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The Journal of Physical Chemistry, Vol. 83, No. 21, 1979
Vieth, W. R.; Howell, J. M.; Hsieh, J. H. J . Mernbr. Sci. 1976, 1 , 177. Vieth, W. R.; Sladek, K. J. J . Colloid. Sci. 1965, 20, 1014. Fenelon, P. J. In "Permeability of Plastic Films and Coatings to Gases, Vapors, and Liquids"; Plenum Press: New York; 1974. Komiyama, J.; Iijima, T. J . Polyrn. Sci., PartA-2, 1974, 12, 1465. Ludolph, R. A.; Vieth, W. R.; Venkatasubramanian, K.; Constantinaes, A. J . Mol. Catal. 1979, 5 , 197. Tshudy, J. A.; von Frankenburg, C. A. J . Poly. Sci., PartA-2, 1973, 1 1 , 2027. Vieth, W. R.; Eilenberg, J. A. J . Appi. folyrn. Sci. 1972, 76, 16.
T. I. Quickenden and G. K. Yim (22) Vieth, W. R.; Frangoulis, C. S.; Rionda, J. A. J. Colloid Interface Sci. 1966. 22. 454. (23) Vieth; W. '3;Tam, P. M.; Michaels, A. S. J . Collold Interface Scl. 1966, 22, 360. (24) Vieth, W. R.; Wang, S. S.; and Gilbert, S. G. Biotechnol. Bloeng. Syrnp. 1972, 3 , 285. (25) Weisz, P. B.; Hicks, J. S. Trans. Faraday SOC.1967, 63, 1815. (26) Weisz, P. B.; Hicks, J. S. Trans. Faraday SOC.1967, 63, 1897. (27) Vieth, W. R.; Amini, M. A,; Constantinides, A,; Ludolph, R. A. Ind. Eng. Chern., Fund. 1977, 16, 82. (28) Paul, D. R.; Koros, W. J. J . Polyrn. Sci. A2 1976, 74, 675.
A Comprehensive Model for Photovoltage Generation at Metal Electrodes In Contact with Solutions of Fluorescent Dyes T. I. Quickenden" and G. K. Yimt Department of Physical and Inorganic Chemistry, The University of Western Australla, Nedlands, W.A., 6009, Australia (Received February 27, 1979)
A comprehensive model involving both photochemical and electrochemicalsteps has been constructed for photoelectrochemical cells comprising identical, inert metal electrodes immersed in a solution containing a fluorescent dye and a nonphotosensitive redox couple. The model allows for the possibility of electrodic electron transfer to or from dye photolysis products formed in the bulk of the electrolyte as well as at the electrode surface. Steady-stateanalyses on the model lead to complicated relationships between open-circuit photovoltage, V,,, irradiance, E , the dark concentrations of the various species, and the various rate constants. When the photolytic concentrationperturbation and the open-circuit photovoltage are small, these relationshipssimplify to V,, = ( R T / F )In (1 + KIE),where K1 is a function of the dark concentrations of the various species and of the rate constants for excitation, fluorescence emission, intersystem crossing, photochemical reaction, and the various types of quenching, diffusion,and electron transfer. When the further and not necessarily identical assumption of low irradiance is made, the logarithmic relationship simplifies to the linear form V, = (RT/F)KIE.
