Methods for the analysis of complex fluorescence decays: sum of Becquerel functions versus sum of exponentials

Ensemble fluorescence decays are usually analyzed with a sum of exponentials. However, broad continuous distributions of lifetimes, either unimodal or multimodal, occur in many situations. A simple and flexible fitting function for these cases that encompasses the exponential is the Becquerel function. In this work, the applicability of the Becquerel function for the analysis of complex decays of several kinds is tested. For this purpose, decays of mixtures of four different fluorescence standards (binary, ternary and quaternary mixtures) are measured and analyzed. For binary and ternary mixtures, the expected sum of narrow distributions is well recovered from the Becquerel functions analysis, if the correct number of components is used. For ternary mixtures, however, satisfactory fits are also obtained with a number of Becquerel functions smaller than the true number of fluorophores in the mixture, at the expense of broadening the lifetime distributions of the fictitious components. The quaternary mixture studied is well fitted with both a sum of three exponentials and a sum of two Becquerel functions, showing the inevitable loss of information when the number of components is large. Decays of a fluorophore in a heterogeneous environment, known to be represented by unimodal and broad continuous distributions (as previously obtained by the maximum entropy method), are also measured and analyzed. It is concluded that these distributions can be recovered by the Becquerel function method with an accuracy similar to that of the much more complex maximum entropy method. It is also shown that the polar (or phasor) plot is not always helpful for ascertaining the degree (and kind) of complexity of a fluorescence decay.