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Photogalvanic cells
J. Electroanal. Chem., 86 (1978) 19--34
19
© Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands
PHOTOGALVANIC CELLS PART 4. THE MAXIMUM POWER FROM A THIN LAYER CELL WITH DIFFERENTIAL ELECTRODE KINETICS
W. JOHN ALBERY
Physical Chemistry Laboratory, South Parks Road, Oxford, OXI 3QZ (England) MARY D. ARCHER
Department of Physical Chemistry, Lensfield Road, Cambridge, CB2 1EP (England) (Received 3rd February 1977; in revised form 28th March 1977)
ABSTRACT The optimal conditions for the production of power from the direct conversion o f solar energy into electrical energy in a photogalvanic cell are found. The cell has a dye couple (A, B) and another redox couple (Y,Z):
hv A+Z~B+Y k
On the illuminated electrode the A,B couple is assumed to be reversible and the Y,Z couple irreversible while on the " d a r k " electrode the Y,Z couple is assumed to be reversible. Analytical solutions are found to the differential equation describing the concentration o f B right across the cell. The output o f power depends on the cell length, the concentrations of A, Y and Z, the kinetics of the homogeneous back reaction B + Y, the extinction coefficient of the dye A, the load on the cell and the difference in standard electrode potentials of the two couples. Conditions for the optimum output o f power are found for all these variables. The maximum possible power output under average isolation (10 - 7 tool cm - 2 s- 1 ) is found to be 5 mW cm - 2 (or power conversion efficiency o f ca. 7%) but the conditions are rather severe and a more realistic estimate would be 0.5 mW cm - 2 (or 0.7% efficiency).
INTRODUCTION
In the previous paper [1; this issue pp. 1--18] we considered the m a x i m u m power that could be obtained from a photogalvanic concentration cell using the following system: Solution
A+Z~ B+Y
Photochemistry
A+Z~ B+Y
k
In this paper we consider the m a x i m u m power t h a t can be obtained from exactly the same system but with one important difference. In the concentration cell the electrodes were made of the same material, but in this paper the two electrodes are made of different materials and have different electrode kinet-
20 ics. Previously the A,B couple was assumed to be reversible at both electrodes and the Y,Z couple was assumed n o t to react at either electrode. In this paper we make a different and complementary assumption a b o u t the electrode kinetics at the " d a r k " electrode; we assume that the Y,Z couple is reversible. We explore t w o different cases for the electrode kinetics of the A,B couple at the " d a r k " electrode: case I, irreversible, and case II, reversible. We therefore can summarize our assumptions a b o u t the electrode kinetics as follows:
Concentration cell [ 1 ] Differential electrode kinetics case I case II
Illuminated electrode a t x = 0 ( × = 0}
" D a r k " electrode a t x = l ( × = 1)
A+e~B ~/
A+e~B ~/
x/ x/
Y+e~Z X
X X
X x/
Y+e*Z
X
x/ x/
The system and notation are exactly the same as in the previous paper [1] and the argument follows exactly the same course until eqn. (16) of that paper. To describe the concentration of the photogenerated B we have to integrate eqn. (14) of the previous paper: (1)
~2u/~×2 + 72p _ u(Tep + ~2) = 0
where Op/O × = --[3p(1 -- u)
(2)
X = x/l
(3)
p =I/I o
(4)
u = b/a D
(5)
3' = l ( ¢ I o e / D ) 1/2 = l / X ~
(6)
1¢ = l ( k y / D ) 1/2 = I / X k
(7)
[3 = lea D = l / X c
(8)
and lower case letters a, b and y have been used to indicate the concentrations o f A, B and Y; aD is the concentration o f A in the dark. THE BOUNDARY CONDITIONS Since A and B react on the illuminated electrode the boundary conditions there are: at x=O,p=
l
and
(au/aX)o -~ m
(9)
where m = li[l/FAa D
(I0)
21
At the " d a r k " electrode for case I the A,B couple does n o t react at all and at × = 1,
(~U/~X) 1 =
(11)
0
For case II the electrode consumes B b y the two electrode reactions B-~A+e and e+Y-~Z Since for case II we have assumed that the electrode kinetics are rapid the electrode quenches the products of the photochemical perturbation and returns the solution back to its unperturbed state. In the dark the concentration o f B has been assumed to be negligible and hence we have the b o u n d a r y condition at×=1,
ul - ~ 0
(12)
We therefore have to solve eqn. (1) with boundary conditions (9) and either (11) or (12) to find Uo. This determines the potential at the illuminated electrode. At the other electrode, where the Y,Z couple is electroactive, we assume, as before, that the concentrations of Y and Z are sufficiently in excess compared to those of A and B, that the concentrations of Y and Z are n o t significantly perturbed b y the photochemical conversion of A to B. The electrode at × = 1 then simply remains at the potential it had in the unilluminated solution and obeys the Nernst equation for the Y,Z couple for both cases I and II. P1 = AP° + In y -- In z
(13)
where, as before [1] (14)
P = F(E -- E~.B)/RT and
(15)
hpO -~ F(Ey,z O -- E ~ , B ) / R T THE " B E E R - L A M B E R T " CASE
As in our previous treatment we can integrate eqn. (1) with boundary conditions (9) and (11) or (12) when u < < 1 a n d p obeys the Beer-Lambert law: p = exp(--flX ) For case I we obtain u -
cosech g / /~a Ic /fl~---~--K2 L ° s h ~(1 -- ×)
I¢ sinh - -fl
\ --cosh ~X e x p ( ~ ) ] - - m cosh ~ ( 1 - X) /
exp(--flX)
(16)
This equation describes the concentration profile o f B right across the cell and we obtain for uo, which describes the surface concentration of B on the illumi-
22 n a t e d electrode:
cotter
uo = - -
-
- - f¢
tanh •
j-- m
}
,.)
The collection efficiency, N , c o m p a r e s the flux of electrons to the flux of photons: N = D(b b/bx)oIo
=
(¢~/3, 2)m
The collection efficiency at short circuit, Nsc, is obtained f r o m eqn. (17) since at short circuit uo is very small: [~ Nsc - f12 ¢~ _~2 -- ~ t a n h ~
/~ exp(--fl)-I cosh K J
(18)
For case II we obtain a set of similar equations; u =
~-
inh ~(1 -- X)" ~ cosh ~: exp(--~X)
cosh(gx) exp(--~) 1 - - m sinh K(i --X)}
u0 -
-- ~ c o t h ~ +
--
K
(19)
(20)
sinh ~ J
and Nsc=
~ 2~b-~~ 2 [ / ~ - ~ c ° t h ~ + ~ sinheXp(--~)-]~ J
(21)
A p p r o x i m a t e solutions for Uo at o p e n circuit (m = 0) and for Nsc are collected in Table 1 for cases I and II. Figure 1 shows typical c o n c e n t r a t i o n profiles for t h e eight d i f f e r e n t Beer-Lambert cases in Table 1. These profiles are similar to the equivalent cases discussed for the photogalvanic c o n c e n t r a t i o n cell [ 1 ] TABLE 1 Approximations for the surface concentration (u 0 ) on the illuminated electrode at open circuit, and for the collection efficiency, Nsc , at short circuit Condition (see Fig. 1)
u 0 at open circuit
NSC
Case I eqn. (17)
Case II eqn. (20)
Case I eqn. (18)
~/2/K2 ~'~ 72/K 2 > ,),2/K2 ~ > ~/21~ ~
72/K 2
~/K q~
~---
>
~
¢ ~b
"~ ~
¢ ~b
Case II eqn. (21)
B e e r - L a m b e r t solu tions
Case A Case B Case C
• > 1 and ~ 1 > ~ and/3 ]~ > 1 > K
Case D
~> K > 1
Bleached solution
See Fig. 2
eqn. (26) 1 =
9'2/2 ,,f2/~ "/21g/~
1
eqn. (27) ~b~/3, =
~/K
~b~17
23 log J<
I0 ~ 1 0
0"0
I -I
L,t ], o,loo BI
C
too
~
! ,0t...... 0"0
,o,L X
I'0
,0L f - - - t , 0 0"0
"X.
