Discrete coordination nanochains based on photoluminescent dyes reveal intrachain exciton migration dynamics

Abstract

Elucidating exciton migration in polymer chains has been one of the major research goals in photophysics for over half a century. While great efforts have been made to understand picosecond phenomena by ultrafast spectroscopy, ambiguous molecular conformations and/or random polymer sequences have hindered the construction of an ideal exciton migration model. Here we present the creation of unique end-capped coordination nanochains and quantitative description of intrachain exciton migration therein. The nanochain features unique molecular architectures in discrete polynuclear complexes, with a linear and rigid structure, the defined number of metal nuclei, and charge neutrality. These features allow well-defined arrangement of emissive dye moieties, making the nanochain a sound platform for studying exciton dynamics. Readily accessible absorption spectroscopy, and photoluminescence lifetime and quantum yield measurements allow the construction of continuous-time Markov chains model, thereby estimating non-trivial exciton migration across the metal center.

Introduction

Over 50 years, researchers have discovered that the random walk theory^1,2,3,4 is applicable to nanoscale photophysical phenomena in polymer chains, estimating their exciton hopping behaviors^{5,6,7,8,9,10,11}. In most of resent study, ultrafast laser spectroscopy plays a key role in disclosing fast exciton dynamics where the time scale of observable phenomena depends on the performance of lasers and detectors. However, the high conformational freedom of typical polymers has hampered the elucidation of quantitative exciton transfer phenomenon. Meanwhile, Wang et al. utilized a designed skeleton of metal-organic frameworks (MOFs) to study exciton diffusion based on a fluorescence resonance energy transfer (FRET) model¹². Therein, donor molecules were incorporated into the MOF framework to study on exciton transfer to acceptor molecules adsorbed in its mesopores, although the arrangement of the acceptor molecules is hard to determine. We have previously developed a random coordination copolymer (P1, Fig. 1a) based on bis(dipyrrinato)zinc(II) complexes by employing two different dipyrrin bridging ligands (BL1 and BL2 as exciton acceptor and donor, respectively)¹³. As the constituents are analogous to well-known FRET-based BODIPY probes^{14,15,16,17,18,19}, the copolymers emit bright photoluminescence (PL). Unlike conventional flexible polymers, this series of coordination wires features a robust skeleton, which was visualized by AFM as straight single strands^13,20,21. This feature is beneficial in the study of exciton migration, as structural fluctuation upon bending and folding, which affects the phenomenon largely, is circumvented. Also, the fixed distance and orientation between the dipyrrin moieties and localization of a photogenerated exciton on a dipyrrin in a chain allow for the calculation of Förster-type intramolecular energy transfer rates. With these structural advantages in hand, we performed, to P1, a combination of steady-state measurements of photophysical properties and a numerical simulation on photoluminescence (PL) efficiencies with a random walk model, revealing intrachain exciton transfer with an estimated hopping rate of 24–43 ns^–1 (23–41 ps as a time constant). It is remarkable that contrary to the previous studies with transient spectroscopies, our strategy can understand picosecond phenomena with a combination of steady-state techniques and an easy-to-access nanosecond spectrometer used for PL lifetime measurement. However, the out-of-control coordination reaction in P1 would generate intrinsically random donor and acceptor arrangements in the copolymer chain, as is similar to other copolymer systems^{13,22,23,24,25,26,27,28,29,30}. In our previous study, such indefinite arrangements actually compelled us to adopt not only mathematical but also statistical procedures and to neglect several hopping paths. In order to build a more realistic and sophisticated random walk model, precise control of polymer sequences is requested.

Fig. 1: Coordination nanochains of this work.

a Synthetic scheme of coordination random copolymer P1¹³. b Synthetic scheme of end-capped coordination nanochains C1_N and C2_N. c Molecular design concept of C1_N for the purpose of the present work.

In this work, end-capped multinuclear complexes (hereafter called nanochains) C1_N (Fig. 1b; N is the number of metal centers, N = 1 – 16) are synthesized and isolated in every N. Instead of two bridging ligands BL1 and BL2 used in the random copolymer (Fig. 1a), end-capping ligand L1 and bridging ligand BL3 are used for the construction of C1_N (Fig. 1b). L1 corresponds to the half structure of BL1, thereby behaving as an exciton acceptor. BL3 is an analog of BL2 with branched alkyl chain groups introduced at the phenylene linker for the improved solubility of the resulting nanochains, serving as an exciton donor. The complexation of these ligands to zinc(II) ions formed a mixture of end-capped multinuclear complexes, and each discrete coordination nanochain was separated by size-exclusion chromatography (SEC). Additionally, nanochains C2_N (Fig. 1b; N = 1 – 10) were prepared in a similar manner by substituting L1 with L2. C2_N lacks the acceptor dipyrrin unit at the terminus, such that it serves as a reference for C1_N in the exciton migration study. From the viewpoint of coordination chemistry, these nanochains are also characterized as an unparalleled molecular architecture in the category of linear multinuclear complexes. Peng^31,32, Shionoya³³ and Murahashi³⁴ reported metal-cluster-type complexes where coordination bonds are used for assembling metal ions and do not influence their chain elongation processes. In contrast, discrete coordination chains have been poorly developed. Their main chains are constructed based on coordination bonds, and the molecular strands have defined lengths depending on the number of metal atoms. To the best of our knowledge, C1₁₆ is the second largest linear multinuclear complex after Re(I) oligomers bearing 20 complex units reported by Ishitani and coworkers in 2008³⁵. However, the one-dimensional structures are intrinsically different with each other: The Re(I) oligomer has winding and deformable structures due to the flexibility of phosphine bridging ligands, while our nanochain has a straight and solid structure thanks to the rigidity of the bidentate dipyrrin bridging ligand (evidenced later). Besides, charge neutrality is another advantage in the nanochain to circumvent electronic perturbation by counterions, while the Re(I) oligomer is cationic. The critical advantage of the nanochain over random copolymer P1 (Fig. 1a) is that the former has a defined sequence (Fig. 1b). Together with the rigid architecture mentioned above, the structural feature of C1_N allows it to serve as an ideal platform for exciton migration study (Fig. 1c; detail discussed in a later section). Here, the movement of an exciton in nanochain C1_N is described with continuous-time Markov chains (CTMC), a generalized random walk. By constructing a CTMC model for the exciton migration, and simulating PL lifetime and quantum yield measurement data thereby, we can estimate non-trivial exciton transfer across the metal center.

