Enhancement of Energy Transfer Efficiency with Structural Control of Multichromophore Light‐Harvesting Assembly

Abstract

Multichromophore systems (MCSs) are envisioned as building blocks of molecular optoelectronic devices. While it is important to understand the characteristics of energy transfer in MCSs, the effect of multiple donors on energy transfer has not been understood completely, mainly due to the lack of a platform to investigate such an effect systematically. Here, a systematic study on how the number of donors (n_D) and interchromophore distances affect the efficiency of energy transfer (η_FRET) is presented. Specifically, η_FRET is calculated for a series of model MCSs using simulations, a series of multiporphyrin dendrimers with systematic variation of n_D and interdonor distances is synthesized, and η_FRETs of those dendrimers using transient absorption spectroscopy are measured. The simulations predict η_FRET in the multiporphyrin dendrimers well. In particular, it is found that η_FRET is enhanced by donor‐to‐donor energy transfer only when structural heterogeneity exists in an MCS, and the relationships between the η_FRET enhancement and the structural parameters of the MCS are revealed.

Introduction

Light‐harvesting systems consisting of multiple chromophores^[1] have recently attracted much interest as building blocks of molecular optoelectronic devices, for example, artificial light‐harvesting assemblies,^[2–7] semiconductor and organic electronic materials,^[8] and photovoltaic solar cells.^[9–11] For the development and optimization of such devices, it is crucial to understand the physics and chemistry of dynamic processes, for example energy transfer, occurring in multichromophore systems (MCSs). In general, a multichromophore light‐harvesting system consists of multiple donors, which can absorb light in a wide spectral range, and multiple acceptors, which can receive the absorbed energy from the donors. For the optimum performance of such a system, it would be desirable to achieve efficient donor‐to‐acceptor (D–A) energy transfer. Energy transfer between a donor and an acceptor is often explained by the mechanism of Förster resonance energy transfer (FRET).^[12] According to the FRET theory, the efficiency of energy transfer is inversely proportional to the sixth power of D–A distance,^[13–16] and therefore a donor–acceptor pair needs to be located within a certain distance for facilitating the transfer. While it would be ideal to achieve coherent energy transfer, it is difficult to precisely locate the donors and acceptors at positions required for coherent energy transfer to occur. In reality, incoherent energy transfer in the weak coupling regime is operational for most of MCSs.

For MCSs, however, there have been reports that the experimentally measured efficiency of excitation energy transfer (η_FRET), which is defined as the probability of the acceptor to receive the energy absorbed by the donor, is much lower than the theoretically predicted efficiency.^[17–22] It has been known that such a discrepancy in the experimental and theoretical η_FRET is mainly caused by structural heterogeneity of the molecular systems,^[22–27] which becomes larger in complex MCSs. As an effort to compensate for the drop of η_FRET due to structural heterogeneity, many researchers have attempted to utilize donor‐to‐donor (D–D) energy transfer as a means of seeking the optimum path for efficient energy transfer toward acceptors.^[7,28] However, the influence of D–D energy transfer on η_FRET in MCSs has been controversial. For example, Olejko and Bald^[29] studied the effect of the number of donors (n_D) on η_FRET using the DNA origami structure as a scaffold for donors and acceptors. By varying n_D from one to four, they found that η_FRET barely changes with n_D although the amount of harvested energy increases with n_D. Buckhout‐White et al.^[28] also investigated how n_D affects η_FRET by using the DNA structures of various shapes as a scaffold for donors and acceptor and observed only little improvement of η_FRET with the increase of n_D. In contrast, Trofymchuk et al.^[7] and Gartzia‐Rivero et al.^[30] showed that, in polymeric nanoantennas composed of thousands of donors and a few acceptors, η_FRET increases with the increase of the weight ratio of donors to the mass of the polymer. Besides, there have been debates on whether the rate of D–D energy transfer influences η_FRET. Vijayakumar et al.^[31] showed that, for self‐assemblies having different sizes of end functional groups, η_FRET increases with the increase of the rate of D–D energy transfer. In contrast, Melinger et al.^[21] reported that, for simple DNA dual‐rail structures containing only one or two donors, η_FRET is enhanced with the aid of D–D energy transfer but only when the rate of D–D energy transfer is faster than the rate of D–A energy transfer. Also, they showed that, as long as the D–D energy transfer is fast enough, η_FRET does not change much with the change of the rate of D–D energy transfer. Since these discrepancies were observed in studies on different MCSs, a more systematic study is called for to establish a general principle that can be used to describe the energy transfer among multiple donors and acceptors.

In this work, to better understand how the presence of multiple donors influences η_FRET in MCSs, we comprehensively investigated energy transfer in a series of model MCSs by combining simulations, chemical synthesis, and time‐resolved spectroscopy. We note that we employed the MCSs with the donors arranged in spherical symmetry, in contrast to the hybrid MCSs with donors arranged linearly as investigated in a previous study.^[32] In the linearly arranged donors, the D–A distance varies depending on the position of each donor. Since the D–A distance is the most critical factor that determines the FRET efficiency, we chose to investigate the spherically symmetric MCSs where the D–A distances are identical for all donors so that we can focus on the effect of homo‐FRET on FRET efficiency. As the first step of such efforts, to account for the effect of d_DA, d_DD, and the orientation factor (κ²) on η_FRET, we theoretically calculated η_FRET with simulations for a series of model MCSs consisting of multiple donors and a single acceptor while independently changing n_D and interchromophore distances, which include the D–A distance (d_DA) and the D–D distance (d_DD). Each simulation consists of Monte Carlo processes to calculate parameters needed to set up multiple differential equations, the solution of the differential equations to determine the time‐dependent excited‐state populations of donor and acceptor, and the calculation of the FRET efficiency. By running a sufficient number of simulations, we obtained η_FRET for various situations. Then, to experimentally check the validity of the calculation results of model systems, we designed and synthesized various multiporphyrin dendrimers whose d_DD and n_D were independently adjusted, and measured the dynamics and efficiencies of energy transfer in the multiporphyrin dendrimers using femtosecond transient absorption (TA) spectroscopy. From our comprehensive study, we found that η_FRET is enhanced with the aid of D–D energy transfer and such enhancement is intricately related to the structural heterogeneity of MCSs, specifically the distributions of d_DA, d_DD, and κ². The simulations presented in our study not only explain the seemingly contradicting results of previous studies but also predict η_FRET of various MCSs.

