sensors Sensors 2002, 2, ISSN by MDPI Michael E.G. Lyons

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1 enor,, enor IN 44-8 by MPI Mediated lectron Traner at Redox Active Monolayer. Part 3: Bimolecular Outer-phere, Firt Order outecy- Levich and Adduct Formation Mechanim Michael.G. Lyon Phyical lectrochemitry Laboratory, epartment o hemitry, Nar Intitute o Advanced Material cience, Univerity o ublin, Trinity ollege, ublin, Ireland. Tel: Fax: mail : melyon@tcd.ie Received: 7 November / Accepted: 3 ecember / Publihed: 4 ecember Abtract: A number o theoretical model decribing the tranport and inetic procee involved in heterogeneou redox catalyi o olution phae reactant at electrode urace coated with redox active monolayer (uch a el aembled alane thiol containing a errocene group) i preented. Thee model are : imple bimolecular outer-phere, generalized outecy-levich, and ubtrate binding/ adduct ormation. For each model a general expreion or the teady tate reaction lux i derived. impliied analytical expreion or the net reaction lux under teady tate condition, are derived or experimentally reaonable ituation, and inetic cae diagram are contructed outlining the relationhip between the variou approximate olution. The theory developed enable imple diagnotic plot to be contructed which can be ued to analye experimental data. eyword: eterogeneou redox catalyi; Redox active monolayer; Monolayer modiied electrode; inetic modeling o el aembled monolayer; Mediated electron traner at monolayer Introduction The chemitry o molecularly deigned electrode urace ha been a major apect o interacial electrochemitry or over twenty year. Tailormade electrode urace at the molecular level provide much added value to the unique eature inherent in electrochemitry where, a recently tated by Murray [], one can employ the electrode potential a a ource or in o pure, uncomplicated electron o lexibly choen ree energy. Thi eature o electrochemitry i readily apparent when

2 enor, 474 the reaction i Nerntian or reverible. owever many redox reaction o practical importance are irreverible. They exhibit inite inetic at the electrode urace, an activation energy barrier at the interace mut be overcome, igniicant overpotential mut be applied to mae the reaction proceed at a inite rate, and the rate o product ormation, and indeed it identity, will depend in a mared manner on the nature o the electrode urace. ence the quet to undertand and control electrochemical reaction inetic and catalyi ha been the motivation o reearch in phyical electrochemitry or decade. The cience o electrocatalyi wa enhanced in the 97 when electrochemically reactive molecular material were attached to electrode urace uing either a ytematic ynthetic trategy or via controlled adorption. The eminal paper o Lane and ubbard [], a well a the early review article authored by Murray [3] and by Albery and illman [4] well illutrate the latter approach. ovalent attachment o redox active group wa accomplihed via peciic chemical coupling trategie uch a urace ilanization, the ue o cyanuric chloride, or in the cae o carbon electrode, direct reaction o the redox moiety with acidic and carbonyl unctionalitie on the electrode urace. urace modiication via adorption reulting in table modiied urace, i well illutrated in the wor o Gorton and co-worer [5] who developed viable amperometric bioenor or NA uing redox dye monolayer. Redox mediation i imple in concept. In thi proce urace immobilized ite may be activated electrochemically via application o a voltage to the upport electrode. The latter ite may then oxidie or reduce other redox agent located in the olution phae adjacent to the immobilized layer, or which the direct oxidation or reduction at the electrode urace i inhibited, either becaue o intrinically low heterogeneou electron traner inetic or becaue cloe approach o the oluble redox pecie to the electrode i prevented. Although the initial ocu o tudie in the ield o chemically modiied electrode wa in the area o molecularly modiied urace o monolayer thicne, interet in the latter coniguration or electrocatalytic purpoe decreaed when it wa realized that igniicant catalatytic advantage could only be obtained i electrode were modiied with many monolayer o redox active ite. The attraction o a three dimenional networ o catalytically active ite localized next to a upport electrode urace, compared with the correponding two dimenional ituation characteritic o a monolayer were clear, and much invetigation wa made on the undamental propertie o electrode urace modiied with electroactive polymer thin ilm [6]. Multilayer polymer ilm were not without their problematic apect with repect to mediated electrocatalyi however. The current repone relecting the net reaction rate wa ound to depend not only on the heterogeneou cro exchange inetic involving the dipered mediator pecie and the ubtrate, and the diuion o ubtrate in the adjacent olution to the redox ite (the latter being pertinent alo to monolayer ytem), but alo rate limitation could involve electron percolation or hopping between adjacent redox ite in the polymer matrix (decribed via a diuion/migration mechanim), and partition and diuion o reactant pecie through olvent containing pore in the polymer matrix. ence, although catalytic eiciency could be enhanced via the adoption o a multilayer trategy, it could be reduced i charge percolation and/or ubtrate tranport procee proved to be inetically igniicant. Furthermore, a ull phyicochemical characterization o immobilized polymer ilm uing the tandard pectrocopic tool o modern chemitry proved eluive. The complex phyical chemitry underlying mediated electrocatalyi in

