Relating Molecular Morphology to Charge Mobility in Semicrystalline Conjugated Polymers

The molecular-level origins of the effects of annealing temperature and molecular weight on hole mobility in P3HT are elucidated using coarse-grained molecular dynamics, quantum chemical calculations, and kinetic Monte Carlo charge-transport simulations on a variety of realistic thin-film morphologies. The zero-field hole mobility is shown to increase as the annealing temperature or average molecular weights of samples are increased, in accordance with experimental results. Crystal structure analysis shows that the annealing temperature dependence of the mobility can be attributed to the size and structural order of the crystallites in both the chain-backbone and π-stacking directions. However, the molecular weight dependence of the mobility cannot be rationalized in the same way. Longer chains are shown to belong to more crystallites in the morphology, suggesting that the crystals become better connected as the molecular weight of the sample increases. We show that engineering samples to have an increase...


INTRODUCTION
Solution-processed conjugated polymers are an important class of semiconductor due to the possibility of affecting their optical and electronic properties through molecular design 1,2 as well as their compatibility with scalable manufacturing processes. 3,4 These versatile materials are attractive alternatives to inorganic semiconductors in applications where low cost, large area, and flexibility are needed. A fundamental property of component polymers is the charge-carrier mobility (μ), which often determines performance in electronic devices ranging from organic thin-film transistors (OTFTs) and 5 light-emitting diodes (OLEDs) 6,7 to photovoltaics (OPVs). 8−11 Indeed, in some devices, low mobility currently prohibits the use of conjugated polymers in industrially important applications. 12 It is therefore vital to exert control over the mobility to enable conjugated polymers to achieve their full potential. As such, the design of high mobility polymers 13−15 and subsequent processing to yield high mobility morphologies 16−18 is an active area of research. A significant remaining challenge is understanding the relationship between morphology on the molecular level and mobility. 19 This is demonstrated by investigations of poly(3-hexylthiophene) (P3HT)a semicrystalline polymer often used in OTFTs 16,20 and photovoltaics. 18,21 As is the case for many conjugated polymers, the electronic properties of P3HT depend sensitively on the morphology. 22 In the amorphous phase, P3HT exhibits a time-of-flight hole mobility of the order μ ∼ 1 × 10 −5 cm 2 V −1 s −1 . 23 However, P3HT chains can self-assemble into crystals when the film is subjected to thermal or solvent vapor annealing, or high pressure, 24,25 with the resultant semicrystalline morphologies exhibiting hole mobilities larger by at least an order of magnitude. 23,26,27 The degree to which crystals form depends also on the properties of the polymer chains such as regioregularity, 24 molecular weight, 28,29 and polydispersity. 30 However, the collective effect of these variables on aspects of morphology, and subsequently the mobility, is complex. Many aspects of molecular morphology have been correlated with increased mobility in P3HT, such as a larger proportion of crystalline material, 30,31 improved crystalline order in the πstacking direction 32 or along the backbone of the chains themselves, 28,33,34 or, more recently, the presence of tie chains within the morphology linking crystals together. 19,35,36 This demonstrates the challenge of determining causal relationships between mobility and morphology, since even with the benefit of a battery of carefully performed experiments on a wellstudied polymer, the underlying physics is still the subject of much debateespecially in the context of crystallinity. 37 Simulations offer a way to examine the relationship between molecular morphology and mobility in greater detail than is generally available by a single experiment. Of course, for such analysis to be meaningful, the descriptions of both the morphology and charge transport must be realistic. While there are a range of morphology simulation techniques, 38,39 molecular dynamics (MD) methods stand out as providing morphologies with atomistic detail for experimentally relevant molecular weights and polydispersities. MD simulations use a calibrated force field to replicate the bulk behavior of chains under different processing conditions and annealing temperatures. 