J . Chem. SOC., Faraday Trans. 1, 1985, 81, 2647-2658 The Rotating Optical Disc-Ring Electrode Part 1 .-Collection of a Stable Photoproduct BY W. JOHN ALBERY* AND PHILIP N. BARTLETT Department of Chemistry, Imperial College, London SW7 2AY AND ANNA M. LITHGOW, JORGE RIEFKOHL, LORRAINE ROMERO AND FERNANDO A. SOUTO Department of Chemistry and Center for Energy and Environment Research, University of Puerto Rico, Mayaguez, Puerto Rico 00708, U.S.A. Received 14th September, 1984 The rotating optical disc-ring electrode consists of a transparent disc surrounded by a concentric ring electrode. Light is shone through the central disc to drive a photoredox system. The current from the photogenerated product is measured downstream on the ring electrode. The convective diffusion equation for this system is solved and analytical solutions are presented for the collection of a stable product.The theory takes into account the bleaching of the solution. Experiments performed on the iron-thionine system are found to be in good agreement with the theory. A useful electrode for the investigation of photoelectrochemical systems is the ‘ rotating photoelectrode’ developed by Johnson and co~orkers.~-~ This electrode consists of a central quartz disc through which light is shone into the electrolyte solution; the disc is surrounded by a concentric ring electrode, and the whole assembly is rotated. Such a ring-disc arrangement is similar to a conventional ring-disc electrode. The downstream ring electrode detects products and intermediates generated photochemicaly by the light shining through the disc.For instance, the electrode has been used to study the photopinacolisation of benzophenone in strongly alkaline alcohol-water mixtures.2 Again there are similarities to the optical rotating-disc electrode (ORDE), where light is shone through a semitransparent disc electrode and the photochemical intermediates and products are detected on the disc electr~de.~-~ Since the term ‘rotating photoelectrode’ does not discriminate between these two types of‘ electrode, we prefer to call the ring-disc system a ‘rotating optical disc-ring electrode ’ (RODRE). The advantage of using rotating-disc hydrodynamics is that one can often calculate how much of the electrochemically or photochemically generated product will be detected by the ring electrode.Hitherto these calculations for the RODRE2 have relied on the simulation method of Feldbergg as developed by Prater and Bardlo for ring-disc systems. In this paper we present an analytical theory for the collection of a stable photoproduct at the RODRE. Experimental results for the iron-thionine system are presented and shown to be in good agreement with the theory. 26472648 RING-DISC ELECTRODES THEORETICAL DISC ZONE We consider the following reaction scheme : disc zone : hv A + B ring electrode : B & ne + products. We assume that in the zone of the disc there is no significant radial variation in the irradiance of the incoming light. Then in the zone of the disc we have the following convective diffusion equation for the generation of B : a2b ab TI, D --u, -+- exp ax2 ax X& where b is the concentration of B, D is its diffusion coefficient, x measures distance normal to the electrode, u, is the velocity in the x direction, I, is the irradiance at the disc surface, cp is the quantum efficiency of the photoredox reaction and X&, the absorption length,5 is given by XE = (&a)-l.(2) In eqn (2) a is the concentration of A and E is its natural extinction coefficient. The absorption length describes the distance over which the light is absorbed in a Beer-Lambert profile. Eqn (1) is the same for the ORDE, but whereas for the ORDE the boundary condition for b on the disc surface is that b = 0, For the RODRE the boundary condition is that at x = 0, (abiax), = 0. (3) The interplay of the three lengths, the absorption length, the diffusion length, X,, and the generation length, XG, is much the same, where X,,, the thickness of the diffusion layer, is given by the Levich equation1' and XG describes the distance A diffuses in the irradiance I,: XG = ( D / ~ I , E ) ~ / ~ .