Atomistic Simulations of P(NDI2OD-T2) Morphologies: From Single

Article pubs.acs.org/JPCB

Atomistic Simulations of P(NDI2OD-T2) Morphologies: From Single Chain to Condensed Phases Claudia Caddeo,*,†,‡ Daniele Fazzi,¶ Mario Caironi,§ and Alessandro Mattoni*,‡ †

Dipartimento di Fisica, Università degli Studi di Cagliari, Cittadella Universitaria, I-09042 Monserrato, Cagliari, Italy Istituto Officina dei Materiali (CNR - IOM SLACS), Unità di Cagliari, Cittadella Universitaria, I-09042 Monserrato, Cagliari, Italy ¶ Max-Planck-Institut für Kohlenforschung, Kaiser-Wilhelm-Platz 1, D-45470 Mülheim an der Ruhr, Germany § Center for Nano Science and Technology@PoliMi, Istituto Italiano di Tecnologia, Via Pascoli 70/3, 20133 Milano, Italy ‡

S Supporting Information *

ABSTRACT: We investigate theoretically the structure, crystallinity, and solubility of a high-mobility n-type semiconducting copolymer, P(NDI2OD-T2), and we propose a set of new force field parameters. The force field is reparametrized against density functional theory (DFT) calculations, with the aim to reproduce the correct torsional angles that govern the polymer chain flexibility and morphology. We simulate P(NDI2OD-T2) oligomers in different environments, namely, in vacuo, in the bulk phase, and in liquid toluene and chloronaphthalene solution. The choice of these solvents is motivated by the fact that they induce different kinds of molecular preaggregates during the casting procedures, resulting in variable device performances. Our results are in good agreement with the available experimental data; the polymer bulk structure, in which the chains are quite planar, is correcly reproduced, yet the isolated chains are flexible enough to fold in vacuo. We also calculate the solubility of P(NDI2OD-T2) in toluene and chloronaphthalene, predicting a much better solubility of the polymer in the latter, also in accordance to experimental observations. Different morphologies and dynamics of the oligomers in the two solvents have been observed. The proposed parameters make it possible to obtain the description of P(NDI2OD-T2) in different environments and can serve as a basis for extensive studies of this polymer semiconductor, such as, for example, the dynamics of aggregation in solvent.

1. INTRODUCTION High-mobility semiconducting polymers1 can be uniformly processed from solution on large areas with printing techniques2,3 on nonrigid substrates, thus enabling a vast range of appealing optoelectronic and sensing applications: nonplanar light-sources,4,5 lightweight and low-cost solarcells, 6−8 along with plastic micro- and nanoelectronic circuits9,10 that may serve as mechanically robust electronics in future flexible displays, conformable digital imagers, and disposable point-of-care devices. The intriguing optoelectronic properties of polymer semiconducting films are inherently linked to the electronic structure of the single π-conjugated segments along polymer chains as these in the solid state are kept together by weak van der Waals forces. It is very well established, for example, that the solid-state packing motif of molecules has a critical role in determining charge-transport properties11 as π-orbitals overlap is required to enable efficient intermolecular transport of charge carriers.12 It is also very well known that processing conditions dramatically influence the film microstructure and, therefore, the density of states (DOS). In order to fully unveil the physics of, for example, charge photogeneration and transport, the nexus between micro© 2014 American Chemical Society

structure and electronic properties is therefore a fundamental prerequisite. This is a field that has to be explored yet. On the one hand, this is because of the rather limited degree of order present in polymer films, especially in the latest generation of πconjugated copolymers lacking substantial long-range order,13,14 thus making it harder to adopt common structural characterization techniques (e.g., X-rays) to access the overall morphology. On the other hand, just a few attempts to directly compute the electronic properties through atomistic models have been done so far;15,16 the development of models capable of fully describing the supramolecular structure of macromolecules in the solid state is therefore an intensive field of research as it would provide powerful tools to finally unveil structure−property relationships. Naphthalenediimide (NDI)based copolymers17−19 have an important role among recently developed π-conjugated polymers because they generally show stable and good electron-transporting properties. In particular poly(N,N′-bis-2-octyldodecylnaphthalene-1,4,5,8-bis-dicarboxiReceived: August 25, 2014 Revised: September 29, 2014 Published: September 29, 2014 12556