Results

Preparation of end-capped coordination nanochains

In order to obtain nanochain C1_N, a dichloromethane (DCM) solution of bridging ligand H₂BL3 and capping ligand HL1 in a ratio of 5:2 was prepared, and then a methanol solution of zinc acetate was added to the solution and stirred overnight. As chelation reaction between dipyrrins and zinc(II) ions proceeds spontaneously without adding any base, the resulting mixture is supposed to consist of various multinuclear complexes (e.g., mononuclear, dinuclear, trinuclear, etc) with the capping ligands at both ends of a linear chain (Fig. 1b,c). Without the capping ligand, a powder of the homopolymer (P2, Supplementary Fig. 7) is obtained from the reaction mixture as reported in our previous studies^13,20,21. Therefore, the capping ligand plays an important role in producing discrete complexes as it stops the chain elongation. The reaction mixture was evaporated under vacuum, and the resulting product was dissolved again in chloroform. This solution was then subjected to SEC to separate and purify each product (Supplementary Figs. 9–11). Figure 2a represents the second cycle of SEC, clearly showing the product contains more than ten compounds. Note that the peaks are located regularly, so the mass differences between two adjacent peaks seem constant. This indicates that both ends of the complexes are capped with dipyrrinato capping ligand, L1, and the number of bridging ligands is the difference between the complexes. Once after the removal of acetic acid, the complexes are quite stable and ligand dissociation or exchange does not happen as demonstrated in previous studies^{13,22,23,24,25}. This feature allowed us to isolate the multinuclear complexes and investigate the properties of the individual products. They are thermodynamically produced considering the yield of each nanochain (Supplementary Fig. 12). Similarly, another set of nanochain C2_N was prepared using the capping ligand HL2 whose dipyrrin structure is the same as those of the bridging ligand. Here all the complex units become homoleptic which is reminiscent of homopolymer P2 (Supplementary Fig. 7).

Fig. 2: Characterization of nanochain C1_N.

a SEC chromatogram of the second cycle of C1_N separation. b Aromatic region of ¹H NMR spectrum of C1₁₀ measured in chloroform-d. c ESI-MS spectrum of C1₁₀ (upper panel) with a calculated spectrum for [M]⁵⁺ (lower panel).

Characterization of coordination nanochains

All the obtained fractions after SEC separation were characterized by ¹H NMR spectroscopy. Figure 2b depicts the ¹H NMR spectrum for C1₁₀ focusing on the aromatic region. In the aromatic region (7.15 – 6.15 ppm), there are peaks derived from the proton of the mesityl group of L1 (H_a, 6.98 ppm), the phenyl proton of L1 (H_b, 7.11 – 7.09 ppm), and two kinds of protons at γ and β positions of the BL3 dipyrrins, H_c (around 6.60 ppm) and H_d (around 6.23 ppm), respectively. Integration of these peaks revealed the ratio of L1 to BL3 is 2 to 9, which is consistent with the ideal ratio for decanuclear complex C1₁₀. Similarly, the N^th fraction (N = 1 – 16) was identified as end-capped dipyrrin-zinc complex C1_N with (N − 1) bridging ligands (Supplementary Fig. 14). Also, electrospray ionization mass spectrometry (ESI-MS) was used for further characterization of the complexes (Fig. 2c). All the fractions were investigated, and mononuclear to decanuclear complexes (C1₁ – C1₁₀) were detected from the corresponding fractions (Supplementary Fig. 15). As the molecular chain becomes longer, it was detected with an increased number of charges. C1₁₀ was detected as a 5+ charged ion. Ionization of larger complexes (C1₁₁ to C1₁₆) was not successful, only detecting their decomposed fragments, probably due to their highly anisotropic structures. However, in addition to the ¹H NMR investigation, the regulated location of the SEC peaks and mass detection of up to decanuclear complexes supported the formation of the end-capped 11- to 16-nuclear complexes. These coordination nanochains are unique among discrete metal complexes in that they have robust and straight chain structures, charge neutrality, and monodispersity. In addition, mononuclear to decanuclear complexes were separated in the case of C2_N. In order to confirm the orthogonal and linear structure of the coordination nanochain, reference nanochain C3_N (N = 1 – 4) was prepared from HL1 and H₂BL4 lacking the long and branched alkyl chain (Supplementary Figs. 16,17). The stability of the coordination nanochains in solution and more rigid structure of the bridging ligand BL4 afforded a single crystal of C3₂ suitable for X-ray diffraction analysis. We note the single crystal of C1_N was not obtained due to the existence of the long alkyl chain. The single crystal structure of C3₂ disclosed the linearity of the nanochain as designed (Supplementary Fig. 18 and Supplementary Table 1).

Coordination nanochain design

A distinctive structural feature of nanochain C1_N is that the relative conformations of all dipyrrin moieties are orthogonal or parallel. To demonstrate this, we conducted DFT optimization for mononuclear model complex MC-1 (Fig. 3a,b, Supplementary Table 2) and dinuclear one C3₂ (Supplementary Fig. 19 and Supplementary Table 3). MC-1 is an asymmetric complex constructed from end-capping ligand L1 and ligand BL3’: The latter corresponds to half of bridging ligand BL3. The dihedral angle of MC-1 exhibited the orthogonality at the zinc center, with a dihedral angle of 89.99°. Meanwhile, in C3₂, the two dipyrrin ligands belonging to the same bridging ligand feature a parallel conformation to each other (average dihedral angle: 0.00°). The orthogonal and parallel conformation was supported further by the single crystal X-ray structure of C3₂, with dihedral angles of 89.42° and 0.00°, although the dipyrrin moiety was perturbed by the crystal packing (Supplementary Fig. 18). Rigidity is another structural feature of C1_N. The bidentate coordination of the dipyrrin moiety to the Zn center prevents free rotation at the Zn-N bond. The introduction of methyl groups at the α-position of all dipyrrin moieties prevents structural changes relating to the Zn coordination sphere. The rigidity of the nanochain is ensured by the fact that the linear structure of a single strand of a coordination polymer that contains the nanochain as a substructure was visualized by AFM²⁰. Furthermore, the charge neutrality of the bis(dipyrrinato)zinc complex motif is also advantageous, as it prevents structural perturbation by the counter ions. The orientation of the transition dipole moment of each dipyrrin site is also critical. The lowest transition in a dipyrrin ligand is a π- π* transition, which is reported to be oriented in the long-axis direction (vide infra)³⁶. Exciton transfer in the nanochain is in principle based on continuous Förster exciton migration processes. In this scheme, the orientation between the transition dipole moments of two dye moieties, is a critical factor. The structural features of C1_N and the directionality of the transition dipole moment of the dipyrrin moiety can greatly simplify the exciton hopping model as follows (Fig. 1c). (1) Förster energy transfer between orthogonal dipyrrin sites is neglected with one exception, while that between parallel dipyrrin sites is taken into account. (2) The rigid structure of C1_N may significantly reduce the number of Förster energy transfer processes to be considered. If molecular chains are flexible, as in common polymers or multinuclear complexes, a huge number of Förster energy transfer processes must be considered depending on possible conformations. (3) The presence of capping ligand L1 in nanochain C1_N is also important: PL from L1 allows us to spectroscopically demonstrate that the exciton reaches the end of the nanochain, which is not possible in the case of nanochain C2_N lacking L1. For these reasons, nanochain C1_N is a suitable molecular platform for elucidating exciton transfer processes based on the CTMC model; such molecular systems are rarely realized.

Fig. 3: Photophysical properties.

a Molecular structure of model mononuculear complex MC-1. The dipyrrin ligand labeled as BL3’ is regarded as the half of bridging ligand BL3. b Optimized structure of MC-1 by DFT calculation (B3LYP/6-31 g + (d)). c Normalized UV/vis absorption spectra of coordination nanochains C1_N (N = 2 – 16) in toluene. d Relationship between the peak ratio of the absorption bands at around 490 nm and 554 nm, and N in C1_N (N = 2 – 16) in toluene. e Kohn–Sham occupied and unoccupied orbitals of MC-1 that chiefly contribute to the S₀→S₂ and S₀→S₆ transitions calculated at the level of B3LYP/6-31 g + (d). f Transition dipole moments of MC-1 for the S₀→S₂ and S₀→S₆ transitions. g, h PL spectra of C1_N (N = 2 – 16) excited at (g) 554 nm and (h) 490 nm in toluene.

