On the theory of the Stern–Volmer coefficient for dense fluids

Abstract

In phenomenological theories of fluorescence quenching in dense fluids, one assumes that the Stern–Volmer coefficient K has a dependence upon a limited number of parameters. Denoting by K0 the value of K when the quencher molecules are infinitely dilute, our starting point is that K0 = K0(D,τ,k,R), where D is the sum of the diffusion coefficients of the fluorophore and quencher, k is a rate constant, τ is the unquenched lifetime of the fluorophore, and R is the range of interaction between fluorophore and quencher. We then find by dimensional analysis that K0 can be simplified to K0 = DRτ⋅F (k/DR,Dτ/R2), where F(x,y) is a function of just two variables. Furthermore, if S is a function specifying the region of configuration space in which fluorophores and quenchers interact [G. Wilemski and M. Fixman, J. Chem. Phys. 58, 4009 (1973)], we find that K0 depends only upon an average of S over the steady state fluorophore–quencher pair correlation function. Under ordinary conditions, k/DR and Dτ/R2 will be large. We note that in this limit K0 approaches the Smoluchowski value 4πDRτ. Correction terms to the Smoluchowski value are explicitly computed, however, when S is a delta function or a step function. Inasmuch as both forms lead to corrections tending in the same direction, it may be difficult to distinguish them by experiment. Thus, we conclude that future theoretical work should focus on a more realistic form for S.