Optical Tweezers: Experimental Demonstrations of the Fluctuation Theorem

Transcription

1 Optical Tweezers: Experimental Demonstrations of the Fluctuation Theorem A thesis submitted for the degree of Doctor of Philosophy of The Australian National University David Michael CARBERRY October 2005

2 ii Statement of Originality I hereby declare that this submission is my own work. To the best of my knowledge and belief this thesis contains no material previously published or written by another person, except where due acknowledgement has made in the text of the thesis. This work has not been submitted preiously, in whole or in part, to qualify for any other academic award David Carberry Date

3 iii Acknowledgements First and foremost I would like to thank Dr Genmiao Wang who has taught me virtually everything I know about Optical Tweezers. Without his assistance and guidance I doubt that these experiments would have been successful. I would also like to thank Dr Edie Sevick, my principal supervisor, for providing guidance with the project when it was sought, yet allowing me to freely pursue my goals at other times. Her implicit trust in my research abilities and supervision method proved to be ideal for turning this student into an independent researcher. Prof Denis Evans dedication and non-stop enthusiasm for his science, willingness and ability to explain difficult concepts, and overall approachability also made this project truly enjoyable. The other members of the Sevick and Evans research groups: James Reid, Matthew Baker, Dr Stephen Williams and Dr Joanne Bright who ensured that my time spent in the lab was not all serious, and that my focus remained wider than my individual project. Additional thanks need to go to Stephen who patiently provided detailed explanations and error checking with many of the derivations in this thesis. I would also like to thank the staff in the RSC workshop: Russell Koehne, Don Pepper, Mike Hill, Peter Devitt, Ray Filardo and Ian Clark. Their assistance building, designing and repairing devices for the Optical Tweezers has ensured that this research could proceed. Working with these people was always entertaining and gave me a chance to do some interesting engineering work. I would also like to thank my family for listening to me complain about this PhD for 3.5 years. They ensured that life remained relatively normal regardless of how my research was going. The ability to visit the farm to relax and do something completely different for a few days proved vital for my sanity. My friends, especially Christine White, Kate and Dave Osborn, Mel Ford, Monica Cooper and Richard van der Male, Belinda Haupt, Angela Ford, Marcus Uzubalis, Sara Barnes, Enesh Seitmuradova and many others have ensured that life outside of uni was truly enjoyable. I d also like to thank the members of the RSC soccer team, who provided a much needed chance to escape the lab during lunch and ensured that I learnt how to manage people. I now understand why people say that organising academics is like herding cats. Finally, I d also like to thank the people at the ANU gym: Charon, Angie, Nina, Emma, and the rest of the bars crowd who kept me pushing my limits while keeping things fun.

4 iv Abstract In the late 19th and early 20th centuries famous scientists like Boltzmann, Loschmidt, Maxwell and Einstein tried, unsuccessfully, to find the link between the timereversible equations of motion of individual molecules and irreversible thermodynamics. The solution to this puzzle was found in 1993, and the link is now known as the Fluctuation Theorem (FT). In the decade that followed theory and computer simulation tested the FT and, in 2002, an experiment indirectly demonstrated the FT. This thesis describes original experiments that demonstrate the FT directly using Optical Tweezers. A related expression, known as the Kawasaki Identity, is also experimentally demonstrated. These experimental results provide a rigorous demonstration that irreversible dynamics can be obtained from a system with timereversible dynamics.

5 Contents 1 Introduction Thesis Goals References The Optical Tweezers Background The Physics of Optical Tweezers The Energy and Momenta of Photons Interactions of Photons with Objects Refraction through a Mie Scattering Particle Reflection from a Mie Particle Photon Flux The Optical Tweezers as a Potential Well Instrumentation The Original, Commercial Setup Trap Asymmetry Modifications to the Optical Tweezers The Quadrant Photodiode System in Detail Optical Trapping and Simple Operation of the Optical Tweezers Calibration of the Optical Tweezers Optics Calibration Quadrant Photodiode Calibration Piezoelectric-stage Translations Trap Strength Calibrations References v

