Rod separation by sawtooth channel

By applying entropic barriers, we present a rod separation mechanism that induces the movement of rods of different sizes in the opposite directions. This mechanism is based on the combination of the saw-tooth channel, a static force, and an oscillating driving force. The asymmetric shape of the channel and the elongated shape of the rod causesa complicated interaction effect between the rods and the channel walls which reduces the accessible configuration space for the rods and leads to entropic free-energy effects.

Figure 1

(Left: side view) Schematic illustration of a rod adjacent to a wall. The coordinates $x,y,z$ and $X$, $Y$ refer to the laboratory and rod body frames, respectively, and $\theta$ is the angle between the direction of the rod ($X$ axis) and the $x$ axis (variable between $+\pi /2$ and $-\pi /2$). $2a$ and $2b$ are rod length and width, respectively. (Right: top view) Schematic illustration of a rod between two parallel walls. $\mathbb{Z}$ is the distance between two walls along the $z$ axis, which has a value of about $2b_{\max }(\mathbb{Z}=2b_{\max }+\delta \mathbb{Z})$, $H$ is the distance between the wall and the edge of the rod, and $N$ is the distance between the wall and the rod center ($N=H+b$).

Figure 2

Schematic illustration of the quasi-two-dimensional channel that restricts the rods' movement. The walls' structure is determined by Eq. (12). The rods are driven by a constant force $f$ and an oscillating force $F\left(t\right)$ (e.g., a square wave) along the main axis of the channel. The dashed blue (red) line represents the limit for the positions of the largest (smallest) rod center inside the channel. $h$ is the half-width of the bottleneck, $c$ represents the location of a point with the maximum width, $L$ is the periodicity of the channel, and $\mathbb{Z}$ is the channel thickness along the $z$ axis.

Figure 3

Illustration of the channel wall, the intersection points, and the limit of the position of the rod center near the wall.

Figure 4

Average current vs the periodic force $F$ for rods of different aspects, for a channel with $m_1=4$, $m_2=1/4$, $h=1.6$, $\mathbb{Z}=0.35$, $L=14$ and the rod with $b=0.1$. For $f<0$ small rods move to the left, whereas large rods move to the right. The critical aspect that determines these two behaviors can be tuned by the values of $F$ and $f$. The critical force that determines velocity inversion depends on the rod length $a$.

Figure 5

Schematic illustration of rods of different sizes inside the channel where they are initially placed in the middle of the channel, and after the forces were applied, the large rods moved to the right and the small rods to the left.

Figure 6

The position of the rods' center with different lengths and the same widths ($b=0.1$) after 60 sec. First, the rods were at $x=0$, and after 60 sec, the rods were separated.

Figure 7

Translational diffusion parallel to the long axis in the unbounded space vs aspect ratio, for three widths, $b=1/6,1/8,1/10\mu \text{m}$.

Figure 8

Translational diffusion perpendicular to the long axis in the unbounded space vs aspect ratio, for three widths, $b=1/6,1/8,1/10\mu \text{s}$.

Figure 9

$XX$ element of a translational diffusion tensor confined between two parallel walls with distance $\mathbb{Z}$, vs aspect ratio, for three widths, $b=1/6,1/8,1/10\mu \text{m}$.

Figure 10

$YY$ element of translational diffusion tensor confined between two parallel walls with distance $\mathbb{Z}$, vs aspect ratio, for three widths, $b=1/6,1/8,1/10\mu \text{m}$.

Figure 11

The element of diffusivity matrix along the main axis of the channel, ${\mathbb{D}}_{xx}$, vs the angle of the rod, for three aspect ratios, $p=6,10,14$ and two widths, $b=1/6,1/10\mu \text{m}$.