To simplify the calculation of η_FRET for an easy application to general MCSs of complicated structures, we employed the ideal dipole approximation (IDA),^[33,34] where the electronic coupling is described as the Coulombic interaction between transition dipole moments of donor and acceptor,^[35] instead of the exact calculation without any approximation. While IDA has limitations compared with the exact calculation,^[36] for the porphyrin dendrimers investigated in our work, it turned out that the η_FRET values calculated with IDA exhibit only ≈3.5% deviation from the experimental η_FRET values.

Results

Calculation of η_FRET for Model Multichromophore Systems

To examine the effect of d_DD, d_DA, and n_D on η_FRET in MCSs, we calculated η_FRET for seven model systems that contain a single acceptor at the center and have multiple donors with various d_DD, d_DA, and n_D values, as shown in Figure 1. In the simulations, it was assumed that only a single donor per MCS was excited and the excitation probabilities of all donors were set to be equal. To effectively compare the energy transfer behavior in these MCSs of different chemical structures and various complexities, we define the normalized interchromophore distances, Δ_DA and Δ_DD, which are obtained by dividing d_DA and d_DD by the Förster radius for the D–A or D–D pair, respectively. To distinguish the donor‐to‐donor and donor‐to‐acceptor energy transfer, we denoted the D‐to‐A and the D‐to‐D energy transfer as “FRET” and “homo‐FRET,” respectively.

1 Figure. Seven model multichromophore systems used in simulations. In these model systems, the acceptor (A, red) is located at the center of multigonal structure formed by multiple donors (D, blue). The efficiency of energy transfer among chromophores is determined by the normalized interchromophore distances (ΔDA and ΔDD) and the orientation factor (κ2). Detailed information on the seven model systems is described in the Experimental Section.

While MCSs tend to have inherent structural heterogeneity, the distributions of structural parameters of those systems were not considered in previous studies.^{[7,28–32,37]} Structural heterogeneity can originate from both (i) the rapid structural fluctuation of orientation factor and interchromophore distance around each of their average values and (ii) the static distribution of structural parameters such as D–A distance and orientation factor of all donors. To consider the structural heterogeneity of MCSs, we performed the simulations for calculating η_FRET in the model MCSs while considering two cases for each of Δ_DA (and Δ_DD) and κ², which are two major parameters that govern η_FRET, resulting in four cases in total. Specifically, for Δ_DA (and Δ_DD), we considered either a single value or a distribution of values. A single value means that all D–A (or D–D) pairs have the same Δ_DA (or Δ_DD) value, and a distribution means that those pairs can have different distance values from each other following a Gaussian distribution. For κ², which is defined in the Experimental Section, we checked two representative cases: the dynamic and static isotropic limit. The dynamic isotropic limit refers to a condition where the transition dipoles rotate much faster than the time scale of energy transfer and, at this limit, κ² has a rotationally averaged value of ⅔.^[23] In contrast, the static isotropic limit describes the condition where the transition dipoles rotate much slower than the time scale of energy transfer^[23] and thus κ² has a distribution of values. To summarize, as can be seen in Figure S14a,d in the Supporting Information, the following four cases were considered: i) a single value of Δ_DA (and Δ_DD) and the dynamic isotropic limit, ii) a single value of Δ_DA (and Δ_DD) and the static isotropic limit, iii) a finite distribution of Δ_DA (and Δ_DD) and the dynamic isotropic limit, and iv) a finite distribution of Δ_DA (and Δ_DD) and the static isotropic limit. Details of the simulations and the model systems are described in the Experimental Section.

