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## Diffusion and separation of binary mixtures of chiral active particles driven by time-delayed feedback

Liao Jing-Jing, Lin Fu-Jun
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• #### 摘要

在二维空间内, 考虑周期性边界条件, 提出了一种用时间延迟反馈分离混合手征活性粒子的新方法. 当系统引入时间延迟反馈时, 手征活性粒子动力学特征发生明显改变. 通过调节外加时间延迟反馈的强度和反馈时间可以控制逆时针旋转(counterclockwise, CCW)粒子扩散受到顺时针旋转(clockwise, CW)粒子扩散的影响程度. 当时间延迟反馈强度和反馈时间较大且系统参数取最优值时, CCW粒子加快旋转角速度, 扩散完全由粒子相互作用决定, 而CW粒子的扩散由自身参数和粒子相互作用共同决定, 在此情况下, CCW粒子容易聚集形成团簇, 而CW粒子加快扩散, 混合手征活性粒子实现分离.

#### Abstract

Considering the periodic boundary conditions, a new prescription for separating binary mixtures of chiral active particles by time-delayed feedback in a two-dimensional square box is proposed. We investigate the angular velocity, the feedback intensity, the delayed time, the rotational diffusion coefficient, the self-propelled speed and the packing fraction as functions of the effective diffusion coefficient and the separation coefficient numerically by the extensive Brownian dynamics simulations. It is found that mixed chiral active particles be separated without time-delayed feedback, but the dynamics of chiral active particles are different obviously and mixed chiral particles can be separated when the time-delayed feedback is introduced. The particle configuration (mixing or demixing) is determined by the dominant factor of particles’ diffusion. We can control the extent to which the diffusion of counterclockwise (CCW) active particles is affected by the diffusion of clockwise (CW) active particles adjusting the strength and the delayed time of the feedback. The response to the feedback for different chiral particles show different behaviors under different system parameters. When the feedback intensity is strong and the delayed time is long enough, the angular velocity of counterclockwise particles is accelerated and the diffusion of which is dominated by the interactions between particles completely. However, the angular speed of clockwise particles change little and the diffusion of which is determined by its parameters and particle interactions jointly. In this case, the counterclockwise particles aggregate to form clusters easily, and the clockwise particles diffuse quickly, therefore, the mixed chirality active particles are separated. When the feedback intensity is weak and the delayed time is short, the chirality difference between different chiral particles modulated by the feedback is smaller than the former case. The diffusions of counterclockwise particles and clockwise particles are both determined by their parameters and particle interactions, and the particles are mixed. Our findings provide novel strategies for the experimental pursuit of separating mixed chiral active particles and could be applied practically in many biological circle swimmers, such as autochemotactic particles, the bacteria in an external light field and sperm cells with vortex motion.

#### 作者及机构信息

###### 通信作者: 蔺福军, fujun012@yeah.net
• 基金项目: 国家自然科学基金(批准号: 11905086, 11804131)、江西省自然科学基金(批准号: 20192BAB212006)和江西省教育厅科技项目(批准号: GJJ191598, GJJ191599)资助的课题

#### Authors and contacts

###### Corresponding author: Lin Fu-Jun, fujun012@yeah.net
• Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11905086, 11804131), the Natural Science Foundation of Jiangxi Province, China (Grant No. 20192BAB212006), and the Foundation of Jiangxi Provincial Educationa Department, China (Grant Nos. GJJ191598, GJJ191599)

#### 施引文献

• 图 1  时间延迟反馈示意图. 当$\tau = 0$时, ${{\varOmega}}(t) = K_{\rm {fb}}$; 当$\tau\rightarrow\infty $$\theta({t}\!-\!\tau ) > \theta(t)时, {{\varOmega}}(t) = 0 ; 当 \tau\rightarrow\infty$$\theta({t}\!-\!\tau ) < \theta(t)$时, ${{\varOmega}}(t) = 2 K_{\rm {fb}}$

Fig. 1.  Schematic diagram of time-delayed feedback. When $\tau = 0$, ${{\varOmega}}(t) = K_{\rm {fb}}$; when $\tau\rightarrow\infty$ and $\theta({t}-\tau ) > \theta(t)$, ${{\varOmega}}(t) = 0$; when $\tau\rightarrow\infty$ and $\theta({t}-\tau ) < \theta(t)$, ${{\varOmega}}(t) = 2 K_{\rm {fb}}$.

