2432
J . Phys. Chem. 1984, 88, 2432-2434
Pt-titania interface. A general representation of this model for Pt-TiO, systems, including Pt dispersed on a titania support, is depicted in Figure 2. The presence of an X-ray pattern for the multilayer TiO,/Pt sample, although weak, shows that this TiO, species exists in the form of crystallities and therefore does not cover the Pt surface entirely, in agreement with the chemisorption experiments. The evidence supporting this model is as follows. The rate-determining step in the methanation reaction over Pt and Pd catalysts appears to be bond rupture of the C O molecu1e.6,23-25The higher activation energy on Pt/Ti02 catalysts indicates the rate enhancement occurs because of a higher preexponential factor caused by an increase in the number of active sites, for example. The reduction of TiO, results in oxygen vacancies and Ti3+cation~:*~”~~ which can reside at the surface near the Pt c r y ~ t a l l i t i e s . ’ Due ~ * ~ to ~ the activation of H2 and its subsequent spillover, such defect sites, especially incompletely coordinated Ti3+cations, could be created in the Pt-TiO, interface region even at low temperatures where migration of TiO, species would not be expected to be facile. These sites could interact with (32) T. Huizinga and R. Prins, J . Phys. Chem., 85, 2156 (1981). (33) J. C. Conesa and J. Soria, J . Phys. Chem., 86, 1392 (1982). (34) S.J. DeCanio, T. M. Apple, and C. R. Dybowski, J. Phys. Chem., 87, 194 (1983). (35) E. L. Kugler and R. L. Garten, US.Patent 4273724 (1981). (36) One reviewer disclosed that a similar experiment had been reported in the patent literature for the TiO,-Ni system. Results were similar to those reported here but activity enhancements were much less p r o n o ~ n c e d . ~ ~
the oxygen end of a C O molecule adsorbed on the Pt surface to facilitate its dissociation. A similar situation could exist involving bare A13+cations (acid sites) thereby explaining the higher activity of Pt on A1203and Si02-A1203 compared to Pt on S O 2 . This model does not require electron transfer or decreased chemisorption as a consequence of HTR to account for higher activity; however, should either occur, an alteration could be induced in surface concentrations under reaction conditions, such as a higher H(,)/CO(,) ratio, which could also enhance activitye6 These interface sites could provide IR-inactive CO; the number of these sites would be dependent upon the metal-support interface area and would be increased by migration of TiO, onto the metal even though total chemisorption could decrease, thereby negating or minimizing any crystallite size effect; the creation of these sites would be dependent upon the ability of the metal to activate hydrogen, but hydrogen would be required to migrate only very short distances; the requirement of both the metal and the support to produce such a site would lead to a variation in rates and activation energies, as observed; and such sites would not necessarily be expected to enhance other hydrogenation reactions or hydrogenolysis reactions. For these reasons, we believe the higher C O hydrogenation rates in Pt-TiO, systems are attributable to special sites created at the metal-support interface by partial reduction of the titania surface.
Acknowledgment. Partial support of this study came from the U S . DOE under Contract No. DE-AC02-77ERO 4463.
Low-Temperature Emission Spectra of Disordered Solids E. W. Knapp and S. F. Fischer* Physik- Department der Technischen Universitat Miinchen, Theoretische Physik, 0-8046 Garching, West Germany (Received: February 8. 1984)
The delayed emission profile of triplet excitons in molecular crystals is investigated under the influence of unidirectional exciton transfer at low temperatures. The one-dimensional exchange and the three-dimensional dipole-dipoletransfer mechanisms are considered in detail.
