On the Supramolecular Packing of High Electron Mobility Naphthalene

Article pubs.acs.org/Macromolecules

On the Supramolecular Packing of High Electron Mobility Naphthalene Diimide Copolymers: The Perfect Registry of Asymmetric Branched Alkyl Side Chains Vincent Lemaur,† Luca Muccioli,‡ Claudio Zannoni,‡ David Beljonne,† Roberto Lazzaroni,† Jérôme Cornil,†,* and Yoann Olivier†,* †

Laboratory for Chemistry of Novel Materials, University of Mons-UMONS, Place du Parc 20, B-7000 Mons, Belgium Dipartimento di Chimica Industriale “Toso Montanari” and INSTM, Università di Bologna, Viale Risorgimento 4, IT-40136 Bologna, Italy

‡

S Supporting Information *

ABSTRACT: The supramolecular organization of the n-type P(NDI2OD-T2) polymer (also known as Polyera Activink N2200), featuring two branched octyldodecyl lateral chains, has been investigated by a combination of quantum-chemical and molecular mechanics techniques coupled to two-dimensional X-ray diffraction simulations. The structures exhibit non fully extended alkyl chains due to a compromise between the size of the monomer unit and the nature and length of the alkyl chains. Interestingly, the supramolecular organization of the polymer chains is only weakly affected by the nature of the stereogenic centers of the branched alkyl chains. The size and shape of the monomer unit and the chemical structure of the side chains appear to be the key elements governing the relationship between the supramolecular organization of P(NDI2OD-T2) polymer chains and their charge transport properties and are as important as the fine-tuning of the electronic properties of the molecular subunits in order to guarantee large electron (and hole) mobility. This study opens new perspectives for the rational design of new n-type and/or p-type polymers for further improvements of organic-based device efficiencies.

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INTRODUCTION Organic conjugated polymers have attracted an increasing interest over the years for use in organic opto-electronic devices such as light-emitting diodes, solar cells, or field-effect transistors as a result of their low cost, lightweight and ease of processing from solution. The development of p-type materials has been extensive over the past decade and hole mobilities reaching or exceeding 1 cm2 V−1 s−1 have been obtained for conjugated polymers.1−7 In contrast, due to their high reactivity with water and O2, the synthesis of n-type polymers with high electron mobility remains a challenging task. However, by carefully choosing a sequence of alternating electron-donating and accepting units in order to stabilize the lowest unoccupied molecular orbital (LUMO),8 high electron mobilities can be reached. In 2009, Facchetti et al. demonstrated for the first time that electron mobility up to 0.85 cm2 V−1 s−1 could be reached with a highly soluble, airstable, n-type copolymer P(NDI2OD-T2) (poly(N,N′-bis-2octyldodecylnaphtalene-1,4,5,8-bis-dicarboximide-2,6-diyl-alt5,5−2,2-bithiophene) in top-gate transistors.9 More recently, the charge carrier mobilities of this polymer measured in fieldeffect and diode configurations have shown a strong asymmetry, with mobility 2 orders of magnitude larger for electrons than for holes.10 In addition, temperature-dependent © 2013 American Chemical Society

electron mobility measurements have shown a small activation energy, pointing toward a very low degree of energetic disorder in this material.11 Since the charge transport properties of conjugated polymers are strongly sensitive to the relative position of the chains in the solid state,12 the morphology of P(NDI2OD-T2) thin films has been the subject of many experimental studies to shed light into the link between the supramolecular organization of the polymer chains and the exceptionally high electron mobility of the material. Initially expected to be amorphous, it has been evidenced using electron microscopy that the morphology of the polymer appears instead to be well ordered on the 10 nm scale and retains longrange correlation up to one micrometer.13 In addition, grazingincidence wide-angle X-ray scattering (GIWAXS) studies have shown that P(NDI2OD-T2) forms lamellae on the substrate with an interlayer distance (d100) ranging from 24.3 to25.0 Å (q ∼ 0.25 Å−1), depending on the processing conditions and thermal treatments.14−16 Within the lamellae, an intense diffraction peak at q = 1.6 Å−1 has been attributed to πstacking. GIWAXS16,17 and vibrational spectroscopy18 experiReceived: May 29, 2013 Revised: September 24, 2013 Published: October 4, 2013 8171