(4) For most systems X& + XD, and we can ignore the exponential term in eqn (1). We can also write a+b = am ( 5 ) where a, is the bulk concentration of A. We solve eqn (1) with boundary condition (3) and the fact that b + 0 as x -+ GO by substitution of eqn ( 5 ) followed by splitting the solution into one part with the diffusion term inside the diffusion layer and a second part with the convective term outside the diffusion layer; the two solutions are joined at the edge of the diffusion layer.This procedure had been shown to work well for the ORDE.5 As shown in Appendix 1, we obtain the following result for the concentration of B on the disc surface : wherew. I. ALBERY et al. 2649 0.0 2 .o 4.0 6.0 Y Fig. 1. Plot of eqn (6) showing the surface concentration of B on the disc electrode as a function of the bleaching parameter y defined in eqn (7). diffusion -- layer 0 9 9 0.OL I I 1 0.0 ID 2 .o 3.0 X l X D Fig. 2. Concentration profile of B for the unbleached case ( y -+ 0) calculated from eqn (A 2) and (A 4). Inspection of eqn (6) shows that when X , >$XG ( y > 1) the solution inside the diffusion layer becomes bleached. The species A cannot survive its passage across the diffusion layer, and the concentration of B then becomes equal to the bulk concentration of A.On the other hand, when X , < X , ( y < 1) then we find that b, = a , y = a, vI0 &Xf,/D. (8) Under these conditions only a small fraction of A is converted into B. The right-hand side of eqn (8) shows the balance between the generation of B from A, vIoe, and the2650 RING-DISC ELECTRODES Table 1. Boundary conditions at x = 0 disc gap ring r < rl rl < r < r2 r2 < r < r3 loss of B by convective dilution DIPD, where1l.l2 XD = 1 .288(D/C)1/3 = 0.643 W-1/2~1/6D1/3 (9) Wis the rotation speed in Hz and v is the kinematic viscosity. Fig. 1 shows a plot of eqn ( 6 ) and the two regions for the bleached and unbleached cases. In fig. 2 we show the concentration profile for B from the disc surface across the diffusion layer and out into the solution when the system is unbleached.GAP ZONE Outside the disc zone the convective diffusion equation for B must include the term for radial convection, but not the photogeneration term : Following the usual ring-disc procedure this equation is transformed with x = (r/r1)(C/D)1/3x and 5, = ~ 4 ~ ) ~ - 11/3 to give Following our usual procedurel37 l4 we now split b into three contributions, bD, bG and b,, corresponding to the disc, gap and ring zones, respectively: The term b, is zero in the zone of the disc and the term b, is zero in both the zone of the disc and the zone of the gap. The boundary conditions at x = 0 for the different contributions are given in table 1. However, the differential equation for b , in the disc zone contains the photo- generation term. Hence b, does not obey eqn ( 1 1 ) ; inclusion of the photogeneration term and transformation with the ring-disc variables gives b = bD-bG-bR.( 1 2) where Y ”) = exp ( y / 2 ) cosh ( ~ l / ~ ) - 1w. J. ALBERY et al. 265 1 and b, is the concentration of B on the disc surface, given by eqn (6). In the unbleached case when y is smallfly) = 1. On the other hand, in the bleached casefly) --+ 0. In order for b to obey eqn (1 1) in the gap and ring zones it is necessary for bG to obey the following differential equation : a2bG abG =x---0.60bofly)(l +3<1)-2/3. ax2 at, In addition to the boundary condition in table 1, we have the two further boundary Taking the Laplace transform of eqn (15) with respect to 5, we obtain15 conditions that b, = 0 both at t, = 0 and as x + GO.where and r is the incomplete gamma function.16 By writing [ = sy3x w = s2/3 and G 1 /ndsl) eqn (16) becomes a 2 W p p - cw = - 1 in. With the boundary conditions the solution to this differential equation is17 w = Gi([)+3-lI2Ai([). On the disc surface [ = 0, and the value of w is17 wo = 2Ai(0)/2/3 = 0.41. Substitution of this value in eqn (17) and (18) gives an expression for 6G,o, which describes the variation of 6 , on the gap surface: For s, > 3 the incomplete gamma function