I-0
Fig. 1. T h e c o n d i t i o n s w i t h r e s p e c t t o K a n d ~ f o r cases A t o D in T a b l e 1. T h e insets s h o w t y p i c a l p r o f i l e s in a s h o r t - c i z c u l a t e d ceil f o r t h e light i n t e n s i t y ( . . . . . . ) and f o r t h e c o n c e n t r a t i o n p r o f i l e s w h e r e o n t h e d a r k e l e c t r o d e a t X = 1, u / u 1 ~ 1 f o r case I ( e q n . 1 6 ) a n d u / u 1 --* 0 f o r ease I I ( e q n . 19).
except t h a t previously B was produced on the " d a r k " electrode. Now for the case I profiles B does n o t react on the electrode, while for the case II profiles B is consumed. Notice that in cases C and D all the light is absorbed and m o s t o f the B reaches the illuminated electrode giving the m a x i m u m collection efficiency at short circuit o f N s c = ¢. In cases A and B either the light is not all absorbed or some B is lost by diffusion and destruction in the dark part o f the cell. Comparison of the case I and case II solutions in Table 1 shows that, as one might expect, the case I electrode kinetics where B is not destroyed on the dark electrode are to be preferred to case II. However, for cases A and D the same approximate solutions are found, and the profiles near the illuminated electrode in Fig. 1 are almost identical. This is because when K > 1, B is being lost by homogeneous destruction in the dark part of the cell and its fate on the " d a r k " electrode is relatively unimportant. For case C, the behaviour at short circuit is similar since the illuminated electrode collects most o f the generated B. However at open circuit, when B does n o t react on the illuminated electrode, it is instead destroyed on the " d a r k " electrode for case II but n o t for case I; this leads to the lower value of Uo for case IIC compared to case IC. We will show below that the m a x i m u m powerpoint o f the differential cell is close to the short circuit condition. Hence for cases A, D and C one m a y conclude that whatever the rate o f the electrode reaction for the A,B couple on the "dark" electrode the same behaviour will be found under conditions o f optimum load for the production o f power. This conclusion does n o t apply to case
24 B but this case produces the least power, and we do n o t explore it further. We now present a very good approximation for the expression for Nsc for case I in eqn. (18). This approximation, which has a m a x i m u m error o f only 14% (for ~ = ~ = 1), is: N s c ~- ~fll(1 + ~: + fl)
(22)
For case II, the error in the approximation is 26% at ~ = fl = 1. Thus the simple eqn. (22) covers all cases except for case IIB. THE BLEACHED CASE The Beer-Lambert case holds when u < < 1 throughout the cell. The complementary assumption is that the light intensity is large enough to bleach the solution close to the illuminated electrode, so that in this region u ~ 1. Then exactly the same argument can be used as for the photogalvanic concentration cell [ 1 ], because, if the solution is bleached, the boundary condition on the dark electrode is irrevelant to the concentration profile close to the illuminated electrode. In solving eqn. (1) the dark electrode boundary condition,eqns. (11) or (12), is replaced with the boundary conditions that the solution is bleached. Hence from eqn. (41) of ref. 1 uo = 72/~ 2 -- m/o~
(23)
and g s c = ¢/3/~
(24)
where ~2 =
72 +
~2
(25)
The value o f u in the photostationary state is given by u ~ 72/~ 2 = 72/(72 + ~2)
This case gives the same answers as case A in Table 1 when g2 > 72 and the solution is n o t bleached. On the other hand when 72 > ~2 the solution is bleached and we find Uo ~
1 -- m / 7
(26)
¢,6/7
(27)
and
Nsc ~
The concentration profile is similar to t h a t in the top left-hand corner of Fig. 1. The boundaries between the bleached solution and the Beer-Lambert solutions are shown in Fig. 2 where, as before [1], we have taken both fl and K to be greater than unity. For the concentration cell the line ~ = 72 was the b o u n d a r y because B had to destroyed homogeneously close to the dark electrode. This meant that at short circuit there was a build up of B at the dark electrode and hence at the illuminated electrode. For the present cell the same boundary is found for N = 0, corresponding to open circuit, because under these conditions no B is removed at the electrode and the bleaching is effective. However when
25
Xk L I0~,.._ BEERLAMBERT C,SE , ,
I
I
If BLEACHI~EI),
,o,
,o \
I
,
~
I I
~
,o k
I\;
0"5
(>0'
io'
I
I
P--~=~I G
J
\
,o
IX, ,a~ = r z
'
'
'
Po
ud'O
uo=0"90 ,,o=0"~ %=1
Fig. 2. T h e r e g i o n s w h e r e t h e Beer-Lambert and bleached solutions hold. For ~ and K b o t h greater than unity the Beer-Lambert cases A and D of Fig. 1 axe considered. In the area between ~ = T and flK = T2 t h e boundary depends on the current at the illuminated electrode. The inset shows a typical current-voltage curve.
current flows B is converted to A at the illuminated electrode, thereby regenerating the dye. At the dark electrode there is no build up in B, since the Y,Z couple is present in larger concentrations and is assumed to be reversible. Hence the concentration o f B at the illuminated electrode can tend to zero. This leads to efficient electrochemical regeneration o f the dye, which in turn absorbs the photons, and hence the boundary between the t w o cases shifts from the line / ~ = 72 to the line ~ = 7. Fig. 2 shows in detail h o w the b o u n d a r y between the bleached solution and the Beer-Lambert solution depends on the collection efficiency, N, and the surface concentration of B at the illuminated electrode, Uo. A typical approximate current-voltage curve for the illuminated electrode is shown in the inset. There is a sharp break between the bleached and the BeerLambert solutions; this arises from the abrupt change in the surface concentrations of A and B at the point where the current is just sufficient to regenerate the dye. An efficient cell operates at values o f N close to unity in the BeerLambert region and we shall show below that the cell with o p t i m u m load has Uo < < 1. Hence we can write for the cell with o p t i m u m load that the b o u n d a r y between the bleached and Beer-Lambert solutions is given by/3 = % The efficient absorption of the photons, the conversion into a flux o f electrons and the regeneration o f the d y e close to the electrode are desirable features for the operation o f the cell. We n o w modify eqn. (22) to include eqn. (24) for the bleached case b y writing Nsc -~ eft/(1 + ol +/3)
(28)
Using the same approximations we m a y also write from eqn. (17) Uo - c- ° t- h ~{
72
-- m )
(29)
26
THE VOLTAGE DEVELOPED BY THE CELL
Since we have assumed that the A,B couple is reversible on the illuminated electrode and since the dimensionless potential scale, P, is measured from E~,B (eqn. 14), P0 = ln(a/b)x=O = ln(uo I --
1)
(30)
The voltage developed by the cell, AP, is found by combining eqns. (13) and
(3o): AP = P1 -- ln(uo 1 -- 1)
(31)
THE POWER DEVELOPED BY THE CELL
We use the same dimensionless function, 03, as before [1] to describe the power, W, from the cell: W = (El --E0)[i[ = (P1 -- Po)(RT/F)(ANFIo) (32)
= OsARTIo
where (33)
03 = CNAP = Cf3mAP/T 2
Substitution from eqns. (28), (29) and (31) gives:
P1
~2 c o t h a
where P1Nsc
" "
~bPlfl/(1 + ~ + ~)
(35)
When the first square bracket is zero, AP, is zero (eqn. 31) and the cell is short circuited, so that no p o w e r is produced. When the second square bracket is zero, the cell is on open circuit and Uo has the value given b y eqn. (29) with m = 0. Again no p o w e r is produced. In b e t w e e n these t w o extremes there will be an o p t i m u m values o f Uo corresponding to the load that produces maximum power. We explore this condition below, b u t for an efficient cell it is to be hoped that conditions can be found where the Uo terms in the square brackets are small compared to unity. Then 08 ~-- P1Nsc
(36)
We therefore start by examining h o w to maximise P1Nsc. Then, having found the best conditions, we return to the question of the o p t i m u m load and the effect of the square brackets.
