Kinetic analysis of reversible electrodimerization reactions by the combined use of double potential step chronoamperometry and linear sweep voltammetry

J Electroanol Chem. 184 (1985) 1-24 Elseviler Sequoia %A., Lausann e - Pnnted in The Netherlands

1

KINETIC ANALYSIS OF REVERSIBLE ELECTRODIFvlEMZATION REACTIONS BY THE COMBINFD USE OF DOUBLE POTENTIAL CHRONOAMPEROMETRY AND LINEAR SWEEP VOLTAMMiXTRY APPLICATION

C AMA-I-ORE,

TO THE REDUCTION

D GARREiAU.

M HAMMI,

OF 9-CYANOANTHRACENE

J ?INSON

a.ld J

M S4VEANT

Loborarorre d%lectrochrmre de I’Umuersrte Pans 7. tqurp~ de Recherche Assocree -Etcctrochtmte Molecu[arre”, 2, Place Jtuseu - 75251 Pms Cedex 05 (France) (Rccelved

STEP

r-rl C IV RS

23rd July 1984)

The formal kmeucs of a reaction mecharusm mvolvmg a reversible dunenzauon follobmg the electron transfer reacuon IS analyzed m the context of DPSC and LSV as a function of the hmeucs of the charge transfer and forward and bachward dunenzatron steps The ensumg trea:ments of expenmental data are apphed to the reductton of 9-cyanoanthracen: m dunethylsuifoxlde lks illustrates the capablhtres of a cGmbmed use of DPSC and LSV teadmg ID a precise assgnment of the reacuon mcchamsms and deternunations of the pertment rate constants

INTRODUCTION

Unul recently the apphcatron of electrochemical techmques to the kinetic analysis of electrochermcal reactrons mvolvmg homogeneous chemical reactrons coupled with electron transfer has mostly been restricted to stmple reaction schemes m whrch the overall kinetrcs are governed by a smgle rate dete rmmmg step (see ref 1 and refs therem). Under these condmons, the charactenstics of the polar?zatton curves are related strarghtforwardly to the kmetxs of the homogeneous chemrcal prccess provrded the rates of the heterogeneous electron transfer reactrons 2re fast enough to mterfere only negligrbly in the overall kinetrcs Dimensronal analysis of the polar~zanon problem then shows that the electrochemrcal response depends upon a smgle dxmensronless kinetic parameter whxh is 2 mezsure of the kmetx competrhcn between the chen-ucal and the diffusion processes. This IS a converuent means for relating the electrochemrcrrl response to the reactron orders of the homogeneous chemxal process [l-5] in the case where the form of the rate law does involve reactron orders. 1 e., in the case where the overall kinetrcs are governed by a single rate dete muning step. However, 2s soon as the homogeneous coupled chemrcal process mvolves two or more competing pathways. the retiction order approach is no !onger applicable since 0022~0728/85/.%3

30

(9 1985 Else\rler Scquola S

A

2

the overall kmetlcs are not only governed by a diffusion-chenucal reaction dunenslonless parameter but also by additional dunensionless parameters measunng the degree of com!Jetltion between the various concurrent pathways. The latter parameters may mvoh e, when the orders of the competmg pathways are afferent, not only concentratloc terms but also terms that are related to the time scale of the expenment (mversion tune III double potential step chronoamperometry, sweep rate m hnear and cychc voltammetry, rotation rate in rotatmg disc and rmg-disc electrode voltammetry, etc.) This IS related to the fact that the “homogeneous” chemical process actually occurs w&in a space-dependent context related to the diffusion of :he various species to and from the electrode surface. Overlookmg these parucular conltlons under which the chemical process acts and the mathematicophyslcal procedures appropnate to solve such problems may lead to erroneous mechamzm analysts as already discussed urlth several examples [1,6] In the last few years strateses for rigorously treatmg these competmg pathway mechamsms have been developed and apphed to several expenmental problems (for a general dlscusslon of tl~s questlon see ref. 1). An early example of a reaction scheme mvolvmg two rate determimng steps was the reversible electrodlmenzatlon mechamsL> [7]: A+e-&B

(standard apparent

2El2C

k-1

potential: standard

(equfibnum

E”,

transfer

rate constant.

constant

K, = k,/k_,

coefhclent:

a,

(0)

I@‘)

)

(I)

However, the problem was only treated m the context of hnear sweep voltammetry (LSV) assummg an mfmtely fast electron transfer [7]. The purpos; of the work described hereafter was to extend the analysis to the case where the latter assumptlon is removed and to give a treatment apphcable m the case of double potential step chronoamperometry (DPSC). The reduction of g-cyanoanthracene wdl serve as an experimental example showmg that the combmed use of DPSC and LSV can be particularly prohtable for mechamsm and rate constant determmatlon in the context of such reaction schemes. Several kmetlc results obtamed for the reduction of 9-cyanoanthracene using this approach have already been described in preliminary notes [8,9]. The present paper provides a detaJed descnption of the procedures that have been employed m tlus

case and can be employed ~t.h other systems. It should be noted that we Jo not dlsccss in the followmg the lssie that m the couphng of two amon radlccls of 9-cyanoanthracene the formalon of the 10-10’ carbon-carbon bond may or may not mvolve the precedmg fast and reversible formalon of a non-bonded tier mtermediate [8-lo]: the kmeuc law bemg the same in both cases, apphcation of electrochemical techmques IS obviously not able to answer ths question Convenuonal use of DPSC [11,12] mvolves the stepping of the potential to the plateau of the catho&c wave (reduction of A) and then steppmg back to the plateau of the anod~c B reoxldation wave. This is a significant advantage of the method since

3

the kmettcs of the A/B ektron transfer reactton do not mfluence the electrochemrcal response, usually taken as the ratro r(26)/.1(0) of the anodtc current at twrce the inverston time (28) over the cathodtc current at the mverston time (6 j. However a lumtauon of the method LS that the attamable chemical rates d.re of the same order of magmtude as the maximal avatlable diffusion rates [11,12] as wtth an4 double step or double scan techniques as also with roratmg rmg disc electrode voltammetry Faster chemical processes ~IU give nse to an undetectable anod~c current The peak potential m LSV ma> still be sensitive to the kmetrcs of the chemical reactton for such fast processes grvmg nse to complete u-reversrbility and to the esrabhshment of a stationary state resultmg from mutual compensatton of diffusion and chermcal reaction (“pure klJXt_iC condrtions”) [1,13]. However the kmettcs o_Pelectrcn transfer may mterfere 111the location of a peak ~omtly wtth that of the che.mcal process and, when too slow, blur out the mfluence of the latter (compare e.g ref 14). The two methods thus appear as complementary makmg theu combined use profitable. FORMULATION

