# Quenching Single-Fluorophore Systems and the Emergence of Nonlinear Stern–Volmer Plots

## TL;DR

This paper explains how fluorescence quenching can lead to nonlinear Stern–Volmer plots by considering small subsystems instead of bulk thermodynamics.

## Contribution

A new equation is derived to describe fluorescence quenching based on local equilibrium in small subsystems.

## Key findings

- The new model explains linear and nonlinear Stern–Volmer plots based on fluorophore properties and quencher concentration.
- The model was validated with 151 experimental plots and achieved an average correlation coefficient of 0.9985.
- The equation applies to dynamic, static, and combined quenching mechanisms, including exciplex emissions.

## Abstract

Reduction in fluorescence
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 nonlinear
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
a finite number of surrounding quencher molecules, effectively partitioning
the (macroscopic) system into many noninteracting small subsystems.
Depending on the fluorophore’s properties, the association’s
strength, and conditions, the resulting behavior exhibits linear dependencies,
upward curvatures, or downward curvatures. More specifically, the
relation reads,
I°/I=1+ZK[Q]T/(1+(1−Z)K[Q]T)
, where K is a steady-state
equilibrium constant for complex formation and [Q]

T

is the total concentration of quencher in
the small subsystem. The dimensionless parameter
Z
has
different expressions for dynamic and
static mechanisms. In the former, it is a ratio between the maximum
rate of quenching and the 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.