UV/vis absorption spectra

Isolated nanochain series C1_N (N = 2–16) were dissolved in toluene, and their UV/vis absorption spectra were measured. The normalized spectra are shown in Fig. 3c. C1_N has two absorption bands in the visible region, with peak tops at around 490 nm and 554 nm, respectively. As N increases, the relative intensity of the absorption at 554 nm decreases. In addition, the intensity of the absorption peak at 490 nm relative to that at 554 nm increases linearly with N–1, the number of bridging ligand BL3 (Fig. 3d). In contrast, the absorption peak at 554 nm is not observed in the C2_N series, which does not have L1 (Supplementary Fig. 20). Therefore, the absorption bands at 490 nm and 554 nm originate from bridging ligand BL3 and capping ligand L1, respectively. In addition, when compared with the spectra of free-base dipyrrin proligands H₂BL3 (439 nm) and HL1 (508 nm), which are the starting materials of C1_N, we notice that both of the two absorption bands of C1₂ undergo redshifts (Supplementary Fig. 21). Such redshifts are characteristic of the formation of a bis(dipyrrinato)zinc complex motif²⁰. This series of absorption spectral results also ensures nanochain C1_N based on coordination bonding is formed and exist stably. Furthermore, the absorption peaks at around 490 nm derived from the bridging ligands are almost identical in C2₂ and C3₂ (Supplementary Fig. 22). The bridging ligand of C3₂ (BL4; Supplementary Fig. 16) has a durene linker, the four methyl groups of which provide orthogonality with the dipyrrin moiety²⁰. From the spectral comparison and previous finding, we may conclude that the two branched alkyl groups at the 2- and 5-positions of the arene linker of BL3 are sufficient to maintain the rigidity of the dipyrrin ligand and the orthogonality with the dipyrrin unit^37,38,39,40.

Next, we investigated the electronic transitions of nanochain C1_N. Here we employed model mononuclear complex MC-1 (Fig. 3a). MC-1 corresponds to the terminal part of nanochain C1_N, thereby being a suitable model for the investigation of interest. MC-1 was synthesized in practice, measuring and comparing its normalized absorption spectrum with that of nanochain C1₂: It is notable that they are almost identical (Supplementary Fig. 23), indicating the validity of MC-1 as a model. Electronic transitions corresponding to the photoabsorption of MC-1 were characterized using time-dependent density-functional theory (TD-DFT) calculations with the level of B3LYP/6-31 G + (d) at the molecular geometry optimized at the ground state with the same level of theory. Figure 3e shows the main molecular orbitals of MC-1 contributing to the S₀-S₂ and S₀-S₆ transitions with large oscillator strengths. The S₀-S₂ transition is a ¹π-π* transition localized in L1, and the S₀-S₆ transition is that localized in BL3’, which corresponds to half of bridging ligand BL3 (Fig. 3a). The contribution of the zinc center to both transitions is negligible. Supplementary Fig. 24 shows the oscillator strengths and transition energies of the S₀-S₂ and S₀-S₆ absorptions estimated by TD-DFT calculations. S₀-S₂ corresponds to the absorption band at 554 nm, and S₀-S₆ corresponds to that at 490 nm. Note that S₀→S₁, S₀→S₃, S₀→S₄, S₀→S₅ possess negligible oscillator strengths (Supplementary Fig. 25). The calculated excitation energies are overestimated compared to the experimental values. We note such overestimation has been frequently reported for dipyrrin-BF₂ complexes (BODIPYs), the main transition of which is the ¹π-π* transition localized at the dipyrrin ligand, similar to dipyrrin-zinc complexes⁴¹. This series of results indicates the dipyrrin site of BL3 in C1_N can be selectively excited with 490 nm light, whereas that of L1 can be done with 554 nm light. Figure 3f shows the transition dipole moments of the S₀-S₂ and S₀-S₆ transitions in MC-1. Both moments are nearly aligned with the long axis of the dipyrrin ligand. In addition to the orthogonality of the two dipyrrin ligands coordinated to the same zinc center (Fig. 3b), the two dipole moments were also shown to be orthogonal to each other. Therefore, the design concept for the orientation of the transition dipole moments of C1_N (Fig. 1c) is also verified. The absorption spectra of C1_N and C2_N were comprehensively recorded in toluene, a mixture of toluene and DCM (1:1 v/v), and DCM (Fig. 3c; Supplementary Figs. 26–30).

Steady-state PL

Supplementary Fig. 31 shows the PL spectrum of homopolymer P2 dispersed in toluene. Therein, P2 was excited at 490 nm, which is the absorption maximum of constituting bridging ligand BL3, to find PL at 512 nm. This PL is derived from either dipyrrin site of BL3²⁰. Similar PL behavior was found in nanochain C2_N (N = 2–10) in toluene (Supplementary Fig. 32), reflecting the structural similarity to P2. We also measured the PL spectra of C2_N in a mixture of toluene and DCM (1:1 v/v), and DCM (Supplementary Figs. 33,34), which again detected PL from BL3. In contrast, nanochain C1_N exhibited more complex PL behavior. When C1_N (N = 2–16) was illuminated with 554 nm light, L1 got photoexcited directly, and PL from L1 was observed at 579 nm (Fig. 3g). On the other hand, when C1_N was irradiated with a 490 nm light source, the excitation light is dominantly absorbed by the dipyrrin moieties of BL3. Nonetheless, as shown in Fig. 3h, the PL spectra of C1_N upon excitation at 490 nm featured a dominant peak at 579 nm, similar to the case of 554 nm excitation. This indicates most portion of excitons generated at BL3 transmitted to L1 located at the terminus of the nanochain before being deactivated at BL3. The efficient exciton transfer from BL3 to L1 was also supported by the excitation spectra of C1_N monitored at 579 nm (Supplementary Fig. 35). Therein, both the 490 and 554 nm bands contribute to the PL, with a spectral shape and N-dependence similar to the absorption spectra of C1_N. When the PL spectra of C1_N upon excitation at 490 nm are enlarged in the range of 500 – 550 nm, we can faintly find PL at 512 nm, the intensity of which slightly grows with N increases (Fig. 3h). This implies the efficient exciton migration from BL3 to L1 is slightly deteriorated as the chain length of C1_N increases. Here, interchain exciton migration needs not considering because an exciton in a nanochain decays before another nanochain comes close (SI Section 4.3). Similar PL spectra were obtained for C1_N in other solvents, DCM and a 1:1 mixture of toluene and DCM, showing the absence of significant solvatochromism (Supplementary Figs. 36–38).

