1 Introduction

Polarons in conjugated polymers have been attracting much attention as nanoscale charged nonlinear excitations in one-dimensional electron systems and are expected to play important roles in manifesting the function such as electroluminescence [1]. The structure of charged π-conjugated chains determines the nature of the charge carriers in electrically conducting and semiconducting oligomers and polymers. This issue is of importance because of the use of these materials, such as oligo and polythiophenes, in electronic and optoelectronic applications [2]. Given the presence of a strong electron–phonon coupling in π-conjugated chains, i.e. the strong dependence of molecular geometry on the charge state, charge self-localization has been predicted, that is, an excess charge produces a structural distortion over a limited section of the conjugated chain and the charge concentrates in this distorted geometry region, even in the absence of any counterion. The charge self-localization primarily alters the carbon–carbon bond alternation pattern along the backbone [3]. Neutral thiophene rings (even though they are considered as aromatic [4]) are characterized by a marked carbon–carbon bond-length alternation (BLA). On the other hand, charged thiophene rings tend to relax to a quinoid geometry, with an inverse BLA pattern. Thus, the appearance of a quinoid structure allows one to determine the location and extension of the polaron. That an excess charge in oligothiophene radical cations leads to the formation of a polaron has recently become a subject of controversy [5]. Further experimentally, important information on the properties of charged species in conjugated oligomers can be obtained from optical absorption spectroscopy. However, a few experiments have been reported concerning the charged conjugated oligomers and polymers in the absence of the counterions. The results of ab initio and semi-empirical self-consistent field (SCF) calculations (with restricted open shell) on oligothiophenes are in good mutual agreement; they indicate the self-localization of spin and charge over five to six rings around the middle of the chain, while the quinoid geometric distortion itself is somewhat more confined [3, 6, 7]. However, density functional theory (DFT) calculations performed at the local density approximation (LDA), generalized gradient approximations (GGA) and hybrid methods predict complete charge/spin delocalization over the whole chain [5, 7–11]. The general conclusion drawn such calculations based on DFT is that no clear self-trapping is observed for the studied oligomers of computationally feasible sizes, except the hybrid BHandHLYP approach that leads to some localization effects [12]. These results are later confirmed by several groups [13, 14].

There are two strategies to model the effect of doping in π-conjugated polymers. The first one consists of the investigation of properly chosen infinite neutral and charged oligomers. This strategy is largely investigated in the literature [9, 15–17]. The second one involves the investigation of charge transfer from the conjugated chain to one or several atoms representing the counterions [15, 18–21]. In the literature, no accurate treatments of electronic structure are done with a large counterion (as p-TSA) [22]. The present paper provides details of these calculations: devoted firstly to determining the structure of single oligothiophene (pTh) chain in the presence of the counterion (p-TSA) and second to the assembly composed of two conjugated chains and two counterions (the solid-state packing). Another motivating factor for the present work is the presence of the experimental data reporting the structure of thin films of PEDOT/p-TSA (polyethylenedioxythiophene/toluenesulfonate acid).

2 Computational methods

In a precedent study dedicated to the oligothiophene radical cations [12], we have compared the predictive abilities of different methods in reproducing the polaron formation. The purpose of that investigation was to select the most effective functional for future DFT studies on doped conducting polymers. As expected, the study demonstrated that BHandHLYP method appears to be reliable tool to study the charged π-systems. Therefore, the latter functional is selected for this study of pTh/p-TSA complex. The choice of this functional is also done by the good results obtained for polarons and bipolarons [23, 24]. DFT calculations for radical ions were spin-unrestricted, which is in accordance with the general spirit of DFT. The spin contamination in DFT, calculated formally from the Kohn–Sham determinant, was found to be insignificant: the solutions are practically pure doublets, with S 2 differing marginally from the correct value of 0.75. Spin contamination in the BHandHLYP solutions is somewhat higher, due to the admixture of an unrestricted Hartree–Fock contribution; however, S 2 never exceeding 0.9. The charge distribution was calculated with the DFT method and was estimated through the Mulliken population analysis.

