4036
J . Phys. Chem. 1985,89, 4036-4039
A Note on the Total Depktlon Method of Measuring Exthrction Coeffklents of Triplet-Triplet Transitions' Ian Carmichael* and G . L. Hug Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556 (Received: March 11, 1985)
The existence is demonstrated of a minimum pulse width, necessary to ensure complete conversion of a known concentration of ground-state molecules to the triplet state, under flash photolysis, regardless of the intensity of the flash. This demonstration does not involve considerations of multiphotonic processes or complicated kinetic schemes. Results derived from a numerical solution of a three-state excitation model are presented for a number of aromatic hydrocarbons in polar and nonpolar solvents. An analytical solution of this model for a square excitation pulse is developed to show the explicit dependence of the maximum achievable conversion on the relevant photophysical parameters. For the square pulse idealization it is shown that to expect 95% conversion to the triplet state, the singlet lifetime, TS, the triplet quantum yield, @T, and the pulse width, rp, must satisfy It is thus generally expected that extinction coefficients for triplet-triplet absorptions, obtained the inequality TS d by pulsed laser photolysis, will be, even if other complications are ruled out, underestimates of the true values. It is shown by specific examples that the appearance of saturation does not necessarily accompany total depletion of the initial ground-state population.
Introduction The extinction coefficient of triplet-triplet absorption transitions is often measured by pulsed optical techniques. In such experiments, in addition to recording an appropriate optical density and correctly attributing it, it is necessary to estimate the concentration of the triplet state present within the path of the monitoring radiation. It is often assumed that by simply increasing the intensity of the exciting pulse complete conversion of a suitably small ground-state concentration to the triplet manifold can be achieved. In early flash photolysis workZ such conditions were generally expected, provided that the quantum yield for intersystem crossing was not negligibly small. With the introduction of short, intense laser pulses and the attendant possibility of high photon fluxes, this expectation has often p e r ~ i s t e d . ~ The quantitative interpretation of the results of high-intensity pulsed laser photolysis is frought with many difficulties such as the participation of multiphotonic processes4including ionization:p6 excited-state a b s ~ r p t i o n , ~and - ~ photochemistry and problems of beam overlap.* Many further complications9such as triplet-triplet annihilation and exciplex and ion-pair formation prevail on the nanosecond time scale of commonly used pulses. Even if one assumes that these factors can be either ignored or accommodated by a suitable choice of experimental conditions, one basic problem remains. The desired conversion to the triplet manifold may be kinetically unattainable. Below we develop a simple kinetic scheme for the photolysis and subsequent relaxation processes and show the limitations inherent in the use of short pulses, however intense. Theoretical Model We employ a three-state model of the photoexcitation-deactivation process. If ['MI is the ground-state concentration, ['M*] (1) This is Document No. NDRL-2688 from the Notre Dame Radiation
Laboratory. (2) Porter, G.; Windsor, M. W. J . Chem. Phys. 1953, 21, 2088. (3) See, for example: Wildes, P. D.; Lichtin, N. N.; Hoffman, M.Z.; Andrews, L.; Linschitz, H. Photochem. Phorobiol. 1977, 25, 21-5. McVie, J.; Sinclair, R. S.; Truscott, T. G. J. Chem. Soc., Faraday Trans. 2 1978.74, 1870-9. Pileni, M. P.; Santus, R.; Land, E. J. Phorochem. Phorobiol. 1978, 27, 671-81. (4) Topp, M. R. Appl. Specrrosc. Reu. 1978, 14, 1-100. (5) Fisher, M. M.; Veyret, B.: Weiss, K.Chem. Phys. Lett. 1!374,28,60-5. (6) Lachish, U.; Shafferman, A,; Stein, G. J . Chem. Phys. 1976, 64, 4205-1 1. (7) Speiser, S.; van der Werf, R.; Kommandeur, J. Chem. Phys. 1973,1, 297-305. ( 8 ) Bazin, M.; Ebbesen, T. W. Photochem. Phorobiol. 1983, 37, 675-8. (9) Birks, J. B. "The Photophysics of Aromatic Molecules"; Wiley-Interscience: New York, 1970.
