Molecular Fluorescence

5.5 Case of motionless molecules with random orientation 135

where K is a proportionality factor. The bars characterize an ensemble average over the N emitting molecules at time t.

Because of the axial symmetry around Oz, a2ðtÞ ¼ b2ðtÞ, and because a2ðtÞ þ

Finally, because g ¼ cos yE, the relation between the emission anisotropy and the angular distribution of the emission transition moments can be written as

5.5

Case of motionless molecules with random orientation

5.5.1

Parallel absorption and emission transition moments

When the absorption and emission transition moments are parallel, yA ¼ yE, the

common value being denoted y; hence cos2 yA ¼ cos2 yE ¼ cos2 y. Before excitation, the number of molecules whose transition moment is oriented within angles y and y þ dy, and f and f þ df is proportional to an elementary surface on a sphere whose radius is unity, i.e. 2p sin y dy df (Figure 5.6).

Taking into account the excitation probability, i.e. cos2 y, the number of excited molecules whose transition moment is oriented within angles y and y þ dy, and f and f þ df, is proportional to cos2 y sin y dy df. The fraction of molecules oriented in this direction is

The denominator, which is proportional to the total number of excited molecules, can be calculated by setting x ¼ cos y, hence dx ¼ sin y dy, and its value is 4p=3. Equation (5.21) then becomes

136 5 Fluorescence polarization. Emission anisotropy

Fig. 5.6. The fraction of molecules whose absorption and emission transition moments are parallel and oriented in a direction within the elementary solid angle. This direction is defined by angles y and f.

It is then possible to calculate the average of cos2 y over all excited molecules

ð2p ðp

cos2 y ¼ df cos2 yWðy; fÞ dy

00

¼3 ð2p df ðp cos4 y sin y dy

4p 0 0

r0 is called the fundamental anisotropy, i.e. the theoretical anisotropy in the absence of any motion. In practice, rotational motions can be hindered in a rigid medium. The experimental value, called the limiting anisotropy, turns out to be always slightly smaller than the theoretical value. When the absorption and emission transition moments are parallel, i.e. when the molecules are excited to the first singlet state, the theoretical value of r0 is 0.46), but the experimental value usually ranges from 0.32 to 0.39. The reasons for these di erences are discussed in Box 5.1.

6)The corresponding value of the polarization ratio is p0 ¼ 0:5.

5.5 Case of motionless molecules with random orientation 137

Box 5.1 Fundamental and limiting anisotropies

The di erence between the theoretical value of the emission anisotropy in the absence of motions ( fundamental anisotropy) and the experimental value (limiting anisotropy) deserves particular attention. The limiting anisotropy can be determined either by steady-state measurements in a rigid medium (in order to avoid the e ects of Brownian motion), or time-resolved measurements by taking the value of the emission anisotropy at time zero, because the instantaneous anisotropy can be written in the following form:

rðtÞ ¼ r0 f ðtÞ

where f ðtÞ characterizes the dynamics of rotational motions (orientation autocorrelation function) (see Section 5.6, Eq. 5.32) and whose value is 1 at time zero.

It should first be noted that the measurement of emission anisotropy is di - cult, and instrumental artefacts such as large cone angles of the incident and/or observation beams, imperfect or misaligned polarizers, re-absorption of fluorescence, optical rotation, birefringence, etc., might be partly responsible for the di erence between the fundamental and limiting anisotropies.

If there is no significant instrumental artefact, any observed di erence must be related to the fluorophore itself. There may be several causes. Firstly, a trivial e ect can be the overlap of low-lying absorption bands, resulting in an anomalous excitation polarization spectrum. Secondly, the oscillator associated with absorption may not be strictly linear, but may be partly two-dimensional or three-dimensional. However, it is generally considered that the di erence between the fundamental and limiting anisotropies is mainly due to torsional vibrations of the fluorophores about their equilibrium orientation, as first suggested by Jablonskia). A temperature dependence of the limiting anisotropy is then expected, and was indeed reportedb,c): at low temperature (i.e. when the medium is frozen), r0 may be considered as a constant quantity whereas at high temperature it decreases linearly with the temperature.

Fast librational motions of the fluorophore within the solvation shell should also be consideredd). The estimated characteristic time for perylene in para n is about 1 ps, which is not detectable by time-resolved anisotropy decay measurement. An ‘apparent’ value of the emission anisotropy is thus measured, which is smaller than in the absence of libration. Such an explanation is consistent with the fact that fluorescein bound to a large molecule (e.g. polyacrylamide or monoglucoronide) exhibits a larger limiting anisotropy than free fluorescein in aqueous glycerolic solutions. However, the absorption and fluorescence spectra are di erent for free and bound fluorescein; the question then arises as to whether r0 could be an intrinsic property of the fluorophore.

