Metallo-supramolecular polymers (MSPs) are composed of defined simple or oligomeric organic molecules capped with chelating end-groups (referred to as unimers) that spontaneously assemble into polymer chains by coordination to metal ions (dubbed ion couplers) (Ciferri, 2002). The ideal MSP should exhibit high thermodynamic stability under operational conditions, but it should be kinetically labile (should significantly dissociate) in solutions and/or at elevated temperature under processing conditions (Constable et al., 1994). Kinetic lability gives to MSPs processing advantages, easier control of the morphology of thin films, and multilayered structures and opens up new possibilities of post-synthesis modifications and tailoring their properties. Polymers showing this so-called constitutional dynamics are referred to as dynamers (Lehn, 2005). Dynamics of MSP chains also allow their structure healing by exchanging and/or reshuffling of their unimeric constituents and/or ion couplers.

During the last decade, MSPs underwent tremendous development because of their unique electronic, photonic, magnetic, and catalytic properties (Hissler, 2019). MSPs containing π-conjugated building blocks gained interest as materials for optoelectronic devices, covering light to electricity conversion, and for polymer light-emitting diodes (PLEDs) (Hissler, 2019). Recent finding of an efficient singlet fission process in thin films of MSP based on α,ω−bis(tpy)terthiophene unimers assembled with Zn²⁺ ion couplers (Rais et al., 2017b) shows that some MSPs could also be potentially exploited for better light harvesting in photovoltaics. The supramolecular approach to polymer chemistry evidently meets the actual needs of new functional materials for advanced electronics. Linear conjugated MSPs described in the literature mostly comprise α,ω−bis(tpy) unimers (tpy stands for 2,2′,6′,2″-terpyridin-4′-yl end-group) with oligo (p-arylenevinylene) (El-Ghayoury et al., 2002), oligo (p-aryleneethynylene) (Burnworth et al., 2008), oligo(fluorene) (Chen and Lin, 2007; Hrma et al., 2017), oligo (phenylene) (Li et al., 2016; Winter and Schubert, 2016), or co-oligomeric (Chiper et al., 2008; Zhang et al., 2019) central units linked into MSP chains via coordination to Zn²⁺, Co²⁺, Fe²⁺, La³⁺, or Eu³⁺ ion couplers (Hissler, 2019). Specific facial-meridian coordination of tridentate terpyridine end-groups to ion couplers (Zhang et al., 2019) gives MSPs well-defined stereochemistry, which is important for reproducible preparation of functional materials. This feature is absent in many other MSPs (Burnworth et al., 2007).

The kinetics of photoexcited states in α,ω−bis(tpy)terthiophenes (unimer T) and their MSPs with Zn²⁺ ion couplers (PT) were measured using pump-probe transient absorption spectroscopy. In solutions, their photophysical properties were found to be strongly influenced by chain dynamics. Relaxation processes running in photoexcited molecules of these unimers and MSPs were identified and characterized, and the impact of disturbed coplanarity of adjacent rings (dihedral angles between planes of rings due to the attached hexyl side groups) on these processes was shown (Rais et al., 2015). Two different physical processes were found in thin films of PT depending on the excitation wavelength (Rais et al., 2017b). Using the wavelength 332 nm, the system showed a fast singlet fission (SF) process with a time constant t =   160   fs . At a longer timescale, the formed triplet state population followed a slow power-law decay. The fast bi-molecular annihilation of singlet excitons was deduced from the explicit dependence of the decay rates on the pump pulse intensity. On the other hand, when these PT thin films were photoexcited at the wavelength 440 nm, no sign of the SF process has been found. The singlet excitons showed a power-law decay rate dependent on the pump pulse fluence. It indicated the role of bi-molecular collision events in the singlet annihilation process. At longer times after photoexcitation, the transient absorption spectra proved a well-discerned formation of excited state absorption (ESA) assigned to triplet states. Interestingly, the rise time of this ESA formation was strongly dependent on the pump fluence, ranging from ca 20 ps at low pump fluence down to ca 3 ps at high pump fluence. These triplet excitations also proved power-law decay. Although the evolution of singlet and triplet exciton populations was experimentally well-characterized in the study by Rais et al., (2017b), the origin of the bi-molecular singlet annihilation and subsequent triplet formation and decay remained unexplained. Meanwhile, we have elaborated a novel mathematical technique (Menšík et al., 2018a; Rais et al., 2018; Menšík et al., 2019), which makes the determination of the time-resolved diffusion coefficient from the transient absorption kinetics of photoexcited species possible. In this article, we applied the developed procedure on the previously acquired experimental data (Rais et al., 2017b) and proved that it is capable of elucidating in detail the process of the bi-molecular singlet annihilation and the subsequent formation and de-excitation of the triplet states.

