Anti-Smoluchowski time dependence of the delayed fluorescence from anthracene in viscous solution due to triplet-triplet annihilation. Effect of Förster energy transfer S₁ + T₁ → S₀ + T_n on the initial spatial distribution of molecules in T₁

The delayed S₁ → S₀ fluorescence from the first electronically excited singlet state S₁ of an aromatic compound in liquid solution is caused by diffusion-controlled triplet-triplet annihilation (TTA) T₁ + t₁ → S₁ + S₀. For a random spatial distribution of triplet state molecules at time t = 0, Smoluchowski's theory for a diffusion-controlled reaction predicts a time-dependent rate constant k_d(t) of TTA with k_d(0) ≫ k_d(∞). If the triplet state is populated by optical excitation S₀→S₁ and subsequent intersystem crossing S₁ → T₁, it is principally impossible to generate a random distribution of triplet state molecules. Since the pair S₁⋯ T₁ is an intermediate during the creation of a triplet pair T₁⋯ T₁, Förster energy transfer S₁ + T₁ → S₀ + T_n may compete with the generation of T₁⋯T₁ at short intermolecular distances. As a consequence, one expects an anti-Smoluchowski behavior of TTA with k_d(0)≪k_d(∞), or with respect to the intensity of the delayed fluorescence, a strong initial rise. The anti-Smoluchowski behavior of a delayed fluorescence has been observed for the first time (with anthracene in a viscous alkane mixture as solvent). The anti-Smoluchowski behavior can be quantitatively described with a simple kinetic model, which contains only four parameters: the diffusion coefficient of molecules in T₁, the Förster radius for the S₁-T₁ energy transfer, and two parameters specifying an exponential distance dependence of the annihilation rate constant for a triplet pair.