CIVIL ENGINEERING 365 ALL ABOUT CIVIL ENGINEERING


Limiting the analysis to finite number of modes P ignores a substantial part of the mode spectrum and seems to be unjustified. On the other hand, the increased stiffness caused by the delocalized π-electron system in P3HT introduces a finite correlation length (dynamic equivalent to the static Kuhn segment), which can be taken into account by adding a fourth-order term, ({p}^{2}+alpha {p}^{4}), to the entropic spring constant ((k=3{k}_{B}T/{ell }^{2}propto {p}^{-2})), with the dynamic stiffness parameter α15,36,37. The modified Zimm scattering model is obtained by replacing the mode dependence, p3v, of τz (in Eq. 1) by ({p}^{3nu }+alpha {p}^{4-nu }) and the corresponding cosine amplitude, ({p}^{2nu +1}), which evolves as ({p}^{2nu +1}+alpha {p}^{4})15. Unlike limiting the number of modes, we now exploit the fact that by increasing the momentum transfer Q, the dynamic structure becomes more sensitive to higher modes. In addition, for a given Q, the calculated S(Q,t)/S(Q) becomes independent of p, beyond a certain threshold (p > pmin). This uses the fact that 2π/Q probes a certain finite length, which limits the number of modes required to describe the experimental data theoretically. As a consequence, the spatial resolution is only determined by the Q dependence of S(Q, t) but not affected by the maximum Q.

The solid lines in Fig. 4a,b compare the result of our analysis with the experimental dynamical structure factor. We can accurately describe our experimental data by simultaneously fitting all the Q’s. From this analysis, we obtain the stiffness parameter α, that decreases with increasing molecular weight and/or temperature, cf. Table 1.

Figure 4

Normalized dynamic structure factor, (S(Q,t)/S(Q)), as a function of Fourier time, t, for P3HT samples with different molecular weight (a) Mn = 90, and (b) 63 kg/mol. Each figure includes two temperatures, T = 313 K (solid) and 353 K (dash). The momentum transfer, Q, is given by, = 0.062 Å−1, = 0.087 Å−1 and = 0.124 Å−1. The solid and dashed lines in (a,b) represent the best fit summing over a large number of modes, pmin, and a finite dynamical stiffness, α.

Based on this result, we can now estimate the minimum number of modes, pmin, that are required to theoretically describe the experimental S(Q, t), within the Q range of our NSE experiments, by solving (S(Q,,t,alpha =0,,N={p}_{min})=S(Q,t,alpha ,N={N}_{infty })). To calculate the mode independent parameter α, above the threshold, p > pmin, we summed over p = 1… 1000. We obtain considerably greater pmin than earlier determined p values. This pmin is the maximum mode numbers which are visible in our experiment.

If we compare the quality of the fits based on the stiffness parameter (Fig. 4) with those calculated assuming a low number of modes (Fig. 3), we observe a similarly good description irrespective of their physical origins. The description of the relaxation of a chain by its mode spectrum assumes a certain number of statistically independent segments, connected by entropic springs. Numerous experiments justified the assumption of an infinite number of modes in case of flexible polymers like poly(ethylene-alt-propylene) or poly(ethylene glycol) (with α = 0)38,39. In the present case the conjugated polymer P3HT, the increased stiffness caused by the delocalized π-electron system introduces a finite correlation length, which decreases the number of statistically independent beads. Thus, the calculation of (S(Q,t)/S(Q)) using a reduced number of modes is formally equivalent to the calculation using a stiffness parameter α (cf. Table 1). The absence of higher order modes elucidates the fact that the chain dynamics is partially frozen. Indeed, this is the first experimental evidence of the existence of single chain glass (SCG) state in a conjugated polymer. As the highest Q is limited in experiments, (S(Q,t)) cannot represent the entire mode spectrum. However, higher Q values probe more local structures. If the stiffness already impacts the smaller momentum transfers, it is very likely that the wider angles would not change this discussion. However, we re-emphasize, if Q-values are reached that start to probe more local dynamics, then additional processes are to be incorporated in the model40,41. However, in the current situation there was no indication that this is the case with P3HT.

