J . Phys. Chem. 1992,96, 566-571
566
well-defined. However, this is somewhat illusionary unless both shift tensors are axially symmetric. In the present case, Kc/KH = (Yc/YH)(A~c/AQH)(~ - r)c(COS 2% - (9/2) sin 2ac)/(3 ~ ~ ( 2ffH ~ 0 (9/2) s sin 2 4 = 16. If the ratio of angular terms is not considered, the observed value of KH implies a proton shift anisotropy of approximately 3 ppm.
Conclusion A relaxation study of a 30% v/v solution of cyclopropenone in acetone-d6 a t -80 O C has been described. For this system, R l d i ~ l a r / R I t= o t 0.82. a1 At first glance, this suggests that the information available from a relaxation study will be limited. However, it has been demonstrated that if relaxation-induced multispin order is monitored, the dipolar interaction serves to amplify rather than obscure weaker interactions. Using well-defined interaction constants, dipolar interferences provided information about the relevant molecular dynamics. With the basic motional features understood, additional interaction constants could be determined. It was possible to deduce the orientation of the shift tensor (the least shielded component is rotated 7 O f 2 O from the perpendicular to the olefinic linkage and lies in the molecular plane) and the sign of the two-bond C-H coupling constant (-6.3 H z a t -30 OC) and to assess the relative importance of the proton and 13C chemical shift anisotropies.
Although it did not prove possible to isolate individually A u and r), it is apparent how these methods could be implemented in conjunction with complementary techniques. It has also been argued that relaxation-induced multispin order can be used to isolate the symmetric from the antisymmetric components of the shift tensor (the present study suggests a shift antisymmetry of 80 f 40 ppm). Although nuclear spin relaxation via the anisotropic chemical shift is recognized as important, relatively few, thorough investigations of this mechanism have appeared in the literature. In most studies, a t least two, and often three, of the parameters, r), 60, or D,, - (Dxx D,,,,)/2,are assumed to be zero for no other reason than convenience. However, as the present study clearly indicates, this can be quite dangerous. It is expected that multispin relaxation will prove essential for further developments in this area.
+
Acknowledgment. We thank Madame Nicole Ratovelomanana for her aid in the synthesis of the cyclopropenone. Note Added in Proof: A recent theoretical calculation (Facelli, J., private communication) suggests that the shielding tensor of cyclopropenone contrasts markedly with other olefinic Carbons and is characterized by a large anisotropy, Au 250 ppm, and a small asymmetry, r) 0.2. The calculated values for 6u and a! are 20 ppm and 14O, respectively.
-
-
Fluorescence Anisotropy of Reversible Interacting Fiuorophores in Solutions. A Theoretical Study Z. LimpouchovQ and K. ProchPzka* Department of Physical and Macromolecular Chemistry, Charles University, Prague, Albertov 2030, 128 40 Prague 2, Czechoslovakia (Received: July 10, 1991; In Final Form: September 16, 1991)
A model for the time-resolved fluorescence anisotropy is presented for a dilute solution containing symmetric-top molecules where rotational and translational diffusion take place simultaneously with a reversible complex formation. The description of the molecular behavior is based on a rotational diffusion model and the Smoluchowskitheory of diffusioncontrolled reactions. A system of differential equations for orientational probability densities was solved both numerically for the time-dependent rate constant of the complex formation and analytically under the simplifying assumption of the time-independent rate constant of the complex formation. Parametric studies of fluorescence, difference, and anisotropy decays were performed for typical values of parameters describing reorientational motion of molecules and kinetics of complex formation and dissociation.
Introduction For n m y years, the reversible kinetics of intermolecular excited complex formation and dissociation as well as other possible reactions of an excited fluorophore has been attracting attention of a number of researches.'-* Formation of complex in a solution is a diffusion-controlled reaction between molecules which undergo simultaneous random translation and rotation. With the exception of a few recent theories of diffusion-controlled reactions do not take into account the orientational factor and consider only the influence of the translational diffusion. Their experimental verification is than based only on the time-resolved fluorescence decay measurements. Fluorescence anisotropy measurements with excited complexes have been reported recently;'*15 however, these experiments did not study kinetics of excited complex formation and dissociation in solutions. The aim of this paper is to include kinetics of excited complex formation and dissociation into a rotational diffusion model for the fluorescence anisotropy in an isotropic solution. The timeresolved polarization spectrofluorimetry may be used to test the derived expressions. Mathematical treatment is presented for *Towhom correspondence should be addressed.
