Principles of Fluorescence Spectroscopy

266 DYNAMICS OF SOLVENT AND SPECTRAL RELAXATION
Figure 7.48. Relationship of the phase angles of the F and R states.
A further interesting aspect of the phase difference or
the demodulation between F and R is the potential of measuring
the reverse reaction rate k 2 . This reverse rate can be
obtained from the phase-angle difference between the R and
F states, or from the modulation of the R state relative to the
F state. For a reversible reaction these expressions are
tan (φ R φ F)
m R
m F
ω
(γ R k 2) ωτ R
γR k2
√(γR k2) 2 1
2 ω √1 ω2τ2 R
(7.39)
(7.40)
These expressions are similar to the usual expressions for
the dependence of phase shift and demodulation on the
decay rates of an excited state, except that the decay rate is
that of the reacted species (γ R + k 2 ). The initially excited
state populates the relaxed state and the reverse reaction
repopulates the F state. Nonetheless, the kinetic constants
of the F state do not affect the measurement of tan (φ R – φ F )
or m R /m F . Measurement of either the phase or modulation
of the reacted state, relative to the unrelaxed state, yields a
lifetime of the reacted state. This lifetime is decreased by
the reverse reaction rate in a manner analogous to the
decrease in the lifetime of the F state by k 1 . If the decay rate
(γ R ) is known, k 2 may be calculated. If the emission results
from a single species that displays one lifetime, and φ and
m are constant across the emission, then φ R – φ F = 0 and
m R /m F = 1.
Returning to the irreversible model, we note an interesting
feature of φ R (eq. 7.35). This phase angle can exceed
90E. Specifically, if ω 2 exceeds γ R (γ F + k 1 ), then tan φ R < 0
or φ R > 90E. In contrast, the phase angle of directly excited
species, or the phase angles resulting from a heterogeneous
population of fluorophores, cannot exceed 90E. Therefore,
observation of a phase angle in excess of 90E constitutes
proof of an excited-state reaction.
7.14.1. Effect of an Excited-State Reaction on the
Apparent Phase and Modulation Lifetimes
The multiplicative property of the demodulation factors
and the additive property of the individual phase angles are
the origin of a reversed frequency dependence of the apparent
phase shift and demodulation lifetimes, and the inversion
of apparent phase and modulation lifetimes when compared
to a heterogeneous sample. The apparent phase lifetime
(τ φ) R calculated from the measured phase (φR ) of the
relaxed state is
tan φ R ωτ φ R tan (φ F φ 0R)
Recalling the law for the tangent of a sum one obtains
τ φ R τ F τ 0R
1 ω 2 τ Fτ 0R
(7.41)
(7.42)
Because of the term ω 2 τ F τ 0R , an increase in the modulation
frequency can result in an increase in the apparent phase
lifetime. This result is opposite to that found for a heterogeneous
emitting population where the individual species are
excited directly. For a heterogeneous sample an increase in
modulation frequency yields a decrease in the apparent
phase lifetime. 145 Therefore, the frequency dependence of
the apparent phase lifetimes can be used to differentiate a
heterogeneous sample from a sample that undergoes an
excited-state reactions. Similarly, the apparent modulation
lifetime is given by
τm R ( 1
m2 1 )
R
1/2
Recalling eq. 7.36, one obtains
τ m R (τ 2 F τ 2 0R ω 2 τ 2 Fτ 2 0R) 1/2
(7.43)
(7.44)
Again, increasing ω yields an increased apparent modulation
lifetime. This frequency dependence is also opposite to
that expected from a heterogeneous sample, and is useful in
proving that emission results from an excited-state process.
In practice, however, the dependence of τ R m upon modulation
frequency is less dramatic than that of τ R φ. We again
stress that the calculated lifetimes are apparent values and
not true lifetimes.
The information derived from phase-modulation fluorometry
is best presented in terms of the observed quantities