Colloidal particles driven across periodic optical-potential-energy landscapes

We study the motion of colloidal particles driven by a constant force over a periodic optical potential energy landscape. First, the average particle velocity is found as a function of the driving velocity and the wavelength of the optical potential energy landscape. The relationship between average particle velocity and driving velocity is found to be well described by a theoretical model treating the landscape as sinusoidal, but only at small trap spacings. At larger trap spacings, a nonsinusoidal model for the landscape must be used. Subsequently, the critical velocity required for a particle to move across the landscape is determined as a function of the wavelength of the landscape. Finally, the velocity of a particle driven at a velocity far exceeding the critical driving velocity is examined. Both of these results are again well described by the two theoretical routes for small and large trap spacings, respectively. Brownian motion is found to have a significant effect on the critical driving velocity but a negligible effect when the driving velocity is high.

Figure 1

(a) Schematic of the experimental geometry. A one-dimensional periodic optical potential energy landscape, $U_{\mathrm{T}}\left(x\right)$, is generated by a line of overlapping optical traps created from focused laser spots, time shared using acousto-optical deflectors, separated by a spacing $\lambda$. A spherical colloidal particle sedimented to the bottom of the sample cell is driven across the landscape by a constant force $F_{\mathrm{DC}}$. (b) The optical potential energy landscape, $U_{\mathrm{T}}\left(x\right)$, corresponds to a trap spacing of $\lambda =3.5\mu \mathrm{m}$ and a laser power per trap of 0.75 mW. The tilted “washboard” potential, $U\left(x\right)$, for $F_{\mathrm{DC}}/\zeta =2.25\mu \mathrm{m}{\mathrm{s}}^{-1}$ is shown in the lower panel. The symbols are the experimental data and the solid black line a fit with sine function. The solid orange line in the lower panel illustrates a hypothetical tilted washboard potential corresponding to a subcritical driving force, which leads to finite barriers in the potential.

Figure 2

Comparison of theoretical approximations for the optical potential energy landscape $U_{\mathrm{T}}\left(x\right)$. () Periodic landscape composed of the sum of 11 individual Gaussian optical traps at trap spacing $\lambda$ given in $\mu \mathrm{m}$ and units of $\sqrt{V_0/k}$, using typical trapping parameters $k=3.8\times {10}^{-7}$ kg ${\mathrm{s}}^{-2}$ and $V_0=90k_{\mathrm{B}}T$. () Sinusoidal landscape: small $\lambda$; () nearest-neighbor landscape: large $\lambda$.

Figure 3

Trajectories of Brownian particles driven over a periodic potential. (a) Individual particle trajectories for cases with differing trap spacings and driving velocities but the same average velocity. Lines are spaced in $t$ for ease of comparison. (b) Particle velocity as a function of time for the four trajectories in (a). (c) Particle velocity as a function of position for the four trajectories in (a). The dashed line in the right panel is at $v=2.7\mu \mathrm{m}{\mathrm{s}}^{-1}$.

Figure 4

Average particle velocity against driving velocity for $\lambda =3.5\mu \mathrm{m}$, compared to the case of no traps. $▵$ experimental data (error bars: standard deviation of repeats). () Fit corresponding to a sinusoidal optical potential energy landscape, $\zeta \overline{v}=\sqrt{F_{\mathrm{DC}}^2-F_{\mathrm{C}}^2}$, with fitting parameter $F_{\mathrm{C}}$. () No traps.

Figure 5

Average particle velocity against driving velocity for varying trap spacing. (, , , , , ) Experimental data (error bars: standard deviation of repeats). () Fit corresponding to a sinusoidal optical potential energy landscape, $\zeta \overline{v}=\sqrt{F_{\mathrm{DC}}^2-F_{\mathrm{C}}^2}$. () No traps. Inset: The effect of Brownian noise on particle velocity close to the critical driving velocity. () Numerical solutions for a Brownian particle driven over a sinusoidal optical potential energy landscape, Eq. (19).

Figure 6

Critical driving velocity as a function of trap spacing. $(\circ )$ Experimental data (error bars: experimental uncertainty). Lines: Theoretical predictions. Sinusoidal potential energy landscape: () deterministic force, Eq. (11); () stochastic force, according to Eq. (19). Nearest-neighbor landscape: () deterministic force, Eq. (15); () stochastic force, according to Eq. (17).

Figure 7

Effect of the trap spacing on the average velocity of a particle at a driving velocity of $5.8\mu \mathrm{m}{\mathrm{s}}^{-1}$. $(\circ )$ Experimental data (error bars: standard deviation of repeats). Sinusoidal potential energy landscape: () deterministic force, Eq. (13); $\left(\begin{array}{c}\text{—}\end{array}\right)$ stochastic force, Eq. (19). Nearest-neighbor landscape: $\left(\begin{array}{c}\text{−}\text{−}\text{−}\end{array}\right)$ deterministic force, Eq. (6); () stochastic force, Eq. (17).