can be expanded18 in terms of the form s;%, which can then be inverted term by term to give The condition s, > 3 corresponds to an assumption that, compared to the radius of the disc, the electrode has a thin ring and a thin gap.As discussed below, the treatment fails when [ ( r a / r ~ ) ~ - 13 2 1 /3. RING ZONE In the zone of the ring electrode b, obeys eqn (1 1) with 5, replaced by c2, where r2 = (r3 - r3/3r;. The usual Laplace transformation gives2652 RINGDISC ELECTRODES The ring current, i,, is given by where t;l = (r: - r 3 / 3 r : . From the boundary conditions in table 1 (ab/Q), = -(ab,/Q),, and the integral in eqn (21) is found by inverting 1/s2 times the Laplace transform of the gradient given in eqn (20). The value of bR,, on the right-hand side of eqn (20) is given by the boundary condition in table 1 : tR, 0 = bob2 - &, 0 (23 1 where b, is given by eqn (6) and &, , is the Laplace transform of bG, , [eqn (19)] with respect to r2. In eqn (19) tl is replaced by t2+ti, where Using the convolution integral we find that r; = (r;/r;-- 1 ) / 3 .( 2 4 ) where I&, q) is the incomplete beta function.l9 (20)-(23) and substitution for XD from eqn (9), gives Application of this lemma to each of the terms in eqn (19), together with eqn THE BLEACHED CASE We start by considering the case where the irradiance is large or the rotation speed is low, and the solution close to the disc electrode becomes bleached. Under these conditions, as discussed above, y is large [eqn (7)], fly) tends to zero [eqn (14)] and b, = a,. In eqn (25) we need only consider the first term in the large parentheses. In our previous workz0 we showed that the ratio of the limiting current at the ring electrode to that at the disc electrode was given by p/3, where = 35;.Inspection then shows that the ring current is identical to that which would be observed for the limiting current from the reaction of A. This is not surprising, since the light is converting all the A into B upstream of the ring electrode. Under these conditions the current will be proportional to the square root of the rotation speed. THE UNBLEACHED CASE At the other extreme the solution may be hardly bleached at all, Then y is small, and fly) + 1 in eqn (25). Substitution of eqn (8) for b, in eqn ( 2 5 ) gives I iR I = nFzr; @I,, @= l(XD, w-l/X,) W-li2M (26) where @ describes the fraction of light transmitted by the neutral density filter andw. J. ALBERY et al. 2653 Table 2. Values of M for common radius ratios ~ ~ ~ ~ ~~~~ r3/r2 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.12 1.14 1.16 1.18 1.20 1.22 1.24 0.147 0.192 0.23 1 0.268 0.302 0.334 0.364 0.393 , 0.42 1 0.475 0.526 0.574 0.621 0.667 0.71 1 0.754 0.147 0.192 0.232 0.269 0.303 0.335 0.366 0.396 0.424 0.478 0.529 0.579 0.626 0.672 0.717 0.148 0.193 0.233 0.270 0.305 0.337 0.368 0.398 0.427 0.482 0.533 0.583 0.63 1 0.678 0.148 0.194 0.234 0.272 0.307 0.339 0.371 0.401 0.430 0.485 0.538 0.588 0.637 0.684 0.149 0.195 0.236 0.273 0.309 0.342 0.373 0.404 0.433 0.489 0.542 0.593 0.642 0.150 0.196 0.237 0.275 0.31 1 0.344 0.376 0.407 0.436 0.493 0.547 0.598 0.648 0.151 0.198 0.239 0.277 0.3 13 0.347 0.379 0.410 0.440 0.497 0.551 0.603 0.152 0.199 0.241 0.279 0.3 15 0.350 0.382 0.413 0.444 0.501 0.556 0.609 0.I53 0.200 0.243 0.282 0.3 18 0.352 0.385 0.417 0.447 0.506 0.561 0.154 0.202 0.245 0.284 0.320 0.355 0.388 0.420 0.45 1 0.510 0.566 W is the rotation speed. In eqn (26) M, the detection efficiency, is a function only of the radii and is given by The ring current is M times the disc current that would be observed using an ORDE under these conditions. The fraction of incoming photons collected is given by X,/X,; the photons have to be absorbed in the diffusion layer for the product to have a chance of reaching the electrode. The geometric factor M is similar to the collection efficiency for a ring-disc electrode. Eqn (26) predicts that the ring current will vary with W-1/2; if the rotation speed is slower, the diffusion layer becomes thicker, more product reaches the electrode and the current is larger.Values of M calculated from