0022-3654/92/2096-566$03.00/0
exciplexes, however, the results are suitable for other reversible interacting fluorophore systems, e.g., for excited complexes, which do not exhibit fluorescence or for excimers under the condition (1) Birks, J. B. Phorophysics of Aromatic Molecules; Wiley: London, 1970. (2) Weixelbaumer, W. D.; Burbaumer, J.; Kauffmann, H. F. J. Chem. Phys. 1985,83, 19980. (3) Lee, S.;Karplus, M. J . Chem. Phys. 1987, 86, 1883. (4) Andre, J. C.; Baros, F.; Winnik, M. A. J . Phys. Chem. 1990,94,2942. ( 5 ) Hauser, M.; Wagenblast, G. In Time-Resoloed Fluorescence Spectroscopy in Biochemistry and Biology; Cundall, R. B., Dale, R. E., Eds.; Plenum: New York, 1983. (6) Sienicki, K.; Winnik, M. A . J . Chem. Phys. 1987, 87, 2766. (7) Martinho, J. M. G.; Winnik, M. A. J . Phys. Chem. 1987, 91, 3640. (8) Vogelsang, J.; Hauser, M. J . Phys. Chem. 1990, 94, 7488. (9) Baldo, M.; Grassi, A.; Raudino, A. J . Chem. Phys. 1989, 91, 4658. (10) e n a b l e , T.; Cranston. D. H.; Soutar, I. Eur. Polym. J. 1989, 25,221. (1 1) Yliperttula, M.; Lemmetyinen, H.; Mikkola, J.; Virtanen, J.; Kinnunen, P. K. J. Chem. Phys. Lett. 1988, 152, 61. (12) Stegemeyer, H.; Hasse, J.; Laarhoven, W. Chem. Phys. Lett. 1988, 137, 516. (13) Fraser, I. M.; MacCallum, J. R. Eur. Polym. J . 1988, 23, 171. (14) Gardette, J.; Phillips, D. Polym. Commun. 1984, 25, 366. (15) MacCallum, J. R. Eur. Polym. J . 1981, 17, 953.
0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 561
Reversible Interacting Fluorophores in Solutions that the depolarization due to energy transfer may be neglected.
Description of the Model It is impossible to obtain an analytical solution for the timeresolved fluorescence anisotropy for the case of a general reversible exciplex under real conditions. A following simplified model system is considered: We assume that the excited fluorophore (which may be either a donor or an acceptor molecule) as well as the excited complex are symmetric-top species. The rotational diffusion of both species is described by the rotational diffusion (RD) model’623 which leads to the following equation:
a
,P(Q,t)
= -mJ(Q,t)
(1)
where P(Sl,t) is the probability that the excited molecule is oriented a t the time t in a narrow region of the Euler angles between Q a?d Q dQ (in the laboratory axes system). The Hamiltonian H has the same form as that of a symmetric rigid rotor:
+
3
fi = CDi(Li)2 i= I
where Li are components of the quantum mechanical angular momentum operator according to and Di are components of the rotational diffusion tensor. In the simplest case ((i) either the absorption or the emission transition dipole moment of the fluorophore is oriented along one of its principal exes of rotation or (ii) the fluorophore and exciplex are spherical-top) the fluorescence anisotropy, r ( t ) , may be expressed in the form
where III(t) and I l ( t ) are the time-resolved emission intensities parallel and perpendicular to the excitation polarization, respectively, S(t) is the total fluorescence decay (which is not affected by the reorientational motion), and D(t) is the difference decay. The rotational correlation time, T,,,, is expressed by a simple scalar rotational diffusion coefficient Drot:rrOt = (6DrOt)-l, and D,,, is given by D,,,= (kbT)/(6V9),kb is the Boltzmann constant, T i s temperature, Vis the volume of the molecule, and 9 is the viscosity of the medium. The following kinetic scheme is used to describe energy relaxation in the studied system: F*
-
exchation F-
1
k~
k l ( 0Cx
discussed in detail in ref 26) was used to describe the exciplex formation. In this theory a two-step process of (i) a slow diffusive transport which is followed by (ii) a rapid chemical reaction is assumed. The time-dependent rate constant of a diffusion controlled reaction is derived in the form
k,W = a + B
/4
(4)
a = 4?rDRC,
(5)
0=4GR2C,
