Study of a CE Mechanism in Cyclic Chronopotentiometry with Spherical Electrodes

Transcription

1 Porugalae Elecrochmca Aca ( PTUGALIAE ELECTCHIMICA ACTA Sudy of a CE Mechansm n Cyclc Chronopoenomery wh Sphercal Elecrodes M. López-Tenés *, Á. Molna, J. M. Molna Deparameno de Químca Físca. Faculad de Químca, Unversdad de Murca. Espnardo, 3. Murca, Span eceved 5 January 3; acceped n revsed form February 3 Absrac The complee heory correspondng o a CE mechansm when applyng cyclc chronopoenomery o a sphercal elecrode of any sze s developed. The nfluence of several varables on he ranson me raos, such as he elecrode radus, rae consans of he homogeneous chemcal reacon and curren densy, s dscussed. A smple and praccal creron based on he varaon of curren densy appled o he elecrode s proposed for he deecon of a CE mechansm. Keywords: CE mechansm, cyclc chronopoenomery, sphercal elecrodes, ranson me raos. Inroducon The sudy of a CE mechansm, n whch he charge ransfer reacon follows a homogeneous chemcal reaco has been repored n he leraure usng several chronopoenomerc echnques such as consan curren chronopoenomery [], chronopoenomery wh programmed curren [-4], and alernang curren chronopoenomery [5], and wh planar and convenonal szed sphercal elecrodes. In hese echnques, only one curren sgnal s appled o he elecrode. However, elecrochemcal echnques n whch he elecrcal perurbaon (poenal or curren s appled more han once are of grea neres boh analycally and knecally [6,7]. Thus, a more exhausve sudy of hs ype of * Correspondng auhor. Fax: ; e-mal: manuela@um.es

2 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( processes wh knec complcaons can be made usng cyclc chronopoenomery, a classcal elecrochemcal echnque conssng of he applcaon of several successve curren seps of alernang sgns o a deermned elecrode whou he balance beng recovered n he elecrodesoluon nerphase. Ths echnque was nroduced by Herman and Bard n 963 [8], who appled o plane elecrodes, and s use has been shown for he qualave and quanave sudy of elecrode processes [9-3]. The echnque was exended o convenonal szed sphercal elecrodes n references [4-6]. In hs paper, we have developed he heory concernng a CE mechansm n cyclc chronopoenomery for sphercal elecrodes of any sze, ncludng planar elecrodes and sphercal ulramcroelecrodes as parcular cases. From hs general heory, we analyse he response obaned n cyclc chronopoenomery for a CE process and we dscuss he nfluence of several varables, such as he elecrode radus, on he ranson me raos ( τsep τ sep, he evoluon of whch wh he number of curren seps appled can be used as a creron o esablsh he presence of knec complcaons. We have proved ha he characersc behavour of he ranson me raos correspondng o a CE process wh he chemcal rae consans can be also aaned by changng he densy curren appled, somehng ha s very easy o do expermenally. We also ndcae how o oban knec nformaon of he process, and we conclude ha cyclc chronopoenomery has real advanages over oher echnques for he sudy of a CE mechansm. Theory The reacon scheme for he CE mechansm can be wren as (see noaon k A - f + n e k kb k (I We wll consder a sphercal sac elecrode of any sze, and we wll analyse he response of CE processes n cyclc chronopoenomery. Ths echnque consss of applyng successve and alernang sgn curren seps n he followng way 56

3 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( I, τ I, τ M M + ( I, τ M M k + ( I, k τ k (II where, for he sake of smplcy, has been supposed ha all he curren seps have he same absolue value, I. In hs scheme, s he me durng whch a curren sep ( = o k s appled, and τ s he me for whch he change n sgn s produced, beng he ranson me correspondng o any reducon of speces (forward ranson mes, τ, τ 3, τ 5, or o any oxdaon of speces (reverse ranson mes, τ, τ 4, τ 6,. Thus, durng he applcaon of he h curren sep, he oal me elapsed from he begnnng of he expermen s gven by = τ+ τ τ + ( Under hese condons, when he h curren sep s appled, he followng equaon sysem mus be solved n order o oban he expressons for he concenraon profles ( c (, r, = A,, of speces nvolved n a CE process ˆ δ c = kc + k c ˆ δ c= kc A kc ˆ δ c = A A A ( wh ˆ δ beng he operaor for he second Fck s law n sphercal dffuson ˆ δ = D + r r r (3 where D s he dffuson coeffcen of speces, and r s he dsance from he cenre of he elecrode o any pon n he soluon. 57

