Chapter 4 Förster resonance energy transfer
4.2 Theoretical model
antibonding modes from the coupling (―hybridization‖) of the surface and cavity modes,
as well as the nonlocal optical response of the metallic shell have also been established [40]. However, while the spherical nanoshells admit high tunability in their bonding and antibonding modes via the adjustment of the size and thickness of the shell, the recently-fabricated (spheroidal) nanorice will be in addition capable of providing much greater enhanced local fields for the FRET process. Furthermore, it may have even higher tunability due to the greater number of geometric parameters (e.g. the aspect ratio) available for adjustment. Hence it will be of interest to calculate the FRET rate at such a nanorice to provide guidance for possible future experimental observations.
4.2 Theoretical model
In our present formulation, we shall model the D and A as harmonic point dipoles interacting directly with each other, as well as via the electromagnetic interaction mediated by a spheroidal metallic nanorice (Figure 4.1).
Figure 4.1 The geometry for the donor (D) and acceptor (A) near the nanorice, with A at various positions oriented normal to the spheroidal surface.
To calculate the FRET rate
K between the D and the A, we follow the original theory [30-31, 38] which applied the Fermi golden rule and obtain the following expression:
9 4 ( )4 ( ) 2
( ) | ( ) |
8
A D
K c U , (4.1)
where A
is the absorption cross section of the acceptor, D
the emission rate per unit frequency of the donor, and U
is the interaction energy between D and A in the presence of the nanorice. In principle, both the absorption and emission of the molecules will be modified by the presence of the nanorice. However, as discussed previously [40], these modifications are of higher order effects. Hence, to first order4.2 Theoretical model 47
perturbation, the presence of the nanorice will only modify U( ) which we shall study in the following adopting the long wavelength (electrostatic) approximation.
FRET on a spheroidal solid particle
Before we consider the nanorice case, let us first review briefly the problem of energy transfer between the D and A near a solid spheroidal particle [38]. Working in spheroidal coordinates
, ,
, the electrostatic potential both inside and outside the particle can beobtained by solving the Laplace equation 2 0 [38]. For a position outside the
is the dipolar potential associated with the isolated donor (D) or acceptor (A), and
, is the potential due to the induced charge (on the particle) caused by the donor or acceptor, where P and nm Q are the associated Legendre functions, and f is the foci of the prolate nm spheroid. Here we shall use the symbol tilde (~) for quantities in the case of solid particle to
distinguish from those for the nanorice. The quantity nm in Eq. (4.4) can be obtained as
which is a dimensionless factor determined by the geometry and the dielectric property of the particle () and that of the host medium (h). The various coefficients in Eq. (4.5) are where
' denotes the derivative with respect to the argument as indicated.FRET on a spheroidal nanoshell
Next we consider the energy transfer process between the D and A in the vicinity of a nanorice modeled as a confocal prolate spheroidal nanoshell (Figure 4.1), where the two interfaces are specified by 1 and 2 in prolate spheroidal coordinates, with the same foci for these interfaces [8]. It is clear that a direct approach to solving the boundary value problem for this case will be very involved. However, in chapter 2 or Ref. [9], we have discovered that an application of a previously-formulated effective medium model [14] can greatly simplify the mathematical description of the dielectric response of a
4.2 Theoretical model 49
spheroidal nanoshell, and it is this approach we shall apply in our following study of the FRET process at a nanorice.
The main idea of this effective medium model is to replace the multilayer shell structure of different media by a homogeneous solid particle (i.e. a solid spheroid) with an
effective dielectric function accounting for both the geometric and dielectric characteristics of the given shell. By requiring this ―effective particle‖ with the shape of the external
surface (dimension 2) to produce the same multipolar polarizability of the original spheroidal shell, in an environment when the outside medium is filled with the material of the shell (dielectric function2) [9, 14], the effective dielectric function can be obtained as:
Using this result, the potential of the D-A system outside the spheroidal shell (Figure 4.1) can simply be obtained from the results in Eqs. (4.2)-(4.6) by replacing the dielectric
and [9] where we have made explicit use of the result in Eq. (4.7).
The electric field E at the acceptor position can then be obtained from the A
Hence the interaction energy for the acceptor in the presence of both the donor and the nanorice is obtained as: To study the energy transfer between the donor and acceptor we only have to consider the first two terms in Eq. (4.12), i.e. UAD UAD ind, with the latter given as in Eq. (4.14). The
4.3 Numerical results 51
last term in (4.12), UAA ind, , is associated with the energy transfer between the acceptor and
the nanorice which leads to modifications on the decay rates of the acceptor molecule [38].
