Lane and band formation of oppositely driven colloidal particles in two-dimensional ring geometries

We study the segregation phenomena for oppositely driven colloidal particles in two-dimensional ring geometries by means of Brownian dynamics simulations without hydrodynamic interactions. The particles interact via a repulsive Yukawa potential and are confined to a two-dimensional circular channel by hard walls, in which half of the particles are driven clockwise and the other half are driven counterclockwise. In addition to lane formation, which is commonly found in oppositely driven systems, we found band formation along the angular direction in channels with a very large radius. This indicates that a formation of lanes is prevented in the limit of channels with an infinitely large inner radius. The dependency of this segregation has been examined for the two control parameters, the interaction strength between the particles and the width of the circular channel.

Figure 1

Schematic representation of the radial tube around a particle $i$ inside of which all particles $j$ contribute to the lane formation parameter. Particle $i$ and the channel wall are represented in black, the small tube around $i$ and the width $d$ of the tube are displayed in green.

Figure 2

Snapshot of the circular channel with radius $R_1=90$ and width $W=20$ with different driving forces. The picture on the far left (a) shows the system with a driving force of $F=15$, in this case there is disorder. Panel (b) shows the system with a driving force of $F=60$ in the critical area. In this area, the system starts to organize itself. The image on the far right (c) was taken at a force of $F=85$. The particles separate completely into two lanes. The red particles run in the clockwise direction, the blue particles run counterclockwise.

Figure 3

Band formation in systems of inner radius $R_1=400$ with width $W=20$ and force (a) $F=35$, (b) $F=50$, and (c) $F=80$. In order to better recognize the particle distribution, a section of the ring was enlarged. The red particles move clockwise against the blue ones.

Figure 4

(a) Order parameter ${\mathrm{\Phi }}_{\text{lane}}$ in dependence on the force for the system size with $R_1=40$ and $W=35$. This curve looks similar for all system sizes. For wider systems the curve flattens slightly, and the steep increase shifts to a higher value. (b) The critical force $F_c$ of widths $W=20, W=35$, and $W=50$ for all simulated radii up to $R_1=100$. The red circles correspond to systems of width $W=20$, the blue triangles to systems of width $W=35$, and the orange squares to systems of width $W=50$. The average of all points is plotted with the three lines in their respective colors.

Figure 5

Particle positions in the system with $R_1=300, W=35$, (a) $V_0=35$, and (b) $V_0=65$ for the driving force $F=85$. The graphic shows the position over the radius and the angular coordinate. Red shows the positions of the particles which are moved in a clockwise direction. The blue particles move counterclockwise. In panel (a) we see band order, whereas in panel (b) the particles break up into lanes.

Figure 6

Particle positions for the systems with $R_1=300, W=20$, 35, 50 for $F=85$ and an interaction strength of $V_0=35$. It can be seen that in this system size the number of bands decreases with increasing width. At a width of $W=50$ the band order smears and the order parameter decreases.

Figure 7

Particle positions for the systems with (a) $W=20$, (b) $W=35$, and (c) $W=50, R_1=300$ and an interaction strength of $V_0=65$ with a driving force of $F=85$. The graphs show that by increasing the width of the system, for every step of $\mathrm{\Delta }W=15$ one additional lane is created.

Figure 8

(a) Local particle density $n_{\mathrm{loc}}$ plotted over the angular coordinate of the system $R_1=300, W=35$, and $V_0=35$ for a snapshot at a driving force of $F=85$. Here the two particle species are counted and plotted as the local density in a $5^{\circ }$ interval. The density in the system is $n_{\text{tot}}=\frac{N}{A}=0.32$. (b) Force dependence of the minimum and the maximum local particle density as a function of the driving force $F$. The minimum and maximum values are represented by the red circles and the blue triangles, respectively. A high maximum value implies the existence of dense cluster, whereas a low minimum value means that there are gaps in the system.

Figure 9

(a) The order parameter for band formation ${\mathrm{\Phi }}_{\text{band}}$ as a function of the driving force for the system with inner radius $R_1=300$, width $W=35$, and potential strength $V_0=35$. The critical force is defined at the point where ${\mathrm{\Phi }}_{\text{band}}=0.25$ is extracted from the fit. In this case the critical force is $F_c=53.4\pm 0.67$. (b) The number of bands $N_{\mathrm{band}}$ of one particle type as a function of the driving force for the specified parameters. A band is counted if there is an increase of the local density of $\mathrm{\Delta }{n}_{\mathrm{loc}}=0.08$.

Figure 10

(a) A system with $R_1=400$, width $W=25$ and potential strength $V_0=59$, and a driving force of $F=60$. This system is transitioning from band formation to lane formation. Evident are the lanes of the two particle types and at the same time accumulations of particles distributed in the ring. The comparison of the two order parameters for (b) band and (c) lane formation shows that in this system both orders exist.

Figure 11

Probability to find the system in the ordered state with band or lane formation for all systems with inner radius $R_1=300$. The probability depending on the width $W$ and potential strength $V_0$ of the system. A value of $P=1$ (blue) represents 100% band formation, and $P=-1$ (red) represents 100% lane formation. For small widths $W$ and small potential strengths $V_0$, band formation predominates. In systems with larger width and higher potential strength, lane formation is found.

Figure 12

(a) Snapshot of a simulation in a linear channel. The channel has a length of $L=3000$ and a width of $W=25$, and the potential strength was set to $V_0=35$. Since the channel is very long and rather thin, only a segment is shown here. The phenomenon of band formation can be clearly seen, and the bands also occur in the whole channel. (b) Closeup of two bands penetrating each other. To the left one can see a small red band that has almost completely passed through the blue band already, and to the right one can see the two fronts of different species hitting each other.