Theory of Triplet Exciton Annihilation in Polyacene Crystals

An improved theoretical formulation of the triplet exciton bimolecular decay constant in organic crystals is developed which avoids the ad hoc introduction of a “weight factor” $w$ used in previous theoretical treatments, allows for explicit calculations of all reaction constants pertinent to mutual exciton annihilation of triplets, and properly accounts for the anisotropy of the interaction responsible for the bimolecular annihilation. A simplified model for the effective density of final electronic states $\mathrm{\rho (\epsilon )}$ is presented and explicit calculations are performed assuming that only the totally symmetric intramolecular C–C stretching modes are active and by including intermolecular vibrations phenomenologically. Calculations of the bimolecular decay constant $\mathrm{(\gamma )}$ and diffusion tensor, both with and without the inclusion of charge‐transfer interactions, are performed for solid naphthalene, anthracene, and tetracene. We obtain $\mathrm{\gamma \; (}\mathrm{naphthalene}\text{) ∼0.1γ (}\mathrm{anthracene})\mathrm{and}\mathrm{\gamma \; (}\mathrm{tetracene}{\text{∼10}}^{\text{−5}}\mathrm{\gamma \; (anthracene)}$ ; only for anthracene are experimental values available and here agreement between theory and experiment is excellent. Specifically, $\mathrm{\gamma \; (}\mathrm{theoretical}{\text{)\hspace{0.17em}=\hspace{0.17em}4\hspace{0.17em}×\hspace{0.17em}10}}^{\text{−11}}{\mathrm{cm}}^3{\sec }^{\text{−1}}$ compared with $\mathrm{\gamma \; (}\mathrm{experimental}\text{)∼(2}\mathrm{to}{\text{5)\hspace{0.17em}×\hspace{0.17em}10}}^{\text{−11}}{\mathrm{cm}}^3{\sec }^{\text{−1}}$ . The dependence of the bimolecular rate constant on the rate of encounter of two localized triplet excitons and the transition rate to final state is quantitatively analyzed. The intermediate states involved in the annihilating process are identified theoretically as charge‐transfer excitons. The inverse of triplet–triplet exciton annihilation is predicted to have nonnegligible probability in pentacene and tetracene crystals and the corresponding transition rates are calculated and correlated with the low fluorescent efficiency of these crystals.