For the molecular orbital expansion, we have used the 6-31G* basis set [25]. It has demonstrated that this basis set is sufficient to predict the electronic properties of neutral and doped polymers [26]. The optimization of pTh/p-TSA and stacking of conjugated chains in the presence of two counterions assemblies is fully performed with Gaussian 03 [27]. No geometrical constraints are imposed to obtain the minimum energy structure of different complexes.

3 Results and discussion

3.1 Structure of charged thiophene chain/p-TSA interaction and the influence of the counterion

The structure of charged thiophene chain/p-TSA (pTh/p-TSA) interaction and the influence of the counterion on the charge distribution of polaronic defects have been investigated. Based on our previous work [22] dedicated to the modelling (molecular simulations) of the solid-state packing of charged chains (PEDOT) in the presence of the counterions, several orientation of the counterion with respect to the conjugated chain are considered: perpendicular, parallel and lateral orientations. So, in the present work, based on computational chemistry, we studied all these configurations.

In the case of perpendicular orientation, a dynamical simulation is done, in which we moved the counterion along the conjugated chain; several local minima are obtained corresponding to the stable structures of pTh/p-TSA complex. The calculations indicated that the most stable structure corresponds to that with the counterion localized in the middle of thiophene oligomer. This is also the spatial domain where most of positive charge carried out by the chain is located. This stability can be explained by a strong influence of attraction of opposite ions. And one moving away from the centre, the stability of the system decreases because the electrostatic interaction becomes increasingly weak (the charge transfer from the conjugated chain towards the counterion decreases). From the three beginning orientations (perpendicular, lateral and side) of the counterion with respect to the conjugated chain, the calculations indicate only two configurations are obtained after the calculations (side and perpendicular configurations, Fig. 1). The counterion positioned in the lateral orientation in the beginning of the calculations shifts towards side configuration after optimization. The interaction energies, defined as differences between the energies of well specified states (E inter = E complex − E cation − E couterion), obtained for side and perpendicular orientations are −65.8 and −63.2 kcal/mol, respectively. When the basis set superposition error (BSSE) correction is taken into account, the interaction energies are −59.9 and −59.1 kcal/mol. The stability difference between the two structures is much lower (<1 kcal/mol) with the correction BSSE.

Fig. 1
figure 1

Structures of pTh/p-TSA complex. a Perpendicular orientation; b side orientation

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These values (corresponding to the interaction energies) are in excellent agreement with those obtained, by Singh-Miller et al. [28], on the interaction between terthiophene cation and PF6 −. However, our results are in different to that obtained in the case of PEDOT/p-TSA [22]. This difference is probably because in the present work, no constraints are used, while in the previous work a rigid system is used and only intermolecular interactions are optimized. The large deformation (twisting) of the conjugated chain by the presence of the counterion in the perpendicular orientation (Fig. 1) is responsible for less stability of this configuration. The DFT results on geometry as well as charge distribution in radical cation oligothiophene are given in Figs. 2 and 3. Note that in the figure depicting the carbon–carbon backbone geometry, the three bond lengths within each thiophene ring are represented by three connected points; an inverse Λ-shape connection corresponds to an “aromatic BLA pattern” within the ring, while a V-shape connection corresponds to “quinoid rings”.

Fig. 2
figure 2

C–C bond length (Å) along the conjugated chains of charged oligomer without and in the presence of the counterion. The Arabic numbers corresponds to monomer units

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Fig. 3
figure 3

Charge distribution of polaron without and in the presence of the counterion

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In the absence of the counterion, Fig. 2 indicates the appearance of quinoid rings in the middle of the chains (even in the units 3 and 6). However, in the presence of the counterion, the quinoid forms appears only on the units 4 and 5. These data allow concluding that the defect is more localized in the presence of p-TSA. Chemical considerations taking into account the valence saturation suggest that the terminal rings should tend to be aromatic.

The presence of the counterion affects considerably the geometry of the conjugated chain. Indeed, as pointed before [12], the main conjugated backbone of the charged thiophene octamer is fully planar. The presence of the counterion introduces some significant deviations from the planar conformation, with a twisting angle between two neighbouring units of 0°; 7°; 9° and 20° from the middle to the chain ends in the side orientation, while in the perpendicular configuration the corresponding twisting angles are 5°; 10°; 14° and 22°. In the last configuration, the main conjugated chain is more distributed (large twisting) and confirms the less stability of this configuration.

