Enhancement of mobility in an interacting colloidal system under feedback control

Feedback control schemes are a promising way to manipulate transport properties of driven colloidal suspensions. In the present article, we suggest a feedback scheme to enhance the collective transport of colloidal particles with repulsive interactions through a one-dimensional tilted washboard potential. The control is modeled by a harmonic confining potential, mimicking an optical “trap,” with the center of this trap moving with the (instantaneous) mean particle position. Our theoretical analysis is based on the Smoluchowski equation combined with dynamical density functional theory for systems with hard-core or ultrasoft (Gaussian) interactions. For either type of interaction, we find that the feedback control can lead to an enhancement of the mobility by several orders of magnitude relative to the uncontrolled case. The largest effects occur for intermediate stiffness of the trap and large particle numbers. Moreover, in some regions of the parameter space the feedback control induces oscillations of the mean velocity. Finally, we show that the enhancement of mobility is robust against a small time delay in implementing the feedback control.

Figure 1

Single-particle transport through a potential with wavelength $a=2.5\sigma$. (a)–(c) Density plots (black, left axis) and potential $V_{\mathrm{DF}}+u$ (gray, right axis). The time $t$ is (a) ${10}^5{\tau }_{\mathrm{B}}$ and (b,c) ${10}^4{\tau }_B$, where the Brownian time ${\tau }_{\mathrm{B}}={\sigma }^2\gamma /\left(k_{\mathrm{B}}T\right)$. Part (d) shows the mean particle position with respect to time for different $\eta$. For strong confinements, an oscillatory behavior emerges.

Figure 2

Velocity $v\left(t\right)$ (a) and width $w\left(t\right)$ (b) as a function of time for different confinement strengths $\eta$ for $a=2.5\sigma$. Oscillatory solutions are characterized by sharp peaks. (c) Amplitude $\mathrm{\Delta }v$ and period $\mathcal{T}$ of the velocity as a function of $\eta$ for $a=2.5\sigma$. Data points are shown only for values of $\eta$ where we find oscillations. The dashed line indicates the time $1/r_K$, where $r_K$ is Kramers' rate.

Figure 3

Single-particle transport: Mobility $\mu$ in dependence on $\eta$ for $a=2.5\sigma$. The mobility is scaled by the mobility ${\mu }_0$ of uncontrolled diffusion in a washboard potential. The feedback control can enhance the mobility by up to $\approx 20\%$. Inset: potential $V_{\mathrm{DF}}(x,t=0)+u\left(x\right)$ for the three values of $\eta$, which are indicated by arrows in $\mu \left(\eta \right)$.

Figure 4

Single-particle transport in the presence of the open-loop potential $V_{\mathrm{openloop}}$ [see Eq. (24)] with $\eta =1k_{\mathrm{B}}T{\sigma }^{-2}$ and $v_0=F/\gamma$ at $t=100{\tau }_{\mathrm{B}}$ and $a=2.5\sigma$. Black line: one-particle density $\varrho (x,t)$. Gray: potential $V_{\mathrm{openloop}}+u$.

Figure 5

One-particle density $\varrho (x,t)$ on the left axis (black lines) and the potentials $V_{\mathrm{DF}}$, $u$, and $V_{\mathrm{int}}$ on the right axis. The parameters show (a) the case in which $u$ is suppressed, (b) the decrease of barrier height by interaction, and (c) the low-$\eta /\mathrm{low}-N$ regime. In detail, (a) shows $N=10$ hard particles forming a chain in a narrow trap ($\eta =10k_{\mathrm{B}}T{\sigma }^{-2},a=2.5\sigma$) at $t=28.9{\tau }_{\mathrm{B}}$, (b) $N=4$ hard particles forming a loose chain in a moderately narrow trap ($\eta =0.7k_{\mathrm{B}}T{\sigma }^{-2},a=2.5\sigma$) at $t=985{\tau }_{\mathrm{B}}$, and (c) $N=4$ ultrasoft particles in a wide trap ($\eta ={10}^{-2}{k}_{\mathrm{B}}T{\sigma }^{-2},a=2.5\sigma$) at $t={10}^5{\tau }_{\mathrm{B}}$.

Figure 6

(a) One-particle density $\varrho (x,t)$ for $N=4$ hard particles subject to the tilted washboard potential $V_{\mathrm{ext}}$ [see (b)] at $\eta =3k_{\mathrm{B}}T{\sigma }^{-2},a=2.5\sigma$ at four times. For clarity, the curves are shifted vertically. Each time is labeled with a symbol, reappearing in parts (c) and (d), which show the width $w\left(t\right)$, given by Eq. (17), and the velocity $v\left(t\right)$, Eq. (21), over time, respectively.

Figure 7

Velocity $v\left(t\right)$, Eq. (21), for $N=10$ hard particles at $a=2.5\sigma$ for a broad range of $\eta$. The velocity $3.7698\sigma /{\tau }_{\mathrm{B}}=F/\gamma$ corresponds to the velocity of free motion under the force $F$. This value is reached at large $\eta$. Oscillations occur for $\eta \ge 0.03k_{\mathrm{B}}T{\sigma }^{-2}$.

Figure 8

Mobility $\mu$ for ultrasoft particles in dependence of $\eta$. Given that enough particles contribute, the mobility can rise to $1/\gamma$, the mobility of free motion. The thick line indicates the mobility in the uncontrolled case. Results from Fig. 3 are included with the notation $N=1$.

Figure 9

Mobility $\mu$ for hard particles in dependence of $\eta$. The mobility reaches $1/\gamma$, the mobility of free motion, if enough particles contribute. The thick line indicates the mobility in the uncontrolled case. Results from Fig. 3 are included with the notation $N=1$.

Figure 10

Effect of time delay on mobility $\mu$ for hard particles. Even for very long delays, the mobility is only reduced by one order of magnitude.