40−42 While some MD simulations model samples atomistically, an alternative method involves grouping together clusters of neighboring atoms into a single "coarse-grained" (CG) site for reduced computational cost. 43,44 This is an especially important issue as computational cost limits the amount of simulated material that can be considered. CG models have been used to successfully reproduce the experimental data of chain conformation and phase behavior in conjugated polymers, 45 the miscibility and interpenetration of OPV blends, 46 and predicting microstructure architectures 47 that can be linked to mesoscopic morphological features. 48 However, to date, the detailed film morphologies generated by these techniques have not been joined with charge transport simulations. Studies have instead focused solely on either examining the film morphologies 44 or charge transport in idealized morphologies. 49,50 In this article, we generate realistic P3HT morphologies using CG MD simulations and use these in a charge transport model to relate structure to mobility. We simulate populations of chains that have commercially available weight-averaged molecular weights (19.4 kDa ≤ M w < 69.9 kDa) and polydispersity indices (1.5 ≤ PDI < 2.4) for a realistic thin film density (ρ = 1.1 g cm −3 ). These chains are then subjected to various annealing protocols to better understand how thin film morphology evolves. These diverse, realistic morphologies are used in kinetic Monte Carlo (KMC) charge transport simulations in which hopping rates are determined by semiempirical quantum chemical ZINDO/S calculations. Using this approach, we recreate the experimentally observed increase in P3HT hole mobility, μ, when the annealing temperature, T a , and molecular weight, M w , are increased. We show that the observed annealing temperature dependence of μ is correlated with increasing volume of crystals, and better intrachain crystal order, although not crystal order in the πstacking direction. Conversely, we find that the observed M w dependence of μ cannot be explained by these aspects of the morphology. Instead, we show the improvement in μ as a function of M w is largely due to the presence of tie chains The Journal of Physical Chemistry C Article linking together crystallites within the morphology, which provide improved charge transport through the amorphous regions. This suggests a model of mobility in semicrystalline conjugated polymers that is jointly determined by transport in the crystal (dependent on crystal size and order) and the availability of connecting tie chains, which are a subset of the otherwise noncrystalline regions of the film. Although P3HT is used as a test case in this investigation, the conclusions are expected to be generally relevant for other semicrystalline polymers such as poly(2,5-bis(3-tetradecylthiophen-2-yl)thieno[3,2-b]thiophene) (PBTTT) and poly(5,5′-bis(3-alkyl-2-thienyl)-2,2′-bithiophene) (PQT), which have been shown to exhibit similar transport activation energies. 19

SIMULATIONS
The coarse-grained (CG) simulation model of regioregular P3HT represents each monomer by three spherical sites that correspond to the thiophene ring (P1), first three hexyl sidechain methyl groups (P2), and last three side-chain methyl groups (P3) ( Figure 1a) and was parametrized to reproduce the local structure of an accurate atomistic P3HT model. 45,51 We note that as three coarse-grained sites are used, the monomers are anisotropic, and so chains are still permitted to self-assemble despite each site being spherical. A closely related CG model of P3HT with the same bonded interactions (which most strongly determine the polymer melt structure), but with nonbonded interactions that describe an implicit anisole solvent, has previously been shown to capture thermodynamic and nanoscale structural properties of solution-phase P3HT well. 45 The CG model of pure regioregular P3HT derived here captures the structural properties of P3HT oligomers in the atomistic model over a wide range of thermodynamic conditions, as shown in Supporting Information section S1.
A database of P3HT chains with polymerization lengths ranging from 2 to 1000 monomers (330 Da−164 kDa) was generated. These chains were permitted to relax within the MD simulations for 1 ns in isolation at 290 K, according to the intramolecular components of the force field, thus mimicking their arrangement within solution. A total of 40 relaxed chains were then selected at random from a custom chain distribution (Supporting Information section S2) to yield weight-averaged molecular weight (M w ) and polydispersity similar to those available commercially (e.g., Sigma-Aldrich, Reike Metals). It was found that the commonly used Schulz−Flory distribution 52 did not adequately describe the combination of large molecular weights and high polydispersities expected in P3HT samples.