6 vi 3 The Fluctuation Theorem Background Derivation of the Fluctuation Theorem Derivation of the FT using Deterministic Dynamics Derivation of the FT using Stochastic Dynamics The Second-Law Inequality The Integrated Fluctuation Theorem The Dissipation Function Steady State FT References The Capture Experiment The Optical Trap Experiment Experimental Expressions for the Dissipation Function Deterministic Derivation of Ω t for Capture Stochastic Derivation of Ω t for Capture Details about Ω t for Capture Results and Discussion for the Capture Experiment Capture Conclusions References Steady State Fluctuation Theorem Experiments Experimental setup Linear Drag Circular Drag Derivation of Expressions for the Dissipation Functions Deterministic Derivation of Ω t Stochastic Derivation of Ω t Results Transient FT using Linear Drag Steady State FT using Linear Drag Steady State FT using Circular Drag Conclusions References

7 vii 6 The Kawasaki Identity and the Fluctuation Theorem Derivation of the KI KI Experimental Procedure Results of the KI Experiment Concluding Remarks References The Fluctuation Theorem in Complex Systems The Viscoelastic Capture Experiment The Dissipation Function for Viscoelastic Capture Results Concluding Remarks References Conclusions and Future Work 119

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9 List of Figures 2.1 Refraction and reflection at an interface Multiple rays refracting through a microscopic particle Various intensity profiles for optical trapping systems A focal point is used to form an optical trap Reflection from a microscopic particle The optical trap as a potential well The Optical Tweezers in June 2002, before most modifications An image of the Cell Robotics laser spot The Optical Tweezers in June 2005, after modifications Flowchart of the visible light path to quadrant photodiode An example of a quadrant photodiode Pictures a particle at various heights within the sample cell A comparison of the calculated position-voltage curve with an experimentally determined curve Graph of Position vs Quadrant Photodiode Voltage Piezoelectric calibration graphs Drag calibration: Photodiode voltage vs time graph Drag calibration: fitting the Drag Force vs optical displacement Example of a PSD calibration A graph showing trap constants determined using various methods A graphical representation of the trajectories used in determining Ω t deterministically A graphical representation of the trajectories used in determining Ω t stochastically The optical trap strength cycle for the capture experiment ix

10 x 4.2 Histograms showing the distribution of the dissipation function, Ω t Experimental and Theoretical FT results Slope of the Least Squares Fit to the FT vs Time for Capture The IFT results for the capture experiment Langevin prediction for the average Ω t compared to the experiment Velocity and position profiles for linear drag Velocity and position profiles for circular drag Histograms of Ω t (Γ) at 50 ms and 3000 ms Direct FT results with Ω t (Γ) for 0 t Transient FT results with Ω t (r) at 50 ms and 500 ms Slope of the Least Squares Fit to the FT vs Time for Ω t (Γ) and Ω t (r) Direct IFT results with Ω t (Γ) and Ω t (r) for 0 t FT results with Ω t (Γ) and for t FT results with Ω t (r) for t Least Squares Fit to FT results vs Time with Ω ss t (Γ) and Ω t (r) for t IFT results with Ω t (Γ) and Ω t (r) for t 20 s Circular drag FT results with Ω ss t (Γ) at 0.25 s and 2.5 s Circular drag FT results with Ω t (r) at 0.25 s and 2.5 s Least Squares Fit to circular drag FT results vs Time with Ω ss t (Γ) and Ω t (r) IFT results for circular drag using Ω ss t (Γ) and Ω t (r) Counter-example showing that Kawasaki Identity doesn t prove the FT The IFT and Kawasaki for a poor set of data Histograms from the KI/capture experiment Time segments from various trajectories showing normal and abnormal behaviour An schematic of the cycle after an exchange event The IFT and Kawasaki function for a good set of data The Kawasaki function is plotted with various numbers of experimental trajectories, as indicated. The horizontal axes are in time (s) The evolution of the Kawasaki function using erroneous results

11 xi 7.1 The power spectral density for data collected in the viscoelastic solvent Histograms of Ω t for the viscoelastic capture 2, 20 and 200 ms after the change in trap constants The FT is shown at 2, 20 and 200 ms after the change in the strength of the optical trap constant The slope of the least squares fit for the FT vs time for k 0 to k 1 and k 1 to k The IFT for the k 0 to k 1 transition and the k 1 to k 0 transition

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