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The limiting diffusion current value on a macroscopically inhomogeneous electrode
THE LIMITING DIFFUSION CURRENT A MACROSCOPICALLY INHOMOGENEOUS V. Yu.
Institute of Electrochemktry, (Received
Academy
26 October
of
VALUE ON ELECTRODE
FILINOWKY Sciences
of the U.S.S.R., Moscow V-71,
U.S.S.R.
1978; in revised form 8 June 1979)
Abstract- The limitingdiffusioncurrentI ,# to an electrode in the form of system of alternatingactive and inactivesiteshasbeencalculated.It is shown thatI., _.. = I? MV. 8). Here 1: is the limitinadiffusion current to the sameelectrode with an all-active surface. The co&&ion factor p(Z?.e) depends on the degree of the surface activity 0 and on the distribution of active sites N along the surface. The factor p(N, 0) does not depend on the intensity of stirring of electrolyte solution.
INTRODUCTION
spite of special methods of preparation, the surface of solid electrodes proves to be inhomogeneous as regards its electrochemical activity. Unlike the microscopic inhomogeneity, which is associated with the phenomena at the zitomic and molecular levels, the macroscopic inhomogeneity can be caused by traces ofimpurities, gas bubbles, fractions of the crystal faces on the surface, formation of insoluble products, etc. Let us assume the characteristic size of the inhomogeneity being considered, I, to be comparable with the thickness of the diffusion boundary layer, S,, formed near the electrode surface during transfer of electrochemically active components. Let us also assume, however, that I is much less than the thickness of the hydrodynamic boundary layer 6,. Therefore, we can consider the pattern of the solution motion near the surface to urrdergo no appreciable change. The appearance on the electrode surface of sites with dimerent electrochemical activity leads to a change (compared with a homogeneous electrode) in the distribution of the reagent concentraiions. There have been several attempts to take quantitative account of the influence of macroscopic surface inhomogeneity on the limiting diffusion current value, I,. The results obtained by different authors are briefly described in[i]. The majority of authors, for example[2,3] proceed from the assumption that there exists a diffusion boundary layer near the electrode surface, which is the same thickness 6, (Nernst’s model) regardless of the transformations occurring on In most
cases, in
0
r;
-.,, _ %I
it. The results of these model calculations depend significantly on the value of 6,, ie the intensity of stirring. The results of model calcula,tions[2,3] are somewhat at variance with those obtained by a consecutive solution for a number of the simplest systems of the problem of the mutual influence of the active surface sites during convective diffusion. Thus, in calculating the interaction of the rotating disc and ring electrodes[4], a rotating two-ring electrode[5-71, two electrodes in a channel, in a tube, on the surface of a cone or a wedge[S] it was established that the mutual influence of two active sites is determined by strictly geometrical factors alone and does not depend on the velocity of motion of the solution. The thickness of the diffusion boundary layer 6, varies along an active surface site and can not be considered the same over the whole electrode. For this reason, it is expedient to reliniquish Nemst’s model with its limitations and to calculate the limiting diffusion current on the electrode surface with a regular macroscopic inhomogeneity. FORMULATION
OF
THE
PROBLEM
For simplicity, let us consider a two-dimensional problem assuming the working electrode oflength L to constitute a system of alternating active and inactive bands. The width of the active sites is I and of the inactive ones d. If N is the total number of inhomogeneity sites on the electrode surface and m is the number ofinhomogeneity sites per unit length, then we have
__-_
~
w I
X.2
!+-lJcdJcl--J
xrn.1
x
Z&2-d4
Fig. 1. The model of the macroseopicallyinhomogeneouselectrode. 309
V. Yu.
310 L = N(I + d) = Lm(i + d).
FILINOVSKY
(1)
= nFk,(x)C,
Denoting by B the fraction of inactive surface, we have e = d/(l + d).
(2)
Let the axis OX pass along the working electrode surface and the coordinates corresponding to the beginning and the end of the active site have values XL and x,,,, ie
’ -I %I - xm - I %I+1
(3a)
d.
