Relation between anisotropy and the absorption and emission dipoles

7.2.1. Parallel absorption and emission transition moments

We will consider a population of N molecules randomly oriented and excited at time 0 by an infinitely short pulse of light polarized along Oz. At time t, the emission transition moments, , of the excited molecules have a certain angular distribution.

Here we assume that the absorption and emission transition moments are parallel.

The orientation of these transition moments is characterized by , the angle with respect to the Oz axis, and by , the angle between the projection of the transition moments in the xy plane and the Ox axis (Figure 4.26)³.

Figure 4.26. System of coordinates for characterizing the emission dipole orientation of one molecule (adapted from ref. 3).

Figure 4.26 shows IVV and IVH being proportional to the projection of the emission transition moment of one molecule onto the axes. This is true because the projection of the transition moment is the same as the projection of the electric field created by the fluorophore. Hence the projection of the field onto the z-axis is proportional to cosθ and the intensity is proportional to cos²θ. Similarly, the field along the y-axis is proportional to sinθsin and the intensity is proportional to sin²θsin²:

IVV  cos² (4.67) and IVH  sin²θsin² (4.68)

151

If we consider now randomly oriented fluorophores, the anisotropy is calculated by performing the appropriate averaged intensities based on excitation photoselection.

For excitation polarized along the z-axis, the population of excited molecules is oriented with values of  from 0 to 2 with equal probability. Hence we can eliminate the  dependence (equation 4.68) by calculating the average value of sin² with the general definition of an averaged quantity¹⁷:

s s

Therefore,

IVV  cos² (4.70) and IVH  sin²θ (4.71) If we assume we are observing a collection of fluorophores that are oriented relative to the z-axis with a probability f(θ), the measured fluorescence intensities for this oriented excited-state population with optical excitation of homogeneous solutions.

Hence the anisotropies are always less than 1.0. Complete loss of anisotropy is equivalent to θ=54.7. This does not mean that each fluorophore is oriented at 54.7,

152 highest probability of excitation (photoselection) and this probability is proportional to cos². For the random ground state distribution, which must exist in a disordered solution, the number of molecules at an angle between θ and θ + dθ is proportional to sin θ dθ. This quantity is proportional to the surface area on a sphere within the angles θ and θ + dθ.

Hence, the probability distribution of molecules excited by vertically polarized light is given by:

f()d = cos²sind (4.76) This probability determines the maximum photoselection that can be obtained using one-photon excitation (more highly oriented populations can be obtained using multiphoton excitation). The emission anisotropy can thus be written as

s θ

r0 is called the fundamental anisotropy, i.e. the theoretical anisotropy that is observed when the absorption and emission dipoles are collinear, and when there are no processes which result in depolarization (in practice, rotational motions can be hindered in a rigid medium). Under these conditions, the excited-state population is preferentially oriented along the z-axis and the value of IVH is one-third the value of IVV

153

(IVV = 3 IVH). This value of 0.4 is considerably smaller than that possible for a single fluorophore oriented along the z-axis (r = 1.0).

The experimental value, called the limiting anisotropy, is always slightly smaller than the theoretical value (normally it ranges from 0.32 to 0.39). This difference is mainly due to torsional vibrations of the fluorophores about their equilibrium orientation⁵⁰. Consequently, a temperature dependence of the limiting anisotropy is noted^51,52: 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.

It is important to remember that there are other possible origins for polarized light.

These include reflections and light scattered by the sample that can interfere with anisotropy measurements: in the case of one-photon excitation, if the measured anisotropy for a randomly-oriented sample is greater than 0.4, we can infer the presence of scattered light in addition to fluorescence (for multiphoton excitation, anisotropy values can exceed 0.4)¹⁷.

7.2.2. Non-parallel absorption and emission transition moments

The situation of non-parallel absorption and emission transition moments occurs when excitation brings the fluorophores to an excited state other than the first singlet state from which fluorescence is emitted. In this case, the absorption and emission transition moments are displaced by an angle β relative to each other. We saw previously that displacement of the emission dipole by an angle θ from the z-axis resulted in a decrease in the anisotropy by a factor (3cos²θ – 1)/2. Similarly, the displacement of the absorption and emission dipoles by an angle β results in a further loss of anisotropy. The observed anisotropy in a vitrified dilute solution is a product of the loss of anisotropy due to photoselection (2/5), and that due to the angular displacement of the dipoles. The fundamental anisotropy of a fluorophore is then given by

s β

where β is the angle between the absorption and emission transition moments³. The fundamental anisotropy value is zero when β = 54.7. When β exceeds 54.7 the anisotropy becomes negative. The maximum negative value (-0.20) is found for

154

β=90. In other words, for any fluorophore randomly distributed in solution, with one-photon excitation, the value of r0 must be within the range from -0.20 to 0.40. The values for both r0 and p0 are summarized in Table 4.5.

Table 4.5. Relationship between the angular displacement of transition moments (β) and the fundamental anisotropy (r0) or polarization (p0)¹⁷.

 (deg) r0 p0 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).

The measurement of the fundamental anisotropy requires special conditions. In order to avoid rotational diffusion, the probes are usually examined in solvents that form a clear glass at low temperature, such as propylene glycol or glycerol. Additionally, the solutions must be optically dilute to avoid depolarization due to radiative reabsorption and emission, or due to resonance energy transfer. One commonly used solvent for measuring fundamental anisotropies is propylene glycol at -60 to -70 C. Under these conditions, the fluorophores remain immobile during the lifetime of the excited state.

Glycerol also forms a rigid glass at low temperature. However, glycerol typically displays more autofluorescence than propylene glycol. At similar temperatures, phosphorescence from the fluorophores seems to be more common in glycerol than in propylene glycol. The anisotropy values (r0) determined in these rigid solutions provide a measure of the angle between the absorption and emission dipoles¹⁷.