dx.doi.org/10.1021/jp5085789 | J. Phys. Chem. B 2014, 118, 12556−12565

The Journal of Physical Chemistry B

Article

Figure 1. Definition of the angles θ (a) and τ (b) on the P(NDI2OD-T2) chemical strucuture. Panel (c) shows the superposition between the DFT and classical structures of the relaxed P(NDI2OD-T2) monomer.

thermodynamic effects, requiring long simulations at finite temperature, which are prohibitive for ab initio methods. However, MPMD calculations require accurate force fields to reproduce the properties of the systems under study, and parameter customization is often needed when modeling nonstandard systems.34,35 In this work, we propose a set of new force field parameters for the description of P(NDI2OD-T2) polymer chain flexibility and morphology. To this aim, we optimize force field parameters against density functional theory (DFT) potential energy profiles (PEPs) in order to correctly reproduce the polymer conformations and, in particular, the torsional angles between the aromatic subunits. The force field is then tested by studying the isolated chain in vacuo, the bulk polymer phase, and two mixed polymer−solvent systems, with the aim to predict the polymer solubility and the miscibility effects. Two different solvents (toluene and chloronaphthalene) have been studied because they provide opposite effects on the final polymer film morphology and microstructure. The simulated polymer crystalline phase, at ambient conditions, is in good agreement with the available experimental data, with planar chain conformation and characteristic distances well reproduced. However, the new torsional parameters describe a chain that is flexible enough to allow for chain folding and selfaggregation in vacuo. Polymer solubility has been investigated within the Flory− Huggins theory,36,37 based on an effective dimensionless parameter χFH, taking into account the specific solvent−solute pair and the temperature. The latter can be calculated by atomistic simulations by following different approaches.38−41 Here, we make use of the same methods applied successfully for conducting conjugated polymers in tetrahydrofuran. The solubility parameters χFH for both toluene and chloronaphthalene are obtained through the evaluation of the enthalphy of mixing by simulating the pure solvent, pure polymer, and the mixed solvent−polymer phase.42 We confirm the experimental observation of a better solubility of P(NDI2OD-T2) in chloronaphthalene. The analysis of the NDI2OD-T2 oligomer in solution shows different dynamics and morphology in the two solvents. The force field parameters proposed in the present work are able to describe P(NDI2OD-T2) as an isolated molecule in aggregated crystalline phase as well as in solution, opening the way to the study of the dynamics of the polymer in solvents.