OF THE

PO LARIZATION

The reactton scheme conststmg set of partial denvattve equations:

PROBLEM

of reacrlon

acdat

= D,a”c/,/~x2

ac,/at

= D&,/ax”

- 2k,c;

acdat

= Dca%c/ax’

+ k,c; - k_,cc

(0) and (I: gwes me

to the foLlowmg

(1) + 2k_,cc

(2) (3)

The diffusion 1s assumed to be lmear and semi-mfunte dependmg therefore only on time, t, and on the distance from the electrode surface, K The c’s are the concentrations and the D’s the diffusion coefftctents of the various reactants For the sake of slmphcity we wtll assume, m a first approxlmauon, that DA. D, and DC are practically equai to their average value, D. The vahcht~ of Gus assdmptlon and Its effects on the results wrll be &scussed LUz later secbon of thx paper. Under these ~ndmons, the conservrtion of matter m the lffusion !ayer as obtamed from integrahon of an aT&ropnate linear combinatton of the three parttal denvatrve equations leads to cc = l/2(

CO- c, - cg)

co being-the bulk concentration above systems are as follows:

t=O, x = 0,

D(acdax)

and

x10 t 3 0:

x=oc),

(acdax)

of A. The initial and boaundary conchtlons

t>,O

+ (acB/ax)

c,=r”, = 0,

= k;P exp( &) [ c, - cB exp( - <)I

for the

c,=c,=O

(4)

acda_K = 0

(5) (6)

wrth 5 = -(F/RT)(E - E’). E bemg the elec?rode potential at trme t, E”, a and kzP being tne standard potential, transfer coefficteni and apparent standard rate

4

(i e , uncorrected for double layer effect) of the A/B couple. The current flowmg throu~ the electrode surface is given by z = F&Wac&3x), where S 1s the electrode surface area. The polarizauon curve, 1.e , the vanations of the current wrth time ill then be obtamed from the integraucn of eqns. (l-3) takmg into acrount the inthaI and boundary condrtrons (4-6). The above forrnulatton stands fcr bo:h LSV ar,d DPSC. The polarizatton curve for each techtuque IS obtamed from the introdaction into above equations of the parttcular dependence of the electrode potenti& upon time tn each case Adimensionahzation of the above equatrons IS obtamed by mtroduction of a senes of dunensionless variables: 7 = t/e (8 = RT/Fv in LSV, t? IS the duration of the catholc steps m DPSC), y = x(OB)-‘~, a = cJc”, b = c,/c’, X = k,c08, K = k_,,/(k,cO) = l/K,c’, A = k:P(O/D)‘n-

constant

aa/aT=aza/ay2 ab/a7= a2b/ay2 -

0) 2x [ b2 - (42)(1-

a - b)]

(8)

wllll. for

~=O,y>0

for

x=O,yaO

and

y=cc,r),O:

a=1

aQ/ay= -ah/ay aa/ay= A exp( a.$)

The dunenstonkss function defmed

poianzauon

curve

b=O

(9)

(10) [ co - 5, exp( -01

1s obtatned

from

(11) the dimensionless

current

as- \k = I/[ I?!&“( D/8)‘/2]

For LSV this set of equations 1s associated with 5 = u + r where u = (F/RT)( E, - E’), (E = E, - vt where v IS the sweep rate and E, the nutral potentral of the scan).In DPSC, E >> 0 for the catholc step (0 c Q G 1) and 5 -=z 0 for the anodic step (T > 1). It IS noted that the cases where C is oxldrzab!e m the avatlable potenttal range, the potential should be stepped anodrcally m between the B and C reoxldatton waves so that the condrtton t < 0 1s fulfilled wrthout appreciable contrrbution of the C oxidation current. Under these conditions, eqn. (11) can be replaced by. y = 0,

O-=r
y=o,

7=-l.

To proceed

a, = 0

MaI

bo=O further the two techniques

@lb) must be dealt ~th

separately_

DOUBLE POTENTIAL STEP CHRONOAhWEROMETRY (DPSC) The most commonly used parameter observable in DPSC is the ratio ]i(2e)/l(e)] of the anodrc current at time 28 over the cathodic current at time 8. In the absence of a follow-up reaction, i.e., for a purely dtffusion controlled system this ratio 1s equal to (1 - 2- l/2) = 0.293 [11,12]. We will use accordingly m the following the

normalized

ratio-

which IS equal to 1 for pure diffusion control. When X and tc do not have partrcuhuly large or smal! values, no snnpbfxation can be made. It is then necessary to have recourse to a F5nite chfference approach for the numerical resolution of the system (eqns 7-10, lla. l!b). We used an explicit method as described m the Appendix. It is however emphastzed that thz asymptotic solutions described in the following are essential for carrymg out the complete fmite difference calculation only in restncted ranges of A and K values in which convergence and accuracy can be mastered wrthout dlfficu!t!- usmg a reasonable amount of computer time and memory occupauon. The results are shown in Fig 1 m the form of R vs log X fcr a senes of values of K. It is first seen that when K a :, in practice as soon as K x=- 2, R does not deviate apprectably from 1 over the whole range of A values. ti derzves from +he fact *hat, under these conditions, dnnertzatron IS essentially m favor of the left-hand side. For smaller values of K (1 e smaller values of the dtmenzation eqrihbrmm and/or of the bulk concentratron), R fust decreases as h mcreases rin increase 111 X may result from an mcrease m k,, and/or co, and/or 8. An rncrense of any of these factors results tn a faster conversion of the Ill0'115zci 2 into the diner C dunng the cathodtc and anodx steps and therefore a smaller anodic current. 731s implies that AK = k_,e should not be too large for C to apprecrably redrssocxate d!_:rmg the