-%#I=
(3b)
Let us assume that the working electrode is located on the wall of a plane-parallel channel far removed from the entrance. If the distance the entrance to the electrode (OX;) is much more than the channel height the electrolyte flow near the electrode has a fully developed laminar regime. The velocity distribution has the Poiseuille profile. For liquids (when Schmidt number SC z+ l), the diffusion effects are concentrated in a very thin layer adjacent to the solid surface. We suppose therefore that the thickness of this layer is much smaller than the channel height. The velocity distribution in this region can be described by a linear profile U(Y) =
(4)
7YlK
where U is the x-component of the flow velocity of the liquid, 7 is the tangential stress of the friction on the channel wall, p is the dynamic viscosity, y is the distance from the wall. For a fully developed huninar flow 7 does not vary along the electrode length. Our results can be used, generally speaking, under the other hydrodynamical conditions, if the tangential stress of the friction z (or the thickness of the hydrodynamical boundary layer a,-,)does not vary too much along the electrode length, the mathematics is rather complicated, but the concluding results will be approximately the same. Let us assume that on active surfacc sites the electrochemically active solution component enters into a fast reaction occurring under diffusion control and accompanied by transfer of n electrons. The generalization to the case of a redox reaction is easy[3,5]. The influence of the limited value of the rate constant of the electrochemical transformation, which is quite considerable near the ends of active sites, requires special consideration. The reagent transfer in the solution is described by the convective diffusion equation*
at
x > 0, y = 0.
(6c)
Here C, C, and Co are the values of the reagent concentration near the electrode, on its surface and in the solution bulk, respectively, D is the diffusion coefficient, i = i(x) is the local value of the current density. The boundary condition (6c) characterizes the kinetics of the electrochemical transformation on the electrode. In the case of inhomogeneous surface under consideration k, = k,(x) varies along the surface: on the active sites k, + 00, on the inactive ones k, = 0. METHOD
OF SOLUTION
In the case of a homogeneous highly active electrode k, = 03 along the entire surface. For this case the
limiting diffusion current distribution was found a long time ago[8] and is the form nFDC’(3)“j
jl” =
l-(#)(Dp/~)“~(x
-
(7)
x’#‘~’
where I’(f) is the gamma-function of the corresponding argument. The total limiting current to a homogeneous electrode is obtained by direct integration of (7) and equal to
s
xi+L
II” =
iO II
dx
=
xi
nFDC0(3)“3 _3 ~z/j. r(~Wcc/W3 2
(8)
It can be readily shown[5,6] that in the case of arbitrary activity distribution along the electrode surface. the current densitv distribution ilxl can be found by solving the integial equation beI& ‘tx)
=
nFk*CO
I
’
-
(W7)l’3 ,,FDC‘,(3)“3r(f)
x
1;
(9)
(:(l$J
To simplify the reduced relations let us write * ~ nFDC0(3)1’3
” = I-($)(Dp/t)"3
I
il0)
’
and introduce the new functions g==i/h;
g, = $/A
= (x - x;)-l’3;
Go = 1,0/A = +Lri3.
(11)
(9) assumes the form
Now
with the boundary conditions c+co
at
x,O.y+cg
0%)
C = CO
at
x = 0, y a0
(6b)
+ We supposehere that the solutioncontains an excess of an indifferentelectrolyte.fn this case the migration term in the convective diffusionequation can he discardedand the equation takes a simple form.
In order to obtain the equations describing the current distribution on the first, second and subsequent active sites (gl, gr,. . . , gn,. . . , respectively) it is sufficient to give the function k,(x) concrete vales on active and inactive sites. Then we shall obtain the following sequence of integral equations :
,b
l-zx
s =
ri
(x
g,dC -
#I3
=
O
at
X>
Xi
The limiting diffusion
current
value
on
a macroscopically
electrode
inhomogeneous
311
and
at
xz
+
s xi
X>
Xi
(12)
gzdt (X _
&y/3
+
. . .
dc-
x -!-
at
x
>
. .
x;.
s &I
In a consecutive solution of (12) it is convenient to use the following formula for reversal of Abel’s integral equationC9) :
at
xr&.
However, integration of (Ma) gives
if
cp(s)= then
s
‘ S(t)dt p (s - ty at
(13)
9m=!L-i
x>x;.
+&
(174
d5
Fog gl(x) we find
at
x>x&.
