Time-dependent concentration depolarization of fluorescence

Abstract

A theory of the time course of fluorescence anisotropy (polarization) after impulsive excitation is presented for viscous solutions of identical molecules. In a point-dipole model of the molecular oscillators, it is shown that the relative anisotropy is a universal function A of the quantity T = c2t/τ, where c is solute concentration (in units of the usual critical concentration), t is time, and τ is actual fluorescence lifetime. The function A(T) is expressed as a contribution A0 from the initially excited dipole plus one from all other dipoles, with the latter shown to be quite small in the region of interest. It is also shown that the representation, A0=Σk=0∞αk Tk/2, is useful for calculations except when T ≫ 1. The kth term in the series corresponds to energy transfer pathways involving k + 1 dipoles. The αk alternate in sign, with α0 = 1 and α1 = − 1.06004 derived analytically; numerically derived values of α2 through α6 are also given. We show that A(T) is the Laplace transform of c2 Ã(c), where Ã(c) is the steady-state anisotropy, and we use that relation to obtain an expansion of Ã(c) in powers of c. The resultant steady-state curve compares favorably with the best previous theoretical curves. The results of the point-dipole model are corrected for the effects of finite molecular volume. Specifically, it is shown that, for an excluded spherical volume v of twice the molecular ``radius,'' the behavior of A(T) for t/τ ≪(v/V0)2 is simply 1 − c(V0/v)(t/τ), which gives a finite decay slope at t = 0 in contrast to the point-dipole case. (V0 is the volume of the sphere of radius R̄0.) In addition to the A(T) theory, a simple derivation is given for the fluorescence-intensity formulas for anisotropy in linearly polarized and unpolarized exciting light. Finally, potential applications of the A(T) theory, particularly to testing the validity of the Förster mechanism in multiple homotransfer, are discussed.