Distance and Orientation Dependence of Excitation Energy Transfer

J. Phys. Chem. B 2009, 113, 1817–1832

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FEATURE ARTICLE Distance and Orientation Dependence of Excitation Energy Transfer: From Molecular Systems to Metal Nanoparticles Sangeeta Saini,† Goundla Srinivas,‡ and Biman Bagchi*,† SSCU, Indian Institute of Science, Bangalore 560012, India, and IBM Almaden Research Center, San Jose, California 95120 ReceiVed: July 24, 2008; ReVised Manuscript ReceiVed: October 21, 2008

The elegant theory developed by Fo¨rster to describe the rate of fluorescence resonance energy transfer between a donor and an acceptor has played a key role in understanding the structure and dynamics of polymers, biopolymers (proteins, nucleic acids), and self-assemblies (photosystems, micellar systems). Fo¨rster theory assumes the transition charge densities of donor and acceptor molecules are point dipoles and hence predicts a 1/R6 dependence of energy transfer rate on center-to-center separation distance, R. In addition, a preaveraging over the orientations of the two dipoles is usually performed. The present review examines the validity of these assumptions in following different donor-acceptor (D-A) systems: (i) dye molecules attached to a flexible polymer chain in solution, (ii) extended conjugated dye molecules in quenched conformation, (iii) dye and a spherical metal nanoparticle of different sizes, (iv) two spherical metal nanoparticles, and (v) two prolate shaped metal nanoparticles at different relative orientations. In the case of dye molecules attached to a flexible polymer chain, we discuss the recent theoretical and computer simulation studies of energy transfer dynamics. It includes an analysis of Wilemski-Fixman (WF) theory of a bimolecular reaction in solution, applied to the excitation energy transfer between two ends of the polymer. We briefly describe the limitation of the WF theory and its generalizations that lead to a better agreement between the theory and the simulation results. The orientational dynamics of dye molecules is found to significantly influence the rate of excitation energy transfer, and may play a “hidden role” in influencing the observed distance dependence. For extended conjugated D-A systems and those involving nonspherical metal nanoparticles, even at intermediate separations, a significant deviation from 1/R6-type distance dependence of the energy transfer rate is found. Surprisingly, however, this distance dependence is robust for D-A systems involving spherical metal nanoparticles. For both spherical and nonspherical metal nanoparticles (MNps), the functional dependence of rate on the surface-to-surface separation distance (d) is quite different, at small to intermediate distances (compared to the size of the MNps). The rate calculations of excitation energy transfer between extended conjugated dye molecules reveal that optically dark states can significantly contribute toward enhancing the energy transfer rate. It is further found that the rate of energy transfer between nonspherical metal nanoparticles exhibits an interesting orientation dependence not anticipated in Fo¨rster’s approach. 1. Introduction The process of excitation energy transfer (EET) from a donor to an acceptor molecule is ubiquitous in nature. The most striking example is the highly efficient light harvesting system (LH) involved in photosynthesis. A considerable amount of effort is being made to develop efficient solar cells based on the phenomenon of EET observed in natural photosynthetic systems. EET is an underlying physical mechanism of biosensors and has already been successfully applied to investigate a wide range of biological processes. Later, we will briefly discuss the insights gained into these processes through EET experiments and the possible technological headways such a molecular level understanding can offer. The widespread applications of EET are attributed to the simultaneous developments in the field of * Corresponding author. E-mail: [email protected]. † Indian Institute of Science. ‡ IBM Almaden Research Center.

optical engineering, chemistry, biology, and physics. In fact, with the further advances in materials science, the process of EET appears to be poised for increasing use in the near future. Figure 1 shows a schematic representation of the EET process. It involves a nonradiative energy transfer from an electronically excited state of a donor molecule (D*) to the ground state of an acceptor molecule (A).1 The total energy of the D-A system is conserved during the EET process. Therefore, EET is often referred to as resonance energy transfer (RET). In experiments, RET is mostly detected either via decrease/increase in the fluorescence intensity of D/A depending on whether the donor or the acceptor is fluorescent. Thus, this technique is also popularly known as fluorescence resonance energy transfer (FRET).1 Historically, EET was first observed around 1925 in a series of fluorescent quenching experiments. Perrin suggested that quenching is caused by dipole-dipole interactions between the donor and acceptor molecules.2 Later, in 1948, Fo¨rster3,4 proposed an elegant theory for the rate of EET in terms of (i)

10.1021/jp806536w CCC: $40.75  2009 American Chemical Society Published on Web 01/07/2009

1818 J. Phys. Chem. B, Vol. 113, No. 7, 2009

Sangeeta Saini did her undergraduate and master study from Panjab University, Chandigarh. She is currently pursuing her doctoral degree at Indian Institute of Science, Bangalore, under the supervision of Prof. Biman Bagchi. Her research focuses on understanding energy transfer in polymers and systems involving metal nanoparticles.

Goundla Srinivas (born 1975) did his masters in chemistry from Osmania University, Hyderabad, India, in 1997. He received his doctoral degree from Indian Institute of Science, Bangalore, in 2002 under the guidance of Prof. Biman Bagchi. From 2002 to 2006, he was a postdoctoral fellow in Professor Michael Klein’s group at University of Pennsylvania, Philadelphia, PA. Presently, he is a postdoctoral fellow at IBM Almaden Research Center, San Jose, CA. His current research interests lie in understanding the large scale self-assembly of nanomaterials and their adsorption behavior on surfaces.

dipolar interaction between the transition dipole moments of D and A and (ii) the spectral overlap between the emission and absorption spectra of D and A, respectively. The transition dipole moments of D and A are assumed to be point dipoles located at the center of geometry of respective molecules. This assumption is referred to as the point-dipole approximation (PDA). As a consequence of PDA, Fo¨rster theory predicts that the rate of EET follows a 1/R6-type distance dependence, where R is the center-to-center distance between D and A. Besides PDA, Fo¨rster theory also assumes EET to be an equilibrium process. These assumptions, in particular PDA, limit the applicability of original Fo¨rster theory. Note that, at small separations where a direct overlap between the electronic wave functions of D and A is possible, the energy transfer (ET) occurs via exchange interactions in contrast to Coulombic interactions that dominate ET at large separations. The former mode of ET is called the Dexter mechanism5 and is predicted to have an exponential dependence on R. In the present study, we restrict ourselves to separations where EET processes are mediated only via Coulombic interactions. Fo¨rster theory as discussed in section 2 predicts that the efficiency of energy transfer strongly depends on the separation

Saini et al.

Biman Bagchi (born 1954) received his undergraduate degree from Presidency College, Calcutta University (W. Bengal, India), in 1976. He obtained his Ph.D. degree from Brown University (Providence, RI) in 1981 with Professor Julian H. Gibbs as his advisor. He was Research Associate at the James Franck Institute, University of Chicago (1981-1983), where he worked with Professors David W. Oxtoby, Graham Fleming, and Stuart Rice, and at University of Maryland (with Robert Zwanzig) before returning to India in the fall of 1984 to join as faculty in Indian Institute of Science, Bangalore. He is a Fellow of the Indian Academy of Sciences, Indian National Academy of Science, and also of the Third World Academy, Trieste. His research interests include statistical mechanics, relaxation phenomena, chemical reaction dynamics, phase transitions, protein folding, and enzyme kinetics.

distance and the relative orientation of the participating molecules. This strong distance dependence of FRET rate has been exploited to develop sensors for various analytes found in biological systems like CO2,6 glucose,7,8 and metal ions.9,10 Further, with a suitable D-A pair (generally dye molecules) attached to a macromolecule, the rate of EET provides information on the configuration and conformations of the macromolecule.1 The developments in basic optical techniques used in FRET experiments and the availability of better dye molecules have made the FRET technique all together more exciting. The combination of FRET with the recently developed single molecule spectroscopy (SMS) has elevated its applicability to new levels.11 In a novel smsFRET experiment, Deniz et al. identified the subpopulations of different conformations based on the distribution observed in FRET efficiency.12,13 Similar results are obtained in a simulation study of polymer folding and unfolding “reaction”.14 Such an ability to distinguish different species and monitor their kinetics proved useful in the study of complex biological processes, for example, enzyme kinetics15,16 and conformational fluctuations of biopolymers like nucleic acids and proteins.17 The use of smsFRET has provided a deeper insight into the thermodynamic and kinetic properties of protein folding and unfolding reactions.18-20 The computer simulation studies can also provide the information at single molecule level. A simple coarse-grained simulation study used a distribution in energy similar to the distribution of FRET efficiencies for investigating the folding of a small protein like HP-36 and could capture some of the qualitative features of the complex protein folding process.21 The knowledge obtained from such experimental and theoretical studies could help to identify the conditions that result in dysfunctional biomolecules. In dilute solutions, with the D-A pair attached to a polymer chain, the distance dependence of rate should be adequately described by Fo¨rster theory. However, in these cases, it is difficult to control or have information about the relative orientation of the participating D-A pair. Therefore, experimentalists often assume a preaveraging over all possible orientations. This averaging is however inaccurate if the rate

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Figure 1. Schematic picture showing the excitation energy transfer between two spatially separated donor (D) and acceptor (A) molecules. The excited states of D and A are represented by D* and A*, respectively. The resonance energy transfer mediated via Coulombic interaction is represented by double sided arrows between D and A states.

