1. Discussion or Conclusions
Optical Microscope and Tweezer Laboratory Bing Xia and Timur Skeini, Department of Physics and Astronomy, Ohio University
Abstract
Materials and Methods
Results
The Optical Microscope was used to successfully observe and record 10 s clips of the Brownian motion of 2 µm latex beads at 0.01% concentration in 5 µl of buffer solution, using Oil (100x), Water (60x) and Air (40x) objectives.
The measured Bolzmann’s constant using the oil objective for 2 beads was 3.72 x J/K, which is 4 orders smaller than the accepted value of 1.38 x J/K, while for water objective for 5 beads was 4.21x10-24 J/K – 1/3 of the accepted value. The difference arise from the statistical fluctuation because of only several particles in the ensemble average calculation, the lack of significant digits in the fluid’s viscosity coefficient and the bead’s diameter, and the effect of adhesion forces when the beads are close to the surface.
The Optical Tweezers were also successfully used to observe and record trapping beads using the oil and water objectives, but no stable trap was found using the air objective. The calculations for water objective yielded the trap stiffness kx = 4.98 x 10-7 N/m and ky = 1.44 x 10-7 N/m.
Setup (see Figure 2)
Inverted Optical microscope with Optical Tweezer (Figure 3)
objectives Oil (100x), Water (60x), Air (40x)
Black/White camera
infrared diode laser, power = 800 mW (no filters used)
Laser traps controlled by Holographic Optical Plate (Arryx Inc)
Sample was prepared using 2 µm-diameter polystyrene beads (Bangs Labs) in buffer diluted twice to .01% concentration by volume.
Viewing chamber was approximately 1 mm gap between strips of tape, filled with about 5 µl sample solution, covered by square cover slips at bottom and round at top, and sealed with nail polish (Figure 4).
Data collection procedure
Viewing chamber placed on microscope
Needed objective adjusted and beads focused via eyepieces first
HOTGui software was used to observe and record 10 sec video at 15 FPS
Data Analysis
Recorded videos were broken down into (grayscale TIFF) frames using VideoMach software
Bead size in pixels was measure in frames using ImageJ software
The positions of beads were then tracked in each frame using RyTrack scripts in IDL software
For Brownian motion videos:
Center of mass locations in each frame were computed via IDL script
Drift was calculated from the slope of the line fit of center of mass locations versus time (1/15fps).
Mean square displacement of beads, <r2>, (relative to center of mass) for each frame was also computed via IDL script
Diffusion coefficient, D, was then calculated from the slope of line fit of <r2> versus time.
Boltzmann’s constant was then calculated using the theory equation with room temperature (T=293K), viscosity of water (η=0.001 Pa.s) and the measured bead radius.
For Trap stiffness videos:
Center of trap in frames was located using ImageJ software
The statistical variance of particles was computed using IDL script and then the stiffness kx=ky (assumed the same) was computed via Equipartion formula, using room temperature (T=293 K) and standard kb.
Brownian motion:
Observed using Oil, Water and Air objectives and recorded 10 sec 15 fps video clips (Figure 5-6 (a))
The analysis procedure was followed to calculate the following for Oil objective:
Drift (Figure 5 (c), Figure 6 (c-d))
Boltzmann’s constant, 3.72 x J/K for oil, 4.21x10-24 J/K for water (Figure 5-6 (b)
Analysis for Water and Air data was not finished.
Trap Stiffness:
Stable trapping was observed only with Oil and Water objectives, and recorded (10 sec 15 fps clips (Figure 7 (a))
The analysis procedure was followed to calculate the stiffness with Oil objective, kx = 4.98 x 10-7 N/m and ky = 1.44 x 10-7 N/m, from one video only.
Other videos were determined not suitable for analysis.
Trap was not stable when its depth position was changed.
Error analysis was not performed.
