Theory of phase-modulation fluorescence spectroscopy for excited-state processes

Theory is presented for the analysis of excited-state reactions by fluorescence phase shift and demodulation methods. Initially, a two-state model with spectral overlap is considered to illustrate most simply the effects of excited-state reactions on the expected phase and modulation values. Secondly, a multistate model is described to illustrate the probable effects of a fluorophore interacting with several solvent molecules. We note the following unique features of phase-modulation data expected from a fluorophore whose emission spectrum shifts during the lifetime of the excited state: (1) The modulation frequency dependence of the apparent phase (τ^p) and modulation (τ^m) lifetimes of the reacted species is opposite to that of a heterogeneous population of fluorophores. (2) For the reacted species τ^p > τ^m. For a heterogeneous sample τ^p < τ^m. (3) The phase angle of the reacted species can exceed 90°. For a heterogeneous sample phase angles are always less than 90°. Thus, phase and modulation measurements can distinguish between time-dependent processes and spectral heterogeneity by observation of any feature described above. Additionally: (4) The lifetime of the product species can be measured directly. (5) Reverse relaxation can be identified, and the reverse relaxation rates calculated. (6) The wavelength-dependent phase and modulation data can be used to resolve the individual spectra from a two-state reaction. (7) And finally, under favorable conditions, a two-state excited-stale process can be distinguished from a continuous multiple-state process. In each instance, model calculations are presented to illustrate the unique potentials of phase-modulation fluorometry for investigations of excited-state processes.