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The persistence lengths calculated in the previous section serve only as one particular measure of conformational freedom of a single molecule. For further investigation, we have also calculated distributions of the end-to-end length for the 8mers, 16mers, and 32mers of fluorene and thiophene. As discussed in Section 1.2, the end-to- end length (Eq. 1.1) and the radius of gyration (Eq. 1.6) measurements both give insight into how folded a given conformation is. Of the two, the radius of gyration is typically more relevant to polymer-scale molecules as it implies a spherical picture of a given chain. For oligomers, the radius of gyration has a less intuitive interpretation. Instead, the end-to-end length is more conceptually suitable as a measure of chain- folding.

Figure 4.8 depicts the distributions of end-to-end lengths for 8mers and 16mers of fluorene (Figure 4.8(a)) and 8mers, 16mers, and 32mers of thiophene (Figure

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalised end-to-end length

8mer 16mer 32mer

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalised end-to-end length

0 2 4 6 8 10 12 Probability density

(a) Fluorene

(b) Thiophene

Figure 4.8: End-to-end length distributions (solid lines) of various lengths of (a) fluorene and (b) thiophene. The dashed lines given for each distribution are the calculated distributions, r2_P

ee(r), obtained using Eq. 4.10 [118, 119] with a curve

given in (a) for a fluorene 32mer based on the calculated persistence length. The end-to-end length is scaled to give each length as a fraction of the fully extended length of each molecule. Legend in (b) applies to both graphs with the 32mer in (a) omitted.

4.8(a)) each obtained from 100 ns simulations in chloroform. In each, the end-to- end length is expressed as a fraction of the total arc length of each molecule for comparability between each length scale and molecule. While converged tangent correlation functions were obtained for the 32mer of fluorene, the distribution of the end-to-end lengths was not suitably converged over similar time-scales and has not been included.

Comparing the different lengths of each molecule, the clear progression of the distributions is towards lower values and larger ranges with increasing length. This follows intuitively given the increased conformational freedom afforded by longer chains. For thiophene, the 32mer has a much wider distribution and a lower peak length fraction (_{' 0.7) than that of the 16mer which is narrower and peaked at} ' 0.85. The 8mer is narrower still and peaked at ' 0.9. For fluorene, the distribu- tions at each length scale are narrower and peaked at higher values when compared to the corresponding length of thiophene with the 16mer peaked at _{' 0.9 and the} 8mer peaked at _{' 0.95. That the thiophene distributions are considerably broader} and peaked at lower end-to-end length fractions than the corresponding lengths of fluorene is consistent with the smaller persistence lengths, and, thus, higher flexibil- ity, of the thiophene molecules.

While experimental measurements of radii of gyration of polymers is possible - and leads to the experimental measurements of persistence length discussed pre- viously - experimental measures of end-to-end lengths are far less available. As such, it is difficult to utilise such a measure as a means of experimental comparison. However, recent advances in utilising end-markers - functional groups generally of high fluorescent quantum yield attached to each end of the molecule - have allowed for such measurements to be possible for molecules in host polymer matrices. One

relevant example is that of Muls et al [116] who, using end-marked hexyl-fluorenes of_{' 42 monomer units in length (with a polydispersity of 1.8) in an inert host poly-} mer matrix, measured end-end length distributions which are centred at a length fraction (based on the fully-extended 42mer) of _{' 0.89. Judging roughly by the} progression of our end-to-end length distributions, it can be expected that a 42mer simulation would be peaked at a length fraction of around _{' 0.7-0.8 and indicates} a more flexible fluorene molecule. Given the significantly different environments of the experiment and our solution-phase simulations, it is difficult to compare these results and we feel that the inert environment of the experiment can be expected to yield an effectively ‘stiffer’ molecule given the reduction of conformational freedom in going from a solution to an effectively solid phase. However, future experimental work in this spirit may yield suitable comparison.

Both the progression of the distributions and the difference in overall spread between fluorene and thiophene can be understood conceptually by considering the increase in conformational entropy with increasing length and is consistent with the persistence lengths calculated previously. One feature of the tangent correlation functions discussed previously is the clear exponential decay which is indicative of a wormlike chain (WLC). As briefly mentioned in Section 1.2.2, the WLC model can yield a number of general results using path integral approaches. (Many of these calculations and details of the method are presented in reference [119].) In particular, a general end-to-end length distribution for a WLC has been calculated by Willhelm and Frey [118]:

Pee(r) =N ∞

X

k=1

(₋₁₎k+1k2π2e−k2π2ξ(1−r), (4.10)

where r is the end-to-end length fraction, ξ = lp/L is the persistence length fraction,

and_{N is the normalisation constant such that}R₀1drr2Pee(r) = 1. In their work, they

found that this expression is in near-exact agreement with Monte Carlo simulations of WLCs. As such, comparison of this distribution function with our obtained distributions serves as an indicator of how much WLC behaviour is exhibited by fluorenes and thiophenes. The calculated distributions are shown as the dashed lines in Figure 4.8 with each value of ξ used taken from the values of np given in

Table 4.5. In all cases, it is seen that there is very close agreement between the MD distributions and the distributions of Eq. 4.10. In turn, this is indicative of fluorene and thiophene exhibiting WLC behaviour in chloroform and, most probably, in other solvents of similar quality. Furthermore, as we have obtained a value of np

for the fluorene 32mer, utilising Eq. 4.10 allows us to predict its end-to-end length distribution as shown in Figure 4.8(a).

From the above, we have seen that the end-to-end length distributions behave in a manner similar to that of a WLC based on their associated persistence lengths. As

chains get longer, more substantial folding occurs which leads to fractional end-to- end length distributions which are peaked at lower values and of significantly higher ranges. As with the persistence lengths, the additional flexibility in thiophene gives it substantially greater conformational freedom than fluorene. Potentially, this could result in significantly different spectral properties due to greater conformational distortion in thiophenes which we shall discuss further in Chapter 5.