The results of simulations are illustrated in Figure 2; and Figure S14 (Supporting Information). Initially, the simulations were performed under the simplest condition, case (i). In particular, we calculated η_FRET while varying Δ_DA or Δ_DD. As shown in Figure 2c, η_FRET increases with the decrease of Δ_DA in the D20A model, which is in good agreement with the one predicted by the Förster's equation for a single D–A pair. Here we define the degree of η_FRET enhancement induced by homo‐FRET as Δη_FRET by taking the difference between η_FRET values in DnA model and D1A model. In case (i), as shown in Figure 2e,f, η_FRET is not enhanced at all, that is, Δη_FRET = 0, with the variation of Δ_DA, Δ_DD, or n_D. Next, the simulations were performed for the other three cases, where Δ_DA (and Δ_DD) and/or κ² have a distribution of values. As shown in Figure 2d; and Figure S14f–h, η_FRET tends to increase with the decrease of Δ_DA, in agreement with case (i). However, in contrast to case (i), η_FRET is enhanced with the variation of Δ_DA or Δ_DD, that is, Δη_FRET > 0, in case (ii)–(iv). Specifically, Δη_FRET becomes maximal at intermediate values of Δ_DA, as shown in Figure 2e; and Figure S14j–l. In addition, Δη_FRET increases with the decrease of Δ_DD, but such increase is saturated below a certain threshold Δ_DD value, as shown in Figure 2f; and Figure S14n,p. Especially, it should be noted that Δη_FRET is positive even when Δ_DD is larger than Δ_DA. In addition, Δη_FRET increases with the increase of n_D, even in the Δ_DD region where the increase of Δη_FRET is saturated. However, as shown in Figure 2f, Δη_FRET does not simply increase with n_D when Δ_DD is larger than a certain limit. For example, Δη_FRET of D20A is smaller than that of D12A at Δ_DD > 0.7, indicating that the effect of n_D on η_FRET is saturated above a certain n_D value. These results of our simulations demonstrate that the presence of distributions for Δ_DA, Δ_DD, and κ² values as in case (ii)–(iv) induces the enhancement of η_FRET (that is, Δη_FRET > 0), indicating that the Δη_FRET is closely related to the structural heterogeneity of MCSs. The results of the simulations are further discussed in detail in the Discussion section and the Supporting Information.

2 Figure. Simulation for calculating ηFRET in model MCSs. a,b) Schematic of distributions of the interchromophore distances and the orientation factor in cases (i) and (iv) used in our simulations. The distributions are shown in the form of heat maps where blue and red indicate the lowest and highest values. c,d) The FRET efficiency (ηFRET) of D20A as a function of ΔDA for various ΔDD values. ηFRET was calculated with given values of ΔDA, ΔDD, and κ2. ηFRET curves for various ΔDD values are plotted following a rainbow color scheme, from red (the largest ΔDD) to purple (the smallest ΔDD). e) ΔηFRET of DnA relative to D1A, that is ηFRET(DnA) – ηFRET(D1A), as a function of ΔDA with the shortest ΔDD (= 0.1) for D2A (red), D4A (orange), D6A (dark yellow), D8A (green), D12A (blue), and D20A (violet). f) ΔηFRET as a function of ΔDD with ΔDA = 0.9, where ΔηFRET is maximized for D20A. In both (e) and (f), results of case (iv) are indicated by filled symbols and those of case (i) are indicated by open symbols. All ΔηFRET curves for case (i) have zero values for all ΔDA and ΔDD regardless of nD. The simulation results of case (ii) and case (iii) are shown in Figure S14 (Supporting Information).

Design of Multiporphyrin Dendrimers

To experimentally test the validity of the results of simulations, we designed and synthesized the MCSs with various n_D and d_DD values. Since it is difficult to experimentally prepare the ideal forms of MCSs as the ones used for the simulations, we instead prepared multiporphyrin dendrimers as the model compounds. The multiporphyrin dendrimers should be an excellent model system because both n_D and d_DD can be adjusted independently by controlling the type, number, and position of linkers while keeping d_DA and the spectral overlap integral for D–A pairs unchanged for each donor.^[6,38] In this context, we synthesized a total of six multiporphyrin dendrimers with various combinations of n_D and d_DD values, as illustrated in Figure 3. For example, SD4A, MD4A, and LD4A have the same n_D value but different d_DD values, and MD2A, MD4A, and MD8A have the same d_DD value but different n_D values. The absorption and emission spectra of dendrimers and their constituent donor and acceptor units are shown in Figure 3c,d. The weak electronic coupling between ground‐state porphyrins in dendrimers was confirmed by the absence of the spectral broadening or shift in their spectra. The Förster radius for the D–A pair was calculated to be 2.43 nm from the static spectroscopic data, as described in the Supporting Information.

3 Figure. Structures of multiporphyrin dendrimers and their structural units. a) Molecular structures of multiporphyrin dendrimers used in this study represented as a function of nD and dDD. The actual structures of the dendrimers are 3D rather than planar, but they are depicted as the planar configurations for the sake of clear presentation. In all of the six dendrimers, each zinc porphyrin donor (D, blue) is connected to a free‐base porphyrin acceptor (A, red) by the linker of the same length. For easier comparison among the dendrimers, the structures of all the dendrimers are drawn on the same frame as SD16A, and the spots without any chromophore are indicated in gray. A dendrimer containing short D–D pairs is labeled as S, a dendrimer containing long D–D pairs without any short D–D pair is labeled as L, and all other dendrimers without any short or long D–D pairs are labeled as M. b) Molecular structures of zinc porphyrin donor (D) unit, free‐base porphyrin acceptor (A) unit, and linkers connecting the D and A units in the multiporphyrin dendrimers. c) Normalized absorption spectra of multiporphyrin dendrimers. d) Absorption and emission spectra of monomeric donor and acceptor.

To estimate the equilibrium structures of the dendrimers, molecular dynamics (MD) simulations were performed for the multiporphyrin dendrimers. The results of MD simulations are shown in Figure 4, and details of MD simulations are described in the Supporting Information. According to the MD simulation, the distributions of d_DA are similar for all donors in a dendrimer, as shown in Figure 4a, and the averaged d_DA distributions for all donors in a dendrimer are similar for all of the dendrimers, as shown in Figure 4b. This result suggests that the mean d_DA of all dendrimers does not change regardless of the variation of d_DD and n_D. Unlike d_DA, however, the averages of d_DD distributions are different for each pair of donors, depending on their geometry in a dendrimer, as shown in Figure 4c. Therefore, in these dendrimers, we can clearly examine the effect of d_DD and n_D on η_FRET, independent of d_DA. The structural information on the porphyrin dendrimers obtained from the MD simulations was used as input for calculating the theoretical η_FRET for the dendrimers using the simulations described above.