(6) where D is the coefficient of mutual translational diffusion and R is a critical distance at which the reaction takes place. In our model we assume that (i) coplanar orientations of exciplex-forming components are needed for exciplex formation (a sandwich complex), (ii) no changes in orientations of molecules take place during formation and dissociation of the exciplex, (iii) concentration C, is constant in time and isotropic in orientation, which is fulfilled in most fluorimetric measurements. System behavior is described via orientational densities of molecules, f;.(Q,t),which are proportional to the probability that the molecule of i type is oriented at time t in a narrow region of the Euler angles between Q and Q + dQ. The densitiesf;:(Q,t) are normalized to the concentrations Ci(t): ci(t)=
Sfi(Q,r)
d~
(7)
Polarized fluorescence intensities may be expressed in the form =
dQ
(8)
I,(t) = jh(Q,r)lP..x(Q)12 dQ
(9)
Ill(t)
Sfi(QJ)lP..l(Q)12
where pe,,(Q), MJQ), P ~ . ~ ( Qare ) components of the emission transition dipole moment of the excited molecule with a given orientation Q . The following system of partial differential equations may be derived for the time dependence of orientational densities of excited fluorophore, fi(Q,t), and exciplex, f2(Q,’t):
a
Tpw= -fil(Q)fl(%t)
+ k2f2(Qrt) - kFfI(Q,t) - kl(t) cxfl(Q,t) (10)
- E* 3
IkE
f i i = EDj(L,)’ F + X
where F and F* represent the fluorophore in the ground and excited state, respectively, X represents the second component of the exciplex in the ground state and E* the excited complex, C, is the concentration of X in the ground state, k l ( t ) is the rate constant of the exciplex formation, k2 is the rate constant of the exciplex dissociation, kF is the rate constant of the fluorophore emission, and kE is the rate constant of the exciplex emission. The theory for diffusion controlled reactions which was formulated by S m o l u c h o w ~ k and i ~ ~ Collins and Kimbal125(and is (1.6) Michl, J.; Thulstrup, E. W. Spectroscopy with Polarized Light; VCH Publishers: New York, 1986. ( 1 7) Fleming, G. R. Chemical Applications of Ultrafast Spectroscopy; Oxford Universitv Press: New York. 1986. (18) Belford, G. G.; Belford, R. L.; Weber, G Proc. Natl. Acad. Sci. U -S . A. -. 1972. -, 69. - , 1392 -- (19) Cross, A. J.; Waldeck, D. H.; Fleming, G. R. J . Chem. Phys. 1983, 78, 6455; J . Chem. Phys. 1983, 79, 3173. (20) Ehrenberg, M.; Rigler, R. Chem. Phys. Lett. 1972, 14, 539. (21) Chuang, T. J.; Eisenthal, K. B. J . Chem. Phys. 1972, 57, 5094. (22) Favro, L. D. Phys. Reu. 1960, 119, 53. (23) Rose, M. E. Elementary Theory of Angular Momentum; Wiley: New York, 1957. (24) Smoluchowski, M. Z . Phys. Chem. 1917, 92, 129. (25) Collins, F. C.; Kimball, G. E. J . Colloid. Sci. 1949, 4 , 425.
i=l
where 4 are rotational diffusion coefficients of the Guorophore (i = 1 ) and exciplex (i = 2), respectively. The terms Hi(Q)f;.(Q,t) describe rotational diffusion; k2f2(Q,t)represents dissociation of the exciplex, kFfl(Q,t)and kEf2(Q,t)describe fluorescence of the fluorophore and the exciplex, respectively, and kl( t ) C.Jl(Q,t) represents the formation of the exciplex. The initial valuefi(Q,O) is proportional to the probability of the excitation of a fluorophore with orientation Q. The initial value off2(Q,0) isf2(Q,0) = 0, as no exciplex is present at the moment of excitation. For an Whitely long time, i.e. for the relaxed system, both orientational densities are zero. Boundary conditions are requirements of continuity of functionsf;.(Q,t) and their derivatives at the boundaries of the Euler angles. Equations 10 and 11 may be transformed to the Birks scheme’ for the time-independent kl or to equations derived by Weixelbaumer et al.* for the time-dependent k , ( t ) by integration over 0. In the case of an irreversible exciplex formation (k2= 0), eqs 10 and 11 reduce to a simple model of rotational diffusion.I8 Functionsf;(Q,t) may be expanded into series of symmetric,rotor eigenfunctions alk,(Q) with time-dependent coefficients, A b ( t ) : (26) Eyring, H.; Lin, S . H.; Lin, S. M. Basic Chemical Kinetics; Wiley: New York, 1980.
