Electronic Energy Transfer on Fractals

PHYSICAL REVIEW LETTERS

VOLUME 52, NUMBER 24

11 JUNE 1984

Electronic Energy Transfer on Fractals U. Even, K. Rademann, ' and J. Jortner Department of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel and

Department

N. Manor and R. Reisfeld of Inorganic and Analytical Chemistry, The Hebrew

University,

Jerusalem, Israel

(Received 23 March 1984)

We advance an experimental method for the characterization of the geometric site distribution of a fractal structure, which rests on the interrogation of direct, long-range, singletsinglet, intermolecular, electronic energy transfer. Time-resolved picosecond spectroscopy was utilized to study energy transfer from rhodamine B to malachite green doped into a porous glass, resulting in a fractal dimension of d = 1.74 + 0. 12 for this irregular structure. PACS numbers:

61.40.-a

Self-similar structures with dilatation symmetry, which are referred to as fractals, ' are of considerable current interest because of their potential to describe a multitude of irregular structures ranging from proteins to galactic structures. ' In the area of the condensed-matter physics it has been suggested recently that a wide range of disordered syssurtems, e.g. , linear and branched polymers, faces, epoxy resins, and percolation clusters, are characterized by a fractal geometry over a miFractal structures are croscopic scale length. described by (at least) three distinct dimensionss 9; the spatial dimension, d, of the embedding Euclidean space, the fractal dimension, ' d, which relates the dependence of the site number N (R) on the radius R [N (R) ~ R d], and the spectral (fracton) dimension, d, which characterizes the density of states' and random-walk processes on fractal ag' The spectral dimension of some gregates. ' fractals was inferred from electron-spin relaxation ' i.e. ferrodoxin and ferricytodata in proteins, , chrome c, and from triplet-triplet annihilation studies in mixed molecular crystals. ' We wish to report on the first determination of the fractal dimension of an irregular structure. Our experimental methods rests on the measurements of direct, long-range, singlet-singlet, intermolecular, electronic energy transfer (EET) from an excited donor molecule to an ensemble of randomly distributed acceptors on a fractal structure. EET processes provide a powerful tool for the characterization of complex structures. This method has already been utilized to probe the morphology of The applicability of EET for the interpolymers. rogation of fractal structures was suggested by Klafter and Blumen, ' who have demonstrated that the direct trapping process is determined by the fractal dimensionality. The survival probability

""

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P(t) of

the excited state of the energy donor at time t is given by'

P (t) = exp[

~ (t/~) ],— y(t/r)—

where

P=d/s,

(2)

v is the radiative decay lifetime of the donor, and s is the order of the multipolar EET rate w (R) ~ R '. The proportionality factor y is

y =xq

(d/d)I'(1 —d/s) (Ro/a

),

(3)

where xz is the fraction of the fractal sites occupied 1), a is an (average) size by the acceptors of a unit cell, and Ro is the "critical" radius for EET, corresponding to w (R o) ~ = l. As is apparent from Eqs. (1) and (2), direct unistep EET on a fractal is determined by the fractal dimensionality. We have studied EET between rhodamine B (RB) and malachite green (MG) doped into a porous "thirsty" glass. ' This glass (Vycor 7930), which is prepared of borosilicate by leaching constitutes a porous open structure (efglass, ' fective surface area A = 200 m'/g; density p =1.5 g cm 3).2 The average diameter is pore r„=40+ 3 A. The volume fraction of the pores is2 0.28, which exceeds the critical volume fraction p, = 0.16 + 0.02 for continuous percolation in three dimensions. ' The pores form topologically connected, intermeshed, random paths in the porous glass, whose stochastic geometry characteristics are amenable to description in terms of a self-similar structure. This fractal structure was lightly doped with the RB energy donor and the MG energy acceptor. Disks of Corning Vycor 7930 porous glass (14.5 mm diam and 1.4 mm thickness) were cut from a glass rod. The disks were dipped for 48 h into methanol solutions containing RB or RB+MG