4 Figure. Distributions of dDA and dDD of multiporphyrin dendrimers obtained from MD simulations. a) Distributions of dDA for each of the four donors in SD4A. b) Averaged distributions of dDA for MD2A, LD4A, MD4A, and SD4A. The mean value of all the averaged dDA distributions for the six dendrimers is 1.69 ± 0.44 nm. c) Averaged distributions of dDD for short D–D pairs (blue), medium D–D pairs (green), and long D–D pairs (orange) in all six dendrimers. The classification into short, medium, and long D–D pairs are defined in the caption of Figure 3a. The mean value of each averaged dDD distribution is indicated by a dashed line. Examples of short, medium, and long D–D pairs are illustrated in (d). d) Average structures of multiporphyrin dendrimers with four donors. The positions of each D–D pair are indicated. For MD4A, the distance between donors attached to different linkers from the acceptor (2.44 nm) is slightly longer than the distance between donors connected with the same linker from the acceptor (2.34 nm) although both pairs of donors are classified into medium D–D pairs.

FRET Efficiencies of Multiporphyrin Dendrimers

To measure the dynamics of energy transfer in the multiporphyrin dendrimers, femtosecond TA spectroscopy^[39] was employed, as described in the Experimental Section. The results of TA measurement for multiporphyrin dendrimers and monomeric zinc porphyrin are shown in Figure 5; and Figure S10 (Supporting Information). In Figure 5, it can be seen that the TA spectra of the monomeric zinc porphyrin exhibit the decay of the 1.3 ns time constant. Considering the characteristic spectral evolution shown in decay associated spectra and that the singlet excited state of zinc porphyrin species mainly decays via intersystem crossing to a triplet state rather than fluorescence emission, we can assign the TA decay of monomeric zinc porphyrin to intersystem crossing.^[40–44] Unlike the spectral evolution of the monomeric zinc porphyrin, the TA spectra of dendrimers show distinct and fast spectral changes. The ground state bleach (GSB) of donor (543 nm) is reduced, while the GSB of acceptor (517 nm) is increased, and the SE of donor (637 nm) is shifted to longer wavelengths. Based on these features, we attribute the spectral changes of dendrimers to the FRET from donors to the acceptor.

5 Figure. Transient absorption (TA) spectra and decay‐associated spectra (DAS) of monomeric zinc porphyrin and MD2A dendrimer. a,b) TA spectra of the monomeric zinc porphyrin (a) and MD2A (b) at various time delays. c,d) DAS of the monomeric zinc porphyrin (c) and MD2A (d) obtained from the global target analysis of TA spectra. The steady‐state absorption spectra (solid line) and emission spectra (dashed line) of the donor (cyan) and the acceptor (magenta) are shown together at the bottom. The vertical dashed lines indicate the peak positions of steady‐state absorption and emission spectra. Temporal decay of the DAS of e) monomeric zinc porphyrin and f) multiporphyrin dendrimers.

To characterize the dynamics of the time‐dependent spectral changes in the TA spectra of various dendrimers, we globally fitted the TA spectra in the entire spectral range of our measurement. The resultant decay‐associated spectra (DAS) of the monomeric zinc porphyrin and MD2A are shown in Figure 5c,d, respectively, and the temporal decays of those DAS are shown in Figure 5e,f.^[45] Since there exist multiple donor–acceptor pairs in each dendrimer and d_DA has a distribution of values rather than a single value, the dendrimers are expected to exhibit complex, parallel energy transfer dynamics, as manifested in Figure 5f.^[46] To reflect such complicated dynamics, we fitted the decay of TA spectra using a stretched exponential function, which is suitable for describing a relaxation process that takes place in a parallel manner with varying time scales [Image Omitted. See PDF]where τ_DA is the time constant of FRET process and β is related to the degree to which the exponential function is stretched. The fit parameters are presented in Table 1, and η_FRET was calculated using Equation (4).^[47,48] It can be seen that, for various dendrimers, η_FRET differs by up to 11%. Especially, η_FRET of MD8A is higher than those of MD4A and MD2A although these three dendrimers have identical mean values of d_DD and d_DA. For the three dendrimers having four donors (SD4A, MD4A, and LD4A), it can also be seen that η_FRET of a dendrimer with shorter d_DD is larger than that of a dendrimer with longer d_DD. These results show that η_FRET of a dendrimer is enhanced by the increase of n_D or the decrease of d_DD.

1 TableSummary of the time constants and β values observed in TA spectra of multiporphyrin dendrimers and calculated FRET efficiency (η_FRET)

	τDAa) [ps]	βa)	Experimental ηFRETb) [%]	Theoretical ηFRETc) [%]	Modified theoretical ηFRETd) [%]
MD2A	338 ± 29	0.73 ± 0.03	74.0 ± 2.2	72.1	74.1
LD4A	302 ± 4	0.83 ± 0.02	76.8 ± 0.3	74.7	76.5
MD4A	273 ± 4	0.75 ± 0.02	79.0 ± 0.3	77.0	78.6
SD4A	248 ± 18	0.76 ± 0.04	80.9 ± 1.4	78.2	79.8
MD8A	238 ± 17	0.85 ± 0.05	81.7 ± 1.3	78.8	80.4
SD16A	194 ± 15	0.72 ± 0.05	85.1 ± 1.1	83.9–85.8	85.2–87.1

a)