4 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( The boundary value problem s gven by =, r r ca(, r = ca (,, r c(, r = c (,, r c(, r = c (, r ( > >, r c (, r = c, c (, r = c, c (, r = c ( = * * * A A (4 c c ( I >, r = r : D = D = r r nfa r= r r= r + (5 c DA A r r= r = (6 where c (, r (= A, or are he soluons for he ( h curren sep appled and r s he elecrode radus. The soluon of equaon sysem ( wh he boundary value problem gven by eqs. (4-(6 has been carred ou n he Appendx of hs paper and he expressons for c (, r have been obaned. From hese, he equaons for he specal case r = r (elecrode surface can be deduced. Thus, we fnd ha he equaons for he surface concenraons of speces parcpang n he CE mechansm correspondng o he h ( = o k curren sep are c (, c r K = ( ( (, + A A { {,,, N CE, S ξ T ξ χ ca + c + K A ( r ( ( ( (, n+ S ξ T ξ χ n=, = ( ( + (, + { {,,, N CE, S ξ KAT ξ χ ca + c + K A (, * ( ( ( (, n+ S ξ + KAT ξ χ n= c r c, n+ = + γn ( CE, S( ξ + ( ( S( ξ ca + c ca + c n= (7 (8 (9 58

5 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( where * c (= A, or are he nal concenraons of speces, K A s he equlbrum consan of he chemcal reacon for he CE mechansm, whch s gven by K k A = = k c c * A * ( and N CE I = nfad ( c c A + ( γ D = D (, = τ + τ (3 n n n, = (4 The funcons expressons ( S ξ and (, T ξ χ (= or are gven by he followng T ( ( ( erfc S ( ξ = exp ξ ξ ξ ξ = ( ξ, χ exp( χ 4 ( χ ξ ( χ exp exp ( ξ ( ξ erfc( ξ ( χ erf ( χ (5 (6, wh varables ξ n and χ gven by equaons (A69 and (A7 n he Appendx by changng n hese equaons by (eq. (3. I s worhwhle hghlghng ha, due o he compac form of eqs. (5 and (6, he soluons presened n hs paper are vald for sphercal elecrodes of any sze, from sphercal ulramcroelecrodes o planar elecrodes. 59

6 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( For planar elecrodes ( r,.e. ξ, eqs. (5 and (6 become, S ξ = n ( π n (( χ (, T ξ, χ = erf ( χ (7 (8 and f χ >>, hen eq. (8 can be wren as T n n ( ξ, χ >> =,, ( k+ k For convenonal szed sphercal elecrodes, when oban χ (9 >>, from eq. (6 we ( ξ, χ r ( k + k D r T >> = r ( k+ k D ( For ulramcroelecrodes ( r and ξ >>, eqs. (5 and (6 are smplfed o n n n ( ξ ( ξ, χ S >> = T >> =,,, D r ( The poenal-me response for he h ( = o k curren sep can be deduced by subsung he expressons obaned for he surface concenraons of he oxdzed and he reduced speces, c ( r, and c ( r, (eqs. (8 and (9, n he Buler-Volmer equaon ( I nfa + = kc( r, kc( r, f b ( Thus, we oban he followng expresson 6

7 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( D αη( + NCE e = ' ( k,,, n+ n, n, n, NCE (, S( KAT(, ( ( S( KAT(, K ξ + ξ χ + ξ + ξ χ + A n= * η( c, n+ n, e + γn ( CE, S( ξ + ( ( S( ξ ca+ c n= (3 wh nf ( ( ( ' E E η = T (4 Eq. (3 can be smplfed n wo lm cases: When ' k (reversble charge ransfer reacon we oba for any curren sep appled, he followng expresson η (,,, n+ n, n, n, NCE (, S( ξ + KAT( ξ, χ + ( (, n S( ξ + KAT( ξ, χ n= = ln * c, n+ ( + KA + γn ( CE, S( ξ + ( ( S( ξ ca+ c n= (5 When k ' << (oally rreversble charge ransfer reacon, he response obaned depends on he sgn of he curren sep appled: a If s odd (he sgn of he curren s posve and a reducon process akes place η ( N ( S ( K T (, ( ( S ( K T (, = ln α,,, n+ n, n, n, CE, ξ A ξ χ ξ A ξ χ n= D ( + KA NCE ' k (6 6