Limpouchovl and Prochlzka
568 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 m
f;(Q,t)
=
k=l
n=l
C C C AM0
130 k=-1 n=-1
*/kn(o)
(13)
Eigenfunctions @/k,,(?) are common for both species, but differ in their eigenvalues &,,. Equations for coefficients Aikn(t)which are obtained from expressions 10, 1 1 , and 13 are linear and homogeneous ordinary differential equations of the first order:
where
-
The time dependence of coefficients all(t)and azl(t)precludes finding an analytical solution in the time region t = 0 to t a. However, it is possible to find an analytical solution under the simplifying assumption that the rate constant k , is time-independent (see later). Fluorescence and difference decay may be expressed as follows: si(t) = & A h ( r )
(20)
where (p;',pt,pzl) are the vector components of the emission transition dipole moment of the fluorophore ( i = 1 ) and the respectively. exciplex ( i = 2 ) in the spherical Further we suppose that the difference decays Di(t) are proportional to the coefficients &(t) only. This assumption holds, when one of the two following conditions is fulfilled: (i) the absorption transition dipole moment and the symmetry axis of the fluorophore are parallel or (ii) either the emission transition dipole moments of both species are parallel to their symmetry axes, or the species are spherical top. The aforementioned assumption simplifies the mathematical treatment; however, it does not impose principal limitations on the model.
A ~ l y t i dSolution for the Time-Independent Rate Constant of Exciplex Formation For the time-independent rate constant of exciplex formation, the solution of eq 14 and 15 is
where hi = -f/z[(-a11- azz) - d(a11 - a d 2 + 4aizazil A2
= -f/z[(-aii - 0 2 2 )
+ d(a11 -~
(24)
+
2 2 ) ~4a12~211( 2 5 )
The respective pairs of coefficients Afkn(t)and A;kn(t)for given indexes lkn represent individual solutions of the set of linear
differential equations ( 1 5)-( 18), which differ for various indexes lkn only in values of the constants E!,,,, E;h and in values of &(O) and A;k,,(O). In each pair, both coefficients (except the case that both are zero) are double-exponential functions with the common coefficients XI and Xz. The coefficient &(t) is always a monotonously decreasing time function (both preexponential factors are positive) and the coefficient &,,(t) shows a maximum (the preexponential factor of the second component is negative). The time dependences of coeffcients A h ( ? )and A&(t), which are proportional to the concentrations of the fluorophore and exciplex, respectively, are identical with Birks relations' (which are derived for concentrations). A more general description of the system used in our model gives not only the time dependence of concentrations but also the timeresolved fluorescence anisotropy (and possibly also anisotropy of other physical quantities). Results and Discussion Parametric studies of fluorescence, difference, and anisotropy decays have been performed for typical values of parameters describing the reorientational motion and the exciplex kinetics: the rate constants of the fluorophore and exciplex fluorescence were chosen to be kF = 5 X lo8 s-l and kE = lo8 s-', respectively; the rate constant of the exciplex dissociation was k2 = lo7 s-l; solvent viscosity was varied from q = 1 to 50 c P the standard temperature, T = 293 K, was used. Radius of the fluorophore ranged from r = m to r = 10 X m, and the concentration of the second exciplex component in the ground state, C,, was in the range 10-2-5 X lo-' M. The ratio of the rotational correlation times of the exciplex to the excited fluorophore, a = ( T & / ( T F ) ~ ~ , was varied from a = 2 to a = 8 in our calculations. Here we present only results for a = 8; as for a = 8 the timeresolved fluorescence anisotropies are strongly affected by the reversible exciplex formation. For lower values of a, the calculated anisotropy decays are more "normal looking". (The other results are available