Fig 1 DPSC Vanauons of R = 11(28)/l (8)1/(2-‘flof K = l/Klco

1) wth log X (A = k,c’B) (the numbers on the c~t-vc are the values of log K)

for a senes of values

6

anodtc step; oxrdation of B which depletes its concentratton at the electrode, thus tends to serve as a driving force for C re-dtssoctation. The latter phenomenon progressively shows up for a further increase m x which, at a grven value of K, results m an mcrease in XK = k_,B, the lonetic parameter of the backward reaction Accordmgly we observe that R stops decreasing and nscs agam to reach 1 ultimately, the value c>rrespontig to pure diffusion control Thus mcreasmg X, the system passes from a first type of dtffuston control (DO) resulttng from the fact that the dnnenzauon reaction is “frozen” within the hme-scale of the expenment to a second type of dtffuston control (DE) where, conversely, the dimertzatton reaction IS so mobile rn both drrecttons, wrthm the tune-scale of the expenment, that B and C undergo pa-alIe dtffuston processes coupled by a drmenzatron process which constantly remams at eqmlibnum When h 1s large enough for this situation to be reached, all of B and C that had been produced dunng the cathodic step have been re-oxrdrzed dunng the anodtc step wtth the exceptron of that which escaped by &ffusron. It IS therefore understandable that the first decrease in R below 1 should be compensated for by a further mcrease wluch overshoots 1 In other words, a part of B is stored tn the form of C at short times and restored at longer times. As expected a smnlar pbcnomenum occurs in the case of a first order follow-up reaction [15] although the fact *hat the calculatton were camed out m a much less extended range of h makes it less apparent. An interestmg hurting situation 1s met, when K is so small that R ftrst decreases to prachcally zero (Fig 1) before mcreasmg agam to reach 1 ultnnately The R-log X working clirve can then be divrded mto two independent portrons, one descending for small X’s and one nsmg curve wrth a maxrmum for large A’s. a

-2

b

0

I

2

log h Fig 2 DPSC Vanatmns of k wth log X (same defimtmn as m hg 1) fo- small values of L (a) rreverslble d!menza?lon (KO zone III the zone diagram of Rg 3), (b) pure kmetic conditions (KE zone m Fig

3)

7

The descendmg branch corresponds to an rrreversrble drmenzatror and 1s therefore the same as already computed [16] (FIN. 2a) The nsing branch corresponds to “pure kmetrc” condrtrons As shown m the Appendix the system of partial derivattve equations, imttal and boundary condttions can be transformed into the followmg mtegral equatron. W(2P

)3’4 = [ b,’ - ( &/2)‘r-]

[ (r+/,)‘fl+

h&/2] “1

(12)

with b; = &,K-“‘,

p = 2~(X/3)*‘~

= ~~_,(/c,c~)-~‘~(L

/3)“3

and a0 = 1 - Ia)

(13)

wtth

Tl-us set of equattons and condrttons apply whatever the elec’roche~ntlc~ techmques In the case of DPSC, eqns. (Ila) and (13) lead to + = (TT)-“’ dur.ng the cathodic step whrle eqns. (lib) and (12) leads to I$ -r p+4’3 = 0

(15)

for the anodtc step. The resulting R-log p workrng curve :s shown rn Frg. 2b. As drsccssed above it 1s seen that R passes through a maximum (R = 1.34) before tending asymptotically towards 1. Figure 3 summanzes, m the form of a log ~-log h kmetx zone diagram, the transltton between the various hmrtmg behaviors as a function of the parameters governmg the system. The hmits between the zones correspond to an uncertarnty of 0.1 111the deterrnmatton of R. The two diffusion contrcllert behavrors are designated by DO and DE respectively. The first corresponds to a “frozen” reaction (A small) and the second to a totally mobile equrhbrium (X large). They both correspond to R = 1 and cannot be distinguished from each other if K is shgbtly larger than one on the basis of current measurements. Note, however, that one cannot pass directly from DO to DE by varymg 0 as can also be seen in FIN. 3. KP corresponds to R = 0 and KO and KE io the workmg curves shown in Fogs. 2a and 2b respectively. KG, the general case, 1s obtamed when K is not too small wluch corresponds to the cases where X does not come close to zero upon incrca.&g h (Frg 1). To obtam the rele-/ant workmg curves, which cannot then be decomposed mto two mdependent branches, a firme drfference resolution of the partral derivative equation system is required (see Appendix). It 1s noted us this conuectron that for CmaIler values of K, the fact that the fmite difference resolutron can be replaced by the resolution of the integral equatton (15) is of great help. Indeed, as X mcreases departure from equihbnum IS restricted wrthm a thinner reaction layer adlacent to the electrode

-0 DE R-_I

--2

--4

KE

\

--6 KO

DO --8

KP R,O

R=l

--10

k 8 k

L

-4

-2

0

2

4

k,,C"

,t5,#,,

8

logi

Fig 3 DPSC Kmetlc zone diagrams For the defimtlon of the zones see text

surface. A y-7 gnd of srr;Jler and smaller mesh should then be used m the fmte CJference resol6on leading rapidly to prohibitive computatron tn-ne. LINEAR

SWEEP VOLTAMMETRY

The kmehcs of the folk+up drmenzauon mechanism have already been mvestigated m the case nhere reactton (I) 1s n-reversible [13] The cychc voltammogram then depends upon two dunensronless parameters: A = k, c’RT/ Fv and A = kgP( RT /DFv)‘~. The kmetlc behavior 1s summarized by the zone lagram m Fig 4a DO corresponds to a pure diffusion control srtuauon where the dirnerizatlon process is slow and the electron transfer 1s fast relatrvr: to the sweep rate. In Kp, electron transfer 1s sttll rapid but X so large that the pure kmetic conditrons are fulfilled. The cyclic voltammogram is lrreversrble and the peak potential depends UPOP v and co accordmg to [2]- &%,/a log v = - 19.4 mV, i3Ep/a log co = 19.4 mV at 2c “C. In IR, the rate controlling step IS, besides diffusion, the mitial electron transfer either because of its mtMSlC slowness (left-hand part of the zone) or because of the rapid consumption of B by the dimerization reaction which prevents the backward electron transfer occumng (right-- hand part of the zone). The transition between any two to these hmitmg behaviors depends upon a single

9

parameter. A for the KO zone (transttion-between DO and KP), .A for the QR zone (transttion between DO and IR), AA-“/3 in the W zone (transitron between KP and IR). The system depends upon two parameters m the genera: case represented by a rather restricted zone (KGl) in the hagram (Fig 4a). If K 1s larger, the reverstbtlity of the dimerization reactton shnulli be taken rnto reactton may interfere kinetically When account, I e., the backward dimenzation K x1 the overall effect of the dunenzatlon reactron varushes for the same reasons as already dtscussed m the case of DPSC. Conversely when K -Z 1, the backward dtmerizatton reaction mterfers for large values of A, i e., in the context of pure kmetic condittons for t: e same reasons as in the case of DPSC The system 1s thus described by the foilowmg sysiems of integral equattons: -

02) (16)

I# berg defined by eqn. (14) and wrth. A* = AK”/~, <* = .$ - (l/2) In K, p = 2 K ( A/3)2’3. The I_.SV peak charactenstrcs thus depend upon two dimensionless parameters, p and A*. Three lmutmg situations are obtamed for extreme values of these _ parameters as shown m the kmetic zone diagram of Fig. 4b. When A* + 0, eqn (16) becomes: (#/A*) exp( -a<*) = 1 - 14 which corresponds to kinetic mntrol by the forward electron transfer (IR zone). With the change of variables 51R = a< + ln(A~!-‘I”), #rR = #~a-‘/~.

