Single-file dynamics of colloids in circular channels: Time scales, scaling laws and their universality

In colloidal systems, Brownian motion emerges from the massive separation of time and length scales associated with characteristic dynamics of the solute and solvent constituents. This separation of scales produces several temporal regimes in the colloidal dynamics when combined with the effects of the interaction between the particles, confinement conditions, and state variables, such as density and temperature. Some examples are the short- and long-time regimes in two- and three-dimensional open systems and the diffusive and subdiffusive regimes observed in the single-file (SF) dynamics along a straight line. In this paper, we address the way in which a confining geometry induces new time scales. We report on the dynamics of interacting colloidal particles moving along a circle by combining a heuristic theoretical analysis of the involved scales, Brownian dynamics computer simulations, and video-microscopy experiments with paramagnetic colloids confined to lithographic circular channels subjected to an external magnetic field. The systems display four temporal regimes in the following order: one-dimensional free diffusion, SF subdiffusion, free-cluster rotational diffusion, and the expected saturation due to the confinement. We also report analytical expressions for the mean-square angular displacement and crossover times obtained from scaling arguments, which accurately reproduce both experiments and simulations. Our generic approach can be used to predict the long-time dynamics of many other confined physical systems.

Figure 1

Microscope image of colloidal particles confined by gravity in a microchannel. Left: Channel radius $R=5\mu \text{m}$ and 5 particles. Right: Channel radius $R=10\mu \text{m}$ and 11 particles. The widths of the channels are $2.5\mu \text{m}$ at the bottom and $5\mu \text{m}$ at the top, with a depth of $2.8\mu \text{m}$. The channel radius is measured from the center of the channel. Scale bar is $10\mu \text{m}$ in both images.

Figure 2

Angular distribution function $g\left(\varphi \right)$ for paramagnetic particles diffusing along a circle measured from video-microscopy experiments (open symbols) and calculated using Brownian dynamics (BD) simulations (solid lines) with the Ermak-McCammon (EM) algorithm for curved manifolds in Eq. (2) for four sets of system parameters, as indicated.

Figure 3

Mean-square angular displacement $\langle {\left[\mathrm{\Delta }\varphi \left(t\right)\right]}^2\rangle$ from experiments (open symbols) and Brownian dynamics (BD) simulations (solid lines). For all sets of curves, we identify the following four temporal regimes: the short-time diffusive regime (I), the subdiffusive intermediate time regime (II), the second diffusive regime (III), and the geometrical saturation regime (IV). The straight gray lines are guides for the eye.

Figure 4

Mean-square angular displacement $\langle {\left[\mathrm{\Delta }\varphi \left(t\right)\right]}^2\rangle$ obtained with the Ermak-McCammon (EM) algorithm in Eq. (3) for paramagnetic colloids diffusing along a circle (a) for different numbers of particles, (b) varying the radius of the circle, and (c) changing the strength of the interparticle repulsion, as indicated. The thin straight lines are guides for the eyes, showing the short-time $2D_{0}t/R^2$ and the geometrical limit ${\pi }^2/3$ values.

Figure 5

Transition times ${\tau }_d,{\tau }_c$, and ${\tau }_G$ for the crossover between the different regimes for paramagnetic colloids diffusing along a circle of radius $R$. In (a) and (b), the straight line is the linear fit with slope $a\approx 0.99\pm 1.74\times {10}^{-2}$. The line in (c) is the best linear fit with slope $a\approx 1.0\pm 2.78\times {10}^{-14}$. Both experimental and simulation results displayed in Figs. 3 and 4 are included.

Figure 6

Mean-square angular displacement for six paramagnetic colloidal systems obtained with the Ermak-McCammon (EM) algorithm in Eq. (2). Bold lines represent Eq. (5) with the mobility factor given by Eq. (11) evaluated using the physical parameters as indicated.

Figure 7

Comparison of the mean-square angular displacement for the systems displayed in Figs. 3 and 3 (dotted lines, orange and red for Brownian dynamics (BD) and black for experiments) and Eq. (17) (continuous lines with the same color code) using the crossover times reported in Fig. 5.