27 OPTIMISATION OF P1 Ns c
It is convenient to reintroduce [ 1] the dimensionless parameters ~"and p, where = X6/Xk
= ~ / 7 = ( k y / ¢ I o e ) 1/2 cc y l / 2
(37)
= fl/'Y = a D ( D e / ¢ I o ) 1/2 o~ aD
(38)
and p = X6/Xe
For solar energy conversion Io will be fixed by the ambient solar radiation and e and D by the optical absorption and transport characteristics of the dye A. However the concentrations of Y and A c a n be varied; ~.2 is proportional to [Y] and p to [A], so ~"and p can be adjusted to some extent. Substitution from eqns. (37) and (38) in (35) gives (P' + In ~-2)p P I N s c - ~ - 1 + (1 + ~-2)1/2 + p
(39)
where P' = A P e + l n ( ¢ I o e / k [ Z ] )
(40)
and from eqn. (6) .y = l ( ¢ I o e / D ) l / 2
Now in eqn. (39), 7 is the only quantity that depends on the cell length. Hence to maximize P 1 N s c , l must be large enough so that ~/-1 is insignificant in eqn. (39). Hence from eqn. (35) a + j3 > > 1
(41)
This shows that cases IB and IIB are less efficient than the other cases. Returning to eqn. (39) we see that holding the other parameters constant and increasing p (~ [A] ) always increases PiNsc, although eventually a limiting value will be reached of P 1 N s c = P' + In ~.2
(42)
Hence we conclude that, if possible, p should be large enough to dominate the denominator. Turning to the variation o f P 1 N s c with ~"we find from eqn. (39) that P i N s c passes through a maximum. Increasing ~-(c~[ y ] l / 2 ) i n c r e a s e s the voltage of the cell through the effect of Y on the potential of the dark electrode. But if there is too m u c h Y, then too m u c h B is destroyed reacting with Y, and the power falls. In algebraic terms the (1 + ~-2)1/2 term is then the dominant term in the denominator of eqn. (39). Differentiation of eqn. (39) with respect to ~"gives the following relationships between the optimum values ~'m and (P1Nsc)m and P' and p: tY = ( P 1 N s c ) m + 2 + 2~'m2 - - In ~2 m
(43)
and p = ½ ~-2(1 + ~-2 )-l/2(P1Nsc)m
(44)
28
Using these equations, we construct the contour diagram in Fig. 3 which shows how (P1Nsc)m and ~m depend on P' and p. The optimised power function, (P1Nsc)m, increases from the bottom left-hand comer (~ 0.2) to the top righthand comer (~50). Large values are found for large values of P', because the greater the potential difference AE °, the more power will be produced. Large values of p on the right-hand side of the diagram correspond to having enough A to trap all the photons close to the illuminated electrode; the flux of photons is thereby converted into a flux of B, all of which is consumed on the electrode. In the top right-hand comer of Fig. 3 the contours for (P1Nsc)m are nearly parallel to the x-axis, showing little dependence on p. This is the limiting behaviour given by eqn. (42). In this region therefore (P1Nsc)m ~ AP and further increase in p does not lead to worthwhile increases in power. In the useful region the contours for ~'2 m are between 1 and 10 and so one may write
~'m~ 2 This is not surprising, since, if ~ < 1, increasing [Y] increases the potential difference without significant quantities of B, but when ~ > 1 then too much B is destroyed. On the far right of the diagram, where p ~ 100, larger values of ~m can be found because the optical length is so short that B still has a good chance of reaching the electrode before being destroyed by Y. In contrast, on the left of the diagram the optical length is longer than the generating length (p < 1). A photostationary state is then established where the concentration of B is given by: b/(a
+
b) = 1/(1 + ~'~')
Under these conditions the optimum performance is found for smaller values of
..~
I
....
!
~o-\
--":~--""'-
.... ~.-
,~
_~_~__~_.-" --~'~ I'O
0
~ ---;"
-0'5
j"
2"0..._."
.-'%
\
s/
s
."~-. ~.~ . ~
,.~'-"
_.~.--
,~/
s
.J
.'.~_ ~
./
i
i
p
~,.
i
s
, t 50,"
i
. ' ~ ' . J
, , " /
to
/
, ZO
o-s
,,
~
a-
Ioo
\ .,,' \,, ,-,,,,,-,,,,'" ,, , J \ ."
0"0
I X 0"5
/I
I'0
/
log p
I 1"5
Fig. 3. C o n t o u x d i a g r a m c a l c u l a t e d f r o m e q n s . ( 4 3 ) a n d ( 4 4 ) t o s h o w h o w ( P 1 N S C ) m ( ) and ~m (. . . . . . ) d e p e n d o n p ' a n d p. T h e p o w e r is a p p r o x i m a t e l y p r o p o r t i o n a l t o P I N S C ( e q n . 3 6 ) a n d as ~ is v a r i e d ( e q n . S 9 ) , P 1 N s c w i l l p a s s t h r o u g h a m a x i m u m ( P 1 N s C ) m at S'm; Fig. 4 s h o w s t h i s v a r i a t i o n f o r t h e p o i n t .. T h e p a r a m e t e r ~" ( e q n . 3 7 ) c o m p a r e s t h e generating length t o t h e kinetic length and c a n b e increased by increasing [Y]. The parameter p (eqn. 38) compares the generating length to the absorption l e n g t h a n d c a n b e i n c r e a s e d b y i n c r e a s i n g [ A ] . T h e parameter P ' ( e q n . 4 0 ) will b e m a i n l y d e t e r m i n e d b y £kP* (= F A E ® / R T ) w h i c h d e s c r i b e s t h e d i f f e r e n c e i n standard e l e c t r o d e p o t e n t i a l s o f the t w o c o u p l e s ; syst e m s c a n t h e r e f o r e b e r o u g h l y l o c a t e d o n t h e d i a g r a m u s i n g A E ° and the r i g h t - h a n d scale.
29 ~"(~ 0.1). The scale on the right-hand side in volts is so that one can obtain rough estimates of P' from known AE ° . From this Figure we conclude that the ideal cell is a case D cell with the following characteristics: P = X G / X c ~ 10
(45)
= XG/Xk ~ 2
(46)
X~ < XK < XG < X, p ' = Ap" + In ~ I ° e ~ 20
k[z]
(47)
or
A E e / V ~ 0.5 to 1.3
giving P 1 g s c ~ 20
(48)
THE OPTIMUM LOAD
Now that we have found the conditions the optimise PiNsc in eqn. (34), we return to the effect of varying the load on the cell, which is described by the two square brackets in that equation. Assuming t h a t the Uo terms are small and so neglecting their product, we write 08 ~ P 1 N s c [1 -- P~-I1 ln(uo 1 -- 1) -- u0(1 + ~-2)112{p + (1 + ~,2)112} ]
where we have also assumed from eqns. (45) and (46) that a ~/3 and hence from eqn. (41) that a > 1
(49)
Differentiation with respect to Uo gives the optimum values Uo.m for Uo and hence the conditions t h a t give the largest value of 03, 03,m:
u0~ ~- [Pl(1 + ~2)1/2 {p + (1 + ~-2)1/2} ]-1 and 03,m ~ P1Nsc [1 + P~-Il(ln Uo~ -- 1)] For the conditions t h a t optimise P1Nsc in eqns. (45) to (48) we find U0,m = bo,m/a D ~ 5 × 10 -8
(50)
and 03.m -~ 0.7 P1Nsc for p = 10 and P1 = 20. The value for Uo,m justifies the approximation made above and the choice of boundary in Fig. 2. Larger values of P1 or smaller values of p give values of 0 8 ~ even closer to P 1 N s c . Hence providing firstly that P1 > 20(E1 - - E ~ $ > 0.5 V) and secondly that p > 10, then at the m a x i m u m power point the function 03 is at least 70% of PiNsc. Therefore for this cell for the o p t i m u m conditions, P1Nsc is a good guide to the
3o power o u t p u t of the cell and we need not evaluate the square brackets in eqn. (34). We merely put them equal to 0.8. ESTIMATES O F T H E M A X I M U M
POWER
From the arguments above and eqn. (32), we therefore obtain for the maxim u m p o w e r Wm Wm ~ 0.8(P1Nsc)m A R T I o 0.8 A E* FNscAIo
(51) (52)
The p o w e r is obtained from a current ~ F A I o ( N s c ~- 1) delivered at a voltage of ~ A E *. Neither the current nor the maximum p o w e r voltage can be significantly larger than these values, so eqn. (52) describes the m a x i m u m possible p o w e r that can be obtained from any photogalvanic cell. If the cell is operated by polychromatic light, then at best only photons of energy greater than e A E ° will cause the photochemical reaction to occur, and eqn. (52) must be re-written.
