Colloidal dynamics over a tilted periodic potential: Nonequilibrium steady-state distributions

We report a systematic study of the effects of the external force $F$ on the nonequilibrium steady-state (NESS) dynamics of the diffusing particles over a tilted periodic potential, in which detailed balance is broken due to the presence of a steady particle flux. A tilted two-layer colloidal system is constructed for this study. The periodic potential is provided by the bottom-layer colloidal spheres forming a fixed crystalline pattern on a glass substrate. The corrugated surface of the bottom colloidal crystal provides a gravitational potential field for the top-layer diffusing particles. By tilting the sample at an angle $\theta$ with respect to the vertical (gravity) direction, a tangential component of the gravitational force $F$ is applied to the diffusing particles. The measured NESS probability density function $P_{ss}(x,y)$ of the particles is found to deviate from the equilibrium distribution $P(x,y)$ to a different extent, depending on the driving or distance from equilibrium. The experimental results are compared with the exact solution of the one-dimensional (1D) Smoluchowski equation and the numerical results of the 2D Smoluchowski equation. From the obtained exact solution of the 1D Smoluchowski equation, we develop an analytical method to accurately extract the 1D potential $U_0\left(x\right)$ from the measured $P_{ss}\left(x\right)$. This work demonstrates that the tilted periodic potential provides a useful platform for the study of forced barrier-crossing dynamics beyond the Arrhenius-Kramers equation.

Figure 1

3D plot of the potential $U_0(x,y)$ generated by Eq. (15) with $A=3$.

Figure 2

(a) 2D contour plot of equilibrium PDF $P(x,y)/P_0$ for the potential $U_0(x,y)$ shown in Fig. 1. [(b) and (c)] Evolution of the NESS-PDF $P_{ss}(x,y)/P_0$ with increasing values of $F$ along the [1,0] direction with (b) $F/F_T=1.5$ and (c) $F/F_T=5.0$. [(d) and (e)] Evolution of $P_{ss}(x,y)/P_0$ with increasing values of $F$ along the [1,1] direction with (d) $F/F_T=1.5$ and (e) $F/F_T=5.0$. The arrows indicate the [1,0] and [1,1] crystalline directions.

Figure 3

Schematic diagram of the sample cell (side view): SC, stainless steel cell; GC, glass cover slip; GA, gravity axis; OA, optical axis; $\theta$, tilt angle of the sample cell; red particles, large silica spheres forming a monolayer crystal on the bottom glass substrate; blue particles, smaller diffusing particles on top of the colloidal crystal; arrow, direction of the force $F$ acting on the diffusing particles.

Figure 4

Microscope image of sample S2. The uniform honeycomb pattern in the background is the optical pattern resulting from the bottom layer colloidal crystal. The bright dots with a nonuniform intensity profile are the diffusing particles in the top layer. The arrows indicate the [1,0] and [1,1] crystalline directions. The scale bar is 10 $\mu \mathrm{m}$.

Figure 5

Three-dimensional plot of the measured potential $U_0(x,y)$ for sample S1 at equilibrium. The plotted area of $U_0(x,y)$ is $7.3\times 7.5\mu {\mathrm{m}}^2$.

Figure 6

[(a) and (b)] Evolution of the measured NESS-PDF $P_{ss}(x,y)/P_0$ for sample S1 with increasing values of $F$ along the [1,0] direction with (a) $F/F_T=1.4$ ($\theta =5.5^{\circ }$) and (b) $F/F_T=5.2$ ($\theta ={20}^{\circ }$). [(c) and (d)] Evolution of the measured NESS-PDF $P_{ss}(x,y)/P_0$ with increasing values of $F$ along the [1,1] direction with (c) $F/F_T=1.4$ and (d) $F/F_T=5.2$. The value of $P_0$ is chosen so $P_{ss}(x,y)/P_0=1$ at the peak positions. The arrows indicate the forcing directions. The scale bar is 1 $\mu \mathrm{m}$.