$I’~ exp( --,rR)

= 1 - I#lR

(17)

b

G-j-

I

lo9 A+

(‘14)

109 (H/z)

4 DO

KO

KP

KE

KP

I I

DE

2 I

I

4

2

0 QR

I

1

-2 I

I

0 I

I

4 [

fog A+ (3/2) log K Fig. 4

LSV

tiermuon

Kmellc zone dmgram (cc= 0 5) for an ureterable cmmpondmg

to pure kmetic ~I~ILIOP~ (II) For the

drmeruatlon (a) and a fast reversible

defiiuon of the zones see text

10

which Iedds to the following peak charactenshcs [13]: [LR = 0.78, [kR - S$p,z = 1.85, $tR = 0.496. When, conversely, A* + co, eqn. (16) converts into the Nemst equation Ieadmg to: b,* = (1 - I+) exp(t*). The system then depends on a smgle parameter, P. A ftrst sub-case corresponds to P + co, i.e., to total mobrhty of the drmertzatton equthbnum (zone DE in the diagram of Ftg 4b). Equation (12) then becomes sunply: b,* = (1$/2)“2 and thus: 14 = ([4

+ exp( -2EDE)]“’

- exp( -EDE))‘/4

(18)

In 2 = 5 - l/2 ln(~/2). NumencaI resolution of eqn. (lf.) with gDE= 5 + (l/2) provrdes: $‘” = 1.04, 6FE - t,; = 1.63, GL, = 0 500. A second sub-case is obtarned when, conversely, p -+ 0 (zone ICP rn the chagramj. b,* + &‘2’3/(21’3p)“2 and thus: @I3

exp( -t”)

= 1 - 14

(19)

leadmg to [7]: .$r = 0.502, &p” - tp”/i = 1.512, qr, with. tKP= & + (1/3)Ln(4h/3), = 0.527. Thrs case corresponds to a rate determinmg n-reversible dtmerization following a fast electron transfer reactton as already discussed in the context of Fig 4a. The transition between any two of these three hrmting situations, I.e., IR (eqn. 17), DE (eqn. 18) or KP (eqn. 19) mvolves a stngle parameter. The transrtton, RI, between IR and ICI-’ corresponds to p - 0 so that: b,* = 1j.~‘/~/(2’~p)‘~ which leads to. + = A exp(&) [l - I$ - (3/4h)‘/392/3 exp( -E)] i.e., and A’ = A(4X/3)-“j3, to: mtroducmg: tK.’ = t + (1/3)Ln(4X/3) #=AhKJ exp(aEKJ)[l

- I+ - +‘I3

exp( -.$“)I

(20)

The dunensionless equatron of the voltammogram therefore depends upon a smgle parameter, AM. IR 1s obtamed for AK-’ --, 0 and KP for AK” --, co as drscussed prevtousiy [ 131. The transrtton, El, between DE and IR corresponds to p + 0 so th2t. b,* = whrch leads, takmg eqn (16) mto account, to: ( 1+/2)“2 9 = A* exp( (x5*) [ 1 - I+ - (&/2)r”

exp( -

#j*)]

(2;)

whch shows that the system depends only upon A* = AK~/‘. The IR and DE ZO-ICS are reached for A* + 0 and AL + cc respectively. The transrtton, ICE, between ICI’ and DE, corresponds to A* j co. The system depends only on p, the dimensionless voltammogram being give by: (2P) -3’4$

= [(l

- 14)

exp(<*)

x { ( 1+/2)]”

-(

1$~/2)‘~]

+ Cl --(1+/Q]

eAp(E*)}rfl

(22)

KP and DE correspond to p + 0 and P + co respectively as discussed prevrously [7]. The boundary Iines separa*ung -&e varrous kineuc zones m Frg 4 were determmed

‘1

0500

----

-I -2

0 log

2

0

0

-2 log

A?

b

2

-2 ’

Fig 5 UV p& charactenshcs for lhe trIISltlOn Lmutmg kmeuc behawors (- - -) DE, (-

WlleS

E.1

for

CX=

’

1 log

A”

2

0

’

’

I

A*

0 5 h* = Axan.

<* = 5 -(1/2)~~

1 IR

on the basis of a &2 mV uncertainty in peak potential measurements Note that a smgle kmetic zone diagram can be constructed for each value of K by simply shdmg Fig. 4b onto Fig. 4a so that the zero of each axis iu Fig. 4b comncldes pllth -(3/2) log K (abscissas) and (-l/4) log K (ordmates) of the axes m Fig. 4a (see e g. Figs lob, c). The vanatlons of the peak charactensucc. with the appropnate dunenslonless parameters A* in the transition zone EI are shcwn in Fig. 5. Those corresponding to the transitions KI and KE have been published already (respectively, refs. 13 ar?d 7). EFFECT OF THE AND DIMER

INEQUALITY

OF THE DIFFUSI3N

COEFFICIENTS

OF THE

MONOMER

The above discussion assumed that all three diffusion coefflclents of A, B and C are equal. This IS a reasonable approxrmation for A and B but may appear questionable for C owmg to the large change m molecular size What could be the effect of this inequahty is now discussed. Let us hrst note that there is no effect as long as we are dealmg ~th situations where C does not interfere in the overall kinetics, i.e., m DO, KP, IR, KO, QR, KI or KG1 (Fig. 4). An effect is expected m the DE, KE, EI and KG2 (Fig. 4) cases, i.e., in cases where the pure kmetic conditions are flulfilled it follows that eqn. (12) sUll apphes provided K is replaced by K* = Kd-ln, where d is the ratio of the Mfusion coefficients of the dimer and monomer, both in the defnubon of p and bg. In the case of DPSC, the general case (KG on the zone diagram of Fig 3) results from an overlappmg of the KO and KE behaviors. In th: frrst case there is no effect of d while in the s@and the effect is descnoed by the replacement of K by ted-‘/Z. In the KG zone the effect of d is thus presumably smaller than in the KE zone. There 1s no major tiflculty for obtammg a quanutative estimation of the effect of d 111the KG zone in the context of a finite Merence resolution provided its value 1s knows

