Tying Together Multiscale Calculations for Charge Transport in P3HT: Structural Descriptors, Morphology, and Tie-Chains

Evaluating new, promising organic molecules to make next-generation organic optoelectronic devices necessitates the evaluation of charge carrier transport performance through the semi-conducting medium. In this work, we utilize quantum chemical calculations (QCC) and kinetic Monte Carlo (KMC) simulations to predict the zero-field hole mobilities of ∼100 morphologies of the benchmark polymer poly(3-hexylthiophene), with varying simulation volume, structural order, and chain-length polydispersity. Morphologies with monodisperse chains were generated previously using an optimized molecular dynamics force-field and represent a spectrum of nanostructured order. We discover that a combined consideration of backbone clustering and system-wide disorder arising from side-chain conformations are correlated with hole mobility. Furthermore, we show that strongly interconnected thiophene backbones are required for efficient charge transport. This definitively shows the role “tie-chains” play in enabling mobile charges in P3HT. By marrying QCC and KMC over multiple length- and time-scales, we demonstrate that it is now possible to routinely probe the relationship between molecular nanostructure and device performance.


ZINDO and DFT Comparison
In this section, we compare the ZINDO calculation of electronic properties used in this 2 investigation, to a more rigorous DFT method to determine the accuracy of semi-empirical frontier 3 molecular orbital energy calculations for P3HT. We use three representative P3HT chromophore 4 pairs selected from an equilibrated, ordered test morphology, visualizations of which are depicted in 5 Figure S1. The DFT calculations were performed using the B3LYP functional [1] and the 6311++g** 6 basis set [2].  The calculated electronic properties of the chromophore pairs are shown in Table S1. ZINDO Figure S2. Distributions of chromophore Voronoi neighbor transfer integrals for the representative 1,000 molecule a) amorphous, b) semi-crystalline, and c) crystalline morphologies. The red line shows the Gaussian filtered distribution shape that was used to determine the cluster cut-off criterion. The black vertical line shows the value of the cut-off criterion, which was automatically determined to be at the minimum for each system -J i,j > 0.562, 0.549, and 0.457 eV for the crystalline, semi-crystalline, and amorphous morphologies respectively.
The transfer integral distributions for each representative system are shown in Figure S2. In all 30 three cases, the distribution has a large spike at very low transfer integrals and a bump at high TI 31 corresponding to pairs within the same P3HT chain. Initially, we set the transfer integral cut-off to 32 the location of the minimum for each morphology, such that only connections with transfer integrals 33 greater than the cut-off are added to the same cluster. It is convenient to set cut-offs to maxima and 34 minima as these can be determined automatically, rather than being calibrated manually for each 35 separate system. For the crystalline, semi-crystalline, and amorphous morphologies, the cut-offs were 36 set to J i,j > 0.562, 0.549, and 0.457 eV respectively. 37 Figure S3. Visualizations of the clusters in the a) amorphous, b) semi-crystalline, and c) crystalline systems with size > 6 monomer units. Clusters were determined based on an automatically-defined transfer integral cut-off for each system based on the distributions in Figure S2.
The resultant cluster visualization in Figure S3 suggests that these cut-off values are too large -38 in all morphologies, hops with J i,j >∼ 0.5 eV are generally only intra-molecular hops (red region in 39 Figure S2). This leads to nearly every chain in the system being considered an individual cluster, with 40 few occurrences of clusters forming between multiple chains. There is no significant difference in the  Figure S4. Distributions of chromophore Voronoi neighbor transfer integrals for the representative 1,000 molecule a) amorphous, b) semi-crystalline, and c) crystalline morphologies. The red line shows the Gaussian filtered distribution shape that was used to determine the cluster cut-off criterion. The black vertical line shows the value of the cut-off criterion, J i,j > 0.2 eV.
We can, for instance, reduce the cut-off to something smaller in order to include higher J i,j 44 inter-molecular hops. This however, has the short-coming in that such a selection will likely be 45 artitrarily chosen, not rather than an automatically identified minimum. Regardless, reducing the J i,j 46 cut-off to 0.2 eV ( Figure S4) provides significantly improved results as now a non-negligible proportion of inter-molecular hops have J i,j > cut-off, thereby, allowing clusters to form between molecules.
48 Figure S5. Visualizations of the clusters in the a) amorphous, b) semi-crystalline, and c) crystalline systems with size > 6 monomer units, given the following clustering criteria: transfer integral > 0.2 eV. Now, we compare the clusters identified with the J i,j cut-off between the three systems. The 49 crystalline morphology shows one large cluster (shown in red) and a few smaller clusters with opposing