mide-2,6-diyl-alt-5,5′-2,2′-bithiophene) (P(NDI2OD-T2)) (see Figure 1) has been the subject of intense studies since its publication in 200914,20,21 and can be considered as a model system due to its rich microstructure and high electron mobility, exceeding 1 cm2/(V s) under high applied fields.22 P(NDI2OD-T2) films have been largely investigated, evidencing a strong interconnectivity of solvent-induced nanofibrillar structures,22,23 where the ordered phase shows a prevailing inplane alignment of crystalline planes, on which the bithiophene unit lays flat, with the NDI unit featuring a dihedral angle of ∼40°.24−26 Very interestingly, thermal-induced, drastic modifications of the local packing in the bulk do not lead to substantial variations in carrier mobility in field-effect devices,27,28 an observation compatible with a hopping transport process of a strongly localized intramolecular polaron.24,28 Furthermore, it is now established that the morphology, and in particular, the chain alignment at the surface (which determines the OFET mobility) can be very different from that in the bulk. In particular, recent NEXAFS measurements by Schuettfort et al. revealed that the polymer chain exhibits an egde-on orientation at the surface, even for a preferential face-on orientation in the bulk.29 The critical role of the processing solvent in film formation has been recently evidenced; it has been observed that P(NDI2OD-T2) has a tendency to preaggregate in specific organic solvents, while nonaggregated chains are predominant if the polymer is dissolved in solvents with large and highly polarizable aromatic cores (such as, e.g., chloronaphthalene).30,31 Preaggregation in the liquid phase has an impact on the thin film morphology at different length scales. For example, organic solar cells obtained from nonaggregated chains have shown highly increased power conversion efficiency (PCE) with respect to those built from preaggregated ones; nonaggregated P(NDI2OD-T2) was used as the electron-accepting component in an organic solar cell with a PCE of 5%, among the best values for all-polymer solar cells, owing to an improved interphase with the donor polymer.30,32,33 A careful choice of the solvents employed for spin-coating leads to the possibility to finely control the electrons mobility over 2 orders of magnitude, enabling liquid-crystalline-like domains that extend for hundreds of micrometers.22 In this frame, the polymer solubility plays a fundamental role in determining the thin film microstructure and device performance. In particular, because the final morphologies are determined by the polymer−solvent interactions at the molecular level, atomistic simulations can aid the experimentalist to understand and predict morphological features that are the key to the fabrication of efficient devices. Model potential molecular dynamics (MPMD) is the appropriate tool to investigate large size systems and to take into account

2. THEORETICAL FRAMEWORK 2.1. MPMD Simulations. MPMD simulations were performed by using the NAMD43 molecular simulations package (v. 2.9). The equations of motion of atoms were integrated by using the Velocity Verlet algorithm with a time 12557



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Figure 2. Snapshots of a nonperiodic P(NDI2OD-T2) chain in vacuo at T = 300 K. Top panels (a) show the evolution of the chain modeled with our reparametrized force field. Panel (b) shows the same chain modeled with the Amber force field; the chain does not change its average conformation during the dynamics. Panel (c) is a zoomed view of the chain (modeled with the reparametrized force field) after a 10 ns simulation, showing that a curvature radius as small as ∼1 nm is possible. Side chains are semitransparent for clarity.

particular attention; most classical force fields in fact describe the dihedral angle between thiophenes as flat because it has been observed that when packed in crystalline structures, polythiophenes adopt a planar conformation.64−66 However, it has been shown that the torsional angle between thiophenes is quite sensible to the number of consecutive thiophenes in the chain and to the environment; in particular, for bithiophene in the gas phase and in solution, the angle is not planar.67−71 In order to assess the accuracy of the Amber force field to reproduce the structural and morphological features of P(NDI2OD-T2), we have tested it against the B3LYP/6311G** PEP for the two dihedral angles calculated on the monomer unit, in which the alkyl chains have been replaced by methyl groups. Starting from the stable structures computed from ref 25, the PEP has been obtained by rigidly rotating the two planes defining either the NDI2OD and T2 unit (for angle θ) or the two thiophenes (for angle τ) and calculating the potential energy for each conformation. The same configurations have been used for both DFT and classical PEP. While the standard Amber force field gives a reasonable agreement with the ab initio PEP for angle θ, this is not true for τ, for which it presents different minima and maxima and predicts very high energy barriers for torsion, thus resulting in a flat and rigid polymer chain. Other force fields have been proposed in the literature to describe τ in thiophene derivatives (see, e.g., refs 72 and 73), but to our knowledge, there are no specific parametrizations available for systems containing NDI groups. Here, we modify the Amber force field parameters for τ and θ to reproduce the DFT PEP of P(NDI2OD-T2). More details on the reparametrization procedure and a comparison between the PEPs are reported in the Supporting Information. The torsional potential is expressed as a sum of harmonic terms