The nearest intermolecular distances between the sulphur atom of the counterion and sulphur atom of thiophene oligomer are found to be 3.56 and 3.74 Å for side and perpendicular orientations, respectively. Again the shorter intermolecular distance in the side orientation favours the attraction of opposite ions on the stability of pTh by p-TSA.

From Fig. 3, the charge distribution has sharp maxima in the middle of the chain, where their magnitude is appreciably higher than the average value and the value close to the chain end. In each orientation, the presence of the counterion induces a substantial increase in charge density on the central part of the oligomer (so-called pinning effect). This effect is particular pronounced for the perpendicular arrangement, for which a large part of the positive charge is located on the units 4 and 5. In order to confirm this behaviour, we have studied the chain length dependence by investigating the oligomers with 10 and 12 rings. The interaction energies (corrected with BSSE) obtained are −62.1 and −64.3 kcal/mol for complexes of oligomers with 10 and 12 rings, respectively. Figure 4 gives the distribution of the positive charge in thio8+, 10thio+ and 12thio+ in the presence of the counterion positioned nearly in the middle of the chain (this position is obtained after optimization). Even with oligomers containing 10 and 12 rings, we obtained as highly localization of the charge in agreement with the notion of polaron formation. And as signalled before, the presence of the counterion tends to increase the charge in the middle of the chain (rings 4–8) and decreases elsewhere.

Fig. 4
figure 4

Distribution of the excess charge in 8thio+, 10thio+ and 12thio+ in the presence of the counterion positioned in side orientation

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The favourable interaction energy of side orientation in comparison with perpendicular one (2.6 kcal/mol) is consistent with the distribution of electrostatic charges. Thus, the charge transferred from the oligomer to the p-TSA is slightly larger in side orientation (0.97 e−) than in perpendicular orientation (0.93 e−). In both cases, the charged oligothiophenes and counterions form ions pairs that interact electrostatically.

The spin density results are summarized in Fig. 5. The electron transfer from the conjugated chain to the counterion generates a difference in spin density, indicating the regions that show significant change during this transfer. Figure 5 indicates the dominant effect of transfer of charge from the conjugated chain (blue regions) to the counterion (red regions). The presence of the counterion allows a strong localization of spin density in the middle of the conjugated chain. The large part of negative charge is shared between the three oxygen atoms of p-TSA. Further, a slightly more localization of spin density for side orientation is observed, thus confirming the more stability of this configuration in comparison with perpendicular orientation. These theoretical results are very interesting and again support qualitatively the experimental data on regioregular poly(3-octylthiophene) showing the spatial extend of the polaron [29].

Fig. 5
figure 5

Spin density distributions for pTh/p-TSA complexes for perpendicular and side orientations. Red regions correspond to the acceptance of the electron, while the blue positions indicate the regions from where the electron moves

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3.2 Comparison between different counterions

In the literature, few groups investigated the interactions between charged oligomer and counterion [28–32]. Different sizes of the oligomers and simple counterions as Cl3 or PF6 are studied. The very interesting work published by Salzner [5] indicates that, in the presence of the counterion, the excess positive charge is localized in the three central rings. Salzner found also that the charge transfer from the thiophene chain to the counterion is 0.96 e− in 9ThioCl3, 13ThioCl3 and 19ThioCl3. These results are in excellent agreement with our findings. Further, the geometry of the cations in the presence of Cl3 indicates that the defect is localized over 11 thiophene rings and the main distortion affects the inner three rings. This agrees qualitatively with our results. However, in the absence of the counterion, a defect delocalization is obtained [5], while a slight localization is obtained from our work. This disagreement is due to the choice of the functional used (B3P86-30% compared with BHandHLYP). As indicated in the methodology, the BHandHLYP localize the excess charge because this functional incorporates 50 % of Hartree–Fock exchange, thus allowing for a significant reduction in the self-interaction error inherent to DFT and an increase in observed electron/hole localization. However, the expectation value for the spin operator is about 0.9 (UBHandHLYP) slightly different to those obtained with B3P86-30% (0.77). In the case of PF6, Singh-Miller et al. [28] studied only short oligomers (up quarterthiophene), using B3LYP and Perdew–Burke–Ernzerhof (PBE) functionals and concluded that the oligomer and counterion form ion pairs that interact electrostatically (strong interaction about 60 kCal/mol, which is in good agreement with our results). This comparison is very interesting and demonstrated the credibility of the BHandHLYP in predicting the properties of the doped oligothiophenes in the presence of the counterions. However, for the charged chains in the absence of the counterions, we expected that B3LYP and especially PBE are more reliable if we considered the expectation values of the spin operator. PBE (and in general pure DFT) does not suffer from spin contamination, So the pure DFT result seems to be more reliable than hybrid functionals. However, BLA is underestimated with pure DFT [32].

