Time-resolved Studies of Intermolecular Electronic Energy Transfer Processes between Molecules in Solution

A summary of the main energy transfer (ET) mechanisms between isolated pairs of molecules in solution is presented. Various ET models for many-molecules systems are discussed in their basic assumptions and their range of applicability. Time-resolved studies of ET in solution allow to determine the predominant ET mechanism and to test 
ET models.


INTRODUCTION
Energy transfer processes are widespread in nature.The intermolecu- lar energy transfer (ET) from excited molecules to unexcited ones plays a major role e.g., in photosynthesis and the visual process. 2 Further, there is a great interest in the technical use of such ET processes because excitation energy can be transported with negligible losses over molecular distances (up to some 100 A) within time inter- vals of less or much less than a nanosecond.In addition, energy can be transferred from one spectral region to another.Examples are the exciton migration in solids 3 or the sensibilization of photophysical 4 and photochemical processes.
In the particular field of laser physics ET processes have been widely used to extend the lasing range, increase the output efficiency and 20 M. KASCHKE AND K. VOGLER influence the spectral and temporal characteristics of the output pulses of energy transfer dye lasers 6'7 or solid state laser materials. 8 '9 Thus, the investigation of ET mechanisms is stimulated on the one hand by the aim to gain principal knowledge on these mechanisms and on the other hand to achieve successful technical utilization of some basic ET processes.
The early studies of ET, after the classical papers of F6rster, 1 Dexter 11 and Galanin, 12 were exclusively carried out with stationary methods, such as concentration quenching of luminescence and fluo- rescence depolarization.
It is, however, very difficult to distinguish between various ET mechanisms by use of these methods only.Time resolved ET studies have opened the possibility to determine the predominant energy transfer mechanism and with increasing accuracy of experimental data to refine ET transfer models.
The purpose of this paper is to give a summary of the main ET mechanisms occurring between interacting molecules in solution and to summarize briefly the substantial assumptions and limitations of the most important models.Further on we will shortly describe some time resolved studies of ET processes and compare their results with the theoretical predictions of relevant ET transfer models.

ENERGY TRANSFER MECHANISMS
Though nearly all ET theories are based on classical descriptions with parameters adapted to experiment, most of them can be derived from quantum mechanical considerations.An atomic system like a mole- cule in solution can be characterized by its Hamiltonian.If there is another independent molecule in the vicinity some kind of mutual interaction can take place, the strength of which depends on the distance and type of interaction energy.In the case of comparatively weak interaction between distant molecules the Hamiltonian of the system can be given by 12[ I2"IDonor + I2"IAcceptor + I2Iint (1) Describing the initial state of the system of two molecules by the the following expression for the matrix element of the interaction Hamiltonian is obtained: The first term is the coulombic interaction, whereas the second term represents exchange interaction and requires a spatial overlap of the electronic clouds of the acceptor and donor molecules and is therefore expected to be significant for small distances of the donor-acceptor pair.The coulombic interaction is of long-range, and the expansion of the interaction Hamiltonian in series yields contributions of dipoledipole, dipole-quadrupole, quadrupole-quadrupole, etc., interac- tions, the strength of which again depends on the mutual distance of donor and acceptor.
For a description of the energy transfer kinetics one could employ the density matrix approach 13 for two coupled two-level systems (Figure 1), without using perturbation theory.However, when taking into account the broad spectral bands and the extremely short trans- verse relaxation times of polyatomic .molecules in solution, time- dependent perturbation theory proves to be more fruitful.This process is given by: D* + A D + A* (no reverse process is assumed to take place).
approach starts from Fermi's Golden Rule for the transfer rate, implying that no reverse transfer is possible: dKET --IHyl 2 6(AE) (4) where AE 0 represents the condition of energy conservation, which is fulfilled by spectral-overlap of electronic and vibrational levels of donor and acceptor molecules (Figure 1).
As it is obvious from Eqs. ( 5) and (7) the transfer rate strongly depends on the intermolecular distance RDA Of the donor-acceptor pair.
It should be noted that several energy transfer mechanisms can act simultaneously in solution.The above-mentioned dipole-dipole and exchange transfer can be combined with a diffusion process of neigh- 15 16   bouring molecules which determines the transfer rate.Also pure radiative energy transfer processes are possible. 17The predominant mechanism is determined mainly by the mean between donor and acceptor and some characteristic lengths (Rr, lDif, R0) (Figure 2).
So far we have only dealt with an isolated donor-acceptor pair.However, quantities measured by fluorescence or absorption spectro- scopic methods in disordered systems are mean numbers of excited donors and acceptors per volume, which can be obtained by averaging with respect to the spatial distribution of both donor and acceptor molecules.
Thus, in modelling energy transfer processes one has to make certain assumptions concerning the spatial distribution of donor and acceptor molecules independent on the predominant transfer mechan- ism.Several ET models are given below.In deriving these models a common rate equation method is used, implying that thermal equili- brium over the vibrational levels of the excited donor molecule is established, and further that coherent effects are neglected.The first condition can be violated, e.g., in intramolecular electronic ET, 8 whereas the latter may not be fulfilled in the case of very strong interaction, leading to transfer times in the fs-time range.

