Delayed fluorescence (DF) is a long-lived luminescence process used in a variety of applications ranging from oxygen sensing in biological tissues to organic Light Emitting Diodes. In common cases, DF results from the de-excitation of the first excited triplet state via the first excited singlet state of the chromophore, which produces a mono-exponential light signal whose amplitude and lifetime give an insight into the probed environment. However, non-linear de-excitation reactions such as triplet-triplet annihilation, which can cause decays to lose their mono-exponential nature, are often neglected. In this work, we derive a global framework to properly interpret decays resulting from a combination of linear and non-linear de-excitation processes. We show why the standard method of using multi-exponential models when decays are not mono-exponential is not always relevant, nor accurate. First, we explain why the triplet de-excitation and light production processes should be analyzed individually: we introduce novel concepts to precisely describe these two processes, namely the deactivation pathway - the reaction which mainly contributes to the triplet state de-excitation - and the measurement pathway - the reaction which is responsible for light production. We derive explicit fitting functions which allow the experimenter to estimate the reaction rates and excited state concentrations in the system. To validate our formalism, we analyze the in vitro Transient Triplet Absorption and DF of Protoporphyrin IX, a well-known biological aromatic molecule used in photodynamic therapy, cancer photodetection and oxygen sensing, which produces DF through various mechanisms depending on concentration and excitation intensity. We also identify the precise assumptions necessary to conclude that triplet-triplet annihilation DF should follow a mono-exponential decay with a lifetime of half the triplet state lifetime. Finally, we describe why the commonly used definitions of triplet / DF lifetime are ill-defined in the case where second-order reactions contribute to the deactivation process, and why the fitting of precise mixed-orders DF kinetics should be preferred in this case. This work could allow the correct interpretation of various long-lived luminescence processes and facilitate their understanding.

Faculty of Science University of South Bohemia České Budějovice Czech Republic

Laboratory for functional and metabolic imaging LIFMET Swiss Federal Institute of Technology Lausanne Switzerland

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