Multifractal detrended fluctuation analysis of photon intensity time series in solvent-filled nano-structured porous media

Abstract

Single molecule fluorescence fluctuation experiments were conducted to uncover complex molecular mass transport dynamics in nanoporous media. Confocal fluorescence microscopy was employed to capture single-molecule photon intensity time series from bulk solution and from porous anodic aluminum oxide (AAO) membranes incorporating 10 and 20 nm diameter pores. The membranes were filled by 20 nM Nile red (NR) dye solution in pure ethanol and pure toluene. Regardless of the solvent, the photon intensity time series revealed clear evidence of dye diffusion along the one-dimensional nanopores, as well as evidence of dye adsorption to the pore surface in the 10 nm membranes. Autocorrelation of the time series revealed that NR diffused through the 10 nm membrane by two distinct mechanisms that could be classified as fast and slow diffusion. The diffusion coefficients, D_f and D_s, yielded values differing by 100-fold. The fast mechanism was attributed to hindered bulk-like diffusion in the central pore cavity, while slow diffusion likely involved absorption and desorption of the dye at the pore surface. Interestingly, only fast diffusion was observed in the 20 nm membrane. While autocorrelation functions frequently reveal the occurrence of multicomponent diffusion, it can be challenging to distinguish Fickian and anomalous diffusion mechanisms. The method of multifractal detrended fluctuation analysis (MF-DFA) from statistical physics was applied to the photon intensity time series as a means to further characterize the diffusion mechanisms. The MF-DFA analysis afforded values for the generalized Hurst exponent, h(q), of ~ 0.5 for times series obtained from bulk solution, consistent with Fickian diffusion. For both the 10 and 20 nm membranes, h(q) > 0.5 was obtained, consistent with super diffusion. Application of MF-DFA methods to photon times series affords additional evidence needed to better characterize the mechanisms of molecular transport in nanoporous media such as in oil and gas shales