Co- and cross-flow extensions in an elliptical optical trap

E. Schonbrun, J. Wong, and K. B. Crozier

Phys. Rev. E 79, 042401 – Published 28 April 2009

Abstract

The extension of a particle (translational deviation) in an isotropic harmonic potential well is linearly proportional and parallel to the applied force. In an anisotropic trapping potential, the extension is instead related to the applied force by a compliance tensor. Using the focal spot of a high numerical aperture zone plate to create an elliptical potential and microfluidics to apply a calibrated force, we measure the two-dimensional extension of a trapped spherical particle. As a function of the orientation of the elliptical potential, the extension sweeps out a circular trajectory, exhibiting extensions both parallel (coflow) and perpendicular (cross flow) to the direction of the flow. The results fit well to a compliance tensor model.

Received 17 November 2008

DOI:https://doi.org/10.1103/PhysRevE.79.042401

Authors & Affiliations

E. Schonbrun¹, J. Wong², and K. B. Crozier¹

¹School of Engineering and Applied Science, Harvard University, Cambridge, Massachusetts 02138, USA
²Schlumberger-Doll Research Center, Cambridge, Massachusetts 02139, USA

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Issue

Vol. 79, Iss. 4 — April 2009

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Images

Figure 1
(Color online) Fresnel zone plate tweezer integrated with microfluidics. (a) Scanning electron micrograph of the gold-on-glass zone plate. (b) Microscope image of a $1.1 μ m$ sphere trapped by the zone plate tweezer. (c) Schematic of the experimental setup. The zone plate is illuminated by a laser beam propagating in $Z$ that slightly overfills the zone plate aperture. The orientation of the elliptical potential is controlled by rotating the incident polarization $P$ , where $X$ is parallel to the fluid flow (coflow) and $Y$ is perpendicular to the fluid flow (cross flow). $P$ is in the $X − Y$ plane.Reuse & Permissions
Figure 2
(Color online) Force coflow extension curves. Extension is measured along the flow direction when the polarization of the laser beam incident on the zone plate tweezer is parallel to the flow in blue (dark) and perpendicular to the flow in red (light). Uncertainty in the applied force arises from uncertainty in the velocity field applied by the syringe pump. Uncertainty in the extension is below 3 nm and is not plotted. The optical compliance at 1.2 pN for parallel polarization is approximately twice the optical compliance for perpendicular polarization.Reuse & Permissions
Figure 3
(Color online) Extension time traces. Coflow extension in blue (dark) and cross-flow extension in green (light) for four different orientations of the elliptical potential, $- 45 °$ , $0 °$ , $45 °$ , and $90 °$ for (a), (b), (c), and (d), respectively. The orientation angle describes the relative angle between the fluid flow and the polarization angle or the major axis of the elliptical potential. At 4 s, the syringe pump is turned on and after reaching equilibrium applies a force of 1.2 pN to the trapped bead. Insets illustrate particle position after equilibrium is reached. For orientations of $45 °$ and $- 45 °$ , a cross-flow extension is resolvable that is pointed in opposite directions. An offset has been added to each extension trace so that each has a mean value of zero.Reuse & Permissions
Figure 4
(Color online) Equiforce equilibrium positions. For a constant 1.2 pN applied force, the equilibrium position as a function of the orientation of the elliptical potential is plotted. Each data point in blue (dark) corresponds to a $15 °$ counterclockwise rotation of the trapping potential, starting at an orientation of $θ = 45 °$ . The data are fit to the optical compliance tensor model which predicts a circular trajectory in red (light). The origin of the two-dimensional extension is the position of the sphere without an applied fluid force. Uncertainty in the extension measurements is below 3 nm, which is the size of the plotted dots.Reuse & Permissions

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