3.3 Stacking of two charged chains in the presence of the counterions

We considered the interaction between two charged oligothiophenes (octamer) in the presence of two p-TSA counterions. The system including a positively charged oligothiophene and negatively charged counterion is electrically neutral. A spontaneous charge transfer within the system leads to positively charged oligothiophenes and negatively charged toluenesulfonate ions. The ions are located on opposite sides of the dimer (stacking oligothiophenes). Based on the results obtained in first part of this work, i.e. singly charged oligothiophene in the presence of the counterion, we fixed one counterion in the middle of the conjugated chain (orientation) and we moved the second counterion along the second conjugated chain in order to search different stable minima of the assembly (two charged chains with two counterions).

Two minima are found corresponding to the structure sketched in Fig. 6. The most stable structure corresponds to the both counterions are placed nearly in the middle of the assembly (Fig. 6a). This structure is more stable than those of one counterion is found in the middle and the second in the edge of the oligothiophene (5.4 kcal/mol in favour of the first assembly). Our theoretical results are very interesting since they are consistent with experimental finding on similar polymers [30, 31], indicating that the conjugated chains stacks to form a perfect packing and that the counterions are placed on each side in the middle of the assembly. This configuration favours the interchain transport and maybe is responsible for good conducting polymer. In both assemblies, a spontaneous charge transfer between p-TSA and oligothiophene takes place. The intermolecular distances between the conjugated chains are nearly similar (3.7 Å for the first assembly and 3.8 Å for the second assembly). These distances are in good agreement with those obtained by Singh-Miller et al. [28].

Fig. 6
figure 6

Doubly charged oligothiophene dimer in the presence of two p-TSA. a The two counterions are found to be nearly in the middle of the assembly: b one counterion is in the middle, while the second is in the edge of the assembly

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Further, it has been shown experimentally [33] that when oligothiophenes are oxidized, they will dimerize through π–π stacking. However, Scherlis et al. [34] have studied the π-stacking in charged thiophene oligomers, and their results do not support the existence of stable dimers of small oligomers in the absence of the counterions (in vacuum), because the π–π interactions between singly oxidized oligothiophenes are not strong enough to overcome the Coulombic repulsion. In the present work, our results indicate that the presence of the counterions (without the solvent) stabilizes the dimerized oligothiophenes. These findings are in good agreement with previous studies [35, 36].

4 Conclusions

Quantum chemical calculations (DFT) are done to investigate the interaction between charged oligothiophene chain and p-toluenesulfonate acid (p-TSA). Starting with several configurations, the calculations indicated that only two stable configurations are obtained: perpendicular and side orientations. For each orientation, the interaction energy of pTh/p-TSA complex is very important; this is due to the strong electrostatic interactions between the charged chain and their counterion. The charge transferred from the conjugated chain to the counterion is at least of 0.93 e−, indicating that oligothiophene cation and counterion form ion pairs that interact electrostatically. We demonstrated also that the presence of the counterion affects considerably the planarity of conjugated chain and allows a more localization of charge and spin density in the middle of the conjugated chain. Further, a preliminary study on the effect of the counterions on the interactions of charged oligothiophenes is reported, and the results obtained here and those from references [28, 34] can serve as an interesting tool to study the stacking of oxidized oligothiophenes both in vacuum and in solution in the presence of the counterions.