8 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( b If s even (he sgn of he curren s negave and an oxdaon process akes place η ( c = ln α c + γn ( S ( ξ + ( ( S ( ξ + * A c, n+ CE, n= D NCE ' k (7 The expressons correspondng o he ranson me of he h curren sep ( > can be obaned by makng c ( r, = n eq. (8 f s odd and c ( r, = n eq. (9 f s even. Thus, we fnd τ odd =,,, n+ NCE (, S( ξ + KAT( ξ, χ + ( ( S( ξ + KAT( ξ, χ n= N CE S( ξ + KAT( ξ, χ (8 τ even = c ( ( ( ( ( c + *, n+ + γn CE, S ξ + S ξ A c n= γnce S( ξ (9 In he parcular case =, from eq. (8 we oban τ = N CE S ( ξ + KAT ( ξ, χ (3 The equaons deduced for he surface concenraons, poenal-me response and ranson me of a CE mechansm become hose correspondng o a smple charge ransfer reacon (E mechansm when K A =. Ths behavour also occurs, for K A, when χ = or χ (see esuls and dscusson. In hese cases, our equaons concde wh hose obaned n reference [4]. 6

9 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( Expermenal In cyclc chronopoenomery, successve curren seps are appled accordng o scheme (II. Ther sgn s alernaely changed a a me whch may be less han he ranson me correspondng o he h curren sep, τ, or equal o. We wll consder ha he curren s reversed when he ranson me τ s reached, whch s acually he mos common case n pracce. Fg..a shows he varaon of he surface concenraon of speces and, parcpang n he elecrochemcal reacon. In a CE mechansm, s he oxdzed speces and s concenraon becomes zero a he surface of he elecrode a odd ranson mes ( τ, τ 3, τ 5,, whle s he reduced speces and s concenraon becomes zero a he surface of he elecrode a even ranson mes ( τ, τ 4, τ 6,. Fgs..b and.c show he ypcal poenal-me response for a CE mechansm n a cyclc chronopoenomerc expermen n he case of a reversble charge ransfer reacon (Fg..b and n he conrary case of an rreversble elecrochemcal reacon (Fg..c. In hese fgures, he values of he ranson mes can be also observed. In cyclc chronopoenomery, s of grea neres o sudy he varaon of he ranson me raos, defned wh respec o he ranson me of he frs elecrochemcal reaco τ, as a τ = τ (3 wh he number of alernang curren seps appled,, snce allows us o characerze he elecrode process. I s useful o plo he ranson me raos vs. for he CE mechansm, as well as hose obaned for an E mechansm, whch s aken as a reference. 63

10 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( c (r,/(c A * +c * (=, (=,, 3, 4, 5,,8,6,4,, 5 a b 5 E/mV c 45 E/mV Fgure.a. Varaon of he surface concenraon of speces (sold lne and (dashed lne wh me n cyclc chronopoenomery ( =,, 3, 4, 5 for a CE mechansm (eqs. (8 and (9. K A =, k+ k = 5 s -, N CE =.5 s -/ 5, D = cm s - *, γ =, c =, r = cm. Fgure.b. Poenal-me curves correspondng o he applcaon of fve curren seps for a CE mechansm wh reversble charge ransfer reacon (Eq. (5. T = 98 K, n =. her condons as n Fg..a. Fgure.c. Poenal-me curves correspondng o he applcaon of fve curren seps ' 5 for a CE mechansm wh rreversble charge ransfer reacon (Eq. (6. k = cm s -, α =.5. her condons as n Fg..b. /s 64

11 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( As can be deduced from eq. (8, he ranson me obaned for a CE mechansm when a reducon process akes place ( odd s gven as a funcon of he equlbrum and rae consans of he chemcal reacon. However, when an even curren sep s appled (eq. (9, he ranson me does no depend explcly on hese parameers. Thus, odd ranson me raos show a behavour que dfferen o ha observed for even ones, and ha s why hey need o be suded separaely. Fg. shows he nfluence of k + k on he varaon of he ranson me raos wh he number of curren seps appled, consderng a planar elecrode and K =. A.. a τ /τ (=, 3, 5, k +k (s - = 5.5. E, -, k +k (s - = 5 b τ /τ (=, 4, 6, E, -, Fgure. Influence of he rae consans on he varaon of τ τ wh for a CE mechansm n a planar elecrode (Eqs. (8-(3 and (7, (8. K A =. The values of k+ k (n s - are shown on he curves. The curve wh label E corresponds o he behavour of a smple E mechansm. a odd; b even. her condons as n Fg..a. 65