12

EXPERIMENTAL

ILLUSTRATION,

REDUCTIVE

DlMERIZATlON

OF 3-Ck’A>:~ANTHR.A-

CENE Products

9-cyanoanthracene was an Aldrich product and was used as received. An authentrc sample of lO,lO’-dicyano-9,9’, lO.lO’-tetrahydro-9,9’-bianthyl was prepared vra a Gngnard reaction accordmg to reported procedures [17,18]. Identifrcatron of the clihydrodtmer obtained through electrolysts of ANCN was based on comparison of rts NMR spectrum wrth that of the arlthentic sample A 50/50 ratro of the two stereoisomers was obtamed NMR = (isomer la) = multrplct 6 8.75-6.75 ppm (aromatrc protons), smglet S 4 5 ppm (equrv. H m posrtrons 10,lO’); singlet S 3 ppm (equrv. H in posrtrons 9,9’); (rsomer lb) = mrlltrplet 6 8.75-6.75 ppm (aromatic protons), two smglets 6 = 5.2 and 4.85 ppm (non equrv. H rn posrtrons 10,lO’); AB spectrum 64, 4 1, 4 45, 4.55 ppm (non eqmv. H m positions 9.9’). H” CN

NC Pa

lb

H’ CN

1 Eiecrroiysu

Electrolysts was performed in a 100 ml au-tight cell under a mtrogen atmosphere, at a mercury pool (ca 40 cm* surface area) electrode The reference electrode was a SCE v&h a bridge of the same cornpositron as that of the catholyte The counter electrode was a platmum gauze. The anodrc compartment was isolated from the catholyte by a Nafron 125 membrane (DuPont de Nemours). At the end of the electrolysrs, ca. 100 ml of deareated water were added to the catholyte to precrprtate the organic products, wluch were then separated and purihed. The solvents (ACN or DMSO) were previously bulled and dried over alumina except when proton donors were added. Transrent electrochemrstry

The solvents (ACN or DMSO) were freshly distilled ucder a dry nitrogen atmosphere and were stored alumina. In some experiments alumina was present in the cell to prevent any water cortammation. In all experiments reported here 0.1 M LCIO, was used as supportmg electrolyte. The reference electrode was a Ag/O.Ol M

13

Ag+ rn the corresponding solvent. The counter electrode was a mercury pool of ca 4 cm2 surface area. The working electrode was a hanging mercury drop. All expenments were performed at 20 OC. The electromc set up was tdenttcal to that prevrously described The potenhostat with positive feedback ohrmc drop compensahon [lo] was drrven by a PAR (model 175) signal generator C~dx voitammetry The voltarnmograms

were recorded wither on a X-Y recorder or on a storage osctlioscope. The peak potentials were measured by a coincrdence method. For comparison,, the same procedure was repeated usmg an analog-dtgrtal conversion of the voltammograms. Reproductbrlity of the E, measurements was ulthm f 2 mV. Double potential step chronoamperometr)

The method consisted of the measurement of r(r = 8) and I(T = 28), m order to evaluate the rauo R = [z(20)/~(0)]/[2-~/’ - 11. For thrs purpose, the data acqmsttlon devtce (8 bits, CAN 4110 Teledyne Plllbnck) was dnven by an external clock allowmg the storage of three sets of 256 data correspondmg to the followtrIg ttme mtervals. [ - 512 ps, 0], [9 - 512 ps] and [ 29 - 512 ps] whatever the value of 9 2 1 ms For each set a linear regression analysts of the last 64 data led to the determmation of the lgital values of I( T = 0), I(T = 8) and I( T = ZB), and the ensumg value of R was computed accordmg toR=

{[I(O)-1(28)]/[I(B)-1(0)]}/(1

-2-lfl)

All the acqursttton process was controlled by 2 F8 rmcroprocessor (6fi K byte) accordmg to the following procedure. Imtmhzation of the acquisttton started the external clock which drove both the analog/d@al converter and the signal generator P,4R 175 to obtam a total synchronizatron of the procedure. At the end of the third acquisitton set, the 768 data (stored in a 1 kbyte buffer memory) were transferred into the F8 nucroprocessor for further treatment (see above) for R determmatron, or for display of the raw results on a pruner (Srlenr 7000) or on a scope (via a dtgrtal/analog converston through an 8 btt CNA 4021 Teledyne Phtlbnck). Owmg to the analog/digrtaf converter used m the set-up the mtrmstc precrsron on R determmatton was better than fO.O1 for R 3 0.1. For lower R values the error was greater owmg to the poorer precrston of the dtfference I(O)--1(219), arismg from the lower signal/noise ratio durmg the anodic step when j1(29 >I -=Kl( 6) The reproducibihty of the overall R measurement was within ca. +O.Ol provrded Iz was > 0.1. RESULTS

AND

DISCUSSION

Most of the prewous electrochemical investlgatlons on 9-cyanoanthracene (ANCN) reductron have been concerned witt its mech‘amsm of drmerizatton [8-10, 203. However, to the best of our knowledge no attempt has been made to isolate the

14

corresponding duner although this has been shown to be posstble for related compounds [10,21]. In dry DMSO or even in the presence of 10% added water the coulometry gives c value far above the theoretical 1 electron molecule predicted for a dimcn-rrtion mechamsm, showing that the mechamsm involved IS certamly more complex (DMSO, n = 2, DMSO + 10% H,O, n = 1.5). Further evidence of the discrepancy between the behavrors observed by transient methods and preparative experiments IS shown by the fact that the electrolysis current never drops to ;1 zero value but remams fixed after sometune at ca. 20 to 30% of the mitial current. Th.ts strongly mdrcates a catalyuc process regenerating ANCN. More evidence is brought by the fact that an electrolyzed solutton of ANCN m DMSO, allowed to stand under a nitrogen atmosphere, shows a progressive restoratton of the ongmal ANCN wave. Similarly the workup of an electroiyzed solutron leads to: 52% 4NCN, 40% anthraqumone and ca. 2% anthracene In order to check thus possible regenerattve mechamsm, the presumable dthydrodrmer 1 (drcyano-10,10’,-tetrahydro-9,9’,lO,lO’-dianthryl-9,9’) was prepared [17,18] and allowed to stand m a DMSO solution under a nttrogen atmosphere. The ANCN wave progressively showed up m ca. 12 h. In the presence of a storcluometnc amount of NBu,OH base the ANCN wave was restored almost mstantaneously In ACN, wtthout added base m the presence of phenol, the drhydrodrmer 1 was stable for ca. 4 days; when NBu*OH was added to an ACN solutton of 1, ANCN was agam obtamed (75% ANCN + 20% anthraquinone after extractton). All the above observatrons thus strongly suggest that although a drmenzation process occurs m the dtffuston layer the resultmg duner 1, or its conJugaled bis deprotonated base, (ANCN);-, undergoes a factle cleavage restoring ANCN. Although the exact mechamsm of thrs last process 1s not known rt appears to be catalyzed by bases (see above)_ Owing to the evolutton of the eleetrolyzed solution a

plawble

mechamsm

could be described

as consistmg

of a ray-d monomenzation.

