Gösele, U.

Hauser, M.

Klein, U. K. A.

English

Gösele U.

Klein U. K. A.

Hauser M.

University of Szeged

LONG-RANGE ENERGY TRANSFER IN SYSTEMS OF DIFFERENT DIMENSIONS By U. GOSELE, U. K. A. KLEIN and M. HAUSER Max-Planck-Institut fur Metallforschung, Stuttgart, and University of Stuttgart, Federal Republic of Germany The time-dependence of long-range energy transfer by dipole-dipole interaction depends on the dimension of the systems considered. This may provide a possibility to distinguish between different molecular arrangements such as spheres, discs or rods. 4 Introduction The statistics of long-range energy transfer by dipole-dipole interaction from excited donor molecules to unexcited acceptor molecules was firstly treated by FÖRSTER [1] and GALANIN [2] for statistically distributed donor and acceptor molecu-les. Later on the influence of additional diffusion on the energy transfer was inves-tigated by a number of authors [3, 4] and, e.g. applied to energy transfer ('excitons') in molecular crystals [5, 6]. In this paper we will extend the previous treatments for rigid solutions to two- and one-dimensional systems, i.e. for example energy transfer on surfaces or along molecular chains, respectively. It is to be expected that different dimension lead to differences in the time behaviour of the energy transfer. The following section deals with statistically distributed donors and acceptors in 'infinite' systems and the last section with single donor-acceptor pairs in finite systems of molecular dimensions. Throughout this paper we always assume that diffusion effects are negligible. Long-range energy transfer in infinite systems of different dimensions The time-dependence of the concentration c*Djn of excited molecules (proportional to the donor fluorescene intensity) after ¡5-excitafion may be derived from [7] -JT= -nDc*Dm-<Psm(t)c*Dm, (1) where nD is the reciprocal lifetime of the donor fluorescence without energy transfer and $ „ ( 0 is given by r m ( 0 = c A f exp [-W (r)t]amrm~1 dr. (2) 0 90 U. GOSELE, U. K. A. KLEIN and M. HAUSER The subscript m in Eq. (2) is the number of dimensions (m = 1, 2, 3) of the system. The superscript s indicates that we deal with statistically distributed molecules. cA is the volume concentration, surface concentration or linear concentration of acceptor molecules for the three-, two- or one-dimensional case, respectively. Accor-ding to FORSTER [1 , 8] the energy transfer rate W(r) for dipole-dipole interaction is given "by • ' ' • *• . , = = ( 3 ) The factors am in (2) are ai = 2, a2 = 2n and a3 — 4TT. (4) Evaluation of (2) and integration of (1) leads for ¿-excitation of the donor molecules in the three-dimensional case to [1, 2, 6, 7] in the two-dimensional case to ct2(t) = c%°exp [ - „ „ / - ^ ( a O 1 ' 3 ] , and in the one-dimensional case to -" c*Dl(t) = c$0exp[-nDt-2.26cA(0Lty16]. The acceptor fluorescences are determined by the expression • ' • = - n A C * A m + ^ U t ) c * D J t ) - . • . (8) which transforms for ¿-excitation of the donor molecules into . i - • • c*Am(ty= txp(-nAt) f exp{nAt')c*Dm(t')<P°m(t')dt', (9) . 0 where c*Am is the concentration of excited acceptor molecules. As shown is ref. [7] the differences between the acceptor fluorescences for the different dimensions are even more distinguished than those of the donor .fluorescences. Long-range energy transfer for single donor-acceptor pairs in finite systems In this section we assume a solution which contains finite systems of molecular dimensions such as spheres, discs or rods. A number of these molecular arrange-ments shall be occupied by single (immobile) donor-acceptor pairs. Diffusion effects and interaction between different pairs (in different finite systems) shall be neg-ligible. This may, e.g. be realized in micellar solutions (9, 10). If the donor-accep-tor distribution in the molecular arrangements is given by a distribution function p(r) with rma \ / p(r)dr = 1 (10) (5) (6) (7) LONG-RANGE TRANSFER IN SYSTEMS OF DIFFERENT DIMENSIONS 91 (rmax: maximum distance between donor and acceptor molecule in the finite system) the donor fluorescence after ¿-excitation ma be evaluated with 'max