step as small as 1.0 fs for constant pressure (NPT ensemble) calculations and 0.5 fs for microcanonical (NVE ensemble) calculations. Multiple time stepping was used, with short-range nonbonded interactions calculated every two time steps and full electrostatics evaluated every 4 time steps. All of the electrostatic contributions were computed by the Particle Mesh Ewald (PME) sum method, with PME grid spacing of 1 Å. The temperature was controlled by a Langevin thermostat with a damping coefficient of γ = 1 ps−1, and the pressure was controlled with the Langevin piston Nosé−Hoover method as implemented in NAMD,44,45 with oscillation and damping times equal to 200 and 100 fs, respectively. 2.2. Force Field Parametrization. The P(NDI2OD-T2) copolymer and the solvents have been modeled using the General Amber Force Field (GAFF)46−48 that includes both bonding (stretching, bending, and torsional) and nonbonding (van der Waals and Coulomb) contributions. The choice of the Amber force field functional form is motivated by its simplicity coupled with its wide range of applicability to organic molecules. Besides its well-known application in biophysics,49−53 it has been successfully applied to study conjugated polymers (in particular, polythiophenes) in the pure polymer phase,54,55 in conjunction with organic56 and inorganic57−59 materials and solvents.42,60 The force field parameters for the relevant dihedrals have been reparameterized against DFT calculations performed on the P(NDI2OD-T2) monomer unit. Partial charges have been assigned according to ESP charges calculated at the DFT level on the full monomer (alkyl chains included). B3LYP, CAMB3LYP, and PBE0 exchange−correlation functionals with 6311G** basis set have been considered. B3LYP/6-311G** results are hereafter reported.61 Gaussian0962 has been used for all of the DFT calculations. For the solvents, atomic partial charges are taken from ref 63. The dispersive (i.e., van der Waals) interactions (both intra- and intermolecular) are described by the sum of two-body Lennard-Jones contributions, with Amber force field parameters. The P(NDI2OD-T2) possesses a number of minima in the PEP, corresponding to different conformations between NDI2OD and T2 subunits. Dihedral angles τ and θ (see Figure 1a,b) are important to define the possible conformations that the polymer chain can assume.25 Angle τ deserves

n

U (ϕ) =

∑ Ai[1 + cos(miϕ − δi)] i=1

(1)

where ϕ is the dihedral angle and Ai, mi, and δi are the potential parameters that have been fitted for the four stable dihedrals governing the angles τ and θ. In Figure 1c, the comparison between the DFT-optimized structure of the P(NDI2OD-T2) monomer and that obtained 12558



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Figure 3. (a) Snapshot of a bulk P(NDI2OD-T2) after a 10 ns simulation. (b−d) Definition of the interlayer (dl), repeat unit (du), and π−π (dπ) distances in the bulk. Side chains are semitransparent for clarity.

by classical simulations using the reparametrized force field is reported. After the validation of the force field, a 16-monomer long polymer chain (oligomer) was built and simulated in vacuo at room temperature in the microcanonical ensemble, to see the possible conformations of the free polymer chain. Figure 2 shows a snapshot of the chain after 10 ns. The standard Amber force field does not allow the chain to bend with small coiling radii, predicting an overall structure that is planar and quite rigid. Conversely, the new model predicts an efficient folding mechanism in vacuo, with coiling angles as small as ∼1 nm (see panel (c) of Figure 2). This is consistent with experiments. Steyrleuthner et al.31 have shown that in pure toluene, P(NDI2OD-T2) forms single-chain aggregates with a radius as small as 10 nm. Because they have considered polymers with average chain lengths of at least ∼100 nm, this corresponds to a coiling radius of ≪10 nm. The new parameters for the force field are given in the Supporting Information.

