Distance-Dependence of Interparticle Energy Transfer in the Near

Distance-Dependence of Interparticle Energy Transfer in the Near-Infrared within Electrostatic Assemblies of PbS Quantum Dots Mohamad S. Kodaimati, Chen Wang, Craig Chapman, George C. Schatz,* and Emily A. Weiss* Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, United States S Supporting Information *

ABSTRACT: This paper describes control of the rate constant for nearinfrared excitonic energy transfer (EnT) within soluble aqueous assemblies of PbS quantum dots, cross-linked by Zn2+, by changing the length of the mercapto-alkanoic acid (MAA) that serves as the cross-linking ligand. Sequestration of Zn2+ by a chelating agent or zinc hydroxide species results in deaggregation of the assemblies with EnT turned “off”. Upon decreasing the number of methylene groups in MAAs from 16 to 3, the interparticle separation decreases from 5.8 nm to 3.7 nm and the average observed EnT rate increases from ∼(150 ns)−1 to ∼(2 ns)−1. A master equation translates intrinsic (single-donor−single-acceptor) EnT rate constants predicted for each ligand length using Förster theory to observed average rate constants. For interparticle distances greater than ∼4 nm, the point dipole approximation (PDA) implementation of Förster theory agrees with experimentally measured rates. At shorter interparticle distances, the PDA drastically underestimates the observed EnT rate. The prediction of the rates of these short-distance EnT processes is improved by ∼20% by replacing the PDA with a transition density cube calculation of the interparticle Coulombic coupling. KEYWORDS: near-infrared, Förster theory, energy transfer, quantum dot assemblies, cross-linking, transition density cube method

E

enough to compete with the excitonic lifetime of the QD; this requirement is usually easily met, even though the long radiative lifetimes of QDs make them inherently slow energy donors,10 because excitons in PbS QDs live for >1 μs. A more stringent requirement is introduced by the prospect of using multiple exciton generation (MEG) to extract multiple electrons or holes from a single high-energy photon, through fission of the high-energy exciton into two (or more) lowenergy excitons.11−23 This strategy breaks the Shockley− Queisser limit for the efficiency of a single-junction solar cell.24 PbS QDs are particularly adept at the MEG process, but the challenge for exploiting MEG is that lifetimes of resultant multiexcitonic states in PbS QDs are typically 20−120 ps due to the Auger recombination pathway and subsequent carrier cooling.25 At least one exciton must therefore be transferred out of the QD by EnT before the Auger process can occur in order to use both excitons as sources of mobile charge carriers.26 Spatial separation of the biexcitonic state onto multiple Si QDs through EnT has recently been demonstrated

xcitonic energy transfer (EnT) is the mechanism by which solar energy is harvested and transported to an interface for exciton dissociation within natural and artificial photovoltaic and photocatalytic systems,1−6 and the rate of energy transfer can serve as a “spectroscopic ruler” to measure the spatial relationships of fluorophore-tagged species in biological imaging. Both of these applications benefit from dyes that absorb and emit in the so-called “near-IR II” window (1−1.4 μm), an underutilized portion of the solar spectrum for organic and quantum dot (QD)-based solar harvesting, and the region of maximum tissue penetration depth when considering both absorption and scattering.7,8 PbS QDs, although not the ideal material for commerical technologies due to heavy metal toxicity, are excellent model systems to study the parameters that control the rate and yield of near-IR Förster resonance EnT (FRET), because, unlike organic near-IR chromophores, they have photoluminescence quantum yields up to 60%,9 a broad, tunable absorption and emission range (850−1800 nm), and easily manipulated surface chemistry that not only controls the dynamics of their excitonic states but also allows us to controllably assemble them in solution or films to study interparticle EnT processes. For energy harvesting applications, the key figure of merit is the rate of interparticle EnT. At a minimum, it must be fast © 2017 American Chemical Society

