Principles of Fluorescence Spectroscopy

192 FREQUENCY-DOMAIN LIFETIME MEASUREMENTS
apparent lifetimes depend on the method of measurement,
and are not true molecular parameters.
The relationship of τ N app and τ m app are most easily seen
by consideration of a double exponential decay. Using eqs.
5.7 and 5.8 one obtains
Nω(α1τ1 α2τ2) α1ωτ 2 1
1 ω 2 τ 2
1
α2ωτ 2 2
1 ω 2 τ 2 2
Dω(α1τ1 α2τ2)
α1τ1 1 ω 2 τ 2
1
α2τ2 1 ω 2 τ 2 2
(5.36)
(5.37)
Using eqs. 5.9 and 5.33, the apparent phase lifetime is given
by
τ app
φ ∑ iα iτ 2 i /(1 ω 2 τ 2 i )
∑ iα iτ i/(1 ω 2 τ 2 i )
Recall that the average lifetime is given by
τ ∑iαiτ2 i
∑i f
∑
iτi
jαjτj α1τ2 1 α2τ2 2
α1τ1 α2τ2 (5.38)
(5.39)
Comparison of eqs. 5.38 and 5.39 shows that in τ N app each
decay time is weighted by a factor α i τ i /(1 + ω 2 τ i 2) rather
than a factor α i τ i = f i . For this reason the components with
shorter decay times are weighted more strongly in τ N app than
in τ. Increasing the modulation frequency increases the relative
contribution of the short-lived component and hence
decreases the value of τ N app. Using similar reasoning but
more complex equations, 41 one can demonstrate that the
apparent modulation lifetime is longer than the average lifetime.
An example of the use of apparent phase and modulation
lifetimes is given in Figure 5.48, for the mixture of
ACF and AFA. This figure shows the phase-angle and modulation
spectra in terms of τ N app and τ m app. The fact that τ N app
< τ m app for a heterogeneous decay is evident by comparison
of the upper and lower panels. Also, one immediately
notices that the apparent lifetime by phase or modulation
depends on modulation frequency, and that higher frequencies
result in shorter apparent lifetimes. Hence, the apparent
lifetimes depend on the method of measurement (phase
or modulation) and on the frequency, and it is difficult
to interpret these values in terms of molecular features of
the sample.
Figure 5.48. Apparent phase (top) and modulation (bottom) lifetimes
for a mixture of ACF and AFA. Revised and reprinted with permission
from [135], Copyright © 1990, American Chemical Society.
It is always possible to interpret the phase and modulation
values in terms of the apparent lifetimes. However, the
use of apparent phase and modulation lifetimes is no longer
recommended. These are only apparent values that are the
result of a complex weighting of the individual decay times
and amplitudes, which depend on the experimental conditions.
Also, one does not actually measure apparent lifetimes.
These values are interpretations of the measurable
quantities, which are the phase and modulation values.
Supplemental Material
5.11. DERIVATION OF THE EQUATIONS FOR
PHASE-MODULATION FLUORESCENCE
5.11.1. Relationship of the Lifetime to the
Phase Angle and Modulation
The equations relating the phase and modulation values to
the apparent lifetimes (eqs. 5.3 and 5.4) are widely known,