J . Phys. Chem. 1990, 94, 8028-8036
a028
DNA's in solution. By using heavy metal counterions and both monovalent thallium ions and divalent barium ions, w e were able to enhance the signal due to the ions to about 12% of the total. This additional contribution enhances the intensity at moderately small Q range and depresses the intensity at large Q range (see Figure l a ) . The net effect is to make the calculated SAXS intensity distribution in better agreement with the measured absolute intensity. There is some evidence that the penetration of positive counterions into the core region needs to be taken into account in a high-precision comparison of the calculation and the experiment. The ion penetration into the major and minor grooves of the double helices is geometrically plausible and natural. In fact, the most recent theoretical calculation of the ion distributions around DNA takes into account the molecular structure of the DNA backbone e x p l i ~ i t l y . ' ~ .High-precision '~ SAXS measurements, therefore, would be very useful for testing the results of
such large-scale computations. Among the possible future improvements on the experimental side are to extend the Q range to cover 0.005 A-' I Q I 0.4.5 A-', to use more dilute solutions of the order of I mg/mL for the measurements, to apply the thermodynamic correction to the Q = 0 intensity, and to use the synchrotron source for gaining more intensity. A series of such SAXS studies, including the synthetic DNA oligomers of different base-pair sequences and at different conformational states, such as Z-DNA and A-, B-, and C-DNA, would be valuable.
Acknowledgment. This research is supported by a grant from N S F administered through Center of Material Science and Engincering and a special fund from the Dean's office of MIT. We are gratcful to Solid State Division of Oak Ridge National Laboratory for providing the SAXS facility for this experiment.
(15) Conrad, J.; Troll, M.; Zimm, B. H. Biopolymers 1988, 27, 171 I .
(16)
Jayaram, B.;Sharp, K . A.; Honig, B. Biopol~vmers1989, 28, 975.
FEATURE ARTICLE Requirements for Ideal Performance of Photochemical and Photovoltaic Solar Energy Converters Mary D. Archer* Newnham College, Cambridge, England CB3 9DF
and James R. Bolton* Photochemistry Unit, Department of Chemistry, The Uniuersity of Western Ontario, London, Ontario, Canada N6A 5B7 (Received: March 19, 1990) The required characteristics of an ideal photoconverter of solar radiation to electrical or chemical energy are summarized. The four unavoidable loss mechanisms inherent in single-junction photoconverters-lack of absorption of sub-bandgap photons, thermalization of ultra-bandgap photons, the difference between the available energy and internal energy of the thermalized excited states, and the small loss of excited states by radiative decay-are quantified. As the Gibbs energy of the product of an irreversible photochemical reaction falls below the ideal limiting value attained by reversible reactions, the maximum energy stored falls rapidly. This is shown in diagrammatic form for three examples: the isomerization of norbornadiene to quadricyclane, and the splitting of water by a 2-electron, 2-photon and a 2-electron, 4-photon process. The radiative lifetime of the excited states in molecular chromophores and semiconductors is linked to the absorption spectra by the appropriate broadband form of the Einstein relation between absorption and emission probability. For molecules, this is the Forster or the similar Strickler-Berg relation, and for semiconductors it is the van Roosbroeck-Shockley relation. The radiative lifetimes predicted by these relations are compared with those required for ideal performance in four molecular and semiconductor systems. We show that in both cases the Einstein relation may reduce the radiative lifetime by a factor of 100 below its ideal value, which causes a moderate decrease in the output voltage or chemical potential. We discuss some of the reasons for the poorer performance of photoconverters based on molecular chromophores as compared with those based on semiconductors, which arise from differences between the two broadband Einstein relations, and between the absorption spectra and transport properties typical of each system.
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I . Introduction Solar conversion devices can be divided into two classes: those that operate by first converting solar energy into heat and those that convert it directly to electricity or stored chemical energy. In this paper we are concerned only with the latter class, which we call-photoconuersion deuices, since they utilize the photon nature of light. There is now general agreement1-I0 that for an ( 1 ) Shockley, W.; Queisser, H. A . 1. Appl. Phys. 1961, 32, 510. (2) Ross, R . T. J . Chem. Phys. 1967, 46, 4590. (3) Ross, R . T.;Calvin, M . Biophys. J . 1967, 7 , 595. (4) Ross. R . T.: Hsiao. T.-L. J . Appl. Phys. 1977. 48, 4783.
0022-3654/90/2094-8028$02.50/0
isotropic 300 K photoconverter exhibiting blackbody absorption above a sharp threshold or bandgap energy, the limiting conversion efficiency in average sunlight (AM 1.5) is -33% at the optimal bandgap energy of 1.34 eV."