Received: March 14, 2017 Accepted: April 11, 2017 Published: April 11, 2017 5041

DOI: 10.1021/acsnano.7b01778 ACS Nano 2017, 11, 5041−5050

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ACS Nano in sputtered solids27,28 and in CdSe nanoplatelet solids,29 but not in zero-dimensional QDs. We therefore must develop strategies for fast extraction of near-IR excitons from QDs. For applications in FRET-based measurements of biomolecular structure, the important parameter is precise knowledge of the distance-dependence of the rate and yield of FRET, and what level of electrostatic theory is needed to model it. This dependence is well-defined for most molecular chromophore− fluorophore systems30 and has been studied in visible lightemitting QDs,31,32 but is not known for near-IR-emitting QDs such as PbS.33,34 Here we describe the dependence of the rate of nonradiative transfer of near-infrared excitons between PbS quantum dots of two different sizes on the distance between the QDs and identify the interparticle distance at which the point-dipole approximation (PDA) implementation of Förster theory fails to predict this rate. The colloidal QDs, we study with timeresolved optical spectroscopy and model computationally are assembled into water-soluble aggregates through a reversible Zn2+ cation-mediated electrostatic coupling of their ligands, mercaptoalkanoic acids (MAAs, S−-(CH2)(n−1)-COO−, n = 3− 16). This coupling chemistry is similar to the Zn2+-mediated coupling of glutathione-coated QDs we recently reported,35 but uses a length-tunable ligand. With decreasing “n”, the average interparticle (center-to-center) distance within the assemblies decreases from 5.8 nm to 3.7 nm, and the average time constant for EnT from donor (smaller) QDs to acceptor (larger) QDs increases from (∼150 ns)−1 to (2 ns)−1. Our measured rate constant of (2 ns)−1 is not yet fast enough to outcompete the Auger process, but it is the fastest measured EnT rate between PbS QDs in film or solution.36−44 While it has been assumed in the QD literature that the PDA is an adequate method to describe interparticle EnT in the near-IR,44−46 we demonstrate that the PDA only sufficiently describes the EnT rate between PbS QDs for interparticle distances greater than 4 nm and drastically underestimates this rate for interparticle distances less than 4 nm (approximately the diameter of our chromophore). We significantly improve the accuracy of our calculations by replacing the PDA with a transition density cube (TDC) method for calculating Coulombic coupling between transition densities, which better accounts for the delocalization of the excited-state wave functions of the QDs throughout the volume of the particle.

separation of the organic and aqueous layers, collected the aqueous layer, and washed it with 5 mL of chloroform to eliminate displaced oleate species. The aqueous layer was then filtered (0.22 μm pore size) to remove QD aggregates formed during the exchange. 1H NMR spectra of the aqueous QD dispersions show that all of the oleate ligands that were initially bound to the QDs are displaced upon addition of 700 equiv of thiols; see the SI, Figure S2. All samples were prepared and stored under an N2 atmosphere to limit oxidation of the thiolate ligands (except during the pH reversibility study; see below). Figure 1A shows the ground-state absorption and photoluminescence (PL) spectra of the DQDs and AQDs after ligand exchange with HS-(CH2)n−1-COOH, n = 12 (“12-MAA”). These spectra are representative of those of QDs capped with all of the MAAs (except n = 3); see the SI, Figure S3. The first excitonic peaks of both DQDs and AQDs shifted to lower energy by ∼40 nm after ligand exchange, but good spectral overlap between the DQD emission and the AQD absorption remained. The 3-MAA QDs exhibited smaller shifts of the first excitonic peak and increased sub-band-gap (“trap”) emission upon ligand exchange, which slightly worsens the spectral overlap between DQDs and AQDs. We coupled the DQDs and AQDs through the addition of divalent Zn2+, Figure 1B. We chose Zn2+ as the cross-linking agent because Zn2+ perturbs the PL quantum yield (QY) of individual QDs less than other tested divalent metal cations (Ca2+, Mn2+, Ni2+, Cd2+, Pb2+); see the SI, Table S1. We formed soluble PbS QD assemblies by adding Zn2+ to solutions composed of 0.67 μM DQDs and 0.33 μM AQDs (hereafter referred to as “D:A QD samples”). We used a 2:1 molar ratio of DQDs:AQDs, so that, at our excitation wavelength (800 nm), we excite 50% DQDs and 50% AQDs, which is as selective for DQDs as we can achieve with reasonable DQD:AQD ratios given the relatively higher extinction coefficient of the AQDs; see the SI. Through dynamic light scattering (DLS) measurements, we observed that the average hydrodynamic diameter of particles in the solution changes from 6.2 ± 0.9 nm for uncoupled QDs to 74 ± 15 nm upon the addition of 700 equiv of Zn2+; see the SI, Figure S4. We also observed, through transmission electron microscopy (TEM), that >85% of the QDs deposit in aggregates upon coupling. A comparison of TEM images of coupled and uncoupled samples (see the SI, Figures S5 and S6) shows that the QD assemblies are not artifacts of the deposition process; furthermore, we have previously published cryogenic TEM evidence that the assemblies we observe form in solution, not on the grid during drying.35 Optical Signatures of Interparticle Energy Transfer. Rayleigh scattering by the QD assemblies resulted in an increase in the baselines of the absorbance spectra of the particles after addition of Zn2+, but the addition of Zn2+ did not shift the position of the first exciton peaks, Figure 2A; this result indicates that the excitons in the QD assemblies are not strongly electronically coupled to each other. Figure 2B shows PL spectra of the D:A QD sample, with QDs coated with 12MAA, before and after the addition of 700 equiv of Zn2+. The black lines correspond to the D:A mixture, and the blue (red) lines correspond to the deconvolved contributions of the DQDs (AQDs) to the total spectrum, obtained by fitting the spectrum of the mixture with two Gaussian functions with peaks fixed at the positions of the individual DQD and AQD spectra. The signature of EnT, observed here and upon