dx.doi.org/10.1021/ma401124r | Macromolecules 2013, 46, 8171−8178

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Figure 1. Representation of the unit cells corresponding to the anti (R,S) (a = 26.29 Å, b = 4.69 Å, c = 14.18 Å, α = 119.8°, β = 72.8°, γ = 104.3°), syn (R,S) (a = 25.10 Å, b = 4.70 Å, c = 14.07 Å, α = 117.3°, β = 78.5°, γ = 103.5°), anti (R,R) (a = 25.57 Å, b = 4.71 Å, c = 14.17 Å, α = 118.5°, β = 74.7°, γ = 103.5°), and syn (R,R) (a = 24.59 Å, b = 4.78 Å, c = 14.04 Å, α = 116.2°, β = 79.2°, γ = 101.7°) conformers. The green lines illustrate the kinked (anti conformers) or curved (syn conformers) alkyl chains. The filled green circles depict the void between the conjugated cores and the alkyl chains.

quantum-chemical and molecular mechanics calculations together with simulated GIWAXS patterns, four polymorphs corresponding to crystals made of stereoregular polymer chains built with monomer units bearing stereogenic centers with the same nature in the branched alkyl chains have been first identified and compared to experimental X-ray patterns. Then, in order to account for the random distribution of stereocenters (stereoirregular polymer chains) generated during the synthesis of the polymers, the supramolecular organization in the bulk of different combinations of monomer isomers has also been investigated. Finally, the high electron mobilities are rationalized by computing the intermolecular transfer integrals governing charge transport for both stereoregular and stereoirregular structures.20

ments also demonstrated that upon thermal treatment, the orientation of the conjugated backbone with respect to the substrate is changing, from mainly face-on to a dominant edgeon orientation, without affecting the field-effect mobility in a top-gate configuration.16 Recently, a detailed NEXAFS study has shown that an edge-on layer forms at the upper surface of the films independently of the post-thermal treatment,17 thereby explaining the orientation-independent mobility. Electron diffraction studies19 demonstrated that thin films of P(NDI2OD-T2) exhibit a microsegregation of the donor and acceptor parts in separate columns for thin films grown on 1,3,5-tricholorobenzene while for some specific patterned substrates (such as poly(tetrafluoroethylene)), donors in a given chain are overlapping with the accepting units of the adjacent chains (i.e., there is a shift of half the monomer size along the long axis of the polymer chain). Unfortunately, in spite of all these morphological studies, it is not yet possible to rationalize the high electron mobilities of P(NDI2OD-T2) as this would require a description of the polymer packing down to the molecular scale, including both the electronically active π-conjugated core as well as the insulating side chains. An additional difficulty toward the identification of the supramolecular organization resides in the lack of stereoselectivity in the chemical synthesis of the NDI2OD-T2 monomer unit, which leads to branched side chains with either the S or R configurations.9 The present work aims at determining at a full atomistic level the supramolecular organization of P(NDI2OD-T2) chains. By combining

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RESULTS AND DISCUSSION The strategy used in this work to probe the three-dimensional structure of the polymer consists in coupling molecular mechanics and molecular dynamics simulations on a unit cell containing a limited number of monomer units that are replicated using periodic boundary conditions to reproduce an infinite system. All calculations have been performed with the Materials Studio (MS) 6.0 package. The force-field is derived from the Dreiding force-field21 in which four torsion potentials (see Figure S1, Supporting Information) have been reparameterized against reference MP2/cc-pvdz calculations (see Computational Details). The atomic charges have been assigned according to the ESP 8172


