Abstract
The particlelike nature of light becomes evident in the photon statistics of fluorescence from single quantum systems as photon antibunching. In multichromophoric systems, exciton diffusion and subsequent annihilation occurs. These processes also yield photon antibunching but cannot be interpreted reliably. Here we develop picosecond timeresolved antibunching to identify and decode such processes. We use this method to measure the true number of chromophores on welldefined multichromophoric DNAorigami structures, and precisely determine the distancedependent rates of annihilation between excitons. Further, this allows us to measure exciton diffusion in mesoscopic H and Jtype conjugatedpolymer aggregates. We distinguish between onedimensional intrachain and threedimensional interchain exciton diffusion at different times after excitation and determine the disorderdependent diffusion lengths. Our method provides a powerful lens through which excitons can be studied at the singleparticle level, enabling the rational design of improved excitonic probes such as ultrabright fluorescent nanoparticles and materials for optoelectronic devices.
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Introduction
In a wide range of fluorescent nanoparticles such as conjugated polymers, semiconductor quantum dots, perovskite nanoparticles, lightharvesting complexes and many other natural or synthetic multichromophoric nanoparticles (mcNP), multiple excitons can exist simultaneously and in close proximity to each other^{1,2,3,4,5,6,7,8,9,10}. The number of chromophores as well as their interactions through exciton diffusion and annihilation processes are key parameters to describe the photophysical characteristics of mcNPs such as brightness^{11,12}, photoluminescence (PL) lifetime, exciton harvesting efficiency^{13} and photostability^{12,14}, all of which are also important for the performance of materials in optoelectronic devices. Photon antibunching has been used to count chromophores^{15,16,17}; however, this is typically not viable when exciton diffusion and singletsinglet annihilation (SSA) occur as illustrated in Fig. 1a. Singlephoton emission from mcNPs has been interpreted as evidence of longrange interchromophore interactions in a number of large multichromophoric systems^{1,2,8,18,19,20,21,22}. However, in these cases information about the number of physical chromophores in the mcNPs is lost. Here, we demonstrate that picosecond timeresolved antibunching (psTRAB) can be used to disentangle information on the number of physical chromophores and exciton diffusion and annihilation processes. psTRAB exploits the fact that exciton diffusion and annihilation are timedependent processes. Fingerprints of these processes are thus concealed in the PL photon stream of antibunching experiments under pulsed excitation^{6,23}.
The degree of single photon emission is commonly measured by two photodetectors in a Hanbury Brown and Twiss (HBT) geometric configuration and is therefore sensitive to twophoton events. With this technique, it is either possible to count the number of chromophores, provided that SSA is neglected, or to measure the SSA rate if the exact number of chromophores is known. In practical situations, neither the number of chromophores nor the SSA rate are usually known for mcNPs, which severely limits the usefulness of this conventional technique.
With psTRAB, we analyse the photon stream of antibunching experiments with pulsed excitation by grouping photons with respect to their arrival time after the laser pulse and crosscorrelating them to determine the probability of consecutive emission of two photons. Immediately after a laser pulse, SSA has not yet occurred and the emitted photons exhibit photon statistics corresponding to the number of physical chromophores present. As exciton diffusion and annihilation begin to dominate, the number of independent emitters decreases. Thus, the timedependence of the photon statistics synchronised by the laser pulse reports on (i) the number of physical emitters present and (ii) the time evolution of exciton diffusion and annihilation.
To demonstrate the psTRAB technique, we have used DNA origami to construct mcNPs with a known number of chromophores and welldefined spacing between them to accurately measure annihilation and benchmark our method. We then measure psTRAB of mesoscopic deterministic aggregates of conjugated polymers—the building blocks of films used in optoelectronic devices^{2}. There we find that during the first 250 ps after excitation, diffusion of excitons mainly occurs between one and two dimensions, both along the polymer backbone and between πstacked chains. The diffusion then becomes threedimensional at later times, with an orderofmagnitude difference in the rate of annihilation between ordered Htype aggregates and disordered Jtype aggregates. We can also extract the exciton diffusion lengths using the unique knowledge psTRAB gives on the number of independent chromophores present.