The comparison of the data with the Zimm model with all modes illustrates that the equivalent flexible polymer relaxes faster. At least two potential reasons can explain why the relaxation appears to be slower: (1) a reduced number of modes (Fig. 3), or (2) damping of the modes (Fig. 4). Apparently NSE data can be described by a finite number of modes (no damping). A decay of (S(Q,t)) sets in, once modes contribute to the relaxation. Therefore, fewer modes result in less relaxation and more modes lead to a faster decay of S(Q, t). However, the momentum transfer corresponds to a certain length-scale, (Q=,2,pi /d). Therefore, the higher the Q the more local the NSE experiment is, which implies higher modes. In a simplified wording, moving to the higher Q’s requires more modes contributing to (S(Q,t)). In this context, we exploit the fact that each Q has a maximum number of modes and increasing the number of modes would not change the calculated (S(Q,t)) at this specific Q* and at every Q < Q*. Obviously, this calculated (S(Q,t)) relaxes faster than the experimental data. However, including damping slows down the decay. Therefore, we have now the opposite description.

This explanation can be rationalized by a simple estimation. For semi-flexible polymers, the number of modes, pmin in Eq. 1, limits the displacement, (cos ({p}_{min}pi m/N)), over (m=N/{p}_{min}) segments. Therefore, we can estimate a dynamic rigid length, ({R}_{rigid}). For distances less than ({R}_{rigid}), the segments are correlated. These modes will be absent in the analysis. Thus, within a bead spring approach ({R}_{rigid}) represents the length of a bead. It is given by: ({R}_{rigid}=ell {(N/{p}_{min})}^{nu }={R}_{ee}{N}^{-nu }{(N/{p}_{min})}^{nu }={R}_{ee}{p}_{min}^{-nu })15. From Table 1, it is evident that the effects of temperature and molecular weight are negligible on Rrigid, and we obtain ({R}_{rigid}) = 4.72 ± 0.1 nm. From the structural standpoint, Rrigid could likely be interpreted as the polymer conjugation length. Conjugation length is a length of a planarized chain segment where π-bonding is maintained over the entire segment, and is a key parameter which determines electronic and optoelectronic properties of conjugated polymers. Indeed, the value of Rrigid corresponds to a bead length of approximately 12 thienyl repeating units, that is within the range of polythiophene conjugation length reported in literature (ranging between 10 and 20 repeating units)42. It needs to be mentioned that the value of Rrigid determined from the dynamic data is substantially higher than the P3HT persistence length (2.9 ± 0.1 nm) determined from wormlike chain modeling of static SANS data25, and reflects the fact that π-electron delocalization in P3HT extends on essentially longer distances than the geometrical persistence length.

It should be noted that, independently of the observed length scale, we obtained two significant parameters, namely, finite global stiffness, α and a finite size of the bead, ({R}_{rigid}). The parameter α describes the damping of the mode relaxation. In the Rouse or Zimm approach, normal coordinates are introduced to solve the Langevin equation by simple exponential functions. The orthogonality of these normal coordinates follows from the uncorrelated random forces. This assumption corresponds to the freely jointed chain model that neglects correlations between bond vectors. In a good approximation, those finite correlations in a real polymer can be neglected if greater distances along the chain contour are considered. This leads to the introduction of ({R}_{rigid}) and similarly to α.