PL quantum yield and lifetime

Steady-state PL spectroscopy revealed a qualitative exciton transfer from bridging ligand BL3 to end-cap ligand L1 in nanochain C1_N. The absolute PL quantum yield (PLQY) and lifetime (τ_PL) measurements were then performed to obtain information for the quantitative analysis of the exciton transfer (Fig. 4a-c, Supplementary Tables 4–6) in nanochain C1_N (model is shown in Fig. 4d). Before moving on to nanochain C1_N, we first focus on nanochain C2_N that lacks L1. Supplementary Figs. 39–41 and Supplementary Tables 7–10 summarize the ϕ_{PL_490} (PLQY upon excitation with 490 nm light) and τ_PL values of C2_N (N = 1-10). As the polarity of the solvent increases, the ϕ_PL value of C2_N decreases. This is because C2_N is composed of homoleptic bis(dipyrrinato)zinc complex units, which were reported to exhibit PL quenching in polar solvents^22,23,25. ϕ_{PL_490} did not show significant N dependence, except for C2₁, which does not have bridging ligand BL3. Neither solvent nor N value dependence was observed for τ_PL (Supplementary Table 10). These results indicate that the oligomerization of bis(dipyrrinato)zinc complex to form the nanochain does not significantly change the luminescence property. However, C2_N does not provide any evidence for the intrachain exciton hopping. Figure 4a summarizes the ϕ_{PL_490} and ϕ_{PL_554} (PLQY upon excitation with 554 nm light) values of C1_N (N = 1–16) measured in toluene. With incident light at 554 nm, π-extended ligand L1 at the end of the nanochain is selectively excited. L1 has a lower enough excitation energy than the dipyrrin site of BL3 (Fig. 3c, e), and as a result, the exciton stays and decays at L1, resulting in ϕ_{PL_554} with a small N dependence (Fig. 4a, orange plots). With 490 nm excitation, an exciton is generated at one of the dipyrrin moieties of BL3. This exciton can transfer to another dipyrrin site of BL3, and by repetitive exciton transfer, a part of the exciton reaches terminal L1, which is detected as PL from L1. We show that in toluene, the ϕ_{PL_490} values are almost the same as ϕ_{PL_554} (Fig. 4a, blue plots). This means that in toluene, the exciton transfer from BL3 to L1 is almost quantitative. In contrast, in the toluene/DCM mixed solvent, the ϕ_{PL_490} value becomes smaller than ϕ_{PL_554} as N increases (Fig. 4b). The N dependence in ϕ_{PL_490} is more pronounced in DCM (Fig. 4c). This series of results implies that in polar solvents, a part of excitons is deactivated on BL3 before reaching the L1 terminus. No significant solvent or N value dependence was observed for τ_PL of C1_N (Supplementary Table 6), but there appeared an important difference in the PL decay curves used to calculate τ_PL. Figure 4e–g show those of C1_N upon BL3 excitation at 470 nm in the three solvents (toluene, toluene/DCM (1:1 v/v), and DCM) for N = 2, 9, and 13. The comprehensive data for all N are assembled in Supplementary Fig. 42. As N increases, the rise and decay of the PL is delayed, which is the most prominent in DCM, and then a mixture of toluene and DCM. Such a delay was not at all observed for C2_N (Supplementary Fig. 43). The delay in the PL decay curve for C1_N is explained qualitatively as follows. When a typical fluorescent molecule is excited with a pulsed light source, PL is observed immediately after excitation. In sharp contrast, when C1_N is excited at 470 nm, an exciton is generated at one of the dipyrrin sites in bridging ligand BL3. It takes more or less time for the exciton to reach end-cap ligand L1. In this case, a delay is observed in the PL decay curve according to the rate of exciton transfer from BL3 to L1. The fact that the delay becomes more pronounced as N increases supports the above explanation. In addition, the delay with increasing N is toluene < toluene/DCM mixed solvent < DCM, suggesting that exciton transfer is slower in polar solvents. We then expect the delay should contain quantitative information of the exciton hopping kinetics in nanochain C1_N (vide infra).

Fig. 4: PL data and exciton migration dynamics.

a–c ϕ_{PL_490} and ϕ_{PL_554} of C1_N (N = 1–16) in toluene (a), toluene/DCM (1:1 v/v) mixed solvent (b), and DCM (c). d Brief illustration of the CTMC model with major exciton decay and transfer processes. Nearest-neighbor exciton migrations over a zinc center k_h-in and k_h-end, Förster-type exciton transfers (k_F[i,j]), radiative (k_r-L1 and k_r-BL3) and non-radiative (k_nr-L1 and k_nr-BL3) decays. See Supplementary Fig. 44 for more detail. e–g PL decay curves of C1_N (N = 2, 9, 16) in toluene (e), mixed solvent (f), and DCM (g). The samples were excited at 470 nm. The PL decay curve is smoothed for clarity (the raw decay curve is available in Supplementary Figs. 42, 50–52). h–j Simulated decay curves by the CTMC model in toluene (h), mixed solvent (i), and DCM (j). The experimental data are overlaid as open shapes with faint colors. k–m Experimental (color squares) and simulated (black dots) ϕ_{PL_490}-N plots of C1_N in toluene (k), mixed solvent (l), and DCM (m). R² is the residual sum of squares.

Quantification of exciton hopping

In the previous two sections, steady-state PL, PL quantum yield, and fluorescence lifetime measurements were performed on nanochains C1_N and C2_N, and qualitatively demonstrated the existence of exciton transfer from bridging ligand BL3 to end-capping ligand L1. In this section, we attempt to quantitatively and mathematically analyze the exciton hopping behavior in C1_N exploiting the experimental data. The outline of the model adopted here is shown in Fig. 4d. C1_N has 2N dipyrrin sites arranged in one-dimension, with the two terminal ones belonging to L1 (Sites 1 and 2 N; red part in Fig. 4d) and the remaining 2(N-1) ones belonging to BL3 (Sites 2, 3…2N-1; green parts in Fig. 4d). Upon excitation at 490 nm, the exciton occupies one of the dipyrrin sites derived from BL3. As time elapses, the exciton undergoes transfer or deactivation following four processes: (1) radiative deactivation (k_r-L1 and k_r-BL3), (2) non-radiative deactivation (k_nr-L1 and k_nr-BL3), (3) distant migration by Förster energy transfer (k_F[i,j]: rate constant for exciton migration from site i to site j; chief ones are illustrated in Fig. 4d; a more detailed description is given in Supplementary Fig. 44), and (4) nearest-neighbor hopping between the dipyrrin sites coordinating to the same zinc ion (k_h-in: between dipyrrin sites both belonging to BL3, k_h-end: from BL3 to L1). Here, k_m is the first-order rate constant for process m. The probability that the exciton moves to another dipyrrin site or disappears by process m during an infinitesimal time Δt is k_mΔt. Two constraints are imposed to the exciton transfer based on experimental and theoretical facts. First, because the difference in excitation energy is sufficiently larger than the room temperature energy (Fig. 3c,e), exciton transfer from L1 to BL3 is prohibited. Second, theoretical requirements dictate that Förster energy transfer between dipyrrin sites with parallel orientation is permitted, while that between dipyrrin sites with perpendicular orientation is forbidden (Fig. 1c and Supplementary Fig. 44). However, exciton hopping is permitted between nearest-neighbor dipyrrin sites coordinating to the same Zn center, even though they are perpendicular to each other. This exception corresponds to Process (4), which is supported by the experimental fact that efficient inter-ligand exciton transfer was observed in asymmetric bis(dipyrrinato)zinc complexes^22,23. Note that the above model requires that excitons do not exhibit coherence because it ignores quantum-mechanical effects such as the superposition of states. The incoherent nature of excitons in the nanochains treated in this study was discussed in the Supplementary information (SI Section 4.5; Supplementary Figs. 45–49).

