J. Phys. Chem. C 2008, 112, 6299-6306
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Excitation Energy Transfer between Non-Spherical Metal Nanoparticles: Effects of Shape and Orientation on Distance Dependence of Transfer Rate Sangeeta Saini,† Vijay B. Shenoy,‡ and Biman Bagchi*,†,§ Solid State and Structural Chemistry Unit, Center for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore 560 012, India, and Department of Chemistry, Stanford UniVersity, Stanford, California 94305-5080 ReceiVed: NoVember 26, 2007; In Final Form: February 11, 2008
Surface plasmon mediated excitation energy transfer (EET) between two nonspherical metal nanoparticles (MNP) is studied through a microscopic theoretical formulation that provides quantitative information about the distance, size, shape, and orientation dependence of the transfer rate. At constant separation, the EET rate increases with increase in the size of the nanoparticle and decreases with increase in the aspect ratio of the particle. The relative orientation of the nanoparticles is found to markedly influence the center-to-center distance (r) dependence of the excitation energy transfer rate. At intermediate separations, we find a significant deviation from 1/r6 dependence (which is robust for spherical particles at these separations), and it becomes more pronounced with increase in the aspect ratio of the particles. Interestingly, the relative orientation of the nanoparticles effect the surface-to-surface distance (d) dependence of the rate to lesser extend in comparison to the r-dependence of the rate. Our results suggest that for nonspherical particles studying EET rate as a function of d (not r) provides more conclusive results.
1. Introduction The distance dependence of excitation energy transfer rate from a donor to an acceptor moiety forms the underlying principle of most commonly employed experimental technique of fluorescence resonance energy transfer (FRET), which is widely used as a diagnostic tool to investigate structure and dynamics of macromolecules.1-4 Because of its popularity, this resonance energy transfer (FRET) method is referred to as “spectroscopic ruler”, or “optical ruler”.5 The rate of excitation energy transfer for conventional resonance energy transfer systems (both donor (D) and acceptor (A) are dye molecules) is given by the well-known Fo¨rster expression6
kFRET DA ) krad
() r0 r
σ
(1)
where the distance exponent, σ, is 6, krad is the radiative rate of the donor dye molecule in absence of the acceptor molecule, r is the center-to-center distance between the donor and the acceptor and r0 is the Fo¨rster radius.7 The Fo¨rster radius provides a measure of the length scale that can be explored by a given donor-acceptor combination and is determined by the spectral overlap between the donor and the acceptor. Despite its great success, FRET has its limitations; it is necessary that the separation r between donor and acceptor be less than 10 nm. To overcome this limitation, a new excitation energy transfer (EET) system has been devised in which a noble metal nanoparticle is used in place of the acceptor dye molecule. The * To whom the correspondence should be addressed. Tel: 91-8022932926. Fax: 91-80-23601310. E-mail:
[email protected]. † Solid State and Structural Chemistry Unit, Indian Institute of Science. ‡ Center for Condensed Matter Theory, Department of Physics, Indian Institute of Science. § On Sabbatical to Stanford University.
dye-nanoparticle EET systems extend the distance that can be measured by a factor of 2 over the conventional EET systems8 (the systems with nanoparticles as donor and acceptor, the separations up to 70 nm have been measured9). The same study8 reports that the rate of energy transfer in dye-nanoparticle system no longer follows Fo¨rster type distance dependence. Instead, surface-to-surface distance (d) dependence of the rate is found to be the same as observed in case of surface energy transfer (SET)10-12 and rate is given by
kSET DA ) krad
() d0 d
σ
(2)
with σ ) 4. Note that here d is the surface-to-surface distance between the MNP and the dye, and d0 is the characteristic distance length.8 In the case of the energy transfer between a dye and a metallic nanoparticle, an earlier study13 found that while the center-tocenter distance (r) dependence of rate remains 1/r6 (in agreement with Fo¨rster formalism), the rate dependence on surface-tosurface separation distance (d) varies as d-σ; the value of the distance exponent σ lies in range of 3-4 at distances comparable but greater than the size of the nanoparticle. This theoretical result13 was thus in partial agreement with the results reported in the experimental study.8 Note that at separations comparable to the size of the nanoparticle, the value of σ for d is expected to be smaller from that of r, as is found in the calculations. At large distances, however, even the d-dependence of rate goes as 1/d6. Deviation from Fo¨rster theory has been reported in other systems also, for example, where either donor or acceptor is an extended conjugated molecule.14 The reason for the breakdown of Fo¨rster expression when either of the D-A system is an extended molecule arises from the breakdown of the point dipole approximation that plays a
10.1021/jp711197x CCC: $40.75 © 2008 American Chemical Society Published on Web 04/02/2008
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Figure 1. Schematic diagram of two interacting spheroidal nanoparticles. The semimajor and semiminor axes are a and b, respectively. The focal length f is as indicated. The surface of the spheroid is described by the equation ξ0 ) 1/e, where e is the eccentricity of the particle.