In order to investigate the scaling behavior between the chain end-to-end distance and the dynamical chain stiffness α, we systematically varied Ree from low to high values. As shown in Fig. 5, we have used five different linearly spaced values above and below the experimentally obtained Ree. This was done for both temperatures and polymer molecular weights. This reveals the dependence of α on the chain length. In addition to our results on P3HT, we have included the stiffness parameter αPNB of polynorbornene (PNB) of different molecular weights in a good solvent15. For a better comparison, we rescaled αPNB by a factor ~ 7. Irrespective of the polymer, molecular weight and temperature, we observe a generic power-law scaling, (alpha propto {R}_{ee}^{-8nu }). As a consequence, the molecular weight dependence of α is attributed to the increase in Gaussian coil dimension, Ree by a factor ~ 1.26.

Figure 5
figure5

Generalized scaling behavior, (alpha propto {R}_{ee}^{-8nu }), of the polymer stiffness, α, as a function of the chain dimension, Ree of P3HT samples (two molecular weights and two temperatures) and polynorbornene (PNB) (three molecular weights from ref. 15). Here α is vertically scaled by a factor c.

We now want to explore how our findings based on the analysis of polymer dynamics, can be translated to macroscopic materials properties of conjugated polymers. As a special important case, we consider the correlation between the large-scale chain dynamics and thermochromism. Polythiophene shows a distinct thermochromic behavior both in solution and in solid state, as the polymer electronic absorption band undergoes reversible hypsochromic shift upon temperature increase43.

Let’s sum up some of the essential facts. (i) The radius of gyration depends on the molecular weight as expected for a Gaussian coil, and increases around 15% with increase in temperature. At the same time, within the Q-range of our SANS experiments the aggregation is nearly independent of molecular weight or temperature. (ii) The bead size, ({R}_{rigid}), is independent of molecular weight and temperature. (iii) The stiffness parameter, α, decreases with increasing temperature and molecular weights. (iv) The absorption spectra of both P3HT samples in DCB-D4 are independent of the molecular weight but show a thermochromic blue shift and an increase in band gap energy, Eg, with increasing temperature, cf. Figure 2 in the supplemental information (SI). These spectroscopic results agree with those found earlier for regioregular P3HT and seem to be common for semiconducting polymers44,45,46,47,48.

As it is widely accepted in the literature, the thermochromic blue shift in the absorption spectra of polythiophenes, including P3HT, upon increasing temperature is related to cooperative static conformational twisting (i.e. planar to non-planar conformational transition) of the π-electron conjugated backbone49,50,51,52. From our analysis, both the conjugation length (as reflected in the value of Rrigid) and our scaling law, (alpha propto {R}_{ee}^{-8nu }), show no dependence on temperature. It elucidates the fact that within the observed temperature range the constant bead size excludes a correlation with the observed changes in the absorption spectra. Also, the static chain end-to-end distance is not associated with the thermochromic blue-shift. Therefore, our results do not support static intramolecular conformational twisting of the π-conjugated backbone, and thus reduction of the conjugation length as a key factor in the observed thermochromic behavior.

The SANS data in Fig. 2 cannot access the bead size since ({R}_{rigid}) = 4.7 ± 0.1 nm corresponds to (Q,=,2pi /{R}_{rigid},)= 0.13 Å−1, which is at the upper Q-limit of the SANS experiment. As the competition between coherent and incoherent scattering may contribute in this region, we abstain from the discussion of weak effects, which may not be related to the structure. Therefore, it impossible to see a structural peak. However, our SANS data at low Q indicate significant aggregation of P3HT (cf. SI) even at higher temperature, we should suggest that temperature-affected changes in the interchain aggregation may be responsible for the thermochromic blue-shift at the higher temperature. This finding emphasizes the unique role of the large-scale dynamics in understanding the fundamental physics of locally stiff polymers and deriving correlations between the chain stiffness and the macroscopic material properties, which has not been explored in the literature so far. We should emphasize that our findings derived from P3HT behavior in dilute solution have been only studied for the narrow temperature range (313 to 353 K). They may not be directly applied to thermochromism in solid state. Nevertheless, they do agree with recent conclusions about rather complicated nature of thermochromic phenomenon in conjugated polymers where multiple contributing factors are responsible for the observed spectroscopic changes53.



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