positions of NDI2OD and T2 groups. Electron microscopy and grazing incidence X-ray diffraction studies from one side have shown that P(NDI2OD-T2) forms lamellae on the substrate, with an exceptional in-plane order and well-defined π-stacking of the NDI conjugated cores.27 On the other side, grazing angle IR spectroscopy has proven the nonplanarity of the NDI2OD and T2 subunits (∼θ = 42°) as well as the polymer chain orientation with respect to the substrate.25 From the computational standpoint, the crystalline phase and structure of P(NDI2OD-T2) has been studied by Lemaur et al.,74 who have investigated four polymorphs differing by the nature of the stereogenic centers of the branched alkyl chains. In particular, they have found that the supramolecular organization of the polymer chains is almost unaffected by the stereochemistry of the chains, and the latter can thus be neglected. In order to reproduce the correct polymer structure in both crystalline and amorphous phases, we have simulated at ambient conditions (T = 300 K, p = 1 atm) bulk crystals of P(NDI2OD-T2) periodically repeated along all directions. Twenty-four chains (made up of 10 monomers each) have been considered for the bulk simulations. In order to assess the influence of the starting configuration on the final crystalline structure, different structures have been simulated by changing the interlayer (dl) and the π−π (dπ) distances, considering interdigitated (starting dl ≈ 24 Å) and noninterdigitated (starting dl ≈ 32 Å) alkyl chain configurations. In all cases, the equilibrium structure presents well-stacked NDI conjugated cores and the characteristic lamellar organization. Depending on the starting configuration, we obtain a highly interdigitated structure as that obtained by Lemaur et al.74 (starting structure fully interdigitated) or a less interdigitated distribution of the alkyl chains (starting structure noninterdigitated), as shown in Figure 3. In both cases, the interlayer, repeat unit (du) and π−π distances in the crystal are in very good agreement with the

3. RESULTS AND DISCUSSION 3.1. Bulk Polymer Properties. Many investigations have been carried out to get insights and understand the strucuture− property relationships of P(NDI2OD-T2) since the very first publications. The high charge (electron) mobility and the mixed amorphous−crystalline film structure, featuring a face-on architecture once deposited, raise a lot of new and open questions in the field of n-type polymeric materials and also in the fundamental understanding of the charge-transport properties of soft matter. Among the properties that have been investigated so far there are: the amount of crystalline versus amorphous phase in the film, the polymer chain structure, the conformation in both crystalline and amorphous phases, the polymer chains orientation with respect to the substrate, and the relative 12559



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because the liquid phase was isotropic), with a lateral size of ∼56 and ∼61 Å for toluene and chloronaphthalene, respectively, corresponding to an equilibrium density ρ very close to the experimental values; see Table 2. An important parameter to be considered when simulating liquids is the enthalpy of vaporization ΔHv, that is, the energy per unit volume necessary to separate molecules into the vapor phase. It can be obtained by the formula79

available experimental values, as reported in Table 1. Both structures (interdigitated and noninterdigitated) are possible in Table 1. Comparison between Our Calculated (First Column DFT, ωB97XD/6-311G*; Second Column MPMD) and Experimental (Third Column) Values for Several P(NDI2OD-T2) Propertiesa dl du dπ ρ CED

DFTb

MPMD

exp

n/a 14.25 Å 3.62 Å n/a n/a

22−25 Å 13.9−14.1 Å 3.4−3.7 Å 1.05 g/cm3 −287 MPa

24.3−25c Å 13.9c Å 3.93c Å 1.18d g/cm3 n/a

ΔH v = RT −

Properties: the interlayer (dl), repeat unit (du), and π−π (dπ) distances and the mass density (ρ) and cohesive energy density (CED). bFrom ref 28. cFrom ref 27. dFrom ref 31.

real polymers, for example, in poly(3-hexylthiophene),57,75−77 depending on the synthesis conditions. The degree of interdigitation of real P(NDI2OD-T2) films has not been resolved experimentally.78 Here, we focus on the noninterdigitated case (see Figure 3a). However, our conclusions on the relative solubility in the two solvents are unaffected by the choice of the reference bulk structure. The mass density and cohesive energy density (CED) of the polymer have been computed consequently (see Table 1). The former is a useful indication about the quality of our model and force field, while the latter is needed to calculate the solubility (see section 3.3). The mass density is calculated as