Our approach exploits the ability of modern timecorrelated singlephoton counting (TCSPC) hardware to record the absolute arrival time of a photon on each detector, both with respect to the start of the experiment, but also with respect to the last laser pulse (denoted as the microtime) as shown in Fig. 1b. As an example, consider a nanoparticle with five physical chromophores as depicted by the white discs in Fig. 1c. Absorption of a short pulse of light will create a Frenkel exciton (blue disc). The exciton can hop from one chromophore to another, e.g. by homoFRET^{24,25}, in a process referred to as exciton diffusion^{26}. Now, if we consider the case where two excitons are created by the same excitation pulse, this hopping allows the excitons to move so that they become adjacent to each other and can annihilate by SSA^{19,20}. This process has a strong distance dependence due to the underlying FRET mechanism by which SSA occurs and is often hard to study in a quantitative manner^{27}. By inspecting individual mcNPs on a confocal microscope with two singlephoton detectors (Fig. 1b) combined with TCSPC we measure the correlation events, N, dependent on the difference in photon arrival times, Δt, between photon events. We are thereby sensitive to the presence of two excitons in the mcNP. A histogram of Δt delay times in integer units of the excitationpulse period T shows the number of photon detection coincidences from either one excitation pulse or from two separate excitation pulses (Fig. 1c, right column). The ratio of the magnitude of the central peak at Δt = 0 to that of the lateral peaks, \(N_{\mathrm{c}}/N_{\mathrm{{\ell}}}\), provides a measure for the number of independent chromophores, n, provided that the background can be accounted for (see Supplementary Eq. 1 for details on the background correction) according to^{16}
By analysing the statistics of the PL photons detected at different time intervals after photoexcitation (panel c, second column), we can construct corresponding picosecondresolved histograms of the photon statistics and thus measure how many independently emitting chromophores exist on a particular timescale. This is illustrated schematically in Fig. 1c for a 5chromophore mcNP. The left column depicts the evolution of randomized typical examples of such independent chromophores after a single laser excitation event, whereas the histograms in the middle and right columns are an accumulation of multiple excitation cycles to show the timeaveraged result. At early times after excitation (panel c, first row), the two excitons contributing to N_{c} events (blue discs) have had no time to interact or move via homoFRET to neighbouring chromophores. From the photon coincidence histogram (right panel) we obtain a value of n = 5 with Eq. (1). At a later time (panel c, second row), an exciton on a neighbouring physical chromophore may have, for example, interacted through SSA, and consequently excitation of such chromophores thus does not contribute to N_{c} anymore, and we obtain n = 4 independent chromophores accordingly. These diffusion/SSA processes continue as a function of time, reducing the number of independent chromophores that could support the second exciton. Ultimately, at late times after the excitation pulse, only single photons can be detected because excitons on any other physical chromophore would have had enough time to diffuse and annihilate, yielding N_{c} = 0 and n = 1 (panel c, last row). This evolution of the photon statistics and the corresponding number of independent chromophores with time gives us a metric for the effective rate of exciton decay and provides direct microscopic insight into exciton annihilation and diffusion in mcNPs.
Results
Exciton annihilation in DNA origami nanoparticles
To explore the fundamental nature of exciton diffusion and SSA it is desirable to have the best possible control over the number of dye molecules and their spatial position in the mcNP. The dyes need to be within distances to each other corresponding to the range of FRET of ~1–10 nm. We have therefore turned to the method of threedimensional DNA origami to construct highly defined mcNPs. Similar structures have been used previously to study motor proteins and to characterize superresolution microscopy techniques^{28,29}, and are modified here for our needs. The sketch in Fig. 2a shows a short section of a 12helix bundle with 6 inner and 6 outer helices. The total length of this DNA origami structure is ~225 nm (transmission electron microscopy (TEM) images and structure are shown in Supplementary Figs. 1 and 2). Five labelling positions separated by ~3 nm each are available in the centre of this modular structure.
Based on this 12helix bundle DNA origami structure, we designed seven different structures with different numbers of dyes and different distances between the dyes (see Supplementary Information for details of DNA origami structures). For the dye we chose ATTO647N, which is highly photostable and bright in the presence of a reducing and oxidizing system (ROXS)^{30}. The origami structures were examined on a custommade confocal fluorescence microscope as described in the “Methods” section (a typical PL transient is shown in Supplementary Fig. 3)^{12}. We begin discussing the mcNP with all five dye attachment positions filled with a dye. Figure 2b displays a histogram of photon arrival times, i.e. microtimes, in steps of 200 ps following pulsed excitation with a 636 nm laser. We note that the step size also defines the timing error on the xaxis of the plot. This value of 200 ps was chosen according to the budget of photons available to construct the histograms of photon statistics in Fig. 2c. It is necessary to make a tradeoff between the timing resolution of the xaxis and the noise in the photon statistics histograms. This tradeoff depends on the experimental circumstances, i.e. the photon budget which is available. The PL decay is single exponential with a lifetime of 4.2 ns, which is typical for this dye attached to DNA and implies that no strong interchromophoric interactions occur^{12}.
