1 August 1997 Ž Chemical Physics Letters 74 1997 171 176 Kinetics of raiationless energy transfer from upper excite states EN Bounov 1, MN Berberan-Santos, JMG Martinho Centro e Quımica-Fısica Molecular, Instituto Superior Tecnico, 1096 Lisboa Coex, Portugal Receive 6 May 1997 Abstract A moel of the kinetics of raiationless energy transfer from upper excite states is presente, where both the possibility of back energy transfer from the excite acceptor to a lower lying level of the onor an the possibility of energy migration over acceptors is consiere The moel leas to a complex variation of the energy transfer quantum yiel with the acceptor concentration This epenence results from the interplay of irect energy transfer, back energy transfer an energy migration Back energy transfer reuces the efficiency of onor acceptor overall transfer efficiency On the other han, energy migration over the acceptors ecreases the efficiency of back energy transfer an results in an increase in the energy transfer quantum yiel The known experimental results of energy transfer from upper excite states are in qualitative agreement with the present treatment q 1997 Elsevier Science BV 1 Introuction The raiationless transfer of electronic energy from an excite onor Ž D to a groun state acceptor Ž A is a well known phenomenon w1, x In the past, it has clearly been emonstrate that energy transfer can occur from excite states other than the first excite singlet or triplet state w,4 x Nevertheless, relatively few works have been publishe on this subject, espite its relevance in photochemistry, raiation chemistry an photobiology A istinctive feature of the process of energy transfer from upper excite states, is that back transfer is likely to occur, even when the onor an acceptor are alike molecules Inee, the acceptor molecule, after receiving the energy from the onor, can normally back transfer it to lower lying levels of the onor molecule, ecreasing, in this way, the overall efficiency of transfer For a high acceptor concentration excitation iffusion over acceptor molecules shoul also be consiere, since it ecreases the probability of back energy transfer As a consequence of these two aspects, the efficiency of energy transfer has a complex epenence on the acceptor concentration Several authors w5 8x have experimentally confirme that the back transfer of excitation from inirectly excite acceptor molecules to lower lying levels of the onor must be taken into account for a correct estimation of the efficiency of energy transfer from D to A A linear epenence of the energy transfer quantum yiel, h, with acceptor concentration, was observe for low concentrations w,5,9,10 x, while for higher concentrations a non-linear epenence was observe w5,9 x Ermolaev et al w5 x, reporte a quaratic epenence of h 1 On leave from Physical Department, Russian State Hyrometeorology Institute, St Petersburg, Russia 0009-614r97r$1700 q 1997 Elsevier Science BV All rights reserve Ž PII S0009-614 97 0069-8
17 ( ) EN BounoÕ et alrchemical Physics Letters 74 1997 171 176 with concentration in a limite acceptor concentration range, this being qualitatively explaine by the competition between excitation migration over acceptors, irect an back energy transfer The aim of this Letter is to present a quantitative kinetic moel that consiers all the above escribe processes, in orer to obtain the correct epenence of the energy transfer quantum yiel on the acceptor concentration Theoretical preictions are then compare with experimental ata Results an iscussion A simplifie scheme of the electronic levels of onor an acceptor molecules taking part in excitation transfer is shown in Fig 1 Upon excitation, the upper electronic state u of the onor is populate The excite onor can then relax to the lower excite state l with rate 1rtu or transfer the energy to the nearest acceptor A 1 with rate w The excite acceptor can ecay to the groun state with rate 1rt A, transfer the energy to another acceptor A with rate w m, or back transfer the energy to the lower excite electronic state l of the onor with rate w b The first acceptor A 1 can