6128
J . Phys. Chem. 1989, 93, 6128-6136
Surface Coverage. The coverage dependence in S E R R S was studied experimentally (for CoPc molecule) and theoretically by Zeman et aL5 In this work the conclusion was that the negative dependence of S E R R S with increase in coverage was due to the quenching of the surface plasmon resonance by the adsorbate. The experimental results of the present work indicate that the coverage dependence of S E R R S is negative with the maximum occurring well below a monolayer coverage. Coverage dependence studies of SERS were hindered by a very large increase in the fluorescence signal with film thickness. Since the PT'CDA and PTCDIMe dyes strongly absorb in the visible region, the present results obtained with the 488-nm laser line seem to support the physical explanation that negative dependence of SERRS intensities with coverage is due to strong damping of the metal particle resonance by the dye.
in the visible region. Raman bands observed as doublets in the R R S spectra of thin solid films of PTCDA and PTCDIMe were assigned to molecular species since they were also seen in SERRS at submonolayer coverage. It was also concluded that R R S frequencies, not present in the normal Raman spectrum, could be assigned to non-totally-symmetric vibrations observed due to Herzberg-Teller contributions to the intensities. The frequency and the intensity of the plasmon absorption of the metal particles were affected by the dye coating. Coverage dependence studies clearly showed that S E R R S enhancement peaks at submonolayer coverage for both molecules. Enhancement factors for submonolayer coverage were estimated to be > lo3. The same coverage dependence was found for Ag island films and Ag-coated Sn spheres.
Conclusions
Acknowledgment. Financial assistance from NSERC of Canada is gratefully acknowledged.
Comparative experimental studies of silver-coated Sn spheres with Ag and Au island films have shown that the Sn/Ag surface may be used as a SERS active substrate for wide frequency range
Registry No. PTCDA, 128-69-8; PTCDIMe, 5521-31-3; Ag, 744022-4; Sn, 7440-31-5; Au, 7440-57-5.
Voltammetry of Semiconductor Electrodes. 2. Cyclic Voltammetry of Freely Dlffusing Redox Species and Rotatlng Semiconductor Disk Voitammetryt Patrick G . Santangelo, Gordon M. Miskelly, and Nathan S. Lewis* Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91 125 (Received: November 16, 1988; In Final Form: March 9, 1989)
A model electrode circuit has been utilized to describe the cyclic voltammetric response of a semiconductor photoelectrode in contact with a freely diffusing reversible redox system. This equivalent circuit formalism also has been used to generate a set of working curves for the steady-state voltammetric behavior of a reversible redox system at a rotating semiconductor disk electrode. The model electrode circuit consisted of an ideal photodiode in series with a metal electrode. The current-voltage solutions have been obtained under varying conditions of illumination, scan rate, or rotation velocity, and the results are presented in a working curve format.
Introduction Although cyclic voltammetry has been used extensively to probe the behavior of semiconductor interfaces,'" at present there is no analytical theory for the cyclic voltammetric response of semiconductor photoelectrodes. Nicholson and Shain's' quantitative analysis of metal electrode cyclic voltammetry hastened the widespread popular adoption of the triangular potential sweep technique,* and it might be expected that a similar theoretical framework would be equally useful for the quantitative study of voltammetry at semiconductor surfaces. In a previous study? we utilized an equivalent circuit for analyzing the governing equations of cyclic voltammograms at semiconductor electrodes. Using this formalism, we obtained analytical solutions for the cyclic voltammetry of semiconductor electrodes in contact with nondiffusing, electrochemically reversible, surface-attached redox species. An advantage of the approach is that it yields voltammetric I-V solutions in a general working curve format. In the present work, we extend this methodology to the cyclic voltammetric behavior of freely diffusing reversible electroactive species and also present general working curves for the previously discussed situation'O*'laof the steady-state voltammetric behavior of reversible redox systems at rotating semiconductor disk electrodes. Tractable solutions to the voltammetric equations have been obtained by using the equivalent circuit for a semiconductor/liquid junction depicted in Scheme I. This circuit is expected to be an accurate physical model for situations in which interfacial kinetics are not the rate-determining recombination step for photogenerated Contribution No. 7869. *Author to whom correspondence should be addressed.