(b)
Figure 6
(b)
(a)
Figure 5
(c)
(a)
(c)
Objective
(d)
Observe and record the Brownian motion and optical trapping of micrometer-size particles, and calculate Boltzmann’s constant and trap stiffness.
Discussion or Conclusions
Introduction
Brownian Motion:
Calculated Boltzman’s constant 4.21x10-24 J/K was about factor of 3 bigger than standard. This error was possibly due to:
Beads being too close to slide surface (not in the middle of the chamber), thus adhesion forces were significant;
Too few beads were analyzed (previous computation using 2 beads showed order 7 difference, thus this factor was significant);
Bead size was measure inaccurately (high error in estimation from frames).
Too few videos were analyzed.
Suggestions:
Observe beads in the middle of the chamber;
Observe at least 5 beads;
Measure bead size more precisely or do an average for many beads;
Analyze several video clips.
Trapping:
Theory ignored the motion of the bead in the z-direction, but the bead was not constrained to move only in x-y-plane. Thus this is possible source of error. Camera video of z-x or z-y planes should solve this issue.
Trapping in Air was not successful due to low index of refraction between sample and objective, resulting in stronger scattering from laser focus than trapping.
Figure 3
Nikon TE200 optical microscope.
Figure 1
Change in momentum of laser beam as it passes though the bead causes a net restoring force on the bead towards the trap center (in this case down).
3D Trapping Mechanism diagram
The Optical Microscope can be used to observe the Brownian motion of micrometer sized particles in water-like liquid, and to calculate the diffusion coefficient to get Bolzmann’s constant. The Optical Tweezer is an add-on to the Optical Microscope, where the laser beam can be used to trap, hold and move micrometer sized particles at its focus, and to calculate the position variance of trapped particles to get the trap’s spring constant.
The Brownian motion was first observed by Robert Brown (1820s) in pollen grains. He argued that their motion was not due to particles being alive. In 1905, Albert Einstein predicted random changes in direction of travel (random walk) in micrometer-sized particles due to random atomic collisions (as practical consequence of atomic hypothesis). The analysis of random walk showed that the mean square displacement, <r2>, as a function of time, was related to the diffusion coefficient, D (Fick’s Law of diffusion) as <r2>1D = 2Dt in 1-dimension, and <r2>2D = 4Dt in 2-dimensions (since x and y are independent). The diffusion coefficient was further related to Boltzmann’s constant as D = kbT/(6πηR), where T was absolute temperature (Kelvin), η – viscosity of fluid medium, R – particle radius. In 1908, Jean Baptiste Perrin measured Avogadro’s number, from the observations of Brownian motion of latex particles in water.
Optical Trapping is associated with devices called Optical Tweezers or Laser Tweezers. They were developed in 1980s from work on laser trapping of atoms, and can trap (“hold”) and manipulate micrometer-sized particles with different index of refraction than medium (see Figure 1). The trap can be modeled as a potential well for small displacements, x, via Hooke’s Law Fx=-kx(x-xo), where kx is spring (trap) stiffness is x-direction, xo – bottom of potential well (center of trap). The stiffness can be further estimated from Equipartition theorem <U(x)> = ½ kx<(x-xo)2)> = ½ kbT, where U(x) is the potential energy.
Figure 7
(b)
(a)
Figure 2
Experimental setup.
References
K.C. Neuman and S.M. Block, Rev. Sci. Instrum., Vol. 75, No. 9, (2004).
Graham Milne, “Optical Sorting and Manipulation of Microscopic Particles”, PhD Thesis University of St Andrews (2007).
Wesley P.Wong and Ken Halvorsen, Optics Express, Vol. 14 No. 25 (2006)
P. Nakroshis, M. Amoroso, J. Legere, and C. Smith, Am. J. Phys., Vol. 71, No. 6, (2003)
K. Svoboda & S. M. Block, Ann. Rev. Biophys. Biomol. Struct., 23: , 1994
Figure 4
Viewing Chamber
Acknowledgements
Paul Ingram for writing the Optical Trapping manual!