also be re-excite by back energy transfer from the acceptor A with rate w b We assume that excitation energy transfer occurs by the ipole ipole coupling interaction mechanism The energy transfer rate is well known after the work of Forster w11x an Dexter w1x for thermally equilibrate vibronic levels of the excite electronic state However, as the lifetime of the onor in the upper excite state is Ž y11 small t - 10 s u energy transfer may occur from a non-equilibrium vibrational istribution of that state Nevertheless, Kaplan an Jortner wx have shown that FD theory still hols for the case of slow vibrational relaxation on the timescale of the lifetime of the excite state In the following, we will assume that energy transfer always occurs in one of the above limiting cases, with rate constants / r / r / AA 6 1 R ws, Ž 1a ž t r u 6 1 R b wbs, Ž 1b ž t A 6 1 R m wms, Ž 1c ž t A where R, R ban Rmare the critical raii of energy transfer for the irect transfer, back transfer an migration processes, respectively Ž see Fig 1, an r an raa are the D A an A A istances We now consier the steay-state kinetics of the process Fig 1 Kinetic scheme for irreversible energy transfer from an upper excite state The onor D is initially excite to state u an may either relax to a lower excite state l or transfer its excitation energy to a nearby acceptor A 1 This, in turn, may ecay, transfer back to the onor D, or transfer to another acceptor A Migration of excitation energy among acceptors may then occur
( ) EN BounoÕ et alrchemical Physics Letters 74 1997 171 176 17 The onor, after being excite into level u, relaxes to the lower excite state l with probability 1rt u P1s, Ž 1rt qw u or transfers the energy to the nearest acceptor with probability w Ps1yP 1s Ž 1rt qw u Back energy transfer to level l can occur with probability w b Pbs, Ž 4 1rt qw ex b where tex is the effective lifetime of the acceptor molecule A 1 This lifetime is etermine by both the internal relaxation processes of the acceptor Ž rate 1rt A an by excitation migration over the acceptors It can be calculate once the Green function, Gt, Ž that escribes the energy migration over the acceptors is known Inee, H t Ž rt GŽ t t 0 t s s G t tsg s < sg ex H Ž Ž ss0 Ž 0, Ž 5 0 Ž rt G Ž t t H 0 G Ž t being the iagonal part of the Green function for the migration process, since we are only intereste in the first excite acceptor molecule Note that t is the Laplace transform of the Green function, G Ž s ex, for ss0 By substitution of Eq Ž 5 in Ž 4, we obtain wg Ž 0 Pbs Ž 6 1qwG b Ž 0 The function G Ž 0 can be calculate by the Gochanour, Anerson an Fayer Ž GAF self-consistent iagrammatic metho w1 x The GAF metho gives w14x t A G Ž 0 s, Ž 7 aq' aa q1 Ž where as1 in the two-boy approximation an as088 in the three-boy approximation, an p 4p as nar m, Ž 8 4' n being the acceptor number ensity Thus, the energy transfer quantum yiel h Ž r A, from the onor to the acceptor A, separate by a istance r, is given by 1 hž r sp Ž 1yP b Ž 9 ŽŽ Ž Ž After substituting expressions, 4 an 7 into Eq 9, we get tuw 1 hž r s Ž 10 1qtuw 1qw G b Ž 0
174 ( ) EN BounoÕ et alrchemical Physics Letters 74 1997 171 176 This equation is erive for a fixe onor acceptor separation, r In orer to obtain the ensemble energy transfer quantum yiel, h Ž r must be average over the onor acceptor istances, weighte by the appropriate onor acceptor separation istribution function Consiering the nearest-neighbour istribution function Žw15x an references therein 4p gž r s4pr na expž y r n A, / Ž 11 we obtain for the energy transfer quantum yiel: c y yy h sh y e 0 y qc y qz c r pr4' cz z q a pr4' cz z q 1, Ž 1 ( Ž Ž Ž b m b m b where 4p ys r n A, Ž 1 4p cs R n A, Ž 14 R b z s, Ž 15 b R R m zms Ž 16 R b Ž y4 For low acceptor concentrations c < 10, Eq Ž 1 leas to a linear variation of h with the reuce concentration c, p 1 h s c, Ž 17 1qz b as experimentally observe in Refs w,9 x At high acceptor concentrations, the quantum yiel eparts from the linear epenence an, for c41, reaches a limiting value, 1 yy h s1yh y e, Ž 18 6 1qbz y where ' 0 m Ž p a q1 bs Ž 19 This kin of behaviour