All FRET time constants (τ_DA) and β values were determined from the global fits of the first and second right singular vectors of TA spectra

b)

Experimental η_FRETs were calculated from the time constants obtained from TA measurements by using Equation (S4) (Supporting Information). We note that the decay time constant of the TA spectra of monomeric zinc porphyrins is 1.3 ns

c)

Theoretical η_FRETs were calculated with the simulations using the structural information of dendrimers obtained from MD simulations. We note that the mean value (1.69 nm) of the d_DA distribution in Figure 4b was used

d)

Modified theoretical η_FRETs were calculated with the simulations using the d_DA value of 1.65 nm, instead of 1.69 nm. The d_DA value of 1.65 nm was obtained, while we vary d_DA to match the theoretical η_FRET with the experimental value. Based on the good agreement of the modified theoretical and experimental η_FRETs, we can infer that the value of 1.65 nm is the effective d_DA in the multiporphyrin dendrimers.

To verify whether the simulations accurately describe the FRET dynamics in MCSs, we predicted the theoretical η_FRET of the multiporphyrin dendrimers with the simulations using the structural information obtained from the MD simulations. Details of the calculation of theoretical η_FRET are described in the Supporting Information. For each dendrimer, the theoretical η_FRET is compared with the experimentally determined η_FRET. As shown in Table 1, the theoretical η_FRETs agree well with the experimental η_FRETs, but overall they are underestimated by ≈3.5% compared with the experimental η_FRETs. These discrepancies might be attributed to the limitation of IDA applied to the Förster's equation and the slight inaccuracy of the d_DA value obtained from MD simulations. Therefore, we varied d_DA to match the theoretical η_FRET with the experimental value and calculated the modified theoretical η_FRETs. The modified theoretical and the experimental η_FRETs were in agreement with each other when the d_DA value of 1.65 nm were used, instead of 1.69 nm. The excellent agreement of the modified theoretical and the experimental η_FRETs implies that our simulations well describes the FRET behavior in MCSs.

Discussion

The excitation energy of donors can be transferred to the acceptor either directly or through multiple steps involving homo‐FRET among donors. If the energy transfer via these detour paths involving homo‐FRET is more efficient than the direct transfer to the acceptor, η_FRET will be enhanced compared with when homo‐FRET does not occur. From the simulations of energy transfer for the seven model systems, we comprehensively examined how various structural parameters (n_D, Δ_DD, Δ_DA, and κ²) of an MCS affects η_FRET. In particular, we emphasize how structural heterogeneity in an MCS affects η_FRET by comparing case (i) and the other three cases.

According to previous studies, structural heterogeneity tends to deteriorate η_FRET.^[22–28] However, as can be seen in Figure 2; and Figure S14 (Supporting Information), η_FRET can be enhanced with the aid of structural heterogeneity caused by the presence of multiple donors and homo‐FRET among donors. As shown in Figure 2f; and Figure S14n,o (Supporting Information), η_FRET is enhanced with the decrease of Δ_DD in case (ii), (iii), and (iv), implying that the increase of the homo‐FRET rate (due to the decrease of Δ_DD) activates more detour FRET paths of high efficiencies. As mentioned in the simulation results, the increase of Δη_FRET with the decrease of Δ_DD is saturated below a certain threshold Δ_DD value, of which the reason is discussed in Note S8 (Supporting Information). Considering that the enhancement of η_FRET is not observed in case (i), we can infer that structural heterogeneity in the MCS plays an important role in the enhancement of η_FRET induced by homo‐FRET. In case (i), η_FRET does not change at all with the variation of n_D or Δ_DD, suggesting that homo‐FRET does not affect the FRET efficiency without any structural heterogeneity. In contrast, in cases (ii), (iii), and (iv) with heterogeneity of Δ_DA (and Δ_DD) and/or κ², Δη_FRET is positive and changes sensitively with the variation of n_D and Δ_DD, indicating the strong influence of structural heterogeneity on the homo‐FRET‐induced enhancement of η_FRET. Thus, according to our simulations, only when there exists structural heterogeneity in an MCS, homo‐FRET enhances η_FRET, thus compensating for the loss of η_FRET induced by the structural heterogeneity to some extent.

While we demonstrated that structural heterogeneity is essential for the homo‐FRET‐induced enhancement of η_FRET, it would be desirable to describe their relationship quantitatively. If we define the FRET efficiency, Φ_FRET, of an individual donor, the structural heterogeneity in an MCS can be manifested as a distribution of Φ_FRET for multiple donors. In other words, when there exists structural heterogeneity, the distribution of Φ_FRET would have a finite width. While η_FRET is enhanced in cases (ii), (iii), and (iv), where Φ_FRET has a distribution of values, the Δ_DA dependences of Δη_FRET for those three cases are different from each other, as shown in Figure 2e; and Figure S14j,k (Supporting Information), because the distribution of Φ_FRET arises from different origins in those three cases. For case (ii), the distribution of Φ_FRET is governed by only the distribution of κ². For case (iii), the distribution of Φ_FRET is governed by only the distribution of Δ_DA. For case (iv), the distribution of Φ_FRET is governed by the distributions of both κ² and Δ_DA. At Δ_DA > 1.5 in case (ii) and at Δ_DA < 0.4 in case (iii), Δη_FRET induced by homo‐FRET is negligibly small. Such limited enhancement of η_FRET in those Δ_DA regions indicates that the distribution of Φ_FRET is too narrow to enhance η_FRET. In fact, it can be seen in Figure S17 (Supporting Information) that Δη_FRET is governed by the width of Φ_FRET distribution for cases (ii) and (iii). In contrast, in case (iv), η_FRET is enhanced in the entire Δ_DA region because the variation of κ² and the variation of Δ_DA induces the enhancement of η_FRET at small and large Δ_DA values, respectively, thus complementing each other. In other words, in case (iv), the distributions of both κ² and Δ_DA contribute to the enhancement of η_FRET, as confirmed by Figure S18 (Supporting Information). Further discussions on the dependence of η_FRET on Δ_DA are described in Notes S7 and S8 (Supporting Information).