For this fivedye sample we select 200 ps time windows from the microtime histogram (coloured bars) and calculate the photon statistics for each bin as shown in Fig. 2c. We used the peak of the instrument response function (see Supplementary Fig. 4) to determine zero microtime in the calculations. According to Eq. (1), we estimate the number of independent chromophores, n, in the first 200 ps after excitation to be ~4.8, very close to the expected starting value of 5. Between 200 and 400 ps, n drops to ~2.8 and reaches ~1.1 between 6400 and 6600 ps. The photons emitted by the fivechromophore structure at the latest times show almost complete antibunching. In total, photon events of 54 individual mcNPs were accumulated to obtain enough correlation events for this analysis. Photobleaching and blinking of individual dye molecules during the measurement period will impact the overall photon statistics. For this reason, only the first 5 s of each measurement were evaluated, and only if the overall PL intensity was constant to within 10% over this time. Additionally, while photobleaching and blinking has an influence on the overall strength of photon antibunching, it has no impact on the decay of n with microtime. For example, we indeed obtain the expected starting value of 5 for early microtimes, implying that the measurement is not affected by photobleaching and blinking. The five histograms in Fig. 2c reveal the timescale on which the excitons annihilate with each other to lower the number of independently emitting chromophores from five to one. We note that the fact that the number of chromophores inferred at the earliest times is slightly lower than the expected value of five can be explained by SSA having already occurred during the first 200 ps. One immediate conclusion of this method is that the number of dyes can be measured in an mcNP directly, even if the dyes are not emitting independently. Such knowledge is crucial in quantitative spectroscopic methods^{17,31}. A further crucial observation is that, in contrast to ensemble measurements^{32}, the PL decay retains its monomolecular singleexponential form even though SSA clearly occurs. This is a particularly important observation because the nonexponentiality of ensemble PL decays, i.e., a bimolecular decay, is generally used to extract exciton encounter rates to infer diffusion lengths. In the ensemble, this approach only works at very high excitation fluences which are far from the population densities relevant to devices. However, it is crucial to realize that SSA always occurs, even at the lowest excitation fluences, because exciton diffusion always occurs. Our photon correlation technique is sensitive precisely and only to these rare events of doublechromophore excitation, which can be reached at very low fluences at the cost of extended integration times. The detection of these rare events is ultimately limited by the background photons, e.g. the dark count rate of the photo detectors.
Having established that we can recover the number of dyes in an mcNP with our method, we now apply this approach to different DNA origami structures to examine the dynamics of the SSA mechanism in detail. Figure 3 plots the number of independently emitting chromophores n for each 200 ps time gate versus the corresponding microtime for seven different DNA origami structures. We start with the simplest model system with only one dye (dark grey dots in panel a). Except for the first two data points, these values are constant at n = 1.02, which is expected for a single dye. This value is close to unity and only limited by the signaltobackground ratio (SBR) as discussed in Supplementary Fig. 5 and ref. ^{15}. The fast decay in the first two data points originates from multiple excitations of the dye within the same laser pulse of ~80 ps width^{33}. Now we introduce a second dye at a distance of ~12 nm (panel b, light grey dots), which should be far enough away to prevent SSA between the excitons. Indeed, the data can be described with a constant n of 1.85 ± 0.01, which is slightly below the expected value of two, most likely because of slightly different PL intensities of the two dye molecules at the different binding sites of the DNA origami structure. Crucially, again, no decay of n is observed for this sample, implying a negligible exciton annihilation rate.
Next, we examine the more interesting cases, where we build structures with two dyes sufficiently close to each other such that SSA can occur. The red and orange dots in Fig. 3b display the data measured on structures carrying two dyes at ~3 and ~6 nm spacing. n starts out slightly below the expected value of two for both samples, and a decay during the first 2 ns down to n = 1.02 is observed for the 3 nm sample. These datasets are accurately described by a singleexponential model of the number of independently emitting chromophores,
with the offset, y_{0}, amplitude, A, and the exciton annihilation rate, k_{SSA} (see “Methods” for a derivation of Eq. 2). The overall number of physical dyes present in the structure is then given by \(n_{{\mathrm{dyes}}} = \left( {y_0  A} \right)^{  1}\). In Fig. 3b, we extract \(k_{{\mathrm{SSA}}} = 1.72\; \pm 0.06\;{\mathrm{ns}}^{  1}\) for the two dyes separated by 3 nm and \(k_{{\mathrm{SSA}}} = 0.06 \pm 0.01\;{\mathrm{ns}}^{  1}\) for the dyes separated by 6 nm, with \(n_{{\mathrm{dyes}}} = 1.8 \pm 0.03\) in both cases. As expected, k_{SSA} drops significantly when doubling the distance between the two dyes, indicating that we are in the important regime where SSA is controlled by FRET and therefore by dye spacing. Subsequently, we placed three dyes separated by ~6 nm each (Fig. 3c, cyan dots). Fitting with Eq. 2 yields \(k_{{\mathrm{SSA}}} = 0.06 \pm 0.01\;{\mathrm{ns}}^{  1}\) and \(n_{{\mathrm{dyes}}} = 2.7 \pm 0.1\), which is consistent because we expect no SSA between the leftmost and rightmost dyes, and the same SSA rate for the neighbouring dyes as in panel b.