3 enor, 475 polymer modiied electrode ytem ha been elucidated by Andrieux and aveant [7] and by Albery and illman [8]. owever the concept that a molecular electrode urace could be developed, whoe reactivity with repect to a target reactant could be reaonably predicted, wa never ully diregarded, and, more recently, metallic electrode coated with adorbed redox active monolayer which are generated via el aembly mechanim [9] have been the ubject o coniderable attention. An organized el aembled monolayer (AM) i a ingle layer o molecule on a ubtrate urace in which the molecule exhibit a high degree o orientation, molecular order and pacing. In the el aembly method, the monolayer pontaneouly orm upon expoure o the ubtrate to a olution containing the molecule. It i clear that ucceul el aembly require a relatively trong bond between the ubtrate and a moiety or atom in the molecule, and an additional lateral interaction between molecule in the monolayer. The trength o the head group- ubtrate bond, the lateral interaction and the denity o pacing reult in good monolayer tability. A peciic ocu ha been on ordered monolayer abricated rom alane- thiol with pendant redox active center uch a Ru(N 3 ) 5 /3 or errocene/erricinium. uch well deined ytem have been ued to examine current theorie o interacial electron traner [], but have alo been ued a model well deined ytem, to increae our undertanding o the tructure o electriied interace [], and to develop novel enzyme baed bioenor []. Indeed the contruction o modiied enzyme ytem covalently attached to electrode urace ha led to a new generation o ophiticated molecular biorecognition ytem [3]. Mediated electron traner o olution phae pecie at electrode urace containing immobilized redox pecie can be examined experimentally uing a number o electrochemical technique. The technique o rotating dic voltammetry i mot oten applied, ince in principle, the procee o olute tranport in olution and olute reaction inetic may be cleanly eparated, by conducting voltammetric experiment over a range o electrode rotation peed. In the preent paper, we dicu the theoretical analyi o mediated electrocatalyi o a olution oluble ubtrate at an electrode urace containing an immobilized monolayer o redox active molecule. The analyi will be cat in the context o rotating dic voltammetry and will expand on previou model developed by Andrieux and aveant [4], Laviron [5], Anon [6] and Murray and co-worer [7]. The paper i tructured a ollow. We irtly briely dicu the thermodynamic apect o mediated electron traner. We then review a imple analyi o mediated electron traner in the context o teady tate rotating dic voltammetry. More detailed apect o thi analyi have been preented in the irt paper o thi erie [8]. Our attention will in the main be ocued on an analyi o the catalytic eiciency expected or redox active monolayer auming that the heterogeneou cro exchange proce can be decribed in term o imple bimolecular inetic, o that the mediator/reactant interaction i o the imple outer phere type. We then proceed to decribe mediated electron traner at immobilized monolayer in the context o a generalied outecy-levich model, and point out the pitall o uch an approach in the analyi o experimental data. Finally, the outecy-levich approach i then compared with a more ophiticated analyi, more in eeping with reported experimental obervation, in which the olution phae ubtrate orm a complex or adduct with the urace immobilized redox pecie. In thi circumtance the ubtrate/redox ite reaction can be decribed in term o Michaeli-Menten inetic. We hall derive theoretical expreion or the

4 enor, 476 normalied teady tate current repone or both the generalied outecy-levich and Michaeli- Menten model and preent inetic cae diagram or each model. We alo decribe a imple method o experimental data manipulation which will erve to ditinguih between the two approache. In the preent paper we ocu on an analyi o the teady tate current repone. We do not conider the tranient repone uch a that obtained uing potential tep chronoamperometry, which i more complex, and wa the ubject o the proceeding paper in thi erie [9]. Thermodynamic o Mediated lectron Traner at Redox Active Monolayer We irt review the pertinent thermodynamic apect o mediated electron traner. The proce o mediated electron traner i decribed via the ollowing reaction equence: A ne B B P A () In the latter reaction equence A i termed the precatalyt, and B denote the catalytically active orm o the urace immobilized redox couple. The catalytic proce involve the heterogeneou reaction between the catalytically active orm B and the ubtrate pecie to orm the product P. The latter heterogeneou reaction regenerate the precatalyt A. The proce i thereore cyclical. We aume that the tandard redox potential or the mediator generation proce i given by A/B, and that or the catalytic reaction i /P. Thee are related to the correponding Gibb ree energie via: G G A / B / P nf nf A / B / P where n denote the number o electron tranerred and F repreent the Faraday contant. The net change in Gibb energy or the catalytic cro exchange reaction i thereore given by G G A / B ΛG / P nf (3) where / P A / B (4) We can evaluate the equilibrium contant or the B/ cro exchange reaction via G nf exp exp (5) RT RT I we conider an net oxidative proce then in order that the equilibrium contant be large then mut be poitive and o /P mut be more poitive than A/B. () Mediated lectrocatalyi Via the Outer-phere Bimolecular Mechanim In thi ection we dicu the mot imple decription o mediated electron traner via a urace immobilized redox couple. A more complete analyi ha been preented in the irt paper o the

5 enor, 477 preent erie [8]. Thi analyi will prove to be ueul when we develop more ophiticated approache in later ection. Three poible rate determining tep are poible. The irt i the heterogeneou electron traner at the electrode urace generating the mediator pecie B. The generation o B can be decribed either by the Nernt equation, or by the Butler-Volmer equation. In imple model propoed previouly the ormer wa aumed. Thi may well be a good approximation. The econd involve the bimolecular reaction between the ubtrate pecie and the mediator pecie B. The cro exchange reaction involving B and i quantiied via a econd order rate contant (unit: cm 3 mol - - ). The third involve the logitic o ubtrate upply to the electrode urace, which i decribed by diuion. The rate o ubtrate diuive tranport to the active ite i decribed via a, where repreent the δ diuion coeicient o the ubtrate pecie in olution and δ i the diuion layer thicne. Typically δ cm and cm, hence cm. The net lux (unit: mol cm - - ) i given by: diuion rate contant (unit: cm - ). The latter i given by i nfa Γ B (6) where Γ B denote the urace coverage o mediator pecie B (unit: mol cm - ) and i the urace concentration o ubtrate. We can ue the Nernt equation to derive a relation between the mediator coverage and the total urace coverage G a ollow: Γ Γ B exp[ ξ ] (7) where Γ Γ Γ and ξ i a normalied potential given by: A B [ A / B ] nf ξ (8) RT where A/B repreent the tandard redox potential or the immobilized redox couple. The quai teady tate approximation may be ued to determine to obtain: (9) ΓB where denote the bul ubtrate concentration. The net teady tate lux i thereore given by: Γ B () ΓB We can introduce a non dimenional lux (or current) given a the ratio o the net lux to the diuive lux, the latter correponding to the diuion controlled reaction o the ubtrate pecie at an unmodiied electrode urace: Ψ ()