As shown in Figure 2e; and Figure S14j,k (Supporting Information), η_FRET is enhanced overall by the increase of n_D, in the order of D2A, D4A, D6A, D8A, D12A, and D20A. Such dependence of η_FRET on n_D can be attributed the increase of donors that are available for homo‐FRET. However, it is noteworthy that the Δη_FRET is not exactly proportional to n_D when Δ_DD is larger than ≈0.7, as shown in Figure 2f; and Figure S14n,o (Supporting Information). For example, at Δ_DD = 1.3, the Δη_FRET increases in the order of D2A, D4A, D8A, D20A, D6A, and D12A, as shown in Figure S21 (Supporting Information). To find the origin of the deviation from the dependence on n_D, we checked how the average homo‐FRET efficiency depends on Δ_DD. At Δ_DD = 0.1, both the average homo‐FRET efficiencies to the nearest and the second nearest donors are close to 100%. By contrast, at Δ_DD = 1.3, the average efficiency of homo‐FRET to the nearest donors is 20%, while that to the second‐nearest donors is only 5% at best. Such a large difference in the homo‐FRET efficiencies between the nearest and the second‐nearest donors means that only homo‐FRET to the nearest donors is effective. Base on this observation, for all model systems, we checked the number of the nearest donors (n_ND), which is 2, 3, 3, 3, 4, and 5 for D2A, D4A, D8A, D20A, D6A, and D12A, respectively. Indeed, the order of the model systems in terms of n_ND exactly matches the observed order of model systems in terms of η_FRET, indicating that η_FRET is governed by n_ND rather than n_D, the total number of donors. To further verify the influence of n_ND on η_FRET, we also performed simulations for pseudomodel systems where all donors are equally spaced from each other, that is, n_ND equals n_D. As shown in Figure S22 (Supporting Information), the simulation results show that the Δη_FRET is proportional to n_D for such model systems. Thus, the Δη_FRET is governed n_ND rather than n_D. Ultimately, the 3D arrangement of chromophores and the extent of structural heterogeneity in an MCS would be important factors that determine the homo‐FRET‐induced enhancement of η_FRET because n_ND would be determined by these factors.

We can compare our simulation results with those of the previous studies, as discussed in Note S9 (Supporting Information). Notably, the simulations performed in a previous spectroscopic study on DNA‐fluorophore MCSs showed that when d_DD is increased by 54% from 13 Å (Δ_DD = 0.25) to 20 Å (Δ_DD = 0.38), η_FRET changes only by 3% and suggested that η_FRET is not much affected by the rate of homo‐FRET, as long as the homo‐FRET rate is significantly faster than the hetero‐FRET rate.^[21] These results are consistent with ours. Such insensitivity of η_FRET to Δ_DD should be attributed to the limited Δ_DD range (0.25–0.38) examined in that study. As can be seen in Figure 2f, our simulations show that Δη_FRET is saturated in that Δ_DD range. On the other hand, based on their own simulation, they inferred that the enhancement of η_FRET induced by homo‐FRET occurs only when homo‐FRET is faster than FRET. By contrast, our simulations suggest that η_FRET can be enhanced even when Δ_DD is larger than Δ_DA (that is, homo‐FRET is slower than FRET), as can be seen in Figure 2f (Δη_FRET > 0 at Δ_DD > Δ_DA). This result of our simulations implies that homo‐FRET still effectively funnels the excitation energy toward the detour FRET paths of high efficiencies even when homo‐FRET is slower than FRET. In agreement with this result of our simulation, our TA measurements show that η_FRET is enhanced in the dendrimers investigated in this work, all of which have smaller Δ_DA (0.7) than Δ_DD (1.0 for SD4A and SD16A, 1.2 for MD2A, MD4A, and MD8A and 1.5 For LD4A). In a similar study on MCSs based on DNA origami structure,^[29] η_FRET was determined from the measured fluorescence lifetime of the donor, while n_D being varied from one to four, and it was found that η_FRET was barely affected by the increase of n_D. This observation can be easily rationalized by noting that these MCSs correspond to conditions where η_FRET is hardly affected by n_D. Similarly, in a study on MCSs consisting of various DNA networks,^[28] it was found that η_FRET does not show clear dependence on n_D. Here we can interpret that such insensitivity of η_FRET with respect to n_D can be attributed to the low formation efficiency of some of the MCSs used in that study. In fact, when only the MCSs of high formation efficiencies are considered, it can be clearly seen that η_FRET is enhanced with the increase of n_D. In addition, for the MCSs of high formation efficiencies, Δ_DA dependence of Δη_FRET is also in good agreement with the results of our simulations. Thus, our simulations can account for the conflicts of the previous studies and serve as a tool that properly describes the η_FRET in MCSs.