eqn (27) for common radius ratios are collected in table 2. The theory was derived for the thin-ring-thin-gap geometry. The series cease to converge when and this provides a limit to the results in table 2. An interesting feature of the results in table 2 is that M does not vary much with the thickness of the gap given by r 2 / r 1 . This is in marked contrast to collection at an ordinary ringdisc electrode where the collection efficiency, N , decreases significantly as the width of the gap increases. We consider that the reason for the difference is that for the ring-disc electrode material is only generated on the electrode, and the bulk of the solution is an efficient sink for this material.For the RODRE the material is generated well out into the solution; only the small fraction of the material photogenerated in the diffusion layer finds the ring electrode; however, the rest of the material flowing past means that the outside edge of the diffusion layer is not such an efficient sink.2654 RING-DISC ELECTRODES When the system is neither completely bleached nor completely unbleached we write where b, is given by eqn (6), fly) by eqn (14), values of M are in table 2 and EXPERIMENTAL APPARATUS The apparatus for these experiments is similar to the d.c. apparatus described previ~usly.~ The photoelectrode, model AFDPOUQTPT, was supplied by Pine Instruments and has been described in detail by J0hnson.l For experiments with high light intensity the electrode supplied by the manufacturers was modified by inserting into the hollow electrode cavity a 12 in.quartz rod to act as a waveguide. The light was focussed onto the top of the rod and a layer of glycerol was used to make optical contact between the rod and the electrode disc. This arrangement increased the irradiance tenfold. The light source was the 180-W tungsten-halogen lamp of the side-illumination accessory of an Aminco DW2a spectrophotometer. This accessory is fitted with a rear parabolic mirror and associated optics, which were used for rough collimation of the light beam or to focus it onto the top of the quartz-rod waveguide. The excitation wavelength was selected with a three-cavity interference bandpass filter of 600 nm nominal wavelength supplied by Ditric Optics.The irradiance was varied using a set of neutral density filters, supplied by the same company, with values of @ of 0.75,0.53,0.37 and 0.14. The electrode assembly was rotated by an ASRPD Pine Instruments variable-speed rotator and an ASR motor controller. The jacketted cell and the reference (SCE) and counter- (Pt) electrodes were supplied by Astra Scientific Co. The potential of the rotating electrode was controlled by a M173 PAR potentiostat/galvanostat and the currents converted to voltages with a M 176 PAR current follower. The cell was thermostatted at 25 "C with a Haake low-temperature circulating bath. CHEMICALS AND SOLUTIONS All water was deionized, doubly distilled and passed through a four-bowl Milli-Q Super-Q millipore water-purification system.Acids were of Ultrex reagent grade supplied by the Baker Chemical Co. Thionine was supplied by Allied Chemical Corp. as the hydrochloride and purified by a series of recrystallisations from dilute HCl and aqueous n-butyl alcohol. The material was recrystallised until it gave a single spot on a thin-layer chromatogram (Analytech 0.25 mm normal phase silica gel, 10 x 20 cm plates, n-propyl alcohol-concentrated ammonia, 2: 1 v/v). The identity of the recrystallised thionine was confirmed at the high-field n.m.r. spectroscopy facility of Purdue University (470 MHz, [2H,]DMSO). Concentrations of thionine were found from the electronic absorption spectra of solutions diluted to ca. 1 pmol dm-3 (A,,, = 598 nm, E = 5.4 x lo4 dm3 mol-1 cm-l, 50 mmol dm-3 H,SO,).The experiments were carried out with 15-70 pmol dmb3 thionine in 50 mmol dmP3 aqueous H,SO, ; the concentration of the quencher, iron(rr1) ammonium sulphate (Fluka Chemical Co.), was > 5 mmol dm-3. This concentration of quencher is enough to trap all the triplet thionine. With no added FelI1 the maximum