The function G(s) below contained in the integrands The remaining functions gz(x), . . . , g,,,(x), . . . are written as
g2(x) =
$
s
x(x -d5w3
xi
G(s)=
,h’
z
dt s 0 @(l
A,*
+ t)
I+$
(1 + PJ3
3 + %iuctan
at x=-xi
29’3
-
J5
1
I
+2
was examined in detail in[lO] and shows the following asymptotic behavior :
3JJ;
G(s) a~ -pz’3
+ ...
3Js G(s) z 1 - 2re~
-l/3
at +
0
-413
+
Wa)
_ )
4tS
at Equations (15) can be given a somewhat different, but equivalent form:
s<
s>> 1.
(18b)
RESULTS AND DISCUSSION
In order to determine the total current IdN to a macroscopically inhomogeneous electrode equal to L I dN = id dx s0
s L
=A
1
at
x>x;.
Integrating (14) and (15), we obtain Sl =
1 (x - x:1”3
at
x>x;
gdx=A
0
(16)
s
5 xmg_dx,
Xl=1
X&#
(19)
it is necessary to find the current distribution on a sequence of active sites. Unfortunately, though consecutive integration of (17) or (17a) theoretically presents no difficulties, it cannot be performed in an analytical form. Therefore it is reasonable to analyze
V. Yv.
312
FILINOVSKY
only the two limiting cases: (1) low activity surface (0 z 1) where the extent of active sites is not large and they are located far apart (1 <.cd), (2) highly active surface (0 << 1) where wide active sites are separated by narrow.inactive ones (I X- d). We shall not dwell here on a successive mathematical analysis of the expressions for g2,. . . , g,,,,. . . , but shall give qualitative estimates for IdN in each of the limiting cases and explain the physical significance of the results. (1) Low uctiuity sur-ce (0 z 1) In this case the current distribution on active sites can be conveniently examined by means of (17). Integrating (16) and (17) along the active sites and expressing distances in terms of 1 and d, we obtain :
s XI
G,
=
g,
dx = ;1213.
(20)
Xi
G2 =
gz dx
,3p3 2
s I
0
s
sl(x;+CJG(l+.:_i)dL
(21)
%I
G,
=
8,
dx
4m
3 =- 12’3 2
gt(x\ + 0
\ \
I
’ -2---I
Fig. 2. The
The deduction of (23b) can be based on the following considerations: the current produced on the mth active site is weakened by the action of (m- 1) preceding active sites. The decrease caused by each active site is proportional to I,, (ie (22)). If the sites are located sufficiently sparsely, their mutual influence should decrease, which leads to appearance in the second term of the factor (l/d)2’3. At the same time, it is clear that the nearest active site exerts the greatest influence, the effect of more remote sites being weaker. All this leads to the dependence of N413 on the total number ofactive sites. The proportionality factor n can be calculated by means of a rigorous quantitative analysis (21). Let (23b) have the form generally used in such calculations
x [l - .N~‘s(1/d)2’s],
(22)
The current value given above (23a) is somewhat too high since no account is taken in it of the gradual current decrease on active sites due to their mutual influence. It can be shown that at I -CCd this decrease is described as follows
[2/3N4’3([/&2’3
_
_. . _
aN1’3(l/d)2’3],
I& zz I,D. N’J3(l/d)zi3
which corresponds to (20) and the first terms in (21). The resulting total current produced by N such sites is
$
213
or
It can be seen from (17) that the current distribution on the second, third, etc, sites downstream is distorted by the preceding active sites. A qualitative picture of this distribution is given schematically in Fig. 2. The broken lines show the current distribution to be observed on a separate active site of the same size. It is evident that each active site, operating indcpendently of the others produces the current
_
d---
current distributionon the lowly active surface.
x [i -
( (m - 1)(1+ d) - C> dC - . . . 1 ,.d~~-: +W(,+:_c)dT. s0 x G
;P,
I
-I’ -T-
d-
5
ItN = I; . N(I/L)
Id1 = A .
\ G \
\
Wb)
(24~
since it can be assumed that N z L/d at d >> 1. According to (2) i/d = (1 - 0)/O. Substituting this relation into (24), we find r;ik z z$
. N”’
1 - ep3
y--x
[I -c~N”“(3’~]_
(25)
As it follows from (25), the mutual influencebfactive sites, described by the second term depends’not only on their relative positions (fl), but also on their total number (N). Even for narrow active sites (0 =Z 1), the depletion of the solution they cause increases downstream and a&&s markedly the value of the total current. Therefore relation (25) is valid only at
<< 1.