of orientational motion of D and A is slower than the rate of EET.22 Eaton et al. have experimentally observed this for a D-A pair attached to the ends of short length polyproline.23 They found that such an averaging overestimates the rate of EET. The effect of the orientational motion of D and A on the rate of energy transfer has been discussed in great detail in section 3 of the present review. Section 3 also discusses the recent theoretical advances made to understand the polymer dynamics in dilute solutions. The mode of excitation energy transfer in systems made of densely packed polymer chains (like thin films, natural light harvesting systems) has been highly debated. The performance of optoelectronic devices based on conjugated polymer films is significantly affected by the film morphology, that is, the conformations of individual polymer chains and the manner in which the polymer chains are packed together in a film. To optimize the performance of these devices, a thorough understanding of the energy transfer mechanism and the factors influencing the rate of EET is required. The detail account of EET in thin films can be found in a recent review by Schwartz.24 In conjugated systems, the excitation energy is strongly determined by the extent of delocalization of π electron density. As a result, in these systems, the energy can be “funneled” away from high energy segments to low energy segments.25-28 The energy transport in thin films is dominated by interchain species because of their close proximity and lower energy.24,29 In a novel experiment, Schwartz et al. encapsulated the conjugated polymer chains of MEH-PPV in mesoporous silica with a single chain per channel.30 The majority of chain segments are aligned inside the channel, while the fewer randomly oriented segments are outside the channel. The time-resolved polarized luminescence experiment showed that the unidirectional energy transport from randomly oriented, high energy segments to the aligned low energy segments (i.e., from segments of small conjugation length to segments of large conjugation length) can be achieved without the involvement of interchain species.30 The efficiency of energy transport in this system is found to be better than in the case of thin films. This type of photochemical funneling of excitation energy transfer is inspired by the light harvesting mechanism operative in photosystems I and II. In these systems, the solar energy absorbed by the antenna chlorophylls is efficiently directed toward the reaction center through EET.31 Scholes and Fleming et al. have recently investigated the molecular mechanism of EET in photosynthetic systems. We direct the readers to a recent review for the details of their study.32

J. Phys. Chem. B, Vol. 113, No. 7, 2009 1819 The applicability of Fo¨rster’s expression to complex systems has long been a subject of concern.33-35 In the case of photosynthesis, this has been discussed in an elegant series of papers by Fleming and Scholes et al.36-39 They found that Fo¨rster’s expression falters on several fronts. (i) In situations where point-dipole approximation breaks down. (ii) The contribution to the rate of EET from optically dark states becomes significant. Fo¨rster theory completely neglects the contribution from the optically dark states, as spectral overlap for these states is negligible.38-40 Actually, (ii) is a direct consequence of (i). The similar results have been obtained for EET in the case of extended conjugated polymeric systems, as discussed in section 4. Fo¨rster theory except at small separations works better for small dye molecules used in a D-A pair. However, such use of dye molecules (both as donor and acceptor) limits the range of distances that can be probed by EET to less than 10 nm.1 This is a serious limitation when one is interested in conformations and dynamics of long and complex macromolecules. Recently, in a novel development, noble metal (e.g., gold, silver) nanoparticles have been used as an acceptor in excitation energy transfer.41 This doubles the range of separations that can be monitored by EET. The use of semiconducting polymer matrixes containing rare-metal ion complexes is known to considerably improve the electroluminescence and photoluminescence properties of polymer based light emitting diodes.42-44 Further, the EET between nanoparticles is being considered as a potential way to efficiently transport the energy in optical waveguides.45 In section 5, we discuss the applicability of Fo¨rster theory in such systems where either acceptor or both donor and acceptor are metal nanoparticles. First, we briefly outline the formulation used to calculate the excitation energy transfer rate and discuss the Fo¨rster theory. 2. Fo¨rster Theory: Basic Considerations Consider a D-A system in which D is in the excited state while A is in the ground state (see Figure 1). In order to describe the excitation energy transfer (EET) in a D-A system, let us introduce independent wave functions for both the molecules as ΨuV(ru;Runuc) ) φuV(ru;Runuc)χuV(Runuc) (u ) D or A; V ) g (ground, S0) or e (excited, S1); ru and Runuc are the electronic and nuclear coordinates, respectively). We denote the vibronic energy levels associated with the ground and the excited states as Nu and Mu, respectively. The initial state in which the excitation is at D is represented by |i〉 ) |ΨDeMD〉|ΨAgNA〉 and characterized by energy, EDeMD + EAgNA. Similarly, the final state with energy EAeMA + EDgND is given by |f〉 ) |ΨAeMA〉|ΨDgND〉. The S0 f S1 transition energies are given by EDeMD + EAgNA EAeMA - EDgND. The energy transfer occurs via de-excitation of the donor and simultaneous excitation of the acceptor, as shown in Figure 1. As discussed earlier, Fo¨rster assumes the D-A system to be in thermal equilibrium with the surroundings. Therefore, the transition rate from an initial state |i〉 to the final state |f〉 can be calculated using the Fermi golden rule.

kDA )

2π p

∑ ∑ f(EDeM ) f(EAgN ) × M

N

D

A

|〈ΨDeMD, ΨAgNA |HI |ΨAeMA, ΨDgND〉| 2 × δ(EDeMD + EAgNA - EAeMA - EDgND) (2.1) Here, the thermal distribution for the initial vibrational states of D and A is denoted by f(EDeMD) and f(EAgNA), respectively.


Saini et al.

HI accounts for the Coulombic interactions between D and A. Assuming that the electronic coupling matrix elements (VDA) are independent of nuclear coordinates (Condon approximation), we can write

In the next section, we describe the effect of polymer dynamics on the rate of EET under different conditions, e.g., at different values of RF and krad. 3. Polymer Dynamics using EET

〈ΨDeMD, ΨAgNA |HI |ΨAeMA, ΨDgND〉 ) VDA〈χDeMD |χDgND〉〈χAgNA |χAeMA〉 (2.2) For a neutral D-A system, the dipolar component dominates the Coulombic interactions. Therefore, in this case, VDA is given by

VDA ) κ

|µ b(D) b(A) eg ||µ eg |

(2.3)

R3

where R is the D-A separation and κ is a dimensionless geometric factor determined by the orientation of donor and ˆ u|φug〉) in space. acceptor transition dipoles b µ(u) eg ()〈φue|µ The rate of EET in Fo¨rster theory is expressed in terms of the overlap between the donor’s emission and acceptor’s absorption spectrum. In order to incorporate this, the delta function of eq 2.1 is rewritten as a product of two delta functions describing the emission and absorption of energy, E, by the donor and acceptor. respectively, i.e.,

δ(EDeMD + EAgNA - EAeMA - EDgND) )

∫-∞∞ dEδ(EDeM

D

- EDgND - E) × δ(E + EAgNA - EAeMA)

(2.4) Using eqs 2.1 and 2.4, the rate of energy transfer in terms of the absorption coefficient of the acceptor (RA) and emission intensity of the donor (ID) is given by

kDA )

9c4κ2 8πn4R6

ID(ω) RA(ω) ∫0∞ dω ω4

(2.5)

where c is the speed of light and n is the refractive index of the medium. κ2 is the orientational factor which in EET experiments is generally assumed to be 2/3. Equation 2.5 can be written in a simpler form as

kDA ) krad

( ) RF R

6

(2.6)

krad ()∫0∞dω ID(ω)) is the radiative rate, which is typically of the order of 108-109 s-1 for commonly used dyes in EET experiments. ID(ω) denotes the emission intensity of D. According to the above equation, Fo¨rster radius,

(

RF )

9κ2c 4τD 8πn4

∫0

∞

dω

ID(ω)RA(ω) ω4

)

1/6

is defined as the D-A separation at which the nonradiative energy transfer rate becomes equal to the radiative decay rate of the donor, krad. τD ) 1/krad is the radiative lifetime of the donor molecule in the absence of the acceptor molecule.

As mentioned before, EET is a powerful technique for studying various aspects of the structure and dynamics of polymers and biopolymers both in solution46-48 and in constrained geometries. In experiments, the separation distance R is determined by measuring the excitation energy transfer efficiency, ET ) 1 - (τDA/τD), where τDA and τD are the donor fluorescence lifetimes (survival probability) in the presence and absence of acceptor molecule, respectively. The availability of the ultrafast laser technique makes it possible to measure the detailed time-dependent survival probability, Sp(t) in addition to ET. For a fixed distance between D and A, as in the case of rigid biopolymers,12,49 the EET experiment directly provides information on the separation distance R between D and A sites. However, for flexible macromolecules in solution, the distance R and the relative orientation of D and A both are fluctuating quantities that can quantitatively modify the time dependence of survival probability Sp(t). The recent experimental studies that combine EET with single molecule spectroscopy of polymers and biopolymers have emphasized the need for a better theoretical understanding of this dependence of survival probability. In fact, the study of long distance energy transfer between two segments in a polymer chain has been a subject of long-standing interest among theoreticians.50,51 This is a nontrivial problem because the Brownian motion of the individual monomeric units of the polymer is correlated even at long lengths due to the connectivity among the monomers. Fortunately, for systems that are near equilibrium at time t ) 0, and also remain so during the reaction, the Wilemski and Fixman (WF)52,53 theory accurately describes the diffusion limited intrachain reaction of a flexible polymer chain. An elegant alternative derivation of WF theory is provided by Portman and Wolynes using the variational approximation method.54 They obtained upper and lower bounds for the survival probability Sp(t) (for a definition, see Appendix I) and showed that, for a rapidly relaxing system, WF theory defines the upper bound on Sp(t). Our simulation results are in complete agreement with their theoretical predictions (see Figure 2). WF theory gives a nearly analytic solution for Rouse dynamics of a polymer chain for any arbitrary sink function, k(R). Often the Rouse model (although rather unrealistic) is used to describe the polymer dynamics because theoretically this limit can easily be treated. Therefore, computer simulations in conjunction with WF theory can be used to understand the role of polymer dynamics in EET. The extensive Brownian dynamics (BD) simulation studies of energy transfer in dilute flexible polymer solutions have been carried out by adopting a Fo¨rster sink function.55 The formulation of the problem and the definitions of relevant parameters are given in Appendix I. Here, we note that the sink, k(R), represents the rate of energy transfer from D to A at separation R. The excitation energy transfer rate is assumed to be given by

k(R) )

kF 1 + (R/RF)6

(3.1.1)

where kF is the rate of EET when the D-A separation is

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Figure 2. Sp(t) is plotted against the reduced time for different values of RF and kF at fixed N ) 50: (a) RF ) 1 and kF ) 1; (b) RF ) 5 and kF ) 1; (c) RF ) 1 and kF ) 10. The results obtained by simulation are shown by symbols, while the theoretical predictions are shown by lines. The reduced Green’s function (RGF) method (full line) results are in better agreement with the simulation results compared to WF theory (dashed line).