This process was repeated to give a range of samples with 19 ≤ M w < 70 kDa, which are named S-19 to S-70, respectively, as shown in Table 1. The selected chains, representing a particular P3HT sample, were then placed in a large, periodically bounded simulation box at random positions and orientations, with sufficient empty space between chains to avoid interchain interactions. This large simulation volume was then compressed over a period of 40 ps at 290 K, until the density (ρ) reached the average P3HT film density of 1.1 g cm −3 . 53 This mimicked the effect of solvent evaporation, producing a CG morphology containing P3HT chains with realistic M w , PDI, and ρ, occupying a cube with sides of the order 10 nm. The exact simulation volumes varied with M w but were all of similar order to thin film thickness in real devices. 18,44 An example morphology is depicted in Figure  1b, with a single chain highlighted to show expected conformation within the force field.
The CG morphologies then underwent a series of annealing and equilibration MD simulations utilizing both the intra-and intermolecular components of the force field. All simulations were performed at constant volume and temperature, maintained using a Nose−Hoover thermostat and barostat 54 and timesteps of 4 fs, using the LAMMPS suite. 55 The morphologies were first simulated at a temperature T a , varying from T a = 290 K (no annealing) to T a = 623 K for 4 ns. Of course, real devices are annealed for much longer; however, we observed that the total potential energy and structural distribution functions reached constant values within this time scale (Supporting Information section S3), suggesting that some level of thermally activated chain reorganization had taken place. The 4 ns annealing time was imposed due to the computational cost of the MD simulation. To mitigate the effect of the reduced annealing time, T a values exceeding those used in experiment were also used to increase the energy transferred to the system during this time, 44,56 although it was found that T a > 623 K resulted in numerical instabilities in the MD simulations for the chosen time step. Annealed films were allowed to cool down to room temperature over a further 4 ns period, before equilibrating at T a = 290 K for a final 4 ns.
The physical conformation of the chains in the equilibrated system was then analyzed to determine the crystal structure of the samples. Crystals in the morphology were identified by determining the proximity of conjugated subunits (i.e., CG site P1the thiophene ring in each monomer) to others in the morphology, given an initial crystal direction described by the vector [hkl]. For each subunit, d [hkl] was defined as the separation to the closest thiophene ring lying within a cone of apex angle 30°protruding both above and below the CG site in the [hkl] direction. This was calculated along both the πstacking axis (d [010] ) and the chain backbone axis (d [001] ), based on the positions of the bonded alkyl side chain and nearestneighbor thiophene rings.  [010] , respectively, where only the narrow, crystalline distribution is considered for the π-stacking direction, as this is the region probed by XRD methods. The crystal proportion (γ) within the morphology was approximated by determining the integral underneath the crystalline Gaussian curve as a proportion of the integral under the whole distribution. Later, we will show that γ can be affected by both M w and T a . However, while the current morphologies have realistic dimensions, it is important to note that the short annealing time will have reduced the opportunity for crystals to form and hence limit γ. The crystals formed here were necessarily smaller than 10 nm in extent and so were smaller than some reports on P3HT experimental films.
Having related the supramolecular structure of the CG morphologies to variations in M w and T a , we now move on to identifying links between these properties and the simulated hole mobility of the morphology. As such, we implemented a geometrical "fine-graining" procedure to approximate the atomic positions based on the CG morphologies (the full details of which are given in Supporting Information section S4). This atomistic representation is required in order to calculate electronic couplings that were ultimately used for the kinetic Monte Carlo model. Briefly, the fine-graining procedure was implemented by manipulating an energetically minimized atomistic template of a thiophene ring to produce a 100% regioregular chain backbone, with intermonomer torsion determined using the positions of the bonded neighboring CG sites. The manipulation was repeated for every monomer in the chain, resulting in an atomistic polythiophene chain with thiophene ring positions and orientations corresponding to the CG MD outputs, an example of which is shown in Supporting Information section S4. Given that the solubilizing alkyl side chains only affect the charge mobility characteristics by constraining the self-organization properties of the backbone itself, 59−61 we omitted the consideration of the side chains in the transfer integral calculations for simplicity.