Wm ~ 0.8 A E * F N s c A ~ E " Ig dE
(53)
where I~ (mol cm - 2 s- 1 eV - 1 ) is the spectral irradiance o f the incoming light. The p r o d u c t AE* f~aa~.--I~.dE is a function both of AE * and of the spectral distribution I~.. For average solar terrestrial radiation, this product has a maximum value when the integral is 10 - 7 mol cm - 2 s- I , which occurs when e A E ° 1.1 eV 2. Substituting this value in eqn. (53), and taking Nsc = 1, we obtain the maximum power that the cell will deliver when operated by average sunlight: Wm/A -~ 5 mW cm -2
(54)
This corresponds to a p o w e r conversion efficiency of ~7%. CONDITIONS FOR THE OPTIMUM CASE N o w we collect together the conditions for the o p t i m u m case to see if they can be realised in practice. Cell length a ~> 1 or l > D1/2/(ky) 1/2
(49)
Optical length and [A] p = XG/XE = aD(De/¢Io) 1/2 ~ 10
(45)
Kinetic length and [Y] = XG/Xk = (kY/~Ioe) 1/z ~ 2
(46)
31 Standard electrode potentials (47)
AE° > 0.5 V Electrode kinetics
A,B couple Y,Z couple
Illuminated electrode at × = 0
Dark electrode at X = 1
Reversible
Reversible or irreversible
( k ° B)o > > D [ X e Irreversible
Reversible
( k ° z ) o y expl{AP*l < < (kO,B)OaD
(k°zh >> D/l
Taking typical values of Io = 10 -7 mol cm - 2 s-1 for terrestrial solar radiation together with e = 10 s cm 2 mo1-1, D = 10 - 5 cm 2 s-1 and ¢ = 1 we find that the generating length XG is given by XG = (D/¢Ioe) ~ =
10 -a cm
(55)
This is the average distance t h a t A diffuses in the light intensity of Io before it is converted to B. The condition for p in eqn. (45) requires that the optical length, Xe, be a tenth of the generating length:
X~ ~ 10-4 cm
(56)
This in turn means aD ~ 0.1 M
(57)
The dye A has to be very soluble to absorb all the radiation close to the electrode. From eqn. (46) we require the kinetic length Xk to be comparable to the generating length and ky ~ 4Ioe ~
40 s-1
(58)
But the concentration of Y must be at least twice the concentration of B to ensure that the dark electrode remains unpolarised, and so
h < 4 × 1 0 aM- i s -~
(59)
The condition for the cell length (eqn. 49), which requires that it be larger than the kinetic length, is easily satisfied. Of the various conditions for the electrode kinetics, since Xe < l, the most severe is the requirement for the rapid reaction of A and B on the illuminated electrode: (k°,B)O > 0.1 cm s-1 Few redox couples have electrode kinetics as fast as this. To summarise, for the o p t i m u m case (p = 10) we require a very soluble dye with improbably fast electrode kinetics and rather sluggish homogeneous kinetics. These last two requirements m a y well be incompatible for electron transfer reactions that obey the Marcus theory [3]. It therefore seems unlikely t h a t the o p t i m u m case can be achieved in practice.
32
8
6
4
2
0
-4
-2
0
to~
2
Fig. 4. V a r i a t i o n o f P 1 N s c , w h i c h d e s c r i b e s the p o w e r o u t p u t , w i t h ~', w h i c h c o m p a r e s t h e g e n e r a t i n g l e n g t h to t h e k i n e t i c l e n g t h ( e q n . 3 7 ) . T h e c u r v e i~ c a l c u l a t e d f r o m e q n . ( 3 9 ) f o r p = 1 a n d P' = 2 0 ; t h i s p o i n t is p l o t t e d in Fig. 3 as -.
A MORE
REALISTIC
CASE
We now consider a more realistic case which could perhaps be achieved. If one takes p = 1 rather than p = 10 then from Fig. 3 the power is roughly half t h a t for p = 10, but the conditions are somewhat easier to satisfy: a D "~
10 mM
k < 4 × 104 M -1 s-1
(60)
kOAB , > 10-2 cm s- I
(61)
In view of the possible incompatibility of the requirements for fast electrode kinetics (eqn. 61) and sluggish homogeneous kinetics (eqn. 60) we explore how critical it is to optimize PiNsc with respect to k. Figure 4 shows a plot of eqn. (39) for p = I, P' = 20 and ~-i = 0; the position of the maximum of this plot is indicated in Fig. 3. It can be seen that a larger value of ~', say 10, does cause a significant reduction in PiNsc from its maximum value of 8.8 to 2.2. The corresponding condition for k (cf. eqn. 60) is eased to k < 4 × 106/l~r- I
S- 1
We therefore conclude that although an impossibly ideal photogalvanic cell would produce 5 mW em-2 a more realistic estimate for the power that might be achieved in practice is 0.5 mW cm-2. C O M P A R I S O N WITH PREVIOUS RESULTS
Gomer [4] has also considered the power efficiency of a photogalvanic cell containing a semipermable membrane, operating so t h a t only the Y,Z couple can reach the dark electrode. At the illuminated electrode, only the A,B couple, present in much smaller concentration, reacts. His cell is therefore identical in principle to the one we have discussed in this paper and his analysis identifies m a n y of the constraints we have also found. He concludes that a cell operated
33 by solar energy would produce a maximum p o w e r of ~ 1 mW c m - " , and that this would require k < 102 cm 3 mo1-1 s -1 and standard exchange current densities for the electrode reactions o f 102--103 A c m tool -1 (i.e. k ° = 10-3--10 -2 cm s-1 ). His conclusions are therefore similar to, though rather more pessimistic than, ours. However his analysis does n o t explore the interplay of the various lengths leading to the different concentration profiles. Instead he makes assumptions a b o u t the transport layers and the width of the illuminated region. Thus our analysis is more general and allows one to explore the optimization of the different parameters. There are few experimental data in the literature on cells of the t y p e we have discussed in this paper. The best d o c u m e n t e d examples are the thin layer (25-80 pm) iron-thionine photogalvanic cells developed by Lichtin et al. These have a SnO2 anode and a cathode of either Pt [5,6] or indium tin oxide [7]. The Fe2÷/Fe 3÷ couple is very irreversible at SnO2, while the thionine/leucothionine couple is moderately reversible. At Pt or indium tin oxide, both couples are electroactive, the thionine couple having faster electrode kinetics than the iron couple [ 7,8]. Thus the cell has one selective electrode and one rather unselective one. The following data on o u t p u t from a single cell have been reported, for 35 mW cm -2 incident white light: open circuit voltage ~ 0 . 1 5 V [5], maxim u m p o w e r voltage 0.05--0.10 V [4], short circuit current ~ 2 × 10 -4 A cm -2 [5], monochromatic quantum efficiency (electrons flowing per p h o t o n absorbed at 578, 589 or 620 nm) ~ 0 . 0 8 % for a 80 #m cell, ~0.15% for a 25/2m cell [7]. Our analysis indicates the following reasons for some of these results. (Lichtin et al. have independently reached similar conclusions.) Firstly AE e is only 0.44 V at pH = 2 and a rather higher value, assuming k remained constant, would be desirable. Secondly the short circuit current and the quantum efficiency are low because Xe is greater than Xk, SO that some of the photochemically generated Fe 3÷ and leucothionine is lost by back reaction in solution (case A rather than case D in Fig. 1). COMPARISON OF CONCENTRATION CELL WITH CELL WITH DIFFERENTIAL ELECTRODE KINETICS The previous paper [ 1] considered the maximum p o w e r that can be obtained from a photogalvanic concentration cell in which on both electrodes the couple Y,Z is inactive while the couple A,B is reversible. In this paper we have considered the differential case, for which the conditions are set o u t in the Introduction. This t y p e o f cell has the following advantages over the concentration cell: (1) The current is delivered at a voltage of ~ A E ' , the difference in the standard electrode potentials of the A,B and Y,Z couple, rather than at a potential o f R T/F. (2) The collection efficiency at short circuit is unity rather than ½. (3) The maximum p o w e r is a b o u t one hundred time larger. (4) Convective m o t i o n caused b y the heating effect of the light may reduce the concentration difference in the concentration cell b u t does not affect the performance of the cell with differential electrode kinetics.
34 The conditions required to obtain the m a x i m u m power from both types of cell are similar. First, all the light must be trapped close to the illuminated electrode. Secondly, the kinetics of the back reaction B + Y must n o t be too fast but should if possible be optimised using the concentration of Y. Thirdly, the kinetics of the electrode reaction, B -~ A + e on the illuminated electrode must be very rapid. Since these conditions hold for both types of cell it is probable that they also hold for cells where the electrode kinetics are intermediate between the two extreme cases discussed in detail. For both types of cell it is unlikely that any real system will fulfil all the conditions requires for their respective o p t i m u m cases, but the type of cell discussed in this paper will still generate a hundred times more power than the corresponding cell. CONCLUSION We conclude t h a t the inclusion of differential electrode kinetics is essential for the successful development of photogalvanic cells. If the necessary conditions can be obtained then one may reasonably expect an o u t p u t of 0.5 mW cm -2. ACKNOWLEDGEMENTS We t h a n k Dr. Stephen Feldberg for interesting discussions. In our nomenclature he has considered the case D cells. We are grateful to Mr. Andrew Foulds for helpful comments and criticisms. REFERENCES 1 2 3 4 5 6 7 8
W.J. Albery and M.D. Archer, Part III of this series, J. Electroanal. Chem., 86 (1978) 1 (this issue). M.D. Archer, s u b m i t t e d to Solar Energy. R.A. Marcus, J. Chem. Phys., 43 (1965) 679. R. Gomer, Electrochim. Acta, 20 (1975) 13. W.D.K. Clark and J.A. Eekert, Solar Energy, 17 (1975) 147. D.E. Hall, W.D.K. Clark, J.A. Eckert, N.N. L i c h t i n and P.D. Wfldes, J. Ceramic Soc., in press. D.E. Hall, J.A. Eckert, N.N. L i c h t i n and P.D. Wildes, J. Electrochem. Soc., in press. M.D. Archer, Appl. Electrochem., 5, (1975) 17.