crucial role in the derivation of the famous Fo¨rster expression.14 The situation for the nano metal particle-dye system is no different because point dipole approximation also breaks down when the dye is not far from the nano metal particle. In addition, several new features appear in the energy transfer dynamics involving nano metal particles, as discussed later. An emerging area of interest in nanobiology and materials science is the use of energy transfer between metal nano particles (MNP). Indeed an assembly of nanoparticles can have potential applications in the design of nanoscale photonic devices. The optical excitation of a nanoparticle by an external field gives rise to intense nonpropagating field associated with the excitation of surface plasmons (SP). It has been shown that for a linear chain of spheroidal Ag nanoparticles the SP near-field interactions between neighboring nanoparticles result in propagation of excitation energy over a length scale of few hundred nanometers.15 Recently, it has been suggested that the surfaceto-surface distance (d) dependence of excitation energy transfer between two spherical nanoparticles can also be non-Fo¨rsterlike.16 While the energy transfer from a dye to a spheroidal nanoparticle has been studied in detail,17 there exists no theoretical study on distance dependence of EET rate between two nanoparticles of nonspherical shape, although many practical systems noted above contain nonspherical particles. Here we develop and apply a quantum mechanical theory of energy transfer mediated by the surface plasmonic modes of spheroidal nanoparticles that allows for the investigation of both shape and size effects on the rate of energy transfer. The donor-acceptor system is shown in Figure 1. We find that for spheroidal nanoparticles both the surface-to-surface (d) and center-to-center (r) distance dependence of the rate of energy transfer show strong deviation from inverse sixth power dependence. In particular, we see that short distance behavior of the rate (i.e., “small” separations that are comparable but greater than the minor axis of the nanoparticle in case of spheroidal particles and the size of the nanoparticle in case of spherical particles) is governed by an apparent distance exponent, σ that depends on the shape and size of the particles, and may be contrasted with the transfer between two spherical particles (of same size) where the center-to-center distance dependence of RET is always of the 1/r6 type.
2. Theoretical Formulation The collective electronic excitations of the nanoparticle are modeled by using a quantum mechanical compressible electron fluid model.13,18 This minimal model allows formulation of an analytical theory to capture the key features of size and shape dependence of the EET rate and the plasmonic wave functions that control the energy transfer. We consider a system of two spheroidal nanoparticles (see Figure 1) and within this formulation Hamiltonian of the nanoparticle can be described using the Bosonic plasmon destruction (creation) operators, al,m (a†l,m) as
H)
† pωl,mal,m al,m ∑ l,m
(3)
where ωl,m are the frequencies of the plasmon modes labeled by the angular momentum quantum numbers. The quantization axis is chosen to be “a”, which is the semimajor axis (see Figure 1) of the nanoparticle. The plasmon modes and their frequencies are obtained by the solution of the well-known Helmholtz equation
r + k2nl,m(b,t) r )0 ∇2nl,m(b,t)
(4)
where k2 ) (ω2l,m - ω2p)/β2, ωp is the bulk plasmon frequency, and β2 ) 3/5V2F where VF is Fermi velocity of the bulk metal. b,t) using prolate The Helmholtz equation is solved for nl,m(r spheroidal coordinates.19 The number density operator for the |l,m〉 mode is given by
r ) nˆl,m(b,t)
( ) pωlmk20 2e2c I
1/2 † Rl,m(c,ξ)Sl,m(c,η)Φ(φ) (al,m + al,m) (5)