c*D . = 4 0 e x p ( - n D i ) / exp [—IV(r)t]p(r)dr. (11). o This has been done in Ref. [7] for various distributions p(r) for both donor and acceptor fluorescences. Here we will only give approximate results for short times based on a simple analogy to the infinite systems. Since the dipole-dipole energy transfer is mainly governed by the probability of finding, an acceptor molecule in the immediate sorroundings of an excited donor molecule we may obtain approximate results by using the formulae for infinite system (5—7) and as an effective accep-tor "concentration" 1 molecule per finite system, e.g., per sphere, disc or rod. As: an first example we treat the case of a sphere (radius R) where the donor molecule is fixed in the centre and,the acceptor molecule is, on the average, equally distributed around it. With cA=l/(4nR3/3) we obtain after• ¿-excitation with (5) for three-dimensional energy. transfer . 1 c £ ^ c S 0 e x p [ - / 7 0 i - ^ ( ™ 0 1 / 2 ] , (12) which should be a good approximation at least for short times. If the donor is fixed on the surface of the sphere the effective acceptor concentration must be divided by two due to obvious geometrical reasons. This leads to 4 % 4 ° e x p [ - n D i - ^ ( 7 r a 0 1 / 2 ] . (13) If both molecules, donor and acceptor, are fixed oh the surface on statistical posi-tions the energy transfer may be approximated at least in the very beginning as a two-dimensional process with an effective acceptor concentration per area cA = = l/(4nR2). So we get with (6) c*D « exp [ - nDt - J ^ (ar)1'»]. ' (14) The same concept yields for a donor fixed in the centre of a disc (radius R) and.the acceptor statistically distributed around it with cA = l/(nR2) for short times cd ~ cd° e x P " c ? ~ ( a i ) l /3] • (15) For a rod (length L) where the donor is fixed in the middle, or at least not too near to an end of it, we obtain with the effective concentration per length cA = l/L c*D % 4 ° exp - n D i - ^ ( « 1 / 6 | . (16) The formulae (12—16) are only valid for the beginning of the energy transfer. After-wards the finite character of the systems essentially influences the kinetics of the luminescence quenching. More thorough calculations based on (11) are given in / 92 U. GÖSELE, U. К . A. KLEIN and М. HAUSER ref. [7] especially for the acceptor fluorescences which are also in the case of finite systems more sensitive to the dimensionality of the energy transfer. As discussed extensively in this paper it is helpful for the determination of the shape of molecular arrangements, e.g., of micelles [10], to measure the time-dependence of the fluores-cences of single donoracceptor pairs affixed to the molecular systems to be inves-tigated. References [1] Förster, Th.: Z. Naturforschung 4a, 321 (1949). [2] Galanin, M. D.: Sov. Phys. — JETP 1, 317 (1955). [3] see Berlman, I. B.: Energy transfer parameters of aromatic compounds (Academic Press, New York, 1973). [4] Cósele, U„ M. Hauser, U. K. A. Klein, R. Frey: Chem. Phys. Letters 34, 519 (1975); Klein, U. K. A., R. Frey, M. Hauser, U. Gösele: Chem. Phys. Letters 41, 139 (1976). [5] see Wolf, H. C.: in "Advances in atomic and molecular physics", Vol. 3, eds. D. R. Bates and I. Estermann (Academic Press, New York, 1967), p. 119. [6] Powell, R. C..Z. G. Soos: Z. Luminescence 11, 1 (1975). ' [7] Hauser, M„ U. K. A. Klein, U. Gösele: Z. Physik. Chemie, Neue Folge (1976), in press. [8] Förster, Th.: Ann. Physik 2, 55 (1948). [9] Kenney—Wallace, G. A., J. H. Flint, S. C. Wallace: Chem. Phys. Letters 32, 71 (1975). [10] Frey, R., J. Steinwandel (to be published). ДАЛЬНОДЕЙСТВУЮШИЙ ПЕРЕНОС ЭНЕРГИИ В СИСТЕМАХ РАЗЛИЧНЫХ РАЗМЕРОВ У. Гёзеле, И. К. А. Клейн и М. Хаузер Зависимость дальнодействующего переноса энергии от времени по механизму диполь-дилольного взаимодействия зависит от размеров исследованных систем. На основе этого воз-можно различить молекулярное расположение в таких системах как шары, диски и стержни. 

Long-range energy transfer in systems of different dimensions

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Long-range energy transfer in systems of different dimensions

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