RESULTS AND DISCUSSION Spontaneous Electrostatic Assembly of PbS QDs. We synthesized two sizes of PbS QDs, with radii of 1.6 nm (“DQDs”) and 1.7 nm (“AQDs”), using an adapted procedure47 described in the Experimental Methods. Figure S1 in the Supporting Information (SI) shows the ground-state absorption spectra of the as-synthesized oleate-capped QDs suspended in chloroform. We prepared water-soluble PbS QDs capped with mercapto-alkanoic acids, of the form S−-(CH2)n−1COO− (n = 3, 4, 6, 8, 11, 12, 16), through a phase-transfer ligand exchange using a method adapted from Deng et al.48 To perform the ligand exchange, we mixed 7 mL of 5 μM oleatecapped PbS QDs dispersed in CHCl3 with 700 equiv of thiol in 5 mL of water (pH 7.2) and shook the mixture rigorously for 1 min until the QDs flocculated. We adjusted the pH of the aqueous layer to 8.5 by adding NaOH to deprotonate the carboxylate groups of the thiol ligands (pKa ≈ 4.8)49,50 and make the QDs negatively charged and water-soluble. We centrifuged the mixture at 7500 rpm for 5 min to facilitate the 5042


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Figure 2. (A) Ground-state absorption spectra of 12-MAA-capped D:A sample before (solid) and after (dashed) the addition of 700 equiv of Zn2+ at pH 8.5. (B) PL spectra of 12-MAA-capped D:A sample before (solid) and after (dashed) the addition of 600 equiv of Zn2+ at pH 8.5. The DQD and AQD PL are deconvoluted from the D:A mixture. The PL spectra were acquired with an excitation wavelength of 800 nm.

coupling of the QDs capped with all of the MAA ligands, is an overall bathochromic shift in the spectrum (solid black to dashed black) due to simultaneous increase in intensity of the AQDs’ PL (solid red to dashed red) and decrease in intensity of the DQDs’ PL (solid blue to dashed blue). This effective shift continues with increasing concentrations of Zn2+ until saturating at 900 equiv of Zn2+ per QD (DQD + AQD); see the SI, Figure S7. Further addition of Zn2+ results in precipitation of the QDs. For all MAA ligands, the observed yield of EnT was ≥70%; see the SI. By decreasing (increasing) the pH of the solution, we activate (inhibit) the formation of QD assemblies, and, consequently, control the yield of EnT. For example, increasing the pH of a solution of cross-linked assemblies from 8.5 (at which it is prepared) to 12 causes the PL spectrum to revert to its original shape (prior to addition of Zn2+), Figure 3A, green to red. The disaggregation at pH 12 and higher is due to sequestration of the Zn2+ in Znx(OH)y(2x−y) complexes.51 If we then decrease the pH of a solution of cross-linked assemblies