19
© Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands
PHOTOGALVANIC CELLS PART 4. THE MAXIMUM POWER FROM A THIN LAYER CELL WITH DIFFERENTIAL ELECTRODE KINETICS
W. JOHN ALBERY
Physical Chemistry Laboratory, South Parks Road, Oxford, OXI 3QZ (England) MARY D. ARCHER
Department of Physical Chemistry, Lensfield Road, Cambridge, CB2 1EP (England) (Received 3rd February 1977; in revised form 28th March 1977)
ABSTRACT The optimal conditions for the production of power from the direct conversion o f solar energy into electrical energy in a photogalvanic cell are found. The cell has a dye couple (A, B) and another redox couple (Y,Z):
hv A+Z~B+Y k
On the illuminated electrode the A,B couple is assumed to be reversible and the Y,Z couple irreversible while on the " d a r k " electrode the Y,Z couple is assumed to be reversible. Analytical solutions are found to the differential equation describing the concentration o f B right across the cell. The output o f power depends on the cell length, the concentrations of A, Y and Z, the kinetics of the homogeneous back reaction B + Y, the extinction coefficient of the dye A, the load on the cell and the difference in standard electrode potentials of the two couples. Conditions for the optimum output o f power are found for all these variables. The maximum possible power output under average isolation (10 - 7 tool cm - 2 s- 1 ) is found to be 5 mW cm - 2 (or power conversion efficiency o f ca. 7%) but the conditions are rather severe and a more realistic estimate would be 0.5 mW cm - 2 (or 0.7% efficiency).
INTRODUCTION
In the previous paper [1; this issue pp. 1--18] we considered the m a x i m u m power that could be obtained from a photogalvanic concentration cell using the following system: Solution
A+Z~ B+Y
Photochemistry
A+Z~ B+Y
k
In this paper we consider the m a x i m u m power t h a t can be obtained from exactly the same system but with one important difference. In the concentration cell the electrodes were made of the same material, but in this paper the two electrodes are made of different materials and have different electrode kinet-
20 ics. Previously the A,B couple was assumed to be reversible at both electrodes and the Y,Z couple was assumed n o t to react at either electrode. In this paper we make a different and complementary assumption a b o u t the electrode kinetics at the " d a r k " electrode; we assume that the Y,Z couple is reversible. We explore t w o different cases for the electrode kinetics of the A,B couple at the " d a r k " electrode: case I, irreversible, and case II, reversible. We therefore can summarize our assumptions a b o u t the electrode kinetics as follows:
Concentration cell [ 1 ] Differential electrode kinetics case I case II
Illuminated electrode a t x = 0 ( × = 0}
" D a r k " electrode a t x = l ( × = 1)
A+e~B ~/
A+e~B ~/
x/ x/
Y+e~Z X
X X
X x/
Y+e*Z
X
x/ x/
The system and notation are exactly the same as in the previous paper [1] and the argument follows exactly the same course until eqn. (16) of that paper. To describe the concentration of the photogenerated B we have to integrate eqn. (14) of the previous paper: (1)
~2u/~×2 + 72p _ u(Tep + ~2) = 0
where Op/O × = --[3p(1 -- u)
(2)
X = x/l
(3)
p =I/I o
(4)
u = b/a D
(5)
3' = l ( ¢ I o e / D ) 1/2 = l / X ~
(6)
1¢ = l ( k y / D ) 1/2 = I / X k
(7)
[3 = lea D = l / X c
(8)
and lower case letters a, b and y have been used to indicate the concentrations o f A, B and Y; aD is the concentration o f A in the dark. THE BOUNDARY CONDITIONS Since A and B react on the illuminated electrode the boundary conditions there are: at x=O,p=
l
and
(au/aX)o -~ m
(9)
where m = li[l/FAa D
(I0)
21
At the " d a r k " electrode for case I the A,B couple does n o t react at all and at × = 1,
(~U/~X) 1 =
(11)
0
For case II the electrode consumes B b y the two electrode reactions B-~A+e and e+Y-~Z Since for case II we have assumed that the electrode kinetics are rapid the electrode quenches the products of the photochemical perturbation and returns the solution back to its unperturbed state. In the dark the concentration o f B has been assumed to be negligible and hence we have the b o u n d a r y condition at×=1,
ul - ~ 0
(12)
We therefore have to solve eqn. (1) with boundary conditions (9) and either (11) or (12) to find Uo. This determines the potential at the illuminated electrode. At the other electrode, where the Y,Z couple is electroactive, we assume, as before, that the concentrations of Y and Z are sufficiently in excess compared to those of A and B, that the concentrations of Y and Z are n o t significantly perturbed b y the photochemical conversion of A to B. The electrode at × = 1 then simply remains at the potential it had in the unilluminated solution and obeys the Nernst equation for the Y,Z couple for both cases I and II. P1 = AP° + In y -- In z
(13)
where, as before [1] (14)
P = F(E -- E~.B)/RT and
(15)
hpO -~ F(Ey,z O -- E ~ , B ) / R T THE " B E E R - L A M B E R T " CASE
As in our previous treatment we can integrate eqn. (1) with boundary conditions (9) and (11) or (12) when u < < 1 a n d p obeys the Beer-Lambert law: p = exp(--flX ) For case I we obtain u -
cosech g / /~a Ic /fl~---~--K2 L ° s h ~(1 -- ×)
I¢ sinh - -fl
\ --cosh ~X e x p ( ~ ) ] - - m cosh ~ ( 1 - X) /
exp(--flX)
(16)
This equation describes the concentration profile o f B right across the cell and we obtain for uo, which describes the surface concentration of B on the illumi-
22 n a t e d electrode:
cotter
uo = - -
-
- - f¢
tanh •
j-- m
}
,.)
The collection efficiency, N , c o m p a r e s the flux of electrons to the flux of photons: N = D(b b/bx)oIo
=
(¢~/3, 2)m
The collection efficiency at short circuit, Nsc, is obtained f r o m eqn. (17) since at short circuit uo is very small: [~ Nsc - f12 ¢~ _~2 -- ~ t a n h ~
/~ exp(--fl)-I cosh K J
(18)
For case II we obtain a set of similar equations; u =
~-
inh ~(1 -- X)" ~ cosh ~: exp(--~X)
cosh(gx) exp(--~) 1 - - m sinh K(i --X)}
u0 -
-- ~ c o t h ~ +
--
K
(19)
(20)
sinh ~ J
and Nsc=
~ 2~b-~~ 2 [ / ~ - ~ c ° t h ~ + ~ sinheXp(--~)-]~ J
(21)
A p p r o x i m a t e solutions for Uo at o p e n circuit (m = 0) and for Nsc are collected in Table 1 for cases I and II. Figure 1 shows typical c o n c e n t r a t i o n profiles for t h e eight d i f f e r e n t Beer-Lambert cases in Table 1. These profiles are similar to the equivalent cases discussed for the photogalvanic c o n c e n t r a t i o n cell [ 1 ] TABLE 1 Approximations for the surface concentration (u 0 ) on the illuminated electrode at open circuit, and for the collection efficiency, Nsc , at short circuit Condition (see Fig. 1)
u 0 at open circuit
NSC
Case I eqn. (17)
Case II eqn. (20)
Case I eqn. (18)
~/2/K2 ~'~ 72/K 2 > ,),2/K2 ~ > ~/21~ ~
72/K 2
~/K q~
~---
>
~
¢ ~b
"~ ~
¢ ~b
Case II eqn. (21)
B e e r - L a m b e r t solu tions
Case A Case B Case C
• > 1 and ~ 1 > ~ and/3 ]~ > 1 > K
Case D
~> K > 1
Bleached solution
See Fig. 2
eqn. (26) 1 =
9'2/2 ,,f2/~ "/21g/~
1
eqn. (27) ~b~/3, =
~/K
~b~17
23 log J<
I0 ~ 1 0
0"0
I -I
L,t ], o,loo BI
C
too
~
! ,0t...... 0"0
,o,L X
I'0
,0L f - - - t , 0 0"0
"X.