0022-365418912093-6128$01SO10
carriers in the semiconductor. Thus, the conclusions obtained from this circuit representation are restricted to the behavior of ~~
~~
(1) (a) Schneemeyer, L. F.; Wrighton, M. S. J. Am. Chem. Soc. 1979,101, 6495. (b) Schneemeyer, L. F.; Wrighton, M. S.; Stacy, A,; Sienko, M. J. Appl. Phys. Lett. 1980, 36,701. (c) Yeh, L A . R.;Hackerman, N. J . Phys. Chem. 1978,82,2719 (d) Kautek, W.; Gerischer, H. Ber. Bunsen-Ges. Phys. Chem., 1980,84, 645. (2) (a) Baglio, J. A,; Calabrese, G. S.; Kamieniecki, E.; Kershaw, R.; Kubiak, C. P.; Ricco, A. J.; Wold, A,; Wrighton, M. S.;Zoski, G. D. J . Electrochem. SOC.1982, 129, 1461. (b) Kubiak, C. P.; Schneemeyer, L. F.; Wrighton, M. S. J . Am. Chem. SOC.1980, 102, 6898. (c) Calabrese, G. S.; Wrighton, M. S. J . Am. Chem. SOC.1981, 103, 6273. (d) Baglio, J. A.; Calabrese, G. S.; Harrison, D. J.; Kamieniecki, E.; Ricco, A. J.; Wrighton, M. S.;Zoski, G. D. J . Am. Chem. SOC.1983, 105, 2246. (e) Simon, R. A.; Ricco, A. J.; Harrison, D. J.; Wrighton, M. S.J. Phys. Chem. 1983,87,4446. (f) Aruchamy, A.; Wrighton, M. S. J . Phys. Chem. 1980, 84, 2848. (g) Tanaka, S.;Bruce, J. A,; Wrighton, M. S.J. Phys. Chem. 1981, 85, 3778. (h) Aruchamy, A.; Bruce, J. A,; Tanaka, S.;Wrighton, M. S.J. Electrochem. SOC.1983, 130, 359. (3) (a) Frank, S. N.; Bard, A. J. J . Am. Chem. SOC.1975, 97, 7427. (b) Kohl, P. A.; Bard, A. J. J . Am. Chem. SOC.1977, 99, 7531. (c) Laser, D.; Bard, A. J. J . Phys. Chem. 1976, 80, 459. (d) Kohl, P. A.; Bard, A. J. J. Electrochem. SOC.1979, 126, 59. (e) Di Quarto, F.; Bard, A. J. J. Electroanal. Chem. 1981, 127, 43. (f) Fan, F. R. F.; White, H. S.; Wheeler, B. L.; Bard, A. J. J . Am. Chem. SOC.1980, 102, 5142. (g) White, H. S.; Fan, F. R. F.; Bard, A. J. J . Electrochem. SOC.1981, 128, 1045. (h) Koval, C. A.; Austermann, R. L. J. Electrochem. Soc. 1985, 132, 2656. (i) Koval, C. A.; Austermann, R. L.; Turner, J. A,; Parkinson, B. A. J. Electrochem. SOC.1985, 132, 613. (4) (a) Singh, P.; Rajeshwar, K.; DuBow, J.; Job, R. J. Am. Chem. SOC. 1980, 102,4676. (b) Bookbinder, D. C.; Bruce, J. A.; Dominey, R. N.; Lewis, N. S.; Wrighton, M. S . Proc. Natl. Acad. Sci., U.S.A. 1980, 77,6280. (c) Lewis, N. S.;Wrighton, M. S. Science 1981,211, 944. (d) Chao, S.; Robbins, J. L.; Wrighton, M. S . J . Am. Chem. SOC.1983, 105, 181. (e) Bookbinder, D. C.; Lewis, N. S.; Wrighton, M. S . J. Am. Chem. SOC.1981, 103, 7656. ( f ) Lewis, N. S.; Bocarsly, A. B.; Wrighton, M. S.J . Phys. Chem. 1980,84, 2033. ( g ) Lewis, N. S.; Wrighton, M. S. ACS Symp. Ser. 1981, 146, 37.
0 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 16, 1989 6129
Voltammetry of Semiconductor Electrodes SCHEME I: Equivalent Circuit Used as a Model for the Semiconductor/Liquid Interfacea
iter
U
w +
Theory Diode Element Behavior. The equivalent circuit in Scheme I implies that the current through the diode element (1,) equals the current through the electrode element (Ie), while the voltage applied to the diode/electrode system (V,) must be the sum of the diode ( Vd) and electrode (V,) voltage drops. V,, Vd, and V, are implicit functions of time, t , for the cyclic voltammetric experiment but are independent of time for steady-state rotating disk voltammetry. For the ideal diode considered in this work, the analytical expression for the I-V response is given by ( l):I3 Id = IL Io[exp(-qVd/Ak7‘) - 11 (1)
+
In this equation, Io is the reverse saturation current, IL is the photogenerated current, and vd is the voltage developed across the diode. The diode quality factor ( A ) is assumed to be unity for our “ideal” diode discussion, although, if desired, the calculation procedures described in this work could be modified to provide for an analogous treatment with A # 1.0. The sign conventions in (1) are appropriate when the diode behaves as a model for an n-type photoelectrode system. Also, anodic currents are considered to be positive, and the voltage across the diode is determined with respect to the metal electrode potential as a reference (Scheme 1). DiodelMetal Electrode Circuit Behavior with a Freely D q fusing Electroactive Redox System. We consider the reversible redox reaction R 0 ne- and, for simplicity, assume that n = 1 and that O(x,t) = 0 for x 2 0, t = 0. The boundary condition at the interface is8
-
a V, is the voltage, applied between the working electrode and the reference electrode, vd is the resulting voltage drop across the photodiode, and V, = V, - vd is the voltage appearing across the electrode/solution interface.