Upon moving the three dyes closer to each other, now only separated by 3 nm (Fig. 3c, blue dots), Eq. (2) is no longer sufficient to describe the time evolution of n since nextnearest neighbour interactions arise. We therefore used an analogous biexponential model of SSA, with a fast rate for neighbouring dyes and a slow rate, which combines direct annihilation of nextnearestneighbouring dyes and exciton hopping with subsequent annihilation of neighbouring dyes, to describe the blue dataset in panel c,
We derive from this dynamics an average amplitudeweighted SSArate \(\left\langle {k_{{\mathrm{SSA}}}} \right\rangle = (A_1k_{\mathrm{SSA,1}} + A_2k_{\mathrm{SSA,2}})/(A_1 + A_2) = 0.9 8 \pm 0.09\;{\mathrm{ns}}^{  1}\) (see Supplementary Information for complete fitting results in Supplementary Table 2) and a number of dyes, \(n_{\mathrm{{dyes}}} = (y_0  (A_1 + A_2))^{  1} = 2.9\; \pm 0.1\). Finally, for the DNA origami structure bearing all five dyes (Fig. 3d, violet dots), we extract \(\left\langle {k_{{\mathrm{SSA}}}} \right\rangle = 0.72 \pm 0.07\;{\mathrm{ns}}^{  1}\) and \(n_{\mathrm{{dyes}}} = 4.7\; \pm 0.2\) by using Eq. (3).
The crucial observation is that at long microtimes, n decays to 1 for all samples with \(k_{{\mathrm{SSA}}} \,> \,0\). This is particularly intriguing for the fivedye sample, where we would anticipate the case in which two excitons remain on the leftmost and rightmost dyes. According to the experiment with two dyes placed 12 nm apart (panel a, light grey dots), no direct SSA should occur in this case. However, the fact that the fivedye sample still decreases down to only one emitting independent chromophore, rather than two, allows us to conclude that exciton hopping, i.e. exciton diffusion, occurs between the five dyes. We note that all measurements of the DNA origami samples were conducted in buffered solution and consequently, the dyes were free to rotate on the DNA origami. We therefore neglect the possibility of a particular preferred orientation of the transitiondipole moments arising. However, this approximation is no longer valid for mcNPs, which are fixed in space, e.g., set inside a solid matrix. Here, the transitiondipole moment orientation can have a significant impact on the SSA rate, i.e., the morphology plays a crucial role on the dynamics of psTRAB. This conclusion offers a motivation to study morphologically different mcNPs in which significant exciton diffusion arises.
Exciton diffusion in conjugated polymer aggregates
To examine exciton diffusion in conjugated polymers in the mesoscopic size regime, aggregates of chains were grown with distinct electronic and structural properties. These structures are formed by two poly(paraphenyleneethynylenebutadiynylene) (PPEB)based conjugated polymers (Fig. 4a). With a small variation of the alkyl sidechains, ordered aggregates with either Htype interchromophoric coupling (PPEB1, lilac) or disordered aggregates with Jtype intrachromophoric coupling (PPEB2, brown) can be grown by solvent vapour annealing^{18}. Samples were prepared as described in ref.18, yielding individual small aggregates isolated in poly(methylmethacrylate) (PMMA) and measured on a confocal fluorescence microscope as reviewed briefly in the “Methods” section and described elsewhere^{34}. 631 single aggregates of PPEB1, each comprising on average approximately 54 chains, and 705 aggregates of PPEB2 (each ~9 chains, see Supplementary Figs. 6 and 7 and discussion thereof in the Supplementary Information), were grown and measured individually. Only the first 5 s of each PL trace were evaluated (see Supplementary Fig. 8 for examples of PL traces of the H and Jtype aggregates), provided that the PL intensity was constant to within 10%. Following the above procedure, n(t) was determined using psTRAB as shown in Fig. 4b (Supplementary Fig. 10 shows the corresponding photon antibunching histograms). We use different widths of timewindows to generate the evolution of n(t), with 3 ps chosen at early times, increasing to 80 ps (in the Htype aggregates) and 160 ps (in the Jtype aggregates) at later times. We observe a clear decay of n with time, signifying excitedstate interactions primarily due to SSA. We note that this measurement is independent of the excitation intensity in this region of excitation densities as discussed in Supplementary Fig. 9. A substantial difference between the decay dynamics exists for the two aggregates. For the Htype aggregates, n drops rapidly over the first 250 ps and then continues before levelling off at ~2000 ps. The Jtype aggregates show a smaller initial fast drop, followed by a slower linear decay before levelling off at a slightly higher value of n at times >2000 ps.