We note that we used IDA for all of the simulations. One of the limitations of IDA is that it cannot account for the energy transfer between orthogonal chromophores, which should be treated by taking into account molecular vibrations and environmental fluctuations.^[49–53] The error caused by IDA generally increases with the decrease of the d_DA distance and the increase of the size of the chromophore.^[36] The porphyrin dendrimers investigated in our work have d_DA of 16.9 Å and the porphyrin unit has a size of ≈7 Å,^[54,55] giving the D–A separation of ≈2.4 in molecular size unit. According to the previous work,^[36] with this D–A separation value, the error is estimated to be ≈7.5%, which is comparable to the ≈3.5% deviation between the η_FRET estimated using IDA and the experimentally measured value in our work. Since such an error due to IDA will apply to all porphyrin dendrimers systematically, it should not affect the relative η_FRET of the porphyrin dendrimers, which is of our interest in this work. In fact, the variation of the calculated η_FRET for different porphyrin dendrimers is in excellent agreement with the trend of the experimentally measured values for those dendrimers (see Table 1). While the correction using the exact approach instead of IDA will reduce the error, our approach using IDA requires only simple calculation and can be easily applied to MCSs of various chemical structures.

Conclusion

Based on the simulations on model MCSs for four different cases and the TA measurements on multiporphyrin dendrimers, we revealed the followings on the energy transfer efficiencies and the structural parameters of the MCSs: i) η_FRET is enhanced by homo‐FRET only when there exists structural heterogeneity in the MCS; ii) the enhancement of η_FRET is directly related to the distribution of Φ_FRET, of which the width is governed by the distribution of κ² at small Δ_DA values and by the distribution of Δ_DA at large Δ_DA values; iii) η_FRET is enhanced by the increase of n_D, especially by the increase of n_ND. Based on these lessons learned from this study, we can explain why the conclusions from previous studies seemingly conflict with each other. The excellent agreement between the experimental and the calculated η_FRETs (see Table 1) demonstrated for the systematically designed MCSs investigated in our study would provide a principle for accurately predicting η_FRET in various MCSs. Our findings can help a better understanding of energy transfer in MCSs, including natural light‐harvesting systems, and provide a design principle for novel optoelectronic devices.

Experimental Section

Calculation of η_FRET for Model Multichromophore Systems

For each model MCS, η_FRET was calculated at given Δ_DA and Δ_DD values by averaging Φ_FRET, the FRET efficiency of an individual donor, obtained from 100,000 simulations, which are large enough to sample the distributions of structural parameters. Each simulation consists of the Monte Carlo processes to calculate the parameters necessary for setting up multiple differential equations to determine the time‐dependent excited‐state populations of donor and acceptor, the solution of the differential equations which provide the excited‐state populations of chromophores as a function of time at given Δ_DA and Δ_DD values, and the calculation of Φ_FRET, by dividing the final excited‐state population of the acceptor by the initial excited‐state population of the donor. For the calculation of Φ_FRET, it was assumed that the intrinsic decay rate of the excited‐state population of the acceptor was negligibly small. Under this condition, the excited‐state population of the acceptor was changed only by FRET from the donor. For each model MCS, such simulations were performed for all four cases discussed in the main text (cases (i)–(iv)). For cases (i) and (ii) where Δ_DA (and Δ_DD) has a single value, 25 Δ_DA values and 25 Δ_DD values were used between 0.1 and 2.5 with the increment of 0.1. For cases (iii) and (iv) where Δ_DA (and Δ_DD) has a finite distribution, a random number was generated for Δ_DA (and Δ_DD) following its Gaussian distribution. The mean value of the Gaussian distribution was increased from 0.1 to 2.5 with an increment of 0.1. The relative standard deviation of the Gaussian distribution was set to be 0.2.

In general, MCSs are composed of various types of chromophores. For simplicity, however, only a single type of energy‐donor chromophore and a single type of energy‐acceptor chromophore were considered for seven ideal model MCSs investigated in the study. In those model systems, an acceptor is surrounded by donors that are located as far apart from each other as possible. Each donor can have a distribution of Δ_DA for cases (iii) and (iv), but the mean Δ_DAs for all donors were set to be identical for all donors so that the effect of the variation of Δ_DA on η_FRET can be excluded. In addition, the mean Δ_DDs between the nearest donors were also set to be identical for all pairs of the nearest donors. For simulations on model MCSs with the number of n_ND set to be the same as n_D, the mean Δ_DDs among all pairs of donors were set to be equal. In the simulations, the excitation probabilities of all donors were set to be equal, but the simultaneous excitation of two or more donors was not considered. The direct excitation of the acceptor was not considered, either. The unidirectional D–A energy transfer as well as the D–D energy transfer were considered.

Energy transfer among multiple chromophores of two different types cannot be described only with a single Förster's equation but can be described with a combination of multiple Förster's equations. Therefore, differential equations that describe energy transfer for a model MCS consisting of n donors and a single acceptor can be expressed as follows [Image Omitted. See PDF]where D_i(t) and A(t) denote the excited‐state populations of the ith donor and the acceptor, respectively; k_DA and k_DD denote the rate constants of FRET and homo‐FRET, respectively; k_D and k_A are the inverse of intrinsic lifetimes of the singlet excited states of the donor and the acceptor, respectively.