OH-

1 ---, ANCN+2

(11)

followed by a slower process

2 2O:-ANCN Compound

+ 2 was

not

identtfmd,

but

PS most

presumably

the 9-10,

dthydro-9-

cyanoanthracene. As described above, reacttons. (II) and (III) appear considerably slower in ACN, leading to an enhanced stabthty of the drhydroduner 1. We thus electrolyzed 0.2 g of ANCN m 100 ml ACN (0.1 M NBu,BF,) m the presence of 0.1 M phenol. Workup of the solution after 2 electron/molecule ware passed led to the recovery of 25 mg of the dthydrodrmer 1, as identrfied by its NMR spectrum bjr comparison wrth that of

15

an authentic sample prepared vra an independent procedure [17,18]. Note that two tsomers (la and lb) were obtamed 111a 50/50 ratro. No further effort has been made to optinuze the yield of 1. Mechamsm of the reductwe couphng of ANCN As lscussed earher [8-10,201, the reductron of 9-cyanoanthracene (ANCN) rn dtmetbylsulfoxide (DMSO) or drmethylformarmde mvolves the reversrble radrcal-radical couphng of two anron-radrcals following the imtml electron transfer reaction provrdmg an rllustratron of the mechamsm &scussed m the preceding ection: ANCN 2

+ e- *

ANcN - e

ANCN

-

(ANCN);-

(0) (I)

We show hereafter how the combined use of LSV and DPSC allows us to estabhsh the adherence of the kinetic behavior predrcted for thrs mechamsm with ‘the experimental kmetics and to determine the values of the dsmellzatron equrhbrium constant and the forward and backward rate constants. Prehmmary LSV approach Figure 6 shows the results obtained 111DMSO m the presence of 0.1 M LiClO, at 20°C m the form of a plot of the catholc (E,‘) and ano&c (E,“) peak potentrals against log of the sweep rate wrth a milhmolar solutton of ANCN At low SC-~ rates, an almost reversible wave 1s observed. Raismg the sweep rate results m an increase m the peak separatton whrch could be viewed as remmiscent of charge transfer hnetic control. That thts 1s not actually the case is shcwn by the disappearance of the anodic wave as the sweep rate is further mcrea s~d(0.2Vs-‘~v~ZVs-‘) For hrgher scan rates ( v > 2 V s- ‘) the observed behavior suggests mrjced kinetrc control by charge transfer and a follow-up chemical reachon, the former prevatling over 300 V s-’ (compare with ref. 14 where similar behavior is obtained for isophthalomtrile in DhJF). In vrew of the kmetrc analysis developed in the preceding sechon, the observed behavior mdrcates a DE --, KP + DO + IR transrtron as v increases. To support thts vrew we superimposed, in Fig. 6, the E,-log v linear vat-rations correspondmg to each of these hrniting behaviors to the experimental pomts. Although the expected trends are quahtatively followed it is seen that detailed quantitative agreement cannot be achieved this way. Furthermore, we note that the avarIable range of sweep rate values, i.e , ca. 5 orders of magnitude, does not allow all mdividual hmrtmg behaviors to be accurately reached successively (compare with Fig 4). Thrs suggests that rather than pure hmitkg situations, transrtron type behaklor is dealt wnh. In this context quantitanve treatment would involve curve htting with four parameters, k,, K,, kgp and (Y. The reliabihty of such an approach being obviously questronable,

EpI v vs AS/&l+

-1 650

DE

Fig 6 Reductmn ~1 ANCN (1 mM) tn DMSO +0 1 IV LICIO~ at 20°C Cychc voltammetnc cathodtc (A) and antic (A) pea potentials as a function of the sweep rate The stmght hnes correspond to the behavlour predlcteJ for the DE, KP, DO and IR hnutmg %tuatlons as mdlcated

it is preferable mterference

to proceed further usmg DPSC transfer kmetrcs.

data which

are not altered

by the

of charge

Anclysrs oj the DPSC data Figure 7 shows the results of a senes of DPSC expenments can-red out wth the same system as m the proceeding section, ANCN 1 mM, DMSO, 0.1 -44 LiClO, at 20°C The shape of the R-log 8 vanatron with its rnmimum matches what IS predicted for the foGow-up reversible dunerizahon mechamsm (Frg. 1). Frttrng of the data wrth the woriang curves involves a two parameter regression. Thrs can be carried out m a sunple manner by notmg that the mmimum and maxunum on the R-log X workurg curves depend only on K (see ref 22 for a srmrlar treatment). The variatrons of R _ and R _ wth log K are given m Fig 8. In the present case, usrng the mmimum determmed vra an harmomc mte_polation of the data leads to K, = k/k_, = 5.7 x lo4 M-l. Usmg the appropnate R-log h working curve, log K = -1.76, (Frg 7) then leads to k, = 2 8 X 10’ M-’ s-’ wrth a grecrsron of ca 2%. k_, = 5 s-l ensues. In the derivation of these results the drffusion coeffrctents of the monomer and dnner were assumed to be the same. This certamly results in a systematrc error m both rate constants. As discussed earlier the largest error is expected to affect the value of k_,. We have no expenmental estunation of d. The monomer and duner can be viewed roughly as thm circular discs and the overall diffusion as an average

17

-2

-3

-1

I

log ( e/s

0

1

Fig 7 DPSC data for the reduction of ANCN (1 mM) m DhlSO+O 1 M LKIO, at 20°C Potential stepped from - 1 094 to - 1 541 V vs SCE and back The sohd lme IS the workmg curve correspondmg to IogK=-176

-6

-4

-2 log

I

0

1

2

I

K

Fig 8 Analysis of DPSC data Vanalons of the muumum, R_. workmg

cuives

w~tb log K

and IMXUIIU~. R_.

