Separation of propeller-like particles by shear and electric field

Separation of one type of chiral particle from mirror-image particles in a racemic mixture by shear flow and rotating electric field is studied theoretically. In shear flow the chiral particles and their mirror-image particles migrate in opposite directions along the vorticity direction. In a rotating electric field, they migrate in opposite directions along the axis of rotation. In each case, the migration velocity of the particles is calculated for a model chiral particle: a propeller-like particle which consists of two disks, at an angle to each other, rigidly connected to a thin rod. Effects of shear rate, field strength, and particle structure on the migration velocity are discussed. It is shown that the chirality of the particle is characterized by different parameters depending on the method of separation.

Figure 1

(a) The propeller-like particle considered in this paper. Two disks of radius $a$ are connected by a thin rod to give a disk-to-disk separation of $2h$. (b) The particle viewed from the ${\mathit{u}}_3$ axis. $\theta$ is the angle between the two disks, and $\varphi$ is the angle between $\mathit{m}$ and ${\mathit{u}}_1$. (c) The shear flow considered in this paper. (d) The rotating electric field considered in this paper.

Figure 2

Migration velocity $〈V_z〉/a\dot{\gamma }$ of a propeller particle in simple shear flow plotted as a function of the Péclet number $\dot{\gamma }/D_r$. Symbols show the results of simulation. The dashed lines show Eq. (36).

Figure 3

Migration velocity $〈V_z〉/a{D}_r$ (normalized by $a{D}_r)$ of a propeller particle in simple shear flow plotted as a function of the Péclet number $\dot{\gamma }/D_r$.

Figure 4

Orientations $〈u_{1x}^2-u_{2x}^2〉$, $〈u_{2y}^2-u_{1y}^2〉$, and $〈u_{3x}^2〉-1/3$ of a propeller-like particle with $\theta =\pi /4$ are shown as a function of the Péclet number $\dot{\gamma }/D_r$ in simple shear flow.

Figure 5

Migration velocity $〈V_z〉/a\dot{\gamma }$ of a propeller particle in simple shear flow plotted as a function of the angle $\theta$. Symbols show the results of simulation. Lines show the curve $〈V_z〉/a\dot{\gamma }=\alpha sin2\theta$, where $\alpha$ is a fitting parameter.

Figure 6

Migration velocity $〈V_z〉/\omega a$ of a propeller-like particle having dipole moment perpendicular to the axis, placed in rotating electric field, plotted as a function of $mE/k_{B}T$. Symbols show the results of simulation.

Figure 7

Migration velocity $〈V_z〉/a\omega$ of a propeller-like particle having dipole moment perpendicular to the axis plotted as a function of the angle $\theta$. Symbols show the results of simulation. Lines show the curve $〈V_z〉/\omega a=\alpha sin\theta$, where $\alpha$ is a fitting parameter.

Figure 8

Migration velocity $〈V_z〉/\omega a$ of a propeller-like particle having dipole moment perpendicular to the axis plotted as a function of the angle $\varphi$. Symbols show the results of simulation. Lines show the curve $〈V_z〉/\omega a=\alpha cos2\varphi$, where $\alpha$ is a fitting parameter.

Figure 9

Migration velocity $〈V_z〉/a\omega$ of a propeller-like particle having dipole moment parallel to the axis plotted as a function of $mE/k_{B}T$. Symbols show the results of simulation.

Figure 10

Migration velocity $〈V_z〉/a\omega$ of a propeller-like particle having dipole moment parallel to the axis plotted as a function of the angle $\theta$ in a rotating electric field. Symbols show the results of simulation. Lines show the curves $〈V_z〉/\omega a=\alpha sin2\theta$, where $\alpha$ is a fitting parameter.

Figure 11

Top views of the propeller-like particles for various twist angles. The reds and blues show disks 1 and 2, respectively. The black arrows show the dipole moments. Particles a-1 through a-4 are without dipole moment. Particles a-1 and a-3, and a-2 and a-4, are completely identical particles. Particles b-1 through b-4 are with dipole moment. All particles from b-1 to b-4 are different.