Segregation of oppositely driven colloidal particles in hard-walled channels: A finite-size study

We investigate segregation phenomena of particles driven in opposite directions with Brownian dynamics simulations. The particles interact via a repulsive potential and are confined in three-dimensional hard-walled pipes with quadratic cross sections. In a systematic finite-size study, the pipe length is varied. Ordering on finite length scales can be observed, but global segregation seems to vanish for infinite channel length, as it was recently found for lane formation in two dimensions. As an additional effect of the finite hard-walled boundary conditions, interface vibrations and longitudinal demixing are found.

Figure 1

Diagram for the calculation of the segregation parameter (top, global; bottom, local). The periodic direction is marked by a dashed line.

Figure 2

Different views of a channel with length $L=30$ and cross section $9.5\times 9.5$ at $\text{Pe}=20$ (top) and $\text{Pe}=80$ (bottom) (for video of the dynamics of the channel see Ref. [28]). Visualization has been created with vmd [29].

Figure 3

Shown for channels with a cross section of $12\times 12$ and different length $L$ are (a) the global segregation parameter $\mathrm{\Phi }$, (b) the local segregation parameter ${\mathrm{\Phi }}_{\text{loc}}(\lambda =20)$ on a local length of $\lambda =20$, and (c) the local segregation parameter ${\mathrm{\Phi }}_{\text{loc}}(\lambda =4)$ on a local length of $\lambda =4$.

Figure 4

Segregation Péclet number as a function of the length of the channel $L$ for the three cross sections.

Figure 5

(a) Global segregation parameter of a channel with length $L=1000$ and cross section $12\times 12$ for both ordered and disordered starting conditions, at short simulation time $t_{\text{max}}=225$, and at longer simulation time $t_{\text{max}}=225$. (b) Segregation Péclet number ${\text{Pe}}_s$ as a function of channel length $L$ for both short and long simulation times as well as a linear fit to the latter data for $L>200$.

Figure 6

(a) Global segregation parameter over time for the channel of length $L=540$ and cross section $9.5\times 9.5$ at Péclet number $\text{Pe}=125$. (b) Interface of the domains throughout the channel at the three points in time that are marked in (a). The positions are projected on the $xz$ plane, while the $x$ axis is highly clinched. The oscillation in the segregation parameter arises from the oscillation of the interface (for a video of the interface vibration see Ref. [28]).

Figure 7

(a) Chronology of a longitudinal demixing in the channel of length $L=300$ and cross section $7\times 7$ at $\text{Pe}=90$. Snapshots at different times from top to bottom. (b) The global segregation parameter for the same channel declines for $\text{Pe}>45$ due to longitudinal demixing.