Introduction Recently, electronic spectra of delayed emission in disordered solids have attracted much Thereby a certain frequency range in an inhomogeneously broadened spectrum is initially excited, and one observes via the delayed emission how the excitation moves in the frequency domain. This spectral diffusion corresponds to a migration of the excitation in real space from an initial donor site to acceptor sites with different transition frequencies. Such inelastic processes are phonon assisted. Therefore, in the absence of phonons at low temperatures the excitation transfer becomes unidirectional toward sites of lower energye5Then the emission profile appears in the low-energy wing of the inhomogeneous spectrum (Figure 1). It contains valuable information on the excitation transfer mechanism. A good model substance to study this type of spectral diffusion is a molecular crystal of 1-bromo-4-chloronaphthalene (BCN),2-5 where the system parameters are well-known. It exhibits disorder with respect to the position of the halogens, whereas the centers and orientations of the molecules are rather regular? The stacking (1) Yen, W. M.; Selzer, P. M., Eds. “Topics in Applied Physics”; Springer-Verlag: West Berlin, 1981. (2) Prasad, P. N.; Morgan, J. R.; El Sayed, M. A. J. Phys. Chem. 1981, 85, 3569. (3) Morgan, J. R.; El Sayed, M. A. J . Phys. Chem. 1983,87, 200. (4) Morgan, J. R., El Sayed, M. A. J. Phys. Chem. 1983,87, 383. (5) Morgan, J. R.; El Sayed, M. A. J. Phys. Chem. 1983,87, 2178. (6) Bellows, J. C.; Prasad, P. N. J . Chem. Phys. 1979,67, 5802. Bellows, J. C.; Stevens, E. D.; Prasad, P. N. Acta Crystallogr., Sect. A 1978, 34, 3256.
0022-3654/84/2088-2432$01 .SO10
feature in BCN is similar to that in 1,4-dibromonaphthalene (DBN), giving rise to one-dimensional triplet excitons.’ The substitutional disorder introduces frequency shifts of the electronic levels of the individual molecules. This is why the spectrum of triplet excitons in BCN is inhomogeneously broadened in contrast to the spectrum of DBN5 (Figure 1). In the ordered substance DBN the triplet excitons are delocalized via the exchange interaction. The substitutional disorder leads to localized excitons which now extend over small molecular aggregates only. These molecular clusters together with their environment will be henceforth treated as sites. They contribute to the excitation spectrum at specific transition frequencies. Detailed studies of the emission profiles have shown that the site energies are essentially uncorrelated and the transfer rates can only weakly depend on the energy mismatch between donor and acceptor sites.5 Hence, the standard theories8-l0 of uncorrelated excitation transfer in randomly disordered solids can be applied. Another reason to study triplet excitations in BCN crystals is that they exhibit the interesting phenomenon of switching between different transfer mechanisms5 At high acceptor concentration the one-dimensional exchange transfer mechanism is operative. This has been confirmed by studying the early portion of the decay (7) Hochstrasser, R. M.; Whiteman, J. D. J. Chem. Phys. 1972,56,5945. ( 8 ) Inokuti, M.; Hirayama, F. J . Chem. Phys. 1965, 43, 1978. (9) Blumen, A.; Manz, J. J. Chem. Phys. 1979, 71, 4694. (10) Blumen, A. J . Chem. Phys. 1980, 72, 2632.
0 1984 American Chemical Societv
Letters
The Journal of Physical Chemistry, Vol. 88, No. 12, 1984 2433 W = A(8 In 2)'12 = 2.VA
(4)
is 64 cm-' 4,5 (Figure 1). To consider excitation transfer, we introduce the residence probability p(w,t). In the absence of radiative decay, it is the probability of finding an excitation placed initially on a site with frequency w still at the same site after time t has elapsed. Now the emission competes with the transfer process, providing the following frequency-dependent emission probability:
The emission intensty is then I
I
20100
20200
I
20300 frequency ~ c m - ' t
I
I
20400
-
Figure 1. The absorption profile (right part) and emission profile (left T transition in BCN are depicted. The solid part) for the lowest S lines are the experimental data taken from ref 3-5. The broad profile (dashed line) is a Gaussian of width (fwhh) 64 cm-I, fitting the low-energy wing of the absorption spectrum. The narrow profile (dashed line) is calculated by assuming a dipole-dipole transfer law to fit the emission spectrum. The model parameters are discussed in the text.