Introduction The search for renewable sources of energy has led to an increasing interesP4 in photoelectrochemical cells because of their possible role as transducers of solar to electrical energy. Although photoelectrochemical studies commenced with the observations of Becquere16in 1839, the mechanisms of photovoltage generation in photosensitive electrode/electrolyte systems are still at an early stage of development. This is particularly true of cells containing fluorescent electrolytes, where photochemical and electrochemical processes combine to produce a substantially complicated system. Although semiconductor electrodes are assuming an increasing importance6$'in photoelectrochemical studies, the mechanisms of the simpler metal/fluorescent electrolyte cells still require considerable development and are the subject of this paper. In particular, the attention herein has been confined to the derivation of relationships between open-circuit photovoltage and irradiance8 as the first step in understanding the complicated interactions between electrochemical and photochemical variables in such cells. Two approaches to photoelectrochemical cells are currently available. On the one hand, Albery and Archerll have provided a good account of the electrochemical kinetics applying to light-sensitive concentration cells containing inert electrodes and two dye couples, one of which is photosensitive. On the other hand, Memming and Kursten3J2 have given attention to the reactive inter'Formerly nBe Tan. 0022-3654/79/2083-2796$0 1.OO/O
mediates produced by photolysis in such cell electrolytes and propose that semioxidized and semireduced forms of the fluorescent dye are responsible for the photoinduced electrodic electron transfers. There are other factors which also make the analysis of photoelectrochemical effects quite complex. Although Memming and Kursten3 have shown from depth-ofpenetration experiments that diffusion of photolysis products from the bulk of the solution to the electrode can be important, it is by no means clear that adsorbed dye does not play a similar and significant role in cells containing fluorescent solutions. A recent in~estigation'~ in fact shows that a layer of red material builds up on the illuminated electrode in cells containing gold electrodes in contact with rhodamine B solution and that, concurrently, the photovoltage from such a cell increases. Furthermore, recent studies14 of the rhodamine B-gold system using a rotating-disk photoelectrode confirm that a nondiffusive component of the photovoltage builds up over several days when a clean gold electrode is immersed in an aqueous electrolyte containing rhodamine B and ferrous and ferric ions. It consequently seems desirable to incorporate the possibility of surface as well as bulk photolysis in any comprehensive model of photoelectrochemical cells containing fluorescent dyes. A further area of neglect is the omission of well-known photochemical reactions and side reactions from the existing kinetic analyses of photoelectrochemical systems. The purpose of this paper is to develop a comprehensive model for the processes associated with photovoltage generation at metal electrodes in contact with solutions 0 1979 American Chemical
Society
A Model for Photovoltage Generation at Metal Electrodes
The Journal of Physical Chemistry, Vol. 83,
No. 27, 7979 2797
one of which is light sensitive. In a typical practical situation, one of these couples (e.g., (2)) may be an inorganic redox system such as Fe3+/Fe2+while the other couple (e.g., (1))may be a photosensitive system comprising a fluorescent dye such as thionine, together with its reduced form. Attention will be confined to the generation of opencircuit photovoltages and, consequently, problems relating to net current flow will not be examined. The kinetic analysis will begin by considering separately the reactions in a nonilluminated and an illuminated cell. T h e Dark Reactions. As well as electron exchange in the bulk of the solution
(b)
Ab + Z b As
Bb + Y b
(3)
2s
Flgure 1. Schematic diagrams of (a) the modified’ model of Hillson and Rideall’ and (b) Albery and Archer’s‘’ model.
of fluorescent dyes and to derive therefrom relationships between open-circuit photovoltage and irradiance.
Existing Models Two kinetic models are currently available for the elucidation of photoelectrochemical effects at inert metal electrodes. The first applies to a photoelectrode coated with a layer of solid dye and immersed in a non-lightabsorbing electrolyte solution. The second applies to a photoelectrode immersed in a solution containing a fluorescent dye and a nonphotosensitive redox couple. ( a ) Model for Electrodes Coated with Adsorbed Dye. This model (Figure la) represents a modification by Quickenden and Yimg of a scheme presented by Hillson and Rideal15 in which fluorescent dye A, in an adsorbed layer is capable of electrodic electron transfer to give a reduced form B, either in the dark or via an optically excited species $*, Photovoltage-irradiance relationships have been derivedg for this system previously. ( b ) Model for Electrodes Immersed in a Fluorescent Solution. This model (Figure l b ) is due to Albery and Archerll and involves a fluorescent dye Ab which converts via an optically excited species Ab* to a reduced form Bb. The species Ab is also capable of conversion to B b via electron transfer to a nonphotosensitive redox couple, Z b / Y b , which commonly comprises inorganic ions. The above species are all located in the bulk of the solution, but diffusion to the electrode surface is possible, where $, B,, Z,, and Y, (the subscripts indicating location at the surface) may undergo electrodic electron transfer. Several special cases are considered by Albery and Archer,ll who derived relationships between irradiance and the photolytic concentration perturbation it produces. Quickenden and Yim’O have extended this work to obtain relationships between the more commonly measured quantities, open-circuit photovoltage, and irradiance.