1984 The American Physical Society

(x„((

VOLUME 52, NUMBER 24

PHYSICAL REVIEW LETTERS

dried for 48 h in a vacuum at and subsequently 30 C until a pressure of 10 Torr was attained. The effective concentrations CD of RB and C& of MG (per unit volume of the glass) were determined from the change of the dye content of the solutions. These effective concentrations were varied in the range CD= (3x10 -6x10 )M and Cz = (1.9 x10 -5.1x10 )M. The porous glass was uniformly doped, as indicated by optical absorption measurements as well as by optical bleaching experiments conducted at high laser intensities. EET in the MG+RB system is expected to be induced via dipole-dipole coupling 26 27 (s = 6). The high fluorescence quantum yield of RB's ( Y= 0.9—1.0) and the low fluorescence quantum yield of MG 10 2 in the in methanol29 and Y ( Y 10 porous glass) enabled us to interrogate the time evolution of the donor fluorescence without in-

=

1I JUNE 1984

LLJ

l— LLI

i3-

0—

=

terference from the fluorescence of the acceptor. The EET for this donor-acceptor pair is characterized by a critical radius of Ro = 90 A, as determined Accordingly, 8 0 somewhat by us in solution. exceeds r„, and the unistep BET process monitors the site distribution of the fractal over a microscopic scale length of r„, which is considerably shorter than the characteristic length of the fractal object. The time-resolved fluorescence decay of RB and of RB+MG mixtures in the porous glass was interrogated by a picosecond photon counting method. RB was excited at 5730 A by a dye laser (Coherent C599) synchronously pumped by a mode-locked Ar-ion laser (Coherent CR12). The dye-laser output (75-MHz pulse train with a pulse width of 5 psec and average power of 250 mW) was used at a reduced intensity (10 mW) to illuminate the doped porous glass, which was maintained in a vacuum The time-resolved chamber at room temperature. fluorescence was detected by a two-stage microchannel-plate photomultiplier whose output pulse was monitored by a fast time analyzer which consists of a pair of constant fraction discriminators The (Tennelec 453) and a microcomputer. response curve, F(t), of the lifetime measurement system (Fig. 1) was characterized by a width (full width at half maximum) of 180 psec. The timeresolved decay of RB was simulated by the convolution Io(t) =F(t) S exp( —t/r), and a mean leastsquares fit of the experimental data was performed to extract the lifetime 7. The decay dynamics of the RB+ MG system was simulated by the convolution of the survival probability, Eq. (1), in the form I(t) =F(t) P(t). With use of the 7 value obtained from lo(t), the best values of p and y were mean least-squares extracted by a multidimensional

—

8

Excitation Pulse Response

0 0

I

2

5

4

5

T ME / I

n

6 sec

7

8

9

l0

FIG. 1. Fluorescence decay curves of solutions of RB and of RB+ MG in methanol. The dots represent the exdata. The lower curve represents perimental the response of the photon counting system to the picosecond laser pulses with the solid line simply connecting the experimental points. The upper curve portrays the time-resolved decay of RB (CD=7.5x10 'M), with the solid line corresponding to the best fit of Io(t) with the timev =2.5+0.1 nsec. The middle curve portrays resolved fluorescence of RB (CD=1.0x10 M) plus MG (Cq = 2.2 x 10 4M), with the solid line corresponding to the best fit of 1(t), with P= 0.50, y =0.75, and ~ = 2.5 nsec.

fit analysis of I(t). To test the reliability of our exof p, perimental procedure for the determination we have obtained some data on EET between RB and MG in methano1 solution. The donor concentration in solution was CD = (10 -2 x 10 )M, while the acceptor concentration was Cz —(10 2.2x10 4)M. From the exponential decay curve (Fig. 1) of RB, we obtain v=2. 5+0.1 nsec. The decay curve for RB+MG (Fig. 1) resulted in the value of p=0. 50+ 0.05, which is in perfect agreement with the Forster result26 p= —, for dipolespace (s =6 and dipole EET in three-dimensional d = 3). The values of y were found to be independent of CD and were proportional to Cq within the of + 5'/o. From the uncertainty experimental y=4m RoC„/3, together with Forster relation value of y = 0.75 + 0.08 for our experimental =2. 2x obtain Ac= 91 + 5 A for the 10 we M, Cz RB+ MG pair.