vanishingly small (i.e., R/RF f 0). The above form of k(R) (eq 3.1.1) is different (somewhat trivially) from the commonly used form of the Fo¨rster rate. The modification is necessary, as otherwise the rate will show a divergence with R f 0, which is allowed for a Rouse chain (note that in the Rouse chain the beads can pass through one another without any hindrance). Thus, the modified form (eq 3.1.1) used in simulations is reasonable.55 For a N monomer long polymer chain, the comparison of the survival probability Sp(t) obtained from simulations with WF theory is shown in Figure 2 for several values of kF and RF. The difference between the results obtained from simulations and the WF theory suggests that for many realistic situations one needs to go beyond zeroth order approximation that forms the basis of WF theory. In agreement with earlier studies,56,57 we find that WF theory works well for a sink with smaller reaction radius. This is because WF theory assumes a local equilibrium, which holds only for a narrow reaction window and small reaction rates but not otherwise. The above limitation of WF theory motivated alternative theoretical treatments. We used a three-dimensional reduced Green’s function method (RGF) to calculate Sp(t). In this approach, a 3N-dimensional equation-of-motion for the polymer (where N is the number of monomer units) is at first reduced to a 3D problem for a distance vector b R between D and A molecules and then to a 1D equationof-motion for distance R. This reduction from 3D to 1D is possible only because the Fo¨rster sink function k(R) does not depend on the orientation of b R. We subsequently used the discretized sink method to solve this 1D problem and calculate Sp(t). The details of our approach can be found in ref 58. Figure 2 also shows the Sp(t) calculated from the reduced Green’s function (RGF) method. We find that the agreement between RGF and WF theories becomes progressively poorer with increasing RF. The RGF results in comparison to WF theory are in better agreement with that of simulations, even at large RF values (see Figure 2b). Nevertheless, the simulation results are not in perfect agreement with the above-discussed RGF method especially for large reaction radii RF (see Figure 2b) and large radiative rate kF (see Figure 2c). Later, Tachiya et al.59 introduced an elegant effective sink approximation based

on a standard time scale separation ansatz. The agreement between theory and simulation was found to be nearly perfect. Ghosh et al.60 also obtained a good agreement using an alternative approach to this problem through the use of a nonMarkovian reaction diffusion equation. 3.1. Effects of DA Pair Orientation on EET Efficiency. According to Fo¨rster theory, EET occurs via dipole-dipole interaction between D and A. The rate of energy transfer given by eq 2.6 can be rewritten as

( )

3 RF kDA(R) ) krad κ2 2 R

6

(3.2.1)

The effect of orientational dynamics of the chromophores enters into the above equation through the orientation factor κ2 defined as

κ2(θA, θD, ΦAD) ) (2 cos θA cos θD - cos ΦAD sin θA sin θD)2 (3.2.2) where θD and θA are the angles which the dipoles of D and A form with the axis joining D and A. ΦAD is a dihedral angle between the planes containing D and A molecules. Note that the orientational preaveraging that replaces κ2 by 2/3 involves unrestricted averaging over all three angles of eq 3.2.2. The validity of this preaveraging is not clear because in principle one should obtain the dynamics of energy transfer at various angles and then perform the averaging. Also, the relative orientation of D and A can change with the variation in separation distance between donor and acceptor, as observed in protein folding and unfolding experiments. In these cases, the distance estimated from Fo¨rster theory using a preaveraged value of orientation factor will lead to an erroneous conclusion. This assumption of preaveraging is however valid when the orientational motion of the chromophores is much faster than the time scale of the EET rate. In general, orientational averaging (〈κ2〉Ω) can become a major source of uncertainty in the analysis.


Figure 3. Schematic representation of the model polymer used in the study of the orientation dependence of the rate of EET. The donor and acceptor chromophores are embedded at the two ends of the polymer (represented by D and A, respectively). The arrows on D and A show the direction of the unit dipole present on the respective chromophores.

Figure 4. The survival time, τFRET or τEET, obtained by Brownian dynamics simulations (filled circles) plotted as a function of rotational diffusion coefficient DR for a pair of mobile donor and mobile acceptor molecules. The horizontal line corresponds to τFRET in the dynamically averaged limit, i.e., κ2 ) 2/3. Here, RF ) 5, kF ) 10, and N ) 40.

Therefore, it is essential to study the effect of rotational dynamics of chromophores’ dipole moment on the rate of EET. Westlund and co-workers61 have studied the effect of distribution in molecular reorientation on the electronic energy transfer between a pair of mobile chromophores. We have used their scheme to describe the reorientation of the dipole on D and A. The details of the model and simulations can be found elsewhere.22 The schematic representation of the polymer model used in the study is shown in Figure 3. We find that the rotational dynamics of the chromophores have a significant effect on the EET rate. Figure 4 shows the survival time τFRET (which is the inverse of the rate, i.e., kDA-1) as a function of the rotational diffusion coefficient DR for a freely rotating donor and acceptor, at given values of RF, kF, and N. The horizontal line in the figure represents the value of τFRET obtained from a separate BD simulation which used κ2 ) 2/3. On going from the static limit (DR ) 0) to the dynamic limit (DR f ∞) of the

Saini et al. rotational diffusion coefficient, the EET rate increases. We find that, in the limit of slow rotation (small DR), the rate of EET is overestimated by the conventional scheme, i.e., κ2 ) 2/3. The preaVeraging oVerestimates the rate by about 20% for a mobile D-A pair. However, exactly the opposite happens for fast rotation. The preaVeraged rate is about 20% smaller than the actual rate. This DR dependence of energy transfer rate can be explained as follows. The probability that donor and acceptor molecules are in an orientational configuration with κ2 < 2/3 is almost the same for configurations with κ2 > 2/3. In the static limit (DR ) 0), EET takes place from the frozen configuration, i.e., fixed θA,θD, and φAD. Since the survival time is the inverse of the energy transfer rate, in the small DR limit, the average rate (the inverse is shown in Figure 4) of EET is dominated by small decay rates. This results in a long survival time compared to that predicted by Fo¨rster theory. However, in the large DR limit, the D-A pair explores the orientational space a number of times before the energy transfer takes place. Therefore, in the dynamic limit, the configurations with large κ values are likely to dominate the energy transfer rate which explains the increase in EET rate. It is worthwhile to point out that even in the absence of the rotational diffusion EET involves several different time scales, e.g., σ2/D0 (σ is the Lennard-Jones molecular diameter and D0 is a single monomer diffusion coefficient), kF-1. Thus, the effects of rotational diffusion of a chromophore pair on the EET rate are likely to vary for different combinations of kF, RF, and D0. However, it is interesting to note that while the calculated rate differs from Fo¨rster’s preaveraged rate at all values of DR, the preaveraged rate falls in between the two limits and is never more than 20% off from the correct one. 3.2. N Dependence of FRET Rate. The dependence of a bimolecular reaction rate on the length of the polymer chain is an important problem in polymer chemistry. The rate of EET between D and A sites located on the polymer chain depends on the D-A separation and hence on the number of monomeric units, N, between the two sites. Figure 5 shows the log-log plot of τFRET, for a mobile D-A pair, as a function of the chain length (L2 ) N) for two different values of RF at fixed values of DR and kF. We find that τFRET increases logarithmically with N.22 Such a linear dependence of the average survival time was predicted earlier by Pastor, Zwanzig, and Szabo55 for a Rouse polymer with a Heaviside sink function. In the case of EET, the rate (kDA ) 1/τFRET) has a power-law dependence on chain length (kDA-1 ∝ NR), with R ≈ 2.6. The N dependence of EET rate is an interesting theoretical problem which deserves further attention. We note that in our study (subsections 3.1 and 3.2) DR is assumed to be an isotropic rotational diffusion constant, but it is more realistic to consider a case where the rotation of the D-A pair is restricted, for example, to a cone.62 4. Excitation Energy Transfer in Conjugated Systems The morphology of thin films of conjugated polymers significantly influences the performance of polymer based optoelectronic devices. An understanding of the spatial and orientation dependence of the rate of EET on the relative internal geometries of D and A is therefore important. Such a study has been carried out in the case of energy transfer from a six-unit oligomer of polyfluorene (PF6) to a tetraphenylporphyrin (TPP) molecule.63 The representative orientations and the structures of the donor and acceptor are given in Figure 6. The full resonance-Coulomb coupling matrix elements and the pointdipole approximation to the coupling were computed using the

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Figure 5. Log-log plot of the N (number of monomers in a polymer chain) dependence of survival time (τFRET) for the mobile donor-mobile acceptor pair. The simulation results are shown by symbols for RF ) 5 (filled circles) and RF ) 7 (filled triangles) at kF ) 10. The straight lines are linear fits to the simulation data. The calculated slopes are 2.63 and 2.56 for RF ) 5 and RF ) 7, respectively.