Figure 1. (A) Normalized steady-state absorption (solid lines) and PL (dashed lines) spectra of DQDs (black) and AQDs (red) after ligand exchange with HS-(CH2)n−1-COOH, n = 12, in H2O. The first excitonic peaks are centered at 970 and 1090 nm for the DQDs and AQDs, respectively, and the emission peaks are centered at 1082 and 1215 nm, respectively. The excitation wavelength used to acquire the PL spectra is 800 nm. (B) Schematic diagram of watersoluble PbS QDs capped with MAAs coupled through coordination of the carboxylate groups of the ligands with Zn2+ at pH 8.5. (C) TEM image of assemblies of 12-MAA-capped QDs, formed by adding 700 equiv of Zn2+ per QD to a mixture of QDs with DQD:AQD = 2:1. Some measured interparticle distances are marked. Inset: Histogram of 240 center-to-center distances measured from 20 separate TEM images for the 12-MAA-capped QDs coupled with 700 equiv of Zn2+. The average center-to-center distance between adjacent QDs is 5.1 ± 0.3 nm, based on a Gaussian fit of this histogram. The SI contains images and histograms for assemblies of QDs with each capping ligand. 5043


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Distance-Dependence of the EnT Rate. We determined the average rate constants for EnT from DQDs to AQDs within the assemblies by measuring the formation of AQD excitons those not created by direct photoexcitation of AQDsusing time-correlated single-photon counting (TCSPC, a dynamic range of 2.5 ns to 100’s of μs) or transient absorption spectroscopy (TA, a dynamic range of 250 fs to 3 ns). The rate constants we obtained with these measurements are, in general, a convolution of several EnT hopping steps within the assemblies rather than a single, nearest-neighbor DQD-toAQD EnT process; we translate between the two in our modeling of this data, as described in detail in the next section. Figure 4A shows the time-dependence of the AQD PL signal, monitored at wavelengths longer than 1200 nm, from the D:A QD sample (n = 12), with and without addition of 700 equiv of Zn2+, at pH ∼8.5. Residual DQD PL in the monitored region accounts for less than 3% of the measured signal. Before coupling with Zn2+, all of the observed PL from the AQDs is present at the first measurable time point of the experiment an indication that it originates from directly excited AQDs and decays (primarily through radiative recombination) with a lifetime of 1.2 μs. Upon addition of Zn2+, we observed an additional growth in the PL of the AQDs due to EnT from DQDs, with an observed EnT lifetime of 79 ns. The PL of the QDs then decays, through radiative recombination and Zn2+induced nonradiative processes,35 with a lifetime of ∼900 ns. Figure 4C shows the average EnT lifetimes for assemblies linked with all of the MAA ligands (n = 3−16), with DQD:AQD = 2:1, pH = 8.5, and coupled with 700 equiv of Zn2+. This series of ligands allows us to achieve a range of center-to-center interparticle distances within the assemblies of ∼3.7 to ∼5.8 nm, as measured by HR-TEM; see the SI, Figure S6. In calculating the average interparticle distance, we only considered the distances between QDs in large aggregates (>5 QDs) to distinguish solution-assembled QDs from unassembled QDs that co-deposit during the drying procedure. We did not attempt, in determining average interparticle distance, to distinguish between DQDs and AQDs in the TEM images, but rather assumed that, since their radii differ by only 0.1 nm, they would distribute homogeneously throughout the aggregates, such that the average overall interparticle distance equals the average DQD−AQD distance. All lifetimes in this plot were measured using the TCSPC technique, except for the n = 3 sample, which was measured with TA, Figure 4B, because its EnT lifetime was faster than the instrument response function of the TCSPC. In TA, we obtained the EnT lifetime by monitoring the growth of the ground-state bleach (GSB) of the AQDs and decay of the GSB of the DQDs upon excitation of the sample. The SI contains all of the kinetic traces and fits used to obtain these lifetimes, Figures S10−S18. The average observed EnT rate constant, kObs, increases from (152 ± 2)−1 ns−1 to (2.1 ± 0.3)−1 ns−1 upon decreasing the average interparticle separation from 5.8 ± 0.3 nm (n = 16) to 3.7 ± 0.3 nm (n = 3), Figure 4C. Our average time constant of 2.1 ± 0.3 ns for EnT between DQDs and AQDs nominally linked by S−-(CH2)2−COO−-Zn2+-−OOC-(CH2)2-S− is the fastest measured for EnT between PbS QDs in a film or solution. There is one example of a sub-100 ps observed time constant for this process in a film of PbS QDs and Au NPs, but this measurement has a high degree of uncertainty, as it was derived from an exciton diffusion length rather than measured directly, and neglects the rate of charge transfer from the PbS QDs to the Au NPs.54