12 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( Fg..a corresponds o odd values of and fgure.b o even. Boh fgures show ha relaons a are concden wh hose obaned for an E mechansm (curves labelled wh E n fgure n wo cases:. For k k + s - (mmoble chemcal equlbrum, speces A and do no nerconver chemcally. Consequenly, A merely acs as a chemcally ner componen of he sysem, and hence we observe a response whch corresponds o an E mechansm wh an nal concenraon of speces equal o c * (bulk concenraon of speces before he frs curren sep s appled.. For k+ k 5 s -, he chemcal equlbrum s oally moble, and he sysem behaves as an E mechansm wh an nal concenraon of elecroacve speces equal o c * + c *. Thus, even hough absolue ranson mes are A greaer han hose n he suaon above, he ranson me raos are dencal n boh cases. For nermedae values of k + k, can be observed n fgures.a and.b ha a are greaer han hose correspondng o an E mechansm for any value of. However, boh fgures show a dfferen behavour. Thus, for even values of, he ranson me raos always ncrease wh he number of curren seps appled (Fg.b, whle, f s odd (Fg..a, he raos a exhb he mos characersc behavour of he CE mechansm: here s a range of values of k + k ( k+ k 5 n Fg..a for whch he ranson me raos frs decrease bu hen ncrease wh he growng number of curren seps appled. As can be observed, hs does no occur wh an E mechansm ( a always decrease wh. Therefore, s possble o characerze a CE mechansm by changng he rae consans, whch can be done by modfyng he expermenal condons (such as ph n he case of he reducon of an acd n a buffered soluon or he concenraon of lgand n he case of he reducon of a meal complex f he chemcal reacon s of pseudo-frs order. 66

13 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( The exsence of a CE mechansm can be shown as well f we change he curren densy appled o he elecrode, ha s, f we change he varable N CE (eq. (. In Fg. 3 we have ploed he ranson me raos correspondng o a CE mechansm (planar elecrode vs. ( odd n Fg. 3.a and even n Fg. 3.b for several values of E mechansm. N CE. As n Fg., he curve labelled wh E corresponds o an,, a τ/τ (=, 3, 5,...,9,8,7,6,5 N CE (s -/ = E,,4,6 N CE (s -/ = 3 b τ/τ (=, 4, 6,...,5, E,, Fgure 3. Influence of he curren densy on he varaon of τ τ wh for a CE mechansm n a planar elecrode (eqs. (8-(3 and (7, (8. k + k = s -. The values of N CE (n s -/ are shown on he curves. a odd; b even. The curve wh label E corresponds o he behavour of a smple E mechansm. her condons as n Fg.. 67

14 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( Ths fgure has been prepared wh a value of he rae consans ( k + k = s - for whch he odd ranson me raos do no show he ypcal behavour of a CE mechansm n he condons of Fg..a. From he analyss of Fg. 3.a we can conclude ha: a The varaon of N CE has an nfluence on he raos a. Ths does no occur n an E mechansm, where he ranson me raos are ndependen on he curren densy appled for planar elecrodes (and a smlar behavour s observed for sphercal elecrodes. b The ranson me raos show he pecular behavour descrbed above for hgh values of N CE, and so we can characerze a CE mechansm by changng he curren densy appled, wha s very easy o do expermenally. Thus, an ncrease of N CE acheves he same effec as a possble or hypohec dmnuon of k + k, and herefore, s a way of exernally modfyng, hrough an expermenal varable, he mobly of he chemcal equlbrum coupled o he charge ransfer reacon. In order o analyse he nfluence of he sphercy, n Fg. 4 we have ploed he varaon of he ranson me raos correspondng o a CE mechansm wh, for several values of r. As can be observed, he raos a decrease always when r dmnshes. The behavour s qualavely he same as ha shown for a planar elecrode, bu, from a quanave pon of vew, he effec exered by he elecrode radus on he ranson me raos s very mporan, and becomes greaer wh he growng number of curren seps appled. Therefore, s specally mporan n cyclc chronopoenomery, and s consequenly of grea neres o have deduced equaons avalable ha ake no accoun he sphercy of he elecrode. 68