where Φ(φ) )
1 ; x2π
for m ) 0
)
1 cos(mφ); xπ
for even m
)
1 sin(mφ); xπ
for odd m
Energy Transfer between Nonspherical Metal Nanoparticles 0 is the permittivity of space, ec is the electronic charge, I is normalization constant that depends on the semimajor axis, a, and the inverse eccentricity, ξ0 (see Figure 1). Rl,m(c,ξ) and Sl,m(c,η) are the radial and angular spheroidal wave functions, respectively, with c ) fk; f is the focal length of the spheroid and ξ and η are the ellipsoidal coordinates, and φ is the polar angle.20 If various |l,m〉 plasmon modes are excited, the total density fluctuation (n(r b,t)) is given by n(r b,t) ) ∑l,m nl,m(r b,t). The values of the plasmon frequency and the parameter c corresponding to the excitation of |l,m〉 mode are calculated from the following dispersion relation obtained by imposing the appropriate boundary conditions18
(ω) ) 1 -
Pml (ξ0)Q′lm(ξ0) - Qml (ξ0)P′ml (ξ0) Rl,m(c,ξ0)
(6)
(ω) represents the frequency dependent dielectric function which is 1 - ω2p/ω2 for the case of the free electron gas. The effect of the interband contributions to (ω) needs to be accounted21,22 for quantitative comparison with experiments. The prime over symbols denotes derivative. We now proceed to consider the incoherent nonradiative energy transfer between two spheroidal nanoparticles of same size and shape with donor nanoparticle in |l,m〉 ) |1,0〉 plasmon mode. Note at separations less than “small” separations, the perturbation may become too large for the first-order perturbation theory, which is the basis of Fermi Golden rule, to provide an accurate description of the rate. In this limit, coherent energy transfer becomes important. Therefore, our theory and the corresponding numerical results are not expected to hold strictly for such separations. We here intend to study the effect of shape on the distance dependence of the rate, and the results are strictly valid for separations larger than the minor-axis of the nanoparticle. Our results point out even the slight deviation from the sphericity leads to deviation from 1/r6 distance dependence of the rate. The energy transfer is mediated via dipole-dipole interactions between two nanoparticles, thereby exciting the acceptor nanoparticle to |1,0〉 plasmonic state. The rate of energy transfer is calculated using Fermi golden rule
kDA )
corresponding to various plasmon modes which in present case are l ) 1, m ) 0 mode for both donor and acceptor nanoparticles. HI, the interaction Hamiltonian that mediates the energy transfer, is given by the Columbic interactions between the charge densities of donor and acceptor nanoparticles
HI )
1 4π0
F(b r )F(b r )
∫d3rD ∫d3rA|RB + Abr A - Dbr D|
(9)
With F(r b,t) ) -ecn(r b,t), the substitution into (eq 9) gives the interaction Hamiltonian from which the rate of energy transfer between two nanoparticles is obtained through eq 7. The delta term in eq 7 can be written as g g - EM - ENe A) ) δ(ENe D + EM A D
Q′ml (ξ0)P′ml (ξ0) - P′ml (ξ0)Qml (ξ0)
R′l,m(c,ξ0)
J. Phys. Chem. C, Vol. 112, No. 16, 2008 6301
2π g f(ENe D) f(EM )|〈χgD;χeA|χeD;χgA〉|2 × |VDA|2 A p MD,NDMA,NA
∑ ∑
g g - EM - ENe A) δ(ENe D + EM A D
(7)
g g ;ψNe A|HI|ψNe D;ψM 〉 VDA ) 〈ψM D A
(8)
Here VDA represents the electronic contribution to the rate of energy transfer while the nuclear contribution is represented by the terms appearing under the summation. The plasmon excitation (the collective excitation of the electron density) is considered against the uniformly distributed positive charge background. χ g/e D/A in Fermi golden rule represents the nuclear wavefunction of the donor/acceptor in the ground/excited-state of the system. Here, the nuclear contribution may be accounted for by including the core polarization effects that results in the shift of plasma frequency; a characteristic property of a nanoparticle of a given material. The presence of delta term accounts for energy conservation. ψ’s represent the wavefunctions (represented by the density modes given in eq 5)