Calculations of the density of states (DoS) for P3HT have shown that the orbital density localization is around seven monomers. 56,62 In our semiclassical charge transport model, we assumed that holes hop between these segments of increased orbital overlap. As such, the polythiophene chains were split into segments by considering the "chain backbone" vector between centers-of-mass of adjacent thiophene rings, starting at one end of the chain. If the angle between the chain backbone vector and the vector to the next thiophene along in the chain was within a tolerance angle θ c , the thiophene units were assumed to be part of the same delocalized segment; otherwise, the thiophene unit was assumed to be part of a new segment. θ c was selected such that the hole was, on average, delocalized over seven monomers for the S-57 sample and then kept constant for all morphologies. This segmentation of the chains is an approximation since the HOMO wave functions are not calculated directly. However, the mean delocalization fits with more rigorous calculations via θ c . 56,63 Segments were then sorted into pairs of hopping partners, i.e., two segments that were within r max = 20 Å of each other including the periodic boundary conditions, which is much larger than the d̅ [010] and d̅ a spacings. Quantum chemical ZINDO/S calculations were then performed on each segment individually, as well as all the hopping partner pairs, in order to calculate the molecular orbitals 64 using the ORCA program suite. 65 These allowed the electronic coupling transfer integral |J ij | to be calculated: where (E HOMO − E HOMO−1 ) denotes the HOMO splitting energy for the dimer containing the two hopping partners and ΔE ij is the difference in energy between the initial and final sites. Like some other quantum chemical calculations, 63 we found the DoS to be approximately Gaussian, but significantly broader than those expected in real devices. 27,33,66 Using these values to calculate ΔE ij led to imaginary transfer integrals which made the charge transport simulations unrealistic. Some studies utilize Koopmans' approximation, 67 in order to set ΔE ij = 0 by assuming that isolated segments have the same energy. 68,69 While Koopmans' approximation provides good agreement with density functional theory results for a variety of systems, 70,71 it nonetheless discards relevant information about the energy levels of the segments, which in turn are dependent on the morphologies of interest and argued to be very important in determining mobility. 19 To resolve this issue, we used an approximate method where the ZINDO/Scalculated HOMO levels were mapped onto the experimentally determined DoS (Supporting Information section S5), such that the sigma within the distribution (e.g., ±1 standard deviation from the mean) was retained. Note that this mapping was only used to determine ΔE ij for the transfer integral and the subsequent Marcus hopping rate in eqs 1 and 2; the HOMO splitting calculation was left unmapped from the simulated molecular orbitals. The transfer integral was then used to determine the rate at which hops could take place, given by the semiclassical Marcus expression: 72 where k B is the Boltzmann constant, T is the absolute temperature, and the reorganization energies, λ ij , were estimated using a model predicted from DFT calculations based on the length of the host segment (Supporting Information section S6). 73 Section S6 also includes a comparison of the transfer integral calculations from other studies, which are shown to match well the values used here. Note that the transfer integrals between all segment pairs with separation r < r max in the morphology were considered, thereby implicitly allowing variable range hopping. 74 The Journal of Physical Chemistry C

Article