Figure 6. Schematic diagram showing the relative orientation of donor chromophore PF6 and the acceptor TPP. The transition dipole moments of D and A are aligned parallel to each other and either are (a) orthogonal to the DA intermolecular axis (cofacial parallel) or (b) parallel to DA intermolecular axis (collinear parallel).

semiempirical quantum chemical method. The details of the computational approach can be found in ref 63. This study brought forward several limitations of Fo¨rster theory. Figure 7 shows the comparison of the calculated distance dependence of the Fo¨rster rate to the full resonance-Coulomb rate. The rates calculated here using the PPP Hamiltonian were in good agreement with the experimental results of Cerullo et al.64 The experimentally obserVed time scale of EET rate was of the order of a few tens of picoseconds, while the Fo¨rster theory predicts the time scale to be in femtoseconds (see Figure 7). The plot clearly demonstrates that the point-dipole approximation results in a gross overestimation of EET rates at small D-A separations. The transitions from donor states to acceptor states having midrange oscillator strength (0.70) are found to dominate the excitation energy transfer rate. However, the Fo¨rster theory assumes that EET is possible only between optically bright states. A separate calculation of EET between optically dark states of D and A also indicates that the contribution to the energy transfer rate from these states is of the same order as that from the optically bright states.63 This suggests that contrary to Fo¨rster assumption optically dark states can mediate excitation energy transfer.38-40,63

Fo¨rster theory also fails to predict the correct orientational dependence of EET rate at small separations. Figure 8 shows the orientation dependence for the cofacial case (Figure 6) at two D-A separation distances, 10 and 100 Å. The angle θ corresponds to the rotation of the TPP acceptor molecule about its transition dipole moment axis (z-axis in Figure 6). Figure 8a shows that while the rate calculated from full Coulombic interactions varies by a factor of ∼2 in going from 0 to 90°, the dipole approximation to the rate shows a negligible dependence. At large separations (Figure 8b), however, the EET rate calculated from both of the approaches shows weaker orientation dependence. To conclude, the study substantiates the fact that Fo¨rster theory fails to explain the correct distance and orientation dependence of EET rate in situations where the transition dipole densities of donor and/or acceptor are distributed on the length scale comparable to D-A separation. 5. Resonance Energy Transfer Involving Metal Nanoparticles A number of theoretical and experimental studies exist on energy transfer from a dye molecule to a planar65-68 or a


Figure 7. Distance dependence of the rate of EET between donor and acceptor is plotted for the cofacial parallel orientation ((0) Fo¨rster and (b) resonance-Coulomb rates). The acceptor states have a midrange oscillator strength of 0.70. The Fo¨rster distance dependence (R-6) is shown by a solid line. The total Fo¨rster (dotted line) and resonanceCoulomb rates (dashed line) are summed over all optically bright and dark states which mediate EET at a particular excitation wavelength.

spherical metal surface.41,69-72 The energy transfer from semiconductor nanoparticles to dye molecules has also been investigated.73-75 These studies showed that the use of nanoparticles in EET experiments has a number of advantages over the conventionally used fluorescent dye molecules. The one such experimental study has reported that the EET rate from a dye to a spherical metal nanoparticle is proportional to 1/d4.41 As discussed earlier, this greatly increases the range of separation distances that can be monitored by these D-A systems. Considering the recent developments, we revisited the problem of EET between a dye and a metal nanoparticle (subsection 5.2).76 We also report our results on energy transfer between a pair of spherical/prolate shaped metal nanoparticles (subsections 5.3 and 5.4).77,78 This study assumes importance as now-a-days; the possibility of using a chain of conducting nanoparticles to transport energy over short distances is being explored. We found that the distance dependence of the rate of energy transfer is highly sensitive to the shape of nanoparticles involved in energy transfer. In the case of a spherical nanoparticle, the rate of EET follows 1/R6 distance dependence. However, with the increase in the asphericity of a metal nanoparticle, a significant deviation from 1/R6 dependence is observed. Further, for a

Saini et al. system of nonspherical nanoparticles, the distance dependence of EET is strongly influenced by the relative orientation of the two nanoparticles. The details of excitation energy transfer in these systems are discussed below. To begin with, we first present a brief outline of theoretical formulation employed to describe the energy transfer involving metal nanoparticles. In the present formulation, we have not taken into account the effect of coherent interactions and the retardation effect which become important at small (less than the size of MNp)79-85 and large (greater than the reduced wavelength of virtual photon)86 separation distances, respectively. The results presented in this section are strictly valid only at intermediate separation distances (a e d e 10a). Note that, for nanoparticles with a diameter >50 nm, the retardation interactions are important even at small separations.87 5.1. Theoretical Formalism. In the limit of weak coupling between donor (D) and acceptor (A), the rate of excitation energy transfer from D* to A is calculated using the Fermi golden rule (eq 2.1). In the present discussion, the acceptor is a noble metal nanoparticle (MNp), while the donor can be either a dye molecule or a MNp. The summation in eq 2.1 is over all of the possible vibrational states of the dye molecule and the other degrees of freedom of the nanoparticle like the interaction with phonons, electron-hole pair interactions which result in the broadening of the optical spectrum of the nanoparticle. VDA, the interaction energy between the donor and acceptor, is given by

VDA ) 〈φDeMD,φAgNA |HI |φDgND, φAeMA〉

(5.1.1)

where HI is the interaction Hamiltonian and ψ denotes the electronic wave function. The surface plasmon mode of noble metal nanoparticles lies in a visible range. Since the donor used in such energy transfer experiments also emit in the visible range of the spectrum, we consider the excitation energy transfer from a donor to one of the surface plasmon mode of a MNp. We have used quantized electro-hydrodynamic theory88 to model the electronic response of a metal nanoparticle toward the approaching donor. The electro-hydrodynamic theory despite its mathematical and numerical simplicity captures the physics of collective excitations semiquantitatively89,90 and provides an elegant, almost analytical, solution to the complex problem studied here. A dye molecule, if a donor, is modeled as a particle in a one-dimensional box and is characterized by the length of the box. The charge density operator for a dye molecule in terms

Figure 8. Orientation dependence of normalized rate plotted for an initial cofacial parallel alignment of D and A transition moments at (a) short (10 Å) and (b) long (100 Å) D-A separations. The Fo¨rster and resonance-Coulomb rates are shown by empty squares and filled circles, respectively.

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of creation (cp†) and annihilation (cp) operators is given by

Fˆ D(b r D, t) ) -

2ec L

Φ(φ) )

p,q

p

where ψ’s are the electronic wave functions which depend on the position vector b rD, ec is the magnitude of electron charge, and 2L is the length of the one-dimensional box. The second term in the expression ensures the overall charge neutrality of the dye molecule. For a given metal nanoparticle, the surface plasmon frequency for a particular |l,m〉 mode is determined by its bulk plasma frequency, the frequency-dependent dielectric constant, and the size and shape of the nanoparticle. Here, l and m are the angular momemtum quantum numbers. In terms of charge distribution, the excitation of the l ) 1 mode corresponds to the dipolar mode, l ) 2 to the quadrupolar mode, and so on. The energy transfer from a donor to a metal nanoparticle can result in excitation of surface plasmons corresponding to different |l,m〉 modes. bA,t) Therefore, the total charge density fluctuation operator Fˆ MNp(r for an acceptor nanoparticle is a sum over all of the charge bA,t) associated with the density fluctuation operators, Fˆ l,m(r bA,t) ) ∑l,m Fˆ l,m(r bA,t). excitation of various |l,m〉 modes, i.e., Fˆ MNp(r bA/D,t) Within the linearized electro-hydrodynamic theory, Fˆ l,m(r is obtained as a solution to the Helmholtz equation (for details, bA/ refer to Appendix II). For a spherical (S) nanoparticle, Fˆ l,m(r D,t) is given by S S Fˆ l,m (b r A/D, t) ) Al,m (t) jl(RrA/D) Yl,m(θ, φ)

[

pωl,mε0R2 2a3

ωl,m2 (2l ωp2

)

1 cos(mφ); for even m √π

)

1 sin(mφ); for odd m √π

)

NS Al,m (t) )

(

pωl,mR2ε0 2I

)

1/2 † (al,m + al,m)

(5.1.7)

I is a normalization constant that depends on the semimajor axis, a, and the aspect ratio of the nanoparticle. Rl,m(c,ξ) and Sl,m(c,η) are the radial and angular spheroidal wave functions, respectively, with c ) fR, f is the focal length of the spheroid, ξ and η are the ellipsoidal coordinates, and φ is the polar angle. Note as c f 0, the spheroidal coordinates go to the spherical coordinates.91 The value of parameter c corresponding to the excitation of a given |l,m〉 mode is calculated from the following dispersion relation

ε(ω) ) 1 -

Plm(ξ0) Qlm(ξ0) - Qlm(ξ0) Plm(ξ0) Rl,m(c, ξ0) m Ql (ξ0) Plm(ξ0) - Plm(ξ0) Qlm(ξ0) Rl,m (c, ξ0) (5.1.8)

Pml (ξ0) and Qml (ξ0) are the associated Legendre function of the first and the second kind, respectively. The prime over symbols denotes the derivative with respect to ξ evaluated at ξ ) ξ0. ε(ω) represents the frequency-dependent dielectric function. The Coulombic interactions between the charge densities of donor and acceptor mediate the energy transfer from D to A. ˆ I is given by Thus, the interaction Hamiltonian H

1/2 † (al,m + al,m) ×

]

+ 1) 2 jl+1 (Ra) - jl-1(Ra) jl+1(Ra) 2

ˆI ) H

-1/2

ε(ω) )

l + 1 jl+1(Ra) l jl-1(Ra)

1 4πε0

Fˆ (b r , t) Fˆ (b r D, t)

∫ dbr D ∫ dbr A |Rb +A br

A

-b r D|

(5.1.9)

(5.1.4)

where a is the radius of the nanoparticle. The value of R is calculated from the following dispersion relation which is obtained by imposing suitable boundary conditions (see Appendix II).

(5.1.5)

bA/D,t) is given by For nonspherical nanoparticles, Fˆ l,m(r NS NS Fˆ l,m (b, r t) ) Al,m (t) Rl,m(c, ξ) Sl,m(c, η) Φ(φ) (5.1.6)

where

; for m ) 0

and

(5.1.3)

where R2 ) (ωl,m2 - ωp2)/β2, with β2 ) 3VF2/5. VF is the Fermi velocity of the electron gas, ωl,m is the surface plasmon frequency corresponding to the |l,m〉 mode, and ωp is the bulk S plasma frequency. Al,m (t) is the “amplitude operator” that can † , be expressed in terms of the plasmon bosonic operators (al,m al,m) as

(

√2π

e

∑ φ/p(br D, t) φq(br D, t)c†pcq + Lc ∑ c†pcp (5.1.2)

S Al,m (t) )

1

where R ) d + a is a center-to-center distance between D and A while d represents the surface-to-surface distance between two MNps or the distance from the center of the dye molecule ˆ I, the rate to the surface of the MNp. Using the above form of H of excitation energy transfer is calculated from eq 2.1 where Lorentzians of finite width replace the delta functions to account for the broadening caused by various thermal degrees of freedom. The nuclear overlap factors in eq 2.1 are assumed to be of the order of unity. In the subsequent subsections, we use the above formalism to calculate the rate of energy transfer for different pairs of acceptor and donors. We study the dependence of energy transfer rate on the distance and relative orientation between D and A. The variation of EET rate as a function of shape and size of the nanoparticles is also addressed. The knowledge of these aspects of energy transfer will prove to be helpful in the further improvement of the design of biological sensors and plasmonic devices.


