Biomedical Engineering Reference
In-Depth Information
quenching constant
K
(
x
) is initially constant but becomes smaller close to a
“critical” molar ratio
x
c
(see Fig.
4.20
c) [
63
,
64
], while the
x
c
value increases
systematically with the diameter of the QD. This can be explained by the fact that
“QD-H
2
P” nanoassemblies are formed in a dynamic equilibrium of attachment and
detachment of porphyrin molecules to the QD surface. It follows from Fig.
4.23
that
the time to reach the initial equilibrium is faster than our experimental observation
time of about 60 s. This implies that there have to be some free or very easily accessi-
ble sites [
28
-
34
] for the dye attachment on the QD surface, since ligand replacement
occurs typically on time scales of minutes or even longer. From the results presented
in Fig.
4.20
c we conclude that assembly formation can be tentatively divided into a
low-molar-ratio and high-molar-ratio range which is separated by a “critical” molar
ratio
x
c.
. Above the critical molar ratio
x
c,
the probability of attaching dye molecules
to the QD surfaces is considerably reduced. In our generalized approach [
63
,
64
],
the Stern-Volmer quenching constant
K
depends on
x
if, e.g. dye attachment is
limited by the number of accessible sites
n
max
on a QD surface. This number can
also vary according to the particular conditions, e.g. if a re-arrangement of the ligand
shell or the surface topography [
48
] has to happen in order to allow for further dye
attachment. Now let us assume that, at low
x
, the QDs have free or easily accessible
attachment sites available for assembly formation with dye molecules. At these
low concentrations, quenching relates to “quasi stable assemblies,” and it is very
similar to doping-dependent transfer processes in molecular crystals [
152
]. In that
low molar ratio range,
K
x
<
x
c
is independent of
x
and can be described (according to
Eq.
4.11
)by
K
x
<
x
c
=
K
c
r
k
q
·
τ
0
.
However, both the absolute value of
K
and the critical molar ratio
x
c
depend
on the size of QDs. For the low-molar-ratio range
x
=
<
x
c
, the dependence of the
K
c
r
on the size is in agreement with the results of the quantum-
mechanical considerations (see Fig.
4.21
c) [
63
]. With respect to
x
c
value, it was
shown [
64
] that the QD surface is far from being totally covered with dye molecules,
but this value represents a landmark where the major processes responsible for
assembly formation change. Based on the generalized Stern-Volmer formalism and
the results of Fig.
4.20
c, these
x
c
values (which should be related to the maximum
number of freely accessible binding sites
n
max
) as well as the modified equilibrium
constants
K
x
>
x
c
=
magnitude of
K
x
<
x
c
=
K
c
for the high-molar-ratio range have been evaluated for a
spherical approximation for QDs of different sizes [
64
] and are presented here in
Fig.
4.24
as a function of the QD surface area.
The linear behavior of
x
c
as a function of the QD surface area for the same
absolute QD concentration (presented in Fig.
4.25
a) reveals that the number of
freely accessible sites is proportional to the surface area of the respective QD. From
the slope of the linear relation we obtain that (normalized by the QD surface unit)
n
0
≤
0.006 H
2
P/nm
2
can freely be attached to the CdSe QD surface. Hence,
the average number of attached molecules
n
max
corresponds via
x
c
to the QD size.
According to this linear relationship,
n
max
is between 0.8 (for the smallest species:
uncapped CdSe QDs with core diameter 3.0 nm) and 5.3 (for the largest CdSe/ZnS
QDs with core diameter 5.2 nm) H
2
P molecules per QD. This is only an upper limit
and the real number of attached dye molecules
n
maybebyfarsmallerthan
n
max
as
0.032
±