I-0
Fig. 1. T h e c o n d i t i o n s w i t h r e s p e c t t o K a n d ~ f o r cases A t o D in T a b l e 1. T h e insets s h o w t y p i c a l p r o f i l e s in a s h o r t - c i z c u l a t e d ceil f o r t h e light i n t e n s i t y ( . . . . . . ) and f o r t h e c o n c e n t r a t i o n p r o f i l e s w h e r e o n t h e d a r k e l e c t r o d e a t X = 1, u / u 1 ~ 1 f o r case I ( e q n . 1 6 ) a n d u / u 1 --* 0 f o r ease I I ( e q n . 19).
except t h a t previously B was produced on the " d a r k " electrode. Now for the case I profiles B does n o t react on the electrode, while for the case II profiles B is consumed. Notice that in cases C and D all the light is absorbed and m o s t o f the B reaches the illuminated electrode giving the m a x i m u m collection efficiency at short circuit o f N s c = ¢. In cases A and B either the light is not all absorbed or some B is lost by diffusion and destruction in the dark part o f the cell. Comparison of the case I and case II solutions in Table 1 shows that, as one might expect, the case I electrode kinetics where B is not destroyed on the dark electrode are to be preferred to case II. However, for cases A and D the same approximate solutions are found, and the profiles near the illuminated electrode in Fig. 1 are almost identical. This is because when K > 1, B is being lost by homogeneous destruction in the dark part of the cell and its fate on the " d a r k " electrode is relatively unimportant. For case C, the behaviour at short circuit is similar since the illuminated electrode collects most o f the generated B. However at open circuit, when B does n o t react on the illuminated electrode, it is instead destroyed on the " d a r k " electrode for case II but n o t for case I; this leads to the lower value of Uo for case IIC compared to case IC. We will show below that the m a x i m u m powerpoint o f the differential cell is close to the short circuit condition. Hence for cases A, D and C one m a y conclude that whatever the rate o f the electrode reaction for the A,B couple on the "dark" electrode the same behaviour will be found under conditions o f optimum load for the production o f power. This conclusion does n o t apply to case
24 B but this case produces the least power, and we do n o t explore it further. We now present a very good approximation for the expression for Nsc for case I in eqn. (18). This approximation, which has a m a x i m u m error o f only 14% (for ~ = ~ = 1), is: N s c ~- ~fll(1 + ~: + fl)
(22)
For case II, the error in the approximation is 26% at ~ = fl = 1. Thus the simple eqn. (22) covers all cases except for case IIB. THE BLEACHED CASE The Beer-Lambert case holds when u < < 1 throughout the cell. The complementary assumption is that the light intensity is large enough to bleach the solution close to the illuminated electrode, so that in this region u ~ 1. Then exactly the same argument can be used as for the photogalvanic concentration cell [ 1 ], because, if the solution is bleached, the boundary condition on the dark electrode is irrevelant to the concentration profile close to the illuminated electrode. In solving eqn. (1) the dark electrode boundary condition,eqns. (11) or (12), is replaced with the boundary conditions that the solution is bleached. Hence from eqn. (41) of ref. 1 uo = 72/~ 2 -- m/o~
(23)
and g s c = ¢/3/~
(24)
where ~2 =
72 +
~2
(25)
The value o f u in the photostationary state is given by u ~ 72/~ 2 = 72/(72 + ~2)
This case gives the same answers as case A in Table 1 when g2 > 72 and the solution is n o t bleached. On the other hand when 72 > ~2 the solution is bleached and we find Uo ~
1 -- m / 7
(26)
¢,6/7
(27)
and
Nsc ~
The concentration profile is similar to t h a t in the top left-hand corner of Fig. 1. The boundaries between the bleached solution and the Beer-Lambert solutions are shown in Fig. 2 where, as before [1], we have taken both fl and K to be greater than unity. For the concentration cell the line ~ = 72 was the b o u n d a r y because B had to destroyed homogeneously close to the dark electrode. This meant that at short circuit there was a build up of B at the dark electrode and hence at the illuminated electrode. For the present cell the same boundary is found for N = 0, corresponding to open circuit, because under these conditions no B is removed at the electrode and the bleaching is effective. However when
25
Xk L I0~,.._ BEERLAMBERT C,SE , ,
I
I
If BLEACHI~EI),
,o,
,o \
I
,
~
I I
~
,o k
I\;
0"5
(>0'
io'
I
I
P--~=~I G
J
\
,o
IX, ,a~ = r z
'
'
'
Po
ud'O
uo=0"90 ,,o=0"~ %=1
Fig. 2. T h e r e g i o n s w h e r e t h e Beer-Lambert and bleached solutions hold. For ~ and K b o t h greater than unity the Beer-Lambert cases A and D of Fig. 1 axe considered. In the area between ~ = T and flK = T2 t h e boundary depends on the current at the illuminated electrode. The inset shows a typical current-voltage curve.
current flows B is converted to A at the illuminated electrode, thereby regenerating the dye. At the dark electrode there is no build up in B, since the Y,Z couple is present in larger concentrations and is assumed to be reversible. Hence the concentration o f B at the illuminated electrode can tend to zero. This leads to efficient electrochemical regeneration o f the dye, which in turn absorbs the photons, and hence the boundary between the t w o cases shifts from the line / ~ = 72 to the line ~ = 7. Fig. 2 shows in detail h o w the b o u n d a r y between the bleached solution and the Beer-Lambert solution depends on the collection efficiency, N, and the surface concentration of B at the illuminated electrode, Uo. A typical approximate current-voltage curve for the illuminated electrode is shown in the inset. There is a sharp break between the bleached and the BeerLambert solutions; this arises from the abrupt change in the surface concentrations of A and B at the point where the current is just sufficient to regenerate the dye. An efficient cell operates at values o f N close to unity in the BeerLambert region and we shall show below that the cell with o p t i m u m load has Uo < < 1. Hence we can write for the cell with o p t i m u m load that the b o u n d a r y between the bleached and Beer-Lambert solutions is given by/3 = % The efficient absorption of the photons, the conversion into a flux o f electrons and the regeneration o f the d y e close to the electrode are desirable features for the operation o f the cell. We n o w modify eqn. (22) to include eqn. (24) for the bleached case b y writing Nsc -~ eft/(1 + ol +/3)
(28)
Using the same approximations we m a y also write from eqn. (17) Uo - c- ° t- h ~{
72
-- m )
(29)
26
THE VOLTAGE DEVELOPED BY THE CELL
Since we have assumed that the A,B couple is reversible on the illuminated electrode and since the dimensionless potential scale, P, is measured from E~,B (eqn. 14), P0 = ln(a/b)x=O = ln(uo I --
1)
(30)
The voltage developed by the cell, AP, is found by combining eqns. (13) and
(3o): AP = P1 -- ln(uo 1 -- 1)
(31)
THE POWER DEVELOPED BY THE CELL
We use the same dimensionless function, 03, as before [1] to describe the power, W, from the cell: W = (El --E0)[i[ = (P1 -- Po)(RT/F)(ANFIo) (32)
= OsARTIo
where (33)
03 = CNAP = Cf3mAP/T 2
Substitution from eqns. (28), (29) and (31) gives:
P1
~2 c o t h a
where P1Nsc
" "
~bPlfl/(1 + ~ + ~)
(35)
When the first square bracket is zero, AP, is zero (eqn. 31) and the cell is short circuited, so that no p o w e r is produced. When the second square bracket is zero, the cell is on open circuit and Uo has the value given b y eqn. (29) with m = 0. Again no p o w e r is produced. In b e t w e e n these t w o extremes there will be an o p t i m u m values o f Uo corresponding to the load that produces maximum power. We explore this condition below, b u t for an efficient cell it is to be hoped that conditions can be found where the Uo terms in the square brackets are small compared to unity. Then 08 ~-- P1Nsc
(36)
We therefore start by examining h o w to maximise P1Nsc. Then, having found the best conditions, we return to the question of the o p t i m u m load and the effect of the square brackets.