The Present Model Because a number of very large equations arise in the course of the following derivation, the more cumbersome of these have been relegated t o the Appendix, which contains a n unabridged derivation. T h e Appendix is available as supplementary material. Following Albery and Archer,ll the system considered comprises two inert, identical electrodes immersed in a liquid electrolyte containing the redox couples A+e-+B (1) and Y +e-+Z
there will be movement between bulk species and species at the surface of the electrode. These processes are laid out in eq 4-7, where species in the bulk of the solution are designated by the subscript b and species at the surface of the electrode are denoted by the subscript s. (4) (5)
3y, Yb
‘k,Y
Zb
‘k,z
Z,
(7)
In some cases the transport step may be followed by an additional step in which the surface species is adsorbed on to the electrode. If both steps are present, processes 4-7 are replaced by
where Q represents any of the species A, B, Y, and Z and the symbol Q with the subscripts b, a, and s corresponds to bulk species, unadsorbed species close to the electrode, and species adsorbed on the electrode, respectivelyamThe triply subscripted k’s in eq 8 represent rate constants for mass transport (with subscript c) or adsorption (with subscript a). Species at the electrode may also undergo mutual electron transfer
A,
+ Z,
k-b
B,
+ Y,
as well as the electrodic electron transfers
and
The rate constants k for the various processes are shown above and below the respective arrows in eq 3-11. It should be noted that k d , k - d , k,y, and k-rsy are electrodic rate constants and are therefore implicit functions of electrode potential. The distinction between bulk species and surface species is made by defining the latter as capable and the former as incapable of electrodic electron transfer. In reality, a sharp distinction does not exist, as the probability of electrodic electron transfer would not be expected to fall
2798
T. I. Quickenden and G. K. Yim
The Journal of Physical Chemistry, Vol. 83, No. 21, 1979 product
intersystem crossing to the triplet excited species 3A* which proceeds by an irreversible reaction with Z to form B and Y. The latter then migrate to the electrode. In sequence 11, migration of the dye to the electrode precedes excitation, intersystem crossing, and the subsequent production of photoproducts B and Y a t the electrode. The subscripts distinguishing the rate constants k in Figure 2 are defined in the Nomenclature section. It will be noted that s denotes surface species and b denotes bulk species. The excited species in Figure 2 are all susceptible to well-known unimolecular and bimolecular deactivating processes. Although there are a number of possible processes of each type, for simplicity only one unimolecular arrow and one bimolecular arrow are shown on the figure. In detail, an excited species A* may undergo the deactivating processes in eq 23, where kf and ki are the
oroducts
I
A t heat \;Ii
A t h u f
Flgure 2. Comprehensive photoelectrochemicalmodel when A is the photoactive species.
as a strictly rectangular function of distance from the electrode, although it does fall steeply. If reactions 3-11 are assumed to be in equilibrium in the dark, then
k
/ A,
kdAz*
A*
/
k u QI'
t
*
2A
+ heat
A t Qi t heat
first-order rate constants for fluorescence emission and internal conversion, respectively. The symbols kx, kq, and h, stand for the respective second-order rate constants for excimer formation, self-quenching, and quenching by other species which are denoted by Qi. In Figure 2 kl = kf + hi (24) and
k[Q1= k x [ A l + k q [ A l + hu[Qil
where [Id denotes the concentration in the dark. Substituting eq 13-16 in eq 17 gives ~ t s ~ k c s A / h - c s A ~ ~ h c s Z / ~ - c s Z ~ ~ A b ~ d= ~Zb~d
(20) Similarly, eq 18 and 19 may be expressed, respectively, h-ts(kcsB/h-csB)(hcsY
/k-csY)[BbId[Ybld
where Qi represents any of B, Y, and Z as well as any impurities present in the cell. When the system is illuminated, the concentrations of the redox species are perturbed from their equilibrium dark values. Mass balances then yield the following equations, where n designates the number of moles of the species denoted by the first subscript and the other subscripts b, s, 1, and d denote bulk, surface, light, and dark, respectively, as used in earlier equations: nAbl
+ nAsl + n B b l + n B d
=
+ nYsl + nZbl + nZsl
=
nAbd
[Abld
= h-rsA(hcsB/k-csB)
[Bbld
(21)
and nYbl
and krsY(kcsY/k-csY)
[Ybld
=
h-rsY(kcsZ/h-csZ)[Zbld