of the R-log

X

18

of two diffusional movements, one with the axis of the &SC parallel to the dtsc axts and the other perpendicular. Then usmg the Stokes-E&tern approximation d can be estrmated roughly as (1/2)‘j3. It follows that the systemattc error in ic__, ?s ca. 10% at the most and presumably much less tn Ir,. Thus confirms, as suggested by the LSV data, that the system “ travels” across the transition zones KO and KE III the avarlable range of 6 variattons. Quarwtame

analyss cf the LSV

data

The LSV data can now be treated knowmg k, and k_, with only two adjustable are shown in Fig. 9 where, parameters, a and k :P. The results of thrs treatment besides the sweep rate, the dependexe on another mportant parameter, the ANCN concentration, was also investigated. The solid lines fitting the experimental pomts were obtamed wrth a = 0.4 and kgpD-‘D = 3 5 X lo* s-‘I’ (leading to kip = 0.8 cm s-1 taking D = 5 x 1o-6 cm2 s-1 )_ The order of magnitude of these values of a and kgp IS certainly correct (compare with ky = 0.32, 0 22, 0.26 cm s-t found for phthalonrtnle, teraphthalonitriie [23] and rsophthalorutnle [l4] in 3MF + 0.1 M NBu,CIO,). However the preclslon of their determmation 1s not expected to be very good. Greater accuracy zould be reached usmg higher sweep rates but this would requtre a mathematical treatment for ehminating the ohmtc drop due to the uncompensated reststance remainmg after posiuve feedback compensation [24]. Another approach would be to adapt the use of ac techniques to the problem described above These techmques whch are the most convenient for analysis of the kinetrcs of fast electron transfer reactions [25] have already been employed m the case of an n-reversible follow-up dunenzatton [26,27]

Ep/Vvs

Ag/Ag+ I

Rg 9 Rehctlon of ANCN (1 mM) m DMSO+O 1 M kCl0, at 20°C Vanauon of the cathodx peak valuesof theANCN concentmhon.co = 0.1 (*), 1 (@), 10 potexltlal.ZP arch the sweep rate. u. for several (A) mM The sohd lmes represent the theoretical &-log u curves for k, = 2 8X -0’ M-’ s-‘, K,=5.7X104M-1,k~,?-‘~=35X102s-‘/2uldcr=04

19

b

a klK 0

f

00

KG

FIN 10 LocatIon of the pomt representrng the system m the zcnz diagram (Ftg 4) for the expe-ments Fig. 9 co = 0 1 (a). l(b), 10 (c) mM See text for (a) and compare Fig 4 for (b) and (c)

of

Figure 10 shows the locauon of the pomt representing the system on th-e zone dtagram denved from Frg. 4 for the set of three expenments (co = 0.1, 1, 13 m&f). The two diagrams shown m Fig. 4 can indeed be combined mto a single one once K = l/K,c’ IS know. The construction of tbs smgle zone diagram 1s &fferent m the three cases smce co, and hence K, vanes from one to the other. For co = 0.1 mM tt I$ seen that charge transfer kmetrcs mterfere only at the highest edge of the sweep rate range. On the other hand, there IS an almost direct passage from the KO to the KE zones with practicaliy no pomts correspondmg to the simple hmrtmg behavior KP. At very low sweep rates the behavior 1s close to DE, t-e, to pure diffus:on control with a totally mobtle dimerrzatrou eqmhbnum. The kinetic zone dragram shown m Fig 10a 1s represented m the log(k,?‘RT/Fu) log(k_,/k,c”) plane smce the interference of the rate of electron exchange (Q’) rs negligtb!e on the overall kinetrcs. Note the similarity of thts kmetac zone dragram estabhshed for LSV measurement with that of Fig 3 corresponding to DPSC expenments. For co = 10 mM, the mfluence of the charge transfer kinetics appears m a much more extended range of sweep rates (the KI zone IS now crossed) due to the fact that the forward drmenzation reaction competes more efhctently with the backward electron transfer due to the fact that its second order rate increases mth concentratron. The mcrease in K resulting from the Lncreztse m co prevents the system reaching the DE reversibtlity at low sweep rates. For co = 1 mM, the srtuation IS between the two behaviors described above, as shown UI Frg lob. CONCL!JSION

The above described results and discussion shows how the combined use of DPSC and LSV can lead to an accurate characterizatron of the reactron mechamsm and determmatlon of the pertinent rate constants. It is worth emphasrzmg that brute force analysts in terms of &!&/a log u and X,/i3 log co slopes may lead to erroneous

20

TABLE

1

Apparent slopes obser.ed m various ranges of u, and co vanauons dewed

u/v s-1

- !aE,/a co=01

lo-2too1 01to10 10 to 10 10 to 10’ 102 to 103 lo&u/v

s-‘)

-20 -15 -10 -05 00 05 10 15 20 25 30

mM

8 15 9 45 10

(aE,/a

from the-data of Fig. 9

log u)/mV c”=lOmM

c”=lOmM

14 5 22 22 11 12

19 20 24 5 23 19

log P)/mV

Ol~PclOmM

lO-=PclOmM

30 31 28 25 21 15 9 3 1 0 0

28 21 20 19 21 225 20 15 5 9 35 1

mechamstic conclusions. In thrs connectron, Table 1 grves the various apparent slopes observed m the various ranges of u, and co vanations denved from the data of Fig. 9 while Table 2 gives the theoretical slopes for the various limiting behaviors. It IS clear that applicatron of the theorehcal slopes from Table 2 to the data of Table 1 neglectmg the posstble transttion behatior would lead to the conclusion that the mechanism 1s more “complex” than stmply mvolvmg reachons (0) and (I). As discussed in detatl elsewhere [l], thrs underhnes the danger of using oversimphfied mechamstic approaches whrle ngorous kmetic analysis is presently avatlable as described here m the case of a follow-up reversible dnnerization.

TABLE

2

Tbeoretxal slopes at 200(’ for the various lmutmg behawors L~rruung behavior

-(x,/a

DO IR KP DE

0 29 O/a 19r 0

log o)/mV

- caE,/a 0 0 194 290

log c”)/mV

21

APPENDIX (I)

Derwation

of eqn

(12)

Equatton (8) being a sunplifted form of eqn. (8’), the following presented for eqn. (8’) The system to be solved 1s then

au/a7

= a*a/ay*

ablaT

= a2s/ay2

ac,fa7

= a2(dc)/ay2

with

-

2h(

analjjs~s wrll be

KC)

6’ -

+ x( b2 - KC)

c = CC/d?