First, we note that, in analogy to the DNA origami model system in Fig. 3d, the decay of n with time constitutes a signature of exciton annihilation mediated by exciton diffusion. Because diffusion is now likely to dominate, however, the dynamics generally cannot be fitted with one fixed k_{SSA} rate. Instead, the annihilation is governed by a rate equation for a secondorder reaction^{35}. The clear difference between the H and Jtype aggregates indicates that the process of exciton diffusion is not the same in both of them. To examine this difference in a quantifiable manner, we plot the evolution with time of the quantity \(\ln \left( {\frac{n}{{n  1}}} \right) \cdot V_{{\mathrm{agg}}}\) as shown in Fig. 4c, where V_{agg} is the calculated aggregate volume (see “Methods” for a full description of this equation and Supplementary Information for how the volumes were obtained). This allows us to quantify and compare exciton diffusion, as data plotted in this manner allows the instantaneous rate of bimolecular exciton annihilation, γ, to be determined from the slope and compared against ensemble equivalents. A linear function signifies a constant, timeindependent γ, whereas curvature implies that γ has a timedependence. Typically, in exciton annihilation measurements, the underlying excitedstate decay has to be accounted for^{36}, complicating analysis in extracting diffusion relevant properties. The advantage of psTRAB is that we directly obtain a measure of the exciton diffusion and are thus uniquely sensitive to weak and slow diffusion. This contrasts with conventional ensemble measurements of the nonexponential decay in PL intensity, which require high exciton densities to see an appreciable effect of annihilation. It is important also to stress that psTRAB offers a unique way to observe the very rare circumstances where two excitons exist in a nanoscale object, and consequently to see how the probability of them coexisting changes on the picosecond to nanosecond timescale as diffusionassisted exciton annihilation occurs. The equivalent ensemble measurements of SSA require appreciable, i.e. measurable, fractions of excitons to annihilate with each other to be distinct from exciton luminescence where no annihilation has occurred. Consequently, as noted, psTRAB allows measurements of weaker and slower processes to be made than would otherwise be possible, with the measured photon coincidences in the PPEB Haggregates typically ~300 parts per million, well below the overall luminescence signal’s shot noise limit. With the data plotted as \(\ln \left( {\frac{n}{{n  1}}} \right) \cdot V_{{\mathrm{agg}}}\) as in Fig. 4c, for both H and Jtype aggregates three regions are identified. At early times (<250 ps) nonlinear behaviour is observed, indicating that γ is timedependent. Exciton diffusion is therefore one or less than twodimensional^{37}. At times 250–2000 ps, both aggregate types show linear behaviour, thus γ is timeindependent and the diffusion threedimensional^{38,39}, with values of γ found to be in the range of 10^{−9} to 10^{−10} cm^{3} s^{−1}, in good agreement with typical conjugated polymers^{32,39,40,41}. Finally, at times >2000 ps, γ = 0, i.e. annihilation has ceased as the exciton density is too low to support continued interactions.