From this and the result in Eq. (4.1), we can now introduce the enhancement factor for the
FRET process at a nanorice defined as:
2 2
which measures the efficiency of the energy transfer between the D and A in the vicinity of the nanorice.
4.3 Numerical results
For all computations in this section, the nanorice is made by a dielectric core (hermatite,
1 ) covered by a silver shell and embedded in the vacuum [8]. The semi-major and
semi-minor axes of the outer surface are fixed at 26.8 nm and 17.9 nm, respectively, and
the dielectric function of silver is described by the following Drude model [8]:
enhancement spectrum into the following three aspects:
a. Effects from the nanoshell geometry
Here we fix the outer shell geometry along with the positions and orientations of the two molecules D and A (see caption of Figure 4.2), but with the inner aspect ratio varied. As this ratio increases, the thickness of the silver shell decreases and one would expect cross-coupling of the plasmons at the two interfaces will become more significant. Indeed, from the results of Figure 4.2, one can clearly draw the following conclusions: (i) as in the spherical case [40], all the shell structures (in contrast to the solid spheroid case) yield resonances split into the low frequency bonding and high frequency antibonding modes, with the splitting more pronounced as the shell gets thinner; (ii) enhancements from the bonding modes are in general greater (up to an order of magnitude) than the corresponding ones from the antibonding models, due to the relative dominance of the outer surface modes over the cavity modes; and (iii) multipolar resonances emerge and can be resolved mostly for the bonding modes except for the thinnest case (aspect ratio 0.6) where they can also be seen among the antibonding modes. All these results are in qualitative consistence with those observed previously for a spherical nanoshell [40].
4.3 Numerical results 53
Figure 4.2 Comparison of the enhancement factor R( ) for different geometries of the nanorice. The foci and the outer (surface) aspect ratio are fixed at 20 nm and 2/3, respectively. The inner aspect ratios are set for three different values, which are 0.2, 0.4, and 0.6. The nanorice has a hematite core (1 ) with the silver shell embedded in the vacuum. The donor and the acceptor are located at the two ―poles‖ at(0, 0, 30nm) and
(0, 0, 30nm) in Cartesian coordinates and aligned along thezdirection.
b. Effects from the orientations of the molecules
In this case, we have the donor fixed at the north pole as in Figure 4.2, but the acceptor moved to the equatorial position at which two orthogonal orientations (i.e., the normal ˆn and tangential ˆz orientations) of it are considered. From the enhancement spectrum in Figure 4.3, we observe that while the multipolar feature in the solid particle case differs appreciably for the two molecular orientations; that for the two split resonances in the nanorice case is less pronounce with both the ˆn and ˆz oriented acceptor yielding rather
similar resonance structure. Since here we consider a rather thin shell (with aspect ratio fixed at 0.6), we observe the multipolar resonance for both the bonding and antibonding modes as revealed in Figure 4.2. The most interesting observation from Figure 4.3 is the general greater enhancement obtained in the case with the D and A in a relatively perpendicular orientation (i.e. A along ˆn in this case). This is generally valid for both the solid and shell particles, except for frequencies close to the plasmon resonances, and can be understood from the electrostatic interaction between two arbitrarily oriented dipoles.
1 2 3 4 5 6 7 configurations of the nanorice (at a fixed inner aspect ratio of 0.6) and the donor are as in Figure 4.2, but the acceptor is moved to the equatorial position at (22.36nm, 0, 0) in Cartesian coordinates. Two different orientations are considered with each along the symmetry axis ( ˆz ) and perpendicular to the surface (i.e.
along ˆn ), respectively. The solid lines indicate the results for the nanorice, where we also show the corresponding results (dashed lines) for a solid spheroid for comparison.
4.3 Numerical results 55
c. Effects from the relative positions of the molecules
Finally we study the enhancement as a function of the relative position between the D and A. Here we have D fixed at the north pole and the position of A varied along a spheroidal surface with =1.5. The orientations of both molecules are all the time in the normal
direction. Positions of A at various angles from the positive z axis are studied as indicated in Figure 4.4, from which one observed that the values of the two resonance frequencies remain roughly unchanged for all the angular positions of A since they are determined largely by the geometry and materials of the nanoshell which remain constant throughout. However, it is worthwhile to note that the enhancement ratio increases as A and D get farther from each other due to the decrease in the strength in the direct transfer between them.
1 2 3 4 5 6 7 8 molecules. The configurations of the nanorice (inner aspect ratio 0.4) and the donor are as in Figure 4.2, but the acceptor is varied in position along a spheroidal surface outside the nanorice. The orientations of both the donor and the acceptor are always perpendicular to the surface and the trajectory of the acceptor is fixed on a spheoridal surface with 1.5.