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fluorescence decay curves The time-resolved from RB and from RB+ MG in the porous glass are portrayed in Fig. 2. The decay of the excited state of RB in the one-component system can be fitted with 7 =3.25+0. 10 we11 by a single exponential nsec. In view of the high fluorescence quantum we assert that the 30% inyield of RB in solution, crease of ~ for RB in the porous glass relative to the solution cannot be blamed on the blocking of the nonradiative decay channel in the former case, but rather is attributed to inner-field effects on the pure radiative lifetime. ' The decay curve of RB for the RB+MG mixture in the porous glass was analyzed (Fig. 2) to yield p=0. 29+0.02. This value of p was found to be independent of CD over the narrow (effective) concentration range used by us, demonstrating that a unistep direct BET is involved. Equation (2), together with s = 6, result in d = 1.74 + 0.12 for the fractal dimension of the porous glass. y was found to be linear within an experimental uncertainty of + 20'/0 on Cz in the (effective) concentration herein, range employed

I— LLJ

1—

0--

11 JUNE 1984

which is in accord with Eq. (3). The experimental values of y can be utilized to extract an estimate of the fraction of the fractal sites occupied by the acceptor molecules. Equation (3), together with the approximate value a 6 A, which corresponds to the molecular radius, results in xz =4.0x10 For C„=5.1 x10 M, we get y= 3.9+0.1 (Fig. 2), which gives xz = 1.6 10 2 for this doped porous glass. From this a posteriori analysis of our experimental data, we infer that the glasses used by 6'/o and us were indeed lightly doped (e.g. , 2'/o for the fraction of occupied donor sites xD in the glass studied in Fig. 2), whereupon 1 and && && multistep energy diffusion xD xz between the donor centers is negligible. The fractal dimension d = 1.74+ 0. 12 for the void structure of the porous glass is considerably smaller than the value of d = 2.6 for a percolation cluster in three dimensions, being close to the value d=1.8-2.0 for a three-dimensional cluster backbone, obtained by erasing the dangling bonds from the percolation cluster. Our value of d is somewhat lower than the value of d =2 expected for convoluted and irregular shapes and forms adopted by random coils in a solid, e.g. , a polymer in a glass. 2 Our work demonstrates the potential and scope of iong-range EET for probing important geometrical details of irregular structures. %e are indebted to Dr. J. Klafter and Mr. I. Rips for stimulating discussions, and to Dr. D. Avnir for information. One of us (K.R.) acknowledges with thanks a Minerva Fellowship for Israel-German Scientific Co-operation. Partial support of this research was provided by the Committee for Basic Research of the Israel National Academy of Sci-

=

&&

— -0.

x„=1.

ences, Jerusalem.

LLJ

1LtJ

ine 8 ite - green

us glass

0 0

I

2

3

4

5

6

7

8

9

10

TIME /nsec

FIG. 2. Fluorescence decay curves of RB and of RB+MG in a porous glass. The upper curve portrays the time-resolved decay of RB (Co = 5.6 x 10 'M), with the solid line corresponding to the best fit of Io(t) with r = 3.25 nsec. The lower curve portrays the timeresolved fluorescence of RB (Co=5.6x10 'M) plus MG (Cq = 5.10&& 10 M), with the solid line corresponding to the best fit of I (t) with P = 0.29, y = 3.9, and r = 3.25 nsec.