The psTRAB results also allow insight into the nanoscale organization of material in the aggregates, as sketched in Fig. 4d. At early times, the timedependent γ indicates that exciton motion is one or less than twodimensional, most likely in the dispersive regime, and is therefore consistent with ensemble observations of annihilation on the timescale of a few picoseconds^{42}. In the context of the Htype aggregate, this motion will be along the chains and across the interchain πstack. This conclusion is in agreement with a high degree of chain alignment, evidenced by the PL intensity modulation depths determined when rotating the polarization of the exciting laser^{18}. The Jtype aggregate also shows timedependent annihilation at early times. Here, however, simple onedimensional motion will be favoured since strong intrachain coupling is dominant as evidenced by the Jtype emission characteristics^{18}. At later times, the timeindependence of γ indicates that exciton motion is threedimensional in both aggregate types. γ is an order of magnitude lower in this time region for the J than for the Htype. This difference relates to the nature of chromophoric coupling and disorder in the aggregates. In Htype aggregates, chains with the smallest degree of disorder will show the strongest interchain electronic coupling, facilitating efficient threedimensional diffusion. In Jtype aggregates, in contrast, which do not show a high polarization anisotropy^{18}, chains are relatively disordered. Poor chain alignment will lead to weak interchain electronic coupling and a lower value of γ. Exciton diffusion is then limited by the random chain alignment that excitons encounter when diffusing. The impact of chain disorder on exciton diffusion can also be examined by comparing the psTRAB of the 9chain Jtype aggregate with a smaller one that comprises of ~6 chains shown in Fig. 4e. In the region where γ is timeindependent and threedimensional diffusion dominates, γ is almost a factor of two higher in the 6chain aggregate, indicating increased order in the smaller aggregate which facilitates effective interchain sitetosite hopping. We also note that at early times (0–125 ps), in the Jtype aggregates a significantly stronger timedependent gradient of the psTRAB functionality is observed, consistent with fast onedimensional exciton motion along the chain. We are cautious with regard to overinterpreting these data, however, since such exciton motion is likely to be much faster than the time resolution of our experiment. Indeed, we would expect the onedimensional exciton motion along the chain in strongly coupled Jtype aggregates to be higher than the twodimensional diffusion alongchain and across πstacks in Haggregates, where intrachain coupling can be weaker^{43,44}.
Finally, at late times where γ → 0, we enter the regime where the exciton density is too low to support continued annihilation. These conditions can be used to obtain a lower limit on the exciton diffusion length, L_{3D}. The rationale for this approach is simple: we know the volume of the aggregate and the number of independent chromophores that the aggregate can support when we can no longer measure annihilation occurring, i.e. when excitons no longer interact with each other. Division yields the volume that a single independent chromophore occupies, equivalent to the volume explored by an exciton. If diffusion is presumed to arise in a spherical volume in three dimensions, a diffusion length, L_{3D}, can be determined. The value will be a lower limit as the length is technically defined as the distance excitons diffuse in their lifetime rather than once the exciton density is too low to support continued interactions, but the difference between these two definitions will be small at these late times. We find lower limits of L_{3D} ≈ 9 nm for the Htype aggregate and L_{3D} ≈ 5.2 nm for the Jtype aggregate, consistent with typical literature values for conjugated polymers^{26,36,39,45}. The unique advantage of our chromophorecounting method is that the calculation of these values contains no presumptions other than the mass density of the aggregate. L_{3D} is derived from simple observables and is only possible because we consider single objects at the discretised level of excitons and the resulting photon correlation.
Discussion
Knowledge of the nanoscale organization of a material, the electronic coupling between chromophores, and energy transfer pathways is important in a wide variety of systems. In this work we have introduced a powerful method to quantify exciton–exciton annihilation and exciton diffusion in multichromophoric mesoscopic objects. This is achieved by resolving the fluorescence photon statistics on a picosecond timescale. Using deterministic DNA origami structures, we position dyes at specific distances from each other and obtain direct measurements of the rate of annihilation between two excitons and the true number of dyes. This accuracy is a direct consequence of utilizing twodetector coincidences that are sensitive to twophoton emission events. Our method can measure the annihilation rate γ in welldefined structures and directly yields the number of physical dyes present in each sample. We stress that such chromophore counting is not possible with standard timeintegrated photoncorrelation measurements. The technique can be expanded to look at nanoparticles grown from multiple single conjugatedpolymer chains. In these polymer aggregates, SSA is governed mainly by exciton diffusion instead of fixed distance FRETbased annihilation between chromophores. In addition, the method offers facile differentiation between J and Htype aggregates, determining valuable material properties such as the exciton diffusion length, the dimensionality of diffusion and the degree of nanoscale disorder in the aggregate. The psTRAB technique therefore offers valuable opportunities to explore the nanoscale organization and excitonic coupling of chromophores in lightemitting materials with unprecedented detail.