Nmol ·M m ⟨V ⟩

(3)

where R is the gas constant, T is the temperature in Kelvin, and ⟨Ecoh⟩ is the average cohesive energy. The results for toluene and chloronaphthalene are reported in Table 2 together with the experimental data. The agreement is good and comparable to other theoretical models.63,80 Because it is of interest to calculate the solubility (see section 3.3), we have also calculated the CED of the two solvents, as reported in Table 2. The theoretical data deviate from experiments by ∼15%. 3.3. Polymer−Solvent Interactions. The polymer− solvent interaction is described by the Flory−Huggins parameter χFH, which can be obtained according to the following equation

a

ρ=

⟨Ecoh⟩ Nmol

χFH =

Vref ΔHm 1 kT Vm ϕϕ s p

(4)

where Vref is the reference volume, taken to be equal to the volume of the smallest molecule in the solution (usually the solvent, Vs), (ΔHm/Vm) is the change in enthalpy upon mixing per unit volume (see eq 5), and ϕs and ϕp are the volume fractions of solvent and polymer, respectively. The smaller the value of χFH, the higher the solubility. We have observed that the volume variation upon mixing is negligible; therefore, the changes in enthalpy and internal energy are approximately identical. This approximation holds also for other mixed polymeric systems.40,42 The change in enthalpy can thus be calculated41,42,85 from the cohesive energy densities of pure solvent, pure polymer, and solvent−polymer mixture86

(2)

where Nmol is the number of molecules in the simulated system, Mm is the molecular mass, and ⟨V⟩ is the average volume calculated at p = 1 atm. The CED is the ratio between the cohesive energy Ecoh and the system volume V, where Ecoh is defined as the energy difference between the chains in the bulk (condensed phase) and separated by an infinite distance (gas phase). We find a value of 1.05 g/cm3 for the mass density, in reasonable agreement with an estimated experimental value of 1.18 g/cm3 for very ordered domains.31 The CED is evaluated as −287 MPa, and to our knowledge, experimental values for the CED of P(NDI2OD-T2) are not still available in the literature. The calculated value, however, is similar to those found for poly(3-alkylthiophene)s having side chains made up of 8 and 12 carbon atoms.42 3.2. Solvent Properties: Chloronaphthalene and Toluene. The solvents were simulated in a box containing 1000 molecules, with starting structures taken from ref 63, at constant temperature (T = 300 K) and pressure (1 atm) in periodic boundary conditions (PBCs). Because the starting structures were already equilibrated, the observables were calculated by averaging during 1 ns long production runs. The simulation cell of the solvent remained cubic (as expected

ΔHm = CEDm − CEDs ·ϕs − CEDp ·ϕp Vm

(5)

The interaction parameter calculated via eq 4 takes into account the specific atomistic interactions between the polymer and the solvent. We have simulated the mixed phase by immersing a 16-monomer long P(NDI2OD-T2) chain in the two solvents. The oligomer−toluene and oligomer−chloronaphthalene systems have been simulated for 10 ns at ambient conditions. A snapshot of the systems is shown in Figure 4. The CED of the mixed phase has been calculated as CEDm =

Um − Ns·Usg − Np·Upg Vm

(6)

Table 2. Solvent Properties solvent toluene chloronaphthalene a

ρa 0.83 1.17

ρ (exp)a d

0.86 1.20f

ΔHvb 8.01 13.37

ΔHv (exp)b d

9.07 15.44g

CEDc

CED (exp)c

−278 −383

−331e −402f

ρ is given in g/cm3. bΔHv is given in kcal/mol. cCED is given in MPa. dFrom ref 81. eFrom ref 82. fFrom ref 83. gFrom ref 84. 12560



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Figure 4. Snapshots of P(NDI2OD-T2)/toluene (top) and P(NDI2OD-T2)/chloronaphthalene (bottom) mixed systems after 10 ns simulations. Toluene molecules are represented in red and chloronaphthalene in green. The insets show the details of the morphology of the chain and of the solvent molecules around the NDI cores. Side chains are not shown in the insets.

