Mesoscale molecular network formation in amorphous organic materials

Abstract

High-performance solution-processed organic semiconductors maintain macroscopic functionality even in the presence of microscopic disorder. Here we show that the functional robustness of certain organic materials arises from the ability of molecules to create connected mesoscopic electrical networks, even in the absence of periodic order. The hierarchical network structures of two families of important organic photovoltaic acceptors, functionalized fullerenes and perylene diimides, are analyzed using a newly developed graph methodology. The results establish a connection between network robustness and molecular topology, and also demonstrate that solubilizing moieties play a large role in disrupting the molecular networks responsible for charge transport. A clear link is established between the success of mono and bis functionalized fullerene acceptors in organic photovoltaics and their ability to construct mesoscopically connected electrical networks over length scales of 10 nm.

Keywords: charge generation; disordered properties; soft materials.

Fig. 1.

The molecular structure and LUMOs of several organic electron acceptors used in organic photovoltaic cells. With the exception of C₆₀, all of the acceptors have saturated hydrocarbon solution-processing substituents that contribute little to the LUMO that mediates charge transport. Moreover, the LUMO topology is essentially identical to that of the unfunctionalized parent molecule: All fullerene derivatives have spherical LUMOs, and all PDI derivatives have planar LUMOs.

Fig. 2.

(A, Left) Generalized graph of vertices (a–h) and edges (lines). The edge thickness corresponds to the strength of the connection between vertices, and vertex radii are scaled according to the number of finite capacity edges per vertex. (A, Right) Pedagogic representation of the network analysis algorithm N acting on the adjacency matrix A for the graph on the Left. (B) Returned networks for finite capacity (A_ij > 0), edge capacity greater than 1 (A_ij > 1), and edge capacity greater than 2 (A_ij > 2). (C, Left) Snapshot of 64 DO-PDI molecules from a MD trajectory. Network analysis is performed using states within 0.3 eV of the DO-PDI LUMO, and a variable electronic coupling threshold, V_T. (C, Right) Returned networks for a range of thresholds: V_T values of 1, 5, and 20 meV. In each case, distinct electrical networks are given a unique color, and the total number of networks is specified beneath each cluster.

Fig. 3.

(A, Left) Number of networks, N_NET in PC₆₀BM-based systems as a function of electronic coupling threshold. (A, Right) Ratio of the radius of gyration of the largest network to the total system radius of gyration, as a function of the electronic coupling threshold. (B, Left) Number of networks, N_NET, in PDI derivatives as a function of electronic coupling threshold V_T. (B, Right) Ratio of the radius of gyration of the largest network to the total system radius of gyration, as a function of V_T. (C) Molecular structures of the substituted fullerenes and PDIs. Many regioisomers are possible for bis and tris-PCBM; see SI Appendix for a full discussion. (D) Subnetworks for a sample snapshot of each system, with a coupling threshold, V_T, of 20 meV. In each case, distinct electrical networks are given a unique color, and the total number of networks is specified beneath each cluster. The pink lines in A and B denote the threshold value visualized in D. PC₆₀BM and bis-PC₆₀BM materials are highlighted as the only materials whose largest electronic network percolates the cluster volume at the 20-meV threshold.

Fig. 4.

(A, Left) Average number of connections per molecule, N_C, in C₆₀ derivatives as a function of the electronic coupling threshold (V_T). (Right) N_C as a function of coupling threshold for the PDI derivatives. Shaded areas indicate nonpercolative conditions. (B) Number of networks (N_NET) versus the percolation ratio (R_NET/R_SYS) compared across all materials. Lines added as a guide to the eye, “spherical” refers to the C₆₀ derivatives (in blue) and “planar” to the PDI derivatives (in red).