Methods
Photon correlation, data analysis, and derivation of Eq. (2)
The psTRAB is computed from raw timestamped TCSPC data using MATLAB. The scripts developed operate similarly to conventional calculations of crosscorrelations^{46}. The following parameters are stored for each photon event: (i) the “macrotime” at which the photon arrived, i.e. the integer multiple of the corresponding excitation laser repetition period T; (ii) the “microtime”, t, which corresponds to the time the photon was detected after the excitation pulse excited the NP; and (iii) the detection channel, i.e. the photon counter A or B. The events are crosscorrelated with respect to their macrotimes, after which the microtimes are evaluated as follows: (i) we store the shorter microtime, t, of each correlation event (e.g. the microtime of channel A) and neglect the longer microtime, t + Δt. (ii) For selected microtime intervals, histograms of correlation events are constructed as a function of the macrotime delay between the channels. Finally, the scripts sum over multiple measurements of individual aggregates to produce an overall psTRAB result. As detailed in the Supplementary Information, we rationalize the number of correlation events, N_{c}(t, t + Δt), for a given delay time Δt < T − t between two photon events arising from the same excitation pulse, as follows:
Here, N_{exc} is the total number of observed laser excitation pulses, P(t) is the probability of detecting the first photon at microtime t and P′(t + Δt) is the probability of detecting the second photon at microtime t + Δt < T. In case the exciton annihilation is determined by a single exponential decay rate k_{SSA}, these probabilities are calculated as
where n_{dyes} is the number of chromophores, p_{0} summarizes the probability of the chromophore being excited by the laser pulse and the probability of detecting the emitted photon, k_{r} and k_{nr} are the radiative and nonradiative decay rates and k_{ET} = k_{SSA}/2 is the energytransfer rate between two excited chromophores. Note that in general P′(t) ≠ P(t) since the exciton emitting the first photon at time t can reside on any one of the n_{dyes} chromophores, while the exciton emitting the second photon resides on one of the (n_{dyes} − 1) remaining chromophores. At microtime delays 0 < Δt < T − t, the number of excitons does not decay any further through energy transfer, since only a single exciton is left. The number of correlation events \(N_\ell (t,\,t + {{\Delta }}t)\), where the second photon is detected at nonzero macrotime delays and thus arises due to a separate laser excitation event, is instead calculated from
where
is independent of energy transfer, since only single excitons are present after each laser excitation. The ratio \(N_{\mathrm{c}}/N_{\mathrm{{\ell}}}\) of central to lateral correlation events is thus directly connected to the number of chromophores in the mcNP and the time dynamics of the annihilation process as
The result is independent of k_{nr} implying that additional quenching processes due to singlet–triplet annihilation or the interaction of singlet excitons with dark states such as chargeseparated states do not impact the ratio \(N_c/N_\ell\). Note that the result is independent of Δt and it can also be calculated from the timeintegrated number of correlations
which significantly reduces the noise associated with experimental event data.
Comparing the derived expression for \(N_{\mathrm{c}}/N_\ell\) with Eq. (1) defining the number of independent chromophores n, we obtain
Equation (11) corresponds to Eq. (2) with y_{0} = 1 and \(A = 1  n_{{\mathrm{dyes}}}^{  1}\). A quantumstatistical description of photon correlations in an nchromophore system, using a master equation approach, is given in the Supplementary Information together with Supplementary Figs. 11–13. Note that the assumption of any specific decay law for singlet–singlet annihilation such as an exponential decay according to \({e}^{  k_{{\mathrm{SSA}}}t}\) is not strictly necessary. To that end, psTRAB \(N_{\mathrm{c}}/N_\ell\) can be used to directly measure the decay law associated with exciton–exciton interactions, which is connected to the mean first passage time of the random walk performed by the excitons. The technique can obviously be extended to higherorder photon correlations, using more than one beam splitter in the Hanbury Brown and Twiss setup, to determine the functional difference between twoexciton interactions and higherorder contributions.
DNA origami microscopy
A custommade confocal microscope based on an Olympus IX71 inverted microscope was used. Multichromophoric DNAorigami structures (see Supplementary Information for details on DNA–origami structures and a complete list of all primers used in Supplementary Table 4) were excited by a pulsed laser (636 nm, ~80 ps fullwidth halfmaximum, 80 MHz, LDHDC640; PicoQuant GmbH) operated at 40 MHz repetition rate. Circularly polarized light was obtained by a linear polarizer (LPVISE100A, Thorlabs GmbH) and a quarterwave plate (AQWP05M600, Thorlabs GmbH). The light was focused onto the sample by an oilimmersion objective (UPLSAPO100XO, NA 1.40, Olympus Deutschland GmbH). The sample was moved by a piezo stage (P517.3CD, Physik Instrumente (PI) GmbH & Co. KG) controlled by a piezo controller (E727.3CDA, Physik Instrumente (PI) GmbH & Co. KG). The emission was separated from the excitation beam by a dichroic beam splitter (zt532/640rpc, Chroma) and focused onto a 50μm pinhole (Thorlabs GmbH). The emission light was separated from scattered excitation light by a 647 nm longpass filter (RazorEdge LP 647, Semrock) and split into two detection channels by a nonpolarizing 50:50 beam splitter (CCM1BS013/M, Thorlabs GmbH). In each detection channel, afterglow of the avalanche photodiode was blocked by a 750 nm shortpass filter (FES0750, Thorlabs GmbH). Emission was focused onto avalanche photodiodes (SPCMAQRH14TR; Excelitas Technologies GmbH & Co. KG) and signals were registered by a multichannel picosecond event timer (HydraHarp 400, PicoQuant GmbH). The setup was controlled by a commercial software package (SymPhoTime64, Picoquant GmbH).
