Analytical approach to sorting in periodic and random potentials.

TLDR: Analytical predictions for the dependence of the angle between the direction of motion and the external force on a number of model parameters for periodic as well as random surfaces are provided.

Abstract: There has been a recent revolution in the ability to manipulate micrometer-sized objects on surfaces patterned by traps or obstacles of controllable configurations and shapes. One application of this technology is to separate particles driven across such a surface by an external force according to some particle characteristic such as size or index of refraction. The surface features cause the trajectories of particles driven across the surface to deviate from the direction of the force by an amount that depends on the particular characteristic, thus leading to sorting. While models of this behavior have provided a good understanding of these observations, the solutions have so far been primarily numerical. In this paper we provide analytic predictions for the dependence of the angle between the direction of motion and the external force on a number of model parameters for periodic as well as random surfaces. We test these predictions against exact numerical simulations.

Figures

FIG. 4. Results for v /U for the one-dimensional potential with forcing U=3, plotted as a function of temperature T. Symbols show exact values; the approximation curves use v N for v, as defined in Eq. 24 , with coding N=1 solid , N dashed , N dotted , N dash-dotted .

FIG. 3. Deflection angle for separable two-dimensional potential 22 with T=0.1 and F=15,25,50 from greater to smaller deflection angles. Dashed curves, first-order approximation 11 ; solid curves, second-order approximation 16 ; dotted line, cutoff for approximation curves in the adjusted truncation; symbols, numerical simulation results.

FIG. 2. The dependence of the deflection angle on the potential particle size parameter B at temperature T=0.1, for F=8 and =0.2. Numerical simulation results solid diamonds agree well with theoretical predictions using M =6 modes solid line in the Fourier series 18 . For comparison, the theoretical prediction using only M =1 modes is also shown dashed line .

FIG. 1. The dependence of the deflection angle on the forcing angle at temperature T=0.1, and for parameters F=8 and B=0.9. Numerical simulation results solid diamonds agree well with theoretical predictions using M =6 modes solid line in the Fourier series 18 . For comparison, the theoretical prediction using only M =1 modes is also shown dashed line .

FIG. 5. Deflection angles in the two-dimensional separable random potential with Gaussian energy spectrum in each direction =5, =4 , temperature T=0.2 and external forcing F=1 lower results and F=2 upper results . The dashed curves show the firstorder approximation, and the solid curves the second-order approximation. The adjusted truncation cuts off both approximations at the dotted line. The numerical simulation results are shown as symbols.