Figure 9. Geometric arrangement of the spherical nanoparticle and the dye molecule in two different orientations, in parallel and perpendicular orientation with respect to the center-to-center distance vector b R. b d is the distance measured from the surface of the nanoparticle.

5.2. Excitation Energy Transfer Rate from a Dye to a Metal Nanoparticle. In this subsection, we summarize the results obtained in the case of excitation energy transfer from a dye molecule to a metal nanoparticle.76 Figure 9 shows the schematic diagram of the orientation of the donor dye molecule with respect to a spherical nanoparticle. We assume that the dye molecule emits at 520 nm. The donor dye molecules used in conventional FRET studies are in general fluorescent, i.e., emitting in the visible range of the spectrum. The acceptor is a spherical nanoparticle with bulk plasma frequency ωp ) 5.7 × 1015 s-1. For small size nanoparticles with radius a e 7 nm, the plasmon frequencies show a strong size dependence. The calculation of surface plasmon frequencies for spherical nanoparticles of different sizes considered here in this subsection shows that the dipolar mode (l ) 1) is a dominant energy accepting mode if a donor dye molecule emits in a range between 500 and 600 nm. For example, for a nanoparticle of radius 10 nm (which is the largest size nanoparticle used in the study), the l ) 1 surface plasmon has an absorption spectrum centered on 590 nm, while the l ) 2 mode is centered on 450 nm. Since, with the reduction in the size of the nanoparticle, the gap between energy levels increases, both of these frequencies shift to smaller wavelengths. This justifies that the l ) 1 mode is a predominant energy accepting mode for all sizes of MNps considered in the present study. 5.2.1. Distance Dependence of Rate of Energy Transfer. The rate of energy transfer from a donor dye molecule to a spherical nanoparticle is calculated using the full Coulombic interaction Hamiltonian HI given by eq 5.1.9.76 Figure 10 shows the rate of energy transfer (kDA) as a function of separation distance d measured from the center of the dye molecule to the surface of the nanoparticle. At large separations compared to the radius of the nanoparticle, the rate of EET is proportional to 1/d6. However, at separations approximately between d ) a and d ) 4a, the rate of energy transfer varies as 1/dσ, where σ lies between 3 and 4, which is in partial agreement with the recent experimental study by Strouse et al.41 They found that the rate of energy transfer from a dye, a FAM moiety, to a Aunanoparticle (diameter ) 1.4 nm) can be fitted to a 1/d4 distance dependence and suggested that this result may be understood in terms of surface energy transfer (SET).66 Note that the theoretical expression for SET was derived for the interaction of a dye with a surface of a metallic “half-space”. Though at intermediate distances we find 1/d4-type dependence but

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Figure 10. Distance dependence of energy transfer rate (kDA) from a dye to a spherical nanoparticle of radius (a) 3 nm. The solid line depicts the rate dependence on distance measured from the center of the dye molecule to the surface of the nanoparticle (d/a), while the dotted line shows the rate dependence on the center-to-center distance (R/a). For spherical nanoparticles, the EET rate follows 1/R6-type dependence at all separations. Here, the rate is calculated for the parallel orientation (see Figure 9).

asymptotically distance dependence is still 1/d6 type. The study of rate as a function of center-to-center distance R shows Fo¨rstertype dependence at all separations. The other studies have also reported the same R dependence.92,93 The difference between R and d dependence of rate is understandable, as the rate has a different functional form dependence on d [1/(d + a)σ] and R [1/Rσ]. This distinction between R and d dependence of rate has caused some confusion among researchers. We have used the term Fo¨rster type to describe 1/R6-type distance dependence of the rate of energy transfer. Although in the case of MNps, the underlying physical mechanism behind energy transfer is the same as in the case of FRET (note in FRET no fluorescence is involved in the energy transfer process), i.e., resonance energy transfer, it is important to note that here the energy is transferred to the plasmonic modes confined to the surface of MNp. In the later part of the section (subsection 5.3), we will see that the distance dependence of rate on R is strongly influenced by the shape and orientation of the nanoparticle in comparison to the rate measured as a function of d.78 5.2.2. Orientation Dependence of Rate of Energy Transfer. In conventional FRET where both donor and acceptor are dye molecules, the normalized rate of energy transfer (kDA/kDA[max]) depending on an angle θ between b R and the dye molecule (see Figure 9) varies from 0 to 1. If the dyes are oriented perpendicular to each other with the dipole of one of them oriented along b R, then there is no energy transfer. On the other hand, when the dyes are parallel to each other, kDA/kDA[max] is either 1 (both θ ) 0°) or 0.25 (both θ ) 90°). Because of the spherical symmetry of the nanoparticle, the orientation dependence of the energy transfer rate in the case of the dye-nanoparticle system is different from that in the two-dye system. Here, there is no orientation that forbids energy transfer. At large separations, the ratio of the largest rate of energy transfer to that of the smallest approaches 4 the same as in the case of the twodye system. Interestingly, the orientation dependence of rate becomes weaker as the separation distance between donor and acceptor decreases (see Figure 11). 5.2.3. Dependence of Energy Transfer Rate on Size of Nanoparticle. The rate of energy transfer from a dye to a nanoparticle depends upon the strength of Coulombic interactions between the two. With the increase in the size of the nanoparticle, such interactions are bound to increase and hence the energy transfer rate. The position (surface plasmon frequency) and width (inverse surface plasmon lifetime) of the absorption spectrum of the nanoparticle are also size dependent. Approximating the width of the absorption spectrum to be size

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Figure 11. Dependence of the rate of energy transfer on the orientation of the dye’s dipole moment with respect to unit vector dˆ. The acceptor is a nanoparticle of radius 1 nm.

Figure 13. Distance dependence of the rate of energy transfer between two Ag spherical nanoparticles. The solid line represents EET rate as a function of d, while the dashed line shows the R dependence of the rate. The radius of both donor and acceptor nanoparticles is taken to be 5 nm. The data is represented in a log-log plot. The inset shows the distance exponent σ as a function of d (solid line) and R (dashed line). σ is defined to be equal to -d(log kDA)/d(log d). Figure 12. Energy transfer rate plotted against the separation distance (d) between the dye and different sized nanoparticles. a represents the radius of a given nanoparticle.

independent, we have calculated the size dependence of the transfer rate as a function of separation distance, as shown in Figure 12. For large nanoparticles (a ∼7 nm), the plasmon frequencies are, to a very good approximation, independent of the size of the nanoparticle. Therefore, the size dependence of energy transfer rate is solely determined by Coulombic interactions. The asymptotic analysis76 for such particles shows that at large distance (d . a) the rate of energy transfer is proportional to a3. 5.3. Excitation Energy Transfer between Two Spherical Metal Nanoparticles. Recently, the shift in the surface plasmon frequency of a metal nanoparticle when it is placed in the proximity of another has been used to deduce the separation distance between the two nanoparticles.79-85 The separations that can be monitored are almost 10 times larger than those measured by conventional FRET.85 The plasmon shift is a consequence of coherent interactions between two nanoparticles. The plasmon frequency shift depends on the direction of polarization and the size of the nanoparticle. For a polarization direction in line with the vector joining the centers of two nanoparticles, a significant red shift in plasmon frequency has been observed. However, for a polarization direction perpendicular to the center-to-center distance vector, a small blue shift in plasmon frequency is observed. This blue shift is insignificant for small size nanoparticles. In our study, the sizes of nanoparticles used (in the present and the next subsection) are small enough for any significant blue plasmon shift to be observed. Nevertheless, since we have not taken into account the coherent interactions, our results are strictly valid only at intermediate separation distances as discussed before. In the present subsection, we consider the excitation energy transfer from a spherical Ag nanoparticle to another. The direction of polarization is perpendicular to center-to-center distance vector, b R. The surface plasmon frequencies are obtained as a solution to eq 5.1.5. The optical frequencies calculated using a free electron gas value of ε(ω) ) 1 - ωp2/ωl,m2 does not agree

with the reported values of surface plasmon frequencies of Ag nanoparticles. Therefore, the experimental data for the sizeindependent dielectric function ε(ω) is used94 where a better agreement between calculated and reported values of plasmon frequencies is obtained. The bulk plasma frequency ωp ) 14.6 × 1015 Hz is used. The width of the Lorentzian is taken to be 180 meV. All of the calculations are carried out with air as a surrounding media. The rate is calculated using the interaction Hamiltonian described by eq 5.1.9. We consider the donor nanoparticle to be in its first excited state corresponding to the l ) 1 mode. The dipolar surface plasmon mode is found to be a predominant energy accepting mode. It is advantageous to use spherical nanoparticles in EET experiments, as no orientation related issues are involved. Figure 13 shows the energy transfer rate as a function of separation distance d. Again, the deviation from 1/d6 behavior is found at short separations. However, as before, the plot of rate of energy transfer as a function of center-tocenter distance (R) obeys a 1/R6-type distance dependence at all the separations. However, in the case of RET systems involving nonspherical metal nanoparticles, it is more appropriate to look at the rate of energy transfer as a function of d instead of R. The next subsection discusses this point in detail. 5.4. Excitation Energy Transfer between Two Nonspherical Nanoparticles. In this subsection, our main objective is to study the effect of shape and orientation on the distance dependence of energy transfer rate in the case of metal nanoparticles.78 Recently, a similar dependence has been reported for excitation energy transfer between semiconductor nanoparticles.95,96 Figure 14 shows the D-A pair consisting of two prolate shaped spheroidal Ag nanoparticles. The same parameters as described in the previous subsection are used for the plasmon frequency and the rate calculations. Rl,m(c,ξ) and Sl,m(c,η) functions are numerically evaluated using Mathematica 6.0. Again, ε(ω) used in calculations is size-independent. The inclusion of size correction to the dielectric constant of the nanoparticle results in a small decrease in the EET rate, but the qualitative trend remains the same. The asphericity of the nanoparticles lifts the degeneracy associated with l modes. For the l ) 1 mode, the m ) 1 mode is found to be of larger


Figure 14. Schematic illustration of the relative orientation of two spheroidal Ag nanoparticles. The semimajor and semiminor axes are represented by a and b, respectively. f is the focal length, and ξ0 ) 1/e describes the surface of the spheroid. Here, e is the eccentricity of the particle.