0022-3654/85/2089-4036$01.50/0
is that of the state reached upon initial excitation and [3M*] is that of the subsequently populated triplet state; then, if one makes the usual assumptions of first-order reactions, a set of coupled kinetic equations for the time variation of these populations may be immediately written down. These are d['M]/dt = -k,,['M]
+ ( k , + kic)['M*] + ~ T [ ~ M *(1)]
d[lM*]/dt = k,,['M] - ks['M*]
(2)
and d['M*]/dt = ki,,['M*]
- k~[~h'f*]
(3)
Here the rate constants, k, are for the usual processes of internal conversion, kic, radiative decay, kf, and intersystem crossing, ki,. kT is the sum of the rates of the triplet-state deactivation pathways, and ks (=kf ki, k,) is that sum for the excited singlet state. The excitation rate is given by
+ +
k,, = 2303cSZp(t)
(4)
where Zp is the excitation intensity (einstein cm-2 s-l) and, in a laser experiment, cs is the extinction coefficient for ground-state absorption (L mol-' cm-I) at the photolyzing wavelength. In conventional flash photolysis this quantity would represent an average over the spectral range spanned by the exciting flash, weighted by the distribution of intensity in the flash throughout that range. Note that no dependency of Zpon penetration depth is invoked; i.e., optically thin samples are assumed. For certain particular choices of temporal pulse shape the set of coupled equations, eq 1-3 may be solved analytically. For example, for a square pulse starting at time t = 0, of length rP, the general solution at some time t within the pulse is ['MI = @(kskT(.- - a+)- C Y - ( ~-Ta+) X ( k s - a+)exp(-a+t) a+(kT - a - ) ( k s - a _ )exp(-a-t)] ( 5 )
+
['M*] = @kcx(kT(~- C Y + )- a - ( k ~- C Y + )exp(-a+t) + a+(kT - a-) exp(-a-t)] (6) [3M*] = @kenkisc((a- a+)- CY- exp(-a+t)
+ a+ exp(-a-t)) (7)
for t 6
rP. Here @ = ['MIo/(a+~-(a-- a+))
(8)
with ['MI, the initial ground-state concentration and CY*
= (x
* Y)/2
where 0 1985 American Chemical Society
(9)
The Journal of Physical Chemistry, Vol. 89, No. 19, 198.5 4037
Extinction Coefficients of Triplet-Triplet Transitions x = ks
+ kT + k,,
(10)
and
y z = x2 - 4(k& + kexkT kexki,) (11) After the pulse eq 1 and 2 decouple and the solution for the excited-state populations may be determined from the corresponding end-of-pulse values, [mM*]rp,as [3M*] = ['M*]rp expl-(t - rp)kT) (kisc/(ks
- kT))['M*lrp(expt(t - r p ) k ~-) exPi-(t
- rp)kd
TABLE I: Photophysical Parameters of Model Compounds molecule solvent k,, ws-' +T rS, ns pyrene naphthalene anthanthrene anthracene
EtOH cyclohexane benzene EtOH
530' 96d 3.88 5.2'
0.38b 0.68e 0.21* 0.79
0.717 7.1 55.3 140
77,ms
10.2' 0.22f O.lli 8.W
Reference 11. Reference 12. Reference 13. Reference 14. 'Reference 15. fReference 16. SReference 17. *Reference 18. Reference 19. j Reference 20.
(12)
I
I
0
and ['M*] = ['M*]r, expi-(t
10-
- J?,)ks]
(13)
for t 2 rP. Of particular interest in the case of the total depletion method is the maximum achievable triplet concentration at the end of the pulse. We define the survival fraction, "At),at a time t of the excited state of multiplicity m and with concentration ["'M*], as, for example
Xt) = [3M*l,/['Mlo Then, taking the limit k,,
+
a
(14)
02-
-
in a suitable fashion gives
Xrp) = kisc(1 - expHkisc + kT)rpl)/(kisc
+ k ~ ) ;k
x
/ /
/ I
0
(15) Note however that even in this limit there will, in general, be a persistent (end-of-pulse) concentration of excited singlet state
'Arp) = (
k+ ~ kisc exd-(kisc
-
+ kT)rpj)/(kisc + k ~ ) ;
I
k,, (16) After the pulse these excited-state populations will decay (and some triplet state will grow in) until, a t some later time, t d 3f(td) = Ikisc/(kT + kisc))((l - exp{-(kT + kisc)rpl) expkkT(td - r p ) l + ( k +~ kisc ~ X P H + kisc)rpI)/(k ~T - k,) X (exP{-(td - exp(-(td - rp)kS))); k x (17)
-
For many molecules, in well-purified solvents, the triplet decay rate will be small. Thus, if one assumes kT
