A General Formalism of Excitation-Dependent Luminescence Properties in Triplet–Triplet Annihilation Systems

Triplet-triplet annihilation (TTA) has promising applications in optical functional devices and technologies due to its efficient exciton utilization and up-conversion. Traditionally, the change in the slope of the dual logarithmic relationship curve between emission and excitation light intensity from 2 to 1 has been regarded as evidence of the occurrence of TTA. However, this characteristic change was not observed in many single-component organic TTA systems in recent experiments. In this work, we develop new models for TTA processes by introducing more electronic states and transitions than those considered in the traditional model and derive a general formalism of excitation-dependent luminescence intensity, <i>I</i>_em = <i>N</i>(<i>Ck</i>_ex + <i>Ak</i>_ex + <i>B</i> - <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>AB</mml:mi><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mrow></mml:msqrt></mml:mrow></mml:math>). This formalism can be applied not only to typical TTA systems but also to systems whose dynamics cannot be accurately described using traditional models. As <i>k</i>_ex increases, the slope of log <i>I</i>_em ∼ log <i>k</i>_ex changes from 1 to <i>n</i> and then back to 1 (where 1 < <i>n</i> ≤ 2) with two distinct turning points, and the corresponding luminescence quantum yield (Φ_em) increases monotonically until it reaches saturation, which are fully confirmed by the steady-state spectrum experiments. The characteristic change in Φ_em is a more suitable universal criterion for judging the occurrence of TTA. These findings provide a valuable novel tool for probing the kinetic processes in TTA systems that are challenging to model.