Excitons that undergo transfer or deactivation on a one-dimensional chain according to the probability set at each site may be described by CTMC⁴². 2N-dimensional vector ${{{\boldsymbol{x}}}}\left(t\right)$, which describes the exciton existence probability at the i th (i = 1, 2, …, 2 N) dipyrrin site at time t, evolves over time according to the following master equation:

$$\frac{d{{{\boldsymbol{x}}}}\left(t\right)}{{dt}}=A{{{\boldsymbol{x}}}}\left(t\right)=\left(\,{A}^{{{{\rm{rad}}}}}+{A}^{{{{\rm{F}}}}}\,+\,{A}^{{{{\rm{hop}}}}}\right){{{\boldsymbol{x}}}}\left(t\right)$$

(1)

Here, A is a time-independent 2N × 2N square matrix that describes all transfer and deactivation in C1_N discussed above. A is the sum of three matrices, ${A}^{{{{\rm{rad}}}}}$ describing radiative and non-radiative deactivation, ${A}^{{{{\rm{F}}}}}$ describing Förster exciton transfer, and ${A}^{{{{\rm{hop}}}}}$ describing the nearest-neighbor hopping. The matrix elements of ${A}^{{{{\rm{rad}}}}}$, ${A}^{{{{\rm{F}}}}}$, and ${A}^{{{{\rm{hop}}}}}$ contain rate constants k_m explained above as (1)-(4) (see Numerical simulation with the CTMC model in Method section for the detail). Among the rate constants, k_h-in and k_h-end are difficult to quantify from the spectroscopic data obtained thus far, thereby being unknown parameters in this study. By solving Eq. 1, ${{{\boldsymbol{x}}}}\left(t\right)$ at any time t can be obtained. From ${{{\boldsymbol{x}}}}\left(t\right)$, the PL decay curve of C1_N during the fluorescence lifetime measurement can be simulated (Fig. 4e-g). Since ${{{\boldsymbol{x}}}}\left(t\right)$ includes k_h-in and k_h-end, they were determined by fitting the fluorescence decay curve. In other words, the determination of k_h-in and k_h-end is the purpose of constructing the exciton model (Fig. 4d) and numerical analysis by CTMC. For the detail of the fitting method, see Numerical simulation with the CTMC model in Method section, and Section 4.6 and Section 5 in Supplementary information. The PL curves in toluene, toluene/DCM mixed solvent, and DCM were well reproduced along with the delay of the rise (Fig. 4h-j; Supplementary Figs. 50–52). Furthermore, with known k_h-in and k_h-end, the ϕ_{PL_490}–N plot (Fig. 4a-c) can be also calculated in the scheme of the CTMC model, and was actually well reproduced when we use the same k_h-in and k_h-end as obtained by the PL decay fitting (Fig. 4k-m). The rate constants of the exciton transfer and deactivation processes in C1₉ in the three solvents are summarized in Table 1 in the form of time constant τ_m (=1/k_m): Comprehensive time constants and Förster radii (used to calculate k_F[i,j], see Numerical simulation with the CTMC model in Method section for detail) for all C1_N series were assembled in Supplementary Figs. 53–63 and Supplementary Tables 11–19. As mentioned above, the unknown parameters in the CTMC model are τ_h-in and τ_h-end. Of these, τ_h-end was shown to be valid by comparing the photophysical parameters of known asymmetric bis(dipyrrinato)zinc complexes (See Supplementary Fig. 64 and Section 5.3 in Supplementary Information). On the other hand, experimental verification of τ_h-in is tough because the donor and acceptor are the same chromophore therein. Paradoxically, we may argue that the construction of the exciton hopping model and CTMC analysis thereof succeeded in estimating τ_h-in, which is difficult to estimate. In the model of particle hopping between discrete sites with certain probabilities, the diffusion constant is defined as D = (Δd)²/2τ. Δd is the site spacing, which in this study is 12.5 Å, the spacing between two adjacent zinc ions^13,20. τ is the time constant of the dominating transfer process. In the case of C1_N, τ_F[3,6] and τ_h-in are dominant, such that the contributions of both are added together to evaluate $D=\frac{{(\Delta d)}^{2}}{2}\left(\frac{1}{{\tau }_{{{\mbox{F}}}[{{\mathrm{3,6}}}]}}+\frac{1}{{\tau }_{{{\mbox{h-in}}}}}\right)$. D = 7.3-9.2 × 10^-4, 3.2-3.3 × 10^-4, and 1.5-1.8 × 10^-5cm²s^-1 in toluene, a mixture of toluene/DCM, and DCM, respectively. These are in the range of diffusion constants for excitons in organic molecular systems reported thus far⁴³. The diffusion lengths of an exciton in C1_N, l_D = (Dτ_PL)^1/2, were calculated using D and the fluorescence lifetime of BL3 to be l_D = 146–164, 93-94, and 18-20 Å in toluene, a mixture of toluene/DCM, and DCM, respectively. These values correspond to ~12, 7, and 2 molecules of bridging ligand BL3. The exciton may travel approximately the diffusion length until it is deactivated, which is consistent with the N dependence of ϕ_{PL_490} (Fig. 4a–c). Thus, a non-polar environment is more favorable for the efficient exciton transport in C1_N.

Table 1 Time constants of C1₉ calculated from the theoretical model

Coordination nanochains and CTMC model for more precise description on exciton migration

In the previous work¹³, the degree of polymerization of copolymer P1 (Fig. 1a), and the arrangement of bridging ligands BL1 and BL2 therein, which act as an exciton donor and an acceptor, respectively, were random. Therefore, we prepared 1000 virtual molecular strands with 2000 dipyrrin sites as a model of P1 with a sufficiently long chain length. Numerical simulation was then applied to each strand, with the number of exciton hopping in the strand as a variable parameter. A critical problem of the previous model is there was no theoretical or mathematical support for the determination of the time constant of exciton transfer between dipyrrin sites coordinating to the same zinc ion (corresponding to τ_h-in and τ_h-end in the present study), only to adopt a plausible but arbitrary value. In the present study, instead of the random copolymer, the strategy of synthesizing nanochains C1_N was adopted. C1_N was isolated with defined chain lengths (N = 2–16), and the number of BL3 acting as exciton donors may be accurately determined. In addition, end-capping ligand L1, which functions as an exciton acceptor, is located only at both ends. It is not realistic to apply the analytical approach based on CTMC employed in the present work to the previous random copolymer. The computational cost of calculating the inverse matrix of A in Eq. 1 is proportional to the cube of the number of dipyrrin sites⁴⁴. Furthermore, because of the random structure of the copolymer, analytical solutions must be obtained for a huge number of copolymer strands with different arrangements, and their average value should be adopted. In other words, applying the CTMC approach to random copolymers incurs huge computational costs. By using nanochain C1_N with the above structural characteristics, we were able to calculate τ_h-in and τ_h-end analytically by the CTMC method. The PL decay curve and ϕ_{PL_490}-N plot are described as functions of τ_h-in and τ_h-end. By fitting the simulated PL decay curve with the experimental ones with additional verification by the ϕ_{PL_490}-N plot, we may determine τ_h-in and τ_h-end, which are difficult to calculate from the spectroscopic data, without the use of ultrafast spectroscopy using fs lasers.