Figure 15. The semilog plot shows the effect of size on the distance (d) dependence of the rate of energy transfer between two nanoparticles of aspect ratio equal to 5. The rate increases with the size of the nanoparticle. The inset shows the distance exponent σ as a function of d.

frequency than the m ) 0 mode. This is because the m ) 1 mode is more confined than the m ) 0 mode due to the geometric constraint of a > b. The l ) 2 case has similar features, as do all higher l modes. With an increase in the size of MNp, calculated values tend to the size-independent but eccentricity-dependent limiting value as set by earlier calculations which do not account for the size effects.97 The prolate nanoparticles with a large value of ξ0 are nearly spherical, while the ones with small ξ0 are needlelike. For a given value of l, the difference between the frequencies of plasmon modes corresponding to different m modes decreases with the decrease in the eccentricity (1/ξ0) of the nanoparticles. In the discussion to follow, we consider the energy transfer from the l ) 1, m ) 0 mode of a donor nanoparticle. 5.4.1. Effect of Size and Shape on Energy Transfer Rate. The rate of excitation energy transfer (EET) is calculated as a function of surface-to-surface separation distance, d, for nanoparticles of different sizes. Figure 15 shows the calculated result. The rate of EET increases with the size of the nanoparticle, but the difference in the rates of different sized nanoparticles decreases with the decrease in the separation. In contrast to spherical nanoparticles, the dependence of the rate on both the

Saini et al.

Figure 16. Log-log plot showing the dependence of EET rate on the aspect ratio of the nanoparticles for a given value of semimajor axis, a ) 5 nm.

Figure 17. Rate of EET as a function of center-to-center distance, R, for nanoparticles of different aspect ratio but with the same value of semimajor axis (a ) 5 nm). The inset shows the dependence of the distance exponent σ. The deviation of EET rate from Fo¨rster distance dependence (R-6) becomes more pronounced with the increase in the aspect ratio of the nanoparticles.

surface-to-surface separation, d, and the center-to-center separation, R, exhibits a pronounced deviation from the sixth inverse power dependence of the rate on distance, even at moderate separations. The shape dependence of rate of energy transfer is shown in Figure 16. The rate decreases with the increase in the aspect ratio of the nanoparticles. However, since the effective volume of a nanoparticle for a given value of a increases with a decrease in the aspect ratio, therefore to confirm that the increase in the rate is not merely the effect of an increase in the size, we calculated the energy transfer rates between nanoparticles at different aspect ratios but for a fixed volume of the nanoparticles. We find that the rate for a smaller aspect ratio is still higher, confirming the intrinsic role played by the aspect ratio. Figure 17 shows the rate as a function of center-to-center distance for different shapes of the nanoparticle characterized by their aspect ratios. The deviation from 1/R6 increases as the particle becomes more and more needlelike. It is clearer from the inset of Figure 17 where the value of the exponent σ is plotted. Note that, even at intermediate distances, the value of the distance exponent σ is noticeably less than 6 for spheroidal nanoparticles. Therefore, we conclude that the shape of the MNp

Feature Article has an important effect on the distance-dependent exponent σ which is crucial in the interpretation of experimental data. 5.4.2. Orientation Dependence of Energy Transfer Rate. The distance dependence of rate of energy transfer between two spheroidal nanoparticles is strongly influenced by the relative orientation of two MNps. We consider the axes of the nanoparticles to be in the same plane and obtain the transfer rate as a function of the angle, θrel, between the principal axes of the two MNps. First, we consider a case in which a line joining the centers of the nanoparticles is perpendicular to the major axis of the donor particle. The rate is maximum when the axes of the two particles are parallel (as expected) and falls to zero when the two nanoparticles are perpendicular to each other. The distance dependence of EET rate for this particular orientation is already discussed in subsection 5.4.1. As discussed in subsection 5.2.2, the orientation dependence of the rate of energy transfer depends on the separation between the two nanoparticles. It becomes weaker with the decrease in the separation distance. We find that the relative orientation of MNp has a significant effect on the R dependence of EET. 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 > 6. In comparison to R dependence, the d dependence of the energy transfer rate remains consistent with σ < 6 for all orientations of MNps. Thus, the asphericity of the nanoparticles considerably changes the R dependence of the energy transfer rate. Therefore, we suggest that, in experiments where it is difficult to control the orientation of the anisotropic particles, it may be better to study the transfer rate as a function of surfaceto-surface separation, d. 6. Conclusion A thorough understanding of distance and orientation dependence of excitation energy transfer from donor to acceptor is important considering the wide range of applications that exploit this photophysical process. The resonance energy transfer is most commonly employed to infer the separation distances between various units of macromolecules. Therefore, in principle, the polymer dynamics in dilute solutions can be understood via the EET process. However, such a study is highly complicated first because 3N degrees of freedom are involved (N is the number of monomers in polymer chain) and, second, the individual motion of monomers is greatly influenced by the interactions with other monomeric units of the polymer chain. We have described the method used to reduce this 3N degree of problem to a 1D problem using the reduced Green’s function (RGF) method. A reasonable agreement was obtained between RGF theory and the simulation results which motivated the further studies in this field. In addition, we found that the use of a preaveraged value of orientation factor in deducing the separation distance via Förster expression can lead to a wrong interpretation of the results. This is because the orientational motion of dipole moments of donor and acceptor results in a significant deviation of EET rate from the rate obtained using a preaveraged value of orientation factor. Further, the efficiency of the energy transfer rate between two parts of a polymer chain depends on the number of intervening monomeric units, N. We found that the EET rate varies as N-R, with R ≈ 2.6. It was interesting to see such a power-law dependence of EET rate on N. A similar study of distance and orientation dependence of EET rate involving extended conjugated polymers in a quenched conformation conclusively showed the breakdown of Fo¨rster

J. Phys. Chem. B, Vol. 113, No. 7, 2009 1829 theory at short distances. We also found that the optically dark states play an active role in EET which is in contrast to Fo¨rster theory where only optically active states are assumed to contribute toward excitation energy transfer. Fo¨rster theory falters here because it assumes transition charge densities of molecules are point dipoles. This point-dipole approximation results in a gross overestimation of EET rate at distances comparable to the size of molecules of the D-A system where a significant deviation from Fo¨rster dependence (1/R6) is found. We also studied the distance dependence of the rate of energy transfer as a function of shape, size, and orientation of acceptor nanoparticle with respect to the donor, which can either be a dye molecule or a nanoparticle. We find the center-to-center distance (R) dependence of rate is strongly influenced by the shape and relative orientation of donor and acceptor nanoparticles. At intermediate separations, even a slight deviation from the sphericity of the nanoparticle considerably changes the R dependence of the rate. At these separations, we find a significant deviation from Fo¨rster dependence for nonspherical nanoparticles. However, for spherical shaped nanoparticles, the energy transfer rate follows 1/R6 dependence at all separations, although the surface-to-surface separation d shows a more complex dependence. The results of our study are strictly valid only for intermediate separation distances (a e d e 10a). The study indicates that, in the case of nonspherical metal nanoparticles, it may be more appropriate to monitor the rate as a function of surface-to-surface distance d instead of center-to-center distance R, as the former is less sensitive to the shape and relative orientation of nanoparticles. We note that in the future one needs to go beyond the Fermi-Golden formulation usually used for rate calculations. It will be interesting to study the nonequilibrium effects on rate of energy transfer. As in many situations, the energy of the donor state changes with time subsequent to excitation. This is because the excitation, assumed to occur at time t ) 0, occurs so fast that the surrounding medium remains trapped in the state which was previously at equilibrium with the ground state of the donor. In addition, the donor state can also undergo vibrational energy relaxation before reaching the steady emitting state. These various relaxation processes do not affect the rate of EET if the rate itself is much slower than the processes. However, when the transfer rate is large, comparable, for example, to ps-1 or more, EET can occur from a nonequilibrium donor state. Such a situation can be particularly important when there is considerable overlap between the vibronic energies of the excited donor and acceptor states. Such nonequilibrium effects shall be manifested in a non-exponential kinetics of the EET, and can be detected by time-resolved studies of the fluorescence decay kinetics of donor or rise kinetics of fluorescence from acceptor. Most of the existing theoretical studies of EET do not include such nonequilibrium effects. This problem needs to be investigated by both theory and experiment. For EET in solution, the rate can be influenced by nonequilibrium solvation, especially if the donor excited state undergoes charge redistribution on excitation. Recently, ultrafast fluorescence studies have found an interesting wavelength dependence of excitation energy transfer in micelles and triblock copolymers.98 Further studies in this area will be useful. The coherent energy transfer between metal nanoparticles is another important problem worth studying, as plasmonic coupling between two nanoparticles at short distances compared to the size of the nanoparticles can be quite large and comparable to the energy gap involved in EET. Also, the excitation energy


Saini et al.

transfer in nanocomposites consisting of nanoparticles and conjugated polymers is of great technological interest, as these systems can easily be tuned to emit at desired wavelengths in the visible and near-infrared region. The further study of energy transfer processes in nanocomposites will pave a way toward their greater applicability in biological assays and optoelectronic devices. Acknowledgment. We thank Professors Paul Barbara, Peter Rossky, Arun Yethiraj, K. L. Sebastian, Vijay B. Shenoy, Harjinder Singh, and Mr. Somnath Bhowmick for collaboration. We also thank Professors Kankan Bhattacharyya, Graham Fleming, and Puspendu Das for discussions. This work has been supported in part by grants from DST and CSIR. S.S. acknowledges CSIR, India for a research fellowship. B.B. thanks DST for support through J.C. Bose Fellowship.