The charge transport simulations themselves consisted of a KMC algorithm that calculated the wait time until the next hopping event using the equation where x is a uniformly distributed random number between 0 and 1 and k hop is the Marcus hopping rate calculated from eq 2. In each simulation, a single hole was injected onto a random segment within the morphology and permitted to hop through the system for a time t max . The mean-squared displacement (⟨x 2 ⟩) was calculated and the process repeated 10 4 times. ⟨x 2 ⟩ was shown to be directly proportional to t max , with proportionality corresponding to the 3D diffusion coefficient D. The related zero-field hole mobility, μ 0 , for the morphology was calculated using where q is the elementary charge. Figure 2 shows the simulated μ 0 for various P3HT samples used in this investigation ( Table 1) as functions of annealing temperature, T a , and weight-averaged molecular weight, M w . First, it is important to note that the absolute values of the mobilities recorded here are several orders of magnitude greater than other time-of-flight measurements obtained from real devices. These tend to report mobilities within the range 2 × 10 −4 −2 × 10 −3 cm 2 V −1 s −1 for highly regioregular pristine P3HT films at zero to low fields (F ≃ 4 × 10 6 V m −1 ), depending on the processing regime. 23,26,27 However, it is reasonably common for ab initio calculations of mobility to differ from experiment by some orders of magnitude due to the significant challenge of modeling a complex, disordered system. 75,76 Our focus here is how molecular morphology influences mobility through M w , T a , and the PDI. Hence, we now move on to consider relative changes in mobility and how this compares with experiment. While the data show some scatter, there is a clear monotonic trend of increasing μ 0 with both M w and T a . The improvement in μ 0 is shown to saturate beyond ∼450 K to an extent that depends on M w , with smaller M w showing more pronounced saturation. T a = 423 K is a common annealing temperature, for which it has been shown that μ 0 is larger by a factor of 2−3 when compared to as-cast films. 77,78 These data agree reasonably well with the predicted improvement in μ 0 by a factor of 3.5 shown here. The mobility is also sensitive to M w , as increasing M w from 19 to 70 kDa increases μ 0 by a factor of 3.2 for unannealed samples and a factor of 2.1 for morphologies annealed at 623 K. Some field-effect transistor measurements on samples of unannealed P3HT have reported that an increase in M w from 11 to 51 kDa results in an increase in saturation mobility of around 1 order of magnitude 79 or a factor of 5 over the range 21 ≤ M w < 61 kDa. 80 For transistors annealed at T a = 370 K, mobilities have been reported to increase by a factor of ∼3 for the range 20 ≤ M w < 76 kDa. 81 However, time-of-flight mobility investigations for unannealed films show that the mobility actually decreases by a factor of 5 as M w is increased from 26 to 72 kDa. 33 It is difficult to be exact in comparing the current data to experiment due to the variety of protocols used; however, it is clear that the general trend of increasing μ with M w is reproduced. The change in μ 0 with M w is smaller than most reports; however, this might be expected due to shorter annealing times. We note again that in our model we have made simplifying assumptions to allow charge transport simulations on realistic thin films. Despite this, we note that the model has recreated the relative changes in μ 0 with temperature and molecular weight, as seen in experiment. In order to determine the reason for the observed mobility trends in Figure 2 and that of previous studies, we now consider the morphological structure of the P3HT film.

RESULTS AND DISCUSSION
The results of the paracrystallinity measurements for the single distributions are shown in Figure 3. Along the polymer backbones, the disorder is shown to decrease as the annealing temperature increases, characterized by a narrowing d [001] distribution. Regular lattice spacings in this direction result in strong coherence between adjoining hole delocalization segments, manifesting as an increase in the average segment length, L (Figure 3c). A combination of these factors could explain the observed increasing mobility with increasing annealing temperature (Figure 2). However, we note that Figure 3a does not show a monotonically decreasing g [001] as a function of increasing molecular weight. The g [010] data shown in Figure 3b lie within 7% ≤ g [010] < 10% and show a slight reduction with increasing T a and no clear trend with M w . This is in agreement with sources in the literature that suggest that P3HT π-stacking disorder is high and mostly independent of degree of polymerization beyond 100 monomer units (M w = 16 kDa) 19 as well as XRD data for semicrystalline polymer Figure 2. (a, b) The zero-field hole mobility for single distribution samples with different molecular weights (as given in Table 1) as functions of (a) annealing temperature (T a ) and (b) molecular weight (M w ).