“reversible” redox systems, i.e., redox systems in which the surface concentrations are evaluated by application of the Nernst equation at the appropriate electrode potential. The separation of the diode and electrode elements also assumes that the individual circuit components act independently in their I-V behavior, so the analysis is only rigorous for semiconductor/liquid junctions in which the open-circuit voltage is independent of the value of the solution redox potential. Within these restrictions, the approach is quite general and yields working curves for the various scenarios of experimental interest. Details of the equivalent circuit formalism and the calculation procedures are found in the next section, and the resulting voltammetric behavior is discussed under Results and Discussion.
--
+
-
PY,
[R T ( V , ( t )- E-’) = 4
1
where Eo’is the formal potential of the redox system. The potential is swept from the initial value ( E i )to the switching potential ( E f )at a scan rate u, and the electrode surface area is A,. The flux at the metal electrode surface ( x = 0) was obtained by a digital simulation technique using the finite difference method of Feldberg14with a modification of the expanding grid methodl5 originally proposed by J ~ s l i n - P l e t c h e r . ~Assuming ~ ~ ~ ~ semiinfinite linear diffusion, while placing the metal electrode at the exterior edge of the first volume element, yields the following additional boundary conditions at the interface:
= DR(RI - R O ) / f i l
(3)
f o = DO(0l - OO)/A%
(4)
fR
fR
(5) (a) Chazalviel, J. N.; Truong, T. B. J . Electroanal. Chem. 1980, 114, 299. (b) Byker, H.; Wood, V.; Austin, A. J. Electrochem. SOC.1982, 12, 1982. (c) Nadjo, L. J . Electroanal. Chem. 1980, 108, 29. (d) Bard, A. J.; Bocarsly, A. B.; Fan, F. R. F.; Walton, E. G.; Wrighton, M. S. J. Am. Chem. Soc. 1980, 102, 3671. (e) Bard, A. J.; Fan, F. R. F.; Gioda, A. S.; Nagasubramanian, G.; White, H. S. Faraday Discuss. 1980, 70, 19. (f) Bocarsly, A. B.; Walton, E. G.; Bradley, M. G.; Wrighton, M. S. J. Electroanal. Chem. 1979,100,283. (g) Bruce, J. A.; Wrighton, M. S. J . Electrochem. SOC.1981, 122, 93. (h) Bocarsly, A. B.; Walton, E. G.; Wrighton, M. S. J . Am. Chem. SOC.1980, 102, 3390. (i) Lewis, N. S.;Wrighton, M. S. J . Phys. Chem. 1984, 88, 2009. (6) (a) White, H. S.; Ricco, A. J.; Wrighton, M. S. J . Phys. Chem. 1983, 87, 5140. (b) Bocarsly, A. B.; Bookbinder, D. C.; Dominey, R. N.; Lewis, N . S.; Wrighton, M. S. J . Am. Chem. SOC.1980,102, 3683. (c) Chyan, 0. M.-R.; Ho, S.-I.,Rajeshwar, K. J . Electrochem. SOC.1986, 133, 531. (d) Szabo, J. P.; Cocivera, M. J . Electroanal. Chem. 1988, 239, 307. (e) Feng, Q.;Cotton, T. M. J . Electrochem. SOC.1988, 135, 591. (7) Nicholson, R.; Shah, I. Anal. Chem. 1964, 36, 706. (8) Bard, A. J.; Faulkner, L. F. Electrochemical Methods; Wiley: New York, 1980. (9) Santangelo, P. G.; Miskelly, G. M.; Lewis, N . S. J . Phys. Chem. 1988, 92, 6359. (IO) (a) Bruckenstein, S.; Rosamilia, J. M.; Miller, B. J . Phys. Chem. 1985, 89, 677. (b) Bruckenstein, S.; Miller, B. J . Electrochem. SOC.1982, 129, 2029. (,I1) (a) Decker, F.; Fracastoro-Decker, M.; Badawy, W.; Doblhofer, K.; Gerischer, H . J . Electrochem. SOC.1983, 130, 2173. (b) Gerischer, H. J . Electroanal. Chem. 1975, 58, 263. ( 12) Morrison, S. R. Electrochemistry at Semiconductor and Oxidized Metal Electrodes, Plenum Press: New York, 1980.