3.3.1. P(NDI2OD-T2) Morphology in Explicit Solvent. During the 10 ns simulation, the 16-monomer chain does not fold in either of the two solvents. We conclude that the folding in vacuo is more efficient, and this is attributed to a screening of the polymer−polymer interactions in solvent. It has been reported that chain aggregation in solvent has a strong dependence on the polymer molecular weight MW; evidence of chain aggregation was reported only for polymers of MW ≥ 85 kDa,30,31 corresponding to chains formed by at least 80 monomers. A recent work by Schuettfort et al.29 on P(NDI2OD-T2) films cast with different techniques, using P(NDI2OD-T2) with MW = 85.2 kDa and a polydispersity index of 4.03, reports that the low molecular fraction of their sample does not aggregate prior to film formation. Similar results on the importance of MW when studying aggregation have been reported for poly(3-hexylthiophene).88 A further investigation, beyond the scope of the present work, would be necessary to better investigate folding as a function of molecular weight. Relevant information on the oligomer behavior in the two solvents can be obtained from the present analysis. To quantify the diffusivity of the polymer in the two

where Um is the internal energy of the mixed phase, Usg and Upg are the internal energies of the solvent and polymer in the gas phase (isolated chain for the polymer), and Ns and Np are the number of solvent and polymer molecules in the mixed phase. The (ΔHm/Vm) and the corresponding χFH are reported in Table 3, indicating a much higher solubility of P(NDI2OD-T2) Table 3. P(NDI2OD-T2) Solubility and Diffusion Coefficients solvent

(ΔHm/Vm) (MPa)

χFH

D (10−7cm2/s)

toluene chloronaphthalene

0.08 −0.36

0.41 −1.48

1.31 ± 0.1 0.93 ± 0.1

in chloronaphthalene, in agreement with experiments.31 The aggregation of naphthalenediimide polymers such as P(NDI2OD-T2) is mainly driven by the strong noncovalent interactions between their large conjugated cores.30,87 Chloronaphthalene is an aromatic molecule with the same conjugated core of P(NDI2OD-T2); thus, an enthalpic gain is expected as a result of the mixing that corresponds to a negative FH parameter. 12561



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solvents, we can calculate the mean-squared displacement in time (MSD(t)) of the center of mass of the chain as 2 MSD(t ) = [ rcm ⃗ (t + t0) − rcm ⃗ (t0)]

(7)

where rcm(t) is the position of the center of mass at time t. Figure 5 reports the MSD(t) of the polymer chain in the two

Figure 6. Average dihedral angle between adjacent NDI cores for the polymer in toluene (red) and chloronaphthalene (green). The inset gives a graphical definition of αi.

solvent molecules; chloronaphthalene has in fact a larger aromatic core, and it can provide the chain with an environment that is more similar to the crystalline polymer with respect to that provided by toluene (see insets of Figure 4). From Figure 6, we also observe that the fluctuations of α are larger in toluene, meaning that the polymer is able to explore a higher number of configurations in the same time in that solvent. This will impact the configuration of longer chains in the two solvents, especially in terms of probability of aggregation. The present force field, by providing a good description of basic physical properties of the polymer (density, flexibility, crystallinity, solubility), opens the way to a comprehensive study of aggregation phenomena in solution.

Figure 5. MSD(t) of the P(NDI2OD-T2) center of mass in toluene (red) and chloronaphthalene (green). Straight lines are the f(t) = 6Dt fit.