concentration of photogenerated FeIII is 140 pmol dm-3. Using the known value of the rate constant for the reaction of leucothionine and Fe"18 we estimate the lifetime of leucothionine to be > 10 s. Therefore, the loss of leucothionine in its passage from the disc to the ring is insignificant. Deoxygenation of the thionine-ironfm) solutions was effected by bubbling ultra-high-purity argon supplied by General Gases. MEASUREMENT OF IRRADIANCE AND PHOTOCURRENTS The flux of photons passing through the disc electrode was measured with a Spectra-Physics laser power meter (model 404, 450-900nm).The light was shone through the transparent electrode onto the collector. The irradiance, I,/mol ern+ s-l, was calculated from I, = F'/L hv A ,w. J. ALBERY et al. 2655 0.0 Fig. 3. Ring currents plotted according to eqn (26) against @/ W1/2 where @ is the transmittance of the neutral density filler and W is the rotation speed of the electrode. Values of W in Hz are as follows: a, 1; ., 2; A? 4; +, 6; V, 8; 0, 12; 0, 16; A, 25; x , 36 and V? 49. The values of @ were 1.0, 0.73, 0.53, 0.37 and 0.13. were F' is the power-meter reading in W and A , is the area of the collector exposed to the light (0.143 cm2).The electrode radii were measured with a travelling microscope and were found to be in good agreement with the manufacturers' specifications (rl = 0.501 cm, r2 = 0.507 cm and Y, = 0.625 cm). The ring currents were recorded on a bright Pt electrode (cleaned with concentrated HNO, and concentrated H2S04) using an 8120 Bascom-Turner digital recorder. They are the average of 500 readings with a sampling interval of 50-100 ms per point. The Pt ring was held at the potential of zero dark current (280-300mV us SCE). At this potential we find that a thionine-coated electrode reduces leucothionine but that there is negligible current from the photogenerated FelI1. RESULTS AND DISCUSSION We start with a set of results obtained using a relatively low irradiance where the RODRE did not have the waveguide inserted.Under these conditions there is no bleaching and so the ring current should be described by eqn (26). Fig. 3 shows a set of 50 results obtained at 5 different irradiances and 10 different rotation speeds. In accordance with eqn (26) the ring current has been plotted against OW-;. A good straight line is obtained. In comparing the gradient of the line in fig. 3 with that calculated from eqn (26) there is a problem with the value of I, to be used. As found by Johnson and coworkers,2 the irradiance is not uniform under the disc. Typical results for our electrode show that the irradiance at the edge of the disc is some 30% less than that at the centre. This result is similar to that found by Johnsod and coworkers.2 The ring electrode is more affected by the situation on the outside edge2656 RING-DISC ELECTRODES Table 3.Calculation of gradient in fig. 3 rl = 0.501 cm q~ = 0.27 lo, + = 0.73 nmol cm-2 s-l D = 5.8 cm2 Ms-l am = 70 pmol dm-3 E = 0.129 pmol-l dm3 cm-' M = 0.742 gradient [eqn (26)] = 1.10 pA Hz'/~ gradient (fig. 3) = 1.26 pA Hz112 -1 I I I -3 -2 -1 log [ (@/W'i2)/Hz-'i2 1 0 Fig. 4. Ring currents plotted according to eqn (29) against log (@/ W ) , where from eqn (3 1) y a @/ W for four different concentrations of thionine. At the higher irradiances and lower rotation speeds the solution becomes bleached. The thionine concentrations in pmol dm-3 were as follows: D, 65; .,45; 0, 29 and 0, 15. of the disc than by that at the centre. Therefore, we have used the value of the irradiance found at the outside of the disc.In table 3 we compare the calculated value of the gradient from eqn (26) with that found from fig. 3. The values agree to within 7%. This is reasonable agreement given the uncertainties about the uniformity of the irradiance. Next we turn to the experiments with the waveguide inserted into the RODRE. The irradiance is now large enough for the solution to be partly bleached, and we havew. J. ALBERY et al. 2657 to use the full form of eqn (25) as rearranged in eqn (28). Experiments were carried out at 4 different concentrations of thionine, 