upon request.) The value a = 8 may be a bit overestimated for complexes in which both components are a p proximately of the same size. However for strongly solvated complexes with the second component X significantly larger than the excited fluorophore the chosen value is quite reasonable. In the latter case,the assumption of the symmetric-top complex may not be strictly fulfilled, nevertheless the shape of the derived curves is still qualitatively correct. Due to the mutual dependence of the rotational and translational diffusion, the rotational correlation time of the fluorophore, ( T ~ ) ~ and ~ , , constants a and /3 were calculated for given values of viscosity, temperature, and the radius of the fluorophore using the following simple formulas:
a=-
8R,T 311
where R, is the gas constant. Comparison of Numerical and Analytical Solutions. Figure 1 shows the fluorophore and exciplex time-resolved fluorescence decays, the difference decays, and the time-resolved fluorescence anisotropies resulting from the numerical solution (the RungeKutta method) for the more general model (which assumes the time dependence of kl according to eq 4), and Figures 2 and 3 show the functions resulting from the analytical solution for the simplified model (for the same set of parameters with kl = a and k l = 4a, respectively). The results obtained from the numerical and the analytical solutions do agree qualitatively. The numerical solution gives a higher maximum intensity in the exciplex emission and a faster decrease in the fluorophore intensity for both fluorescence and
s(t)r/ The Journal of Physical Chemistry, VO~. 96, NO. 2, 1992 569
Reversible Interacting Fluorophores in Solutions 10
-.
05
.-.._ - _- _
--._
‘‘0
5
10
20
15
t,ns
t,ns
t,ns
t,ns
t,ns
Figure 1. Time-resolved fluorescence decays (numerical simulation)
based on the theoretical model with the time-dependent rate constant k,(r): (a) total fluorescence decays, S(t);(b) differencedecays, D(t), and (c) time-resolved anisotropies, r(t). The following set of parameters was used: molar concentration C, = 10-1M; rate constants kF = 5 X lo8 s-’, kE = lo8 s-I, k2 = lo’s-]; fluorophore radius, r = 6 X m; the ratio of rotation correlation times of exciplex-to-fluorophore,a = 8; solvent viscosity, 7 = 5cP temperature, T = 293 K. The solid curve is for fluorophore and the dashed curve is for exciplex.
0.41
0.0
I
--- - _ _ 5
10
15
20
t,ns
Figure 2. Time-resolved fluorescence decays (analytical solution) based on the simplified model with the time-independent rate constant kl = a; (a) total fluorescence decays, S(t);(b) difference decays, D(& and (c) time-resolved anisotropies, r(r) for the same set of parameters as in Figure 1. The solid curve is for fluorophore, the dashed curve is for exciplex, and the dotted curve shows the theoretical decays of fluorophore in the absence of the second component X.
difference decays. A better agreement is achieved if an “effective” timeindependent rate constant of exciplex formation is used which is higher than the limiting value a! (seeeq 27). For the “effective” rate constant, kl = 4a,the same maximum intensity in the exciplex emission was obtained as in Figure 1. However, the rise time in the exciplex fluorescence and the decay time in the fluorophore emission obtained by the numerical solution are still shorter than those obtained analytically. Such a behavior is the immediate and simple consequence of fact that k , ( t ) decreases fast upon excitation (the limiting value a may be of 1 order of magnitude lower than that for t 0). The differences between numerical and analytical solutions have the origin in the transient component in k,(r) and decrease with decreasing value of 8. For 8 = 0, the corresponding numerical and analytical solutions are identical.
-
Figure 3. Time-resolved fluorescencedecays (analytical solution) based on the simplified model with the time-independentrate constant kl = 4a. The other specifications are the same as in Figure 1.