Figure 3. (A) PL spectra of a DQD/AQD mixture, where the QDs were capped with 12-MAA, at different pH values, before (black) and after (red, green, blue) addition of 800 equiv of Zn2+ at pH 8. The blue curve was obtained for the same sample by readjusting the solution pH to 5 after the addition of 800 equiv of Zn2+ at pH 8. The red curve was obtained for the same sample by readjusting the solution pH to 12 after the addition of 800 equiv of Zn2+ at pH 8. (B) Plot illustrating the ability to cycle EnT “on” and “off” by changing the pH of a 12-MAA-capped D:A QD sample coupled with 700 equiv of Zn2+ at pH 8.2. At high pH values, we observe less EnT (the emission peak lies at shorter wavelengths). At low pH values, we observe increased EnT (the emission peak lies at longer wavelengths).

below 12, and as low as 5, we release Zn2+ and the QDs reaggregate, Figure 3A, red to blue. The PL intensity also reversibly fluctuates during pH cycling, as the PL QY of individual QDs is pH-dependent52,53 due mainly to (1) a change in the binding strength of the thiolate ligands and (2) quenching of QD PL through hole transfer to hydroxide ions. We observed some hysteresis in the plot of titration step vs pH (Figure 3B) due to irreversible aggregation of the QDs, because we carried out the titrations in ambient conditions (in order to measure the pH). We suspect this aggregation, which is accompanied by a small, irreversible decrease in the overall PL QY of the samples as the titration progresses (see Figure S9 of the SI), is due to oxidation of thiolate ligands to form the corresponding disulfides, which lowers the solubility of QDs. Introducing ethylenediaminetetraacetic acid (EDTA), which chelates Zn2+, reverses the aggregation of the QDs and turns off EnT; see the SI, Figure S8. 5044


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We observe that the amount of Zn2+ added to the mixture samples does not change the observed rate of EnT, but rather changes the fractional contribution of the EnT component in the overall dynamics of the AQD exciton (Figure S10 and Table S2 in the SI). We therefore conclude that the amount of added Zn2+ changes only the size of the QD assemblies, not the average interparticle distance within the assemblies. Comparison of EnT Rates to Those Predicted by Two Implementations of Förster Theory. The intrinsic EnT rate constant predicted by the Förster equation is only valid for EnT between a single donor and a single acceptor; therefore, in order to compare our experimentally observed rate constants to those predicted by Förster theory, we need a method for converting between intrinsic single-donor−single-acceptor EnT rates and the measured ensemble rate constants, kObs. We accomplish this conversion using a rate-equation (RE) model governed by eq 1, similar to our previous work on EnT in QD assemblies.35,37 ⎧ ⎪ d 1 Pi(t ) = Pi(t )⎨− − ⎪ dt τ ⎩ i

⎫ ⎪

∑ ki ,j⎬⎪ + ∑ kj ,iPj(t ) j≠i

⎭

j≠i

(1)

