)(A2+k2DY)Jl/2For instability, We get the following c~nciitions:

B-(l+A2)-k2(Dx+Dy)>0 , } R{(I +k2Dx) (A2+k2Dy)-Bk2Dy}> 0and

, .v,. {B-{1.-A2)-;-k2(Dx+Dy})2<4(1+k?D.)') +

(A2+k2Dy)-4Bk2Dy .

when the Wave number k = 0 then ·these inequa-lities reduce to;

B>1+A2{B-(1+A2)}2<4A2 ... (28)

If B=1+A2, then 4A2>O and sustained limitcycle type oscillations would occur around thesteady state values as long as A is present. Dampedoscillations would occur when B"" 1+ A2•

The. equations (R) can be written in the followingform; B'(c)<B<B"(c) as determined by Prigogineet al», where

B'(c) = B-(1+A2)-k2(D .•+Dy)B"(c) = (1+k2Dx)(A2+Dy)lk2Dy

In the presence of electric field instability would

occur when ex.=ia±Ut(p+Vp2+q2)]I/2 ... (29)

Conclusions

The present study shows that the formation ofdissipative structures in oscillatory phenomenonin the absence of electric field and ionic migration,is not possible, although, the chemical oscillationsmay be obtained. However, in the presence ofelectric field, the imaginary part of the eigen value,i.e. B will give the periodicity of the oscillationswhich will strongly depend on the electric field.

Acknowledgemnt

Thanks are due to Prof. R. P. Rastogi andDr M. C. Gupta for fruitful discussions and tothe CSIR, New Delhi, for financial assistance.

APPENDIX

Equation (13) can be written in the followingform:

... (13)

I

LetP = r cos 8q = r sin 8

Therefore,

r = Vp2+q2; sin 8 = V; 2and cos 8 = V ~ 2P +q P +q

Putting the value P and q in Eq. (13), we have2M" = a+ib ± r1/2[cos 8 +isin 8]1/2 ... (13.1)or

= a+ib ± rl/2[cos 8/2+i sin 8/2J ... (13.2)We know that 2cos2 8/2 = 1+cos 8 and 2 sin8/2 cos 8/2 = sin 8Therefore,

J1+COS 8 . 8 sin 8cos 6/2 = 2 and SIll '2 = v2(1 +cos 8)

Putting the values of sin a/2, cos 8/2 cos 8, sin 8and r in Eq. (13.2) we get:

2Ak" = a ±(P2+q2)1/4[t(P+Vp2+q2)]1/2/(P2+q2)1/4

+i[b+ q ]+1 ... (13.3)- {2(p+V p2+q2)}1/2Equation (13) can be written in the following

form:Ak"= oc+i~ ... (13.4)where

References1. NICOLlS, G. & PARTNOW, J., Chem: Rev., 73 (1973).2. NOYES, R. M. & FIELD, R. J., Ann. Rev. phys. Chem,

25 (1974), 95.3. NOYES, R. M., FIELD, R. J. & KOROS, E., ]. Am. chem;'

Soc., 94 (1972), 8649; 96 (1974), 200t.4. ZHABOTINISKII, A. M., Biofisika, 9 (1946), 306.5. RASTOGI, R. P., YADAV'A, K. D. S., Indian J. Chem.,

12 (1974), 687.6. RASTOGI, R. P., YADAV'A, K. D. S. & PRASAD, K., Indian

J. Chem., 12 (1974), 794. .7. RASTOGI, R. P. & KUMAR, A., J. phys. Chem., 80 (1976),

2548.8. GLANSDROFF & PRIGOGINE, 1., Thermodynamic theory of

structure stability and fluctuations (Wiley-Interscience,New York), 197t.

9. RASTOGI, R. P. & KUMAR, A., Indian J. cu«; 14 (A)(1976): 1.

10. JACOB JORNE, J. theor, Biol., 43 (1974), 375.11. LOTKA, J. phys. cu«, 14 (1910), 271;]. Am.tchem: Soe.,

42 (1920), 1595.12. RASTOGI, R. P. & GUPTA, M. C., Indian J. Chem., (com-

municated).

23