(2)
= -fO
Here, R , and 0, are the concentrations of species R and 0 in the first volume element, D, and Do are the diffusion coefficients for species R and O,fR(t) and f o ( t ) are the fluxes of species R and 0 to the electrode at time t , and is the distance between the position of the average concentration in the first box and the position of the plane of the e1e~trode.l~ Solving (2)-(5) yields a finite difference expression for the flux of electroactive species:
ax,
fR
=
4R1-
0 1
AX14
- +DO
(6)
DR
The concentration profile throughout the remaining volume elements was obtained by the conventional finite difference expressions of Fick’s law^.'^^'^ (13) (a) Sze, S. M. Physics of Semiconductor Devices, 2nd ed.; Wiley: New York, 1981. (b) Fonash, S. J. Solar Cell Deuice Physics; Academic Press: New York, 1981. (c) Fahrenbruch, A. L.; Bube, R. H. Fundamentals of Sofar Cells; Academic Press: New York, 1983. (14) (a) Feldberg, S. In Electrochemistry Calculations, Simulations and Instrumentation; Mattson, J., Ed.; Marcel Dekker: New York, 1972. (b) Feldberg, S. In Electroanalytical Chemistry, Bard, A,, Ed.; Marcel Dekker: Vol. 3, N e w York, 1969. (15) Feldberg, S. W. J . Electroanal. Chem. 1981, 127, 1. (16) J o s h , T.; Pletcher, D. J . Electroanal. Chem. 1974, 49, 171.
6130
The Journal of Physical Chemistry, Vol. 93, No. 16, 1989
Reference to (2)-(5) indicates that the calculations for the flux at a given electrode potential ( Ve(t))in the semiconductor/metal equivalent circuit are identical with those for a simple metal e1e~trode.l~ However, since Ve(t)is not necessarily linearly related to the elapsed time t , as would be the case for a single electrode system (where dVe(t)/dt = Av), a stepwise calculation is necessary to obtain V,, Ve,and the flux at time t . Substituting (2)-(5) into the constraint equation
v,= v, + V,
fR = 0.5(a - [ a 2 - 4bl1J2)
(8)
where I o + IL +-+FA,
is the projected surface area of the electrode, and 9 is defined in (2). Thus, the only system unknown in (12)-(14) needed to calculate I, (=I,) is the potential of the electrode, V,. Combining (12) with the circuit constraint, (7), yields a quadratic equation for I, in terms of V,. The solution is as follows:
I, =
- I, 2l1(
IO
)
+ I, + * + Il,a,m -
(7)
and solving for the specific case of interest where n = 1 and A = 1, a quadratic equation is obtained for the flux f R ( t ) . The solution to this quadratic is
RlDR a=AR,
Santangelo et al.
IoOR @FA,Do
(9)
*
where = (Do/DR)2/3@, with defined by (1 1). For each voltage V,, this allows determination of I, in terms of the experimental parameters Io, I,, v, w , DR, and Do. V, is then obtained from (l), and the calculation of the I-Vrelationship is then complete. For the situation in which Io l S ) , an anodic current plateau develops in the I-V response of the semiconductor electrode. This plateau occurs because the diode is driven into reverse
+
1 0.0
-0.4
0.4
Vs-Eo’ (V)
(V)
Figure 3. (a) Effect of photocurrent on cyclic voltammograms for a freely diffusing redox species at a metal electrode in series with an illuminated photodiode. The values for all other parameters are given in Table IA. The cyclic voltammogram of the electrode without the photodiode in the circuit is also shown (curve e). Curve a: IL = 1 .OO X A, 8 = 1.20 X Curve b: IL = 1.00 X lo4 A, 8 = 1.20 X IO-I. Curve c: IL = 1.00 X IO-’ A, 8 = 1.20. Curve d: IL = 7.50 X IOd, 8 = 1.60. (b) Plot of the voltage drop across the photodiode, Vd, as a function of the applied voltage, Vs,for the voltammograms shown in (a) All parameters are identical with those given for (a).
