Lyons Bartlett MHC 1

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J. Electroanal. Chem., 261 (1989) 51-59Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands

Microheterogeneous catalysis in modified electrodes

M.E.G. Lyons and D.E. McComackPhysical Chemistry L&oratory Vnrversity of Dublin, Trinity College, Dublin 2 (Ireland)P.N. BartlettDepartment of Chemrstryhe University of Warwick, Coventry, West MidIan& CV4 7AL (Great Britain)(Recei ved 21 July 1988; in revised form 28 October 1988)

ABSTRACT

Analytical expressions quantifying the transport and kinetics in conducting polymer modifiedelectrodes containi ng a homogeneous distrib ution of spherical microparticulate catalysts are presented. Inparticular the dependence of the f lux on the number of catalytic particles per unit volume, the layerthickness, the substrate concentration, the particle radius and the electrode potential are outlined.

INTRODUCTION

The electrocatalytic activity of polymer modified electrodes has been the subjectof considerable study for some time. Most systems utilized to date have requiredthat the electrochemically active centre within the coating exhibit a dual purpose,that of being an efficient electron transfer mediator in addition to displaying goodelectrocatalytic activity. This dual requirement is too restrictive and a betterapproach is to utilize an integrated system where the functions of charge transportthrough the film to the catalytic site and catalytic activity are carried out bydifferent components in the layer. Recent reports in the literature have describedmodified electrode systems in which the electron conduction and catalytic functionshave been provided by different components within the surface deposited coating.In one approach the catalytic centres were incorporated into a conducting polymermatrix [1,2]. In another approach described by Anson et al. [3,4], immobile catalyticmolecules and electrochemically reversible inorganic redox couples (the latter actingas electron transfer mediators) were incorporated into an inert polymeric matrix.Provided that good electrical contact is maintained between the catalytic particlesand the conducting polymer, it would be expected that the conducting polymermatrix would exhibit a better charge mediating ability than that of an inorganic0022-0728/89/$03.50 0 1989 Elsevier Sequoia S.A.


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redox couple. Very recently, Holdcroft and Funt [2] examined the electrocatalyticactivity of suface deposited polypyrrole layers containing platinum micropar-ticulates with respect to oxygen reduction, and noted that the catalytic current waslimited by the rate of oxygen permeation in the film. However, no theoretical modelwas proposed by these workers. Consequently, the transport and kinetics in thesetypes of integrated modified electrode systems is described in the present communi-cation, and various simple diagnostic parameters are proposed which can be used asan aid towards mechanistic analysis in further experimental studies in this area.THE MODELMacroscopic approachIn the following model (outlined in Fig. 1) we assume that the microheteroge-neous catalytic particles are dispersed in a uniform manner throughout the conduct-ing polymer film. It is also assumed that the catalytic particles are in intimateelectrical contact with the conducting polymer. Furthermore, we assume initially noconcentration polarization in the solution and that the substrate concentration swithin the layer is given by KS,, where K is the partition coefficient and s, denotesthe bulk substrate concentration. Let L denote the layer thickness. Under steadystate conditions the diffusion equation takes the form:D, d2s/dx2 - ks = 0 (1)where k denotes an effective first order rate constant for the substrate productreaction at the surface of the catalytic particle. This equation must be solved subjectto the boundary conditions:x=0 ds/dx = 0 (2)x=L s = KS00 (3)The diffusion equation admits the general solution:s(x) =A exp(x/X,) + B exp( -x/X,) (4

E L E CT R O D E

Oo 0 S O L UT I O N0 0 0 S = Ks ,

OtiyooI--O N DU C T I N G P O L Y ME R0 0 0 MA T R I XFig. 1. Model and notation for modified electrode.


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where X, denotes ax, = ( Q/k)2The reaction length

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kinetic length defined as(5)

X, describes the distance that the substrate S will diffuse in thelayer before being destroyed by reaction with the catalyst. The boundary conditionsare now utilized in order to evaluate the constants A and B . Application of eqn. (2)results in the assignation A = B. Furthermore substitution of the condition outlinedin eqn. (3) into eqn. (4) and simplifying, results in an expression for A of the formA = KsJ2 cosh( L/X,) (6)Consequently, the substrate concentration in the layer is given bys(x) = { Ks,/cosh( L/X,)} {cosh(x/X,)}The flux j is given by the relationj = D,(ds/dx),,,

(7)

(8)Differentiation of eqn. (7) and utilizing eqn. (8) results in the following expressionfor the fluxj = (D,Ks,/X,) tanh( L/X,) (9)Two rate limiting situations can be considered. Firstly when L >> X, the fluxexpression reduces toj = D, Ks,/X, (10)and the overall reaction in the layer is transport controlled. In this case S isconsumed in a reaction layer of thickness X, at the surface of the film. Conse-quently, the flux j is independent of the film thickness.However, if L Ed X,, the flux admits the form:j = D, KS , L/X,,? = Kk s, L (11)and the overall reaction is kinetically controlled. Now S is consumed by reactionthroughout the film and hence increasing the film thickness, L , leads to an increasein the flux, j.Microscopic approachWe now consider the pseudo-first-order rate constant k (units s-l) for thesubstrate reaction at the catalyst particle. In order to examine the latter quantity indetail, it is necessary to adopt a microscopic approach and consider explicitly thespherical geometry of the catalyst particle [5]. Substrate diffusion to the latter isdescribed by the following equation

(12)


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This equation is solved most readily if we set u = sr, where r denotes the radialdistance. Once this substitution is effected, the differential equation takes the formDd2U=O dr2

(13)This equation has the solutionu=rr+6which transforms to

(14)

s=E+S/r (15)The following boundary conditions pertain. When r = r,, at the surface of theparticle, s = s0 and DSds/dr = k,s,. Note that the quantity k, describes a potentialdependent rate constant of the formk,=k exp(+@)=kO exp(f&(E-EO)/RT) (16)In this latter expression, k o and cz are fundamental electrochemical parameters andare termed the standard rate constant and transfer coefficient respectively. Thequantity E o is the standard potential which will reflect the thermodynamics of thesubstrate reaction at the catalytic particles surface. The other symbols have theirusual meanings. Also we must define a distance r, which describes a sphere ofaction of each catalytic particle [5], such that when r = rl, s = s(x). The relationbetween r, and the number of particles per unit volume N is 4aris/3 = l/N.Application of the boundary conditions results in the assignationsE = s(x) + k,s,ri/DSr, (17)6 = - k,s,ri/D, (18)From eqns. (15), (17) and (18) we find that the substrate concentration at thesurface of the spherical catalytic particle is given bysO=s(x)/{l + (&VQ) - (k&/D,ri)} (19)The substrate/catalyst reaction can be considered formally as a second orderreaction with the rate of consumption of S given byds/dt = -k,s(x)c (20)where c denotes the concentration of catalyst particles. Alternatively we can writean expression for the rate of reaction of S which contains the heterogeneouselectrochemical rate constant k, explicitly,ds/dt = - k,NAs, (21)where A denotes the surface area of the spherical particles (i.e. 4vrr:) and N is thenumber of catalyst particles per unit volume. Comparison of eqns. (20) and (21)shows that the second order rate constant is given byk, = k,NAs,/cs( x) (22)


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We note that the pseudo-first-order rate constant described previously can berelated to this latter quantity via the relation k = k,c, provided that the quantity c isa constant, corresponding to a uniform distribution of catalyst particles throughoutthe film. Consequently, the pseudo-first-order rate constant k is given byk = kENAs,,/.+) (23)We note from eqn. (19) that

1$j = 1 + (k,r,,/D,) - ( kEr;/Dsrl) (24)Hence, substitution of eqn. (24) into eqn. (23) results, upon inversion, in thefollowing useful expression for the pseudo-first-order rate constant:

(25)Since r, x== ,, then the term r,/Dsr, on the r.h.s. of eqn. (25) can be neglected, andconsequently the expression for the pseudo-first-order rate constant reduces to1-=&($+:I (26)

It is important to note that macroscopic or planar diffusion effects can beseparated from microscopic or spherical diffusion effects provided that theparticles are small compared to the film thickness (r. -=s L) and that 4ro -SCX,. Th i scondition is discussed in detail in the appendix.DISCUSSION

We can now combine eqns. (5), (9) and (26) to obtain a master equation for theflux:

We can identify four limiting cases from this equation. We begin by takingL(4m$N[ DsJ$:k,]}2


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the particle/substrate interface. When k, B D,/r,, case I, the reaction at theparticle surface is diffusion controlled and eqn. (28) becomesj, = 4mroNDsKs,L (29)On the other hand, if k, -+z D,/r,,, case II, so that the electrode kinetics at theparticle surface are rate limiting, then the flux simplifies to:j,, = 4vriNk, Ks, L (30)Turning to the remaining pair of cases, whenL (4m$N( Ds~f~k,)}2 > 1then eqn. (27) reduces toj = Ks_,D*{ 4nr,iN( Ds~


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j = ksr,Nk,Ks,LI

L,NFig. 2. Case diagram ill ustrating various rate limiting situations. D and K denote diffusion and kineticcontrol respectively.

TABLE 1M~han~sti c indicators for microheterogeneous catalysis in modified electrodesCase Reaction order w.r.t.

L '0 c (or N) k,1 1 1 1 1 02 1 1 2 1 13 1 0 0.5 0.5 04 1 0 1.0 0.5 0.5

case there is a unique set of dependences of the flux on the experimental parameterss m, L, ro, v,nd the electrode potential (expressed through the electrochemical rateconstant kE). These are given in Table 1. It is clear that on the basis of thesedependences, it should be possible to identify the particular case clearly.

ACKNOWLEDGEMENTS

The support of the Trinity Trust, the Science Faculty, Trinity College Dublin,and the Commission of the European Communities Stimulation Action Program(Grant Number 86300283FRBPUJUl) is gratefully acknowledged. D.E.McC.acknowledges receipt of a Trinity College postgraduate award.


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58APPENDIX

In this appendix we consider the conditions under which it is valid to treat themacroscopic, planar diffusion of the reactant S independently of the microscopic,spherical diffusion to the catalyst particles. In Fig. 3, a single catalyst particle ofradius r, and its sphere of action of radius ri is outlined. Firstly, for cases I and II,the concentration of substrate S in the film is uniform, and the particles see thesame concentration of S in all directions. Under these circumstances we can clearlyseparate the planar and spherical components. In cases III and IV, the concentra-tion of S varies through the film. Now our treatment is most likely to break downwhen the reaction of S at the catalyst particle surface is diffusion controlled (caseIII). Under these conditions the particle perturbs the concentration of S over adistance r, from its surface [5]. Our approximation is therefore valid provided theconcentration of S does not vary significantly over a distance 4r, corresponding tothe difference between the front face of the catalyst particle, and the back face.Thus our approximation is valid provided that:4r,(ds/dx) +z s(x) (Al)where the quantity 4r,(ds/dx) is the change in concentration between the front andback of the particle. Now differentiation of eqn. (7) yields that:ds KS-= m {taiMx/Xk))dx X,

so substituting from eqns. (7) and (A2) in (Al) and simplifying, we find that:$ sinh( x/Xk) +Z cosh( x/&)

k(A31

Fig. 3. Schematic representation of a single catalytic particle of radius r,, and its sphere of action ofradius r,.


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Rearranging and noting that 1anh(x/&) < 1, we obtain:4r, -=x X,Substituting for X, from eqns. (5) and (26):

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(Ad)

(A51Now the particles only perturb the concentration of S significantly when r,/D, z+l/k,. Under these circumstances eqn. (A5) becomes:4r, < {1/4mrON}2 tA6)But we recall that:l/N = 4rrr:/3 (A7)Hence3.3r,

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