(26)
In the opposite case of importance are the consequent terms in the expansion (24). 2. Highly active sur$ace (f3 -x 1) In this case the appearanon the electrode surface of sparsely located inactive sites changes only slightly
The limiting diffusion current value on a macroscopically inhomogeneous electrode
Z dN =
313
ZH dN -ILdN Iff
+
(291
IiN9
and has the correct asymptotic behaviour in the cases of low and high active surfaces. Using (25) and (28) for the values of the total current liN (at 0 z 1) and I& (at 0 -SC1) we obtain IjN~‘I(!$~”
.,;(I
_
77
I dNF~~N11.(~)113+Z~(~-~)
x3
x3
Fig. 3. The current distribution on the highly active surface. the general pattern of current distribution over the electrode surface. A quantitative description of the change in the current density over the electrode can be obtained by means of (17a). Figure 3 gives a qualitative picture of current distribution in the case under consideration. As can be seen from Fig. 3, immediately after an inactive site the current density increases due to accumulation of matter in the stream. The perturbations in current distribution are local and disappear quickly. The current on the active site is calculated by integrating (17a)
J
d
+
dL
0
G, =
J
1
(27)
gm- dx6 + WC
0
J
d
+
dl.
0
Evidently, the total current to a highly active inhomogeneous electrode Z,” should be less than the current to a homogeneous high active electrode Zj. This decrease seems to be prc$ortional to the current which would flow through inactive sites. For the mth inactive site located downstream this current is z hd/(ml)“3. It should he taken into account that the influence of the inactive site depends on its position on the electrode surface and on the relation between d and 1.Taking into consideration the above factors, we can write
This interpolation formula can be written in the other form which shows directly the dependence of the total current on the velocity of convective flow :
&,N = ~,OPW,W
(30)
Here the function p(N, 0) is :
dN,f’) =
CONCLUDING REMARKS As can be concluded from (31), the reduction of the total current due to the inactive sites on the electrode surface is described by the function p(N, 6). The value of p depends on the fraction of inactive surface (0) and the number and location of active sites on the electrode surface. The function p(N, 0) characterizes the mutual influence of the active sites. Figure 4 gives the functions p(N, 0) calculated for several values of N. It is necessary to emphasize that the dependence of the total current to a macroscopically inhomogeneous electrode, Id&.,on the velocity ofaconvective flow is the same as the total current to a homogeneous highly active electrode, It. In the above case, for instance, the limiting diffusion current I,” to a homogeneous electrode is proportional to @, where t is the tangential stress of the friction on the channel wall. The same dependence of the total current I,, on the value of r remains in the case of a inhomogeneous electrode. Inactive sites on the electrode surface change only the numeral factor.
or
IH 17%’ In deducing (28) it was assumed that N B L/1 and tIz d/l. The proportionality factor %?was evaluated by direct analysis of (27) and was found to be W’ss0.4. For intermediate values of 8 an interpolation formula can be used which is based on the two limiting values of (23a) and (28). The interpolation formula for total current is written as follows
0.5
a
Fig. 4. The behavior of the p(N.0) function at different number of the active sites N.
V. Vu. FILINOVSKY
314 REFERENCES
1. V. Ju. Filinovsky
and Ju. V. Pleskov
in Progress in Surface and IMembrane Science (Edited by P. Danielli) Vol. 10, pp. 27-116, Pergamon, New York (1976). 2. R. Landsberg and R. Thiele, Elecrrochim. Acta 11, 1243 (1966). 3. E. Levart, D. Schuhmann and 0. Contanin, J. elmmoanal. Chem. 70, 117 (1976). 4. B. G. Lcvich. Physicochemicol Hydrodynamics, PrenticeHall, Engelwood Cliffs, New Jersey (1962).
5. H. Matsuda, J. electroanal. Cbem. 16, 153 (1968). 6. V. Ju. Filinovsky, I. V. Kadija, B. 2. Nikolic and M. V. Nakic, J. electroam/. Chem. 54, 39 (1974). 7. A. Suwono. M. Daguenet and D. Bodiof int. J. Heat Mass Tmnsfir 19, 239 (1976). 8. M. A. Leveque, Annis Mines Carbur., Paris 13,201(1928). 9. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Vol. 1, Cambridge University Press (1927). 10. W. J. Albery and S. Bruckenstein, Trans. Faraday Sot. 62, 1920 (1966).