-
-0.6
~
Figure 4. (a) Effect of photocurrent on cyclic voltammograms for a freely diffusing redox species at a metal electrode in series with an illuminated photodiode. The values for all other parameters are identical with those in Figure 3 and are given in Table IA. The cyclic voltammogram of the electrode without the photodiode in the circuit is also shown (curve e). Curve a: I , = 5.00 X 10” A, 8 = 2.40. Curve b: I , = 3.00 X 10” A, 8 = 4.00. Curve c: IL = 1.60 X 10” A, 8 = 7.51. Curve d: I , = 9.00 X lo-’ A, 8 = 1.33 X 10’. (b) Plot of the voltage drop across the photodiode, vd, as a function of the applied voltage, V,, for the voltammograms shown in (a). All parameters are identical with those given for Figure 3a. I
I
I
I
u
0 0.2
-0.4
I
0.0
\
0.5
1.5
1.0
2.0
0 Figure 5. Plot of the errors caused by using AEi/z,AEp, and AEg as estimates of V, from the cyclic voltammograms for a freely diffusing redox species at a metal electrode in series with an illuminated photodiode (see Table 11). Curve a: AEw - lVwl. Curve b: AE,/z- IVJ. Curve C: AEpa- IV,l.
bias for sufficiently large V, (Figures 3b and 4b). Under these conditions, the transport impedance of the diode limits the current through the series circuit. In contrast, the cathodic wave does not have as strong a light intensity dependence as the anodic wave, because the impedance of the diode in the forward biased direction is much less than the impedance in the reverse biased direction. This can be seen from Figures 3a and 4a, as well as from the data in Table 11. The absolute error in V, resulting from the use of AE,, AE,, or AE,,2 is displayed in Figure 5. When 0 < 1 , AEi/, is the best approximation to V,, whereas aE, gives a more accurate estimate at larger values of 0. This is in accord with the arguments de-
6134
The Journal of Physical Chemistry, Vol. 93, No. 16, 1989
E Ei
l.O
I=-
0.5
-
Santangelo et al.
E 1.0
4
d
0.5 r
0.0-
0.0
-
-0.5
0.0 n
>
--0.2 0.0-
-0.4
I
C
b
h
> --0.2
-
-0.6
1
3
I
I
>v
-0.8 -0.4-
1
-0.6
L -0.8
-0.4
V,-Eo‘
0.0
0.4
(V)
Figure 6. (a) Effect of scan rate on cyclic voltammograms for a freely diffusing redox species at a metal electrode in series with an illuminated photodiode. The values for all other parameters are given in Table IA, except that I , = 1 X IO-’ A. The cyclic voltammogram of the electrode without the photodiode in the circuit is also shown (curve f ) . Curve a: u = 0.01 V/s, 6’ = 3.80 X IO-’. Curve b: u = 0.10 V/s, 6’ = 1.20. Curve c: li = 0.30 V/s, 6’ = 2.08. Curved: u = 0.70 V/s, 6’ = 3.18. Curve e: u = I .OO X 10’ V/s, 0 = 1.20 X IO’. (b) Plot of the voltage drop across the photodiode, Vd, as a function of the applied voltage, V,, for the voltammograms shown in (a). All parameters are identical with those given for (a).
scribed above regarding the increased diode impedance in reverse bias relative to that in forward bias. The data in Figure 5 also can be considered to be working curves, because they do not explicitly depend on the values of the “real” variables EO’, etc., but are only a function of the dimensionless parameter 8. An additional point of interest in the voltammograms of Figures 3 and 4 concerns the time dependence of the anodic current for V, >> Epa. In this region, when 8 < 0.6 the anodic current for the semiconductor decays as P I 2 , as would be found for a metal electrode.8 For 8 > 0.6, this relationship does not hold, and the system impedance is a combination of the mass-transport impedance of the redox couple and the transport impedance of the diode. There is no simple functional form for V, vs t at these large 8 values, and the wave shape in this region is best obtained by comparison with Figures 3 and 4. Semiconductor Cyclic Voltammetry- Variation with Scan Rate. On “real” axes, the I-Vchanges that result from a variation in I L are not identical with those that result from a change in the scan rate, even if these changes produce identical 8 values. This is because variation in ILalso produces