Quenching Single-Fluorophore Systems and the Emergence of Non-linear Stern-Volmer Plots

Reduction in fluorescence's intensity upon addition of quencher molecules is often quantified by the Stern-Volmer equation. Central to the underlying model is the formation of an adduct between quencher and excited-state (dynamic-quenching), or ground-state (static-quenching), fluorophore at steady-state conditions. Assuming a thermodynamic behavior, that is, a system with large numbers of fluorophore and quencher molecules, the resulting dependency of the ratio between fluorescence intensities, with and without quencher, on quencher's concentration is linear. Yet, alongside abundance reports confirming this linear behavior, numerous observations indicate the dependency can also be non-linear with either upward, or downward, curvature. By maintaining the same physical mechanisms for quenching, we derive in this paper an alternative equation to describe fluorescence quenching. Here however, we assume a local equilibrium (steady-state) between a single-fluorophore and finite number of surrounding quencher molecules, effectively, partitioning the (macroscopic) system into many non-interacting small subsystems. Depending on fluorophore's properties, association's strength, and conditions, the resulting behavior exhibits linear dependencies, upward curvatures, or downward curvatures. More specifically, the relation reads, $I_\circ/I = 1 + {\cal Z}K[Q]_{_T}/(1 + (1-{\cal Z})K[Q]_{_T}) $, where $K$ is a steady-state equilibrium constant for complex formation and $[Q]_{_T}$ is total concentration of quencher in the small subsystem. The dimensionless parameter ${\cal Z}$ has different expressions for dynamic and static mechanisms. In the former it is a ratio between maximum rate of quenching and rate of fluorophore excitation, whereas in the latter, it is a function of the fraction of excited fluorophore. Intriguingly, this relation applies also for systems with exciplex emissions. We tested the validity of this model on 151 experimental fluorescence quenching plots, taken from the literature, operated by dynamic, static, and combined mechanisms. The results of the fitting are excellent with an average correlation coefficient of 0.9985.