15 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( a.9 τ/τ (=, 3, 5, r (cm = r (cm = b τ /τ (=, 4, 6, Fgure 4. Influence of he sphercy on he varaon of τ τ wh for a CE mechansm (Eqs. (8-(3. k + k = s -. The values of r (n cm are shown on he curves. The curve wh r corresponds o a planar elecrode. a odd; b even. her condons as n Fg.. The sudy of he ranson me raos can also be used o calculae he rae consans of he chemcal reacon coupled o he charge ransfer reaco as has been poned ou n reference [9]. The rae consans can be deermned by comparng heorecal and expermenal τ τ vs. curves. Ths mehod s more advanageous han ha descrbed n references [3, 4, 6], where consan curren and curren reversal chronopoenomery are used, because n he frs case we oban all he knec nformaon from only one ranson me measuremen ( τ, and n he case of curren reversal, he rao τ τ s ndependen on knec 69

16 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( parameers. However, n cyclc chronopoenomery we can apply as many curren seps as we wan o. Acknowledgemens The auhors grealy apprecae he fnancal suppor provded by he Dreccón General Cenífca y Técnca (Proec No. BQU-3, and by he Fundacón Séneca (Proecs 696/CV/99 and A 8-698/FS/. Also J. M. Molna hanks Fundacón Séneca for he gran receved. Noaon and defnons k f, k b heerogeneous rae consans of forward (reducon and backward (oxdaon charge ransfer processes ' ' k apparen heerogeneous rae consan of charge ransfer a E α charge ransfer coeffcen k, k rae consans of he homogeneous chemcal reacon K A equlbrum consan of he chemcal reacon ( = k k r dsance from he cenre of he sphercal elecrode o any pon n he soluon r elecrode radus of he sphercal elecrode c (, r concenraon profle of speces (= A, or when a curren sep s appled c ( r, surface concenraon of speces (= A, or when a curren sep s appled * c bulk concenraon of speces (= A,, or me elapsed beween applcaon of he frs and he h curren sep ( = τ+ τ me elapsed beween applcaon of he nh and he h curren sep ( = τ n + τ n me durng whch a curren sep s appled ( τ ranson me raos a τ ranson me of he h curren sep n number of he elecrons ransferred n he elecrochemcal reacon F Faraday consan A area of he elecrode I absolue value of he curren sep appled D dffuson coeffcen of speces ( = A, or and DA = D = I nfad ( c + c N CE A 7

17 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( γ = ( D D ξ dmensonless parameer of sphercal dffuson ( = D r n, χ dmensonless parameer referrng o he chemcal reacon ( = ( k+ k n, E ( me-dependen poenal ' E formal poenal of he elecrode reacon E ' = E ( E eferences. P. Delahay, T. Berzns, J. Am. Chem. Soc. 75 ( Kan, S.K. angaraa J. Elecroanal. Chem. 65 ( A. Molna, M. López-Tenés, Collec. Czech. Chem. Comm. 56 ( A. Molna, M. López-Tenés, C. Serna, J. Elecroanal. Chem. 346 ( M.L. Alcaraz, A. Molna, M. López-Tenés, Elecrochm. Aca 4 ( D.D. McDonald, Transen Technques n Elecrochemsry, Plenum Press, New York, 977, chaper A.J. Bard, L.. Faulkner, Elecrochemcal Mehods, Wley, New York,, chaper H.B. Herma A.J. Bard, Anal. Chem. 35 ( H.B. Herma A.J. Bard, Anal. Chem. 36 ( ; 36 ( H.B. Herma A.J. Bard, J. Elecrochem. Soc. 5 ( A. Molna, M.L. Alcaraz, F. Saavedra, J. González, Elecrochm. Aca 44 ( L.M. Abranes, J. González, A. Molna, F. Saavedra, Elecrochm. Aca 45 ( A. Molna, J. González, F. Saavedra, L.M. Abranes, Elecrochm. Aca 45 ( A. Molna, J. González, C. Serna, L. Camacho, J. Mah. Chem. (