a shift along the voltage axes relative to a fixed reference potential, e.g., E”. Additionally, for the system and changing the scan rate affects the value of Ip,m therefore affects the normalization parameter used to generate the working curves in Figure 1. These effects can be seen by contrasting the cyclic voltammetry behavior for a changing light intensity, Figures 3a and 4a, to the voltammetry for a changing scan rate, Figure 6. Apart from the difference in “real axes” mentioned above, the quantitative aspects of the voltammetry as a function of 8 are identical for a variation in scan rate and for a variation in light intensity. In Figure 6a, when 8 < 0.4, it is seen that the I-V response of the equivalent circuit is very similar to the I-Vresponse of the ideal metal/solution redox couple system. This is as expected from our previous discussion of the dependence of the voltammetry on 8. When the scan rate is increased so that 8 > 0.4, the anodic peak width, AEPl2,increases, while the normalized
-0.4
0.0
0.4
Vs-E liz,m(V> Figure 7. (a) Effect of photocurrent on the steady-state current-voltage response for a freely diffusing redox species at a metal rotating disk electrode in series with an illuminated photodiode. The values for all other parameters are given in Table IB. The steady-state current-voltage response of the electrode without the photodiode in the circuit is also shown (curve 1). Curve a: I , = 1.00 X A, 6’ = 2.09 X lo-’. Curve b: I , = 1.00 X lo-’ A, 6’ = 2.09 X Curve c: I , = 1.00 X A, 6’ = 2.09 X IO-’. Curve d: 1, = 2.00 X IO-’ A, 6’ = 1.05. Curve e: I , = 1.75 X lo-’ A, 6’ = 1.20. Curve f I , = 1.50 X IO-’ A, 6’ = 1.40. Curve g: I , = 1.25 X lo-’ A, 6’ = 1.67. Curve h: IL = 1.00 X IO-’ A, 6’ = 2.09 Curve i: I , = 6.50 X 10” A, 6’ = 3.22. Curvej: I , = 4.00 X 10” A, 6’ = 5.23. Curve k: I , = 5.00 X lo-’ A, 6’ = 4.19 X 10’. (b) Plot of the voltage drop across the photodiode, Vd, as a function of the applied voltage, V,, for the voltammograms shown in (a). All parameters are identical with those given for (a). anodic peak current decreases and ultimately reaches a limiting value (Figure 6). This is also consistent with the behavior expected when the reverse bias impedance of the diode dominates the I-V response of the equivalent circuit. In general, it is seen that the behavior at low scan rates in Figure 6 corresponds to the behavior at high light intensities in Figure 4a, as must be the case if the values of 8 are identical. Steady-State Voltammetry at a Rotating Semiconductor Disk Electrode-Effect of Light Intensity. The simulated steady-state I-V characteristics of the model rotating semiconductor disk electrode are depicted in Figure 7. For this figure, the variation in 0 was achieved by changing I, and holding all other parameters constant. For 8 C 1, the anodic limiting current was found to be independent of 0 and was equal to the anodic mass-transport(1 3). Decreases limited current of an ideal metal electrode, Ip:m in 8 below 1.0 only shift the I-V curve relative to E1/2,m.This latter effect occurs as a result of the increased V, produced by higher ILvalues. When placed in a 0 format, these results are in excellent agreement with previously presented diode/electrode voltammograms obtained for variation in light intensity under the assumption that I, 1, Ip = (I,, + IL)/2. This transition from metal/electrolyte control over I , to diode control over I , causes an abrupt change in the rate of change of El12with variation
The Journal of Physical Chemistry, Vol. 93, No. 16, 1989 6135
Voltammetry of Semiconductor Electrodes
B
1.0
i
0 4
X
0.5
n
3 u
v
0.0
so
b) 0.0-
k
n
b-0.5
--0.2
3
0
a
w
d
-
> 0.0
10.0
20.0
30.0
40.0
-0'41
50.0
-0.6
0 IVJ, caused by using AE1,~as Figure 8. Plot of the error, AE, estimates of V, as a function of 0 from the steady-statecurrent-voltage response for a freely diffusing redox species at a metal electrode in series with an illuminated photodiode (see Table 111).