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charges calculated for a NDI2OD-T2 dimer. This force field customization is often crucial to improve the accuracy of the atomistic model and to reach quantitative agreement when simulation results are compared to experiments.22,23 From the MP2/cc-pvdz torsional energy curves obtained for the isolated small monomer (see Figure S1), it appears that: (i) two twisted isoenergetic conformers are found with respect to the torsion between the naphthalene diimide (NDI) and thiophene (T) units and no planarization in the solid state is expected for this torsion due to the large energy barrier (about 4 kcal/mol) at 0 and 180 deg; (ii) the two thiophene units should adopt the usual anti conformation; and (iii) the branched alkyl chains should orient perpendicularly to the acceptor units, a typical structure found in rylene crystal structures.24 As starting point for the conformational search, many different structures have been built by considering all different torsion potential minima (see Figure S1), by varying the cell parameters, relative orientations and shifts of the conjugated cores and alkyl chains, and by interdigitating or not the alkyl chains. In addition, the impact of the configuration (S or R) of the stereocenters has been addressed. For stereoregular polymer chains, the unit cells contain only one monomer unit and two distinct cases were investigated: either the monomer is bearing two R stereocenters (denoted here as (R,R)) or one R and one S stereogenic center (R,S). (S,S) and (S,R) configurations were not explicitly investigated here since they are expected to yield mirror-symmetry arrangements of the (R,R) and (R,S) configurations, respectively. For nonstereoregular systems, the unit cells contain two monomers with all possible combinations of stereocenters (see below). In all cases, the procedure to extract the most stable structures involves four different steps: (i) the starting structure is optimized at the molecular mechanics level; (ii) 100 ps Molecular Dynamics simulations with periodic minimizations (or “quenched dynamics”, NPT, T = 300 K, quench frequency = 1 ps) are then performed on each optimized structure until the energy between two consecutive runs no longer decreases; (iii) on the most stable structures obtained at step ii, 100 psquenched dynamics are performed at higher temperature, successively at 600 and 1000 K; (iv) and finally, longer quenched dynamics (t = 500 ps) using, as starting points, the most stable structure of the last quenched dynamics in step iii, are performed at increasing temperature (300, 600, and 1000 K) following the procedure developed in steps ii and iii. In each of these different steps, Ewald summation25 was used to describe nonbonded interactions. The conformational search of the stereoregular polymer chains in the bulk matches the calculated MP2 torsion potentials in gas phase. Indeed, the four most stable structures displayed in Figures 1 and Figure S2 correspond to the anti and syn conformers (with respect to the torsion potential between NDI and T) with torsion angles close to those obtained in the gas phase (41° and 135° in bulk and 45° and 121° in the gas phase for anti and syn (R,S) conformers, respectively, and 46° and 131° in bulk and 45° and 121° in the gas phase for anti and syn (R,R) conformers, respectively). Given the torsion between the NDI and T units as well as the perpendicular orientation of the alkyl chains with respect to the NDI conjugated cores, the donor [acceptor] fragments of a given chain are overlapping with the donor [acceptor] units of the adjacent chain, leading to a microsegregation of the donor and acceptor fragments, as