27 OPTIMISATION OF P1 Ns c
It is convenient to reintroduce [ 1] the dimensionless parameters ~"and p, where = X6/Xk
= ~ / 7 = ( k y / ¢ I o e ) 1/2 cc y l / 2
(37)
= fl/'Y = a D ( D e / ¢ I o ) 1/2 o~ aD
(38)
and p = X6/Xe
For solar energy conversion Io will be fixed by the ambient solar radiation and e and D by the optical absorption and transport characteristics of the dye A. However the concentrations of Y and A c a n be varied; ~.2 is proportional to [Y] and p to [A], so ~"and p can be adjusted to some extent. Substitution from eqns. (37) and (38) in (35) gives (P' + In ~-2)p P I N s c - ~ - 1 + (1 + ~-2)1/2 + p
(39)
where P' = A P e + l n ( ¢ I o e / k [ Z ] )
(40)
and from eqn. (6) .y = l ( ¢ I o e / D ) l / 2
Now in eqn. (39), 7 is the only quantity that depends on the cell length. Hence to maximize P 1 N s c , l must be large enough so that ~/-1 is insignificant in eqn. (39). Hence from eqn. (35) a + j3 > > 1
(41)
This shows that cases IB and IIB are less efficient than the other cases. Returning to eqn. (39) we see that holding the other parameters constant and increasing p (~ [A] ) always increases PiNsc, although eventually a limiting value will be reached of P 1 N s c = P' + In ~.2
(42)
Hence we conclude that, if possible, p should be large enough to dominate the denominator. Turning to the variation o f P 1 N s c with ~"we find from eqn. (39) that P i N s c passes through a maximum. Increasing ~-(c~[ y ] l / 2 ) i n c r e a s e s the voltage of the cell through the effect of Y on the potential of the dark electrode. But if there is too m u c h Y, then too m u c h B is destroyed reacting with Y, and the power falls. In algebraic terms the (1 + ~-2)1/2 term is then the dominant term in the denominator of eqn. (39). Differentiation of eqn. (39) with respect to ~"gives the following relationships between the optimum values ~'m and (P1Nsc)m and P' and p: tY = ( P 1 N s c ) m + 2 + 2~'m2 - - In ~2 m
(43)
and p = ½ ~-2(1 + ~-2 )-l/2(P1Nsc)m
(44)
28
Using these equations, we construct the contour diagram in Fig. 3 which shows how (P1Nsc)m and ~m depend on P' and p. The optimised power function, (P1Nsc)m, increases from the bottom left-hand comer (~ 0.2) to the top righthand comer (~50). Large values are found for large values of P', because the greater the potential difference AE °, the more power will be produced. Large values of p on the right-hand side of the diagram correspond to having enough A to trap all the photons close to the illuminated electrode; the flux of photons is thereby converted into a flux of B, all of which is consumed on the electrode. In the top right-hand comer of Fig. 3 the contours for (P1Nsc)m are nearly parallel to the x-axis, showing little dependence on p. This is the limiting behaviour given by eqn. (42). In this region therefore (P1Nsc)m ~ AP and further increase in p does not lead to worthwhile increases in power. In the useful region the contours for ~'2 m are between 1 and 10 and so one may write
~'m~ 2 This is not surprising, since, if ~ < 1, increasing [Y] increases the potential difference without significant quantities of B, but when ~ > 1 then too much B is destroyed. On the far right of the diagram, where p ~ 100, larger values of ~m can be found because the optical length is so short that B still has a good chance of reaching the electrode before being destroyed by Y. In contrast, on the left of the diagram the optical length is longer than the generating length (p < 1). A photostationary state is then established where the concentration of B is given by: b/(a
+
b) = 1/(1 + ~'~')
Under these conditions the optimum performance is found for smaller values of
..~
I
....
!
~o-\
--":~--""'-
.... ~.-
,~
_~_~__~_.-" --~'~ I'O
0
~ ---;"
-0'5
j"
2"0..._."
.-'%
\
s/
s
."~-. ~.~ . ~
,.~'-"
_.~.--
,~/
s
.J
.'.~_ ~
./
i
i
p
~,.
i
s
, t 50,"
i
. ' ~ ' . J
, , " /
to
/
, ZO
o-s
,,
~
a-
Ioo
\ .,,' \,, ,-,,,,,-,,,,'" ,, , J \ ."
0"0
I X 0"5
/I
I'0
/
log p
I 1"5
Fig. 3. C o n t o u x d i a g r a m c a l c u l a t e d f r o m e q n s . ( 4 3 ) a n d ( 4 4 ) t o s h o w h o w ( P 1 N S C ) m ( ) and ~m (. . . . . . ) d e p e n d o n p ' a n d p. T h e p o w e r is a p p r o x i m a t e l y p r o p o r t i o n a l t o P I N S C ( e q n . 3 6 ) a n d as ~ is v a r i e d ( e q n . S 9 ) , P 1 N s c w i l l p a s s t h r o u g h a m a x i m u m ( P 1 N s C ) m at S'm; Fig. 4 s h o w s t h i s v a r i a t i o n f o r t h e p o i n t .. T h e p a r a m e t e r ~" ( e q n . 3 7 ) c o m p a r e s t h e generating length t o t h e kinetic length and c a n b e increased by increasing [Y]. The parameter p (eqn. 38) compares the generating length to the absorption l e n g t h a n d c a n b e i n c r e a s e d b y i n c r e a s i n g [ A ] . T h e parameter P ' ( e q n . 4 0 ) will b e m a i n l y d e t e r m i n e d b y £kP* (= F A E ® / R T ) w h i c h d e s c r i b e s t h e d i f f e r e n c e i n standard e l e c t r o d e p o t e n t i a l s o f the t w o c o u p l e s ; syst e m s c a n t h e r e f o r e b e r o u g h l y l o c a t e d o n t h e d i a g r a m u s i n g A E ° and the r i g h t - h a n d scale.
29 ~"(~ 0.1). The scale on the right-hand side in volts is so that one can obtain rough estimates of P' from known AE ° . From this Figure we conclude that the ideal cell is a case D cell with the following characteristics: P = X G / X c ~ 10
(45)
= XG/Xk ~ 2
(46)
X~ < XK < XG < X, p ' = Ap" + In ~ I ° e ~ 20
k[z]
(47)
or
A E e / V ~ 0.5 to 1.3
giving P 1 g s c ~ 20
(48)
THE OPTIMUM LOAD
Now that we have found the conditions the optimise PiNsc in eqn. (34), we return to the effect of varying the load on the cell, which is described by the two square brackets in that equation. Assuming t h a t the Uo terms are small and so neglecting their product, we write 08 ~ P 1 N s c [1 -- P~-I1 ln(uo 1 -- 1) -- u0(1 + ~-2)112{p + (1 + ~,2)112} ]
where we have also assumed from eqns. (45) and (46) that a ~/3 and hence from eqn. (41) that a > 1
(49)
Differentiation with respect to Uo gives the optimum values Uo.m for Uo and hence the conditions t h a t give the largest value of 03, 03,m:
u0~ ~- [Pl(1 + ~2)1/2 {p + (1 + ~-2)1/2} ]-1 and 03,m ~ P1Nsc [1 + P~-Il(ln Uo~ -- 1)] For the conditions t h a t optimise P1Nsc in eqns. (45) to (48) we find U0,m = bo,m/a D ~ 5 × 10 -8
(50)
and 03.m -~ 0.7 P1Nsc for p = 10 and P1 = 20. The value for Uo,m justifies the approximation made above and the choice of boundary in Fig. 2. Larger values of P1 or smaller values of p give values of 0 8 ~ even closer to P 1 N s c . Hence providing firstly that P1 > 20(E1 - - E ~ $ > 0.5 V) and secondly that p > 10, then at the m a x i m u m power point the function 03 is at least 70% of PiNsc. Therefore for this cell for the o p t i m u m conditions, P1Nsc is a good guide to the
3o power o u t p u t of the cell and we need not evaluate the square brackets in eqn. (34). We merely put them equal to 0.8. ESTIMATES O F T H E M A X I M U M
POWER
From the arguments above and eqn. (32), we therefore obtain for the maxim u m p o w e r Wm Wm ~ 0.8(P1Nsc)m A R T I o 0.8 A E* FNscAIo
(51) (52)
The p o w e r is obtained from a current ~ F A I o ( N s c ~- 1) delivered at a voltage of ~ A E *. Neither the current nor the maximum p o w e r voltage can be significantly larger than these values, so eqn. (52) describes the m a x i m u m possible p o w e r that can be obtained from any photogalvanic cell. If the cell is operated by polychromatic light, then at best only photons of energy greater than e A E ° will cause the photochemical reaction to occur, and eqn. (52) must be re-written.