5 different irradiances and 7 different rotation speeds. The results of the 140 experiments are plotted in fig. 4. Using eqn (6) and (14) we have further rearranged eqn (28) to give where and from eqn (7) The function g(y) depends only on the geometry and the dimensionless parameter y, which describes the balance between bleaching and convective dilution.When y is small, corresponding to no bleaching, g(y) x M ; on the other hand when y is large for the bleached case g(y) x P/”y. Experimental data are plotted according to eqn (29) in fig. 4, where from eqn (31) y varies with @/W. Good agreement is found with theoretical curves calculated from eqn (29). In particular, the vertical separation between the curves for the different concentrations is correctly described by the log a, term, and as required the bleaching becomes significant for each concentration at the same value of y. We conclude that these experiments show that our theoretical description of the RODRE is correct.The experimental part of this work was sponsored by the U.S. Department of Energy, Office of Basic Energy Sciences under contract no. DE-AS05-82ER12088. We thank the U.S. National Science Foundation, the S.E.R.C. and the Atlantic Richfield Foundation for further financial support, and we acknowledge the assistance of the Biological Magnetic Resonance Facility of Purdue University. APPENDIX In this appendix we solve eqn (1) for the concentration of B in the zone of the disc. We first assume that X, is sufficiently large for the exponential term to equal unity throughout. We normalise the distance x with the Levich diffusion length,” X,, defined in eqn (9): cy = X/X,. v, = ex2 In the convective term5 where C = 8.03 W/2v-1/2.We also write Then eqn (1) becomes u = b/a,. where y is defined in eqn (7). u + 0 as cy -, 00 gives Outside the diffusion layer, ty > 1, we ignore the diffusion term in eqn (A 1); integration with (A 2) In (1 - u) x - y/21y. In particular, at the edge of the diffusion layer where cy = 1, u1 x l-exp(-y/2) 872658 RING-DISC ELECTRODES Inside the diffusion layer, ty < 1, we ignore the convective term in eqn (A 1); integration, with the boundary condition au/i3x = 0 at ty = 0, gives 1 - u = (1 - ul) cash (y1/3 ty)/cosh (y”‘). (A 4) Substitution of eqn (A 3) in eqn (A 4) gives the result at y = 0 for uo = bo/a, in eqn (6). 1 2 3 4 J 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 D. C. Johnson and E. W. Resnic, Anal. Chem., 1972, 44, 637. J. R. Lubbers, E. W. Resnick, P. R. Gaines and D. C. Johnson, Anal. Chem., 1974,46, 865. P. R. Gaines, V. E. Peacock and D. C. Johnson, Anal. Chem., 1975,47, 1393. W. J. Albery, M. D. Archer, N. J. Field and A. D. Turner, Faraday Discuss. Chem. Soc., 1973, 56, 28. W. J. Albery, M. D. Archer and R. G. Egdell, J. Electroanal. Chem., 1977, 82, 199. W. J. Albery, W. R. Bowen, F. S. Fisher and A. D. Turner, J. Electroanal. Chem., 1980, 107, 1. W. J. Albery, W. R. Bowen, F. S. Fisher and A. D. Turner, J. Electroanal. Chem., 1980, 107, 11. W. J. Albery, P. N. Bartlett, W. R. Bowen, F. S. Fisher and A. W. Foulds, J. Electroanal. Chem., 1980, 107, 23. S. W. Feldberg, in Electroanalytical Chemistry, ed. A. J. Bard (Marcel Dekker, New York, 1969), vol. 3. K. B. Prater and A. J. Bard, J. Electrochem. Soc., 1970, 117, 207. V. G. Levich, Physicochemical Hydrodynamics (Prentice Hall, Englewood Cliffs, N.J., 1962), W. J. Albery and M. L. Hitchman, Ring-Disc Electrodes (Clarendon Press, Oxford, 1971), p. 15. W. J. Albery, B. A. Coles and A. M. Couper, J. Electroanal. Chem., 1975, 65, 901. W. J. Albery, R. G. Compton and A. R. Hillman, J. Chem. Soc., Faraday Trans. I , 1978,74, 1007. G. E. Roberts and H. Kaufman, Tables of Laplace Transforms (W. B. Saunders, New York, 1966), p. 22. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 260. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 448. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 263. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 944. W. J. Albery, S. Bruckenstein and D. T. Napp, Trans. Faraday Soc., 1966, 62, 1932. pp. 63-69. (PAPER 4/ 1590)