Discussion

Discrete multinuclear complexes, or coordination nanochains were prepared by mixing a terminal ligand and a bridging ligand with a zinc ion. Thanks to the stable complex motif of bis(dipyrrinato)zinc(II), each molecular nanochain could be isolated from the reaction mixture by SEC, affording up to a 16-nuclear complex. They are characterized as unique multinuclear complexes due to their discrete straight-chain structures, orthogonal arrangement of the dye planes, and charge neutrality. The defined ligand sequences of the multinuclear nanochains allowed the construction of a realistic theoretical model for mathematical analysis of exciton behaviors. Analytical calculation based on a CTMC model afforded time constants for exciton migrations in the molecular chains including the non-Förster process. Through this work, we made a large leap in our strategy for evaluating photophysical dynamics in molecular nanomaterials. As our approach is not limited by the performance of lasers and detectors for ultrafast spectroscopy, it is expected to be the key to the new era of photochemistry and photophysics.

Methods

Synthesis of coordination nanochains C1_N

A 10 mL of methanol solution of zinc(II) acetate (19.78 mg, 107.8 μmol) was added to a DCM solution (30 mL) of bridging ligand H₂BL3 (52.66 mg, 81.9 μmol) and capping ligand HL1 (17.41 mg, 31.8 μmol). After stirring the mixture overnight, the solvent was removed in vacuo. The crude material was diluted with chloroform and separated by SEC to give C1_N (N = 1 – 16). Each coordination nanochain was subject to the second SEC separation for further purification.

Synthesis of coordination nanochains C2_N

A 10 mL of methanol solution of zinc(II) acetate (42.3 mg, 231 μmol) was added to a DCM solution of HL2 (52.3 mg, 111 μmol) and H₂BL3 (41.2 mg, 64.1 μmol). After stirring for 48 h, the solvent was removed in vacuo. The crude material was diluted with chloroform and separated by SEC to give C2_N (N = 1 – 10). Each coordination nanochain was subject to the second SEC for further purification.

Numerical simulation with the CTMC model

Here we consider exciton migration in nanochain C1_N with 2N dipyrrin sites based on the CTMC model. The distribution of the exciton on a single nanochain ${{{\boldsymbol{x}}}}\left(t\right)=({x}_{1}\left(t\right),\,\ldots,\,{x}_{2N}\left(t\right))$ evolves with time as:

$${\frac{d{{{\boldsymbol{x}}}\left(t\right)}}{{dt}}}=A{{{\boldsymbol{x}}}}\left( {t} \right).$$

(2)

Exciton transition probability matrix $A$ is expressed as the sum of three matrices, ${A}^{{{\mbox{rad}}}},\,{A}^{{{\mbox{hop}}}}$, and ${A}^{{{\mbox{F}}}}$.

${A}^{{{{\rm{rad}}}}}$ is a matrix describing the dissipation related to radiation. Its elements are written as:

$${A}_{i,j}^{{{\mbox{rad}}}}=-\!\left({k}_{{{{\rm{r}}}},i}+{k}_{{{{\rm{nr}}}},i}\right){\delta }_{i,j}$$

(3)

where k_r,i and k_nr,i are the rate constants for radiative and non-radiative relaxation at the i th dipyrrin unit, and ${\delta }_{i,j}$ is Kronecker delta. k_r,i and k_nr,i are defined as follows:

$${k}_{{{{\rm{r}}}},i}=\,\left\{\begin{array}{ll}{k}_{{{{\mbox{r-}}}}{{{\bf{L}}}}{{{\bf{1}}}}}\hfill &(i=1,\,2N)\\ {k}_{{{{\mbox{r-}}}}{{{\bf{BL}}}}{{{\bf{3}}}}}&\left({\mbox{otherwise}}\right)\end{array}\right.$$

(4a)

$${k}_{{{{\rm{nr}}}},i}=\,\left\{\begin{array}{ll}{k}_{{{{\mbox{nr-}}}}{{{\bf{L}}}}{{{\bf{1}}}}}\hfill &(i=1,\,2N)\\ {k}_{{{{\mbox{nr-}}}}{{{\bf{BL}}}}{{{\bf{3}}}}}&\left({\mbox{otherwise}}\right)\end{array}\right.$$

(4b)

where k_r-L1 and k_nr-L1 were obtained from the PLQY and PL lifetime of C1_N, while k_r-BL3 and k_nr-BL3 from the PLQY and PL lifetime of C2_N. Since C2_N could be isolated from N = 1 to 10, the measurement was not performed for N = 11 to 16. Therefore, to gain the k_r-BL3 and k_nr-BL3 values of N = 11 to 16, the data for N = 2–10 were extrapolated linearly. Comprehensive rate constants for the radiative and non-radiative relaxation with errors for all C1_N series (as time constants) are assembled in Section 5.4 in Supplementary Information and Supplementary Figs. 53–56, and Supplementary Tables 11–14.

${A}^{{{\mbox{hop}}}}$ is a matrix describing the exciton hopping between the dipyrrin unit coordinating to the same zinc center. Its off-diagonal elements are written as:

$${A}_{i,i+1}^{{{\mbox{hop}}}}=\left\{\begin{array}{ll}{k}_{{{{\mbox{h-}}}}{{{\rm{end}}}}}\hfill &(i=1)\\ {k}_{{{{\mbox{h-}}}}{{{\rm{in}}}}}\hfill &(i=2..2N-2)\\ 0\hfill &(i=2N-1)\end{array}\right.$$

(5a)

$${A}_{i+1,i}^{{{\mbox{hop}}}}=\left\{\begin{array}{ll}0\hfill &(i=1)\\ {k}_{{{{\mbox{h-}}}}{{{\rm{in}}}}}\hfill &(i=2..2N-2)\\ {k}_{{{{\mbox{h-}}}}{{{\rm{end}}}}}&(i=2N-1)\end{array}\right.$$

(5b)

where ${k}_{{{{\rm{h}}}}-{{{\rm{end}}}}}$ is the rate constant of the nearest-neighbor exciton hopping from BL3 to L1, both of which coordinate to the same zinc center, while ${k}_{{{{\rm{h}}}}-{{{\rm{in}}}}}$ is that between two BL3 ligands. ${k}_{{{{\rm{h}}}}-{{{\rm{end}}}}}$ and${k}_{{{{\rm{h}}}}-{{{\rm{in}}}}}$ are unknown parameters that are hardly determined by the experimental photophysical data. The other off-diagonal elements are 0. The diagonal elements of ${A}^{{\mbox{hop}}}$ are written as:

$${A}_{i,i}^{{\mbox{hop}}}=-{\sum}_{j\ne i}{A}_{j,i}^{{{\mbox{hop}}}}$$

(6)