∞ dE δ(ENe ∫-∞
D
g g - EM - E) δ(EM - ENe A + E) (10) D A
The first term in Fo¨rster formalism leads to emission spectrum of the donor while the second to the absorption spectra of the acceptor. Here, to account for the broadening of the spectra caused by various other degrees of freedom the sharp resonance lines are replaced by Lorentzians. For Ag nanoparticles, the width of the Lorentzian is taken to be 180 meV.23 Note that all the calculations are carried out with air as a surrounding media. 3. Results and Discussion A. Calculation of Optical Spectra. Figure 2 shows a plot of the frequencies of plasmon modes calculated from eq 6 for Ag nanoparticles (pωp ) 3.3 eV) of different sizes but with a fixed eccentricity. In all cases studied, the plasmons are “blueshifted”, that is, the excitation energies increase with decrease in size due to the confinement effects. For the l ) 1 mode, the m ) 1 mode is found to be of larger frequency than the m ) 0 mode; this is again due to the fact that the m ) 1 mode is more confined than the m ) 0 mode because of the geometric constraint of a > b. The l ) 2 case has similar features, as do all higher l modes. Furthermore, with increase in the size of MNP calculated values tend to the size independent but eccentricity dependent limiting value set by earlier calculations that do not account for the size effects.24 We have also studied the shape dependence (for a fixed a) of the plasmon frequencies as shown in Figure 3. For the l ) 1 mode, we see that the m ) 0 mode “softens” with increasing eccentricity (1/ξ0), while the m ) 1 mode “hardens” with increasing eccentricity. Again, the physics of this phenomenon can be traced to the confinement effect; the m ) 1 mode is more confined with increase in eccentricity, while the effect is opposite on the m ) 0 mode. B. Effect of Size and Shape on Energy Transfer Rate. In the rate of energy transfer calculations, the surface plasmon frequencies were obtained by considering the dielectric function (ω) of Ag as measured experimentally25 and ωp ) 14.6 × 1015 Hz. Through the use of the experimental frequency dependent dielectric constant, the calculated plasmon frequencies are found to be in fairly good agreement with the plasmon peaks obtained from the discrete dipole approximation (DDA) calculations.26,27 These frequencies are used for the rate calculations. Rl,m(c,ξ) and Sl,m(c,η) functions required for the rate calculations are numerically evaluated using Mathematica 6.0. Note (ω) considered here is size-independent. The inclusion of size correction to the dielectric constant28,29 of the metallic nanoparticle results in a small decrease in the EET rate. However, the magnitude of the rate almost remains the same. Figure 4 shows the plot of the calculated rate of excitation energy transfer (EET) as a function of surface-to-surface
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Figure 2. The dimensionless plasmon frequencies of spheroidal MNP (with aspect ratio ) 1.1) corresponding to various l, m modes. (i) For l ) 1; m ) 0, 1 modes. (ii) For l ) 2; m ) 0, 1, 2 modes. Note that in the large size limit, the frequencies approach the ratios (marked by NCE) corresponding to those obtained from a treatment (ref 24) that does not include electron compressibility effects.
Figure 3. The plasmon frequencies for a nanoparticle (a ) 5.0 nm) as a function of ξ0 (increase in the value of ξ0 ) 1/e corresponds to decrease in the aspect ratio of the nanoparticle). The large values of ξ0 correspond to spherelike particles while the small values correspond to needlelike particles.