observed recently by electron diffraction experiments on thin films.19 Interestingly, the conformational search reveals that the supramolecular organization is only weakly sensitive to the nature of the stereogenic centers; the orientation of the conjugated cores and alkyl chains and the cell parameters (see Figure 1) are similar among the different structures. The most striking feature of the conformational search is the specific orientation of the alkyl chains. Indeed, the two most stable (R,S) structures exhibit interdigitated alkyl chains, with the octyl chains in an all-trans conformation while the decyl chains are bent; the first five carbon atoms lie parallel to the polymer chains while the remaining segment is bent in such a way that it gets parallel to the octyl chain of the next monomer unit of the polymer chain (see Figure 1). For the (R,R) structures, the alkyl chains adopt a very similar organization, except that one octyl chain only adopts an all-trans conformation after the first carbon atom (see blue lines in Figure S2) which implies that one decyl chain is kinked after six carbons instead of five. This orientation of the alkyl chains contrasts with previous studies on p-type polymers, such as PBTTT26,27 and CDT-BTZ28 for which extended alkyl chains were found. This unusual shape of the alkyl chains can be explained considering that P(NDI2ODT2) has a slightly longer monomer unit compared to those polymers, which allows in principle for the interdigitation of three fully extended alkyl chains, but considering the branching, four are available per monomer unit, while only two linear side chains are present in PBTTT and CDT-BTZ. Therefore, two of the side chains have to bend in order to fulfill the size criterion mentioned above and efficiently pack the branched alkyl within a limited space. Interestingly, the molecular structure of branched alkyl chains is crucial since the length of the alltrans chain (C8) appears to match nicely the length of the two kinked decyl chains that are pointing toward one another, giving rise to a dense, ordered interlayer region. This very specific interdigitation of the alkyl chains, in concert with the insensitivity of the supramolecular organization with respect to the nature of the stereocenters of the alkyl chains are key factors favoring high intermolecular order. The slight difference in the orientation of the alkyl chains, is observed through the interlayer spacing. This parameter ranges from 24.93 to 24.27 Å for the anti and syn (R,S) conformers and from 24.47 to23.93 Å for the anti and syn (R,R) conformers, in good agreement with the position of (100) experimental reflections;14−16,19 this spacing only amounts to 19.81 Å for a highly energetic polymorph with four fully extended alkyl chains, and to 21.98 and 28.21 Å for two noninterdigitated (R,S) structures (see Figure S3 in Supplementary Information). Whatever the stereoregularity, the conformation of the decyl chains is slightly different for the anti and syn conformers. They are smoothly curved for the syn conformer, whereas a clear kink is observed for the anti conformer (see Figure 1). Interestingly, while the MP2 torsion potential suggests isoenergetic structures in the gas phase (see Figure S1), our simulations show that the anti conformers are more stable than the syn conformers (by 2.11 and 1.82 kcal/mol per (R,S) and (R,R) monomer unit, respectively) due to solid-state effects. This can be understood from the less favorable nonbonded interactions for the syn conformer, due to the presence of voids between the first carbon of the alkyl chains and the closest thiophene ring (see the difference in size of the green filled circles in Figure 1). On the other hand, the lattice parameter of the c axis of the unit cell, which corresponds to the polymer chain axis, amounts to 8173


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Figure 2. Simulated GIWAXS patterns of the anti (R,S) (top left), anti (R,R) (top right), syn (R,S) (bottom left) and syn (R,R) (bottom right) conformers.

two main characteristic distances corresponding to the conjugated backbones that are found in a range between 3.5 and 4.2 Å. In each case, they are located close to the experimentally measured characteristic distance of 3.93 Å, in particular for the most stable anti structures. The latter displays a short π-stacking distance of 3.5 Å between the naphthalene diimide or thiophene units. This distance does not exactly correspond to a q value of 1.6 Å−1 (3.93 Å) since the π-stacking distance does not take into account the relative shift of the chains and the fact that the conjugated cores are tilted by +23° and −30° with respect to the vector normal to the lamellar plane for the naphthalene diimide or thiophene units, respectively (see Figure 1). In order to characterize the charge transport properties of stereoregular crystals of P(NDI2OD-T2), the HOMO and LUMO intermolecular transfer integrals29 have been computed for the four structures using the ADF package, following the procedure described in ref 30. On the basis of the equilibrium geometries of two tetramers, the calculations show that, whatever the conformer and the nature of the stereocenters of the alkyl chains, the LUMO transfer integrals are larger than the HOMO transfer integrals (106 meV [89 meV], 106 meV [93 meV], 94 meV [46 meV], and 99 meV [43 meV] for the LUMO [HOMO] transfer integrals of the anti (R,S), anti (R,R), syn (R,S), and syn (R,R) structures, respectively); this is

