Sorting on Periodic Surfaces

Particles moving on crystalline surfaces and driven by external forces or flow fields can acquire velocities along directions that deviate from that of the external force. This effect depends upon the characteristics of the particles, most notably particle size or particle index of refraction, and can therefore be (and has been) used to sort different particles. We introduce a simple model for particles subject to thermal fluctuations and moving in appropriate potential landscapes. Numerical results are compared to recent experiments on landscapes produced with holographic optical tweezers and microfabricated technology. Our approach clarifies the relevance of different parameters, the direction and magnitude of the external force, particle size, and temperature.

Figure 1

A finite portion of the much larger two-dimensional periodic potential Eq. (3) for the case of wells with $A=5$ and $B=0.8$.

Figure 2

Trajectories for a force applied at different angles $\theta$. These angles are represented by dotted lines, and correspond to $\tan \theta =0.2$ (a), 0.4 (b), 0.6 (c), and 0.8 (d). Other parameters are: $A=5$, $B=0.7$, $\gamma =20$, $F_0=8$, and $\mathcal{T}={10}^{-4}$.

Figure 3

Deflection angle vs field direction for $F_0=8$ and different values of the potential parameter $B$ (increasing $B$ is associated with increasing particle size) for two temperatures. Upper panel $\mathcal{T}=0.01$; lower panel $\mathcal{T}=0.1$. Data correspond to $B_1=0.5$ (solid lines), $B_2=0.7$ (dashed lines), and $B_3=0.9$ (dotted lines).

Figure 4

Plateaus in the absolute velocity angle for $F_0=8$ and different values of the parameter $B$ for two temperatures. Upper panel $\mathcal{T}={10}^{-4}$; lower panel $\mathcal{T}={10}^{-2}$. Data correspond to $B=0.5$ (solid lines), 0.7 (dashed lines), and 0.9 (dotted lines).

Figure 5

Deflection angles ${\alpha }_i$ for different particles $B_i$ as a function of $F_0$ for a fixed force direction $\tan \theta =0.25$ and $\mathcal{T}=0.01$. Squares: $B_1=0.5$, $\beta \equiv {\alpha }_1$. Triangles: $B_2=0.7$, $\beta \equiv {\alpha }_2$. Circles: difference between deflection angles of the two types of particles, $\beta \equiv {\alpha }_2-{\alpha }_1$.