The material chemistry, thin film preparation, X-ray diffractograms (XRDs) of PT films, and transient absorption (TA) spectroscopy experiment were described in details by Rais et al. (2017b) and the references therein. Here, we just briefly mention the following: 1) The procedure of synthesis of compounds of α,ω-bis(tpy)terthiophene (T) and its zinc-bridged supramolecular polymer PT (to keep the same notation as given by Rais et al. (2017b) (cf. Figure 1) in a powder form was described by Svoboda et al. (2011), Bláhová et al. (2014). 2) PT films deposited on 4 × 5 cm rectangular quartz substrates were prepared from a hexafluoroisopropanol solution of an equimolar mixture of T and zinc acetate by spin-casting at 3,000 rpm. 3) The XRD of the as-prepared PT powder samples of PT showed a dominant peak at about 26 ° , which corresponded to the inter-planar distance of 3.4 Å and a band of amorphous halo. This indicated a highly ordered state of 1d anisotropy of the PT samples stacked in the face-to-face direction. 4) For the TA spectroscopy, described in detail by Rais et al., (2015), we used the linearly polarized excitation and detection laser pulses at the so-called “magic” angle α   =   54.7 ° to avoid the anisotropy effects induced by the photoexcitation (Fleming et al., 1976).

The transient differential absorption spectra Δ A ( λ , t ) were monitored in the interval from 550 to 800 nm (corresponding to the ESA of the PT films photoexcited by a 440-nm pump pulse) and analyzed using the following equation (Rais et al., 2017b): Δ A ( λ , t ) = a S ( λ ) c S ( t ) + a T ( λ ) c T ( t ) . (1) Here, a S ( λ ) and a T ( λ ) are known basis spectral functions of singlets and triplets, respectively. Because they were obtained up to a constant scaling factor, the evolutions of singlets ( c S ( t ) ) and triplets ( c T ( t ) ) could be determined only in arbitrary units (Rais et al., 2017b). However, their population fractions with respect to the total number of unimer units can be recalculated from known kinetics of the ground-state bleach at early times and the steady-state absorption spectrum as it is shown below.

In Figure 2, we show the evolutions of the populations of the singlet excitons in the PT thin film after excitation by the pump pulse at the wavelength 440 nm, as obtained from Eq. 1. Note that the singlet population equals the ground-state bleach immediately after the excitation because only singlets are directly created by the pump pulse. It is known that the probe at 500 nm provides a nearly exclusively ground-state bleach (GSB) signal, that is, the sum of singlet and triplet excitons with equal weights, without an overlap with the positive ESA signal, which contributes to the spectrum from ca 550–800 nm. Therefore, the measured intensities of the GSB signal can be considered directly proportional to the sum of the concentrations of singlet and triplet excitons. Considering that a singlet exciton is located on a single unimeric unit due to relatively weak mutual interaction of the frontier orbitals of neighboring units, the ratio of the initial TA signal at 500 nm and the steady-state absorption at 500 nm directly shows the fraction of excited unimers in the sample. The absolute values of the singlet populations were obtained from their relative values in Eq. 1, assuming that exclusively singlet excitons contributed to the GSB signal immediately after the photoexcitation.

The obtained time dependences of the fraction of singlet excited unimers S ( t ) , referred to as singlet exciton population further in the text, were fitted for different pump fluences according to the following formula S ( t ) = A 1 + B ( t − 0.2 ) δ + C , (2) where time t is expressed in ps, and the values of the parameters A, B, C, and δ for different pump fluences are shown in Table 1.

It is seen from Figure 2 that the term A 1 + B ( t − 0.2 ) δ represents the annihilation of singlets on the time scale of hundreds of picoseconds, where the decay of singlets is dominant, while the “additive term” in the right-hand side of Eq. 2 corresponds to the residual longer-living singlet population decaying on time scale significantly longer than nanoseconds. Since the time course at longer times could not be reliably distinguished due to the poor signal-to-noise ratio, it was approximated by constant. Data in Figure 2 also show that the singlet decay rate increases with the pump fluence. This indicates that the decay of the singlet exciton volume density s ( t ) = N 0 S ( t ) (here N 0 is the unimer volume density) is controlled by the mutual exciton–exciton annihilation process according to the equation (Engel et al., 2006; Menšík et al., 2018a) ∂ ∂ t s ( t ) = − γ ( t ) s ( t ) 2 . (3)