图 2  CCW粒子(红色)和CW粒子(蓝色)的混合物分布　(a)$K_{\rm {fb}} = 0, ~\omega = 0$; (b)$K_{\rm {fb}} = 10.0, ~ \tau = 10.0, ~\omega = 0$; (c)$K_{\rm {fb}} \!=\! 10.0, \tau \!=\! 10.0, \omega \!=\! 2.2$; (d)$K_{\rm {fb}} = 10.0,~ \tau = 10.0, ~\omega = 4.2$. 其他参数设置为$v_0 = 2.5$, $D_{\theta} = 0.001$, $\phi = 0.5$

Fig. 2.  The snapshots of mixture of CCW particles (red) and CW particles (blue): (a)$K_{\rm {fb}} = 0, \omega = 0$; (b)$K_{\rm {fb}} = 10.0, \tau = 10.0, \omega = 0$; (c)$K_{\rm {fb}} = 10.0, \tau = 10.0, \omega = 2.2$; (d)$K_{\rm {fb}} = 10.0, \tau = 10.0, \omega = 4.2$. The other parameters are $v_0 = 2.5$, $D_{\theta} = 0.001$, and $\phi = 0.5$.

图 3  (a) CW粒子和CCW粒子的最大团簇粒子数占各自总粒子数的比例P随角速度$\omega$的变化. 图中a, b, c, d四点的构型图分别对应图2(a), 图2(b), 图2(c), 图2(d); (b)在不同$\omega$下, $t = 2\times10^4$时, 相对径向分布函数$g_{\rm {AB}}(r)$. 图中标注的圆圈为第一个零根, 代表单种粒子的团簇尺寸. 其他参数设置为$v_0 = 2.5$, $D_{\theta} = 0.001$, $\phi = 0.5$, $K_{\rm {fb}} = 10.0$, $\tau = 10.0$

Fig. 3.  (a) The ratio of the particle number in maximum cluster of CW particles and CCW particles to the total number of particles respectively as a function of $\omega$. The points a, b, c, d are corresponding to Fig. 2(a),Fig. 2(b),Fig. 2(c),Fig. 2(d), respectively; (b) relative radial distribution function $g_{\rm {AB}}(r)$ for different value of $\omega$ at $t = 2\times10^4$. The first non-trivial root (marked by circles) denotes the cluster size of the single particle species. The other parameters are $v_0 = 2.5$, $D_{\theta} = 0.001$, $\phi = 0.5$, $K_{\rm {fb}} = 10.0$, and $\tau = 10.0$.

图 4  (a)在不同$K_{\rm {fb}} $$\tau 值下, CCW粒子和CW粒子的有效扩散系数D随角频率 \omega 的变化; (b)在不同 K_{\rm {fb}}$$ \tau$下, 分离系数S随角频率$\omega$的变化. 其他参数设置为$v_0 = 2.5$, $D_{\theta} = 0.001$, $\phi = 0.5$

Fig. 4.  (a) The effective diffusion coefficient D of CCW and CW particles as a function of $\omega$ for different $K_{\rm {fb}}$ and $\tau$; (b) the separation coefficient S as a function of $\omega$ for different $K_{\rm {fb}}$ and $\tau$. The other parameters are $v_0 = 2.5$, $D_{\theta} = 0.001$, and $\phi = 0.5$.

图 10  (a)在不同填充率$\phi$下, 分离系数S随时间t的变化; (b)在不同时间t下, $\phi = 0.5$时, 相对径向分布函数$g_{\rm {AB}}(r)$. 图中标注的圆圈为第一个零根, 代表单种粒子的团簇尺寸. 其他参数设置为$v_0 = 2.5$, $D_{\theta} = 0.001$, $\omega = 2.1$, $K_{\rm {fb}} = 10.0$, $\tau = 10.0$

Fig. 10.  (a) The separation S as a function of t for different $\phi$; (b) the relative radial distribution function $g_{\rm {AB}}(r)$ for different t at $\phi = 0.5$. The first non-trivial root (marked by circles) denotes the cluster size of the single particle species. The other parameters are $v_0 = 2.5$, $D_{\theta} = 0.001$, $\omega = 2.1$, $K_{\rm {fb}} = 10.0$, and $\tau = 10.0$.

图 5  在　(a) $\tau = 0.01$, (b) $\tau = 1.0$, (c) $\tau = 10.0$时, CCW粒子和CW粒子的有效扩散系数D随反馈强度$K_{\rm {fb}}$的变化; (d)在不同$\tau$下, 分离系数S随反馈强度$K_{\rm {fb}}$的变化. 其他参数设置为$\omega = 2.1$, $v_0 = 2.5$, $D_{\theta} = 0.001$, $\phi = 0.5$

Fig. 5.  The effective diffusion coefficient D of CCW and CW particles as a function of $K_{\rm {fb}}$ at (a) $\tau = 0.01$, (b) $\tau = 1.0$, and (c) $\tau = 10.0$; (d) the separation coefficient S as a function of $K_{\rm {fb}}$ for different $\tau$. The other parameters are $\omega = 2.1$, $v_0 = 2.5$, $D_{\theta} = 0.001$, and $\phi = 0.5$.