Institute of Electrochemktry, (Received
Academy
26 October
of
VALUE ON ELECTRODE
FILINOWKY Sciences
of the U.S.S.R., Moscow V-71,
U.S.S.R.
1978; in revised form 8 June 1979)
Abstract- The limitingdiffusioncurrentI ,# to an electrode in the form of system of alternatingactive and inactivesiteshasbeencalculated.It is shown thatI., _.. = I? MV. 8). Here 1: is the limitinadiffusion current to the sameelectrode with an all-active surface. The co&&ion factor p(Z?.e) depends on the degree of the surface activity 0 and on the distribution of active sites N along the surface. The factor p(N, 0) does not depend on the intensity of stirring of electrolyte solution.
INTRODUCTION
spite of special methods of preparation, the surface of solid electrodes proves to be inhomogeneous as regards its electrochemical activity. Unlike the microscopic inhomogeneity, which is associated with the phenomena at the zitomic and molecular levels, the macroscopic inhomogeneity can be caused by traces ofimpurities, gas bubbles, fractions of the crystal faces on the surface, formation of insoluble products, etc. Let us assume the characteristic size of the inhomogeneity being considered, I, to be comparable with the thickness of the diffusion boundary layer, S,, formed near the electrode surface during transfer of electrochemically active components. Let us also assume, however, that I is much less than the thickness of the hydrodynamic boundary layer 6,. Therefore, we can consider the pattern of the solution motion near the surface to urrdergo no appreciable change. The appearance on the electrode surface of sites with dimerent electrochemical activity leads to a change (compared with a homogeneous electrode) in the distribution of the reagent concentraiions. There have been several attempts to take quantitative account of the influence of macroscopic surface inhomogeneity on the limiting diffusion current value, I,. The results obtained by different authors are briefly described in[i]. The majority of authors, for example[2,3] proceed from the assumption that there exists a diffusion boundary layer near the electrode surface, which is the same thickness 6, (Nernst’s model) regardless of the transformations occurring on In most
cases, in
0
r;
-.,, _ %I
it. The results of these model calculations depend significantly on the value of 6,, ie the intensity of stirring. The results of model calcula,tions[2,3] are somewhat at variance with those obtained by a consecutive solution for a number of the simplest systems of the problem of the mutual influence of the active surface sites during convective diffusion. Thus, in calculating the interaction of the rotating disc and ring electrodes[4], a rotating two-ring electrode[5-71, two electrodes in a channel, in a tube, on the surface of a cone or a wedge[S] it was established that the mutual influence of two active sites is determined by strictly geometrical factors alone and does not depend on the velocity of motion of the solution. The thickness of the diffusion boundary layer 6, varies along an active surface site and can not be considered the same over the whole electrode. For this reason, it is expedient to reliniquish Nemst’s model with its limitations and to calculate the limiting diffusion current on the electrode surface with a regular macroscopic inhomogeneity. FORMULATION
OF
THE
PROBLEM
For simplicity, let us consider a two-dimensional problem assuming the working electrode oflength L to constitute a system of alternating active and inactive bands. The width of the active sites is I and of the inactive ones d. If N is the total number of inhomogeneity sites on the electrode surface and m is the number ofinhomogeneity sites per unit length, then we have
__-_
~
w I
X.2
!+-lJcdJcl--J
xrn.1
x
Z&2-d4
Fig. 1. The model of the macroseopicallyinhomogeneouselectrode. 309
V. Yu.
310 L = N(I + d) = Lm(i + d).
FILINOVSKY
(1)
= nFk,(x)C,
Denoting by B the fraction of inactive surface, we have e = d/(l + d).
(2)
Let the axis OX pass along the working electrode surface and the coordinates corresponding to the beginning and the end of the active site have values XL and x,,,, ie
’ -I %I - xm - I %I+1
(3a)
d.