18 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( J. González, A. Molna, F. Marínez-rz, C. Serna, J. Elecroanal. Chem. 44 ( A. Molna, J. González, M. López-Tenés, J. Mah. Chem. 3 ( J. Kouecký, Czech. J. Phys. ( A. Molna, C. Serna, L. Camacho, J. Elecroanal. Chem. 394 ( Appendx. Applcaon of he frs curren sep When a sphercal elecrode of any sze s consdered, he mass ranspor o he elecrode surface when he frs curren sep s appled s descrbed by he dfferenal equaon sysem (eq. ( wh = ˆ δ c = kc + k c ˆ δ c = kc k c δ A A A A ˆ c = (A where ˆ δ (= A, or s he operaor for he second Fck s law gven by eq. (3. The boundary value problem (Eqs. (4-(6 s gven by: =, r r (A c = c, c = c, c = c > * * * A A, r c c I >, r = r : D = D = r r nfa r= r r= r (A3 c DA A r r= r = (A4 By nroducng he varables: and wh he assumpon ( r, c ( r, c ( r, ζ = + (A5 A ( k+ k ( A A (, (, (, φ r = c r K c r e (A6 DA = D D (A7 7

19 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( he dfferenal equaon sysem (A and he boundary value problem (eqs. (A- (A4 are ransformed no: ˆ ˆ ˆ c δζ = δφ = δ = (A8 =, r r (A9 ζ = c + c, φ =, c = c > * A, r ζ c I >, r = r : D = D = r r nfa r= r r= r (A r φ ( k+ k = KAe r= r I nfad (A Ths problem can also be solved by nroducng he varables u ζ r = c + c r A (A φ r v = c + c r A (A3 In such a way, he dfferenal equaon sysem (A8 becomes ˆ ˆ ˆ u v c δ = δ = δ = (A4 where ˆ δ s now ˆ δ = D r (A5 By supposng ha ζ, φ, and c have he form: ζ (, r = ( c + c + ρ ( s ( ξ ( χ (A6 p q A p, q pq, = φ (, r = δ ( s ( ξ ( χ (A7 p q+ pq, pq, = c (, r = c + σ ( s ( ξ ( χ (A8 * p q p, q pq, = 73

20 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( where r r s = D (A9 ξ = D r (A χ = ( k+ k (A and usng he dmensonless parameers mehod [7] o solve he dfferenal equaon sysem (A8, we oban he followng soluons: ( s ρ = pq, unless q = (A ρ ρ ( s, ρ ρ ( s, ( s ( s NCE ( ca + c Ψ = ( k+ k p ( s N ( c + c Ψ ( s + CE A, = Ψ ( k k ( s p ( s p ( s NCE ( ca + c Ψ3 Ψ Ψ = + ( k+ k p3 4 CE A ( s ( s ( s ( s N ( c c 5 Ψ ( k+ k 3 + 3, = Ψ4 Ψ + (A3 (A4 (A5 (A6 M δ Ψ KANCE ( ca c q+, q( s = q!( k+ k pq+ + K N ( c + c q + ( s A CE A ( s = Ψ ( s Ψ ( s δ, q q+ q q!( k+ k q+ (A7 (A8 ( ( q + K N ( c + c 4 ( ( ( + A CE A δ, q s = Ψq+ 3 s pqψ q+ s + pqψq s 4 q!( k k pq+ 3 (A9 74

21 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( δ KANCE ( ca + c 3, q( s 4 q!( k+ k 3 = 8q 3q + 4q+ 5 Ψ ( q+ 4( q+ + Ψ + + Ψ Ψ (A3 M ( s ( 6q 4 ( s ( 6q ( s q ( s q+ 4 q+ q q σ ( s γ ρ ( s = (A3 p, p, where γ s gven by eq. ( and funcons ρ,( p s are gven by eqs. (A- (A6 by changng s by s (eq. (A9. Ψ ( s (=, are he Kouecký funcons and p =Γ (+ m Γ (( + m. m. Applcaon of he second curren sep When he second curren sep I s appled (eqs. (-(6 wh =, as hs problem s lnear, we assume ha he expressons of he concenraon profles can be wren: where (see eq. ( c (, curren sep, and c ( r, (, (, (, c r = c r + c% r (A3 A A A (, (, (, c r = c r + c% r (A33 (, (, (, c r = c r + c% r (A34 = τ + (A35 r (= A, or are he soluons obaned for he applcaon of he frs % are he new unknown funcons. Therefore, f we defne he new varables ζ = c + c = ζ + % ζ (A36 A k+ k τ+ ( ca KAc e ( ( φ = = φ + % φ (A37 75