Prior to solving the multiple differential equations shown in Equation (2), the rate constants (e.g., k_D, k_DaA, k_DaDb) included in the equations must be determined. Each rate constant is determined by k_D, κ², and Δ_DA (and Δ_DD), as shown in the following equation [Image Omitted. See PDF]where κ² is the orientation factor, which is the relative orientation between the transition dipole moments of the donor and acceptor chromophores; d_DA and d_DD are the D–A and D–D distances, respectively; Δ_DA and Δ_DD are normalized interchromophore distances obtained by dividing d_DA and d_DD by the Förster radius for the D–A or D–D pair, respectively; R̄₀ is Förster radius when κ² is ⅔; n is the refractive index of the solvent and was set to be a constant; N_A is Avogadro's number; Q_D is the quantum yield of the donor emission; J is the spectral overlap integral between the emission spectrum of the donor and the absorption spectrum of the acceptor. Since FRET is a mechanism for describing energy transfer in the regime of weak electronic coupling, the electronic interactions between all of the chromophores were assumed to be weak and therefore Q_D and J were set to be constants. In contrast, d_DA, d_DD, and κ² were adjusted as major variables in the simulations to examine the effect of n_D and d_DD (and d_DA) on η_FRET. k_D was set to be 1000 ps⁻¹ in the simulations. The orientation factor (κ²) is defined as follows [Image Omitted. See PDF]where the angle θ represents the angular position of the acceptor relative to the donor's transition dipole vector, and ω is the angle between the acceptor transition dipole vector and the electric field of the donor at the location of the acceptor. Since these angles are assumed to be randomly distributed at the isotropic limit, κ² for each FRET pathway can be represented by two random numbers. For cases (i) and (iii) corresponding to the dynamic isotropic limit, κ² value was set to be ⅔, which is the average value of the isotropic distribution. For cases (ii) and (iv) corresponding to the static isotropic limit (Figure S12, Supporting Information), random numbers were generated for the two angles (that is, θ and ω) within the range of 0–2π, and calculated κ² with Equation (4). The approach of calculating η_FRET at the dynamic and static isotropic limits used in the simulations is similar to the one used in previous studies on the FRET behavior at those limits.^[21,23]

Frequency‐Resolved Femtosecond Transient Absorption Spectroscopy

TA spectra were measured with femtosecond laser pulses using a visible pump–broadband probe scheme. The femtosecond pulses at the center wavelength of 800 nm were generated from a Ti:sapphire amplified laser (Coherent Legend Elite) and split into pump and probe beams. On the pump arm, the 800 nm pulses were converted into the 540 nm pulses with the bandwidth of 16 nm using a home‐built, all‐reflective‐optic noncollinear optical parametric amplifier. Since, at 540 nm wavelength, the absorption of the donor is at least 8 times larger than that of the acceptor, the pump wavelength was selected of 540 nm to preferentially excite donors. The pump pulses were sent through a pair of Brewster‐cut fused‐silica prisms to precompensate for the dispersion obtained from transmissive optics and compressed to near‐transform‐limited pulses at the sample position. The pulse energy of the pump pulses was varied from 50 to 200 nJ to check the dependence of transient absorption signal on the excitation energy. It should be noted that, to prevent the excitation of multiple donors, the pump pulses with the pulse energy at the minimal level (50 nJ per pulse) were used for the measurement of TA spectra. On the probe arm, the laser pulses at 800 nm were sent into a sapphire window of 3 mm thickness and converted into white light continuum. The visible portion (460–725 nm) of the white light continuum was used as broadband probe pulses without further compensation of the dispersion. The probe pulses were time‐delayed with respect to the pump pulses using a motorized translation stage (Newport, M‐ILS150HA). By recording the “pump‐on” and “pump‐off” probe spectra, the differential transmission (ΔT/T) spectra were obtained as a function of time. The spectra of transient signal and the reference were detected by a spectrometer (Andor, SR303i) equipped with a CCD (Andor, DU420A). In all the TA measurements, the polarization of the pump pulses was set to be at the magic angle (54.7°) relative to the probe polarization in order to prevent the contribution of anisotropic components of the TA signal. For the TA measurement, porphyrin dendrimers were dissolved in tetrahydrofuran (THF). The porphyrin dendrimer solution in THF was prepared with an optical density of ≈0.4 at its absorption maximum of 543 nm in THF in a glass cuvette of 1 mm thickness.

Statistical Analysis

Microsoft Excel and OriginPro were used for the statistical analysis of the data presented in this work. The preprocessing and data presentation were performed in the following way. Data in Figure 2c,d represent the averaged η_FRET of an individual donor, obtained from 100,000 simulations. The standard error of the mean (SEM) of the η_FRET is less than 0.05%. The mean ± SD of data in Figure 4a are 1.69 ± 0.45 nm. Figure 4b shows mean ± SD (1.69 ± 0.44 nm). The mean ± SD of data in Figure 4c are 1.92 ± 0.40 nm (top panel), 2.34 ± 0.61 nm (middle panel), and 2.85 ± 0.76 nm (bottom panel). Table 1 shows mean ± SD for time constants and β values, which were determined from the global fits of the first and second right singular vectors of TA spectra. Table 1 also shows mean ± SD of the experimental η_FRET. The mean values of theoretical FRET efficiency and modified theoretical η_FRET are shown in Table 1, and their SEMs are less than 0.05%.

Acknowledgements

The authors thank Dae Won Cho and Jaeyoung Sung for their initial help and discussion on the project. This work was supported by the midcareer research project (No. 2020R1A2C3004520) of the National Research Foundation of Korea. This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT, and Future Planning (No. NRF‐2016R1E1A1A01941978). This work was supported by Institute for Basic Science (No. IBS‐R004).

Conflict of Interest

The authors declare no conflict of interest.