6 enor, 478 From eqn.7, eqn. and eqn. we can readily obtain an expreion or the current/potential curve expreed in non dimenional ormat: ηf Ψ η ( ξ ) F( ξ ) Γ where we introduce a parameter η which relate the rate o the urace cro exchange reaction to that o ubtrate diuion in olution and the potential dependence i governed by the unction F ( ξ ). It i readily hown that eqn. may be written in the ollowing orm: exp ξ η Ψ η exp [ ] [ ξ ] η exp[ ξ ] () (3) he normalied voltammogram correponding to variou value o the competition parameter η are illutrated in igure. We note that when η i large then Γ >> and the rate o the cro exchange reaction between B and i not rate determining. The hape o the normalied voltammogram tend to that characteritic o a imple diuion controlled proce and eqn.3 reduce to Ψ. A limiting current plateau given by Ψ exp[ ξ ] L or, L i obtained when the normalied potential ξ i large. In contrat when η i mall Γ <<, ubtrate diuion to the active ite i at and the net lux i controlled by the bimolecular inetic o the cro exchange reaction. Under uch circumtance auming that η >> we obtain Ψ and or large η exp[ ξ ] value o the normalied potential a limiting current repone i given by Ψ L η or, L Γ relecting pure inetic control. We alo note rom igure that the hal wave potential i a trong unction o the competition parameter η, and become le poitive a the latter parameter increae.. Ψ / η. η. η η η 5 η ξ F(- )/RT Figure. Normalized voltammogram or heterogeneou redox catalyi at an immobilized monolayer.

7 enor, 479 We note that the ratio o the ubtrate reaction lux at the monolayer modiied electrode to the ubtrate diuion lux given by eqn.3 i alo a direct meaure o the catalytic eiciency o the monolayer modiied electrode under condition where the heterogeneou cro exchange reaction i o an outer-phere bimolecular type. I we aume that the normalized potential ξ i large then the urace coverage o active mediator pecie Γ B Γ and eqn.3 tae the orm: Ψ L, L Γ Γ η η We will now how that the maximum catalytic eiciency cannot be greater than unity or a monolayer modiied electrode. The bimolecular rate contant or the mediator/ubtrate reaction at the monolayer can be etimated rom the Marcu relation []: AB P G (5) where AB, P repreent the correponding homogeneou olution phae bimolecular el exchange rate contant or the A/B and /P redox couple, i the equilibrium contant deined in eqn.5 and G i a contant, uually cloe to unity and i given by [] : (4) ( ln ) G exp AB 4ln Z P (6) where Z denote a characteritic colliion requency between reactant. Typically, Z, AB P M cm mol (the maximum value o the latter rate contant will be the diuion controlled value which typically ha the value M - - ). Auming that n and T 98, then i we et. V we can compute that 4, G. 58 and hence the heterogeneou cro exchange rate contant i etimated rom eqn.5 and eqn.6 to be 8 4 M. ince the urace coverage o active ite correponding to a monolayer i Γ mol cm, then, or a bul ubtrate concentration o ca. mm -6 mol cm -3, the 5 heterogeneou cro exchange lux i given by Γ 4 mol cm. Thi value hould be compared with the ubtrate diuion lux to the unmodiied electrode 9 mol cm. ence under the circumtance conidered we ee that <<, and rom eqn.4 we ee that the catalytic eiciency will approach unity. A the ratio η get larger, L η we note rom eqn.4 that the catalytic eiciency ΨL. ence or η the imple cenario outlined here (Nerntian mediator generation, irreverible cro exchange reaction), it i clear that under all condition the catalytic eiciency o a monolayer modiied electrode will not exceed that obtained or direct ma tranport controlled ubtrate reaction at the unmodiied electrode urace. Thi act wa pointed out by Andrieux and aveant [4] ome time ago. A imilar concluion pertain i a more complete analyi i perormed. I we aume that mediator generation i inetically irreverible and obey the Butler-Volmer equation, and that the cro

8 enor, 48 exchange reaction ha a large equilibrium contant, then impliication o the general teady tate expreion obtained in eqn.9 o the irt paper o the preent erie [8] reult in the aignment that the net reaction lux at the monolayer coated electrode i: Γ Γ To compute the catalytic eiciency we compare the latter reaction lux with that obtained or the ubtrate at an unmodiied electrode urace: (7) (8) The latter expreion wa obtained uing a teady tate analyi and conidering the rate o reactant diuion and reaction at a bare electrode urace []. We have introduced potential dependent heterogeneou electrochemical rate contant or both mediator generation ( ) and direct ubtrate reaction ( ). Both quantitie are given by Butler-Volmer expreion: β F exp RT β F exp RT ( ) A / B ( ) / P In the latter expreion β, β denote the ymmetry actor or the mediator generation proce and the direct ubtrate reaction repectively, and are the tandard rate contant or mediator, (9) generation and direct ubtrate reaction []. The catalytic eiciency i given by the ratio Ψ cat. Now, auming a ingle tep mechanim or mediator generation on the urace and direct ubtrate reaction, the ymmetry actor will both be in the range, and i both o the activation energy barrier are ymmetric then to a good approximation we can et β β β. 5. From eqn.7- eqn.9 we obtain ater ome manipulation: F Ψ exp[ - ] Ф cat / P A/ B RT () where we note that Φ exp Γ and exp [ β ( θ θ )] [ β ( θ θ )] A / B F where we deine θ, θ A RT / P F RT / B A / B, and β β [ β ( θ θ )] [ θ θ ] / P A / B exp / P A / B. Alo the parameter Φ i given by θ F RT / P / P. We now ue the relationhip: exp () ()