solvents during the simulation. From the MSD(t) of the center of mass, we can calculate the three-dimensional diffusion coefficient D of the chain in the two solvents,89,90 by fitting MSD(t) with the function f(t) = 6Dt. Details of the calculation can be found in the Supporting Information. The values are reported in Table 3, where the errors have been calculated by comparing the results obtained with different methods and are on the order of five times the asymptotic standard error. The value of D is about 40% smaller in chloronaphthalene than that in toluene; although, to our knowledge, experimental values of the diffusion coefficient of P(NDI2OD-T2) in these solvents are not available for comparison, this is expected given the smaller molecular size and density of toluene with respect to chloronaphthalene. Although the predicted values of D may underestimate the actual experimental ones,91 there is a clear indication of a higher diffusivity of P(NDI2OD-T2) in toluene. The local morphology of the chain in the two solvents is also different; in particular, when immersed in chloronaphthalene, P(NDI2OD-T2) shows a flatter backbone with respect to the case of toluene (see insets of Figure 4). If we consider two adjacent NDI cores, we can define the NDI twisting angle αi as the dihedral angle between the two planes specified by the conjugated NDI cores (see the inset of Figure 6), and we can calculate its average value α for the chain during the simulation; the vector vi⃗ connects the nitrogen atoms in the conjugated NDI core, and NU is the number of periodic units of the polymer. ⎡ 1 α = arccos⎢ ⎢⎣ NU

NU − 1

∑ i=1

⎡ vi⃗ · vi⃗ + 1 ⎤ ⎥ = arccos⎢ 1 ⎢⎣ NU |vi⃗| ·|vi⃗ + 1| ⎥⎦

NU − 1

∑ i=1

4. CONCLUSIONS We have studied via fully atomistic simulations the morphological features of P(NDI2OD-T2) copolymer in vacuo, in its bulk phase, and in two solvents. A new set of parameters has been proposed to model the P(NDI2OD-T2) copolymer, with special attention to the torsional angles along the backbone. In order to correctly reproduce the structural and morphological features of the polymer, we have modified the torsional potentials of some dihedral angles from the Amber force field, validating them against DFT calculations. The copolymer has been simulated for the first time in two solvents, namely, toluene and chloronaphthalene, relevant for the film casting procedure and ultimately to the device performance. P(NDI2OD-T2) interactions with these two solvents is of particular importance because it has been shown that it can induce opposite effects in terms of preaggregation in solution, thus affecting the device charge transport properites. We are able to reproduce the bulk structure of P(NDI2OD-T2) and also its high flexibility, which can give rise to folded chains in vacuo. Furthermore, we correctly predict a better solubility of P(NDI2OD-T2) in chloronaphthalene with respect to toluene. Finally, we provide a detailed analysis of the morphology of P(NDI2OD-T2) oligomers in the two solvents, showing that (i) the chain is able to explore a higher number of configurations in toluene due to a higher diffusivity in this solvent and (ii) that the chain backbone is more planar in chloronaphthalene due to the larger aromatic core of this solvent with respect to toluene. These two characteristics are likely to have an impact on long chain aggregation, suggesting a higher probability of folding in toluene. Our new force field parameters accurately describe P(NDI2OD-T2) both in the

⎤ cos(αi)⎥ ⎥⎦ (8)

Figure 6 reports the average twisting angle (α) calculated along the 10 ns trajectory. The angle fluctuates in time, but its average value is always higher in toluene than that in chloronaphthalene, thus indicating, on average, a more planar conformation in chloronaphthalene. The differences in twisting angle observed in the two liquids can be attributed in a first attempt to the shape and size of the 12562



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pure polymer phase and in solution, and they serve as a basis for future research on long chain aggregation in solvent.

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ASSOCIATED CONTENT

S Supporting Information *

Parameters, atomic partial charges of the optimized force field for P(NDI2OD-T2), and full citation for package Gaussian 09. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (C.C.). *E-mail: [email protected] (A.M.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been funded by Istituto Italiano di Tecnologia (IIT) under Project SEED “POLYPHEMO” and IIT platform “Computation”, by Regione Autonoma della Sardegna under L.R. 7/2007 CRP-24978 and CRP-18013, and by Consiglio Nazionale delle Ricerche (Progetto Premialità RADIUS). We acknowledge computational support by CINECA through ISCRA Initiative (Projects SWING and SPASS). D.F. thanks the Alexander von Humboldt foundation for a fellowship.

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REFERENCES

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Atomistic Simulations of P(NDI2OD-T2) Morphologies: From Single

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