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One short-coming of the previous clustering algorithms is that it considers charge transport 61 between two chromophores in isolation. However, in the KMC algorithm, hops to all neighboring 62 chromophores are considered and the preferential hop (based on the hopping rate between i and j and 63 the random number x) will be chosen. As such, a "good" hop may not occur because there is a better 64 hop. Figure S6. Distributions of the frequencies with which carriers hop between chromophore Voronoi neighbors for the representative 1,000 molecule a) amorphous, b) semi-crystalline, and c) crystalline morphologies. The red line shows the Gaussian filtered distribution shape that was used to determine the cluster cut-off criterion. The black vertical line shows the value of the cut-off criterion, which was automatically determined to be at the final minimum of the frequency distribution: a total of 3264, 1566, and 1635 hops for the crystalline, semi-crystalline, and amorphous systems respectively.
As such, defining clusters based on regions in which charges will freely move is prudent, however,

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we must still identify a sensible cut-off in hopping frequency to separate these regions. The distributions 67 of total hole hops between chromophore pairs in the three representative systems are shown in 68 Figure S6. Note that the x-axis in these plots is logarithmic, leading to quantization of the bins on 69 the left-hand side of the plot. In all three systems, a second peak appears at high hop frequencies.

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This leads to a local minimum at 3264 hops in the crystalline case, 1566 hops in the semi-crystalline case, and 1635 hops in the amorphous case. We therefore use these values as the clustering criteria 72 -only chromophores with connections that are used more than this number during the simulation 73 will be added to the same cluster. We note that the exact values of the cut-off criteria are strongly 74 dependent on the duration of the KMC simulation; the value may change significantly if fewer carriers 75 iterations are performed or if simulation times are reduced. In this study, all three systems used the 76 same simulation time-scales for KMC and the same number of carriers were averaged over in order to 77 obtain the charge transport properties.
78 Figure S7. Visualizations of the clusters in the a) amorphous, b) semi-crystalline, and c) crystalline systems with size > 6 monomer units. Clusters were determined based on an automatically-defined hopping frequency cut-off for each system based on the distributions in Figure S6. The cluster visualizations using the hop frequency cut-off are shown in Figure S7 are very

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In our investigation, we record the location history of every carrier as it hops through the system.

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Using the carrier hopping history, we can construct network connectivity diagrams ( Figure S8 highlight preferred carrier transport routes through the morphology.

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The differences in structure between the three classes of morphology are clearly evident in 104 Figure S8. The amorphous network graph ( Figure S8a) shows that no crystallites have formed in the 105 system. There are several high-traffic nodes spread homogeneously throughout the system, explaining 106 the highly isotropic carrier trajectory presented in the main text. The crystalline network graph conclusions are supported by the cluster maps presented in Figure S8, as well as the cluster properties 129 presented in Table 1; the crystalline and amorphous systems are dominated by a single, well-connected 130 cluster of chromophores permitting a high mobility, whereas the semi-crystalline system is composed 131 of many clusters with differing grain orientations. The visualizations of the network in Figure S8 serve 132 to provide additional evidence as to why the clusters described in the main text form within these 133 morphologies. Here we present how we generate polydisperse P3HT simulations. This can be broken into two 136 steps: first creating a dictionary of P3HT oligomers of varying lengths from 1 to 50 monomers long.

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Second is using a distribution to determine the amount of each chain length to place into the simulation.

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To produce chains of arbitrary length, we use the open-source program mBuild in which a polymer 139 can be easily created using monomer building blocks. We limit the chain length used in this study at 140 50 monomers long as to avoid unphysical interactions of chains feeling themselves across periodic 141 boundaries. To generate the distribution of chain-lengths, we use the Schulz-Flory distribution which 142 is a commonly used mathematical description for polymer lengths in the form [3]: in which P L is the probability of seeing a chain of a given length, D P is the degree of polymerization of 144 a particular chain, and α is a tunable parameter which affects the shape of the distribution. The value 145 for α used in this study was 0.1 and was chosen as this value produces polydispersities of ∼ 1.8.

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To create the actual distribution of chain lengths we utilize a simple Monte Carlo algorithm. In 147 this algorithm we select a random chain length between 1 and 50 and a random number (x) associated 148 with this chain length between 0 and 1. If x is less than the probability of seeing a chain of that length 149 P(L) we accept the chain otherwise the chain is rejected. In addition to this, to ensure that we have the