9 enor, 48 β to note that exp[ β θ ] [ β ] Φ β / P exp θ A / B to obtain: β exp [ β ( θ θ )] exp A / B [ β ( θ θ )] A / B An expreion or the catalytic eiciency o the monolayer modiied electrode may now be obtained uing eqn. and eqn.3: Ψ cat β exp exp [ β ξ ] [ β ξ ] where we have introduced the normalized potential ξ a ξ θ θ A / B. ence we note that the catalytic eiciency in general depend on the applied potential (via the ξ term), the equilibrium contant or the heterogeneou cro exchange reaction between urace immobilized mediator and olution phae ubtrate, and on the inetic lux term decribing mediator generation, heterogeneou cro exchange, direct ubtrate reaction and inally, on the lux or ubtrate diuion to the monolayer. The reult contained within eqn.4 may be better appreciated by reorting to a graphical analyi. In igure 4 we examine the manner in which the catalytic eiciency varie with normalized potential ξ or variou value o the equilibrium contant and or peciic limiting inetic cae determined by the value o the characteritic lux term. In all computation we aume that the ymmetry actor β.5, which i a reaonable aumption. In all cae we have et. and 5 ξ which are realitic range. In igure the catalytic eiciency i plotted a a unction o normalized potential or the ituation where.,. and. ence we note that >>, and o we conider the ituation where mediator generation and heterogeneou inetic are low, the direct reaction o ubtrate at the unmodiied electrode urace i acile and ubtrate diuion to the redox monolayer i acile. When the equilibrium contant or the cro exchange reaction i mall (.) the catalytic activity rie rapidly with increaing electrode potential which relect the increaingly acile generation o bound catalytically active ite on the electrode urace. Thi behaviour i alo een in the curve correponding to and. In contrat, when the equilibrium contant i large ( ), the catalytic activity i een to decreae lightly with increaing electrode potential rom an initial value o unity to a limiting value cloe to.8, at large ξ, which i alo attained by the other curve correponding to lower value. In igure 3 we illutrate the ituation where mediator generation i low and rate determining. ere 4 we have et,, and. ence we examine the ituation where < <. For all aumed value o the equilibrium contant the catalytic eiciency i low and le than., When i mall (.) the catalytic eiciency rie lightly with applied electrode (3) (4)

10 enor, 48 potential. In contrat, or larger value o the catalytic eiciency remain contant (at a low value) over much o the potential range and decreae lightly at elevated value o normalized potential.. Ψ cat Figure. Variation o catalytic eiciency with normalized potential or heterogeneou redox catalyi via the bimolecular mechanim or the cae o low generation o mediator pecie and low heterogeneou inetic. In igure 4 we examine the ituation where ubtrate diuion to the redox monolayer i low and rate determining, and where mediator generation and direct ubtrate reaction at the unmodiied electrode urace are greater than the heterogeneou cro exchange reaction inetic between the 4 immobilized mediator and the ubtrate pecie. We have aumed that,,, and o >> >>. For all value o the equilibrium contant (.-) the catalytic eiciency i very large and cloe to unity. ξ.. Ψ cat Figure 3. Variation o catalytic eiciency with normalized potential or heterogeneou redox catalyi via the bimolecular mechanim or the cae o low generation o mediator pecie. ξ

11 enor, 483 We inih thi ection o the paper by drawing attention to a method o data analyi much ued in the literature. In many ituation the technique o rotating dic voltammetry i ued to probe the procee o mediated electron traner between urace immobilized redox couple and olution phae reactant pecie, ince the matter tranport o the latter i well deined uing the rotating dic coniguration. Indeed the diuive rate contant or the olution phae ubtrate pecie i well characterized at the rotating dic due to the act that the diuion layer thicne δ i a well deined unction o electrode rotation peed. We note that uing the Nernt diuion layer approximation the diuive rate contant i given by where denote the diuion coeicient o the δ ubtrate in olution and δ i the diuion layer thicne [3]. For the rotating dic electrode it i well etablihed [4] that the diuion layer thicne i a well deined unction o electrode rotation peed 3 6 / ω.643 / / δ υ ω, where υ denote the inematic vicoity o the olution...99 Ψ cat Figure 4. Variation o catalytic eiciency with normalized potential or heterogeneou redox catalyi via the bimolecular mechanim or the cae o low ubtrate diuion to the redox active monolayer. Returning to eqn. we note that when the normalized potential ξ i large the limiting lux i given by:, L nfa i Γ L and the conecutive procee o diuive tranport and heterogeneou cro exchange inetic are cleanly eparated. From eqn.5 we can readily derive the experimentally ueul outecy-levich equation: Lω / I L (6), L Thu a plot o invere limiting lux veru invere root o the rotation peed i linear with a lope ξ (5)

12 enor, 484 ( ) / 3 / L υ and an intercept I L Γ rom which the heterogeneou rate contant or the cro exchange reaction can be derived. The expreion preented in eqn.6 ha been ued many time in the early literature which ocued on chemically modiied electrode urace o monolayer thicne. In many cae the outecy-levich expreion will alo apply to mediated electrocatalyi at polymer coated electrode o igniicant thicne [5]. owever it ha not been generally realized that the outecy-levich equation mut be applied with care, and that i ome circumtance it ue will provide an erroneou reult, epecially i an adduct pecie i ormed between the immobilized mediator and the olution phae ubtrate. A a conequence, in the next ection o the paper, we examine the generalized outecy-levich mechanim or mediated electrocatalyi at a redox active monolayer and then examine extended verion o the baic equation preented in eqn.5, which tae peciically into conideration the precie nature o the interaction between the ubtrate in olution and the active ite molecule. A Generalied outecy-levich Analyi o Mediated lectron Traner at Immobilized Monolayer. It i not oten realied that the outecy-levich analyi i trictly applicable only to irt order urace reaction. Levich clearly tate thi limitation in hi eminal monograph on phyicochemical hydrodynamic [6]. In thi ection o the paper we extend the tandard outecy-levich analyi and aume a mechanim o the ollowing type: * P ere we aume that all procee are irt order, any o which can be rate determining. We have ma tranport o the ubtrate to the electrode urace (quantiied by a rate contant a beore), preactivation o the ubtrate to orm an activated ubtrate * (quantiied by the rate contant and - ) via procee uch a adorption, binding or diociation, and inally electron traner between * and the electrode urace decribed by the heterogeneou electrochemical rate contant which will be potential dependent. In thi model we peciically introduce a irt order activation proce and include the poibility that electron traner between the active pecie and the underlying electrode may determine the net rate. We apply the quai teady tate approximation to the reaction cheme above and obtain: ( - ) - * ` * (7) We can readily how that: (8) and alo that:

13 enor, 485 (9) and o the net lux i given by: (3) ( ) We invert the latter expreion to obtain: where we have introduced the equilibrium contant (3). Again all poible rate limiting procee are cleanly eparated: activation, electron traner and ma tranport. In act: or any equence o conecutive irt order procee it i poible, uing the quai teady tate approximation, to obtain a eparation o the pertinent rate limiting tep i an expreion or the reciprocal o the net lux at teady tate i developed. Noting that the diuive rate contant i proportional to the quare root o the electrode rotation peed or a rotating dic electrode geometry, eqn.3 can readily be tranormed into an expreion o the outecy-levich type a outlined previouly in eqn.6. ence a plot o reciprocal lux (or indeed reciprocal current) veru invere quare root o the rotation peed i linear with a lope L B where /3 / 6 B.55 υ, and an intercept given by I L. The general experimental approach i outlined chematically in igure 5., L L ω / I L L B I L B /3 / 6.55 υ I L lope yield inetic parameter & will exhibit potential dependence. / 8 exp [ β ξ ] Figure 5. ata analyi protocol or the generalized outecy-levich mechanim.

14 enor, 486 I a outecy-levich type mechanim i operative a plot o the outecy-levich intercept veru invere ubtrate concentration hould be linear and pa through the origin a preented in igure 5. The lope o the latter diagnotic plot hould contain the inetic parameter or the activation proce and hould alo depend on the applied electrode potential via the term. We alo note rom eqn.3 that the teady tate current repone will be alway directly proportional to the bul concentration o ubtrate. The latter prediction are eay to chec experimentally. Again it prove ueul to derive an expreion or the normalied lux Ψ a deined previouly in eqn.. We can readily how that eqn.3 may be tranormed into the ollowing non-dimenional orm: ρ λ Ψ λ ρ λ λ where we have introduced the competition parameter: urace reaction to that o ubtrate tranport in olution and (3) ρ which give the ratio o the rate o the λ which compare the rate o urace electron traner with the rate o de-activation. The latter parameter will be trongly potential dependent. xamination o eqn.3 ugget that the parameter ρ and λ are uitable axe or a inetic cae diagram. Furthermore impliied expreion or the normalied lux may be obtained depending on the magnitude o λ and on the product ρλ. The inetic cae diagram i illutrated in igure 6. qn.3 adopt the ollowing limiting orm when the parameter λ << : ρ λ Ψ ρ λ (33) wherea when λ >> we note that: ρ Ψ ρ (34) ρλ ρ Log ρ low MT Ψ ρ Ψ ρ ρ λ Ψ ρ λ low T urace activation at equilibrium Ψ ρ λ Ψ ρ Fat MT Log λ λ low urace activation ρ λ Ψ λ ρ λ λ Figure 6. inetic cae diagram or the general outecy-levich mechanim.

15 enor, 487 Now taing the cae when λ << we obtain two ditinct rate limiting ituation depending on the magnitude o the product ρλ. Firtly when ρλ < then eqn.33 reduce to: Ψ ρ λ (35) or in term o the net lux: (36) In thi ituation urace activation i at equilibrium and urace electron traner i low and rate determining. The lux will be potential dependent via a Tael relationhip becaue o the exponential dependence o the electrochemical rate contant on applied potential. We label thi ituation a cae I. On the other hand when ρλ > eqn.33 reduce to: Ψ (37) and the lux i given by it ma traner controlled value ince: (38) The net lux will be independent o potential. We label thi ituation cae II. In the inetic cae diagram the line ρλ will eparate cae I and II. Now turning to the cae when λ >> we examine eqn.34 and conider two ituation which depend on the magnitude o the parameter ρ. Firtly, when ρ << eqn.34 reduce to: Ψ ρ (39) and the net lux i given by: (4) ere the urace activation tep i low and rate determining and both electron traner and ubtrate diuion are at. Thi ituation i deignated cae III and will be located in the lower right hand quadrant o the inetic cae diagram. On the other hand when ρ >> eqn.34 reduce to Ψ again and inetic cae II i regained. In igure 7 and igure 8 we plot eqn.3 or variou value o λ and ρ. In igure 7 we indicate the way that the normalized catalytic current varie with the parameter ρ over the range. -, or variou value o the parameter λ.when the parameter ρ i mall the normalied current Ψ will adopt numerical value coniderably le than the ma traner controlled value o unity over the entire range o λ value examined. When both ρ and λ are mall the rate i determined by low electron traner (cae I) When λ i mall the normalized lux only increae lowly with increaing ρ. Thi i to be expected ince the potential applied to the electrode to tranorm the activated * pecie to product will be mall. In contrat when λ i large, we begin in a cae III ituation correponding to low urace activation in which the generation o the active pecie * i rate determining, and we note that the normalized lux increae much more rapidly with increaing ρ value. Thi i brought about by the act that the ytem i now ubjected to a ar larger electrochemical driving orce via the applied potential. A ρ get larger Ψ gradually attain the value characteritic o diuion control over the range o λ value examined (. ).

16 enor, Large.8 Ψ cat mall λ. λ λ λ 5 λ. log Figure 7. Variation o the normalized catalytic current with the parameter ρ which compare the rate o the urace reaction to the rate o ubtrate diuion to the monolayer urace, according to the theoretical expreion developed in eqn. 3 o the text... Large Ψ cat ρ. ρ ρ ρ 5 ρ.. mall. log Figure 8. Variation o the normalized catalytic current with the parameter λ which compare the rate o the urace electron traner to the rate o deactivation o the active pecie *, according to the theoretical expreion developed in eqn. 3 o the text. In igure 8 we how the way that the normalized catalytic current varie with the parameter λ over the range. or variou value o the ρ parameter. When ρ i mall at a value o. ay, the normalized catalytic current remain at a low value over all o the range o λ value examined. Thi will correpond either to the cae o low electron traner involving the active urace pecie * (cae I) valid when λ i mall, or cae III, when λ i large, correponding to low urace activation in which the active pecie * i generated. Alo or any given value o ρ the normalized current Ψ attain a contant value once value o λ > are attained. In contrat when ρ admit large value (5, ) the

17 enor, 489 value o the catalytic current i cloe to unity over much o the range o λ value examined in the computation. Thi can be undertood by examination o the cae diagram in igure 6, where cae II, the region deining diuion control i valid or a wide range o λ value when ρ i large. The drop in normalized current when λ get le than unity oberved in igure 8 correpond to the changeover rom rate determining cae II to inetic cae I involving low reaction o active pecie * to orm product which will be potential dependent. A Precuror/ucceor and omplex Formation/iociation Model In the previou ection o the paper we developed model in which the teady tate current repone varie in a linear manner with the bul concentration o the ubtrate pecie in olution. Thi reult will pertain when either imple bimolecular reaction between ubtrate and immobilized mediator pecie i aumed, or i an activated complex involving the ubtrate pecie undergoe urace electron traner to yield product. In many experimental ituation a more complex relationhip between the teady tate amperometric current repone and the bul ubtrate concentration i oberved. A typical current veru concentration repone proile might exhibit a irt order relationhip at low ubtrate concentration and diplay a current repone independent o ubtrate concentration at high value o ubtrate concentration. At intermediate value o the latter a non linear repone pertain. Thi type o behavior i o quite wide applicability. ome peciic example may be quoted. The wor o Albery and Bartlett [7] on the direct oxidation o the biologically important coactor NA and the enzyme catalyed oxidation o glucoe at conducting organic alt electrode uch a TTF.TNQ (TTF tetrathiaulvalene, TNQ tetracyanoquinodimethane) and NMP.TNQ (NMP N-methylphenazinium ion) and that reported by Gorton and co-worer [8] on NA oxidation at redox dye modiied electrode peciically preent characteritic dog leg or biphaic current/concentration plot obtained rom batch amperometric experiment uing rotating dic electrode. Organic electro-oxidation at conductive hydrated metal oxide urace provide another example [9]. In more recent publication biphaic inetic data are reported or glucoe oxidation at thin electropolymerized poly(phenol) ilm containing immobilized glucoe oxidae [3] and at alanethiol el aembled monolayer containing adorbed glucoe oxidae [3], and or both NA [3] and acorbate [33] oxidation at poly(aniline)/poly(vinylulonate) compoite modiied electrode and acorbate oxidation at an electrode modiied by a Tolex ionomer ilm containing erricyanide [34]. Biphaic inetic may be readily explained uing a Michaeli-Mentin mechanim [35] which i already well etablihed in enzyme inetic [36]. The ey apect i that the ubtrate will orm a ditinct precuror complex with the immobilized mediator pecie which can then tranorm via intramolecular electron traner into a ucceor complex which will ubequently diociate to regenerate the urace bound pre-catalyt pecie and the product. ence ey inetic parameter in the mechanim will quantiy the binding reaction to orm the precuror complex, and the diociation rate o the ucceor complex. We conider the ollowing reaction equence:

18 enor, 49 A B B [ B] - [B] [AP] - 3 [AP] A P P P (4) In the latter reaction equence, P denote the bul reactant and product pecie, wherea and P denote the correponding pecie at the monolayer/olution interace. We will aume that the product i abent rom the bul olution initially and o product inhibition can be neglected. A beore, repreent the diuive rate contant o reactant (unit: cm - ). Alo i a irt order (unit: - ) heterogeneou electrochemical rate contant or generation o the active orm o the mediator pecie B within the monolayer. We note that denote the bimolecular rate contant (unit : cm 3 mol - - ) or precuror adduct ormation involving the mediator and the ubtrate pecie, and - i the correponding irt order rate contant or precuror diociation (unit: - ) to reorm mediator and ubtrate. Furthermore, and - are irt order rate contant or the intramolecular electron traner reaction which tranorm the precuror complex B to the ucceor complex AP, and 3 i a irt order rate contant or diociation o the ucceor complex to orm the pre-catalyt and the product. All o the latter rate contant have unit o -. We can aume that ucceor diociation i irreverible i product inhibition can be neglected. We again apply the quai-teady tate approximation to the latter reaction equence to obtain an expreion or the net reaction lux (unit: mol cm - - ): ( - ) Г A Г B - - Г B (4) Г B - - Г AP Г 3 AP The total urace coverage i given by: Γ Γ Γ Γ Γ (43) A B B AP A ueul expreion or the teady tate reaction lux can be obtained i expreion or the unnown quantitie, Γ A, Γ B, Γ B and Γ AP can be determined. ince there are ive unnown and ix deining expreion (preented in eqn.4 and eqn.43) the latter aim can be readily accomplihed.

19 enor, 49 I we initially neglect the diuion o the ubtrate pecie to the monolayer urace, we can readily how ater ome algebra that the invere teady tate reaction lux i given by : Γ M (44) c c qn.44 conit o three term, each o which relect a poible rate limiting ituation. The irt term on the rh decribe the electrode inetic or the A/B tranormation within the monolayer; the econd relect unaturated binding inetic and the third relect adduct tranormation procee to generate product. In the latter expreion we have ollowed the approach o Albery and nowle [37] and introduce the Michaeli contant M and the catalytic rate contant in term o the undamental rate contant or the conecutive tep in the reaction equence a ollow: U M 3 3 where we have written j j and nowle [37]. Note that the quantity 3 (45) j. The igniicance o thee equation have been dicued by Albery U which ha the dimenion o a bimolecular rate M contant, provide a meaure o the rate o capture o the ubtrate pecie by the immobilized catalyt B to orm the adduct B. The term Michaeli contant and catalytic rate contant are well etablihed in the ield o enzyme inetic. The Michaeli contant M provide a meaure o the binding ainity or adduct ormation ability o the ubtrate pecie or the immobilized mediator. Alternatively, it deine the maximum value o the ubtrate concentration or which the catalytic inetic are irt order with repect to ubtrate concentration. The catalytic rate contant i a irt order rate contant, quantiying the rate o decompoition o the urace adduct pecie to orm product. We note rom eqn.45 that both U and are compoite quantitie, and may be conidered to be internal parameter connected with procee occurring within the monolayer region, a oppoed to external parameter which include the electrochemical rate contant and the diuive rate contant. Both quantitie, when expreed in reciprocal ormat, conit o three eparated term. onidering the component term irt, we note that correpond to low rate determining intramolecular electron traner involving the tranormation o the precuror adduct B to the ucceor adduct AP. The term 3 correpond to the cae where the precuror/ucceor adduct tranormation i at a pre-equilibrium ollowed by a low rate determining decompoition o the ucceor adduct to orm product. Finally the term correpond to low rate determining ucceor adduct decompoition. The mallet o 3 thee term will be the major contributor to the net catalytic rate contant. imilarly, i we examine the U component term we note that the term relect rate determining adduct ormation involving

20 enor, 49 the bimolecular reaction between and B. The econd and third term or involve 3 either ingle ( ) or multiple ( ) pre-equilibria ollowed by low intramolecular electron traner between the adduct B and AP within the monolayer ( ) or low ucceor adduct diociation ( 3 ). A chematic ree energy proile illutrating the ree energy dierence aociated with each o the poible rate limiting tep aociated with the internal parameter i preented in igure 9. It i clear rom eqn.44 that by neglecting ubtrate diuion eect in olution, we can obtain a tranparent expreion in which the important inetically limiting tep may be cleanly eparated rom one another. ubtrate/mediator binding Intramolecular electron traner T T3 ucceor adduct decompoition T B [B] [AP] A P 3 U M Figure 9. nergy proile or adduct ormation mechanim. Our analyi to date ha neglected the external proce o ubtrate diuion to the monolayer. owever in experiment mediated electrocatalyi i probed uing rotating dic voltammetry and the urace concentration o ubtrate will dier rom that o it bul value `. Thi imple obervation ha a mared conequence or the expreion or the net reaction lux. Noting that the ubtrate urace and bul concentration and ` are related via: (46) ubtituting eqn.46 into eqn.44 reult in the more complete expreion: Γ U (47)

21 enor, 493 We ue the ollowing reult ( ) in eqn.46 to obtain: Γ U (48) We can readily recat eqn.48 into the ollowing ueul orm: Γ Γ U (49) where we have recaled the unaturated rate contant actor to tranorm into a peudo irt order rate Γ contant via U. qn.49 decribe the net reaction lux when ubtrate diuion in the olution M to the monolayer urace i taen into account. The important point to note rom eqn.49 i that the net reaction lux appear on both ide o the expreion and o each o the component o the expreion are not eparated in a clear manner uing the trategy o deriving an expreion or the invere o the net reaction lux. The term which caue the complication i. I the net lux become cloe to the limit impoed by ubtrate tranport through the olution diuion layer given by, then the latter diuive tranport reult in the immobilized receptor pecie being le aturated than one would expect rom the value o the bul concentration `. Thi eect i decribed by the term. We can ee that it preence reduce the igniicance o the mediator generation inetic, and the unimolecular decompoition o urace adduct to orm product. I we can aume that << then and eqn.49 reduce to: Γ Γ U and we note that all the poible rate limiting term are cleanly eparated. Thi expreion ha been oten utilized in the literature in conjunction with the outecy-levich equation to analye data obtained via ue o the rotating dic electrode. It i important to note that the expreion will only be valid when the net current (or reaction lux) i much le that the diuion controlled value. The aumption implicit in the ue o eqn.5 i that the monolayer urace and hence the immobilized mediator ite i expoed to the bul concentration o the ubtrate pecie. I, on the other hand, a i oten the cae, the experimentally oberved current i the ame order o magnitude a the ma tranport limiting value, there will be a coniderable concentration gradient in the Nernt diuion layer and the urace and bul concentration o ubtrate will dier coniderably. The eect o thi concentration gradient i that the immobilized mediator will be expoed to lower value o ubtrate than expected, and thi will lead to the prediction that the urace bound mediator pecie will be more eicient than it i in reality. ence the ull expreion preented in eqn.49 mut be ued in data analyi i the experimentally oberved teady tate current repone in a enor ytem approache that o the diuion controlled value. Thi very undamental point ha not been realized in much o the data analyi preented in the literature. (5)