MODELLING OF INTERMOLECULAR ET
In the following we would like to summarize some features of the most important models for intermolecular energy transfer.

Radiative energy transfer
In this case the ET consists of two independent intramolecular pro- cesses taking place successively in time: the emission of the photon of the primary excited donor and the absorption of it at the acceptor located in a certain distance.The absorption (= ET) probability is mainly determined by the optical density in the direction of observa- tion.The self-absorption of fluorescence is an important case of radiative ET between identical molecules.It is very difficult to give a general description of self absorption because results strongly depend on the particular experimental geometry of observation. 13However, it can be stated that reabsorption strongly influences the measured decay time, which is generally prolonged.In the case of strong self- absorption the decay time measured from the ensemble of excited molecules depends not only on cavity dimension and geometry of excitation and detection, but also on wavelength of observation of the fluorescence light.
Under the assumption of weak self-absorption the measured decay time rm is independent of wavelength 19 and the true molecular fluore- scence decay time : can be derived from it by: 2 '21 Z'm '(1 rift" a) -a (8)   where with rm(C 0) r.Here fl, f(Z), e(Z) and c are the fluorescence quantum yield, the normalized fluorescence spectrum, the molecular extinction coefficient and the concentration of the emitting molecules, respectively.These formula can be used in a rough estimation for transverse observation of a point-like excited volume.The light penetrates a length x within the solution in the direction of observation (Figure 3).r///.a is the combined probability of emission followed by an absorption of light along the pathway of propagation x.

Diffusion controlled collisional energy transfer
In the diffusion model the excited donor molecule moves towards the unexcited acceptor and an encounter of both within a certain transfer (or encounter radius Rr) causes a de-excitation of the donor with a excitatio cuvette Figure 3 Point-like excitation in a semi-infinite cuvette containing molecules with weak self-absorption of fluorescence.
certain probability.The diffusive motion is assumed to be incoherent and isotropic among homogeneous distributed molecules and the mutual donor acceptor interaction is restricted to the small encounter sphere (Rr 3 10/22,23).The diffusion length /Dif which the donor travels during its excited state lifetime ro is determined by the solvent viscosity r/ /Dif-" /DDif "t'D (10)   where DDif is the relative diffusion constant of both molecules in the solvent 1 DDif--DD + Da o:- For molecules in solution D is of the order of 10 -6 cm 2 s -1 10 -4 cm 2 s -1 and ET by encounter is likely for rDA < lDif.
The theory starts with the diffusion equation equation for the probability of a donor to be in the excited state at point r and time t 15 pz(r, 0 DDifAzpD(r, t)-,-po(r, t).3t ZD (12)   Spatial averaging of the one-donor solution leads to the following equation for the number of donors still excited after a delta function- like excitation pulse at t 0. 22 D(t) D(t O)e-t/D.exp {--4DDifRTNA t 8yI/2R2TNA(DDif t)I/2} (13)   On the other hand the increase of the excited acceptors is given by (14) with the abbreviations" o 4ZDDifRTNA N and are the acceptor density in cm -3 and the acceptor lifetime in the excited state, respectively.
Equations (13 and ( 14) represent the temporal behavior of an ensemble of excited donors and acceptors, respectively.Thus Eqs. ( 13) and ( 14) also describe the donor and acceptor fluorescence after excitation by a very short light pulse.Besides the shortening of the exponential decay of an isolated donor exp (-.TD t)=> exp-(+ kr)t (17) with kr 4DDifRTNA a characteristic nonexponential term with appears.
The diffusion theory can also be applied if there is no real motion of a molecule but only a migration of excitation energy, as it occurs, e.g., in molecular crystals by excitons.Then, of course, the parameter DDi has to be replaced by the exciton diffusion constant which strongly depends on the type of excitons and which can take values as large as Dx (10 -3 10 +2 cm 2 S-I).Thus, diffusion length up to some microns can be obtained clearly demonstrating effective ET over large distances by excitons. 13similar case exists for molecules in solution if the excitation energy is firstly spread among the highly concentrated donor ensemble (ND > NA) by repeated transfer steps (D7 + D--Di + Dr) until an encounter with an acceptor molecule occurs, Dn* + A --D + A*.
Thus, the donor excitation is quenched only in a final act (trap).The single step of such a cascade might be a real collision, a coulomb interaction, or an exchange interaction.

