Comparing Optical Trap Stiffness Measurement Methods

Adarsh Ravishankar, Shengshee Thor

University of Minnesota - School of Physics and Astronomy

Minneapolis, MN 55455

Introduction

Since optical tweezers use light to manipulate micron to sub-micron sized particles, they have become important characterization tools for single-molecule biophysics experiments. Under the Mie regime, particle size is larger than the wavelength, λ, of light being used and thus ray optics can be employed to understand the physics of optical trapping.

Mie Regime Trapping:

Refraction of the most intense light rays of the Gaussian beam always exerts a net restoring force directed towards the beam center, effectively “trapping” the bead like a Hookean spring with stiffness, k. In this study, we build and characterize k of an optical tweezer that uses a 17 mW HeNe laser (λ = 633 nm). Two common methods of past MXP projects to determine k are Brownian motion and Stoke flow. Here, we compare both methods to determine which method will best suit future undergraduate optical tweezer setups similar to ours.

Brownian Motion

According to the Equipartition theorem, a particle at thermal equilibrium has an average energy of ½ kBT for each degree of freedom. Therefore, the energy of a trapped bead oscillating in the harmonic potential of a Hookean spring system can be modeled by Equation (1):

Equation 1:

where is the mean-squared displacement (MSD) of the bead from its average position.

Stoke Flow

For small displacements from the beam center, the optical trap can be modeled as a Hookean spring system with a spring constant of k. When the trapped bead is subject to laminar fluid flow, the viscous force due to the fluid and can be modeled by Stoke’s flow equation:

where v is the velocity of the water, η is the viscosity of the water (8.90x10-4 Pa-s), and R is the microbead’s radius. When the viscous force and optical trap balance, a linear relationship between fluid velocity and bead displacement can be developed as:

Equation 2:

Experimental Setup

Optical Tweezer Setup

The setup of the optical tweezers constructed in this study is shown below in Figure 1. The design consists of a 633 nm, 17mW Helium-Neon laser, a reflective mirror (M1) for beam redirection, dichroic mirrors for imaging and beam redirection (M2 and M3), lenses for bead expanding (L1,L2, and L3), an objective lens, a CCD camera, and a white light source.

The optical trap is the strongest when the laser beam is the same diameter as the objective that it enters. If the laser beam has a smaller diameter than the entrance of the objective, then the size of the focused beam will have a larger diameter, which decreases the strength of the trap. If the laser beam has a larger diameter, then not all of the light will enter through the objective, which also weakens the trap. Since the diameter of the beam was measured to be 2 mm, and the objective’s was measured to be 6 mm, lenses L1 and L2 were used to triple the initial beam’s diameter, with L2 having about three-times the focal length of L1. After the bead is redirected using mirrors M1 and M2, the diameter of the beam is fine-tuned using lens L3. Since a diverging beam would make unwanted changes to the beam’s diameter, the distance between L1 and L2, along with the distance between L3 and the objective, are equal to the sum of their respective foal lengths (See: Figure 5). The focal lengths for the lenses are listed in the appendix.

The purpose of the dichroic mirrors (M2 and M3) is to reflect the beam on one side of the mirror, while reducing the beam’s intensity on the other side. The white light source will illuminate the optically trapped bead back through the objective lens, through the dichroic mirrors an into the CCD camera for visualization. The resulting image will show the bead in the solution, along with the image of the focused laser beam. Additional specifications for the setup are listed in the appendix.

Figure 1 - Tweezer Setup

Flow Apparatus Setup

In order for the bead-trap system to experience a fluid force, a controlled fluid velocity had to be created using a flow cell apparatus (See: Figure 2). The apparatus consists of two reservoirs (Reservoirs 1 and 2) filled with a solution consisting of 6µm-diameter polystyrene beads and distilled water. Each reservoir is connected to a 27mm x 5 mm x .5mm flow cell, where the bead-trapping occurs. The flow cell and the reservoirs are connected using Tygon tubing with a 1/16” internal diameter. Reservoir 1 is placed at a height ∆h above Reservoir 2 in order to produce the initial fluid flow. A flow valve was also connected between Reservoir 1 and the flow cell, as to completely stop or start the flow. Although changing ∆h would change the velocity of the fluid, finer control over the flow was achieved by placing a clamp on the tubing between Reservoir 1 and the flow cell. By rotating the screw on the clamp, the tubing can be constricted or expanded to obtain the preferred fluid velocity. Since the fluid flow in the cell can be turned off whenever needed, this apparatus was used for both Stoke’s flow and Brownian motion measurements. For both measurements, the temperature of the water was measured using a mercury thermometer.

Figure 2 - Flow Cell Apparatus

Experimental Method

Pixel-to-Distance Calibration

Once water-bead solution settled in the flow cell, some beads became stuck to the glass surface of the flow cell. These immobilized beads were tracked for pixel-to-distance calibration. The flow cell was translated a known distance (measured by the Vernier micrometer of the translation stage), and the immobilized bead's pixel displacement was measured. Measured distance and measured pixel displacement values were correlated to give a calibration curve that yielded a pixel-to-distance conversion factor of 12.7(±0.4) pixels/μm.

Stoke flow

Once a bead was successfully trapped, flow was induced until a noticeable and constant bead displacement could be visualized on the video feed. The laser was then turned off so that the bead could flow with the fluid such that bead position could be tracked over time to determine the induced fluid velocity.

Brownian Motion

Six beads were successfully trapped and allowed to fluctuate in position as the videos were recorded. For each video, an ImageJ plugin, SpotTracker2D, was used to track the bead center as it fluctuated within the trap. The MSD was calculated and used to compute a value for k. All six k values were then used to calculate a weighted mean of k.

Results

Final Results:

Flow Data Results:

Conclusion

A 2-sample z-test on both values of k suggests with 95% confidence that they are indeed statistically similar. Therefore, it can be concluded that measuring k via Brownian motion leads to less error for a sufficient but weak optical trap such as ours. The results here do not invalidate the Stoke flow method, but rather suggests that it is more suited for optical traps of higher strength. This hypothesis is supported by Equation (2) which shows that k is proportional to fluid velocity. Stronger traps that can hold beads against a wider range of velocities would eliminate the uncertainties we encountered for our stoke flow k value.

Future experiments should increase optical trap strength by reducing the laser path length. Stronger optical traps should then be useable in experiments probing biological materials, such as DNA or molecular motors.

References

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Appendix

• • Table 1: