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Effect of diffuse layer on the rate of electron transfer across an electrolyte vb electrolyte solution interface
if:
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,
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ELSEVIER
JOO~L
OF
Journal of Electroanalytical Chemistry 396 (1995) 391-396
Effect of diffuse layer on the rate of electron transfer across an electrolyte lelectrolyte solution interface 1 Hajime Katano, Kohji Maeda, Mitsugi Senda * Department of Bioscience, Fukui Prefectural University, Matsuoka-cho, Fukui 910-11. Japan Received 21 February 1995; in revised form 20 March 1995
Abstract
The basic equations of the kinetics of electron transfer across an electrolyte [electrolyte solution interface, in which the effect of diffuse layers on both sides is taken into account on a simplified model of electric double layer, are presented. Two ideal processes, an inner-layer rate-determining process and a diffuse-layer rate-determining process, are discussed. Theoretical predictions derived from the basic equations are discussed. Some abnormal transfer coefficients can be explained by the theory. Keywords: Diffuse-layer effects; electron transfer; electrolyte lelectrolyte solution interface
1. Introduction In previous papers [1,2] we have derived the basic equations for the rate of charge (ion or electron) transfer across an interface between two immiscible electrolyte solutions, i.e. an oil [water (O [W) interface, in which the effect of the diffuse (double) layer on the apparent or observed rate constants of the charge transfer kinetics was taken into consideration. The electric double layer is assumed to consist of an inner layer sandwiched between two diffuse layers on each side of the interface, and the charge transfer reaction is assumed to take place between two chemical species at the planes of contact of the inner layer with the two diffuse layers. Steady-state ionic transport in the diffuse layers (Levich correction) was also assumed. The equations derived were discussed for two ideal processes: the inner-layer rate-determining process (Frumkin correction) and the diffuse-layer rate-determining process. Basic equations for the rate of charge transfer across the O IW interface have been presented by Samec [3], who discussed the effect of the diffuse layer on the kinetics in terms of the Frumkin theory. Samec et al. [4] have determined the rate constants of electron transfer between ferrocene/ferrocene + in the nitrobenzene (NB) phase and
Fe(CN) 4-/Fe(CN)63- in the aqueous (W) phase at the NB [W interface. Recently, Schiffrin and coworkers [5-8] have reported kinetic data for the interfacial electron transfer between a redox couple in the 1,2-dichloroethane (DCE) phase (i.e. the lutetium diphthalocyanine redox couple [Lu(pc)~"/2 ÷ (O)], the bis-(pyridine) (meso-tetraphenyiporphyrinato)ruthenium(II) redox couple [Ru(tpp)(py)°/÷ (O)] or the 7,7,8,8-tetracyanoquinodimethane redox couple [TCNQ-/°(O)]) and a redox couple in the water phase, (i.e. Fe(CN) 4 - / 3 - ) when the redox couple in the W phase is present in excess compared with the redox couple in the O phase. One of the interesting results of these studies is that the transfer coefficients of the charge transfer kinetics when they are expressed in terms of an equation of the Butler-Volmer type are not constant but change appreciably as the applied interfacial potential difference, is changed, The purpose of this paper is to report some theoretical results, based on what we consider to be reasonable assumptions, predicted by the fundamental equations previously derived [2] for the apparent or observed rate constants of interfacial electron transfer. It is shown that some of the abnormal apparent transfer coefficients can be explained by the present theory. 2. Theoretical
i Dedicated to Professor K. Honda, Professor H. Matsuda and Professor R. Tamamushi on the occasion of their 70th birthdays. * Corresponding author. 0022-0728/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved SSDI 0 0 2 2 - 0 7 2 8 ( 9 5 ) 0 4 0 3 6 - 6
We consider the polarized O IW interface illustrated in Fig. 1. A layer of solvent molecules (the inner layer
H. Katano et al. / Journal of Electroanalytical Chemistry 396 (1995) 391-396
392 ..tW
The rate constants k'e,b and k'e,f c a n be expressed by a Butler-Volmer type equation ( a + fl = I): k;, b = k;.bs exp[ 13(nF/RT)(AW ~b2)] k'.e = k'.fs e x p [ - a ( n F / R T ) ( AW~b2)] t
(5)
t
In these equations ke.bs and ke.fs are the rate constants at aw = - 6o = 0.
"~0
xo
xo xw
Now, if steady-state ionic transport in the diffuse layers (Levich correction) [9,10] is assumed, the flux Je can be expressed by the kinetic equation with respect to the surface concentrations of Red(O) and Ox(O) at x = x °, as given by [2]
x ow
>X Fig. 1. Electric double layer across a polarized O l W interface. The potential differences are defined by At4, w -_ 4'0w - 4'0, act = 4'2o - 4,0o and a 6 w = 4'w _ 4'w.
between x ° and x w where x is the coordinate perpendicular to the interface) separates two ionic space-charge regions (the diffuse layers between x ° and x ° in the O phase and between x w and x0w in the W phase) on each side of the interface; x ° and x0w are the coordinates of the planes, beyond which (i.e. at x < x ° in the O phase and x0w < x in the W phase) the ionic transport is diffusion controlled. Also, we let the potential ~b and the concentration Ci of ion i be th°, th°, tkw and q~0 w and cOo, , cO, , cw /,2 and c Wi.0at x °, x °, x w and x~v respectively. We assume (l) that there is sufficient supporting electrolyte on both sides of the interface and (2) that the structure of the double layer is not affected by the flow of current or ionic transport across the interface. We consider an electron transfer process across an O lW interface between two redox couples Red(O)/Ox(O) and Red(W)/Ox(W) at x ° and x w respectively: Red(O) + Ox(W) = Ox(O) + Red(W)
(1)
Let the charge numbers, including the sign, of the chemical species (ions) Red(a) and Ox(ct) be z~ and zg respectively (et = O, W), and let the number of electrons transferred in Eq. (1) be n. Then we have Z° --ZR° = Z ~ --Z w = n
(2)
In the following, we restrict our discussion to the case where the concentrations of the ions Red(W) and Ox(W) are much higher than those of Red(O) and Ox(O), so that the concentration distribution of Red(W) and Ox(W) can be assumed to remain constant during the electron transfer process of Eq. (1). This is actually the case that is adopted in many experimental studies. Thus Eq. (1) can be replaced by the pseudo-first-order reaction: Red(O) = Ox(O) + ne- (W)
(3)
and the flux Je of electrons transferred across the interface with respect to the surface concentrations of Red(O) and Ox(O) at x = x ° is given by the kinetic equation t O p O J~ = ke,b%,2 - ke,eCo, 2
(4)
Je "= ( ke,b)appCR, ' o 0 -- ( k e, , f ) a p p C oo. 0
(6)
where the apparent or observed rate constants (k'e,b)app and (kle,f)app are given by ( k'e,b)Fo
(Ze,0app =
1 + [k;.b/( -A°R.2 )] + [k;.J( -A°o.2 )] (k;,,)Fc
(ke'f)app
1+
t O [ke.b/(--AR.2)
t O ] + [ke.f/(-Ao.2) ]
(7)
where !
(ke,b)Fc
__ --
!
ke,bRe
(k'e, )Fc = k'e,Re
(8)
and R e = exp[( z ° + fin)( F / R T ) ( - A d ~
°)
+ ( zw -/3n)(F/RT)(-A4~W)]
(9)
In Eq. (7)
a°, 2 = ( u°Rr/Z? )
z°F/Rr)A 6 ° ]
exp[(
( i = O, R)
(10)
where
z°= JxOfX°exp[ ( z°F/RT)4~] dx
(ll)
and u ° (i = R,O) is the mobility of ion i in the O phase. The terms A~b° ( = ~b° - th° ) and Aft w ( = th~v - ~b0 w) are the diffuse-layer potentials in the O phase and the W phase respectively. By introducing a factor A° defined by At = [ k,e.b/( --AR,2)] O + [ k'e , f / / ( - A°0 , 2 ) ]
(12)
we can rewrite Eq. (7) as (k'e.b)ap p = ( k ' e , b ) F c / ( 1
'
-
(ke,f)ap p -
+ A° )
k' (
e,f)vJ(
(t3) 1 +
A° )
The factor A° represents the ratio of the rate constant of electron transfer across the inner layer to that of mass transfer (diffusion and migration) of ionic species across the diffuse layer. Thus we have two ideal processes depending on whether )t° << 1 or )t° >> 1.
H. Katano el al. / Journal of Electroanalytical Chemistry 396 (1995) 391-396
393
2.1. Inner-layer rate-determining process (Frumkin theory) When A° << 1, i.e. the second and third terms in the denominators are negligibly small compared with unity, we obtain the rate equations for the case where the charge transfer across the inner layer is rate determining. Clearly, this is the case of the Frumkin theory for the double-layer ~,orrection in electrode processes [3,10,11]. Then the appar,nt rate constants are given by (ke,b)Fc and (ke,f)pc, as defined by Eq. (8), which are rewritten as
% a b
.
C
k'e.b)ap p = (k'e.b)Fc = k'e,s exp(
fln~)R e
0
(14)
i k'e.f)ap p = ( k ' e , f ) F c = k'e,s e x p ( - a n ¢ ) r e
F / R T ) ( A wo * - Aw 4,~')
(15)
.rod k'e,s is the rate constant of electron transfer at Aw 4, = W o~ -1o 4, e, where AWo 4 'o¢, is the standard potential of electron xansfer represented by Eq. (3). Also, the equilibrium ?otential ((Aw4')eq!~at which Je (defined by Eq. (4)) is zero, is related to A o 4' e by (Aw4')eq = Aw4,°'e +(RT/nF)ln(c°.o/C°,o)
(16)
2.2. Diffuse-layer rate-determining process When A° >> 1, i.e. the second and third terms in the denominators are much larger than unity, we obtain the rate equations for the case where the ion transfer across the diffuse layer is rate-determining. Then we have k;,b)app = (k;,b)Lc exp( nsr ) ( - e x p ( n~" ) / a ° 0) + ( - 1 / a ° . 0 ) k;.f)ap p
=
(ke~f)Lc
1
= [ - exp( n¢ )/A°.o] + ( - 1/A°o)
(17)
4
6
8
A~¢p~
Fig. 2. Diffuse layer potential A~b ° vs. potential difference AW~b. Curves a, b and c are defined in the text.
potential gradient throughout the diffuse layer between x ° and x ° to evaluate A°i,2 (see below). In this study we assume a simplified model of the electric double layer in which the surface charge density o"w on the W phase of the interface changes linearly with Av~4, and the potentials of the outer Helmholtz planes are given by Gouy-Chapman theory without specific adsorption. Furthermore, on assuming that the potentials of outer Helmholtz planes are equal to A 4,0, we obtain curves of A4, ° vs. (A~4,-AW4,pzc ) curves, where AW4,pzc is the potential of zero charge. The results are shown in Fig. 2, where curves a, b and c are the curves calculated for the concentrations of a 1:1 supporting electrolyte in the O phase with c ° values of 4 mM, 16 mM and 400 mM respectively for tr w --4 IxC cm -2 at (F/RT)(Aw4, AW4,pz¢) = 40 and where the dielectric constant ¢o of the O phase is 10.37 at 25°C. On assuming a linear potential gradient in the diffuse layer with a diffuse layer potential of A 4,0 and a thickness of l / K °, we obtain the following equation for A°i,2"•
At: (u°RT// at i,2 - 1 / K O j I I _ e x ' ~
)
(19)
-a°)
where A°,,0 = a ° , ,2 e x p [ - ( z ° F / R r ) ( a 4 , ° ) ]
2
(FIRTX,dWo¢ -
.vhere "= (
0
(18)
where
a ° = ( z i ° F / R T ) ( A 4 , °)
( i = R , O)
(20)
Accordingly, after rearrangement Eq. 12 can be rewritten as
3. D i s c u s s i o n In evaluating the effects of the diffuse layer on the apparent rate constants of electron transfer across an O [W interface the mass transfer rate constant A°i,2 (i --- O,R), as defined by Eq. (10) with Eq. (11), must be calculated. When the diffuse layer is predicted by Gouy-Chapman theory, Eq. (11) is transformed into a convenient form [2] and A°i,2 is given as a function of the diffuse layer potential A4, ° and the Debye reciprocal length K° as well as the ionic mobility u ° and the number of charged ions z °. Another approach to this problem is to assume a linear
Ao =
k'~
u° RTK °
exp ¢tn~' + fl
--A4, ° + A4, w
1 - exp(-
x
aO
+exp[-an~'
x
aO
- a
+ A 4,w)]
394
H. Katanoet al. / Journal of ElectroanalyticalChemistry396 (1995) 391-396 2
-2
-1
J
,
0
1
j
2
-2
Fig. 3. Apparent rate constant (k'e,b)Fc vS. potential difference ~" for A~° ~o, _ A *° ~ = _ 2 R r / F. See text for expla.atio..
where u ° = (u°)"(u°) t3 is the average mobility of ionic species in the O phase and ~" = ~"- ( R T / n F ) l n ( u ° / u ° ) . Thus the factor A° can be evaluated from the ratio of k'e,s to u°RTK o, i.e. the ratio of the standard rate constant of electron transfer across the inner layer to the rate constant of diffusion across the diffuse layer. In the following we take u ° = u ° = u ° so that ~ " = ~'.
3.1. Inner-layer rate-determining process (Frumkin theory) The apparent rate constants (kre,b)Fc defined by Eq. (14) and Eq. (9) were calculated for the interracial electron transfer processes of four redox couples zr~/z o o ((A) 1 + //2 + , (a) 0//1 + , (C) 1 - / / 0 and (D) 2 - / / 1 - ) in the O phase and a single redox couple zR//Zo w w =4-/3in the W phase. Curve b in Fig. 2 is for A 4)°, while curve c is for A~bw; curve c can be regarded as representing the potentials of the outer Helmholtz plane in the W phase when the concentration c w of 1 : 1 supporting electrolyte is 53 mM, t r w = 4 /xC c m - : with (F//RT) (AWqbAWq~pzc)=40, and 8 w =78.38 at 25°C. The resulting (k'~.b)vc values for fl = 0.5 are plotted for A~b °' - Av~~bpzc values of - 2 RT//F, zero and 2 R T / F in Figs. 3, 4 and 5 respectively. These results indicate that when the plots are fitted to the Butler-Volmer type equation
-1
0 z2~"
1
2
Fig. 4. Apparent rate constant (k'e,b)~c vs. potential difference ~' for A~ ~0, _ Aw ~pzc = 0. See text for explanation.
fact, negative flapp is predicted in certain systems that are associated with positive zrt/Zo o o and (large) negative W W ZR/Zo (curves A in Figs. 3, 4 and 5). The formation of ion pairs or ion associates should result in a reduced ion charge number.
3.2. Diffuse-layer rate-determining process The apparent rate constants (k'e,b)Lc defined by Eqs. (17) and (18) were calculated for the interfacial electron transfer processes of four redox couples ZR/Zo o o ((A) 1 + //2 + , (B) 0//1 + , (C) 1 - / / 0 and (D) 2 - / / l - ). Curve b in Fig. 2 was taken for Ark °. The resulting (kte,b)Lc values for Aw ~b°' - A~ q~pzc values of - 2 RT//F, zero and 2 RT//F are plotted in Figs. 6, 7 and 8 respectively. These results indicate that when the plots are fitted to the Butler-Volmer type equation ( kte,b)Lc = ( kte,s)Lc,app × exp[( fl,ppnF/RT)(AWo~b - A~ tk°')]
1
(23)
D
( kte,b)Vc ~---( kte,s)Fc.app
×exp[( flappnF//RT)( AWoq~- AWodp°')]
(22)
the apparent transfer coefficient flapp changes appreciably with the charge number pairs on the redox couples zR//Zo° o and ZR/Zo w w as well as the diffuse layer potentials A th° and A thw. Usually, Ark ° increases and A~bw decreases with increasing A~b. Under these conditions flapp decreases as the charge number for redox couples in the O phase changes from a large negative value to a large O O positive value, i.e. as ZR/Zo changes in the order 2 - / 1 - > 1-/0>0/1 + > 1 +//2+. The opposite behaviour is observed for redox couples in the W phase. In
"
"
2
Fig. 5. Apparent rate constant (ke.b)Fc VS. potential difference ~ for a'~4'°' - v.va~ ~'-,c = 2 R r / r . See text for explanation.
H. Katano et al. / Journal of Electroanalytical Chemistry 396 (1995) 391-396
the apparent transfer-coefficient flapp changes appreciably with Aw
1
~
395
'
'
'
D
0
C. B
O
'~' -2
"~-2
I
I
I
-1
0 n£
1
2
Fig. 8. Apparent rate constant (k'e,b)Lc VS. potential difference ~" curves when Aw~b°' - Aw ~bpzc = 2RT/F. See text for explanation.
1
,
i
'
D
0
,_q -1
6~ 0
"22
I
I
I
-1
0
1
2
Fig. 9. Apparent rate constant (k'e,b)ap p vs. potential difference ~ for Aw ~o, _ Aw ~pzc = 0. See text for explanation.
-3
-2
-1
0
1
2
Fig. 6. Apparent rate constant (k'e,b)Lc VS. potential difference ~ for A~'~&°' - &w ~bp,c = _ 2 RT/F. See text for explanation.
1
%
A
n~ for k'e.,/u°RTK°= 1/5, A~b° = 0.5 A4, ° (curve b in Fig. 2, A~bw = 0 (negligibly small) and Awqb°' - AWthpzc = 0 . Values of ZR/Zo o o of 1 + / 2 + , 0 / 1 + , 1 - - / 0 and 2 - / 1 - were assumed for curves A, B, C and D respectively. In conclusion, the oversimplifications involved in the physical model of the system used in the present paper can be removed whenever necessary. However, before this is done, it seems important to obtain data for charge transfer kinetics and for the electric double-layer structure and to compare them using the simple treatment discussed above.
.ta
Acknowledgements
"5-2
I
I
I
-1
0 nL"
1
2
Fig. 7. Apparent rate constant (k'e.b)Lc VS. potential difference ~ curves when AV~ q~°' - Aw ~bpzc = 0. See text for explanation.
One of us (MS) gratefully acknowledges Professor D.J. Schiffrin, University of Liverpool, for his interest in this study at the Heyrovsky Discussion '94 in Trieste. This work was supported in part by a grant from the Ministry of Education, Science and Culture of Japan.
396
H. Katano et al./ Journal of Electroanalytical Chemistry 396 (1995) 391-396
References [l] [2] [3] [4] [5] [6] [7] [8]
M. Senda, Anal. Sci., 10 (1994) 649. M. Senda, Electrochim. Acta, in press. Z. Samec, J. Electroanal. Chem., 99 (1979) 197. Z. Samec, V. Marecek, J. Weber and D. Homolka, J. Electroanal. Chem., 126 (1981) 105. G. Geblewicz and D.J. Schiffrin, J. Electroanal. Chem., 244 (1988) 291. V.J. Cunnane, D.J. Schiffrin, C.A. Beltran, G. Geblewicz and T. Solomon, 247 (1988) 214. Y. Cheng and D.J. Schiffrin, J. Electroanal. Chem., 314 (1991) 153. Y. Cheng and D.J. Schiffrin, J. Chem. Soc. Faraday Trans. 89 (1993) 199.
[9] V.G. Levich, Dokl. Akad. Nauk SSSR, 67 (1949) 309; 124 (1959) 869. [10] R. Parsons in P. Delahay (Ed.), Advances in Electrochemistry and Electrochemical Engineering, Vol. 1, 1961, p. 52. [11] A.N. Frumkin, Z. Phys. Chem., 164 (1933) 121. [12] Y.I. Kharkats and A.G. Volkov, J. Electroanal. Chem., 184 (1985) 435. [13] A.M. Kuznetsov and Y.I. Kharkats in V. Kazarinov (Ed.), The Interface Structure and Electrochemical Processes at the Boundary Between Two Immiscible Liquids, Springer-Verlag, Berlin, 1987, p. 11. [14] M.A. Marcus, J. Phys. Chem., 94 (1990) 1050, 4152. [15] M.A. Marcus, J. Phys. Chem., 95 (1991) 2010.
,
,
.
ELSEVIER
JOO~L
OF
Journal of Electroanalytical Chemistry 396 (1995) 391-396
Effect of diffuse layer on the rate of electron transfer across an electrolyte lelectrolyte solution interface 1 Hajime Katano, Kohji Maeda, Mitsugi Senda * Department of Bioscience, Fukui Prefectural University, Matsuoka-cho, Fukui 910-11. Japan Received 21 February 1995; in revised form 20 March 1995
Abstract
The basic equations of the kinetics of electron transfer across an electrolyte [electrolyte solution interface, in which the effect of diffuse layers on both sides is taken into account on a simplified model of electric double layer, are presented. Two ideal processes, an inner-layer rate-determining process and a diffuse-layer rate-determining process, are discussed. Theoretical predictions derived from the basic equations are discussed. Some abnormal transfer coefficients can be explained by the theory. Keywords: Diffuse-layer effects; electron transfer; electrolyte lelectrolyte solution interface
1. Introduction In previous papers [1,2] we have derived the basic equations for the rate of charge (ion or electron) transfer across an interface between two immiscible electrolyte solutions, i.e. an oil [water (O [W) interface, in which the effect of the diffuse (double) layer on the apparent or observed rate constants of the charge transfer kinetics was taken into consideration. The electric double layer is assumed to consist of an inner layer sandwiched between two diffuse layers on each side of the interface, and the charge transfer reaction is assumed to take place between two chemical species at the planes of contact of the inner layer with the two diffuse layers. Steady-state ionic transport in the diffuse layers (Levich correction) was also assumed. The equations derived were discussed for two ideal processes: the inner-layer rate-determining process (Frumkin correction) and the diffuse-layer rate-determining process. Basic equations for the rate of charge transfer across the O IW interface have been presented by Samec [3], who discussed the effect of the diffuse layer on the kinetics in terms of the Frumkin theory. Samec et al. [4] have determined the rate constants of electron transfer between ferrocene/ferrocene + in the nitrobenzene (NB) phase and
Fe(CN) 4-/Fe(CN)63- in the aqueous (W) phase at the NB [W interface. Recently, Schiffrin and coworkers [5-8] have reported kinetic data for the interfacial electron transfer between a redox couple in the 1,2-dichloroethane (DCE) phase (i.e. the lutetium diphthalocyanine redox couple [Lu(pc)~"/2 ÷ (O)], the bis-(pyridine) (meso-tetraphenyiporphyrinato)ruthenium(II) redox couple [Ru(tpp)(py)°/÷ (O)] or the 7,7,8,8-tetracyanoquinodimethane redox couple [TCNQ-/°(O)]) and a redox couple in the water phase, (i.e. Fe(CN) 4 - / 3 - ) when the redox couple in the W phase is present in excess compared with the redox couple in the O phase. One of the interesting results of these studies is that the transfer coefficients of the charge transfer kinetics when they are expressed in terms of an equation of the Butler-Volmer type are not constant but change appreciably as the applied interfacial potential difference, is changed, The purpose of this paper is to report some theoretical results, based on what we consider to be reasonable assumptions, predicted by the fundamental equations previously derived [2] for the apparent or observed rate constants of interfacial electron transfer. It is shown that some of the abnormal apparent transfer coefficients can be explained by the present theory. 2. Theoretical
i Dedicated to Professor K. Honda, Professor H. Matsuda and Professor R. Tamamushi on the occasion of their 70th birthdays. * Corresponding author. 0022-0728/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved SSDI 0 0 2 2 - 0 7 2 8 ( 9 5 ) 0 4 0 3 6 - 6
We consider the polarized O IW interface illustrated in Fig. 1. A layer of solvent molecules (the inner layer
H. Katano et al. / Journal of Electroanalytical Chemistry 396 (1995) 391-396
392 ..tW
The rate constants k'e,b and k'e,f c a n be expressed by a Butler-Volmer type equation ( a + fl = I): k;, b = k;.bs exp[ 13(nF/RT)(AW ~b2)] k'.e = k'.fs e x p [ - a ( n F / R T ) ( AW~b2)] t
(5)
t
In these equations ke.bs and ke.fs are the rate constants at aw = - 6o = 0.
"~0
xo
xo xw
Now, if steady-state ionic transport in the diffuse layers (Levich correction) [9,10] is assumed, the flux Je can be expressed by the kinetic equation with respect to the surface concentrations of Red(O) and Ox(O) at x = x °, as given by [2]
x ow
>X Fig. 1. Electric double layer across a polarized O l W interface. The potential differences are defined by At4, w -_ 4'0w - 4'0, act = 4'2o - 4,0o and a 6 w = 4'w _ 4'w.
between x ° and x w where x is the coordinate perpendicular to the interface) separates two ionic space-charge regions (the diffuse layers between x ° and x ° in the O phase and between x w and x0w in the W phase) on each side of the interface; x ° and x0w are the coordinates of the planes, beyond which (i.e. at x < x ° in the O phase and x0w < x in the W phase) the ionic transport is diffusion controlled. Also, we let the potential ~b and the concentration Ci of ion i be th°, th°, tkw and q~0 w and cOo, , cO, , cw /,2 and c Wi.0at x °, x °, x w and x~v respectively. We assume (l) that there is sufficient supporting electrolyte on both sides of the interface and (2) that the structure of the double layer is not affected by the flow of current or ionic transport across the interface. We consider an electron transfer process across an O lW interface between two redox couples Red(O)/Ox(O) and Red(W)/Ox(W) at x ° and x w respectively: Red(O) + Ox(W) = Ox(O) + Red(W)
(1)
Let the charge numbers, including the sign, of the chemical species (ions) Red(a) and Ox(ct) be z~ and zg respectively (et = O, W), and let the number of electrons transferred in Eq. (1) be n. Then we have Z° --ZR° = Z ~ --Z w = n
(2)
In the following, we restrict our discussion to the case where the concentrations of the ions Red(W) and Ox(W) are much higher than those of Red(O) and Ox(O), so that the concentration distribution of Red(W) and Ox(W) can be assumed to remain constant during the electron transfer process of Eq. (1). This is actually the case that is adopted in many experimental studies. Thus Eq. (1) can be replaced by the pseudo-first-order reaction: Red(O) = Ox(O) + ne- (W)
(3)
and the flux Je of electrons transferred across the interface with respect to the surface concentrations of Red(O) and Ox(O) at x = x ° is given by the kinetic equation t O p O J~ = ke,b%,2 - ke,eCo, 2
(4)
Je "= ( ke,b)appCR, ' o 0 -- ( k e, , f ) a p p C oo. 0
(6)
where the apparent or observed rate constants (k'e,b)app and (kle,f)app are given by ( k'e,b)Fo
(Ze,0app =
1 + [k;.b/( -A°R.2 )] + [k;.J( -A°o.2 )] (k;,,)Fc
(ke'f)app
1+
t O [ke.b/(--AR.2)
t O ] + [ke.f/(-Ao.2) ]
(7)
where !
(ke,b)Fc
__ --
!
ke,bRe
(k'e, )Fc = k'e,Re
(8)
and R e = exp[( z ° + fin)( F / R T ) ( - A d ~
°)
+ ( zw -/3n)(F/RT)(-A4~W)]
(9)
In Eq. (7)
a°, 2 = ( u°Rr/Z? )
z°F/Rr)A 6 ° ]
exp[(
( i = O, R)
(10)
where
z°= JxOfX°exp[ ( z°F/RT)4~] dx
(ll)
and u ° (i = R,O) is the mobility of ion i in the O phase. The terms A~b° ( = ~b° - th° ) and Aft w ( = th~v - ~b0 w) are the diffuse-layer potentials in the O phase and the W phase respectively. By introducing a factor A° defined by At = [ k,e.b/( --AR,2)] O + [ k'e , f / / ( - A°0 , 2 ) ]
(12)
we can rewrite Eq. (7) as (k'e.b)ap p = ( k ' e , b ) F c / ( 1
'
-
(ke,f)ap p -
+ A° )
k' (
e,f)vJ(
(t3) 1 +
A° )
The factor A° represents the ratio of the rate constant of electron transfer across the inner layer to that of mass transfer (diffusion and migration) of ionic species across the diffuse layer. Thus we have two ideal processes depending on whether )t° << 1 or )t° >> 1.
H. Katano el al. / Journal of Electroanalytical Chemistry 396 (1995) 391-396
393
2.1. Inner-layer rate-determining process (Frumkin theory) When A° << 1, i.e. the second and third terms in the denominators are negligibly small compared with unity, we obtain the rate equations for the case where the charge transfer across the inner layer is rate determining. Clearly, this is the case of the Frumkin theory for the double-layer ~,orrection in electrode processes [3,10,11]. Then the appar,nt rate constants are given by (ke,b)Fc and (ke,f)pc, as defined by Eq. (8), which are rewritten as
% a b
.
C
k'e.b)ap p = (k'e.b)Fc = k'e,s exp(
fln~)R e
0
(14)
i k'e.f)ap p = ( k ' e , f ) F c = k'e,s e x p ( - a n ¢ ) r e
F / R T ) ( A wo * - Aw 4,~')
(15)
.rod k'e,s is the rate constant of electron transfer at Aw 4, = W o~ -1o 4, e, where AWo 4 'o¢, is the standard potential of electron xansfer represented by Eq. (3). Also, the equilibrium ?otential ((Aw4')eq!~at which Je (defined by Eq. (4)) is zero, is related to A o 4' e by (Aw4')eq = Aw4,°'e +(RT/nF)ln(c°.o/C°,o)
(16)
2.2. Diffuse-layer rate-determining process When A° >> 1, i.e. the second and third terms in the denominators are much larger than unity, we obtain the rate equations for the case where the ion transfer across the diffuse layer is rate-determining. Then we have k;,b)app = (k;,b)Lc exp( nsr ) ( - e x p ( n~" ) / a ° 0) + ( - 1 / a ° . 0 ) k;.f)ap p
=
(ke~f)Lc
1
= [ - exp( n¢ )/A°.o] + ( - 1/A°o)
(17)
4
6
8
A~¢p~
Fig. 2. Diffuse layer potential A~b ° vs. potential difference AW~b. Curves a, b and c are defined in the text.
potential gradient throughout the diffuse layer between x ° and x ° to evaluate A°i,2 (see below). In this study we assume a simplified model of the electric double layer in which the surface charge density o"w on the W phase of the interface changes linearly with Av~4, and the potentials of the outer Helmholtz planes are given by Gouy-Chapman theory without specific adsorption. Furthermore, on assuming that the potentials of outer Helmholtz planes are equal to A 4,0, we obtain curves of A4, ° vs. (A~4,-AW4,pzc ) curves, where AW4,pzc is the potential of zero charge. The results are shown in Fig. 2, where curves a, b and c are the curves calculated for the concentrations of a 1:1 supporting electrolyte in the O phase with c ° values of 4 mM, 16 mM and 400 mM respectively for tr w --4 IxC cm -2 at (F/RT)(Aw4, AW4,pz¢) = 40 and where the dielectric constant ¢o of the O phase is 10.37 at 25°C. On assuming a linear potential gradient in the diffuse layer with a diffuse layer potential of A 4,0 and a thickness of l / K °, we obtain the following equation for A°i,2"•
At: (u°RT// at i,2 - 1 / K O j I I _ e x ' ~
)
(19)
-a°)
where A°,,0 = a ° , ,2 e x p [ - ( z ° F / R r ) ( a 4 , ° ) ]
2
(FIRTX,dWo¢ -
.vhere "= (
0
(18)
where
a ° = ( z i ° F / R T ) ( A 4 , °)
( i = R , O)
(20)
Accordingly, after rearrangement Eq. 12 can be rewritten as
3. D i s c u s s i o n In evaluating the effects of the diffuse layer on the apparent rate constants of electron transfer across an O [W interface the mass transfer rate constant A°i,2 (i --- O,R), as defined by Eq. (10) with Eq. (11), must be calculated. When the diffuse layer is predicted by Gouy-Chapman theory, Eq. (11) is transformed into a convenient form [2] and A°i,2 is given as a function of the diffuse layer potential A4, ° and the Debye reciprocal length K° as well as the ionic mobility u ° and the number of charged ions z °. Another approach to this problem is to assume a linear
Ao =
k'~
u° RTK °
exp ¢tn~' + fl
--A4, ° + A4, w
1 - exp(-
x
aO
+exp[-an~'
x
aO
- a
+ A 4,w)]
394
H. Katanoet al. / Journal of ElectroanalyticalChemistry396 (1995) 391-396 2
-2
-1
J
,
0
1
j
2
-2
Fig. 3. Apparent rate constant (k'e,b)Fc vS. potential difference ~" for A~° ~o, _ A *° ~ = _ 2 R r / F. See text for expla.atio..
where u ° = (u°)"(u°) t3 is the average mobility of ionic species in the O phase and ~" = ~"- ( R T / n F ) l n ( u ° / u ° ) . Thus the factor A° can be evaluated from the ratio of k'e,s to u°RTK o, i.e. the ratio of the standard rate constant of electron transfer across the inner layer to the rate constant of diffusion across the diffuse layer. In the following we take u ° = u ° = u ° so that ~ " = ~'.
3.1. Inner-layer rate-determining process (Frumkin theory) The apparent rate constants (kre,b)Fc defined by Eq. (14) and Eq. (9) were calculated for the interracial electron transfer processes of four redox couples zr~/z o o ((A) 1 + //2 + , (a) 0//1 + , (C) 1 - / / 0 and (D) 2 - / / 1 - ) in the O phase and a single redox couple zR//Zo w w =4-/3in the W phase. Curve b in Fig. 2 is for A 4)°, while curve c is for A~bw; curve c can be regarded as representing the potentials of the outer Helmholtz plane in the W phase when the concentration c w of 1 : 1 supporting electrolyte is 53 mM, t r w = 4 /xC c m - : with (F//RT) (AWqbAWq~pzc)=40, and 8 w =78.38 at 25°C. The resulting (k'~.b)vc values for fl = 0.5 are plotted for A~b °' - Av~~bpzc values of - 2 RT//F, zero and 2 R T / F in Figs. 3, 4 and 5 respectively. These results indicate that when the plots are fitted to the Butler-Volmer type equation
-1
0 z2~"
1
2
Fig. 4. Apparent rate constant (k'e,b)~c vs. potential difference ~' for A~ ~0, _ Aw ~pzc = 0. See text for explanation.
fact, negative flapp is predicted in certain systems that are associated with positive zrt/Zo o o and (large) negative W W ZR/Zo (curves A in Figs. 3, 4 and 5). The formation of ion pairs or ion associates should result in a reduced ion charge number.
3.2. Diffuse-layer rate-determining process The apparent rate constants (k'e,b)Lc defined by Eqs. (17) and (18) were calculated for the interfacial electron transfer processes of four redox couples ZR/Zo o o ((A) 1 + //2 + , (B) 0//1 + , (C) 1 - / / 0 and (D) 2 - / / l - ). Curve b in Fig. 2 was taken for Ark °. The resulting (kte,b)Lc values for Aw ~b°' - A~ q~pzc values of - 2 RT//F, zero and 2 RT//F are plotted in Figs. 6, 7 and 8 respectively. These results indicate that when the plots are fitted to the Butler-Volmer type equation ( kte,b)Lc = ( kte,s)Lc,app × exp[( fl,ppnF/RT)(AWo~b - A~ tk°')]
1
(23)
D
( kte,b)Vc ~---( kte,s)Fc.app
×exp[( flappnF//RT)( AWoq~- AWodp°')]
(22)
the apparent transfer coefficient flapp changes appreciably with the charge number pairs on the redox couples zR//Zo° o and ZR/Zo w w as well as the diffuse layer potentials A th° and A thw. Usually, Ark ° increases and A~bw decreases with increasing A~b. Under these conditions flapp decreases as the charge number for redox couples in the O phase changes from a large negative value to a large O O positive value, i.e. as ZR/Zo changes in the order 2 - / 1 - > 1-/0>0/1 + > 1 +//2+. The opposite behaviour is observed for redox couples in the W phase. In
"
"
2
Fig. 5. Apparent rate constant (ke.b)Fc VS. potential difference ~ for a'~4'°' - v.va~ ~'-,c = 2 R r / r . See text for explanation.
H. Katano et al. / Journal of Electroanalytical Chemistry 396 (1995) 391-396
the apparent transfer-coefficient flapp changes appreciably with Aw
1
~
395
'
'
'
D
0
C. B
O
'~' -2
"~-2
I
I
I
-1
0 n£
1
2
Fig. 8. Apparent rate constant (k'e,b)Lc VS. potential difference ~" curves when Aw~b°' - Aw ~bpzc = 2RT/F. See text for explanation.
1
,
i
'
D
0
,_q -1
6~ 0
"22
I
I
I
-1
0
1
2
Fig. 9. Apparent rate constant (k'e,b)ap p vs. potential difference ~ for Aw ~o, _ Aw ~pzc = 0. See text for explanation.
-3
-2
-1
0
1
2
Fig. 6. Apparent rate constant (k'e,b)Lc VS. potential difference ~ for A~'~&°' - &w ~bp,c = _ 2 RT/F. See text for explanation.
1
%
A
n~ for k'e.,/u°RTK°= 1/5, A~b° = 0.5 A4, ° (curve b in Fig. 2, A~bw = 0 (negligibly small) and Awqb°' - AWthpzc = 0 . Values of ZR/Zo o o of 1 + / 2 + , 0 / 1 + , 1 - - / 0 and 2 - / 1 - were assumed for curves A, B, C and D respectively. In conclusion, the oversimplifications involved in the physical model of the system used in the present paper can be removed whenever necessary. However, before this is done, it seems important to obtain data for charge transfer kinetics and for the electric double-layer structure and to compare them using the simple treatment discussed above.
.ta
Acknowledgements
"5-2
I
I
I
-1
0 nL"
1
2
Fig. 7. Apparent rate constant (k'e.b)Lc VS. potential difference ~ curves when AV~ q~°' - Aw ~bpzc = 0. See text for explanation.
One of us (MS) gratefully acknowledges Professor D.J. Schiffrin, University of Liverpool, for his interest in this study at the Heyrovsky Discussion '94 in Trieste. This work was supported in part by a grant from the Ministry of Education, Science and Culture of Japan.
396
H. Katano et al./ Journal of Electroanalytical Chemistry 396 (1995) 391-396
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