-%#I=
(3b)
Let us assume that the working electrode is located on the wall of a plane-parallel channel far removed from the entrance. If the distance the entrance to the electrode (OX;) is much more than the channel height the electrolyte flow near the electrode has a fully developed laminar regime. The velocity distribution has the Poiseuille profile. For liquids (when Schmidt number SC z+ l), the diffusion effects are concentrated in a very thin layer adjacent to the solid surface. We suppose therefore that the thickness of this layer is much smaller than the channel height. The velocity distribution in this region can be described by a linear profile U(Y) =
(4)
7YlK
where U is the x-component of the flow velocity of the liquid, 7 is the tangential stress of the friction on the channel wall, p is the dynamic viscosity, y is the distance from the wall. For a fully developed huninar flow 7 does not vary along the electrode length. Our results can be used, generally speaking, under the other hydrodynamical conditions, if the tangential stress of the friction z (or the thickness of the hydrodynamical boundary layer a,-,)does not vary too much along the electrode length, the mathematics is rather complicated, but the concluding results will be approximately the same. Let us assume that on active surfacc sites the electrochemically active solution component enters into a fast reaction occurring under diffusion control and accompanied by transfer of n electrons. The generalization to the case of a redox reaction is easy[3,5]. The influence of the limited value of the rate constant of the electrochemical transformation, which is quite considerable near the ends of active sites, requires special consideration. The reagent transfer in the solution is described by the convective diffusion equation*
at
x > 0, y = 0.
(6c)
Here C, C, and Co are the values of the reagent concentration near the electrode, on its surface and in the solution bulk, respectively, D is the diffusion coefficient, i = i(x) is the local value of the current density. The boundary condition (6c) characterizes the kinetics of the electrochemical transformation on the electrode. In the case of inhomogeneous surface under consideration k, = k,(x) varies along the surface: on the active sites k, + 00, on the inactive ones k, = 0. METHOD
OF SOLUTION
In the case of a homogeneous highly active electrode k, = 03 along the entire surface. For this case the
limiting diffusion current distribution was found a long time ago[8] and is the form nFDC’(3)“j
jl” =
l-(#)(Dp/~)“~(x
-
(7)
x’#‘~’
where I’(f) is the gamma-function of the corresponding argument. The total limiting current to a homogeneous electrode is obtained by direct integration of (7) and equal to
s
xi+L
II” =
iO II
dx
=
xi
nFDC0(3)“3 _3 ~z/j. r(~Wcc/W3 2
(8)
It can be readily shown[5,6] that in the case of arbitrary activity distribution along the electrode surface. the current densitv distribution ilxl can be found by solving the integial equation beI& ‘tx)
=
nFk*CO
I
’
-
(W7)l’3 ,,FDC‘,(3)“3r(f)
x
1;
(9)
(:(l$J
To simplify the reduced relations let us write * ~ nFDC0(3)1’3
” = I-($)(Dp/t)"3
I
il0)
’
and introduce the new functions g==i/h;
g, = $/A
= (x - x;)-l’3;
Go = 1,0/A = +Lri3.
(11)
(9) assumes the form
Now
with the boundary conditions c+co
at
x,O.y+cg
0%)
C = CO
at
x = 0, y a0
(6b)
+ We supposehere that the solutioncontains an excess of an indifferentelectrolyte.fn this case the migration term in the convective diffusionequation can he discardedand the equation takes a simple form.
In order to obtain the equations describing the current distribution on the first, second and subsequent active sites (gl, gr,. . . , gn,. . . , respectively) it is sufficient to give the function k,(x) concrete vales on active and inactive sites. Then we shall obtain the following sequence of integral equations :
,b
l-zx
s =
ri
(x
g,dC -
#I3
=
O
at
X>
Xi
The limiting diffusion
current
value
on
a macroscopically
electrode
inhomogeneous
311
and
at
xz
+
s xi
X>
Xi
(12)
gzdt (X _
&y/3
+
. . .
dc-
x -!-
at
x
>
. .
x;.
s &I
In a consecutive solution of (12) it is convenient to use the following formula for reversal of Abel’s integral equationC9) :
at
xr&.
However, integration of (Ma) gives
if
cp(s)= then
s
‘ S(t)dt p (s - ty at
(13)
9m=!L-i
x>x;.
+&
(174
d5
Fog gl(x) we find
at
x>x&.
The function G(s) below contained in the integrands The remaining functions gz(x), . . . , g,,,(x), . . . are written as
g2(x) =
$
s
x(x -d5w3
xi
G(s)=
,h’
z
dt s 0 @(l
A,*
+ t)
I+$
(1 + PJ3
3 + %iuctan
at x=-xi
29’3
-
J5
1
I
+2
was examined in detail in[lO] and shows the following asymptotic behavior :
3JJ;
G(s) a~ -pz’3
+ ...