of the donor emission with time.3,5 The exchange coupling is a short-range interaction and becomes therefore ineffective at low acceptor concentration. The late portion of the decay of donor emission with time corresponding to a low acceptor concentration fits a three-dimensional dipole-dipole transfer m e ~ h a n i s r n . ~ , ~ The steady-state emission profile in the low-energy wing of the inhomogeneous spectrum has been analyzed with a simple oneparameter model accounting for the interaction volume of the various possible transfer mechanisms. It yields a good fit of the emission profile, where the parameter value is reasonable for the dipole-dipole interaction but not for the exchange i n t e r a c t i ~ n . ~ - ~ In this letter we apply a model which more explicitly accounts for the details of possible transfer mechanisms. In section 2 a model description for the emission profile is introduced. It is applied to the one-dimensional exchange and three-dimensional dipole-dipole transfer mechanisms in section 3. In the last section the experimental data are compared with the emission profiles calculated on the basis of the two transfer mechanisms. 2. The Model The excitation transfer of triplet excitations in BCN crytals depends only weakly on the energy mismatch provided the final state has a lower energy. As a result, a given donor site populates the acceptor sites according to their probability of occurrence in the inhomogeneous spectrum. The acceptor sites can emit or serve as new donor sites. A steady-state emission profile obtained this way is related to a experimental setup in which the delayed donor emission after a short laser pulse is recorded. If one collects the donor emission for all delay times, the steady-state emission profile is recovered if the laser scans the inhomogeneous spectrum. In the following we will introduce and discuss our model with respect to the latter experimental setup. In the absence of excitation transfer the time-dependent probability of delayed donor emission collected in the time interval from 0 to t is given by
e(t) = x t d t e-t '/ r, = 1 -
(1)
where 1 / is~the~ rate of radiative decay. The emission intensity depends on the frequency w, where the excitation has been placed according to E(w,t) = Z(w) e(t)
(2)
It is equal to the original inhomogeneous spectrum Z(w). For BCN its low-energy part can well be approximated by a Gaussian Z(w) = (27rA2)-I/*exp(-w2/2A2)
whose width (fwhh), given by
E(w,t) = I ( w ) e(w,t) (6) In the present case the excitation transfer from a donor site of frequency w can only occur to acceptor sites of lower frequency wa. If the transfer does not depend on the energy mismatch w - w,, the transfer rate is proportional to the fraction of sites of lower energy:
(7) For the inhomogeneous Gaussian spectrum it can be written in terms of a complementary error function" c(w) =
y2 erf c (-0/A2l/~)
(8)
The residence probability for infinite delay time as used in ref 4 and 5 is given by P(W)
= (1 - c(w))"
(9)
where n is the number of acceptor sites within the interaction volume. 4. Application of the Model Let us first consider the one-dimensional exchange transfer mechanism. According to ref 1 0 at a small acceptor concentration the residence probability of the excitation is given by
where g, can be expressed by the exponential integral E,(x)" and Eulers constant CE = 0.5712: gl(x)
= El(x)
+ In
(x)
+ CE
(1 1)
The rate of nearest-neighbor transfer is 1 / ~ and ~ , y d is a range parameter. The ratio of nearest-neighbor to next-nearest-neighbor transfer is given by eTd. The dimensionless constant a, accounting for the lattice geometry, is close to unity. In the long time limit the residence probability simplifies tolo
p,,.dw,t) = exp(-CEq(w))
[(rt/t)e-yd]q(")
(12)
where the abbreviation q ( w ) = ac(w)/yd
(13)
has been introduced. For a = (Tt/TI)e-yd
> T~ the above expression is practically valid for all delay times, and the probability of delayed emission can be written as ef,-(w,t) = exp(a(4r/3)c(w)) X { 1 - eQo2(")-Q12(") + a1/2Qo(w)eQo2(")[erf(Qo(o)) - erf (Ql(w))]) (19)