Solution of which is a function s ( t ) = s ( 0 ) 1 + s ( 0 ) ∫ 0 t γ ( τ ) d τ . (4)

In Eqs 3, 4, γ ( t ) determines the rate of bimolecular collisions leading to exciton annihilation. The bimolecular mechanism can be experimentally confirmed by the power-law dependence of the population decay, easily seen when plotting it in the log–log scale (Engel et al., 2006; Marciniak et al., 2011), and by the explicit dependence of the population decay rate on the excitation pump fluence. Both these effects were found by Rais et al. (2017b). The linear decay term (originated e.g. in the exciton annihilation by fluorescence emission) could be also added, but it does not play a role within the time scale discussed in this article. It would have an observable effect only at longer times for sufficiently low singlet population and could be seen when plotting time course of the population decay in the semilogarithmic scale, which was not the case of the discussed experimental data. We see that the algebraic structure of Eq. 4 is similar to the term A 1 + B ( t − 0.2 ) δ in Eq. 2 that dominantly describes the singlet decay, before the long-tail dependence (represented by the additive term) becomes relevant. Eq. 4 can also be equivalently cast into the form 1 s ( t ) − 1 s ( 0 ) = ∫ 0 t γ ( τ ) d τ . (5)

Eq. 5 can be also expressed in terms of the singlet exciton population S ( t ) as 1 S ( t ) − 1 S ( 0 ) = N 0 ∫ 0 t γ ( τ ) d τ . (6)

Substituting S ( t ) from Eq. 2 for to the left-hand side of Eq. 6, we directly obtain the kinetics of the integral of the rate N 0 γ ( t ) of the exciton–exciton annihilation. This annihilation can, in principle, be controlled by either the Förster transfer or the diffusion process.

The Förster transfer is controlled by the long-range dipole–dipole interaction of two, in this case both excited, unimers. Within the discussed recombination process one of them becomes de-excited, while the other one is excited to a higher excited state followed by the subsequent fast Kasha de-excitation from higher to lower excited state or, alternatively, by singlet fission. As a result, there is either only one unimer remaining in the singlet state or two unimers in the triplet state. Such a process can be formally expected due to a significant overlap between the luminescence spectrum and the excited state absorption (ESA) spectrum at wavelengths between 600 and 800 nm (cf. Figure 4-4 of Rais et al. (2017b)).

The diffusion (hopping) proceeds, in turn, by subsequent exciton hopping between the nearest neighbor molecules, facilitated by the overlap of their electronic orbitals. It is, in principle, supported by certain regularity of the arrangement of PT units in thin films characterized by the inter-planar distance of 3.4 Å. It allows getting excitations to the nearest distance where they can recombine.

The dependences 1 S ( t ) − 1 S ( 0 ) obtained from fitting the evolution of singlet population S ( t ) , according to Eq. 2, were plotted in Figure 3, with the singlet populations normalized to unity at early times for all pump fluences. We compared them with theoretical dependences calculated according to Eq. 6, assuming that the excitation annihilation is controlled by either 1d diffusion with a constant diffusion coefficient ( N 0 ∫ 0 t γ ( τ ) d τ ∼ t 1/2 , Ref (Engel et al., 2006)), 3d Förster transfer ( N 0 ∫ 0 t γ ( τ ) d τ ∼ t 1/2 ) or 1d Förster transfer ( N 0 ∫ 0 t γ ( τ ) d τ ∼ t 1 / 6 ) (Marciniak et al., 2011).

Experimental data of the triplet population decay for different pump fluences are shown in Figure 5. For the sake of clarity, we show them as two independent sets: 1) derived from Figure 2 by subtracting the ground-state bleached population G S B ( t ) and the singlet exciton population S ( t ) and 2) taken from the spectral analysis (SA) of transient kinetics reported by Rais et al. (2017b). In the first case, the ground-state bleached population GSB ( t ) was obtained from the ratio of the Δ A ( λ , t ) kinetics and the steady-state absorbance at 500 nm (GSB region); the singlet state population S ( t ) , which is proportional to c S ( t ) in Eq. 1, was normalized to the initial value at time t = 0 as S ( 0 ) = GSB ( 0 ) . The triplet population dependence was, thus, obtained partly from the kinetic trace at the GSB region and partly by the spectral decomposition of the transient absorption in the ESA region.