${A}^{{\mbox{F}}}$ is a matrix describing Förster exciton transfer. Its elements are written as:

$${\,A}_{j,i}^{{{{\rm{F}}}}}={k}_{{{\mbox{F}}}[i,j]}=\left\{\begin{array}{cl}\left({k}_{{{{\rm{r}}}},i}+{k}_{{{{\rm{nr}}}},i}\right){\alpha }_{i}{\beta }_{i,j}{\left(\frac{{R}_{{{{\rm{F}}}},j}}{{r}_{{ij}}}\right)}^{6}&(i \ne j)\\ -{\sum}_{m\ne i}{A}_{m,i}^{{{{\rm{F}}}}}\hfill&(i=j),\end{array}\right.$$

(7)

where ${k}_{{\mbox{F}}[i,j\ne i]}$ is the Förster exciton transfer rate constant from i ^th to j ^th dipyrrin units, and ${R}_{{\mathrm{F}},j}$ is the Förster radius when the j ^th dipyrrin unit serves as an acceptor. The Förster radius between a donor dipyrrin site and an acceptor one is calculated from the experimental spectroscopic data using the following equation⁴⁵;

$${R}_{{\mathrm{F}}}=0.211{\left[\frac{{\kappa }^{2}{\phi }_{{{{\rm{D}}}}}\,J}{{n}^{4}}\right]}^{\frac{1}{6}}$$

(8)

where R_F is the Förster radius in Å, κ is the transition dipole orientation factor and set to 2 (parallel) or 0 (perpendicular), ϕ_D is the quantum yield of the donor, n is the refractive index of the solvent, and J is the spectral overlap integral determined by Eq. 9;

$$J=\int {f}_{{{\mbox{D}}}}\left(\lambda \right){\epsilon }_{{{\mbox{A}}}}\left(\lambda \right){\lambda }^{4}d{\lambda }$$

(9)

where f_D(λ) is the emission spectra of the donor (i.e., C2_N with the corresponding number of nuclei) normalized to an area of 1, and ε_A(λ) is the molar extinction coefficient of the acceptor in M^-1 cm^-1 (L1 analog or half of BL3), which was extracted from previous reports^20,23. λ is in nm. The J values for the exciton transfer from the dipyrrin site of BL3 to that of L1 ranged 4.27–4.34 × 10¹⁵, 3.46–3.56 × 10¹⁵, and 2.13–2.20 × 10¹⁵M^-1 cm^-1 nm⁴ in toluene, a mixture of toluene and DCM (1:1 v/v), and, DCM, respectively. The J values for the exciton transfer between the two dipyrrin sites of BL3 ranged over 5.62–6.30 × 10¹⁴, 3.01–4.28 × 10¹⁴, and 2.36–2.69 × 10¹⁴M^-1 cm^-1 nm⁴ in toluene, a mixture of toluene and DCM (1:1 v/v), and, DCM, respectively. Comprehensive Förster radii (${R}_{{{\mathrm{F}}},{{{\bf{B}}}}{{{\bf{L}}}}{\bf{3}}}$ and ${R}_{{{\mathrm{F}}},{{{\bf{L}}}}{\bf{1}}}$ where the dipyrrin sites of BL3 and L1 are the acceptor, respectively) with errors for all C1_N series are assembled in Section 5.4 in Supplementary Information and Supplementary Figs. 57, 58, and Supplementary Tables 15, 16. ${\alpha }_{j}$ in Eq. 7 is 0 for $j=1\,{{{\rm{or}}}}\space2N$, and otherwise 1: This condition prohibits an exciton transfer from L1 to BL3. ${\beta }_{i,j}$ is the orientation factor and equal to 1 or 0 when the i ^th and j ^th dipyrrin sites with are parallel or perpendicular, respectively. For instance, the two dipyrrin sites in BL3 are parallel to each other, while the two dipyrrin sites that coordinate to the same zinc ion are perpendicular (Fig. 1c). ${r}_{{ij}}$ denotes the distance between the i th and j th dipyrrin sites. Based on our previous report on bis(dipyrrinato)zinc coordination polymers^13,20, we set the distance between two dipyrrin sites coordinating to the same zinc atom to 4.7 Å, and that between two dipyrrin sites belonging to the same BL3 to 7.8 Å. These distances were extracted from an optimized nanochain structure from DFT calculation assuming that the center of dipyrrin is located at the midpoint between the two β-carbon atoms. Thanks to the periodic and rigid structure of C1_N, ${k}_{{{\mbox{F}}}[i,j]}$ may be simplified: For example, ${k}_{{{\mbox{F}}}[{{\mathrm{2,3}}}]}={k}_{{{\mbox{F}}}[{{\mathrm{4,5}}}]}=$ ${k}_{{{\mbox{F}}}[{{\mathrm{6,7}}}]}$, ${k}_{{{\mbox{F}}}[{{\mathrm{3,6}}}]}={k}_{{{\mbox{F}}}[{{\mathrm{7,10}}}]}$. Chief Förster exciton transfer rate constants ${k}_{{{\mbox{F}}}[i,j]}$ with errors for all C1_N series (as time constants) are assembled in Section 5.4 in Supplementary Information and Supplementary Figs. 59–63, and Supplementary Tables 17–19.

For example, in the case of N = 3, the full expression of exciton transition probability matrix A is as follows:

$$A=\left(\begin{array}{cccccc}-{k}_{{{\mbox{r-}}}{\bf{L1}}}-{k}_{{{\mbox{nr-}}}{\bf{L1}}} & {k}_{{{\mbox{h-end}}}} & 0 & {k}_{{{\mbox{F}}}\left[4,1\right]} & {k}_{{{\mbox{F}}}\left[5,1\right]} & 0\\ 0 & -{k}_{{{\mbox{r-}}}{\bf{BL3}}}-{k}_{{{\mbox{nr-}}}{\bf{BL3}}}-{\sum }_{i\ne 2}{A}_{i2} & {k}_{{{\mbox{F}}}\left[3,2\right]} & 0 & 0 & 0\\ 0 & {k}_{{{\mbox{F}}}\left[2,3\right]} & -{k}_{{{\mbox{r-}}}{\bf{BL3}}}-{k}_{{{\mbox{nr-}}}{\bf{BL3}}}-{\sum }_{i\ne 3}{A}_{i3} & {k}_{{{\mbox{h-in}}}} & 0 & 0\\ 0 & 0 & {k}_{{{\mbox{h-in}}}} & -{k}_{{{\mbox{r-}}}{\bf{BL3}}}-{k}_{{{\mbox{nr-}}}{\bf{BL3}}}-{\sum }_{i\ne 4}{A}_{i4} & {k}_{{{\mbox{F}}}\left[5,4\right]} & 0\\ 0 & 0 & 0 & {k}_{{{\mbox{F}}}\left[4,5\right]} & -{k}_{{{\mbox{r-}}}{\bf{BL3}}}-{k}_{{{\mbox{nr-}}}{\bf{BL3}}}-{\sum }_{i\ne 5}{A}_{i5} & 0\\ 0 & {k}_{{{\mbox{F}}}\left[2,6\right]} & {k}_{{{\mbox{F}}}\left[3,6\right]} & 0 & {k}_{{{\mbox{h-end}}}} & -{k}_{{{\mbox{r-}}}{\bf{L1}}}-{k}_{{{\mbox{nr-}}}{\bf{L1}}}\end{array}\right)$$

(10)

Among the matrix element of A, k_h-end and k_h-in were unknown parameters, which were shared among different N but not shared between different solvents. k_h-end and k_h-in were determined by the procedure below.