22 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( where ζ and φ have already been obaned for he prevous sep, he boundary value problem has now he followng, smplfed form n erms only of he new unknown funcons % ζ and % φ : =, r r % = % = % = > r = r ζ φ c, (A38 % ζ c% I (A39 >, r = r : D = D = r r nfa r= r r= r % (A4 φ ( k+ k( τ+ I = KAe r nfad r= r As can be observed, he boundary value problem whch ζ mus fulfl n he frs curren sep (eq. (A s smlar o ha correspondng o % ζ n hs second curren sep (eq. (A39, changng I by I, whle he boundary value problem ha φ mus fulfl n he frs curren sep (eq. (A s analogous o ha fulflled by % φ n he second curren sep (eq. (A4, excep for he value of he consan ( k + e k τ. By supposng ha % ζ ( r, = ρ ( s ( ξ ( χ (A4 p q pq, pq, = % φ (, r = δ ( s ( ξ ( χ (A4 p q+ pq, pq, = c% ( r, = σ ( s ( ξ ( χ (A43 p q p, q pq, = and proceedng as n he prevous curren sep, we fnd ha ρ ( s = ρ ( s (A44 pq, pq, δ ( s = e δ ( s (A45 ( k+ k τ pq, pq, σ ( s = γ ρ ( s (A46 pq, pq, where 76

23 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( s r r = (=, D (A47 D ξ = (=, r (A48 χ = ( k+ k (A49 and funcons ρ pq, ( s and δ pq, ( s have he form gven by equaons (A- (A3, by subsung s by s. The mahemacal reamen used for he frs and he second curren seps can be easly generalsed by nducon for any number of curren seps. Thus, for he h curren sep ( >, by applyng he superposon prncple [4, 6, 8] we can express he soluon for he dfferenal equaon sysem ( n he form where (, (, (, c r = c r + c% r (A5 A A A (, (, (, c r = c r + c% r (A5 (, (, (, c r = c r + c% r (A5 m (, = (, + % (, m (= A,, (A53 m= c r c r c r Thus, f we defne he funcons ζ = c + c = ζ + % ζ (A54 A wh ( ca KAc e ( k + k φ = = φ + % φ (A55 ζ ζ ζ m ( r, = ( r, + % ( r, m= φ φ φ m ( r, = ( r, + % ( r, m= m m (A56 (A57 where s gven by (eq ( 77

24 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( = τ + τ + + (A58... from eqs. (A36-(A4 can be demonsraed ha he boundary value problem has he generalsed form: =, r r > r = r % = % = % = ζ φ c, (A59 % ζ c% + I (A6 >, r = r : D = D = ( r r nfa r= r r= r % I (A6 φ ( k+ k = ( KAe r nfad r= r As can be deduced from eqs. (A38-(A4 and (A59-(A6, he paral soluons ~ ~ r, r, c r, ~ r, ~ φ r, and ζ (, φ ( and ( ( r, % are formally dencal o ζ (, ( c%,.e. he superposon prncple s fulflled. Therefore, f we suppose ha % ζ (, r = ρ ( s ( ξ ( χ (A6 p q p, q pq, = % φ (, r = δ ( s ( ξ ( χ (A63 p q + p, q pq, = c% (, r = σ ( s ( ξ ( χ (A64 p q p, q pq, = s clear ha we can express he soluons n he general form ρ ( s = ( ρ ( s (A65 + pq, pq, δ ( k+ k τ + l= pq, s = e δ pq, s ( ( ( (A66 σ ( s = γ ρ ( s (A67 pq, pq, where s = (=, r r D (A68 78

25 M. López-Tenés e al. / Porugalae Elecrochmca Aca ( D ξ = (=, r (A69 χ = ( + (A7 k k Thus, he concenraon profles (eqs. (A5-(A5 are oally deermned. From he expressons for he concenraon profles, we can deduce hose correspondng o he surface concenraons of speces nvolved n a CE mechansm, whch are gven by eqs. (7-(9 n heory. 79