appears to correspon to the experimental results given in Refs w,9 x We now give some results for the quantum yiel of intermeiate acceptor concentrations, numerically calculate using the general formula, Eq 1 As can be seen from Fig, the energy transfer quantum yiel increases initially with the acceptor reuce concentration c, then reaches a maximum, an ecreases afterwars Ž curves B an D It is seen that the quantum yiel globally ecreases an the maximum shifts to lower concentrations when the back energy transfer rate parameter z b increases On the other han, the increase in the excitation migration over acceptors, as measure by z m, leas to a quantum yiel increase an shifts the curve maximum to higher concentrations Ž note that for curves A an C maxima are not visible in the concentration range isplaye These
( ) EN BounoÕ et alrchemical Physics Letters 74 1997 171 176 175 Fig Energy transfer quantum yiel h Ž Eq 1 as a function of the acceptor reuce concentration c, for selecte values of the reuce parameters z an z Ž Eqs 15 an 16 : z s05, z s Ž curves A, z s05, z s1 Ž curves B, z s1, z s1 Ž curves C b m b m b m b m, an zbs1, z s05 Ž curves D Dashe curves: -boy GAF metho; soli lines: -boy GAF metho The limiting values of h Ž c are Ž m Eq 18 in the -boy approximation : 08 Ž case A, 05 Ž cases B an C, an 00 Ž case D results may explain experimental results where a ecrease in energy transfer efficiency with acceptor concentration is observe The quaratic epenence of h with c observe in Ref wx 5, is also preicte in the framework of the evelope theory, when the acceptor concentration is low, c( 005, but above the linear epenence range, as shown in Fig Fig Energy transfer quantum yiel h as a function of the acceptor reuce concentration c for zbs5 an zms1, calculate by the -boy approximation of the GAF metho A quaratic epenence is apparent The insert shows that for low concentrations the epenence is linear
176 ( ) EN BounoÕ et alrchemical Physics Letters 74 1997 171 176 Conclusions A moel of the kinetics of raiationless energy transfer from upper excite states was presente, where both the possibility of back energy transfer from the excite acceptor to a lower lying level of the onor an the possibility of energy migration over acceptors is consiere The moel leas to a complex variation of the energy transfer quantum yiel with the acceptor concentration, Eq 1 This epenence results from the interplay of irect energy transfer, back energy transfer an energy migration Back energy transfer reuces the efficiency of the onor acceptor overall transfer efficiency On the other han, energy migration over the acceptors ecreases the efficiency of back energy transfer an results in an increase in the energy transfer quantum yiel Three ifferent kins of concentration epenence of the energy transfer quantum yiel Žlinear, quaratic an asymptotic are preicte The known experimental results of the energy transfer from upper excite states are well accounte for by the present formalism A complete comparison between theory an experiment was, nevertheless, not possible because the ata neee for the estimation of the parameters z b, zm an c are not available Acknowlegements Ž ENB is grateful to JNICT Portugal for an Invotan fellowship References wx 1 JB Birks, Photophysics of Aromatic Molecules, Wiley-Interscience, New York, 1970 wx MD Galanin, Luminescence of Molecules an Crystals, Cambrige International Science Publishing, Cambrige, 1996 wx I Kaplan, J Jortner, Chem Phys Lett 5 Ž 1977 0 wx 4 M Terazima, T Azumi, J Phys Chem 94 Ž 1990 4775 wx 5 VL Ermolaev, AA Krasheninnikov, AV Shablya, Sov Phys Dokl 48 Ž 1979 768 wx 6 VL Boganov, VP Klochkov, Opt Spectrosc 48 Ž 1980 18 wx 7 VA Lyubimtsev, Opt Spectrosc 55 Ž 198 64 wx 8 VA Lyubimtsev, Opt Spectrosc 6 Ž 1987 6 wx 9 I Kaplan, J Jortner, Chem Phys Ž 1978 81 w10x JE Hunt, JJ Katz, A Svirmickas, JC Hinman, Chem Phys 8 Ž 198 41 w11x Th Forster, Ann Phys Ž 1948 55 w1x DL Dexter, J Chem Phys 1 Ž 195 86 w1x CR Gochanour, HC Anersen, MD Fayer, J Chem Phys 70 Ž 1979 454 w14x EN Bounov, Opt Spectrosc 74 Ž 199 11 w15x MN Berberan-Santos, Am J Phys 54 Ž 1986 119