PPEB aggregate microscopy
Single polymer aggregates were measured on a customdesigned confocal microscope as described elsewhere^{34}. For excitation, the frequencydoubled output of a Ti:Sapphire oscillator (~100 fs, 80 MHz, 810 and 880 nm) (Chameleon, Coherent) was used, centred at 405 nm for PPEB1 and 440 nm for PPEB2. Femtosecond excitation was required to ensure that double excitation of the aggregates did not occur, because the excited state lifetime for the Jtype coupled PPEB2 aggregates is significantly shorter than for the DNAorigami dyes^{18}, preventing the use of conventional picosecond laser diodes. The laser was spatially expanded, spectrally cleaned and coupled into the microscope base (IX71, Olympus Deutschland GmbH), where it filled the backplane of a ×60 1.35 NA objective (UPLSAPO60XO, Olympus Deutschland GmbH). The sample was placed on a piezo stage (P527.3CL, Physik Instrumente GmbH, Germany), which was scanned to generate microscope images and locate individual aggregates. The PL was detected using two singlephoton detectors (PD25CTE, Micro Photon Devices S.r.l., Italy) connected to a multichannel picosecond event timer (HydraHarp 400, PicoQuant GmbH, Germany) allowing TCSPC and crosscorrelations to be performed. The piezo stage and photon counting hardware were controlled using a customized code in LabVIEW (National Instruments).
Exciton diffusion in PPEB aggregates
Bulk exciton–exciton annihilation by SSA is conventionally described by a simple secondorder reaction equation, \(\frac{\mathrm{d}}{{{\mathrm{d}}t}}\rho _{{\mathrm{exc}}} =  \gamma (t)\rho _{{\mathrm{exc}}}^2\), where ρ_{exc} is the exciton density and γ(t) is the diffusioncontrolled annihilation rate. In the context of our psTRAB method, differentiation of Eq. (11) ultimately leads to
for the number of independent chromophores. This function is the correct form of the secondorder reaction equation in cases where the number of reactants is low, since the reaction rate of change is proportional to the number of pairs that can be chosen. The psTRAB measurements thus resolves SSA on the singlenanoparticle level in a form that can be thought of qualitatively as tracking the mutual annihilation of independent chromophores by bimolecular interaction. From Eq. (12), we derive the following linear form governing the exciton annihilation rate γ = k_{SSA} V_{agg}, where V_{agg} is the aggregate volume:
See the Supplementary Information for details on how V_{agg} is obtained by simply invoking knowledge of the mass and mass density of the polymer chain and the number of chains in the aggregate. Thus, plotting \(\ln \left( {\frac{n}{{n  1}}} \right) \cdot V_{{\mathrm{agg}}}\) as a function of t as in Fig. 4c, e allows γ to be determined from the gradient by straightline fitting.
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All relevant data are available from the authors.
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All relevant codes to analyse the data are available from the authors
Change history
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Acknowledgements
T.E. thanks the Deutsche Forschungsgemeinschaft (German Research Foundation) for funding through Collaborative Grant No. 319559986. F.J.H. thanks the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for funding through the postdoctoral “booster program” of—ProjectID 314695032—S.F.B. 1277, Project B03. P.T., F.S. and T.S. thank the European Union’s Horizon 2020 research and innovation programme under grant agreement No 737089 (Chipscope) and the DFG under Germany´s Excellence Strategy—EXC 2089/1—390776260 for financial support. We thank Dr. Florian Selbach for TEM imaging of the samples.
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G.J.H., F.S., F.J.H. and J.V. devised the psTRAB methodology and developed the technique. T.S., J.V. and P.T. designed the DNA origami structures. T.S. prepared, measured and analysed the DNA origami structures and data. D.L., K.R. and S.H. designed and synthesized the conjugated polymers. G.J.H., F.S. and T.E. measured and analysed the PPEB aggregate data. S.B. developed the analytic and quantumstatistical treatment of psTRAB. G.J.H., T.S., F.S., S.B., P.T., J.M.L. and J.V. contributed to manuscript writing.
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Hedley, G.J., Schröder, T., Steiner, F. et al. Picosecond timeresolved photon antibunching measures nanoscale exciton motion and the true number of chromophores. Nat Commun 12, 1327 (2021). https://doi.org/10.1038/s4146702121474z
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DOI: https://doi.org/10.1038/s4146702121474z
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