When X is large, reactton (I) is m raped equrhbnum outside a thin reaction !ayer adJacent to the electrode. Inside this layer (y G p) pure lonetic condmons apply to B and C, whereas B and C are under diffusion control with b2 = KC outside the kmetic layer. The above system can then be wntten as: d2b/dy2

Y
= 2A(b’-

wrth the bouudary (b + 2dc),, b,=

conltions

= (be + 24,)

-(qj--l~

at y = u: and

)~T(dWdy),(r

b; = KCp and Combmatior,

KC)

= K(dc/d

of the con&tron =

+ Zd(dc/d

y)$ = -#

- +t”drl

2b,(db/dy),

bp + 2d’*K-‘b;

(db/dy),

y>P

at y = ~1, lends to.

-‘1L

(Al)

and (db/dy),=

-+L/‘l

+(4d,‘fc)b,]

Smce under these co.xlitions, to the wave, eqn. (Al) yields

642) b, + 0 and I+ 1s finite 111 the region correspondmg

bp = ( K*1#/2)1’2 K* = Kd-“2, which in turn -E & in the knew -+d-‘/2(2tc’/14)1/2 be solved can be wntten finally as

With

y
d’b/dy2=2X[b2-(tc*/2d’fl)b-

together

with the boundary

y = 0:

(db/dy),

= -$

conditions.

gives

from

eqn.

zones concerned (K*&/2)]

(A2): (db/d y ), = here. Thus the system to

=2x16*-(K*1+/‘2)]

22

and _JJ= CL: (db/dy), Introducmg d2b*/,jy*’

=

= 0, b* =

b, = (~*1$/2)~~

b(K*)-lny*

=y(K*)-I”,

[b*‘-

[sii(K*)-3’2]

giWS

1+/2]

\lth: 1 ,* = 0. y*-,co.

(db*,‘dy*)o (db*/dy*)

Integratmn 9’ = [ (4h/3)( =

---, 0 and b* --, (II/J/~)-‘/”

of tlus system results K*)“‘]

Reorgamsatlon K*

= -r’l

{

then m.

bz3 - (3/2)b,*(

of Gus last equation

I+)

+ 2( 1#/2)3/2}

finally results in eqn. (12)

when d = 1 (i.e.

K).

(II) Nutnerrcal resoluiion of eqn (I5) Smce I) =

+/W/p

(77~)-‘/’ tar 0 -C T G 1 eqn. (ll),

= 1+ (r-‘r-

for T > 1 can be wntten

as:

)j;-+(a~)-“2](~--)-1/2d~

I e: -~4’3/p=(1/2)+~-‘arcsm[(2-~))/~~

+“-1/‘~rJ/(~)(7--)-1r-dq

(A3)

1

Standard procedures for solving integral equatrons are based on the fact that tne unknown function IS negligible in the first stages of the computation loops: thus such methods appear inappropriate for chronoamperometry smce # usually takes extreme values unmedrately after each potential step. For example eqn. (A3) shows that $I- -p314 when T--, l+ lks problem may be avoided by performmg the change of vanable: +(T)=~(7)/((11-~‘2)[7-1fl-(7-1)-1’2])

W)

which normahzes the current versus its lu~:t m a pure Mfusion the followmg mtegral equatron (from eqn. A3) (~{1-[T/(T-l)]1fl}(~T)-“2)4’3/p=

case. Thus leads to

-(1/2)-(7r-r)arcsin[(2-T)/T] - (n-‘)i&~(r

- v)] +dv

+(r-‘)J;T~!(~-1)(7-~)]-1/Zd~ (A5)

23

Such an equaticn, together wrth eqn. (15) shows that +(rj - 0 when T - 1 tid $X(T) -+ 1 when 7 + co, whatever the p value. The effect of p is that +(T) vanes from 0, for r = 1, to the order of unity in a tune mterval S,, of ca (P-~/‘/V). For longer Unes, the vanations of $J(T) are smoother and +(T) is of the order of umty. Thus although eqn. (A5) is m a form appropnate to Its resolution by standard procedures, it is necessary that the tune mterval ]l,l + a,] be drscretlzed *mth a sufficient accuracy. Owmg to tune consumptron and memory occupation thrs becomes a problem at htgh p values, especrally when it 1s considered that a coarser *tie grid 1s suffrcrent at times longer than 1 + 6, Thus a time dependent gnd was used. Thrs tune grid was generated m the following way. The length L = [l - exp( - l/6,)] was divided into N = 300 eqtiil intervals, A. The tune grrd was then defined by the senes 1 - SJn(1 - nA) = 1 - &Ln{l - (n/N)[l of 300 nodes so th,ltTV = exp( - l/6,)]}, P’ = 300. Equatron (A5) mvoIv~s two mtegrak, J1(~) = /$s[~(T ~)]-‘/‘d~ and J2(r) = j+[(n - l)(~ - v)]-1/2d q which are of the kind J(T) = _@&(T, q)dq. For T = T, these integrals can be wrltten m :he form (~c = 1, C#J,= +(T,))’ &(rm)

=

2 /% +g,(%, m=l %#-I

q)ds

=m$r(

““+2’m-1)Jr,

L-1

g,(%

q)dn

i.e., smce @c = 0

which yields-

+

c m=l

hn/2){=4(%t1+1

-

EhI’l[

+

c t&n/2){=4(2Tm ??I=1

(27,4

-‘k)/%]

-

7,)/T,]

}

and:

+I

-

TA)ibA)]

Introductron of these two approximate expressions 111 eqn aigebrarc equation -yrelding @” for each value of 7,. TINS equatron &’

(A5) leads to an is of the kmd:

+ A,+,, = B,,

where A,, and B, have known values dependent procedure startmg wrth cp,_l as the first entry.

on

T,,

and

was

solved by an iterative

24

(III)

Fhte

a>ffeerenceresolutron of eqn: . (7)-(11)

The system m eqns. (7) and (8) associated with boundary conditions m eqns. (9)-(11) was solved accordmg to an explicit finite difference procedure [28]. This was done up to X = i0’ smce above this value the asymptotic solution ik eqn. (13) (Fig 2b) proved to be sufficiently accurate [AR/R G 10-3) for tlus latter solution to be used instead of the general fmik difference procedure. Note that when it is considered that the Mfuslon coefficient of the tier C, may differ from the average tifuslon coefficients, D, of A aqd B, eqn. (8) is replaced by:

ab/aT=a2b/ay2-2A(b2-4 &z/a7

= f.Wc/ay2

+ X(b2 - KC)

w1t.b c = cc/co and d = D,/D. This new set of eqns. is solved together the boundary conltion m eqns. (9)-(11) and

\nth eqn (7),

ZJc/ay=Ofor’y=0,7),0 accordmg

to the same fmte

tiference

procedure

REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 3-S 26 27 28

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