1944
J . Phys. Chem. 1990, 94, 1944-1948
indicate a single, slow, first-order removal process, the nature of which could not be ascertained.
Materials Sciences Division of the U S . Department of Energy under Contract No. DE-AC03-76SF00098.
Acknowledgment. This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences,
83-7.
Registry No. H, 12385-13-6; U02,m,,1344-57-6; U02,036r 109676-
Forward and Reverse Energy Transfer in Langmuir-Blodgett Multilayers K . Sienick? DPpartement de Physique-COPL, UniuersitP Laual, QuPbec, P.Q., Canada G I K 7P4 (Received: April 10, 1989; In Final Form: September 27, 1989)
The transport of electronic excitations among chromophores randomly distributed in Langmuir-Blodgett multilayers is described. In this theoretical analysis, it has been assumed that excitation energy can be transferred forward and reverse between two-dimensional layers. Fluorescence decays have been calculated for each layer. A numerical analysis of fluorescence decays is given in order to show the influence of reverse energy transfer on photophysical properties of Langmuir-Blodgett multilayers. The discussion of results in connection with recent and future experimental studies is presented.
(1) Introduction One of the most important functions of the antenna pigments is the collection of light energy from the sun. There are several different types of antenna pigments; chlorophyll a and b in green plants, chlorophyll c in some algae, and bacteriochlorophyll a, b, or c in bacteria. Another important class of pigments is represented by such accessory pigments as carotenoids and phycobiliproteins. It is well-known that, after light absorption by a pigment molecule, the electronic excitation energy is transferred until it is trapped by a reaction center. There has been a large effort devoted to the understanding of mechanisms of energy transfer and trapping by the antenna pigments.l Recently, this effort was enhanced by the search for photochemical molecular devices and an artificial analogue of the biological antenna capable of harvesting solar light. The main objective of supramolecular photochemistry* was to construct an appropriate assembly of molecular components capable of performing light-induced functions. There are several functions that can be realized by photochemical molecular devices, between them (i) generation and migration (or transfer) of electronic excitation energy, (ii) photoinduced electron transport, and (iii) photoinduced conformational changes. Yamazaki et al.3” have studied excitation energy transfer in Langmuir-Blodgett multilayers. In a recent paper3 of those authors, the studies of sequential excitation energy transfer in Langmuir-Blodgett films consisting of sequences of a donor layer (D) and three acceptor layers (Al, A,, A,) have been presented. In the analysis of fluorescence decays mentioned, the authors assumed that two-dimensional energy transfer for four layers is PC APC Cha.’ This moving in one direction, Le., PE assumption is only partially fulfilled for this system because the Forster critical is 63 %, between PE and P C and 61 A between PC and APC. However, the critical radius for reverse energy transfer is 13 A between PC and PE and 44 A between APC and PC. The question arises-is the reverse energy-transfer process really negligible? One of the preconditions for transfer of excitation energy from an excited donor molecule to acceptor molecule is a partial overlap between the donor fluorescence band and the acceptor absorption, which is usually expressed by the spectral overlap integral9
- - -
SCHEME I
where I is the wavenumber, is the molar decadic excitation coefficient of the acceptor, andfD(ij) is the spectral distribution of the donor fluorescencenormalized to unity. The spectral overlap integral is related to the critical distance RODA, corresponding to the donor-acceptor separation at which the probability of emission is equal to the probability of energy transfer, and is given by
where ( x 2 ) is a factor depending on the mutual orientation of the transition moments of the interacting molecules, a,, is the quantum yield of donor fluorescence, and N’is the number of molecules per millimole. One can easily see that for some molecules a similar precondition described as above could be also fulfilled for reverse energy transfer from acceptor to donor molecule. As an example, the respective critical radii for PE and PC molecules have been given above. Recently, we have theoretically studied the problem of forward and reverse energy transfer in the presence of energy migration.I0J1 ( 1 ) Biological Events Probed by Ultrafast Laser Spectroscopy; Alfano, R. R., Ed.; Academic: New York, 1982. (2) Supramolecular Photochemistry; Balzani, V., Ed.; Reidel: Dordrecht, The Netherlands, 1987. ( 3 ) Yamazaki, I.; Tamai, N.; Murakami, A,; Mimuro, M.; Fujta, Y. J . Phys. Chem. 1988, 92, 5035. (4) Tamai, N.; Yamazaki, T.; Yamazaki, I. J . Phys. Chem. 1987, 91, 841. ( 5 ) Tamai, N.; Yamazaki, T.; Yamazaki, I. Chem. Phys. Lett. 1988, 147, *c
42.