Both models predict a complex time dependence of the fluorescence anisotropy for excited fluorophore, as will be demonstrated below. For practical applications of the presented models it is necessary to keep in mind that the measurements of the timeresolved fluorescence anisotropies are meaningful in the time region up to several units of the rotational correlation time. The relative difference of the timedependent rate constant k,(t) from its long-time limiting value a is 170% for t = and 54% for t = 107,. As rotational and translational diffusions are mutually dependent phenomena (both depending on temperature, viscosity, and radius of the fluorophore), it is then impossible to find experimental conditions which would allow neglect of the effect of translational diffusion on the rate constant of the exciplex formation and treatment of only the rotational diffusion of molecules. For a correct quantitative treatment of the data, the numerical solution should be used. However for a semiquantitative evaluation of the experimental data, the analytical formulas seem to be sufficient. Numerical Solution for the “General” Modd In this paragraph we analyze the influence of the size and concentration of components involved in the exciplex formation, solvent viscosity, etc., on the time behavior of measurable fluorescence characteristics of the system. As a,and 8 depend on the viscosity-totemperature ratio, v / T , the temperature dependence of the derived quantities is not discussed separately. Fluorescence and difference decays and fluorescence anisotropies of the fluorophore and the exciplex are shown in Figure 4 as functions of time and viscosity for T = 293 K. Fluorescence decays (Figure 4a) contain information on the global energy relaxation of the system only. An increase in the solvent viscosity leads to a slower translational diffusion. Thus the rise time of the exciplex emission and the decay time of the fluorophore emission become longer and the maximum intensity of exciplex fluorescence is lower in viscous solvents. Difference decays (Figure 4b) depend on both the energy and the reorientational relaxation and have essentially the same properties as the fluorescence decays. From an experimental point of view, it is interesting to mention that while the rotational correlation time increase with increasing viscosity, the rate of exciplex formation decreases. It means that in a viscous solvent, the exciplex concentration is low and its influence on the fluorophore emission anisotropy is unimportant. This hinders the tempting idea to test experimentally the derived equations in very viscous media. Fluorescence anisotropies are presented in Figure 4c. The exciplex emission anisotropy decreases monotonously with time. The fluorophore emission anisotropy has a more complex shape.
570 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 F l u o r e s c e n c e decC1v
a
Limpouchovl and Prochlzka Fluorescence decay
a
Fluoroohore
Fluorophore
Exciplex
Exciplex
a
\,’#,
, I
\
4
Difference d e c a y
b
A,,,?,\j
rr-
b
Difference d e c a y
Exciplex
Exciplex 7 QI
Fluorescence anisotropy
Fluorescence dnisotropy
C
Exciplex
hb
Figure 4. (a) Time-resolved total fluorescence decays, S(r). (b) difference decays, D(r),and (c) time-resolved anisotropies, r(r), for fluorophore and cxciplex as function o f t and C,. The other parameters are the same as in Figure I .
Figure 5. (a) Time-resolved total fluorescence decays, S(r), (b) difference decays, D W , and time-resolved anisotropies, r(t), for fluorophore and cxciplex as function of t and 7. The other parameters are the same as i n Figurc I .
J . Phys. Chem. 1992, 96, 571-576
Curves for various viscosities decrease relatively fast at short times (especially for low viscosities) and go through a local minimum. At intermediate times, they increase slowly and a small local maximum is reached. At long times, they decrease again to zero (for high viscosities the relaxation to zero was not reached on the time-scale used). Total fluorescence decays, S(t),difference decays, D(t), and the time-resolved anisotropies, r(t), are shown in Figure 5 as functions of t and C,. As may be expected, the exciplex concentration and its influence on the fluorescence properties of the system increases with increasing concentration C,. The influence on S(t) is very clear and straightforward. Kinetics of the exciplex formation and dissociation and therefore the total energy relaxation in the system depend directly on concentrations of the exciplexforming molecules. The reorientational motion of the noninteracting fluorophore does not depend on C, a t all. The fact that the time-resolved fluorescence anisotropies calculated on the basis of our model depend on concentration C, indicates that the time-resolved anisotropies of the fluorophore emission (as well as that of exciplex) monitor the relatively complicated reorientational behavior of the system which includes also an indirect effect of the kinetics of the exciplex formation and dissociation. A pronounced complex shape of r ( t ) vs t with local extremes (see Figures 4c and 5c) is obtained for exciplex-forming systems with kE < kFand ( T ~ >) ( ~T ~~ ) ~At ~ ~short ~ . times, the emission intensity of the directly excited fluorophore prevails (due in part to a higher fraction of this component and to a higher kF,e.g. to its shorter lifetime and therefore a higher fractional fluorescence intensity of this component as compared with exciplex) and the anisotropy monitors the behavior of this compnent. At long times, the concentration of the excited fluorophore is low. Emission occurs from excited fluorophores which are produced by exciplex dissociation and the time-resolved anisotropy monitors the reorientational motion of exciplex. Qualitatively, this behavior is similar to that of a single fluorophore distributed in two different microenvironments: (i) where both lifetime and rotational correlation time are shorter and (ii) In such a case, r ( t ) monitors where both times are (27) Kiserow, D.; Prochlzka, K.; Ramireddy, C.; Tuzar, Z.; Munk, P.; Webber, S. E. Fluorimetric and QELS Study of the Solubilizationof Nonpolar Low-Molar Mass Compounds into Water-Soluble Block-Copolymer Micelles. Submitted to Macromolecules.