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(') Permanent address: Institut fur Physikalische Chemic, Freie Universitit Berlin, D-1000 Berlin 33, %est Germany. iB. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1982). H. J. Stapleton, J. P. Allen, C. P. Flynn, D. G. Stimson, and S. R. Kurz, Phys. Rev. Lett. 45, 1456 (1980). J. P. Allen, J. J. Colvin, D. G. Stimson, C. P. Flynn, and H. J. Stapleton, Biophys. J. 3$, 299 (1982). ~J. P. De Gennes, J. Chem. Phys. 76, 3316 (1982). R. Rammal and G. Toulouse, J. Phys. Lett. 44, 213

(1983). 6D. Avnir, D. Farin, and P. Pfeifer, Nature (London) 308, 261 (1984). S. Alexander, C. Laermans, R. Orbach, and H. M.

VOLUME 52, NUMBER 24

PHYSICAL REVIEW LETTERS

Rosenberg, Phys. Rev. B 28, 4615 (1983). Y. Gefen, A. Aharony, B. B. Mandelbrot, and S. Kirkpatrick, Phys. Rev. Lett. 47, 1771 (1981). S. Alexander and R. Orbach, J. Phys. Lett. 43, 65

(1982) . tOY. Gefen, A. Aharony, and S. Alexander, Phys. Rev. Lett. 50, 77 (1983). "P. G. De Gennes, C. R. Acad. Sci. 296, 881 (1983). t2S. Alexander, Phys. Rev. B 27, 1541 (1983). J. Klafter and A. Blumen, J. Chem. Phys. 80, 875 (1984) . J. Klafter, G. Zumofen, and A. Blumen, J. Phys. Lett. 45, L49 (1984). A. Blumen, J. Klafter, and G. Zumofen, Phys. Rev. B 2$, 6112 (1983). 6In the pioneering work of Stapleton and co-workers (Refs. 2 and 3) no distinction was drawn between the fractal dimension d and the spectral dimension d. The spin-relaxation process in ion proteins is dominated by phonon Raman scattering, which is determined by the density of phonon states. Accordingly, the relevant dimension involves d rather than d. From the structural data analysis, in conjunction with the spin-relaxation (Ref. 3), it appears that for proteins d d. P. W. Klymko and R. Kopelman, J. Phys. Chem. 87,

=

4565 (1983). tSP. Evesque, J. Phys. (Paris) 44, 1217 (1983). The utilization of triplet exciton collisions for the interrogation of the fractal structure of a percolation aggregate in isotopically mixed crystals (Refs. 17 and 18) requires that the triplet energy transfer in the impurity

11 Jt NE 1984

band be described in terms of classical percolation. It is still an open question whether quantum tunneling effects

characteristics of short-range triplet energy transfer; N. F. Mott, Philos. Mag. 29, 613 (1974); J. Klafter and J. Jortner, Chem. Phys. Lett. 49, 410 (1977). 206. H. Fredickson and C. %. Frank, Macromolecules will not erode the classical percolation

16, 1198 (1983). 2tF. A. Schwertz, J. Am. Ceram. Soc. 32, 390 (1944). E. Nordberg, J. Am. Chem. Soc. 27, 299 (1944). Technical information from Corning Glass %'orks.

22M.

24R. D. Zaiien, The Physics of Amorphous Solids (Wiley, New York, 1983). I. %ebman, J. Jortner, and M. H. Cohen, Phys. Rev.

B 14, 4737 (1976). 2~T. Forster, Discuss. Faraday Soc. 27, 7 (1959). 7Retardation of the orientational motion of the donor and acceptor does not modify the nature of the coupling responsible for EET in viscous solvents [D. P. Miller, R. J. Robbins, and A. H. Zewail, J. Chem. Phys. 75, 3644 (1981)] and in glasses [R. Reisfeld, Structure and Bonding (Springer, Heidelberg, 1976), Vol. 30, p. 65]. At the low MG+RG concentrations employed herein, the EET both in solution and in the porous glass is induced by dipole-dipole coupling. 2sI. Berlman, Fluorescence Spectra of Organic Molecules (Academic, New York, 1966). 29E. P. Ippen, C. V. Shark, and A. Bergman, Phys. Rev.

Lett. 3$, 611 (1976). 3 A. B. Myers and R. R. Birge, J. Chem. Phys. 73, 5314

(1980).