or ef,,(w,t) = exp(a(4~/3)c(w)) X (1
- F(Qo(w)) + eQo""'-Q'2'"'[(Q~(w)/Qi(~))F(Qi(w))
- 111
(20) where the function" F(Q) = a1/2Qe@erf c(Q)
(21)
and the abbreviations Qo(w) = c(w)a(4a/3)(a7,/~t)'i'
(22)
and Q I ( ~= ) Qo(w)
+ (t/~,)'/'
(23)
have been introduced. The expressions 19 and 20 of the emission probability are both equivalent. However, the first is appropriate for an evaluation at short delay times, the other for an evaluation at long delay times. In the latter case for z >> T,, it simplifies to ef,-(wJ+m) = e x ~ ( 4 4 ~ / 3 ) ~ (11 w) )F(Qo(w))l
(24)
4. Results and Discussion The present model is employed to calculate the delayed emission profile of the donor sites sampled over variable delay times. This emission profile is equivalent to the stationary emission profile of the acceptor sites if the transfer to lower energies does not
Figure 3. Position (---) and line width (fwhh, cm-I) (-) of the stationary emission profile are depicted as a function of l / Q = (l/u)(Tt/ .,)'I2, the ratio of the dipole-dipole transfer time T~ and the radiative lifetime 7,. The emission profile are calculated according to eq 24 by using a Gaussian spectrum of width 64 cm-'. They do not differ from the results derived with the exact expression (eq 17). The horizontal lines denote the experimental values of ref 4 and 5.
depend on the energy mismatch. Figure 2 displays the line shift relative to the center of the inhomogeneous spectrum and the line width of the stationary delayed emission profile under the influence of the one-dimensional exchange transfer mechanism. The emission profile agrees with the experimental spectrum (horizontal lines in Figure 2) at the range parameter y d = 0.03. This value indicates a long-range behavior. It is 2 orders of magnitude smaller than the value typical for many aromatic systems.I2 It is also rather close to the value found with the more elementary treatment in ref 4 and 5 . The large discrepancy of the range parameter y d excludes the exchange transfer mechanism. For the same reason a superexchange transfer mechanism is rather unlikely. But it cannot be ruled out completely before this transfer mechanism is studied for energetically disordered ~ y s t e m s . ~ In Figure 3 position and line width of the stationary delayed emission profile are displayed as a function of 1/Q = (l/U)(t/Tr)'/' by assuming a three-dimensional dipole-dipole transfer mechas are nism. The parameter values T~ = 0.02 s and T~ = obtained from independent experiment^.^ If one assumes the geometry parameter to be a = 1.0 the value of Q becomes 100(2)1/2.The corresponding emission profile is depicted in Figure 1. Its width is 26 cm-' and the line shift is 81 cm-I as compared to the experimental values of 22 and 76 cm-I, respectively. Varying the Q parameter does not improve the agreement, since line width and shift cannot be fitted well with the same Q value. The present treatment supports the conclusions taken in ref 5 . Namely, at low acceptor concentration the three-dimensional dipole-dipole interaction is responsible for the triplet exciton transfer in BCN. The still-existing mild discrepancy with the experimental emission profile may have several reasons. (1) The inhomogeneous spectrum is not exactly Gaussian far in the low-energy wing, where it may be related to rather large molecular aggregates. (2) The transfer rate depends on the energy mismatch between donor and acceptor sites. (3) Sequences of multiple transfer populate the emitting sites at different times and change the stationary composition of the emitting sites. All these points require a more detailed theoretical treatment. The last one can, however, be ruled out by observing the delayed donor emission directly instead of the stationary emission profile. Acknowledgment. We thank Professor M. El Sayed for several stimulating discussions which initiated this work. Support by the Deutsche Forschungsgemeinschaft within the project SFB 143C2 is also gratefully acknowledged. Registry No. BCN, 53220-82-9. (12) Blumen, A.; Silbey, R. J . Chem. Phys. 1978, 68, 1879.