571
the reorientational motion of the short-living component at short times and at longer times that of the long-living component. The nonmonotonous shape of r(t) in exciplex-forming systems seems to occur generally for the aforementioned choice of lifetimes and rotational correlation times. However in many systems (e.g., for a low concentration C,), extremes in r(t) occur at times when fluorescence intensities are too low for accurate measurements. As mentioned earlier, a similar approach could be applied for excimer-forming systems under the assumption that the donorto-donor energy migration and the subsequent fluorescence depolarization may be neglected. In most excimer-forming systems (especially a t higher fluorophore concentration) a fast depolarization of the emission due to energy migration cannot be neglected and it is why we propose our model mainly for exciplex-forming systems.
Conclusions In this paper we have developed a theoretical model for the time-iesolved polarization spectrofluorimetry for a system containing symmetric-top fluorescent molecules in a solution, where the rotational and translational diffusion motion of molecules take place simultaneously with a reversible exciplex formation. The description of the molecular behavior is based on a rotational diffusion model and a Smoluchowski theory for diffusion-controlled reactions. A system of differential equations for orientational probability densities was solved both numerically for the time-dependent rate constant of the exciplex formation and analytically under the simplifying assumption that the rate constant of the exciplex formation does not depend on time. Parametric studies for both the analytical and the numerical solutions show that (i) the fluorescence anisotropies monitor both the reorientational and energy relaxation and (ii) the timeresolved fluorescence anisotropies may have a complex and nonmonotonous shape. Acknowledgment. Enlightening discussions with Prof. V. Fidler are gratefully acknowledged. (28) Birch, D. J. S.; Holmes, A. S.; Imhof, R. E. Proceedings of VI. ICEET; Prague, 1989, 1, 172. (29) Ludescher, R. D.; Peting, C.; Hudson, S.; Hudson, B. Biophys. Chem. 1987, 28, 59.
Resonance Raman Deenhancement Caused by Excited-State Potential Surface Crossing Christian Reber and Jeffrey 1. Zink* Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90024 (Received: July 1 1 , 1991; In Final Form: August 30, 1991)
Resonance deenhancement, the decrease of the intensity in narrow regions of broad resonance Raman excitation profiles, occurs because of interference between two or more electronic excited states. The known examples of deenhancement have been quantitatively calculated and explained in terms of destructive interference between the real or imaginary parts of the scattering cross section for two states. These terms are added and then squared to calculate the Raman intensity. A second type of interference is discussed in this paper. The interference is most readily apparent from the time-dependent theoretical point of view where the simultaneous motion of two wavepackets on two coupled surfaces is the source of the interference. This type of interference is named population interference in this paper. The origins are examined, the new spectral signatures are delineated, and the results are compared with experimentally determined resonance deenhancement spectra. 1. Introduction
Resonance Raman excitation profiles are a powerful tool for obtaining information about the distortions that molecules undergo in excited electronic states and for providing a basis for constructing excited-state electronic potential energy surfaces. The Raman profdes are frequently more useful than absorption spectra. The latter are often congested, resulting in a broad envelope,
whereas the former "filter out" excited-state information through the speclfc normal mode being examined. Excited-state distortions of large molecules in condensed media have been ~~~
(1) Zink, J. I.; Shin, K. S. K. Molecular Distortions in Excited Electronic States Determined from Electronic and Resonance Raman Spectroscopy In Aduances in Photochemistry; Wiley: New York, 1991; Vol. 16.
Q022-3654/92/2096-511%03.QO/Q 0 1992 American Chemical Society