3Js G(s) z 1 - 2re~
-l/3
at +
0
-413
+
Wa)
_ )
4tS
at Equations (15) can be given a somewhat different, but equivalent form:
s<
s>> 1.
(18b)
RESULTS AND DISCUSSION
In order to determine the total current IdN to a macroscopically inhomogeneous electrode equal to L I dN = id dx s0
s L
=A
1
at
x>x;.
Integrating (14) and (15), we obtain Sl =
1 (x - x:1”3
at
x>x;
gdx=A
0
(16)
s
5 xmg_dx,
Xl=1
X&#
(19)
it is necessary to find the current distribution on a sequence of active sites. Unfortunately, though consecutive integration of (17) or (17a) theoretically presents no difficulties, it cannot be performed in an analytical form. Therefore it is reasonable to analyze
V. Yv.
312
FILINOVSKY
only the two limiting cases: (1) low activity surface (0 z 1) where the extent of active sites is not large and they are located far apart (1 <.cd), (2) highly active surface (0 << 1) where wide active sites are separated by narrow.inactive ones (I X- d). We shall not dwell here on a successive mathematical analysis of the expressions for g2,. . . , g,,,,. . . , but shall give qualitative estimates for IdN in each of the limiting cases and explain the physical significance of the results. (1) Low uctiuity sur-ce (0 z 1) In this case the current distribution on active sites can be conveniently examined by means of (17). Integrating (16) and (17) along the active sites and expressing distances in terms of 1 and d, we obtain :
s XI
G,
=
g,
dx = ;1213.
(20)
Xi
G2 =
gz dx
,3p3 2
s I
0
s
sl(x;+CJG(l+.:_i)dL
(21)
%I
G,
=
8,
dx
4m
3 =- 12’3 2
gt(x\ + 0
\ \
I
’ -2---I
Fig. 2. The
The deduction of (23b) can be based on the following considerations: the current produced on the mth active site is weakened by the action of (m- 1) preceding active sites. The decrease caused by each active site is proportional to I,, (ie (22)). If the sites are located sufficiently sparsely, their mutual influence should decrease, which leads to appearance in the second term of the factor (l/d)2’3. At the same time, it is clear that the nearest active site exerts the greatest influence, the effect of more remote sites being weaker. All this leads to the dependence of N413 on the total number ofactive sites. The proportionality factor n can be calculated by means of a rigorous quantitative analysis (21). Let (23b) have the form generally used in such calculations
x [l - .N~‘s(1/d)2’s],
(22)
The current value given above (23a) is somewhat too high since no account is taken in it of the gradual current decrease on active sites due to their mutual influence. It can be shown that at I -CCd this decrease is described as follows
[2/3N4’3([/&2’3
_
_. . _
aN1’3(l/d)2’3],
I& zz I,D. N’J3(l/d)zi3
which corresponds to (20) and the first terms in (21). The resulting total current produced by N such sites is
$
213
or
It can be seen from (17) that the current distribution on the second, third, etc, sites downstream is distorted by the preceding active sites. A qualitative picture of this distribution is given schematically in Fig. 2. The broken lines show the current distribution to be observed on a separate active site of the same size. It is evident that each active site, operating indcpendently of the others produces the current
_
d---
current distributionon the lowly active surface.
x [i -
( (m - 1)(1+ d) - C> dC - . . . 1 ,.d~~-: +W(,+:_c)dT. s0 x G
;P,
I
-I’ -T-
d-
5
ItN = I; . N(I/L)
Id1 = A .
\ G \
\
Wb)
(24~
since it can be assumed that N z L/d at d >> 1. According to (2) i/d = (1 - 0)/O. Substituting this relation into (24), we find r;ik z z$
. N”’
1 - ep3
y--x
[I -c~N”“(3’~]_
(25)
As it follows from (25), the mutual influencebfactive sites, described by the second term depends’not only on their relative positions (fl), but also on their total number (N). Even for narrow active sites (0 =Z 1), the depletion of the solution they cause increases downstream and a&&s markedly the value of the total current. Therefore relation (25) is valid only at
<< 1.