in 8, and this is seen by reference to the unusual behavior found for AEi12- IV,l vs 8 in Figure 8. A similar argument can be used to explain the discontinuity in the correlation between 8 and Ep3/4 - EpiI4in Table 111. Steady-State Voltammetry at a Rotating Semiconductor Disk Electrode-Effect of Rotation Velocity. The I-V behavior for steady-state voltammetry as a function of rotation velocity is very similar to the response to changing light intensity. As was the case for cyclic voltammetry, the change in rotation velocity affects the normalization parameter Ip,,,, whereas changes in I L effect a shift along the "real" voltage axes due to a changing V,. However, the general behavior predicted by reference to the working curve of Figure 2, or to the AE1l2- IV,l vs 8 curve of Figure 8, or to the light intensity dependence in Figure 7 is evident in the voltammograms of Figure 9. A more detailed analysis of the dependence of these curves on 0 is straightforward using the concepts developed in the previous sections. The important point is that the 8 parameter does precisely describe these curves, and that the dependence of the I-Vcurves on rotation velocity can IL, I,) be used to determine important current parameters (Ip,,,, of the semiconductor electrode system. General Trends in Semiconductor Voltammetry. All of the voltammetric cases treated to date exhibited similar qualitative trends in a variety of areas. In part, this occurred because of the existence of 8 for all of these scenarios. An important consequence of this was that, for sufficiently small 8 values, all of the systems exhibited voltammetry that closely resembled the properties of a metal electrode. Additionally, each system contains several variables that are useful in determining the open-circuit voltage of the semiconductor electrode. Establishing the relationship between these parameters (AElI2:AEF, etc.) and V, should help resolve the controversy regarding interpretations of cyclic voltammetric data in terms of barrier heights,l band-bending values,*' or open-circuit voltages.3d The anisotropic behavior of the anodic and cathodic portions of the cyclic voltammograms may be responsible for the voltammetry of systems in which similar behavior has been attributed to different values of anodic and cathodic interfacial heterogeneous rate constant^.^'^"^^ The working curves contained herein should allow a quantitative analysis of these situations, provided that the voltammetry is obtained over a wide range of 8 values. Additionally, the absolute errors in V, as a function of 8 displayed in Figures 5 and 8 will allow application of the simulations even in cases where the voltammetric data exhibits a complicated wave shape. At a given 8, the agreement between these quantities and their predicted values is also an excellent test of the applicability of the theory to a particular semiconductor electrode system. The simulations have only been performed assuming a oneelectron redox system and a diode quality factor of unity. The more general case can be readily simulated by using the procedures
i -a
-0.8
-0.4
0.0
0.4
Figure 9. (a) Effect of rotation velocity on the steady-state currentvoltage response for a freely diffusing redox species at a metal rotating disk electrode in series with an illuminated photodiode. The values for all other parameters are given in Table IB, except that IL = 5.00 X lod A. The steady-state current-voltage response of the rotating disk electrode without the photodiode in the circuit is also shown (curve k). Curve a: w = 1.00 rpm, 8 = 1.32 X lo-'. Curve b: w = 45.0 rpm, 8 = 8.88 X lo-'. Curve c: w = 65.00 rpm, 8 = 1.07. Curve d: w = 80.00 rpm, 8 = 1.18. Curve e: w = 100.00 rpm, 8 = 1.32. Curve f w = 150.00 rpm, 8 = 1.62. Curve g: w = 250 rpm, 8 = 2.09. Curve h: w = 500 rpm, 8 = 2.96. Curve i: w = 1000.00 rpm, 8 = 4.19. Curve j: u = 10000.00 rpm, 8 = 1.32 X 10'. (b) Plot of the voltage drop across the photodiode, V,, as a function of the applied voltage, V,, for the voltammograms shown in (a). All parameters are identical with those given for
(4. described in this work. It is expected that changing the values of n and A will noticeably affect the working curves for the different voltammetric techniques. Thus, if an experimental system can be shown to exhibit I-V properties that are in accord with the working curves at several different values of 8, this should provide excellent evidence that n = 1 and A = 1 in that system. Alternatively, deviations from the predicted behavior could be consistent with values of n or A different from unity, and these should then be determined independently and used to generate appropriate working curves before evaluation of the voltammetric behavior. An important caveat to the application of these curves is the assumption that the diode and electrode elements act independently in the circuit. This assumption may not hold for the numerous semiconductor/redox combinations where V, depends on the redox potential of the solution. Consequently, the simulations contained in our work should not be directly applied to such systems. The complicated boundary conditions for these situations require a far more sophisticated model for carrier transport and recombination than has been employed in this work. In this respect, it is curious that the qualitative cyclic voltammetric behavior seems to be very similar for redox reported in the couples in the different regions of V , vs E(A+/A). The reasons for this will only become clear after a more detailed analysis of the voltammetry in each of these different regions of photoelectrode behavior. It is hoped that the above analysis will provide a foundation for the quantitative investigation of cyclic voltammetry and rotating disk voltammetry at semiconductor electrodes. The theory is relatively simple and can be readily tested with appropriate experiments. The extent to which it models real semiconductor/liquid interfaces must await the availability of quantitative experimental voltammetric data for a variety of semiconductors in contact with numerous redox couples. We hope that the existence of the theory will serve to stimulate the collection of this data and will thereby establish a quantitative basis for voltammetry at semiconductor surfaces.
