to exciton formation: 

The coefficients and the rates are dehned and further discussed below. The exciton formation process is described as a special type of hopping process. When a hole is located on site i and an electron is located on a neighboring site j , an exciton formation event can occur and an exciton may be formed on either site i or j . The general structure of the exciton formation term is that of the Langevin recombination rate formula: the formation rate is proportional to the product of the electron and hole occupation probabilities on the two sites. At equilibrium, detailed balance is preserved by the thermal generation term , in which < ¾ and Cj are the equilibrium hole and electron concentrations at site i and j , respectively. The electron and hole concentrations at equilibrium are calculated self-consistently by solving the lD-Poisson equation combined with Fermi-Dirae statistics. In this work, we assume that the excitons formed by electron-hole recombination decay immediately to the ground state. Excitonic interaction processes (such as exciton diffusion, exciton- exciton annihilation, exciton-polaron quenching, etc.) are thus not included. We intend to include such interactions in future work. 

From the Langevin expression, the exciton formation rate is expected to be pro portional to the sum of the electron and hole mobilities. The exciton generation term is therefore taken to be proportional to the sum of the local electron and hole hop rates, modified by taking into account that upon exciton formation these rates are affected by the exciton binding energy, E$_{exc}$ 

with 

For disordered small-molecule organic semiconductors, E$_{ex}$$_{c}$ ,b can be as large as 1 eV, or more. E$_{ex}$$_{c b}$ depends on the exciton spin (singlet, triplet). For notational sim plic ity, we have omitted this dependence in Eq. (5). From the Langevin expression, it 