Exchange interaction
Starting from the matrix-element of exchange interaction (Eq. 3) Dexter 11 has derived the expression for the transfer rate of this kind of ET for an isolated donor acceptor pair (Eq.7).This formula holds for distances R > L where the overlapping electronic wavefunctions can be assumed to decline exponentially.A measurable quantity is again only the statistical average of the individual donor decays caused by this transfer rate.This spatial averaging over an infinitely large number of acceptors which are randomly distributed in space leads to the following donor fluorescence decay function: D(t)= D(t O)e -t/vo exp {-4 R3TNAy_3g(eY.t/.D)} (18) Here R r is the critical distance at which the exchange transfer occurs with the same probability as the spontaneous decay in an isolated donor, 7 2Rr/L, and g(z) is given by: g(z) -z exp (-z.y)(ln y)3 dy (19) 24 and results from the spatial averaging (z e't/rD).For any z > 0 the integration can be performed term-by-term leading to a Taylor series which can be expressed for sufficiently large z-values by: g(z) (In z) 3 + 1.732(ln z) 2 + 5.934(ln z) + 5.445 + O(e-Z(ln z)3z-2) (20) 24 For other nonuniform spatial distributions of donors and acceptors the spatial averaging will yield mean donor decay functions differing from Eq. 18 by magnitude as well as time behavior. 53.4.F6rster model of long-range energy transfer and related models In this model the donor transfers its excitation energy to one acceptor of its surrounding via a dipole-dipole interaction (DDI).Further assumptions with respect to the spatial distributions of the molecules are: i) The donor is surrounded by an homogeneous distribution of acceptor molecules which are fixed in space during the time of inter- action.
ii) No mutual influence between donors exists.iii) Each donor possesses its own acceptor environment.
These assumptions imply directly that the number of excited donors is much less than the number of unexcited acceptors (No " NA), a condition obviously fulfilled in steady-state experiments but clearly not in the case of donor excitation by ultrashort light pulses.We will refer to this in Section 4, taking first the assumptions to be valid.Starting again from the rate equation for the deactivation of an excited donor d p(t) 1 p(t) -' aa , /. IET(RDA)PD(t) (21)   dt "D i=1 where p(t) is again the probability of finding the donor still excited and kaaT(RDA) is given by (5).The sum extends over all acceptors in the vicinity of the donor under consideration.
Figure 4 shows the time behaviour of the donor decay (Eq.22) and the rise of acceptor excitation (Eq.23) due to F6rster model after -function-like excitation pulse of the donor ensemble.The optical transitions which form the overlapping integral of Eq. ( 6) can be of singlet or triplet type. 7en DDI is weak because of a forbidden optical transition in the donor or acceptor higher terms of interaction (quadrupole) must be considered 13'26 resulting in smaller R0 values.
But large transfer distances are possible even for forbidden optical transitions of the donor if the quantum yield D is high (phosphorescence) and the acceptor optical transition is allowed (large ez). 1 The critical ET radius R0 can be related to a critical concentration where dipole-dipole ET becomes efficient.
3 30002 (:r3/2NR)-a (24) NAO ---(73/2R03)-1 or CAO (Here Nzo is in cm -3 and CAO in mol 1-1; N is Avogadro's number of molecules per mol). 27ere are some specific spatial configuration to which F6rster-type ET can be applied under the restriction mentioned above.First there is the two-dimensional ET in monolayers, 28 where the donor decay becomes the form D(t) D(t= 0) exp F "q" \-o' J (25)   VD Here q denotes the ratio q rtA/nAO and r/A, nAO are the two- dimensional concentration and critical concentration nzo 1/zR in cm -2 and F(2/3) 1.354.Second in a quasi-one-dimensional system the donor decay function after spatial averaging has the form 13 (26)   and IA is the linear density in cm-1.