We create excitons in random locations within assemblies of 216 ± 40 QDs (on average, 1% of the QDs within the aggregate are photoexcited) and then use the master equation, eq 1, to propagate the exciton within the assembly for its lifetime, which may include multiple hops. In eq 1, Pi(t) and Pj(t) are the photoexcited populations of each QD i or j (DQD or AQD) in the assembly, τi is the measured time constant for relaxation of the AQD excited state without EnT, and ki,j are the rate constants for EnT between adjacent QDs i and j.42,43 We allow for incoherent EnT (with zero intermolecular wave function overlap) between a DQD and another DQD, an AQD and another AQD, and a DQD and an AQD, with rate constants given by the Förster equation (eq 2). 2π kEnT = |J |2 +EnT (2) ℏ DA We assume just one population of DQDs and one population of AQDs exist in the mixture; this assumption is justified since there are no changes in the PL spectrum or kinetics of a DQDonly or AQD-only sample upon cross-linking with Zn2+ (Figures S19−22 of the SI) and therefore no contribution of DQD-to-DQD or AQD-to-AQD EnT to the observed changes in the PL spectra or kinetics of the mixtures upon cross-linking. The two key parameters in eq 2 are +EnT , the combined density of states for EnT, and Jml, the electronic coupling between chromophores D and A, primarily related to Coulombic interactions between the “transition densities” centered on each system, ρD,t(r) and ρA,t(r′). The transition density for a molecular system is defined as the overlap of the ground state, ψi,g, and excited state, ψi,e, wave functions, integrated over the electronic spin, eqs 3a and 3b.55

Figure 4. (A) Normalized kinetic traces of the PL of AQDs (monitored at >1200 nm) in the 12-MAA D:A QD sample without Zn2+ (black) and with 700 equiv of Zn2+ (blue). The excitation wavelength was 800 nm. The red lines are the exponential fits convoluted with the instrument response function. (B) Kinetic traces, extracted at 900 nm (blue, 89% DQDs) and 1030 nm (red, 93% AQDs), from the TA spectrum of the n = 3 D:A QD sample with 700 equiv of Zn2+, after pumping at 790 nm; see the SI, Figure S16. The decay in the DQD GSB and growth in AQD GSB bleach, both with τ = 2.1 ns, do not exist in uncoupled samples (SI, Figure S16) and are due to EnT. The excitation wavelength was 790 nm. (C) Plot of the observed average EnT rate vs the center-to-center separation, R, determined through HR-TEM. We show the average EnT rate (points) and range (vertical error bars) over three D:A samples for each ligand using different batches of ligand-exchanged QDs. Horizontal error bars are calculated from 100 measurements for each ligand. The black trace is a prediction of these rates using the PDA (eq 3a). The blue trace is a prediction using the TDC method.

JDA =

∬

ρi ,t (r ) =

ρ D,t (r ) ρ A,t (r′) dr dr ′ |r − r′|

∫s ψi ,g(r) ψi*,e(r) ds

(3a) (3b)

At distances much greater than the physical size of the chromophores, this Coulombic interaction can be approximately evaluated with a point dipole model. In the point dipole approximation, the transition density is represented by a Taylor 5045


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Table 1. Measureda and Calculatedb Rate Constants for EnT from DQDs to AQDs at a Series of Average Interparticle Distances center-to-center distance 5.8 5.1 4.9 4.3 4.1 3.8 3.7

± ± ± ± ± ± ±

0.3 0.3 0.4 0.6 0.5 0.3 0.3

nm nm nm nm nm nm nm

a −1 kexpt ens /ns

(152 (80 (59 (30 (18 (3.6 (2.1

± ± ± ± ± ± ±

−1

2.2) 0.6)−1 1.1)−1 0.8)−1 1.6)−1 1.0)−1 0.3)−1

b −1 kPDA ens /ns

c kPDA /ns−1 i,j

−1

b −1 kTDC ens /ns

−1

(142) (71)−1 (57)−1 (27)−1 (21)−1 (13)−1 (12)−1

−1

(563) (260)−1 (204)−1 (93)−1 (70)−1 (44)−1 (37)−1

(110) (52)−1 (42)−1 (19.4)−1 (14)−1 (8)−1 (6)−1

c kTDC /ns−1 i,j

(433)−1 (187)−1 (145)−1 (63)−1 (56)−1 (24)−1 (19)−1

a

Measured ensemble (average) rate constants of EnT from DQDs to AQDs within assemblies (plotted in red in Figure 4C). bCalculated ensemble (average) rate constants of EnT from DQDs to AQDs within assemblies using either the PDA (plotted in blue in Figure 4C) or TDC (plotted in blue in Figure 4C) combined with eq 1. cCalculated rate constants for EnT from one DQD to one adjacent AQD (ki,j in eq 1) using either the PDA or TDC methods.