Figure 7

Measured trajectories (white curves) of a top-layer particle for sample S2 under the influence of a gravitational pulling force $F$ along the [1,0] direction toward the left. The amplitude of $F$ is (a) $F/F_T=22.5$ ($\theta ={14}^{\circ }$) and (b) $F/F_T=46.8$ ($\theta ={31}^{\circ }$).

Figure 8

Measured trajectories (white curves) of a top-layer particle for sample S2 under the influence of a gravitational pulling force $F$ along the [1,1] direction toward the left. The amplitude of $F$ is (a) $F/F_T=22.5$ ($\theta ={14}^{\circ }$) and (b) $F/F_T=46.8$ ($\theta ={31}^{\circ }$).

Figure 9

[(a) and (b)] Evolution of the measured NESS-PDF $P_{ss}(x,y)/P_0$ for sample S2 with increasing values of $F$ along the [1,0] direction with (a) $F/F_T=22.5$ ($\theta ={14}^{\circ }$) and (b) $F/F_T=46.8$ ($\theta ={31}^{\circ }$). [(c) and (d)] Evolution of the measured $P_{ss}(x,y)/P_0$ with increasing values of $F$ along the [1,1] direction with (c) $F/F_T=22.5$ and (d) $F/F_T=46.8$. The value of $P_0$ is chosen so that $P_{ss}(x,y)/P_0=1$ at the peak positions. The arrows indicate the forcing directions. The scale bar is 1 $\mu \mathrm{m}$.

Figure 10

Superposition of the background image of the bottom colloidal crystal layer (black and white) and 2D plot of the measured $P_{ss}(x,y)$ (red) with brighter regions indicating higher values of $P_{ss}(x,y)$. Positions $A$, $B$, and $C$ (green dots) indicate the locations of the potential wells in $U_0(x,y)$. Positions $A^{\prime }$, $B^{\prime }$, and $C^{\prime }$ (green circles) indicate the peak positions of the measured $P_{ss}(x,y)$. The arrow indicates the direction of the force $F$ along the [1,0] orientation.

Figure 11

2D plot of the averaged $P_{ss}(x,y)$ over the repeated unit cells with brighter regions indicating higher values of $P_{ss}(x,y)$. Positions $A^{\prime }$ and $B^{\prime }$ indicate the peak positions of the measured $P_{ss}(x,y)$. The red line $\overline{A{B}^{\prime }}$ has a tilt angle of $\pi /6$ with respect to the [1,0] crystal orientation, as illustrated by the lower triangle. The red line $\overline{A{A}^{\prime }}$ has a tilt angle of $5\pi /6$ with respect to the [1,0] orientation, as illustrated by the upper triangle. The intersection point A indicates the position of the potential well in $U_0(x,y)$.

Figure 12

(a) Measured one-dimensional equilibrium PDF $P_{ls}\left(X\right)$ as a function of the normalized transition path $X/\lambda$ (red squares). The solid line is a smooth-fitting curve representing the measured $P_{ls}\left(X\right)$. [(b)–(d)] Evolution of the measured one-dimensional NESS-PDF $P_{ss}\left(X\right)$ (solid symbols) with increasing values of $F$ along the [1,0] direction with (b) $F/F_T=22.5$, (c) $F/F_T=37.0$, and (d) $F/F_T=46.8$. The solid lines are the numerical results obtained using Eq. (5) together with the measured quasi-1D potential $U_0\left(X\right)/k_{B}T=-ln\left[P_{ls}\left(X\right)\right]$ from (a).

Figure 13

Reconstructed 1D potential $U_0\left(X\right)$ from the measured $P_{ss}\left(X\right)$ and $J_{ss}$ using Eq. (8). The symbols are obtained with increasing values of $F$ along the [1,0] direction for $F/F_T=22.5$ (black squares), 37.0 (blue circles), and 46.8 (red triangles). The black solid line is the directly measured $U_0\left(X\right)/k_{B}T=-ln\left[P_{ls}\left(X\right)\right]$ from Fig. 12.