Principles of Fluorescence Spectroscopy

PRINCIPLES OF FLUORESCENCE SPECTROSCOPY 511
known value of 26.1 Å, then χ R 2 = 1152 (Table 15.1). This
highly elevated χ R 2 value indicates the significant influence
of diffusion in this system. When the correct model is used
(eq. 15.8) the variable parameters in the analysis are R 0 and
the mutual diffusion coefficient D. Least-squares analysis
yielded R 0 = 27.8 Å and D = 2.64 x 10 –5 cm 2 /s (Table 15.1),
which are reasonably close to the expected values in this
solvent.
If the acceptor concentration is known the time-resolved
data can be used to determine the Förster distance.
In propylene glycol the recovered value of 24.9 Å is in good
agreement with the calculated value of 24.3 Å. For the D–A
pair in methanol it is interesting to consider the value of R 0
obtained if diffusion is neglected, which is accomplished by
setting D = 0 during the least-squares analysis. In this case
the apparent value of R 0 is larger, 37 Å. This result provides
a useful hint. If the time-resolved data yield a larger than
expected R 0 value, the cause could be diffusion during the
donor excited-state lifetime. It should be noted that we have
assumed that the distance of closest approach (r C ) is zero. If
r C is a significant fraction of R 0 , then eqs. 15.1–15.8 are not
appropriate.
The effect of diffusion on the donor intensity decay can
be seen in the time-domain data. The impulse response
functions for the indole–dansylamide mixtures are shown in
Figure 15.5. It is visually evident that the dansyl acceptor
results in a non-exponential indole decay in propylene glycol.
In methanol the decay is still heterogeneous, but is
closer to a single exponential. This trend towards a singleexponential
donor decay in the presence of diffusion is well
known in the literature, 14 and the donor decay becomes single
exponential in the rapid diffusion limit (Section 15.6).
Figure 15.5. Reconstructed time-domain intensity decays of indole
without acceptor (D) with 5 mM dansylamide in methanol (dashed)
and with 12 mM dansylamide in propylene glycol (solid).
15.2. EFFECT OF DIMENSIONALITY ON RET
In the previous section we saw that a random distribution of
an acceptor molecules around a donor resulted in a characteristic
donor decay. For randomly distributed acceptors the
form of the donor decay depends on the dimensionality of
the acceptor distribution. Different donor decays are
expected for acceptor distributions in a volume, in a plane,
or along a line. Planar distributions are expected for donors
and acceptors in membranes, and linear distribution are
expected for dyes intercalated into double helical DNA
(Figure 15.1). Assuming no diffusion, no excluded volume,
and a random distribution of donors and acceptors in two
dimensions or one dimension, the intensity decays are
known in analytical form. For a random two-dimensional
distribution, 15–16
where
and
(15.9)
(15.10)
(15.11)
Hence C/C 0 is the number of acceptor molecules in an area
equal to πR 0 2, that of a circle with diameter R 0 .
For a random one-dimensional distribution of acceptors
the donor intensity decay is given by
where
and
IDA(t) I0 D exp [ t
2β ( τD t ) τD 1/3
]
β Γ(2/3)
2
δ Γ(5/6)
2
C
, Γ(2/3) 1.354177 ...
C 0
IDA(t) I0 D exp [ t
2δ ( τD t ) τD 1/6
]
C
, Γ(5/6) 1.128787 ...
C 0
C 0 (πR 2 0) 1
C 0 1
2R 0
(15.12)
(15.13)
(15.14)