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( 5 ) Ross, R. T. J . Chem. Phys. 1980, 72, 4031. ( 6 ) Haught, A. F. J . Solar Energy Eng. 1984, 106, 3 . ( 7 ) Bolton, J. R.; Haught, A. F.; Ross, R . T. In Photochemical Conuersion
and Storage o f s o l a r Energy; Connolly, J . S., Ed.; Academic Press: New York, 1981; p 297. (8) Bolton, J . R. Science 1978, 202, 705. (9) Bolton, J. R.; Strickler, S. J.; Connolly, J. S. Nature 1985, 316, 495. (IO) Porter, G .J . Chem. Sac., Faraday Trans. 2 1983. 7 9 , 473.
0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94,No. 21, 1990 8029
Feature Article The question of what further limitation there are, if any, on the conversion of solar energy into a pure high-energy chemical product of appreciable lifetime has not been fully resolved. Consideration of this question is the first purpose of this paper. We adopt a new perspective involving “thermodynamic” and “kinetic” losses and calculate the maximum energy storage efficiency as the Gibbs energy of the chemical product falls below that of the excited state. Our second purpose is to enquire whether the Einstein relation, as extended to broadband molecular chromophores by ForsterI2 and Strickler and BergI3 and to semiconductors by van Roosbroeck and Shockley,I4predicts for a material of given absorption characteristics an excited-state radiative lifetime that is sufficiently long to allow ideal performance. The general principles of photochemical conversion are now well established. DuysensI5 examined the limiting efficiency of photosynthesis by applying a Carnot-like argument but grossly overestimated the efficiency by neglecting the energy lost by the thermalization of excited states. Ross2” developed the concept of the excess chemical potential of a superequilibrium concentration of excited molecules of limited lifetime and showed that his photochemical (PC) approach was equivalent to, but more general than, the treatment of the ideal photovoltaic (PV) cell by Shockley and Queisser.’ Porterlo introduced an alternative approach based on flux-chemical potential relations and the loss in chemical potential consequent on carrying out any chemical process at a finite rate. Energy-storing photochemical reactions are usually multistep, light absorption being followed by a sequence of irreversible chemical reactions that produce a reasonably long-lived product which stores a moderate amount of energy. The energy and work losses incurred in such irreversible steps can be related to the likely product lifetime by transition-state t h e ~ r y . ~ , ~ Warner . ’ ~ , ~ ~and Berry’* have given equations for energy storage efficiency using the concept of the equivalent temperature of sunlight, but these do not lend themselves easily to numerical evaluation. 11. The Ideal Single-Junction Photoconverter We consider the two fundamental types of photoconverter: PV cells, in which the ureactionn may be written
sc
-sc + I hu
hvB+ + ecB-
(1)
where hvB+and eCB- denote holes in the semiconductor (SC) valence band and electrons in the conduction band, respectively; and energy-storing PC reactions, which we write in the general form nC hu
rR
7P + qQ
AGRP/~~
(2)
where R represents the reactant(s), P the desired high-energy product (e&, hydrogen), Q any other products ( e g , oxygen), and AGRpI9is the Gibbs energy change of the forward ground-state reaction as written. R may itself be the chromophore or the reaction may be sensitized by another chromophore Z. Many fuel-forming reactions are multistep, multiquantum processes initiated by one or more photochemical electron-transfer reactions. and in these cases it is useful to define two further ( I I ) In principle, these limits can be exceeded by non-isotropic nonblackbody photoconverters; see: Green, M. A. High Efficiency Silicon Solar Cells; Trans Tech SA: Switzerland, 1987. ( I 2) Forster, Th. Fluoresrenr Organischer Verbindungen; Vandenhoeck und Ruprecht: Gottingen. FRG. 1951. (13) Strickler, S. T.; Berg, R. A. J . Chem. Phys. 1962, 37, 814. (14) van Roosbroeck, W.; Shockley, W. Phys. Reu. 1954, 94, 1558. ( I 5 ) (a) Duysens. L. N. M. In The Photochemical Apparatus, Its Structure and Funcrion; Brookhaven Symposia in Biology; Brookhaven Natl. Lab.: Upton, NY; Vol. 1 1 , 1959; p IO; (b) Plant Physiol. 1962, 37, 407. (16) Almgren, M . In Topics in Photosynthesis; Barber, J., Ed.; Elsevier: Amsterdam, 1977; Vol. 2, Chapter 3, 397. (17) Adamson, A. W.; Namnath, J.; Shastry, V. J.;Slawson, V . J . Chem. Educ. 1984, 61, 221. (18) Warner, J. W.; Berry, R. S. J . Phys. Chem. 1987, 91, 2216. (19) Here and elsewhere, the first subscript to a thermodynamic quantity denotes the initial state and the second subscript the final state.