In this respect, Johanssone) reported that r0 is the same (within experimental accuracy) for fluorophores belonging to the same family, e.g. perylene and perylenyl compounds ð0:369 G0:002Þ, or xanthene derivatives such as rhodamine

5.5 Case of motionless molecules with random orientation 139

Fig. 5.7. Definition of angles a and c when the absorption and emission transition moments are not parallel.

Consequently, the theoretical values of r0 range from 2=5 ð¼ 0:4Þ for a ¼ 0 (parallel transition moments) and 1=5 ð¼ 0:2Þ for a ¼ 90 (perpendicular transition moments)7):

A value close to 0:2 is indeed observed in the case of some aromatic molecules excited to the second singlet state whose transition moment is perpendicular to that of the first singlet state from which fluorescence is emitted (e.g. perylene). r0 varies with the excitation wavelength for a given observation wavelength and these variations represent the excitation polarization spectrum, which allows us to distinguish between electronic transitions. Negative values generally correspond to S0 ! S2 transitions. As an illustration, Figure 5.8 shows the excitation polarization spectrum of perylene.

The case of indole and tryptophan is peculiar because the low-lying absorption bands overlap. Box 5.2 shows how the indole absorption spectrum can be resolved into two bands from the combined measurement of the excitation spectrum and the excitation polarization spectrum.

Particular cases should also be mentioned:

.For planar aromatic molecules with a symmetry axis of order 3 or higher (e.g. triphenylene: D ), r cannot exceed 0.1.

.The fluorescence emitted by fullerene ðC60Þ is intrinsically totally depolarized because of its almost perfect spherical symmetry ðIhÞ.

7)For the polarization ratio: 1=3 ap0 a1=23h 0

140 5 Fluorescence polarization. Emission anisotropy

Fig. 5.8. Excitation polarization spectrum of perylene in propane-1,2-diol at 60 C.

5.6

E ect of rotational Brownian motion

If excited molecules can rotate during the excited-state lifetime, the emitted fluorescence is partially (or totally) depolarized (Figure 5.9). The preferred orientation of emitting molecules resulting from photoselection at time zero is indeed gradually a ected as a function of time by the rotational Brownian motions. From the extent of fluorescence depolarization, we can obtain information on the molecular motions, which depend on the size and the shape of molecules, and on the fluidity of their microenvironment.

Quantitative information can be obtained only if the time-scale of rotational motions is of the order of the excited-state lifetime t. In fact, if the motions are slow with respect to tðr Ar0Þ or rapid ðr A0Þ, no information on motions can be obtained from emission anisotropy measurements because these motions occur out of the experimental time window.

A distinction should be made between free rotation and hindered rotation. In the case of free rotation, after a d-pulse excitation the emission anisotropy decays from r0 to 0 because the rotational motions of the molecules lead to a random orientation at long times. In the case of hindered rotations, the molecules cannot become randomly oriented at long times, and the emission anisotropy does not decay to zero but to a steady value, r (Figure 5.10). These two cases of free and hindered rotations will now be discussed.

142 5 Fluorescence polarization. Emission anisotropy

where r0a and r0b are the limiting anisotropies corresponding to the 1La and 1Lb states when excited independently. In the long-wavelength region of the spectrum (305–310 nm), the 1La level is exclusively excited and the limiting anisotropy for this state is thus r0a ¼ 0:3 (instead of 0.4, for the reasons given in Box 5.1). Then, using Eq. (5.28), in which 0.4 ð2=5Þ is replaced by 0.3 and a is equal to 90 , we obtain

r0b ¼ 0:3 3 cos2 a 1 ¼ 0:15 2

The fractional contributions of the 1La and 1Lb excited states to the emission anisotropies are given by

Finally, the contributions IaðlÞ and IbðlÞ of the 1La and 1Lb bands to the excitation spectrum IðlÞ are

Fig. B5.2.2. Resolution of the excitation spectrum of indole (reproduced with permission from Valeur and Webera)).

5.6 Effect of rotational Brownian motion 143

IaðlÞ ¼ faðlÞIðlÞ

IbðlÞ ¼ fbðlÞIðlÞ

Figure B5.2.2 shows the resolution of the excitation spectrum. The 1Lb band lies below the broader 1La band and exhibits a vibrational structure.

a)Valeur B. and Weber G. (1977) Photochem. Photobiol. 25, 441.

5.6.1

Free rotations

The Brownian rotation of the emission transition moment is characterized by the angle oðtÞ through which the molecule rotates between time zero (d-pulse excitation) and time t, as shown in Figure 5.11.

Using the same method that led to Eq. (5.27), it is easy to establish the rule of multiplication of depolarization factors: when several processes inducing succes-

sive rotations of the transition moments (each being characterized by cos2 ziÞ are independent random relative azimuths, the emission anisotropy is the product of the depolarization factors ð3 cos2 zi 1Þ=2:

Fig. 5.9. Rotational motions inducing depolarization of fluorescence. The absorption and emission transition moments are assumed to be parallel.