14.18 and 14.07 Å for the anti and syn (R,S) conformers and to 14.17 and 14.04 Å for the anti and syn (R,R) conformers, in good agreement with experimental X-ray data.14−16,19 The GIWAXS patterns have then been calculated from the X-ray powder diffractogram data, as described in the Computational Details section for the four structures (Figure 2). Experimentally, the GIWAXS patterns of the edge-on phase, which is responsible for the high in-plane mobility achieved in top-gate field effect transistors,17 exhibit three main features:14,16,17 an intense peak reflecting interlayer spacings ranging from 24.3 to 25.0 Å, a broad halo at q = 1.4 Å−1 corresponding to the alkyl chains and a peak at q = 1.6 Å−1 attributed to the π-stacking of the conjugated backbones. As discussed above, the interlayer spacings are in good agreement with experimental findings:14,16,17 they are ranging from 23.93 to 24.93 Å, depending on the conformer considered. At high q values, the experimentally measured halo at q = 1.4 Å−1 and the peak at q = 1.6 Å−1 are well reproduced in our simulated GIWAXS patterns, in particular for the most stable anti structures. An in-depth analysis of the simulated X-ray patterns, including disentangling the contributions from the conjugated cores and alkyl chains (see Figure S4), confirms that the halo is indeed mainly due to the branched alkyl chains while both the alkyl chains and conjugated cores are contributing to the intense peak at q = 1.6 Å−1 (3.93 Å). Table S1 summarizes the 8174


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Table 1. Cell Parameters of the Stereoregular (Entries from 1 to 4) and Stereoirregular (Entries from 5 to 10) Polymer Chains 1 2 3 4 5 6 7 8 9 10

anti (R,S) anti (R,R) syn (R,S) syn (R,R) anti (R,S) anti (R,S) anti (R,S) anti (R,R) anti (R,R) syn (R,S)

anti (R,S) anti (R,R) syn (R,S) syn (R,R) anti (R,R) syn (R,S) syn (R,R) syn (R,S) syn (R,R) syn (R,R)

a (Å)

b (Å)

c (Å)

α (deg)

β (deg)

γ (deg)

26.29 25.57 25.10 24.59 25.84 25.73 25.32 25.49 24.61 24.85

4.69 4.71 4.70 4.78 4.72 4.70 4.72 4.72 4.74 4.74

28.36 28.34 28.14 28.08 28.37 28.25 28.17 28.26 28.09 28.12

119.8 118.5 117.3 116.2 119.9 118.8 117.4 118.1 114.8 116.8

72.8 74.7 78.5 79.2 73.3 76.8 77.1 76.7 79.0 78.8

104.3 103.5 103.5 101.7 102.7 103.3 102.8 103.0 97.4 102.5

stereoregular structures (see Figure 3). In addition, the characteristic distances corresponding to the π-stacking diffraction peaks reported in Table S2 for the stereoirregular systems are very similar to the distances calculated for the stereoregular structures. Even though the supramolecular organization of the stereoirregular systems does not differ significantly from the stereoregular systems, it is well-known that tiny changes in the relative position of the molecules can have a significant impact on the charge mobilities.12 The impact of these stereocenters randomly distributed along the chains on the charge transport properties has been estimated in first approximation by calculating the transfer integrals between tetramers extracted every 0.1 ns from the second half of a 1 ns-dynamics (NPT, P = 1 atm, T = 298 K) performed on unit cells containing four polymer chains. The average HOMO and LUMO transfer integrals for each of the considered structures are listed in Table 2. The LUMO transfer integral averaged over all considered structures exceeds the HOMO transfer integral (49 vs 42 meV for the LUMO and HOMO transfer integrals, respectively), in line with earlier estimates based on model systems.10,31 As the reorganization energy is smaller for negative compared to positive polarons, our calculations suggest that P(NDI2ODT2) has a higher propensity for conducting electrons than holes. Though this is qualitatively consistent with the experimental results, a much more in-depth analysis of the charge transport properties accounting, among others, for the interplay between intra- and interchain motion and the presence of (static) energetic disorder and traps is definitively needed to rationalize the 2 orders of magnitude difference in electron vs hole mobility.10