-
tI
R’
conducllon band
T
I
R valence band
(a)
(b)
Figure 1. Energy levels, energy bands, and bandgap energy CJg in (a) a semiconductor; (b) a solution of the chromophore R.
quantities: l , the number of photosystems involved (which we assume to have the same bandgap energy) and n, the minimum number of times each photosystem must operate to drive the reaction as written. The number of electrons transferred in the overall reaction is then n and the minimum number of photons required is nl, which we term the photon equivalent of the reaction. The maximum work available per absorbed photon by reversing the fuel-forming reaction 2 is then AGRp/n[. The required characteristics of an ideal single-junction photoconverter have been considered by many authors;l-I0 we summarize them below for reference. 1. The device has a single bandgap or threshold energy U, separating a ground-state band of levels from an excited-state band, as shown in Figure 1. In the PV converter (Figure l a ) these correspond to the valence and conduction bands of the semiconductor; in a PC reaction (Figure lb) they are the ground and electronically excited states of the active chromophore. 2. The device is a perfect step absorber; Le., it absorbs all photons with energies above U, but none with energies below U,. The absorption coefficient is not important in itself, but the device must be deep enough for all light to be absorbed. 3. The excited states, created in unit quantum yield by photon absorption, undergo very rapid intraband thermalization (as indicated by the curly arrows in Figure 1) but relatively slow interband thermalization. The distribution, but not the number, of particles in the excited-state band under illumination is therefore described by equilibrium (usually Boltzmann) statistics. Except in specialized devices involving hot carriers20 which we do not consider here, intraband thermalization is complete before either interband thermalization (Le., decay of the excited states to the ground state) or the work-producing event occurs. 4. The excited states are subject only to radiative decay to the ground state. The absence of nonradiative decay maximizes their lifetime, and hence their concentration, in the photostationary state. The ideal single-junction converter is subject to four inherent sources of inefficiency which are unavoidable and reduce the limiting efficiency to the level of -33% quoted above. We list these below; the number in parentheses following each is the percentage efficiency loss contributed by that factor in the optimal ideal converter of U, = 1.34 eV.21 1. Photons of energies U < U, are not absorbed and hence their energy is lost to the photoconverter (30.6%). 2 . All photons of energies U U, are absorbed, but the initial excited states are rapidly degraded to the energy -U, in the intraband thermalization process and the excess energy - ( U U p )is lost as heat (22.9%). 3. The available energy of an ensemble of thermalized excited states, which under conditions of constant temperature and (20) Ross, R. T.; Nozik, A. J. J . Appl. Phys. 1982, 53, 3813. (21) Numerical values depend on the assumed solar spectral distribution.
All calculations in this paper have been made for the standard global AM 1.5 solar spectrum ( 1 Sun irradiance 967 W m-2) of Hulstrom, R.; Bird, R.; Riordan, C. Solar Cells 1985, 15, 365.
8030 The Journal of Physical Chemistry, Vol. 94, No. 21, 1990 pressure is the Gibbs energy, is significantly less than its internal energy because the excited states are present in small mole fraction in a sea of ground states, and hence there is a significant entropy of mixing.? We shall call the difference between the internal and Gibbs energy per thermalized excited state the thermodynamic loss XI&!. This loss dictates that the voltage of an ideal Pv cell must be less than U , and similarly that the output chemical potential of a reversibfe PC reaction must be less than Ug(12.1%). 4. At maximum power, most of the excited states contribute to the output work (i,e., the quantum yield of work production is nearly unity), but a small fraction must decay radiatively. If the work producing pathway is very much faster than the rate of radiative decay, the concentration of excited states would tend toward zero and hence their chemical potential would be very low. Some radiative decay must be allowed to maintain an effective chemical potential so as to maximize the power (flux times chemical potential) output (1.1%). The maximum-power conversion efficiency q,,(ideal) of an ideal single-junction photoconverter conforming to the above description is given by2.43'
Archer and Bolton R'
I UP
Ll R
R'
L R
((I)
P
pmp(ideal)q$P(ideal)j2 q,,(ideal)
=
where pmp(ideal)(J molecule-I) is the ideal maximum-power Gibbs energy output per photon [p,,(ideal) = eV,,(ideal), where V, (ideal) is the maximum-power voltage of an ideal PV cell]; qYj(idea1) is the ideal maximum-power current or flux efficiency, Le., the fraction of the excited states that contribute to the output work rather than decaying radiatively to the ground state; jf (photons m-* s-I) is the incident flux of solar photons of energy U 3 Ug;and E! ( W m-*) is the total incident solar irradiance. The term ",(ideal) in eq 3 is calculated from2 p,,(ideal)
= pw(ideal)
+ k T In ( 1
- qrp)
(4)
where Fw(ideal) [ =eV,(ideal)] is the ideal open-circuit chemical potential under conditions when no current or chemical product is withdrawn, given by2 p,(ideal) = k T In [ 1
+ j!/jkb]
(5)
jib
where is the (minute) incident flux of photons of energies U 3 Ug in blackbody radiation at the temperature of the photoconverter and its environment. qPP(idea1) in eqs 3 and 4 is obtained from the maximum-power condi t ion4,5J F ~ideal) ( /kT
= {q$P(ideal)/[l - q$P(ideal)]) - In [ I - qrP(ideal)] (6)
Photovoltaic cells approach ideality much more nearly than photochemical converters. In part this is because the transport of holes and electrons in semiconductors is much more rapid and efficient than the transport of molecules in fluid systems. We allude to this difference again later. There is another important difference, namely that a photovoltaic cell is not required to store anything to be judged efficient, but a photochemical reaction must produce a high-energy chemical product of reasonable lifetime. We consider the constraints produced by this requirement in the next section. 111. Photochemical Conversion and Storage Reactions
We first consider a simple PC converter in which the following unimolecular sequence of reactions, constituting a simplified version of reaction 2, occurs: kP
R &U R * S P k+
R'
P
PZ*
..........