EXAMPLES OF TIME RESOLVED ET STUDIES
The simplest observable manifestation of ET is the quenching of fluorescence due to interaction between excited donors and unexcited acceptor, or conversely, the sensitization of fluorescence of a primarily unexcited acceptor.
Stationary measurements of concentration quenching of the fluore- scence 29 or the concentration depolarization 3 qualitatively indicate the existence of an ET process and permit to some degree the determination of characteristic transfer parameters.
Experimentally an exact distinction between the various ET mechanisms is only possible by means of time-resolved spectroscopic methods.Fortunately recent developments of lasers have delivered an almost ideal excitation source for time-resolved investigations of ET processes, which can produce monochromatic, short duration (nearly 6-like) pulse excitation.'33 This is necessary, because various mechanisms of ET differ mainly in the time domain shortly after excitation.
Until now, most of the time resolved investigations of ET of molecules in solution have been made by single-shot or synchroscan streak camera and single photon counting measurements of fluore- scence (e.g. . By means of excite-and-probe-beam spectroscopy using a ps-or fs-continuum the ET process can be observed in a broad spectral region, and even the excitation of non-fluorescent acceptors via ETprocesses can be studied with high time resolution (e.g.Ref. 37).

Confirmation of F6rster's /-Iaw
There are a lot of experimental data which well agree with the theoretical predictions deduced from the common F6rster model.In particular, in an intermediate acceptor concentration range of some CA 10 -2 10 -3 molthe validity of F6rster's decay law (Eq. 17) is well proved 33'34"38 in rigid solutions.
An example is given in Figure 4 which shows the fluorescence decay and rise of a donor and acceptor molecule according to Eqs. ( 22) and (23), respectively.The time resolved fluorescence was detected by a streak camera system. 17nother confirmation of a F6rster model of the donor decay in a time interval ranging from 10ps up to several nanoseconds was reported by Tredwell et al. 38 In the experiments the acceptor concen- tration range from 10 -2 to 10 -3 moll -1 in a rhodamine 6G (donor)-malachite green (acceptor) mixture in ethanol,     5 Fluorescence decay of rhodamine 6 G (10 -4 mol/l) in ethanol with (A) 0 mol/l, (B) 10 -3 mol/1, (C)2.5-10 - mol/l, (D)5.10 -3 mol/1 malachite green in ethanol streak camera records.3s For all concentrations a unique transfer radius of R0 52 + 1 A was obtained from a plot of log I versus t 1/2 (Eq.22).