(26)
In the opposite case of importance are the consequent terms in the expansion (24). 2. Highly active sur$ace (f3 -x 1) In this case the appearanon the electrode surface of sparsely located inactive sites changes only slightly
The limiting diffusion current value on a macroscopically inhomogeneous electrode
Z dN =
313
ZH dN -ILdN Iff
+
(291
IiN9
and has the correct asymptotic behaviour in the cases of low and high active surfaces. Using (25) and (28) for the values of the total current liN (at 0 z 1) and I& (at 0 -SC1) we obtain IjN~‘I(!$~”
.,;(I
_
77
I dNF~~N11.(~)113+Z~(~-~)
x3
x3
Fig. 3. The current distribution on the highly active surface. the general pattern of current distribution over the electrode surface. A quantitative description of the change in the current density over the electrode can be obtained by means of (17a). Figure 3 gives a qualitative picture of current distribution in the case under consideration. As can be seen from Fig. 3, immediately after an inactive site the current density increases due to accumulation of matter in the stream. The perturbations in current distribution are local and disappear quickly. The current on the active site is calculated by integrating (17a)
J
d
+
dL
0
G, =
J
1
(27)
gm- dx6 + WC
0
J
d
+
dl.
0
Evidently, the total current to a highly active inhomogeneous electrode Z,” should be less than the current to a homogeneous high active electrode Zj. This decrease seems to be prc$ortional to the current which would flow through inactive sites. For the mth inactive site located downstream this current is z hd/(ml)“3. It should he taken into account that the influence of the inactive site depends on its position on the electrode surface and on the relation between d and 1.Taking into consideration the above factors, we can write
This interpolation formula can be written in the other form which shows directly the dependence of the total current on the velocity of convective flow :
&,N = ~,OPW,W
(30)
Here the function p(N, 0) is :
dN,f’) =
CONCLUDING REMARKS As can be concluded from (31), the reduction of the total current due to the inactive sites on the electrode surface is described by the function p(N, 6). The value of p depends on the fraction of inactive surface (0) and the number and location of active sites on the electrode surface. The function p(N, 0) characterizes the mutual influence of the active sites. Figure 4 gives the functions p(N, 0) calculated for several values of N. It is necessary to emphasize that the dependence of the total current to a macroscopically inhomogeneous electrode, Id&.,on the velocity ofaconvective flow is the same as the total current to a homogeneous highly active electrode, It. In the above case, for instance, the limiting diffusion current I,” to a homogeneous electrode is proportional to @, where t is the tangential stress of the friction on the channel wall. The same dependence of the total current I,, on the value of r remains in the case of a inhomogeneous electrode. Inactive sites on the electrode surface change only the numeral factor.
or
IH 17%’ In deducing (28) it was assumed that N B L/1 and tIz d/l. The proportionality factor %?was evaluated by direct analysis of (27) and was found to be W’ss0.4. For intermediate values of 8 an interpolation formula can be used which is based on the two limiting values of (23a) and (28). The interpolation formula for total current is written as follows
0.5
a
Fig. 4. The behavior of the p(N.0) function at different number of the active sites N.
V. Vu. FILINOVSKY
314 REFERENCES
1. V. Ju. Filinovsky
and Ju. V. Pleskov
in Progress in Surface and IMembrane Science (Edited by P. Danielli) Vol. 10, pp. 27-116, Pergamon, New York (1976). 2. R. Landsberg and R. Thiele, Elecrrochim. Acta 11, 1243 (1966). 3. E. Levart, D. Schuhmann and 0. Contanin, J. elmmoanal. Chem. 70, 117 (1976). 4. B. G. Lcvich. Physicochemicol Hydrodynamics, PrenticeHall, Engelwood Cliffs, New Jersey (1962).
5. H. Matsuda, J. electroanal. Cbem. 16, 153 (1968). 6. V. Ju. Filinovsky, I. V. Kadija, B. 2. Nikolic and M. V. Nakic, J. electroam/. Chem. 54, 39 (1974). 7. A. Suwono. M. Daguenet and D. Bodiof int. J. Heat Mass Tmnsfir 19, 239 (1976). 8. M. A. Leveque, Annis Mines Carbur., Paris 13,201(1928). 9. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Vol. 1, Cambridge University Press (1927). 10. W. J. Albery and S. Bruckenstein, Trans. Faraday Sot. 62, 1920 (1966).