IDA =
Jmf~(V)
t ~ ( D ) 3 -dt, ~
(1.1)
’
Present address: DEpartement de Chimie, Universite du Montreal, Montreal, Quebec. Canada H3C 337.
0022-3654/90/2094- 1944$02.50/0
( 6 ) Yamazaki, I.; Tamai, N.; Yamazaki, T. J . Phys. Chem. 1987,91,3572. (7) Abbreviations: PE, phycoerythrin; PC, phycocyanin; APC, allophycocyanin; and Chla, chlorophyll a. For more description, see ref 8. (8) Grabowski, J.; Grantt, E. Photochem. Photobiol. 1978, 28, 39. (9) Forster, Th. 2. Naturforsch. A . 1949, 4 , 321.
0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 5, 1990 1945
Energy Transfer in Langmuir-Blodgett Multilayers
In those studies, we were considering three-dimensional energy migration and transfer, and it has been shown that forward energy transfer from the acceptor to donor molecule leads to slower donor decay. Additionally, it has been shown that the long-time approximation reverse energy transfer from the acceptor leads to a nonexponential donor decay (see ref 8, Figure 1). A similar deviation of donor decay has been observed experimentally by Yamazaki et aL3 (see ref 3, Figure 1). In this paper, we will present a theoretical model describing donor-acceptor kinetics of two-dimensional forward and reverse energy-transfer in stacked multilayers. (2) Theoretical Model To develop a theoretical model, we will assume-following Yamazaki et aL3-that excitation energy does not migrate in the donor or acceptor layers and that the layers represent a two-dimensional medium for luminescently active molecules. Let us assume that a multilayer consists of four layers, which are represented in Scheme 1. The rate coefficeints for energy transfer between layers i, j (i # j ) are represented by k . . ( t )and rate constants for fluorescence from layer i are denotedl by kFi. The time-dependent fluorescence intensity after first excitation by a 6-pulse from layer i is given
with the result
i,( I ) ( s)
i,(s) = 1
(2.7)
-PI
where
The situation presented in Scheme I suggests that the concentration of excited molecules in layer 3 can be calculated in a similar way as for layer 1. One can obtain
and
The total concentration of excited acceptors in layer L3 can be calculated from the relation n
i 3 ( s ) = Iim CL3(k)(s) k-m
where ri = 1/kFi andh(0) is the concentration of excited molecules in layer i at t = 0. The concentration of initially excited donors in layer i = 1 is Li'lYt) = ra(t) @ f i ( t )
h(8) d e
After the first energy-transfer step from layer L , to L,, excitation energy can be transferred an infinite number of times between layers L2 and L3and can return from layer L2 to L,. Thus, the equivalent equation for the total concentration of excited donors in layer L , after the first forward and reverse energy transfer from L2 has the form @ (1 + k23(t) f2(t) @ k32(t) f3(t)12 + @ k21(f) f2(t) (2.3) Taking the Laplace transformation of eq 2.3 and noticing that k 2 3 ( t )f 2 ( t )@ k32(t) h(t)< 1, one can obtain
Ll(2)(t)
k32(t)
Ll(2)(f)
=
Ll(l)(t)
@ k12(t) f l ( t )
h(t)+ [k23(t) f 2 ( t )
@
= L , ( 1 ) ( s ) I ( k l ~ ( 2 ) f , ( t ) ) I ( k 2 1f( t2) ( W [1 - L(k23(t)fi(t))L(k32(t)f3(t))I (2.4)
where the circumflex and i(s)
I denote the Laplace transformation
L(g(s)) =
J,
with the result (2.13)
(2.2)
with the assumption thatfi(0) = 1. In eq 2.2, Za(t) is the intensity distribution of excitation pulse and @ denotes the convolution integral g(t) @ h ( t ) = &'g(t-8)
(2.12)
k=l
where
In the case of the second layer, excitation energy can be transferred forward and reverse between layer L2 and layers L1 and L3. One can see that after n steps of forward and reverse energy transfer the concentration of excited molecules in layer 2 is L,c")(t) = L l ( W f2(t)
k12(0fl(0 @ f 2 ( 0 + + + L3(')(t) @ k32(t) f3(t) &(')(t)
'1.
@
k32(t)h(t) @f3(t)
e3'"(s)/