In the second case, the triplet population kinetics was directly obtained from Eq. 1, where it is proportional to c T ( t ) , and from known basis spectral functions of singlets and triplet excitons. As in this case, the triplet exciton population is exact up to a normalization constant; we normalized it to give identical values as the first set of the triplet excitation dependences for the delay time of 10 ps.

We see in Figure 5 that both methods provide identical profiles at longer times, when the excited-state manifold is already thermalized, while they differ for short times, when the hot-state kinetics of the excited-state manifold can play a role.

We also see that for low pump fluence of 200 nJ (initial singlet exciton population S ( 0 ) = 0.0083 ), the triplet population T ( t ) initially increases with time. At a delay time of ca 20 ps, the triplet exciton population reaches the maximum value and decreases at longer times. We can see that with increasing intensity of the pump fluence, the T ( t ) reaches its maximum faster. For the pump fluence of 400 nJ ( S ( 0 ) = 0.029 ) , the maximum of the triplet population is reached at ca 5–10 ps, while for the pump fluence of 1,000 nJ ( S ( 0 ) = 0.093 ) , it is reached already within 1–3 ps. At longer times, the population decay asymptotically approaches the power-law dependence, with the exponent dependent on the pump fluence. Both of these facts indicate that the triplet population decay proceeds via the diffusion-controlled triplet–triplet annihilation.

We should note that the power-law decay is typically observed for diffusion processes (see discussion below). The rate of the triplet population decay depends on the pump pulse fluence, which also determines the initial density of triplet state population.

In the study by Rais et al. (2017b), it was shown that the SF process occurs after the excitation to the S₂ state and not to the S₁ state because the energy of the triplet pair is higher than the energy of S₁. In reverse, the triplet–triplet annihilation can also yield partly S₁ singlets. It would also explain the decreasing rate of the singlet population decay at very long times.

Based on the abovementioned discussion, the triplet population kinetics can be described by the following equation ∂ ∂ t T ( t ) = G ( t ) − γ 1 d T ( t ) T ( t ) 2 . (9) Here, G ( t ) stands for the rate of triplet generation. As triplets are formed from the fast decaying singlet states, the function G ( t ) becomes very small at longer times. Below, we will discuss two possible ways of the triplet exciton formation from singlet excitons.

In the second term in Eq. 9, the term γ 1 d T ( t ) represents the rate of the triplet–triplet annihilation. By fitting experimental data, we found that the power-law ansatz γ 1 d T ( t ) = γ 0 t − r , with 0 < r < 1 , well-describes the long-time triplet power-law ( T ( t ) ∼ T 0 t − q ) asymptote (cf. Figure 5). It can be simply checked that if the triplet decay in Eq. 9 is controlled only by the triplet–triplet annihilation (at longer times), we directly obtain for parameters of the fit the relation 0 < q = − r + 1 = γ 0 T 0 < 1 . The exact analytical solution to Eq. 9 is unknown; however, we could easily find an upper limit for its solution T ¯ ( t ) . It coincides with the exact solution for an early time limit. Namely, T ¯ ( t ) = ∫ 0 t G ( τ ) d τ   for   t ≤ t 1 , (10) which asymptotically describes the early time “triplet pumping” when the triplet–triplet annihilation could be neglected due to their low populations.

For longer time limit, the experimental data approach the asymptote T ¯ ( t ) = T 0 t − q   for   t ≥ t 1 , (11) which follows from Eq. 9 if the “pumping rate” G ( t ) is depleted.

The abovementioned two asymptotes describe the evolution at early and long-time limits, respectively. At the same time, it is evident that T ( t ) ≦ T ¯ ( t ) over the whole experimental time interval.

We can define the point t 1 as the time when these two asymptotes intersect. It can be shown that this point t 1 is located near the time when the triplet population takes the maximum value. As the time dependence of the triplet population is slowly varying the function around its maximum, we can derive ∂ ∂ t T ( t 1 ) ≈ 0 . Then, from Eq. 9, we get G ( t 1 ) ≈ γ 1 d T ( t 1 ) T ( t 1 ) 2 . We can compare this value with the mean value of the rate G ( t )   in the interval, 0 ≤ t ≤ t 1 , G ¯ ( t ) = 1 t 1 ∫ 0 t 1 G ( t ) d t : G ( t 1 ) 1 t 1 ∫ 0 t 1 G ( t ) d t ≈ γ 1 d T ( t 1 ) T ( t 1 ) 2 1 t 1 ∫ 0 t 1 G ( t ) d t = γ 1 d T ( t 1 ) T ( t 1 ) 2 1 t 1 T ¯ ( t 1 ) = t 1 γ 1 d T ( t 1 ) T ¯ ( t 1 ) ( T ( t 1 ) T ¯ ( t 1 ) ) 2 = = t 1 γ 0 t 1 − r T 0 t 1 − q ( T ( t 1 ) T ¯ ( t 1 ) ) 2 = q ( T ( t 1 ) T ¯ ( t 1 ) ) 2 < 1. (12)

We could see that the rate of triplet population generation at time t ₁ is already lower than the mean rate of triplet generation in the interval (0, t ₁). Thus, near the maximum of the triplet population also the “pumping rate” of triplet states decreases. It also proves that the triplet generation is directly bound to singlet exciton population decay.