separation distance, d, for nanoparticles of different sizes. The rate of EET increases with the size of the nanoparticle, as expected, but the difference in the rates of different sizes decreases with the decrease in the separation. Interestingly, for d smaller than a, the trend is reverse. The d-dependence of the rate of EET becomes weaker as the separation distance between
two nanoparticles decreases (see inset of Figure 4). We observe that for a given separation, a small size nanoparticle has stronger dependence on d as compared to larger nanoparticles and eventually at separation d/a ≈ 1; for a large size particle, the d-dependence of rate becomes much weaker than that for the smaller size, resulting in a reversal of the trend. Another
Energy Transfer between Nonspherical Metal Nanoparticles
J. Phys. Chem. C, Vol. 112, No. 16, 2008 6303
Figure 4. Effect of size on distance (d) dependence of the rate of energy transfer for the nanoparticles with an aspect ratio equal to 5. Note the semilog scale used. The rate increases with the size of the nanoparticle. However, there is a reversal of rates at distances, d/a ≈ 1. The symbol d denotes surface to surface distance. Inset shows the distance exponent σ as a function of d, σ ) -d(log kDA)/d(log d).
important aspect is the absolute value of the rate of excitation energy transfer, which can be even larger than 1015 s-1 for large particles, at close surface-to-surface separation. However, note that these calculations are done with air as surrounding media. The dependence of energy transfer rate on the size of the nanoparticle is mainly influenced by three factors, namely, (i) the strength of the Coulombic interaction between donor and acceptor nanoparticles, (ii) the position, and (iii) the width (inverse surface plasmon lifetime) of the surface plasmon band (SPR). For large size particles, the surface plasmon frequencies are independent of the size of the nanoparticle. Therefore, for large sized particles, the size dependence of energy transfer rate is solely determined by the size dependence of Coulombic interactions. At large separations, two metal nanoparticles can be approximated as two interacting dipoles. In this limit, the size and the shape dependence of energy transfer rate (kDA) can be related to the corresponding dependence of interacting dipole moments as kDA ≈ |µA|2|µD|2/r6. Now, the dipole moment (µ bA/D) of a given nanoparticle can be evaluated from the corresponding charge density (n(r bA/D,t)) as b µA/D ) -ec ∫r bA/Dn(r bA/D,t)dr bA/D. The asymptotic analysis (large c, see eq 5) shows for large size needlelike particles, the strength of the dipole moment is related to the geometric parameters of the particle as
[ ]
( )
4πpωp0 ξ20- 1 a µA/D ≈ |k| (Q01(ξ0))1/4 ξ20
1/4
≈
| ( )|
-(πpβ0)1/2(ab)1/2 ln
b a
-1/4
(11)
where Q01(ξ0) is associated legendre function of second kind, and a and b are semimajor and semiminor axis of the particle, respectively. The analysis uses the asymptotic expansion of Rl,m(c,ξ) and Sl,m(c,η) functions.30,31 Note that the above result is derived for l ) 1, m ) 0 mode of excitation. In Figure 5, we show the shape dependence of the rate of EET between two MNPs. The rate decreases with the increase
in the aspect ratio of the nanoparticles implying the rate of EET to be highest for the spherical particles (for a given a). However, with the decrease in the aspect ratio (keeping a constant), the effective size of the particle increases. To confirm that the increase in rate is not due to an increase in the effective size of the particle, we calculated the rates for nanoparticle at different aspect ratio but with the same volume. We find that the rate for smaller aspect ratio is still higher confirming the intrinsic role of the aspect ratio and that it is not merely the effect of the size. As mentioned earlier, a previous study on the rate of EET transfer between a spherical nanoparticle and a dye molecule13 found that at small separations the rate shows a significant deviation from 1/d6 distance (d) dependence, but the dependence of the EET rate on the center-to-center distance remains 1/r6. However, the situation turned out to be quite different for excitation energy transfer between these spheroidal nano metal particles. Here the dependence of the rate of energy transfer on both; the surface-to-surface separation, d, and the center-tocenter separation, r, exhibit a pronounced deviation from the sixth inverse power dependence of the rate on distance, even at moderate distances. As shown in Figure 6, this deviation from 1/r6 increases as the particle becomes more and more needlelike. It is seen more clearly in the inset of Figure 6 where the value of the exponent σ is shown. Note that at distances comparable to the size of the participating MNP, the value of the distance exponent σ is noticeably less than 6 for spheroidal particles. Thus the shape of the particle has an important effect on the distance dependent exponent, σ and could possibly be crucial in the interpretation of experiments. C. Effect of Orientation on Energy Transfer Rate. We next study the effects of relative orientation of two MNP on the rate of energy transfer. We consider the axes of the nanoparticles to be in the same plane and obtain the transfer rate as function of the angle θrel between the two principal axes (Figure 7). Thus the line joining the centers of the particles is
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Figure 5. The shape dependence of the EET rate for a given size of the nanoparticle, a ) 5 nm. The data is presented in the log-log plot.