The Journal of Physical Chemistry C Article films 50,82 such as poly(bithiophene-alt-thienothiophene) (PBTTT). 83,84 GIWAXS measurements on thick (∼100 nm) films annealed at 413 K have shown that P3HT generally exhibits a slightly lower paracrystallinity g ∼ 5%, 85 suggesting that our morphologies are more disordered than equivalent experiments but still lie within the semicrystalline regime. 82 Although we find correlations between g [001] and T a , and hence μ 0 , there is no clear dependence of g [001] or g [010] on M w , even though μ 0 does vary with M w . This may be exacerbated by the short annealing times that are used in the current investigation, but it does seem to indicate that not all of the mobility trends observed in Figure 2 can be solely explained by the paracrystallinity along either axis. This again is in agreement with morphological studies that have suggested that the πstacking paracrystallinity does not affect the mobility. 19,49 The d [010] distributions shown in Figure 1b also allow an estimation of the volume occupied by the crystal regions. To do this, we measure the "crystal cutoff" (d cut ) for the morphology as the value of d [010] corresponding to the minimum value for the combined bimodal distribution (around 6 Å for all morphologies). Thiophene spacings d [010] < d cut are considered as belonging to a crystal, whereas thiophene with d [010] > d cut belong to the amorphous region of the morphology. We then define the crystal proportion, γ, as the integral of the d [010] distribution up to d cut as a fraction of the integral of the whole distribution. Figure 4 shows γ as functions of both T a and M w . The absolute values of γ are around 0.2 lower than those reported from melting enthalpy calculations, 30 again likely because those devices were annealed for a much longer period than the current simulations. In general, Figure 4 shows that γ increases linearly with annealing temperature as expected. By contrast, γ shows no clear dependence on M w .
Differential scanning calorimetry (DSC) measurements of the enthalpy of fusion have shown that the expected degree of crystallinity remains approximately constant for 17 ≤ M w < 93 kDa at room temperature. 86 A different study of smaller molecular weights showed that the volume occupied by crystals increased from 4.5% at M w = 2.6 kDa to 18% at M w = 27 kDa. 31 Perhaps then it is unsurprising that for the current data in the range 19 ≤ M w < 70 kDa and considering the short annealing times, γ varies by a maximum of 5% for unannealed devices and even less for films annealed at increased T a . Again, as with the paracrystallinity measurements, we find that increasing T a is correlated to an increased crystal proportion in the film, which may partly explain the observed increase in μ 0 with respect to T a . However, we also find no clear trend relating γ to M w for the examined conditions even though μ 0 is shown to clearly increase with M w .
It has been argued that high mobility is caused not only by the size of well-ordered crystals but also by the presence of "tie chains" that form efficient connections between them. 19 To show the impact of tie chains in the current data, Figure 5 shows the number of crystals each chain participates in (ζ) as a function of polymerization, D p . We observe that increasing M w increases the likelihood that individual chains will participate in more crystals. Furthermore, it can also be seen that this trend is largely unaffected by the annealing temperature. We therefore propose that the observed M w dependence of μ 0 , which is not explained by the crystalline order or crystallite size, is instead due to the population of long polymer chains which link together many crystallites, in agreement with Noriega et al. 19 Implicit in this argument is that the non-crystalline regions of the film are divided into tie chains, which are members of many crystals and benefit mobility, and shorter chains, which are members of fewer crystals and therefore do not benefit mobility. This suggests semicrystalline conjugated polymer mobility can be optimized by increasing the number of tie chains in the noncrystalline phase.