The experimental PL decay curves of C1_N (Fig. 4e-g and Supplementary Figs. 50–52) were recorded upon 470 nm excitation, i.e., the excitation of either dipyrrin site of BL3. As can be seen from the steady-state PL spectra (Fig. 3h and Supplementary Figs. 36b, 37b), the contribution of the emission from L1 at 579 nm is dominant in this case. This is due to the efficient exciton transfer from the dipyrrin site of BL3 to L1, and the PLQY of the dipyrrin site of BL3 is much smaller than that of L1 (Supplementary Tables 4,5,7–9). Therefore, the PL intensity at time t is proportional to the probability of the exciton presence at dipyrrin sites 1 and 2 N, ${x}_{1}(t)$ and ${x}_{2N}(t)$. Also, due to the symmetry of the nanoarray C1_N, ${x}_{1}\left(t\right)={x}_{2N}(t)$, such that we focused on ${x}_{1}\left(t\right)$ here. We assume that the initial state of the exciton distribution in C1_N is ${{{\boldsymbol{x}}}}\left(t=0\right)=\,\frac{1}{2N-2}\,{({{\mathrm{0,1,1}}},\cdots,{{\mathrm{1,1,0}}})}^{{{{\rm{T}}}}}$, and that the exciton initially appears at any of the 2N − 2 dipyrrin sites of BL3 with equal probability, being an ensemble average of all possible initial states. ${x}_{i}\left(t\right)$ at any time t can be obtained by solving Eq. 2. To save computational resources, Eq. 2 was decomposed as follows:

$$\displaystyle\frac{d}{dt}\left(\begin{array}{c}{x}_{1}(t)\\ {X}^{ \rightharpoonup } (t)\\ {x}_{2N}(t)\end{array}\right)=\left(\begin{array}{ccc}{A}_{1,1} & \overset{\rightharpoonup }{a^{{\rm{T}}}} & 0\\ 0 & A^{\prime} & 0\\ 0 & \overset{\rightharpoonup }{a^{\prime {{\rm{T}}}} } & {A}_{2N,2N}\end{array}\right)\left(\begin{array}{c}{x}_{1}(t)\\ {X}^{ \rightharpoonup }(t)\\ {x}_{2N}(t)\end{array}\right)$$

(11)

Here, A’ is a submatrix of A consisting of A_i,j (2 ≤ i, j ≤ 2N-1). A’ is a symmetric matrix because it denotes only the reversible exciton transfers within the BL3 chain, except for its diagonal elements. ${X}^{ \rightharpoonup } (t)=({x}_{2}\left(t\right),\,\ldots,\,{x}_{2N-1}\left(t\right))$. ${\overset{\rightharpoonup }{a^{{\rm{T}}}}}=({A}_{{1,2}},\,\ldots,\,{A}_{{1,2}N-1})$. ${\overset{ \rightharpoonup }{a^{\prime{T}} }}=({A}_{2N,2},\,\ldots,\,{A}_{2N,2N-1})$. Due to the symmetry of A’, an orthogonal matrix P satisfies PA’P^T = J. Here, J is a diagonal matrix whose diagonal elements are the eigenvalues of A’. Given the centrosymmetry of A and x(0), ${x}_{1}\left(t\right)={x}_{2N}\left(t\right)$ is calculated as follows.

$${x}_{1}\left(t\right)={x}_{1}\left(0\right){e}^{{A}_{1,1}t}+{\overset{ \rightharpoonup }{a^{{\rm{T}}}}}{P}^{T}\frac{{e}^{{Jt}}-{e}^{{A}_{1,1}t}I}{J-{A}_{1,1}I}P{X}^{ \rightharpoonup }\left(0\right)$$

(12)

Here I is the identity matrix. A normalized PL decay curve may be simulated by convoluting ${x}_{1}\left(t\right)$ with the instrument response function of the PL lifetime measurement apparatus (Supplementary Fig. 65). We simulated the PL decay curves for N = 2–16 and compared them with the normalized experimental data. Since ${x}_{1}\left(t\right)$ includes k_h-end and k_h-in, these were set as fitting parameters, and k_h-end and k_h-in that best reproduced the fluorescence decay curves for all N were obtained by the least squares method (Supplementary Figs. 50–52). The errors for k_h-end and k_h-in were set to include the range in which the residual sums of squares were less than or equal to 1.05 times the minimum values (Supplementary Fig. 66). The series of simulation and fitting was performed independently for three solvents (toluene, a mixture of toluene/DCM (1:1 v/v), and DCM).

The ϕ_{PL_490} value, the PLQY of C1_N excited at 490 nm, is simulated as follows:

$${{{{\rm{\phi }}}}}_{{{{\rm{PL}}}\_}490}= {\int }_{0}^{{\infty }}{dt}{{\boldsymbol{k}}}^{{{{\rm{T}}}}}{{{\boldsymbol{x}}}}\left(t\right) \\= {\int }_{0}^{{\infty }}{dt}{{\boldsymbol{k}}}^{{{{\rm{T}}}}} \exp \left[{tA}\right]{{{\boldsymbol{x}}}}\left(t=0\right) \ \\= -{{\boldsymbol{k}}}^{{{{\rm{T}}}}}{A}^{-1}{{{\boldsymbol{x}}}}(t=0)$$

(13)

where ${{\boldsymbol{k}}}^{{{{\rm{T}}}}}=({k}_{{{{\mbox{r-}}}}{{{\bf{L}}}}{{{\bf{1}}}}},{k}_{{{{\mbox{r-}}}}{{{\bf{BL}}}}{{{\bf{3}}}}} \ldots {k}_{{{{\mbox{r-}}}}{{{\bf{BL}}}}{{{\bf{3}}}}},\,{k}_{{{{\mbox{r-}}}}{{{\bf{L}}}}{{{\bf{1}}}}})$ is a 2N-dimensional row vector of the radiative decay rate constants of dipyrrin sites 1, 2…2 N. The initial state of the exciton distribution in C1_N is ${{{\boldsymbol{x}}}}\left(t=0\right)=\,\frac{1}{2N-2}\,({{0,1,1},\cdots,{1,1,0}})^{{{{\rm{T}}}}}$, the same as the simulation of the PL decay curve. Using Eq. 13, we calculated ϕ_{PL_490} for N = 2–16 via k_h-end and k_h-in that were obtained so that the PL decay curves are best reproduced. Note that experimental ϕ_{PL_554} obtained when L1 was directly excited at 554 nm, showed a slight N dependence (Fig. 4a-c; Supplementary Tables 4 and 5), such that we considered this dependence also in the simulation of ϕ_{PL_490}. The simulation of ϕ_{PL_490} was performed independently for the three solvents (toluene, a mixture of toluene/DCM (1:1 v/v), and DCM). We confirmed that the optimal values of k_h-end and k_h-in obtained by the PL decay curve fitting also reproduced the experimental ϕ_{PL_490}–N plot in every three solvent (Fig. 4k-m).