where Green’s function G(R1,R2,t) apart from the positions of reactants (R1 and R2) also depends on F(t), the normalized time b(0) · b correlation function of the end-to-end vector defined as 〈R R(t)〉/ 〈R2〉. Peq(R) is the equilibrium end-to-end distribution function. Note that Veq (in eq A.I.4) is the rate when the distribution of polymer ends is at equilibrium. After averaging over all of the angles, the sink-sink time correlation function D(t) is given by

D(t) )

( )( 3 2πL2

3

1 [1 - F2(t)]3/2

)∫

∞

(

0

4πR12S(R1) dR1 ×

3(R

2

+ R 2)

∫0∞ 4πR22S(R2) dR2 exp - 2L2(11 - F22(t)

)

sinh{[3F(t)R1R2]/(L2(1 - F2(t))]} [3F(t)R1R2]/(L2[1 - F2(t)])

Appendix I The many-body nature of polymer dynamics can be described by a joint, time-dependent probability distribution P(r bN,t), where b rN denotes the position of all N polymer beads, at time t. The time dependence of the probability distribution P(r bN,t) is described by the following reaction-diffusion equation53

∂ r N) P(b r N, t) - k(R) P(b r N, t) (A.I.1) P(b r N, t) ) LB(b ∂t

×

(A.I.6)

Once the choice of the sink function is specified, it is straightforward to calculate the survival probability by utilizing the above set of equations. WF’s choice was the Heaviside sink function. For the Fo¨rster sink function, the comparison of eq 3.1.1 in the main text with k0S(R) shows that D(t) for this particular sink function can be obtained by substituting RF6/(RF6 + R6) for S(R) in eq A.I.6. Appendix II

where LB is the full 3N-dimensional diffusion operator given by N

LB(b r N) P(b r N, t) ) D

∑ ∂r∂ j Peq(br N) ∂r∂ j Peq(1br N) P(br N, t)

In the following appendix, we will briefly discuss some of the details of the formalism described in subsection 5.1. The following set of nonrelativistic linearized electro-hydrodynamic equations describes the motion of electron gas.

j)1

(A.I.2) the subscript “eq” denotes equilibrium, R is the scalar distance between the two ends of the polymer chain, and D is the diffusion coefficient of a monomer. The last term on the righthand side of eq A.I.1 is a sink term. Wilemski-Fixman (WF) introduced a sink function of the form k0S(R) to describe the chemical reaction. Here, k0 is a constant for a particular choice of reactants and S(R) accounts for the chemical reaction. The survival probability Sp(t) is given by

Sp(t) )

∫ P(br N, t) dbr 1 dbr 2.......dbr N

(A.I.3)

In order to obtain Sp(t), WF made a closure approximation, according to which the Laplace transform of Sp(t) is approximated as

kVeq 1 Sˆp(s) ) - 2 s ˆ (s)/Veq) s (1 + kD

(A.I.4)

where k is the momentum transfer variable and s is the Laplace ˆ (s) is a Laplace transform of sink-sink transform variable. D correlation function D(t) given by

D(t) )

∫0∞ d3R1 ∫0∞ d3R2 S(R1) S(R2) G(R1, R2, t) Peq(R2) (A.I.5)

-

1 ∂b β2 b eb φ+ ∇ F j )- ∇ F0 ∂t me F0 ∇2φ )

(A.II.1)

-F ε0

b · bj ) - ∂ F ∇ ∂t Here, F, φ, and bare j the fluctuations in electron charge density, electrostatic potential, and current density, respectively. F0 is the mean electron charge density. We express the force equation (the first equation of eq A.II.1) completely in terms of electron charge density fluctuation F to obtain the following Helmholtz equation

r t) + R2Fl,m(b, r t) ) 0 ∇2Fl,m(b,

(A.II.2)

To derive eq A.II.2, we used the identities FMNp(r b,t) ) b,t) and Fl,m(r b,t) ) Fl,m(r b)e-iωl,mt. The solution of eq A.II.2 ∑l,m Fl,m(r gives the expression for fluctuation in charge density for the spherical (eq 5.1.3) and prolate spheroidal (eq 5.1.6) nanoparS NS (t) and Al,m (t), we express ticles. To determine the constants Al,m the Hamiltonian HNP of a nanoparticle given by eq A.II.3 in second quantization form.

HMNp )

∫

[

]

meβ2 2 1 me 2 j + Fφ + F db r (A.II.3) 2 e 2n ec2n0 c 0

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Here, n0 is a mean electron number density and F0 ) -ecn0. In † ) and annihilation (al,m) operators, the terms of creation (al,m Hamiltonian of eq A.II.3 becomes

ˆ MNp ) H

† al,m ∑ pωl,mal,m

(A.II.4)

l,m

In the process of quantization, we obtain the expressions for S NS Al,m (t) and Al,m (t), as given by eqs 5.1.4 and 5.1.7 in the main text. Note that in order to rewrite eq A.II.3 in the form of eq A.II.4, we also make use of a boundary condition that perpendicular to the surface of a metal nanoparticle the current density bj vanishes. Next, we solve the dispersion relations given by eqs 5.1.5 and 5.1.8 to determine the parameters R and c which for a particular |l,m〉 mode depend on the shape and size of a given nanoparticle and hence do the surface plasmon frequencies. The dispersion relation is obtained by imposing the boundary conditions that at the surface of the metal nanoparticle (i) the electrostatic potential is continuous; (ii) the normal derivative of the electrostatic potential is continuous; (iii) the normal component of current density vanishes. The third condition implies that at the surface of the nanoparticle we have

meβ2 dF dφ ) dr ecF0 dr

(A.II.5)

The first two boundary conditions are used to obtain the expression for fluctuation in electrostatic potential φ inside and outside the metal nanoparticle, while the third boundary condition leads to the dispersion relation. References and Notes (1) Lakowicz, J. R. Principles of Fluorescence Spectroscopy; Plenum: New York, 1983. (2) Perrin, J. C. R. Acad. Sci. (Paris) 1927, 184, 1097. (3) Fo¨rster, Th. Ann. Phys. (Leipzig) 1948, 2, 55. (4) Fo¨rster, Th. Delocalization and excitation transfer. In Modern Quantum Chemistry, Istanbul Lectures, Part III: Action of Light and Organic Crystals; Sinanoglu, O., Ed.; Academic Press: New York, 1995. (5) Dexter, D. L. J. Chem. Phys. 1953, 21, 836–850. (6) Sipior, J.; Bambot, S.; Romauld, M.; Carter, G. M.; Lakowicz, J. R.; Rao, G. Anal. Biochem. 1995, 227, 309–318. (7) Lakowicz, J. R.; Maliwal, B. Anal. Chim. Acta 1993, 271, 155– 164. (8) Rolinski, O. J.; Birch, D. J. S.; McCartney, L. J.; Pickup, J. C. Phys. Med. Biol. 2001, 46, N221–N226, and references therein. (9) Birch, D. J. S.; Holmes, A. S.; Darbyshire, M. Meas. Sci. Technol. 1995, 6, 243–247. (10) Rolinski, O. J.; Birch, D. J. S.; Holmes, A. S. J. Biomed. Opt. 1998, 3, 346–356. (11) Weiss, S. Science 1999, 283, 1676–1683. (12) Deniz, A. A.; Dahan, M.; Grunwell, J. R.; Ha, T.; Faulhaber, A. E.; Chemla, D. S.; Weiss, S.; Schultz, P. G. Proc. Natl. Acad. Sci. U.S.A. 1999, 96, 3670–3675. (13) Deniz, A. A.; Laurence, T. A.; Beligere, G. S.; Dahan, M.; Martin, A. B.; Chemla, D. S.; Dawson, P. E.; Schultz, P. G.; Weiss, S. Proc. Natl. Acad. Sci. U.S.A. 2000, 97, 5179–5184. (14) Srinivas, G.; Bagchi, B. Phys. Chem. Commun. 2002, 5, 59–62. (15) Zhuang, X.; Kim, H.; Pereira, M. J. B.; Babcock, H. P.; Walter, N. G.; Chu, S. Science 2002, 296, 1473–1476. (16) Zhuang, X.; Bartley, L. E.; Babcock, H. P.; Russell, R.; Ha, T; Herschlag, D.; Chu, S. Science 2000, 288, 2048–2051. (17) Laurence, T. A.; Kong, X.; Ja¨ger, M.; Weiss, S. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 17348–17353. (18) Lipman, E. A.; Schuler, B.; Bakajin, O.; Eaton, W. A. Science 2003, 301, 1233–1235. (19) Leite, V. B. P.; Onuchic, J. N.; Stell, G.; Wang, J. Biophys. J. 2004, 87, 3633–3641.

