When is a Single Molecule Heterogeneous? A Multidimensional Answer and Its Application to Dynamics Near the Glass Transition

Even for apparently simple condensed-phase processes, bulk measurements of relaxation often yield nonexponential decays; the rate appears to be dispersed over a range of values. Taking averages over individual molecules is an intuitive way to determine whether heterogeneity is responsible for such rate dispersion. However, this method is in fundamental conflict with ergodic behavior and often yields ambiguous results. This paper proposes a new definition of rate heterogeneity for ergodic systems based on multidimensional time correlation functions. Averages are taken over both time and molecules. Because the data set is not subdivided, the signal-to-noise ratio is improved. Moment-based quantities are introduced to quantify the concept of rate dispersion. As a result, quantitative statements about the fraction of the dispersion due to heterogeneity are possible, and the experimental noise is further averaged. The practicality of this approach is demonstrated on single-molecule, linear-dichroism trajectories for R6G in poly(cyclohexyl acrylate) near its glass transition. Single-molecule averaging of these data does not provide useful conclusions [C. Y. Lu and D. A. Vanden Bout, J. Chem. Phys. 125, 124701 (2006)]. However, full-ensemble, two- and three-dimensional averages of the same data give clear and quantitative results: the rate dispersion is 95% ± 5% due to heterogeneity, and the rate exchange is at least 11 times longer than the mean rotation time and possibly much longer. Based on these results, we suggest that the study of heterogeneous materials should not focus on "ensemble" versus "single-molecule" experiments, but on one-dimensional versus multidimensional measurements.