Deviations from simple M-Iaw
In low viscosity solvents, deviations from the F/Srster's decay law (Eq.22) have been observed by additional motion of the molecules during the time interval of ET.
There are several attempts to take into account an additional diffusion of excitation energy [39][40][41] which are all based on an expansion of the ratio of diffusion length to the energy transfer radius /Dif/ R0.39,42, 49 Slight deviations from the dominating time behaviour according to Eq. ( 22) have been found experimentally for acceptor concentrations CA > 1 10 -3 in liquid solutions (e.g. in methanol DDif 2. 10 -5 cm 2 s-1).43,44 The experimental curves can be fitted well by modified equations for the donor decay, which take into account motion of the molecules (Eq..3.ET under the influence of inhomogeneous spatial distribution Despite the various confirmations of the F6rster model there have also been reported several experimental deviations from the F6rster-X/law, which cannot be explained by additional diffusion 5,35'37 in the low concentration range. Even at such low concentrations of 10 -4 10 -5 mol 1-1 a very efficient ET was observed, suggesting an increase of the critical transfer radius R0 in this concentration range which is in contradiction to its physical meaning (Table I).
The efficient donor-acceptor ET has also been demonstrated by the surprisingly fast rise of acceptor excitation in ps-excite-and-probe beam experiment (Figure 6).The rise time to the maximum of the acceptor excited state population amounts for different donor acceptor pairs: rhodamine 6 G (donor, cresyl violet, DOTCI, oxacine 725 (acceptors) tr 20 ps, 40 ps and 50 ps, respectively, and shows no significant dependence on acceptor concentration or solvent vis- cosity. 37Similar short transfer times have been observed for rhoda- mine 6 G to cresyl violet and malachite green. 36hese short rise times are also in discrepancy with theoretical Dimer in H20 Co 1 10-3 mol/l Co 1" 10--3 mol/l Co Eq. ( 6) Eq. ( 6) Eq. ( 6) Eq. ( 6)  6(a) Spectral changes in optical density at various delay times after excitation.Curves obtained by a ps-excite-and-probe spectrometer with ps-continuum probe pulse; rhodamine 6 G donor, cresyl violet acceptor, in ethanol; Ro 52 + 2 A. 37 -AOD 0.5 0,4 0,2 x [nm] 540 Rho --------623 CVf= Figure 6(b) Excitation density of donor and acceptor molecules versus delay time.., rhodamine 6 G direct excitation by a 5 ps SHG ( 532 nm)--laser pulse- cresyl violet excitation via ET from rhodamine 6 G co 1 10 -4 mol/l, ca 5 10 -5 mol/l. 37redictions of 400 ps (at CA 10 -2 mol 1-1) as derived from F6rster's formula (Figure 4). 34is contradiction can be removed by the assumption of a gen- eralized F6rster-model taking into account a possible inhomogeneous spatial distribution of acceptor molecules around the donor molecules.Charged dye molecules in solution are very likely interacting by weak static van-der Waals forces described by a Lenard-Jones potential, e.g., Figure 7, causing a mutual attraction of these molecules.In this way an inhomogeneous spatial distribution of acceptor molecules can be produced.Assuming a simple step-like sphero-symmetrical inho- mogeneous distribution spatial averaging according to this inho- mogeneity leads to an analytical expression with a modified time behaviour of the donor and acceptor ensemble: 37 For a b and R1 0 this equation approaches normal F6rster-like donor decay for a homogeneous acceptor distribution.The dependence on R1 (the radius of forbidden volume) is not very significant, if it is small enough (R1 10 A).
By fitting an experimental curve with the numerical calculations of A(t) one can derive the larameters of the model (Figure 8), e.g., a/b 25, R1 10 A, R2 20 A for an ET of rhodamine 6 G to cresyl violet in ethanol with R0 50 + 2/.37 Thus from experimental data one may get not only information about the efficiency of ET(R0) but also about the distribution of the molecules (a/b, R1, R2).Equation ( 27) can also be derived in a modified form for two- dimensional planar and also spherical geometry of the donor-acceptor assembly. 25These models have been successfully applied in describing energy transfer processes in microstructured systems (micelles, bilayers) see, e.g., Refs.45, 46 and 47, which present model systems of photosynthesis.

CONCLUSIONS
We have given a survey of the main ET mechanisms occurring between interacting molecules in solution, and have outlined their range of occurrence.Further we have summarized some of the most common models together with their substantial assumptions and limitations.It has been shown that these models, within their range of applicability, can be used to describe energy transfer processes in solution over a wide range of concentration ratios, solvents and molecular structures adequately.
We have not, for sake of shortness, gone into the problem of mutual donor-donor interaction as outlined in Refs.48 and 49.This problem is being treated in a forthcoming paper in detail, together with a detailed investigation of energy transfer processes in microstructured systems (Micelles, Vesicles, Bilayers). 25

Figure ]
Figure ] Simpfified ener8y k]evel scheme of donor and acceptor molecules.The ET

Figure 4
Figure 4 Time resolved fluorescence decays indicating an ET from rhodamine 6 G (a) m Figure

Figure 7
Figure 7 Inhomogeneous spatial distribution of acceptors around one excited donor a, b increased and bulk density of acceptor molecules; R1, R2 radii of sphere of raised acceptor concentration and forbidden volume.37

Table I
Critical transfer radius R0 calculated from the decay of the donor fluorescence in dependence on the acceptor concentration C A.