The fact that near the point of the maximum of triplet population, the function ∫ 0 t G ( τ ) d τ should saturate can be derived also from the analytical examination of the second derivative of the triplet population.

Taking into account the function used for fitting experimental data for the triplet annihilation rate γ 1 d T ( t ) = γ 0 t − r (see the text above) and differentiating Eq. 9, we get directly ∂ 2 ∂ t 2 T ( t ) = ∂ ∂ t G ( t ) − ( ∂ ∂ t γ 1 d T ( t ) ) T ( t ) 2 − 2 γ 1 d T ( t ) T ( t ) ∂ ∂ t T ( t ) = ∂ ∂ t G ( t ) − ( ∂ ∂ t γ 1 d T ( t ) ) T ( t ) 2 − 2 γ 1 d T ( t ) T ( t ) ( G ( t ) − γ 1 d T ( t ) T ( t ) 2 ) = { ∂ ∂ t G ( t ) − 2 γ 1 d T ( t ) T ( t ) G ( t ) } + T ( t ) 2 { − ∂ ∂ t γ 1 d T ( t ) + 2 T ( t ) ( γ 1 d T ( t ) ) 2 } = { ∂ ∂ t G ( t ) − 2 γ 0 t − r T ( t ) G ( t ) } + γ 0 T ( t ) 2 { r t − r − 1 + 2 γ 0 T ( t ) t − 2 r } . (13)

We see that the second derivative ∂ 2 ∂ t 2 T ( t ) consists of two terms denoted by curly brackets {}. The first term, containing the monotonically decreasing pumping rate G ( t ) , is always negative and thus contributes to the “concavity” of the time course of the triplet population, while the second term, formed only by the triplet–triplet annihilation, is always positive and thus contributes to the convexity of the time dependence of the triplet population. The first term is dominant for short times; however, when the integral   ∫ 0 t G ( τ ) d τ saturates, then also the monotonically decreasing function G ( t ) → 0 and also ∂ ∂ t G ( t ) → 0 . Then, the first term in Eq. 13 will disappear, and the time course of the triplet population will attain the convex profile.

The experimental data show that not only the position of the maximum of the triplet population but also the transition time between the concavity and convexity of the triplet population time course shifts to shorter time with increasing pump fluence. It means that also the saturation of the “pumping rate integral” ∫ 0 t G ( τ ) d τ should occur at shorter time when the pump fluence is increased. This conjecture will be also used for the analysis of the mechanism of the triplet exciton generation below. We will discuss two physical mechanisms of the triplet formation: 1) inter-system crossing and 2) singlet–singlet collision.

The detailed analysis of the kinetics of the photoexcited states in thin films of metallo-supramolecular polymers with ditopic thiophene-bridged terpyridine ligands showed that singlet excitons S 1 are formed upon photoexcitation at the wavelength 440 nm. The pump fluence dependent rate of the S 1 singlet exciton population decay proved the exciton–exciton annihilation nature of the decay mechanism. Our theoretical modeling showed that the exciton–exciton annihilation process is controlled more by the Förster energy transfer of singlets than by their diffusion. We have also shown that mutual collisions of singlet excitons can partially yield triplet excitons via singlet fission of higher excited singlet states formed within these collisions. It explains the dependence of the rate of triplet state formation on the pump fluence. We have shown that the triplet state population decays via mutual annihilation and found that the diffusion coefficient follows the power-law time dependence t − p with the exponent p increasing almost linearly with the pump fluence. The faster decrease of the diffusion coefficient with time for higher pump fluence was assigned to the increased disorder due to the dissipated excitation energy.

The reported model is generally applicable to systems in which the energy of triplet pairs falls within the energy levels of S₁ and S₂ states. In systems where the triplet diffusion is slower, the excitation energy can be temporarily stored in these triplets and can be restored as S₁ energy later, which is manifested by prolonged lifetime of S₁ excitons, observable, for example, as delayed fluorescence.