图 6  在(a)$K_{\rm {fb}} = 1.0$, (b)$K_{\rm {fb}} = 2.5$, (c)$K_{\rm {fb}} = 10.0$时, CCW粒子和CW粒子的有效扩散系数D随反馈时间$\tau$的变化; (d) 在不同$K_{\rm {fb}}$下, 分离系数S随反馈时间$\tau$的变化. 其他参数设置为$\omega = 2.1$, $v_0 = 2.5$, $D_{\theta} = 0.001$, $\phi = 0.5$

Fig. 6.  The effective diffusion coefficient D of CCW and CW particles as a function of $\tau$ at (a) $K_{\rm {fb}} = 1.0$, (b) $K_{\rm {fb}} = 2.5$, and (c) $K_{\rm {fb}} = 10.0$; (d) the separation coefficient S as a function of $\tau$ for different $K_{\rm {fb}}$. The other parameters are $\omega = 2.1$, $v_0 = 2.5$, $D_{\theta} = 0.001$, and $\phi = 0.5$.

图 7  在(a)$K_{\rm {fb}} = 0.0$, (b)$K_{\rm {fb}} = 2.5, \tau = 1.0$, (c)$K_{\rm {fb}} = 10.0, \tau = 10.0$时, CCW粒子和CW粒子的有效扩散系数D随转动扩散系数$D_{\theta}$的变化; (d) 在不同$K_{\rm {fb}} $$\tau 下, 分离系数S随转动扩散系数 D_{\theta} 的变化. 其他参数设置为 \omega = 2.1 , v_0 = 2.5 , \phi = 0.5 Fig. 7. The effective diffusion coefficient D of CCW and CW particles as a function of D_{\theta} at (a) K_{\rm {fb}} = 0.0 , (b)K_{\rm {fb}} = 2.5, \tau = 1.0, and (c) K_{\rm {fb}} = 10.0, \tau = 10.0 ; (d) the separation coefficient S as a function of D_{\theta} for different K_{\rm {fb}} and \tau . The other parameters are \omega = 2.1 , v_0 = 2.5 , and \phi = 0.5 . 图 8 (a)在 K_{\rm {fb}} = 10.0 , \tau = 10.0 时, 不同自驱动速度 v_0 下, 均方位移MSD = \left\langle{{{\left| {\Delta {{{r}}_i}(t)} \right|}^2}}\right\rangle 随时间t的变化; (b)在不同 K_{\rm {fb}}$$ \tau$下, 分离系数S随自驱动速度$v_0$的变化. 其他参数设置为$\omega = 2.1$, $D_{\theta} = 0.001$, $\phi = 0.5$

Fig. 8.  (a) The mean square displacement MSD $= \left\langle{{{\left| {\Delta {{{r}}_i}(t)} \right|}^2}}\right\rangle$ as a function of t for different $v_0$ at $K_{\rm {fb}} = 10.0$ and $\tau = 10.0$; (b) the separation coefficient S as a function of $v_0$ for different $K_{\rm {fb}}$ and $\tau$. The other parameters are $\omega = 2.1$, $D_{\theta} = 0.001$, and $\phi = 0.5$.

图 9  (a) CCW粒子和CW粒子的有效扩散系数D随填充率$\phi$的变化; (b) 分离系数S随填充率$\phi$的变化. 其他参数设置为$v_0 = 2.5$, $D_{\theta} = 0.001$, $\omega = 2.1$, $K_{\rm {fb}} = 10.0$, $\tau = 10.0$

Fig. 9.  (a) The effective diffusion coefficient D of CCW and CW particles as a function of $\phi$; (b) the separation coefficient S as a function of $\phi$. The other parameters are $v_0 = 2.5$, $D_{\theta} = 0.001$, $\omega = 2.1$, $K_{\rm {fb}} = 10.0$, and $\tau = 10.0$.

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##### 出版历程
• 收稿日期:  2020-04-07
• 修回日期:  2020-07-08
• 上网日期:  2020-11-09
• 刊出日期:  2020-11-20

## 混合手征活性粒子在时间延迟反馈下的扩散和分离

• 1. 江西理工大学理学院, 赣州　341000
• 2. 江西理工大学应用科学学院, 赣州　341000
• 3. 华南师范大学物理与电信工程学院, 广州　510006
• ###### 通信作者: 蔺福军, fujun012@yeah.net
基金项目: 国家自然科学基金(批准号: 11905086, 11804131)、江西省自然科学基金(批准号: 20192BAB212006)和江西省教育厅科技项目(批准号: GJJ191598, GJJ191599)资助的课题

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