I
Figure 2. Gibbs energy diagrams for photochemical reactions. (a) Solution of R in the photostationary state when no P is formed; (b) ideal converter in which P is formed in equilibrium with R* with AGeR.p = 0; (c) optimal ideal converter producing P in its standard state; (d)
nonideal irreversible reaction. to yield the high-energy product P, which may or may not be formed in the same phase as R. This photochemical reaction is already partly idealized in that all ultra-bandgap photons are absorbed, R is subject only to radiative decay and the only route for the back reaction of P is through R*. We now enquire what relative values of the internal and Gibbs energy of R, R*, and P are consistent with ideal operation; this requires that at maximum power the product P must be produced at the chemical potential p,,(ideal) given by eq 4. In this ideal operation R* and P must therefore be coupled (Le., in equilibrium) so that the reaction R* P proceeds reversibly. Work Available from the Excited Molecule R*. Case a. Consider first the energy level scheme shown in Figure 2a in which P is not formed, i.e., kp = k+ = 0. When the solution of R is in the dark, virtually all the R molecules are in the ground state; for example, for Ug = 1.34 eV there are only N A exp(-Ug/k7') = 20 molecules of R* present at 300 K in 1 mol (NAmolecules) of R. If the solution is exposed to light absorbed by R, a significant excess population of R* molecules is established. Since we have assumed that the R* molecules thermalize we can consider them to be a new thermodynamic entity. For condensed systems, it is a good approximation to take A V B R R , = 0, so that A U 8 R R ' = A H e R R . . We shall also assume that A S e R R * = 0; this is a good approximation for moderately large chromophores, since it is equivalent to assuming that the partition functions of R and R* are the same.22 With these assumptions we see that AGeRR* = U,, as indicated in Figure 2a. If a means existed for extracting work from the ensemble of R* molecules as they returned to the ground state, the maximum work available per molecule of R* for the reaction
-
(7)
We assume that the chromophores R absorb sunlight uniformly, forming thermalized excited states R* at the rate g (molecules m-3 s-l). R* molecules may decay radiatively to R at the rate u (molecules m-) s-I) with a corresponding first-order radiative decay rate constant k!'), or they may react with rate constant k ,
(b)
...........
(3)
E8
P
WNI)
-
R*(N")
(8)
where Nu and Nl (molecule n r 3 ) are the photostationary con~
~~
~~
(22) Even if this IS not so, the effect of unequal partition functions Q is likely to be minor. These contribute the term kT In QR.!QR to OR,*, which is only 0 118 eV when the ratio of the partition functions IS 100
The Journal of Physical Chemistry, Vol. 94, N o . 21. 1990 8031
Feature Article TABLE I: Thermodynamic Loss xw under Open-circuit (oc) and Maximum-Power (mp) Conditions, and Corresponding Values of the Mole Fraction xlkl of Excited States in an Ideal Photoconverter js/IOzl
photons U,/eV 1.00 1.34r 2.00 3.00
n i 2 s-I x&Ia/ev 0.236 1.08 X IO4 2.88 4.46 X 2.10 0.259 0.302 8.45 X 10" 0.88 4.64 X 0.11 0.377
xP&,'/ev
xPLlb
0.324 0.356 0.411 0.496
3.61 X IO" 1.05 X IOd 1.25 X 4.65 X
'Calculated from eq 4-6 and 1 1 for AM 1.5 S u n (see ref 21) assuming T = 300 K. bCalculated from eq IO. CBandgapenergy for maximum efficiency of an ideal single-junction photoconverter. centrations of R* and R, would be equivalent to the (positive) Gibbs energy change AGRR. p = AGRR* = AGeRR.
+ k T In (NJNJ ug
=
+ k T In
(9)
Thus when the concentration ratio attains its limiting value (Nu/N1)jdcal (which is always