Wm ~ 0.8 A E * F N s c A ~ E " Ig dE
(53)
where I~ (mol cm - 2 s- 1 eV - 1 ) is the spectral irradiance o f the incoming light. The p r o d u c t AE* f~aa~.--I~.dE is a function both of AE * and of the spectral distribution I~.. For average solar terrestrial radiation, this product has a maximum value when the integral is 10 - 7 mol cm - 2 s- I , which occurs when e A E ° 1.1 eV 2. Substituting this value in eqn. (53), and taking Nsc = 1, we obtain the maximum power that the cell will deliver when operated by average sunlight: Wm/A -~ 5 mW cm -2
(54)
This corresponds to a p o w e r conversion efficiency of ~7%. CONDITIONS FOR THE OPTIMUM CASE N o w we collect together the conditions for the o p t i m u m case to see if they can be realised in practice. Cell length a ~> 1 or l > D1/2/(ky) 1/2
(49)
Optical length and [A] p = XG/XE = aD(De/¢Io) 1/2 ~ 10
(45)
Kinetic length and [Y] = XG/Xk = (kY/~Ioe) 1/z ~ 2
(46)
31 Standard electrode potentials (47)
AE° > 0.5 V Electrode kinetics
A,B couple Y,Z couple
Illuminated electrode at × = 0
Dark electrode at X = 1
Reversible
Reversible or irreversible
( k ° B)o > > D [ X e Irreversible
Reversible
( k ° z ) o y expl{AP*l < < (kO,B)OaD
(k°zh >> D/l
Taking typical values of Io = 10 -7 mol cm - 2 s-1 for terrestrial solar radiation together with e = 10 s cm 2 mo1-1, D = 10 - 5 cm 2 s-1 and ¢ = 1 we find that the generating length XG is given by XG = (D/¢Ioe) ~ =
10 -a cm
(55)
This is the average distance t h a t A diffuses in the light intensity of Io before it is converted to B. The condition for p in eqn. (45) requires that the optical length, Xe, be a tenth of the generating length:
X~ ~ 10-4 cm
(56)
This in turn means aD ~ 0.1 M
(57)
The dye A has to be very soluble to absorb all the radiation close to the electrode. From eqn. (46) we require the kinetic length Xk to be comparable to the generating length and ky ~ 4Ioe ~
40 s-1
(58)
But the concentration of Y must be at least twice the concentration of B to ensure that the dark electrode remains unpolarised, and so
h < 4 × 1 0 aM- i s -~
(59)
The condition for the cell length (eqn. 49), which requires that it be larger than the kinetic length, is easily satisfied. Of the various conditions for the electrode kinetics, since Xe < l, the most severe is the requirement for the rapid reaction of A and B on the illuminated electrode: (k°,B)O > 0.1 cm s-1 Few redox couples have electrode kinetics as fast as this. To summarise, for the o p t i m u m case (p = 10) we require a very soluble dye with improbably fast electrode kinetics and rather sluggish homogeneous kinetics. These last two requirements m a y well be incompatible for electron transfer reactions that obey the Marcus theory [3]. It therefore seems unlikely t h a t the o p t i m u m case can be achieved in practice.
32
8
6
4
2
0
-4
-2
0
to~
2
Fig. 4. V a r i a t i o n o f P 1 N s c , w h i c h d e s c r i b e s the p o w e r o u t p u t , w i t h ~', w h i c h c o m p a r e s t h e g e n e r a t i n g l e n g t h to t h e k i n e t i c l e n g t h ( e q n . 3 7 ) . T h e c u r v e i~ c a l c u l a t e d f r o m e q n . ( 3 9 ) f o r p = 1 a n d P' = 2 0 ; t h i s p o i n t is p l o t t e d in Fig. 3 as -.
A MORE
REALISTIC
CASE
We now consider a more realistic case which could perhaps be achieved. If one takes p = 1 rather than p = 10 then from Fig. 3 the power is roughly half t h a t for p = 10, but the conditions are somewhat easier to satisfy: a D "~
10 mM
k < 4 × 104 M -1 s-1
(60)
kOAB , > 10-2 cm s- I
(61)
In view of the possible incompatibility of the requirements for fast electrode kinetics (eqn. 61) and sluggish homogeneous kinetics (eqn. 60) we explore how critical it is to optimize PiNsc with respect to k. Figure 4 shows a plot of eqn. (39) for p = I, P' = 20 and ~-i = 0; the position of the maximum of this plot is indicated in Fig. 3. It can be seen that a larger value of ~', say 10, does cause a significant reduction in PiNsc from its maximum value of 8.8 to 2.2. The corresponding condition for k (cf. eqn. 60) is eased to k < 4 × 106/l~r- I
S- 1
We therefore conclude that although an impossibly ideal photogalvanic cell would produce 5 mW em-2 a more realistic estimate for the power that might be achieved in practice is 0.5 mW cm-2. C O M P A R I S O N WITH PREVIOUS RESULTS
Gomer [4] has also considered the power efficiency of a photogalvanic cell containing a semipermable membrane, operating so t h a t only the Y,Z couple can reach the dark electrode. At the illuminated electrode, only the A,B couple, present in much smaller concentration, reacts. His cell is therefore identical in principle to the one we have discussed in this paper and his analysis identifies m a n y of the constraints we have also found. He concludes that a cell operated
33 by solar energy would produce a maximum p o w e r of ~ 1 mW c m - " , and that this would require k < 102 cm 3 mo1-1 s -1 and standard exchange current densities for the electrode reactions o f 102--103 A c m tool -1 (i.e. k ° = 10-3--10 -2 cm s-1 ). His conclusions are therefore similar to, though rather more pessimistic than, ours. However his analysis does n o t explore the interplay of the various lengths leading to the different concentration profiles. Instead he makes assumptions a b o u t the transport layers and the width of the illuminated region. Thus our analysis is more general and allows one to explore the optimization of the different parameters. There are few experimental data in the literature on cells of the t y p e we have discussed in this paper. The best d o c u m e n t e d examples are the thin layer (25-80 pm) iron-thionine photogalvanic cells developed by Lichtin et al. These have a SnO2 anode and a cathode of either Pt [5,6] or indium tin oxide [7]. The Fe2÷/Fe 3÷ couple is very irreversible at SnO2, while the thionine/leucothionine couple is moderately reversible. At Pt or indium tin oxide, both couples are electroactive, the thionine couple having faster electrode kinetics than the iron couple [ 7,8]. Thus the cell has one selective electrode and one rather unselective one. The following data on o u t p u t from a single cell have been reported, for 35 mW cm -2 incident white light: open circuit voltage ~ 0 . 1 5 V [5], maxim u m p o w e r voltage 0.05--0.10 V [4], short circuit current ~ 2 × 10 -4 A cm -2 [5], monochromatic quantum efficiency (electrons flowing per p h o t o n absorbed at 578, 589 or 620 nm) ~ 0 . 0 8 % for a 80 #m cell, ~0.15% for a 25/2m cell [7]. Our analysis indicates the following reasons for some of these results. (Lichtin et al. have independently reached similar conclusions.) Firstly AE e is only 0.44 V at pH = 2 and a rather higher value, assuming k remained constant, would be desirable. Secondly the short circuit current and the quantum efficiency are low because Xe is greater than Xk, SO that some of the photochemically generated Fe 3÷ and leucothionine is lost by back reaction in solution (case A rather than case D in Fig. 1). COMPARISON OF CONCENTRATION CELL WITH CELL WITH DIFFERENTIAL ELECTRODE KINETICS The previous paper [ 1] considered the maximum p o w e r that can be obtained from a photogalvanic concentration cell in which on both electrodes the couple Y,Z is inactive while the couple A,B is reversible. In this paper we have considered the differential case, for which the conditions are set o u t in the Introduction. This t y p e o f cell has the following advantages over the concentration cell: (1) The current is delivered at a voltage of ~ A E ' , the difference in the standard electrode potentials of the A,B and Y,Z couple, rather than at a potential o f R T/F. (2) The collection efficiency at short circuit is unity rather than ½. (3) The maximum p o w e r is a b o u t one hundred time larger. (4) Convective m o t i o n caused b y the heating effect of the light may reduce the concentration difference in the concentration cell b u t does not affect the performance of the cell with differential electrode kinetics.
34 The conditions required to obtain the m a x i m u m power from both types of cell are similar. First, all the light must be trapped close to the illuminated electrode. Secondly, the kinetics of the back reaction B + Y must n o t be too fast but should if possible be optimised using the concentration of Y. Thirdly, the kinetics of the electrode reaction, B -~ A + e on the illuminated electrode must be very rapid. Since these conditions hold for both types of cell it is probable that they also hold for cells where the electrode kinetics are intermediate between the two extreme cases discussed in detail. For both types of cell it is unlikely that any real system will fulfil all the conditions requires for their respective o p t i m u m cases, but the type of cell discussed in this paper will still generate a hundred times more power than the corresponding cell. CONCLUSION We conclude t h a t the inclusion of differential electrode kinetics is essential for the successful development of photogalvanic cells. If the necessary conditions can be obtained then one may reasonably expect an o u t p u t of 0.5 mW cm -2. ACKNOWLEDGEMENTS We t h a n k Dr. Stephen Feldberg for interesting discussions. In our nomenclature he has considered the case D cells. We are grateful to Mr. Andrew Foulds for helpful comments and criticisms. REFERENCES 1 2 3 4 5 6 7 8
W.J. Albery and M.D. Archer, Part III of this series, J. Electroanal. Chem., 86 (1978) 1 (this issue). M.D. Archer, s u b m i t t e d to Solar Energy. R.A. Marcus, J. Chem. Phys., 43 (1965) 679. R. Gomer, Electrochim. Acta, 20 (1975) 13. W.D.K. Clark and J.A. Eekert, Solar Energy, 17 (1975) 147. D.E. Hall, W.D.K. Clark, J.A. Eckert, N.N. L i c h t i n and P.D. Wfldes, J. Ceramic Soc., in press. D.E. Hall, J.A. Eckert, N.N. L i c h t i n and P.D. Wildes, J. Electrochem. Soc., in press. M.D. Archer, Appl. Electrochem., 5, (1975) 17.