fully consistent with the fact that this polymer shows higher mobilities for electrons than for holes. Moreover, we have shown earlier that another parameter governing charge transport, i.e., the reorganization energy is always smaller for negative polarons than positive polarons whatever the conformation of the monomer (anti or syn), thereby also favoring electron over hole transport.10,31 In order to estimate in first approximation the impact of thermal fluctuations on the charge mobilities,32 an unit cell containing four polymer chains has been built and submitted to a 1 ns-molecular dynamics simulation (NPT, P = 1 atm, T = 298 K). The transfer integrals have been calculated for the four different stereoregular systems on structures made of two tetramers extracted from the unit cell every 0.02 ns after an equilibration time of 0.6 ns. From these calculations, P(NDI2OD-T2) appears to be a better electron carrier at room temperature since the LUMO transfer integral averaged over the four structures is larger than the thermally averaged HOMO transfer integral (54 and 36 meV, for the LUMO and HOMO transfer integrals, respectively). In more details, while the transfer integrals associated with the (R,S) structures follow the same trends at room temperature as in the equilibrium geometry (thermally averaged LUMO [HOMO] transfer integrals of 75 meV [64 meV] and 73 meV [37 meV] for the anti and syn structures, respectively), larger deviations are observed for the (R,R) structures. The anti (R,R) conformer exhibits larger HOMO transfer integrals (37 and 16 meV for thermally averaged HOMO and LUMO transfer integrals, respectively) while the syn (R,R) conformer should transport electron carriers rather than holes since it has a much larger thermally averaged LUMO transfer integral compared to the thermally averaged HOMO transfer integral (53 vs 7 meV). Since the nature of the stereocenters is not controlled during the synthesis of the polymer chains, they are randomly distributed along the chains. Therefore, the impact of the nonstereoregularity along the chains on the supramolecular organization and charge transport properties has been investigated by considering structures where polymer chains built from monomer units with different stereocenters coexist. Interestingly, the intrinsic characteristics of each monomer (kinked or bent alkyl chains and the conformation of the octyl chain) are preserved within the stereoirregular systems. The cell parameters are very close to those of the stereoregular structures (see Table 1), indicating that the supramolecular organization of P(NDI2OD-T2) is not highly sensitive to the nature of the stereocenters and hence favor systematically ordered assemblies of the polymer chains in bulk.13 This is confirmed through the analysis of the simulated GIWAXS patterns that do not strongly differ from those obtained for the

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CONCLUSIONS Using quantum-chemical and molecular mechanics calculations coupled with the simulations of 2D X-ray diffraction patterns, the supramolecular organization of the n-type P(NDI2OD-T2) polymer has been elucidated at an atomistic level. The most stable structures exhibit a specific interdigitation of the branched alkyl chains (the octyl chain is mainly extended while the decyl chain is bent) fully compatible with the interlamellar distance measured by GIWAXS experiments. The packing revealed by our calculations originates from very specific nonbonded interactions between the nonplanar conjugated polymer backbone and the branched alkyl side chains. Interestingly, the subtle interplay between the relative size of the monomer unit and the nature (asymmetrically branched) and length of the alkyl chains guarantee dense and stable structures whatever the distribution of the nature of the 8175


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Figure 3. Simulated GIWAXS patterns of two stereoregular structures as well as of all the stereoirregular systems under study.

mobility, which calls for more detailed investigations of charge transport accounting for both intrachain and interchain motion, the presence of energetic and positional disorders. With this study, we have highlighted that the size of the monomer unit and the structural characteristics of the side chains are as important factors as the fine-tuning of the electronic properties of the molecular subunits in order to guarantee large electron (and hole) mobility. This study offers new insights for the

stereocenters along the polymer chains, which is a prerequisite to promote efficient charge transport properties. On the other hand, the LUMO transfer integral averaged over all structures considered (stereoregular and stereoirregular) is larger than the averaged HOMO transfer integral, which confirms the higher propensity of P(NDI2OD-T2) for conducting electrons rather than holes. However, the difference cannot strictly rationalize the 2 orders of magnitude difference in electron vs hole 8176


