The dynamics of colloidal particles at infinite dilution, under the influence of periodic external potentials, is studied here via experiments and numerical simulations for two representative potentials. From the experimental side, we analyzed the motion of a colloidal tracer in a one-dimensional array of fringes produced by the interference of two coherent laser beams, providing in this way an harmonic potential. The numerical analysis has been performed via Brownian dynamics (BD) simulations. The BD simulations correctly reproduced the experimental position- and time-dependent density of probability of the colloidal tracer in the short-times regime. The long-time diffusion coefficient has been obtained from the corresponding numerical mean square displacement (MSD). Similarly, a simulation of a random walker in a one dimensional array of adjacent cages with a probability of escaping from one cage to the next cage is one of the most simple models of a periodic potential, displaying two diffusive regimes separated by a dynamical caging period. The main result of this study is the observation that, in both potentials, it is seen that the critical time t*, defined as the specific time at which a change of curvature in the MSD is observed, remains approximately constant as a function of the height barrier U0 of the harmonic potential or the associated escape probability of the random walker. In order to understand this behavior, histograms of the first passage time of the tracer have been calculated for several height barriers U0 or escape probabilities. These histograms display a maximum at the most likely first passage time t′, which is approximately independent of the height barrier U0, or the associated escape probability, and it is located very close to the critical time t*. This behavior suggests that the critical time t*, defining the crossover between short- and long-time regimes, can be identified as the most likely first passage time t′ as a first approximation.

Wormlike micelles are long semiflexible cylindrical polymer structures formed by amphiphiles. In solution, these linear micelles percolate in multiconected entangled networks, where cross-links can break and recombine dynamically. Technological applications of wormlike micellar fluids include tunable encapsulation/delivery of molecules or colloids in biomedicine, oil industry, and/or cleaning processes. In this work, we propose that the experimental activation energy, the spatial confinement, and the mean first passage and escape times of a spherical tracer immersed in wormlike micellar network, in which caging effects are observed, can be estimated from economic Brownian dynamics simulations of a single particle interacting with an effective one-dimensional cosine-like potential of amplitude U0 and periodicity L. The proposed one-fitting parameter method has been used to characterize the long-time dynamics of wormlike micellar solutions formed by the self-assembly of a mixture of zwitterionic and anionic surfactants at several temperatures and different concentrations of surfactant and brine. The amplitude U0 has displayed a good agreement regarding the corresponding experi- mental activation energy at different temperatures. The periodicity L has shown to be an upper bound of the mesh size ξ and of the same order of magnitude regarding the entanglement length le, obtained from rheology and microrheology experiments. The escape time of the tracer in the effective potential τescape and the time t∗, at which a change of curvature in the mean square displacement occurs, are upper and lower limits, respectively, of the experimental relaxation time. Our method is simple and fast, and we foresee that it should be applicable to model the long-time behaviour of tracers in other polymer systems, in which caging effects are present.