To test this hypothesis, films were generated in which there existed an increased population of tie chains in an otherwise small molecular weight sample. Proportions of the S-19 and S-70 samples were mixed together to produce systems of similar M w to the single distributions, but with polydispersities ranging between 2.2 ≤ PDI < 2.4. These samples are denoted  Table 1 by M-X, where X is the wt % of S-70 incorporated into the S-19 morphology. Figures 6a,b show that the population of high-ζ tie chains has been successfully increased by adding a small number of chains from the S-70 morphology to the S-19 sample, as predicted by experiment. 19,35,36 The tie chains in the mixed morphologies generally belong to more crystals than those within single-distribution morphologies of the same M w . As with the single-distribution samples, there is no T a dependence of ζ, suggesting that the effect of added chains is largely to join crystals together rather than change crystal properties.
Figures 6c,d show the mobilities for the resultant mixed morphologies, along with corresponding single distributions of equivalent M w . Increasing the proportion of long chains within the morphology by mixing is shown to increase the mobility, and indeed, mixed morphologies are shown to have a comparable mobility to single-distribution samples of the same M w . This suggests that it is possible to make a low M w sample behave like a high M w sample through the addition of longer chains, which "tie" the crystallites together. In turn, the data indicate that increasing the polydispersity of semicrystalline polymers can have beneficial effects on the electrical performance. While tie chains have been introduced here by engineering the distribution of M w , we note that similar effects could be achieved in conjugated polymers with increased persistence length and rigidity. 87

CONCLUSIONS
To conclude, we have used coarse-grained molecular dynamics simulations of realistic samples of P3HT chains to obtain thin film morphologies for a variety of annealing temperatures. The morphologies then underwent a fine-graining procedure, returning them to the atomistic representation so that the semiempirical ZINDO/S method could be used to determine the charge transport characteristics of the thin films. Hole mobilities at zero field were calculated and showed that the mobility increases both when the annealing temperature and weight-averaged molecular weight of the sample were increased, in accordance with experimental results. The crystal structure of the morphologies was then analyzed to help explain these dependencies.
Both the crystalline order along the chain and the volume proportion of crystals within the morphology, γ, were shown to increase with increasing annealing temperature, suggesting that as devices are annealed at higher temperatures, more and better ordered crystals are produced, increasing the hole mobility. The π-stacking crystalline order was shown to increase slightly with increasing temperature, indicating that although well-ordered πstacking crystals can contribute slightly to the observed mobility dependence on temperature, it is less important than the other crystal properties, as determined by some experimental investigations.
Conversely, the molecular weight dependence of the mobility could not be explained by along-chain or π-stacking disorder, or γ. It was found that as the molecular weight increases, the  The Journal of Physical Chemistry C Article average number of crystals that each chain belongs to also increases, resulting in better connected crystals. We propose that the bulk, long-range mobility in semicrystalline polymers is limited below the short-range crystal mobilities by the availability of tie chains within the system. This, in turn, means that the noncrystalline sections of a semicrystalline morphology perform a useful function, although not all of the noncrystalline regions are useful tie chains.
As an example to show how the utility of the noncrystalline parts of the morphology can be improved, we engineered a P3HT sample with small molecular weight to include a proportion of longer chains to act as tie chains. We showed that adding long chains resulted in improved mobility which was comparable to a single-distribution sample of the same average molecular weight. The added chains were shown to have improved crystal connectivity, supporting the conclusion that tie chains are the main contributor to the molecular weight dependence of the mobility and simultaneously showing that a high polydispersity is not necessarily detrimental to charge transport within the active layer. These conclusions need not be specific to the test system of P3HT and may be generally true for other semicrystalline polymers.

* S Supporting Information
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.5b11511.
Parametrization of coarse-grained regioregular P3HT model, chain length distribution, potential energy and morphological evolution, fine graining methodology, Gaussian mapping of the density of states, and determination of the reorganization energy and transfer integrals. All underpinning data used in the investigations can be found at DOI: 10.15128/8k71nh08w. (PDF)

Notes
The authors declare no competing financial interest.  ■ REFERENCES