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Table 2. Thermally-Averaged Calculated HOMO/LUMO Transfer Integrals (in meV) Associated to Two Tetramers Made by Alternating the Nature of the Monomer Units (Horizontal and Vertical Entries) anti (R,S) anti (R,R) syn (R,S) syn (R,R)

anti (R,S)

anti (R,R)

syn (R,S)

syn (R,R)

64/75

47/37 37/16

54/78 54/46 37/73

56/64 39/7 24/38 7/53

Article

AUTHOR INFORMATION

Corresponding Authors

*E-mail: (J.C.) [email protected]. *E-mail: (Y.O.) [email protected]. Author Contributions

V.L., L.M., and Y.O. have performed the MM/MD and quantum-chemical calculations, V.L. and Y.O. have written the article. All the authors have designed the study, discussed the results and reviewed the article. Notes

The authors declare no competing financial interests.

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design of new n-type copolymers with further enhanced charge transport properties.

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ACKNOWLEDGMENTS The collaborative work between Mons and Bologna was funded by the European Commission within the ONE-P Project (NMP-LA-2008-212311). In addition, the work in Mons was supported by the European Commission through the SUPERIOR Project (FP7-PEOPLE-ITN-2008-238177), by the European Commission/Rég ion Wallonne (FEDER, Smartfilm RF Project), the Interuniversity Attraction Pole program of the Belgian Federal Science Policy Office (PAI 7/ 05), Programme d’Excellence de la Rég ion Wallonne (OPTI2MAT Project) and FNRS-FRFC. D.B., Y.O., and J.C. are FNRS Research Fellows.

COMPUTATIONAL DETAILS

Force Field Parametrization. The Dreiding force field21 has been reparameterized against quantum-chemical calculations; in particular, the torsion terms that are expressed as: 6

ETor =

∑ 0.5Bn(1 − dn cos(nφ)) n=1

(1)

Using eq 1, we have set up a fitting procedure where Bn, dn, and n are adjustable parameters in order to best reproduce the minima as well as the energy barriers close to these minima (see Figure S1). All geometry optimizations have been performed with Gaussian09 version B01. GIWAXS Pattern Calculations. When generating the GIWAXS patterns, we have defined the x−y plane as the (001) and (010) directions, respectively. The z direction therefore corresponds to the axis perpendicular to the lamellar plane. The angular position of the different spots represented in Figure 3 are calculated by comparing the orientation of the different crystallographic planes as obtained from the Materials Studio Reflex module with respect to the x−y plane while the radial distance with respect to the origin characterizes the interplane distances. However, in thin films, all crystallites do not have the same orientation with respect to the substrate and thus the spots are broadened depending on the amount of disorder present in the films. In our methodology, the intensity of a plane oriented with an φn angle with respect to the x−y plane and corresponding to a peak at 2θn is pondered by a Gaussian function whose standard deviation σ can be varied in order to reproduce the different degrees of disorder in the film. The pondered intensity In is written as

In = In0

⎛ φ2 ⎞ 1 exp⎜⎜− n 2 ⎟⎟ σ 2π ⎝ 2σ ⎠

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(2)

An instrumental broadening of the peaks In was then introduced by a Lorentzian function independent of 2θ, in such a way that the intensity I of the pattern at 2θ is

I=

∑ n

1

I (2θ − 2θn)2 n Δ2

(3)

The broadening is adjusted by the parameter Δ to match the experimental peak width.

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REFERENCES

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ASSOCIATED CONTENT

* Supporting Information S

Ab initio torsion potentials of the P(NDI2OD-T2) monomer, representation of the monomer in the unit cell, and high-energy conformers of P(NDI2OD-T2) polymer chains. This material is available free of charge via the Internet at http://pubs.acs.org. 8177


Macromolecules

Article

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On the Supramolecular Packing of High Electron Mobility Naphthalene

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