(20) Schuler, B.; Lipman, E. A.; Eaton, W. A. Nature 2002, 419, 743– 747. (21) Srinivas, G.; Bagchi, B. J. Chem. Phys. 2002, 116, 8579–8587. (22) Srinivas, G.; Bagchi, B. J. Phys. Chem. B 2001, 105, 9370–9374. (23) Schuler, B.; Lipman, E. A.; Steinbach, P. J.; Kumke, M.; Eaton, W. A. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 2754–2759. (24) Schwartz, B. J. Annu. ReV. Phys. Chem. 2003, 54, 141–172. (25) Kersting, R.; Mollay, B.; Rusch, M.; Wenisch, J.; Leising, G.; Kauffmann, H. F. J. Chem. Phys. 1997, 106, 2850–2864. (26) Hayes, G. R.; Samuel, I. D. W.; Phillips, R. T. Phys. ReV. B 1995, 52, R11569–R11572. (27) Hu, D. H.; Yu, J.; Barbara, P. F. J. Am. Chem. Soc. 1999, 121, 6936–6937. (28) Yu, J.; Hu, D. H.; Barbara, P. F. Science 2000, 289, 1327–1330. (29) Beljonne, D.; Pourtois, G.; Silva, C.; Hennebicq, E.; Herz, L. M.; Friend, R. H.; Scholes, G. D.; Setayesh, S.; Mu¨llen, K.; Bre´das, J. L. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 10982–10987. (30) Nguyen, T-Q.; Wu, J.; Doan, V.; Schwartz, B. J.; Tolbert, S. H. Science 2000, 288, 652–656. (31) Nelson, N.; Yocum, C. F. Annu. ReV. Plant Biol. 2006, 57, 521– 565. (32) Scholes, G. D. Annu. ReV. Phys. Chem. 2003, 54, 57–87. (33) Colbow, K.; Danyluk, R. P. Biochim. Biophys. Acta 1976, 440, 107–121. (34) Causgrove, T. P.; Brune, D. C.; Blankenship, R. E. J. Photochem. Photobiol., B 1992, 15, 171–179. (35) Sumi, H. J. Phys. Chem. B 1999, 103, 252–260. (36) Krueger, B. P.; Scholes, G. D.; Fleming, G. R. J. Phys. Chem. B 1998, 102, 5378–5386. (37) Scholes, G. D.; Fleming, G. R. J. Phys. Chem. B 2000, 104, 1854– 1868. (38) Scholes, G. D.; Jordanides, X. J.; Fleming, G. R. J. Phys. Chem. B 2001, 105, 1640–1651. (39) Jordanides, X. J.; Scholes, G. D.; Fleming, G. R. J. Phys. Chem. B 2001, 105, 1652–1669. (40) Hsu, C. P.; Walla, P. J.; Head-Gordon, M.; Fleming, G. R. J. Phys. Chem. B 2001, 105, 11016–11025. (41) Yun, C. S.; Javier, A.; Jennings, T.; Fisher, M.; Hira, S.; Peterson, S.; Hopkins, B.; Reich, N. O.; Strouse, G. F. J. Am. Chem. Soc. 2005, 127, 3115–3119. (42) Harrison, B. S.; Foley, T. J.; Bouguettaya, M.; Boncella, J. M.; Reynolds, J. R.; Schanze, K, S.; Shim, J.; Holloway, P. H.; Padmanaban, G.; Ramakrishnan, S. Appl. Phys. Lett. 2001, 79, 3770–3772. (43) Chowdhury, P. S.; Saha, S.; Patra, A. Chem. Phys. Lett. 2005, 405, 393–397. (44) Bettencourt-Dias, A. Dalton Trans. 2007, 2229, 2241. (45) Maier, S. A.; Kik, P. G.; Atwater, H. A.; Meltzer, S.; Harel, E.; Koel, B. E.; Requicha, A. A. G. Nat. Mater. 2003, 2, 229–232. (46) Cantor, C. R.; Pechukas, P. Proc. Natl. Acad. Sci. U.S.A. 1971, 68, 2099. (47) Haas, E; Wilchek, M.; Katchalaski-Katzir, E.; Stenberg, I. Z. Proc. Natl. Acad. Sci. U.S.A. 1975, 72, 1807. (48) Ha, T.; Ting, A. Y.; Liang, J.; Caldwell, W. B.; Deniz, A. A.; Chemla, D. S.; Schultz, P. G.; Weiss, S. Proc. Natl. Acad. Sci. U.S.A. 1999, 96, 893. (49) Latt, S. A.; Cheung, H. T.; Blout, E. R. J. Am. Chem. Soc. 1965, 87, 995. (50) Wang, J.; Wolynes, P. G. J. Chem. Phys. 1999, 110, 4812. (51) Doi, M. Chem. Phys. 1975, 11, 107. (52) Wilemski, G.; Fixman, M. J. Chem. Phys. 1974, 60, 866. (53) Wliemski, G.; Fixman, M. J. Chem. Phys. 1974, 60, 878. (54) Portman, J. J.; Wolynes, P. G. J. Phys. Chem A 1999, 103, 10602– 10610. (55) Srinivas, G.; Yethiraj, A.; Bagchi, B. J. Chem. Phys. 2001, 114, 9170. (56) Pastor, R. W.; Zwanzig, R.; Szabo, A. J. Chem. Phys. 1996, 105, 3878. (57) Doi, M. Chem. Phys. 1975, 9, 455. (58) Srinivas, G.; Sebastian, K. L.; Bagchi, B. J. Chem. Phys. 2002, 116, 7276. (59) Barzykin, A. V.; Seki, K.; Tachiya, M. J. Chem. Phys. 2002, 117, 1377. (60) Bandyopadhyay, T.; Ghosh, S. K. J. Chem. Phys. 2002, 116, 4366. (61) Westlund, P.-O.; Wennerstrom, H. J. Chem. Phys. 1993, 99, 6583. (62) Szabo, A. J. Phys. Chem. 1989, 93, 6929; J. Chem. Phys. 1984, 81, 150. (63) Wong, K. F.; Bagchi, B.; Rossky, P. J. J. Phys. Chem. A 2004, 108, 5752–5763. (64) Cerullo, G.; Stagira, S.; Zavelani-Rossi, M.; De Silvestri, S.; Virgili, T.; Lidzey, D. G.; Bradley, D. D. C. Chem. Phys. Lett. 2001, 335, 27–33. (65) Chance, R. R.; Prock, A.; Silbey, R. AdV. Chem. Phys. 1978, 37, 1–65. (66) Persson, B.; Lang, N. Phys. ReV. B 1982, 26, 5409–5415.

1832 J. Phys. Chem. B, Vol. 113, No. 7, 2009 (67) Weber, W. H.; Eagen, C. F. Opt. Lett. 1979, 4, 236–238. (68) Alivisatos, A. P.; Waldeck, D. H.; Harris, C. B. J. Chem. Phys. 1985, 82, 541–547. (69) Gersten, J.; Nitzan, A. J. Chem. Phys. 1981, 75, 1139–1152. (70) Ruppin, R. J. Chem. Phys. 1982, 76, 1681–1684. (71) Dulkeith, E.; Morteani, A. C.; Niedereichholz, T.; Feldmann, J. Phys. ReV. Lett. 2002, 89, 203002 (1-4). (72) Sen, T.; Sadhu, S.; Patra, A. Appl. Phys. Lett. 2007, 91, 043104. (73) Clapp, A. R.; Medintz, I. L.; Mauro, J. M.; Fisher, B. R.; Bawendi, M. G.; Mattoussi, H. J. Am. Chem. Soc. 2004, 126, 301–310, and references therein. (74) Alphandery, E.; Walsh, L. M.; Rakovich, Y.; Bradley, A. L.; Donegan, J. F.; Gaponik, N. Chem. Phys. Lett. 2004, 388, 100–104. (75) Chowdhury, P. S.; Sen, P.; Patra, A. Chem. Phys. Lett. 2005, 413, 311–314. (76) Bhowmick, S.; Saini, S.; Shenoy, V. B.; Bagchi, B. J. Chem. Phys. 2006, 125, 181102 (1-6). (77) Saini, S.; Bhowmick, S.; Shenoy, V. B.; Bagchi, B. J. Photochem. Photobiol., A 2007, 190, 335–341. (78) Saini, S.; Shenoy, V. B.; Bagchi, B. J. Phys. Chem. C 2008, 112, 6299–6306. (79) Jain, P. K.; Huang, W.; El-Sayed, M. A. Nano Lett. 2007, 7, 2080– 2088. (80) Nordlander, P.; Oubre, C.; Prodan, E.; Li, K.; Stockman, M. I. Nano Lett. 2004, 4, 899–903. (81) Tamaru, H.; Kuwata, H.; Miyazaki, H. T.; Miyano, K. Appl. Phys. Lett. 2002, 80, 1826–1828. (82) Rechberger, W.; Hohenau, A.; Leitner, A.; Krenn, J. R.; Lamprecht, B.; Aussenegg, F. R. Opt. Commun. 2003, 220, 137–141.

Saini et al. (83) Su, K. H.; Wei, Q. H.; Zhang, X.; Mock, J. J.; Smith, D. R.; Schultz, S. Nano Lett. 2003, 3, 1087–1090. (84) Gunnarsson, L.; Rindzevicius, T.; Prikulis, J.; Kasemo, B.; Ka¨1l, M.; Zou, S.; Schatz, G. C. J. Phys. Chem. B 2005, 109, 1079–1087. (85) So¨nnichsen, C.; Reinhard, B. M.; Liphardt, J.; Alivisatos, A. P. Nat. Biotechnol. 2005, 23, 741–745. (86) Andrews, D. L.; Juzeliunas, G. J. Chem. Phys. 1992, 96, 6606– 6612. (87) Kottmann, J. P.; Martin, O. J. F. Opt. Lett. 2001, 26, 1096–1098. (88) (a) Ritchie, R. H.; Wilems, R. E. Phys. ReV. 1969, 178, 372–381. (b) Barberan, N.; Bausells, J. Phys. ReV. B 1985, 31, 6354–6359. (89) Ritchie, R. H. Surf. Sci. 1973, 34, 1–19. (90) Barton, G. Rep. Prog. Phys. 1979, 42, 963–1016. (91) Startton, A. Spheroidal waVe function; Chapman and Hall Co.: London, 1956. (92) Carminati, R.; Greffet, J.-J.; Henkel, C.; Vigoureux, J. M. Opt. Commun. 2006, 261, 368–375. (93) Swathi, R. S.; Sebastian, K. L. J. Chem. Phys. 2007, 126, 234701 (1-5). (94) Lynch, D. W.; Hunter, W. R. In Handbook of optical constants of solids; Palik, E. D., Ed.; Academic Press: London, 1985. (95) Curutchet, C.; Franceschetti, A.; Zunger, A.; Scholes, G. D. J. Phys. Chem. C 2008, 112, 13336–13341. (96) Schrier, J.; Wang, L. W. J. Phys. Chem. C 2008, 112, 11158–11161. (97) Moussiaux, A.; Ronveaux, A.; Lucas, A. Can. J. Phys. 1977, 55, 1423–1433. (98) Ghosh, S.; Dey, S.; Adhikari, A.; Mandal, U.; Bhattacharyya, K. J. Phys. Chem. B 2007, 111, 7085–7091.

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Distance and Orientation Dependence of Excitation Energy Transfer

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