into a three-dimensional grid of cells composed of transition charges, defined in eq 5, where dk

expansion of the multipolar transition moments, from which only the leading dipole term is used. Within the dipole approximation, JDA is defined by eq 4, in which μD and μA are the magnitudes of the transition dipoles, κ is the orientation factor, R̂ is the unit vector connecting the centers of the QDs, and ri ̂ JDA

ρD (x , y , z) ≅

1 Vd

∑ ∑ ∑ qx ,y ,z x

y

z

(5)

defines the grid size of the density cube (in our case dk = 0.1 Å) and Vd is the volume of the element, (dk)3. The Coulombic coupling in eq 2 is then defined in eq 6, where i,j are the elements

1 κ |μD ||μA | ≈ where κ = rD̂· rÂ − 3( rD̂ ·R̂ )( rÂ ·R̂ ) 4π ϵ0 R3 (4)

are the unit vectors of the transition dipoles. Previous work on EnT between QDs typically assumes the PDA to be valid.39,40,45,56−61 To determine what range of interparticle distances (if any) this assumption is true, we calculated the intrinsic EnT rates for each combination of QDs (the set of ki,j) using the PDA (eq 4) and then input those ki,j values, along with all of the other measured optical and structural parameters (see the SI), into eq 1 (the RE model) to calculate the observed average EnT rates for our ensembles. The results of this simulation, using the PDA for each interparticle distance, are shown as the black trace in Figure 4C, and the measured and calculated time constants for EnT are summarized in Table 1. The SI contains the details of the simulation. We observed that, for average interparticle distances greater than 4 nm, the EnT rates follow the prediction of the PDA model. At distances shorter than ∼4 nm, however (the n = 4 and n = 3 samples), our measured EnT rates deviate strongly from those predicted by the PDA model. This failure of the PDA is a known issue in molecular systems when the interparticle distance approaches the length (diameter) of the chromophores.62,63 On the basis of this literature, we expect a breakdown of the PDA in our system at interparticle distances of ∼3.5 nm, consistent with our observations (Figure 4C).64 In order to more accurately calculate the Coulombic interaction between transition densities, and therefore better model the observed EnT rates at short distances, we implemented a transition density cube method where the transition density was calculated using time-dependent density functional theory (TD-DFT) calculations.63 We performed the TD-DFT calculations using NWChem65 on a PbS cluster (Pb256S256) using the hybrid Perdew−Burke−Ernzerhof (PBE0) exchange−correlation functional with LANL2DZ basis sets;66 see the SI. We used a PbS cluster as a model system for the transition density of a full PbS QD because calculations on the full QD (∼1000 atoms) are too computationally costly. We then split the continuous transition density

JDA ≅

∑ i,j

qiqj 4π ϵ0ri , j

(6)