J . Phys. Chem. 1989, 93, 6136-6141
6136
Acknowledgment. We thank the Department of Energy, Office of Basic Energy Sciences, and the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this work. N.S.L. also acknowledges support as a Dreyfus Teacher-Scholar and as an A. P. Sloan Foundation Fellow. We are indebted to Dr. Stephen Feldberg of Brookhaven National Laboratory for a critical reading of the manuscript and for numerous helpful discussions. We also acknowledge helpful suggestions regarding computing methodology from Professor Hans Andersen of Stanford University.
diode quality factor electrode surface area, cm2 bulk concentration of reduced species, mol/cm3 bulk concentration of oxidized species, mol/cm3 diffusion coefficient of reduced species, cm2/s diffusion coefficient of oxidized species, cm2/s initial potential, V final potential, V formal potential of redox species, V potential where I = Ip/4, V potential where I = IP/2, V potential where I = (3/4)Ip, V (EP + E,)/2 for cyclic voltammetry, V, or Eo’ + (2kT/3q) In (DR/Do) for rotating disk voltammetry, V EIl2for a metal electrode, V EIl2for a semiconductor electrode, V E,-- Epl2, V E,(metal) - E,(semiconductor), V E,(metal) - E,(semiconductor), V Eii2(metal) - l?l/2(semiconductor),V E, - E,, V flux of reduced species at electrode surface at time I , mol SKI cni2 flux of oxidized species at electrode surface at time I , mol s-I
current through electrode element, A photogenerated current of diode, A limiting anodic current at metal RDE, A reverse saturation current of diode, A peak current of diode/electrode circuit for rotating disk simulations, A anodic peak current of diode/electrode circuit for cyclic voltammetric simulations, A cathodic peak current of diode/electrode circuit for cyclic voltammetric simulations, A peak current at metal electrode for cyclic voltammetric simulations, A , or limiting anodic current at metal RDE, A current through diode/electrode system, A dimensionless current variable I/(I, + IL) concentration of oxidized species at distance x at time t , mol/cm3 concentration of reduced species at distance x at time 1, mol/cm3 concentration of oxidized species in first box at time I , mol/cm3 concentration of reduced species in first box at time I , mol/cm3 time, s voltage relative to the open-circuit voltage, V voltage across diode, V voltage across electrode, V open-circuit voltage, V voltage applied across diode/electrode circuit, V scan rate, V/s distance from electrode surface, cm distance between the average concentration of redox species in the first box and the plane of the electrode, cm total coverage of electroactive material, mol/cm2 dimensionless current ratio; Ip,,,/(IL + I,) =O(O,t)/R(O,t)= exp((nF/RT)(V,(t) - EO’)) expWIRT)(V, - EO’)l kinematic viscosity, cm2/s ( D ~ / D ~ ) ~ / ~ ~ (DO/DR)’’~@
current, A current through diode element, A
angular frequency of rotation, s-I
Activation Energy of the Catalytic Oxidation of Methylbenzenes over Metal Oxides Ragnar Larsson* and Bo Jonsont Group of Catalysis Research, Inorganic Chemistry 1 , Chemical Center, P.O. Box 124, S-221 00 Lund, Sweden (Received: November 21, 1988; In Final Form: February 21, 1989)
A model for the dynamics of catalysis is applied to the oxidation of methylbenzenes (toluene and xylenes). The stepwise change of activation energies found for data from literature is used to determine the frequency of that vibration mode that relates to the reaction coordinate. From this value, about 1040 h 50 cm-I, one can conclude that the distortion of the methyl rocking mode (IR absorption at 1042 f 4 cm-I) is of prime importance for the rate-determining step of the catalytic reaction. The anharmonicity parameter, x, is estimated for two cases from these data; the first one is by neglection of the contribution to the apparent activation energy from the heat of adsorption. This gives x = -2 cm-I. In the other case correction is made for the heat of adsorption, resulting in x = -5 f 1 cm-I.
Introduction The use of activation energy data to describe details of the dynamics of heterogeneous catalysis has for a long time been in disrepute. This situation undoubtedly is caused by the complexity of adsorption equilibria that gives a relation between the true activation energy and the heat of adsorption that may be quite comp1icated.I Recently, however, it has been pointed out by one of usz that for many reactions in heterogeneous catalysis one can observe a
’Present address:
Glass Research Institute, Box 3093, 35003
Vaxjo,
Sweden.
0022-3654/89/2093-6 136$01.50/0
certain stepwise change of activation energies within a series of similar catalysts operating on one and the same substrate. This empirical result has been further demonstrated, e.g., for the hydrocracking of alkanes in zeolite catalysts3 and NzO decompositi01-1.~ ( 1) Boudart, M.; Dj6ga-MariadassouKinetics of Heterogeneous Catalytic Reactions; Princeton University Press: Princeton, NJ, 1984. (2) Larsson, R . Z . Phys. Chem. Leipzig 1987, 268, 721. (3) Larsson, R. Proc. X X . Jahrestreffen Katalytiker DDR; Reinhardsbrunn, 1987;Catal. Today 1988, 3, 387. (4)Larsson, R.Proc. 20th Swedish Catal. Lund,1987; Catal. Today 1989, 4, 235.
0 1989 American Chemical Society