(22)
The Light Reactions. Either A or Z may be photoactive. For clarity, these two cases are treated separately. When the Electron Acceptor A Is the Photoactive Species. This is the situation in the iron-thionine system. The reduced form of the dye (species B) is present in relatively small amount and most of the light is absorbed by A, the unreduced dye. The Y / Z couple in this case is Fe3+/Fe2+.Although this couple may absorb some light, it will be substantially photolytically inert at the wavelengths normally used. Figure 2 shows the proposed reaction scheme for the illuminated system. The scheme contains two basic photochemical sequences which lead to electron transfer at the photoelectrode. In sequence I, the dye A is excited to a singlet excited state lA* which is then converted by
+ nAsd + n B b d + nBsd (26)
as hrsA(hcsA/k-csA)
(25)
nYbd
t n Y s d + n Z b d + nZsd (27)
In addition, since no current is flowing, the total number of moles of oxidized species must remain constant. Thus nAbl
+ nAsl + n Y b l + nYsl
=
nAbd
+ nAsd + n Y b d + n Y s d
(28)
In the presence of light, the concentration profile of each species can be obtained from steady-state analyses which yield
The Journal of Physical Chemistry, Vol. 83, No. 21, 1979 2799
A Model for Photovoltage Generation at Metal Electrodes
and
In the above equations, the functions gb (=k*b[AbIlE) and g, (=h*,[A,IIE) are the rates of photochemical reaction in the bulk of the solution and at the electrode surface, respectively. If, it is assumed, as Albery and Archer'l assume, that the irradiance, E , is constant over small distances close to the electrode, then h*b = 2.303&q, and k,, = 2.303&q, providing the higher powers in the expansion of the decadic term in Beer's law are neglected. 4 is the quantum efficiency of excitation and is the molar decadic extinction coefficient for the fluorescent dye. The subscripts distinguish bulk and surface species, but the optical parameters thereof would not be expected to differ unless the latter species were subject to substantial surface forces, as for adsorption. The operator Trb-s in eq 29-32 gives the rates of transference16 of the bracketed species, from the bulk of the solution to the electrode surface. As previously implied by eq 8, the rate of transference may simply be the rate of mass transport or, when surface adsorption occurs, will be the resultant rate of the conjoined mass transport and adsorption processes. In the case of static electrodes in unstirred solutions, a steady-state mass transport situation can nevertheless be expected, due to the formation of a static diffusion layer on the surface of the electrodes which results from the well-known17coupling of diffusion with thermal convection. A t this stage, it is useful to define the variable p which was introduced by Albery and Archer'l to represent the steady-state photolytic perturbation to species concentration when a dark electrolyte is subject to continuous illumination. Their approach assumes that the time required to set up a steady state is short compared with the duration of photoelectrochemical observations, which themselves occupy a time period in which irreversible photochemical decomposition (e.g., k, in (23)) is not significant.'* If pmrepresents the perturbation in the bulk of the solution and p o represents the perturbation at the electrode surface and if the lifetimes of all the excited species are short compared with the half-lives for their production and their photochemical reactions with ground-state species, then [Ab11 = [Abld - p m (33)
in the Appendix, which is available as supplementary material. Equations in the Appendix are designated as A l , A2, etc. The following outlines the major steps in the derivation and gives the important intermediate equations. Steady-state analyses on [lA,*]l and [3As*]1yield [3As*l~= k*&isE[Asli/(k~s + k~,[Qs11+ kis)
X
(h,' + ~ ~ ' [ Q S I I+ k p s [ Z ~ 1 1 ) (41)
In contrast, a steady-state analysis carried out on [Ballgives a complicated relationship (eq A2, Appendix) between an exponential function of open-circuit photovoltage, V,,, a cubic function of po, and linear functions of pmand E. This relationship, however, simplifies to (1+ [(kcsBPm - P O k c s B + k-ta[(kcsB/k-csB) [Bbld + (kcsY /k-csY) LYbldl + krsA exp(-aVo$/RT) + k r s A exp((1 - a)VocF/RT)I
voc
= (RT/F)
+ hpsh*,kisE(kcsA/h-csA) (kcsZ/k-csZ) h s [ Q s I d + kis)(kl; + b i [ Q s I d
[Abld[Zbld/((kls
+
hps(hcsZ/h-csZ) [zbld)})(exp(.vo$/RT))1/ k-rgA(kcsB/k--csB)LBbld) (42) under the conditions