of the TDC for systems D and A, respectively, and rij is the distance between i and j. In the limit of infinitesimally small transition density cubes (and accurate molecular wave functions), eq 5 approaches the full Coulombic coupling between the two systems. This model does, however, neglect electronic exchange/overlap between the two systems, the interactions that promote Dexter EnT.67 Dexter-type contributions should not affect our measured EnT rates, as they require overlap of the QDs’ wave functions, and there is no optical indication of such overlap in our systems. Due to the delocalization of the transition density over the entire cluster volume (see the SI, Figure S23), the calculated Coulombic coupling between Pb256S256 clusters underestimates the coupling between two experimentally sized QDs. We account for this underestimation by radially scaling the transition densities of the Pb256S256 clusters to match the physical size of the DQDs and AQDs used experimentally (r = 1.6 and 1.7 nm, respectively); see the SI. While radially scaling the transition densities neglects fine atom-based structure in the density, it accounts for the increased delocalization of the exciton with increasing QD volume (akin to the particle-in-abox model). Figure S24 in the SI plots the Coulombic terms for two Pb256S256 clusters at various interparticle distances, calculated using both the TDC and PDA methods. We observe significant deviations in the Coulombic coupling between the PDA and TDC methods with decreasing interparticle distance (1000 atoms), rather than performing it on a smaller cluster. Finally, the accuracy of the calculation will be improved by accounting for emission from near-degenerate thermally accessed defect states that are more localized than the band-edge exciton of the QD.70 Two approaches to extract near-IR excitons at a rate competitive with Auger recombination are (i) the creation of QD superlattices that undergo Dexter EnT, which requires strong electronic coupling between the donor and acceptor, but is not hampered by the long radiative lifetime of PbS QDs, and (ii) replacing one (or both) components of the system with a material with exceptionally strong transition dipoles. This material might be a PbS nanoplatelet, which is predicted to have a “giant” oscillator strength and well-defined transition dipole orientations,71 or a cyanine dye J-aggregate, which has been shown to extract excitons from PbS QDs in as fast as 90 ps.72

The TDC method better predicts the rate of EnT between particles at short distances than does the PDA model, in part, by partially accounting for the delocalization of the exciton throughout the entire QD volume in our strongly quantumconfined PbS QDs. An alternative method for accounting for the exciton delocalization is to consider the dependence of the EnT rate on the edge-to-edge distance between QDs using unscaled Pb256S256 clusters in the calculation. The result of this calculation is an upper bound for the TDC prediction of the EnT rate for experimentally sized QDs; as expected, we observe that it overestimates the experimental rates; see Figure S25 of the SI.

CONCLUSION In summary, we controlled the average rate constant for nearinfrared EnT within soluble aqueous assemblies of PbS QDs, cross-linked by Zn2+, by changing the length of the mercaptoalkanoic acid that serves as the cross-linking ligand. EnT occurs with high yield (>70%) within these assemblies, and the time constant of 2.1 ± 0.3 ns that we measure for EnT between DQDs and AQDs linked S−-(CH2)2-COO−-Zn2+-−OOC(CH2)2-S− is the fastest measured for EnT between PbS QDs in a film or solution. Through the sequestration of Zn2+ by a chelating agent (EDTA) or zinc hydroxide species (formed by adjusting the solution pH), the assemblies disaggregate and EnT is turned “off”. We use a master equation to simulate the propagation of excitons through the assemblies (with measured structural and optical parameters as inputs), in order to translate between the observed average rate constants and the intrinsic (single-donor−single-acceptor) rate constants for EnT, which we predict using Förster theory. Using this approach, we evaluate the validity of the PDA for modeling EnT between PbS QDs. We find good agreement between the experimentally measured rates and those predicted by the PDA for interparticle distances greater than ∼4 nm. At shorter interparticle distances (specifically, 3.7 and 3.8 nm), the PDA drastically underestimates the observed EnT rate. We improve our prediction of the rates of these short-distance EnT processes by ∼20% by replacing the PDA with a transition density cube calculation of interparticle Coulombic coupling. This work is an experimental verification of the distancedependence of EnT rate between PbS QDs31 and experimentally identifies the point of breakdown of the PDA for EnT between QDs, the interparticle distance where higher order multipoles of the transition density become important. We show that, in contrast to measurements of the yield of EnT between PbS QDs (which is typically high regardless of its rate because of the long excited-state lifetime of PbS QDs), the rate constant for EnT, as measured by time-resolved PL or absorption, can serve as a “spectroscopic ruler” for measuring interparticle distance with angstrom-level resolution, if the Coulombic coupling between the QDs is properly modeled. The PDA, typically used in such applications of FRET, is probably sufficient for biological applications of a QD-based near-IR spectroscopic ruler, such as tracking the protein dynamics or measuring intracellular metal concentrations.64,68,69 At interparticle distances comparable to or smaller than the size of the chromophore itself (here

Distance-Dependence of Interparticle Energy Transfer in the Near

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