Figure 6. The rate of EET as a function of the center-to-center distance, r, for a given size of the nanoparticle, a ) 5 nm. Note the deviation from straight line with the increase in the aspect ratio. The data is presented in the log-log plot. Inset shows the dependence of the apparent distance exponent σ. Note that the inset uses semilog scale.
perpendicular to the axis of the donor particle. The main result is shown in Figure 7. The rate is maximum when the axes of the two particles are parallel (as expected) and falls to zero when the two particles are perpendicular to each other. The matrix element connecting the m ) 0 plasmon modes of the two particles goes to zero when the particles are perpendicular. In this configuration, there is still a nonzero matrix element between the m ) 0 plasmon of the donor particle, and the m ) 1 plasmon of the acceptor particle. A transfer of energy is not allowed between these two modes because they have different energies. However, for particles that are nearly spherical, a transfer could still be possible provided that the width of the plasmon mode is of the order of the separation between the frequencies of m ) 0 and m ) 1 modes.
The orientation dependence of the energy transfer in fact depends on the separation between the particles. When the axes of the particles are nearly parallel, the orientation dependence of energy transfer rate is stronger at larger distances. For any given orientation, both r and d distance dependencies of the rate show a deviation from the sixth power behavior as mentioned in previously. We find that although the d-dependence of the transfer rate remains consistent with σ < 6, the r-dependence of the transfer rate is remarkably influenced by the orientation of the two nanoparticles. For particles with parallel major axes and the center-to-center vector lying along the two axes, we find that the r-dependence of the rate has a value of σ greater than 6. Our calculations therefore suggest that in experiments where it is difficult to control the orientation
Energy Transfer between Nonspherical Metal Nanoparticles
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Figure 7. The orientation dependence of the transfer rate at different separations (r) between two nanoparticles with a ) 5.0 nm and aspect ratio equal to 3. The angle θrel denotes the relative orientation of the major axes of the two particles as shown (the donor and acceptor nanoparticles are superimposed on each other to clearly show θrel.). Both of the major axes are in the same plane, and the vector joining the centers is perpendicular to the major axis of the donor particle. The energy transfer rate (kDA) for a given separation is normalized with the corresponding maximum in the transfer rate (kmax DA ).
of the anisotropic particles, it may be more meaningful to measure the transfer rate as a function of surface-to-surface separation, d, instead of center-to-center distance, r, for arriving at conclusive results as is usually done. 4. Conclusions In conclusion, we have developed a microscopic theory to investigate the distance dependence (in terms of both r and d) of the rate of energy transfer between two spheroidal nanometal particles. The microscopic theory describes electrostatic interaction between the two MNPs in terms of Coulombic interaction between fluctuating charge densities (the surface plasmon modes). The theory allows study of energy transfer rate as a function of their shapes, sizes, and relative orientation. The theory provides quantitative prediction about the value of the energy transfer rate as a function of the size and the aspect ratio of the ellipsoidals of revolution. Numerical calculations show that for a ) 5 nm, as the aspect ratio increases from unity (sphere) to 5 (needlelike), the rate of EET at small separations decreases from 1015 s-1 to 1012 s-1. The rate of energy transfer increases with the increase in the size and decreases with increasing eccentricity (aspect ratio) of the particles. The r-dependence (center-to-center distance) of the energy transfer rate is found to be strikingly dependent on the shape (eccentricity) of the particles and is distinctly different at different relative orientations of the acceptor and the donor nanoparticles, and in some orientations the short distance dependence even goes as 1/rσ with σ > 6. On the other hand, the dependence of rate on the surface-to-surface separation distance (d) shows deviation from 1/d6 form with σ always
