As was the case with obtaining DFT potentials in Section 3.2, obtaining FF po- tentials for molecules with side-chains results in significant complications. In the particular case of thiophene, which has side-chains close to the intermonomer junc- tion, optimisation procedures are complicated by the additional steric contributions and relative freedom of the side-chains. While we have found that differences in the dihedral potentials of thiophene resulting from the side-chains are almost entirely the result of steric contributions, which would be expected to be represented by the force-field already, it is imperative to properly examine whether this implied transferability of the potential holds.
0 30 60 90 120 150 180 Dihedral Angle (º) 0 2 4 6 8 10 12 Energy (kJ/mol) DFT (C2) Unsub. Fit + C2 FF C1 Fit + C2 FF C2 Fit + C2 FF 0 30 60 90 120 150 180 Dihedral Angle (º) -15 -10 -5 0 5 10 15 20 25 Energy (kJ/mol) DFT Profile FF Contribution Subtraction Fitted Profile Fit + FF Contribution
(a)
(b)
Figure 4.3: (a) Subtraction potentials for C2-thiophene. (Labelling follows that
of Figure 4.1.) (b) Total force-field potentials for C2-thiophene comprised of the
sum of the C2-thiophene FF contribution and subtraction potentials fitted from
unsubstituted (unsub.) thiophene, C1-thiophene, and C2-thiophene. Comparison is
given to the C2-thiophene dihedral potential.
fluorene 2mer, we found that the dihedral potential resulting from the scan is effec- tively identical to that of fluorene. As is seen in Table 4.1, values of ∆Em obtained
from minimisations of C1-fluorene are in agreement with the DFT values to within
0.1 kJ/mol. Performing the same test with C2-fluorene also showed the same be-
haviour. This follows as would be expected from the invariance found in performing the DFT calculations and shows that the fluorene dihedral potential is transferable to the full class of dialkyl-fluorene molecules.
For thiophenes, we have performed the subtraction procedure for C1 and C2-
thiophene with the resulting curves for C2-thiophene shown in Figure 4.3(a). It is
noted that this procedure results in a far less smooth fit than that of thiophene shown in Figure 4.1(b). This behaviour is an example of the inconsistency in a geometry optimisation scheme which comes from the freedom of the side-chains and is amplified as side-chain length is increased. To minimise this inconsistency, taking the initial scan geometries from the DFT calculations is of particular importance. This being said, it is not the case that this measure leads to a perfect correspondence in the side-chain conformations between the DFT and FF scans. While the error in the fitting procedure impacts the direct comparison of the fit with the DFT potential, in calculating ∆Emof the C2-thiophene 2mer (Table 4.1), we find agreement between
the DFT and our FF fit to within ' 0.1 kJ/mol.
In Section 3.1, we argued that steric interactions, responsible for large changes in dihedral potential for alkyl-thiophenes, will be incorporated by the force-field and, thus, the potential fitted from thiophene should be transferable in the same manner as with fluorene. However, the FF potential obtained from C2-thiophene
differs drastically from that obtained from thiophene. In Figure 4.3(b), we show the effective FF potential resulting from the FF contribution of C2-thiophene and
Table 4.2: Energetic barriers at 0° (∆E0) and 180° (∆E180) of thiophene 2mers with
various side-chains using fitted potentials obtained from scans using different side- chain lengths. The labels Cx (Cy) denote the energies of a 2mer with a side-chain of
x carbons using the potential obtained from a fit of the 2mer with a y carbon side- chain with ‘unsub.’ denoting the unsubstituted case. The DFT values shown are those from the dihedral scans of the Cx-thiophene 2mer. Each barrier is calculated
relative to the closest local minimum (i.e. the trans minimum for ∆E0 and the cis
minimum for ∆E180).
∆E0 (kJ/mol) ∆E180 (kJ/mol)
DFT FF DFT FF unsub. (unsub.) 0.59 0.33 2.02 1.75 C1 (unsub.) 1.20 5.35 4.65 7.21 C1 (C1) 1.20 1.33 4.65 5.00 C2 (unsub.) 4.02 11.07 6.71 12.00 C2 (C1) 4.02 6.67 6.71 9.18 C2 (C2) 4.02 3.90 6.71 7.66
the expected DFT potential, we see that the assumption of transferability leads to a significant overestimation of the planar energetic barriers. Quantitatively, by defining the difference in energy between each unrestrained minimal configuration and their nearest planar configurations, defined as ∆E0 and ∆E180 for the trans and
cis sides respectively, we find that the error introduced in the planar barriers of C2-
thiophene from utilising the C1-thiophene potential can be as large as ' 2.5 kJ/mol
(' kBT ). Using the unsubstituted thiophene potential, this error increases approxi-
mately two-fold. These energies are summarised in Table 4.2. This is one of the key results of this chapter as it highlights that one cannot assume that the FF dihedral potential is transferable purely from the argument that the conjugated component of the dihedral potential is transferable (as was shown in Section 3.2.2).
One possible reason that the transferability of the dihedral potential with re- spect to increasing side-chain length in thiophenes implied by the DFT calculations does not hold is due to the use of the Lennard-Jones 12-6 potential in the OPLS force-field. Dubay et al [93], have shown that the OPLS force-field produces an overstatement of the planar barriers in C2-thiophene and highlighted that this may
be remedied, in part, by utilising a buffered 14-7 potential [173]. It may be the case that using such a potential may replicate the expected behaviour and signifies that an entirely generic force-field for conjugated polymers (e.g. one built of interchange- able conjugated moieties such as amino acids in current protein force-fields) may require such a modification. However, as it currently stands, we have shown that careful consideration of the thiophene side-chain by performing subtraction with the ethyl side-chain leads to an appropriate force-field representation of the dihedral potential without any modification to the Lennard-Jones definitions of the OPLS force-field.
Given the inconsistency observed in the C2-thiophene fitted potential resulting
from the side-chain degrees of freedom, performing similar calculations for longer side-chains becomes impractical due to increasingly noisy fits. However, as shown in Figure 3.7(b), the difference in DFT dihedral potential between C2 and C3-thiophene
is ' 0.5 kJ/mol (0.2 kBT ) at the planar barriers which is slight compared to those
between C2and C1, and C1 and the unsubstituted case. As such, we feel it is justified
that the C2-thiophene potential be used as a general potential for longer lengths of
side-chain.
We have found that the subtraction procedure is capable of reproducing the expected form of the dihedral potential and respective energetic minima even for the complicated C2-thiophene 2mer. To conclude this section, we now discuss how
we modify equilibrium bond lengths and angles in order to further enhance the correspondence obtained from the subtraction procedure.