22 enor, 494 We can develop a variant o eqn.49 which i imilar to the ane equation ued in the analyi o enzyme baed ytem. We can how that: M M (5) which can be ubequently impliied to: M M M (5) In the latter expreion M repreent the eective electrochemical rate contant (unit: cm - ) or the monolayer modiied electrode at low ubtrate concentration, and M denote the Michaeli contant or the monolayer modiied electrode. Both o thee quantitie are total parameter in the ene that they are compoite quantitie not only relecting the binding and inetic within the monolayer but alo include the external variable related to ubtrate diuion in the olution and mediator generation via an external applied potential. We can readily how that: Γ Γ Γ Γ Γ M M M M U M M (53) We note that M hould exhibit a potential dependence via the term. We can tranorm the ane type expreion preented in eqn.5 into a quadratic equation in the net reaction lux to obtain: ( ) M M M (54) which ha the olution: ( ) ( ) / 4 M M M M (55) We can ollow the previou ection o thi paper and introduce a normalized lux or eective catalytic eiciency o the modiied electrode a Ψ and rom eqn.55 obtain the ollowing dimenionle mater expreion: / / ρ, 4ρ Ψ { } {-(- ) } { } {-(- ) } ( ) u u T u u u (56) where we have introduced

23 enor, 495 ρ u M M (57) and we note that 4ρ u ρ (58) T, u ( u) The irt parameter in eqn.57 compare the electrochemical rate contant or the modiied electrode with the diuive rate contant o the ubtrate, and the econd compare the bul ubtrate concentration with the Michaeli contant o the modiied electrode ytem. We now examine the term T ρ,u deined in eqn.58. We irtly note that ρ. We can plot the T term veru u or typical value o ρ. Thi i done in igure or three value o ρ (,.5 and.). We note rom thee plot that when u i very mall (u <.) or very large (u > ) the T - unction approache zero. Thi will be o or all value o ρ... T,u ρ ρ.5 ρ u/ M Figure. Variation o the unction 4ρ u ρ with u. T, u ( u) I T i mall then or all ρ we have { T } peciically i u i large then or u >> we note ( u) u eqn.56 or the normalized lux or catalytic eiciency reduce to: ρ Ψ u u T / ρ, u ρ, u. 4ρ u ρ u ( u) ( u) ρ and u / and o { Tρ, u } u l arg e in the limit o large u where ` >> M. We can now how uing eqn. 59 that the net reaction lux when ` >> M i given by: (59)

24 enor, 496 M M M M M (6) Recalling rom eqn.53 the deinition o M in term o the more undamental inetic and binding parameter we obtain: Γ Γ The pertinent parameter in eqn.6 i the ratio, which compare the rate o mediator generation (via the term) with the rate o adduct decompoition (via the term). Firtly i << then we have the ituation where mediator generation i low and rate limiting and urace adduct decompoition i rapid. In thi particular ituation eqn.6 reduce to: (6) Γ which we will label cae III. On the other hand, i >> then the proce o mediator generation will be rapid and the urace adduct decompoition to orm product will be low and rate determining. In thi circumtance (cae IV) we obtain the ollowing limiting expreion or the net reaction lux: (63) Γ We note that both or cae III and cae IV we are concerned with the limit where ` >> M. In both cae the net reaction rate i independent o the bul ubtrate concentration ` and depend directly on the total urace coverage o the mediator pecie in the monolayer. For cae III the net rate hould be potential dependent via the Tael relation becaue o the term. For cae IV no uch potential dependence i expected. We now ocu attention on the ituation where the bul ubtrate concentration i much le than the Michaeli contant o the modiied electrode, when ` << M. When u i mall u << or /u >> and u. Under thee circumtance eqn.56 or the normalized lux reduce to: o ( ) { ( 4ρ u) } / Ψ (64) u Now ince ρ and u << then the term 4ρu will be mall. Under thee condition / ( 4ρ u) ρ u and eqn.64 reduce to: Ψ ρ (65) Again, thi i a ueul limiting expreion, valid when << M. Noting the deinition o ρ we can how that: M (66) I we recall the deinition o the modiied electrode rate contant provided in eqn.53 we can how that (6) U U (67)

25 enor, 497 Again thi i a ueul expreion. The pertinent parameter here i the ratio U which repreent the balance between ubtrate diuion in the diuion layer and chemical reaction involving the urace bound adduct. Two limiting cae may be enumerated. Firt, when U << we have the ituation o low rate determining ma tranport o ubtrate in the olution to the modiied electrode urace and at unaturated inetic involving ubtrate and mediator pecie. Under uch condition eqn.67 reduce to: (68) which we label cae I. ere the teady tate current repone will vary in a linear manner with bul concentration o the ubtrate. econd, when U >>, we have the cae o acile material tranport and low rate determining chemical reaction between immobilized mediator and ubtrate. ere eqn.67 reduce to: U (69) which we denote a cae II. Again, the teady tate current repone will vary in a linear manner with bul ubtrate concentration. We can dierentiate between cae I and II by noting that in the ormer, the current i independent o urace coverage o redox mediator wherea in the latter, it depend linearly on Γ via the U term. evelopment o a inetic ae iagram It i deirable to derive a relationhip between the overall quantity M and the more undamental Michaeli contant M. We recall that important parameter which deine rate limiting ituation are ζ and ξ. Uing the latter deinition and the expreion preented in eqn.53 we can how, U ater ome algebra that the Michaeli contant or the modiied electrode i related to the Michaeli contant decribing the adduct ormation proce via: ξ ζ M M ζ ξ We can deine a urther normalized variable which quantiie the degree o unaturation o the ytem and introduce v. We can relate the normalized concentration variable u introduced M earlier to the latter quantity via: ( ξ ) ξ u v M ( ) M (7) ξ ζ ζ ence the three pertinent parameter which will deine a inetic cae diagram are ζ, v, and ξ. We can, i we o wih, contruct a three dimenional cae diagram involving the latter parameter. owever intead, to aid viualization, we contruct the two dimenional analogue, where we preent plot o log v veru log ζ or the limiting cae o ξ << and ξ >>. We chooe log ζ a the M (7)