5.4 The light potential and the driving force
5.4.3 Calculation of the experimental driving force
In the previous section we tested successfully that our experiments can be described by a static sawtooth potential and a constant driving force. Therefore, we could stop here our considerations, if there would not still remain an open question. From the combination of equations 1.2 and 1.3 (chapter 2) we would expect the driving force FDriv(x) to be proportional to the intensity I squared, FDriv(x)∝ v(x)∝I2(x). But, since we are driving our system with the same laser trap that we used to create the–for the particles–static sawtooth potential, the intensity, and therefore the driving force, should vary along the circular line. In this section we will try to calculate the the real, i.e. the not constant driving force of our experiment to find out why it seems not to disturb our model imagination of a constant driving force. For this reason, we will extent the equation 1.2 by modulation functionM(ξ) of equation 4.4. From earlier-mentioned velocity measurements without sawtooth potential for the highest and lowest intensity used in the experiments, we know that the intensity varies at most ±12.5%. So we get instead of equation 1.2, now with the mean potential depth hVF ocusi:
FDriv(x)≈6πηav(x)≈ 1
Figure 4.26 shows the contribution of the ’real’ driving force, FDriv(x), to the energy that is transferred to the particle-solvent system (Rx
0 FDriv(x0)dx0). The red line would be the contribution of the constant driving force we used so far in our theoretical model, i.e. FDriv·x. We see that the effect of the ’real’ driving force should be very close to that of a constant force; this is why our earlier model works so well. The reason is that we modulate the laser intensity of the single trap only with a small amplitude in comparison to its mean value. But it is its mean value which does the driving. If we subtract from the expected ’real’ driving the so far supposed constant driving, we get the error we did not consider in the identification of the static light potential (q.v. figure 4.24).
Owing to the substraction, we get the blue dashed line in figure 4.27. This ’error’ in the potential adds to the static sawtooth potential we calculated earlier. Remember, the equations which led to the predictions of static sawtooth potential in figure 4.24(b) are independent of the considerations of this section. At this point, if we add the calculated static sawtooth potential with a chosen depth of 60kBT (black-dashed line) to the ’potential distortion’ of the non-constant driving, we get the blue solid curve in figure 4.27. The blue curve describes also very well the experimental potential (black squares), i.e. the not-constant driving has almost no effect on the asymmetry of the static sawtooth potential. It just changes the potential depth. The red curve in figure 4.27–the same as in figure 4.24(b)–is the 100kBT deep static sawtooth potential, predicted from our earlier comparison between experiment and the constant driving model.
This section gave us a second, more complex model for the sawtooth potential and the driving force in our measurements which better compares to our experimental realization. The depth of the sawtooth potential we need in this model to explain our experimental deduced potential is even closer to our earlier estimated depths (q. v. figure 4.24(b)). But, we also showed that our results can be described sufficiently well, if we rely upon the simpler model of a constant driving force, which was used in the previous sections of this chapter.
In this chapter we have used a colloidal model system to study driven, non-equilibrium dynam-ics. In smooth channels we have found an interesting dynamic steady-state (limit cycle), while in structured channels we have discovered a caterpillar-like motion which helps particle cluster to surmount potential barriers.
5.4. THE LIGHT POTENTIAL AND THE DRIVING FORCE 109
Figure 4.26. Influence of the calculated, ’real’ driving force FDriv(x) of our experiment onto the single particle potential (black line). In comparison, the red line is the effect of the constant driving force in our earlier, simpler model.
110 CHAPTER 5. NON-EQUILIBRIUM SYSTEMS
Figure 4.27. Comparison of an experimental deduced sawtooth potential (black squares) with the predictions of the constant driving force model (red solid line) and of the model of this section, which takes into account the varying driving force in our experiments (blue solid line). The blue lined potential consists of a static sawtooth potential of depth 60KBT (black dashed line) and the ’potential distortion’
owed to the non-constant driving (blue dashed line).
Conclusion
In this thesis we employed colloidal suspensions to study the structure and the dynamics of one-dimensional systems in and out of thermal equilibrium.
In chapter 3, we have presented a new and very effective method of evaluating the pair dis-tribution function in order to gain the particle pair-potential in a system. The derivation of the pair-potential did not imply any theoretical approximations which are necessary for theories in higher-dimensional systems. Approaches like Ornstein-Zernike can lead even to false results, although knowing the complete experimentally determined pair-distribution function. For the method we applied, it is enough to know a small part of the pair-distribution function to predict precisely both the pair-potential and the entire equation of state.
In chapter 4, we investigated the diffusive behavior of particles in a single-file channel. Our data reproduced the known square-root dependence of the mean-square displacement in time.
Yet, we obtained the first experimental results that resolve the complete transition between initial normal diffusion and single-file diffusion. Only on the basis of these results we were able to apply an analytical theory that can make not only qualitative, but also quantitative predictions on the long-time particle mobilities in single-file systems –in comparison to all previous approaches. We revealed with the help of this new theory the dependence of single-file mobilities on short-time density fluctuations, and demonstrated the validity of the theory even for very small particle systems. Due to our experiments, it is now possible to determine the long-time particle mobilities in single-file systems from short-time measurements.
With the presentation of a thermal ratchet in the beginning of chapter 5, we established the ability to structure our channels in closer resemblance to real atomic or molecular structures and to study non-equilibrium systems. In the following sections 5.2 and 5.3, we examined two other systems out of thermal equilibrium. First, we moved particles along a smooth channel with a constant driving force by means of a rotating single optical trap. For three particles, we found an interesting steady-state where the two-particle cluster motion was enhanced due to hydrodynamic interactions. This was the first experimental demonstration of the previously predicted limit cycle.
Then, we produced a corrugated channel by modulating the intensity along our circular light channel to form a sawtooth potential. In these structured channels, instead, we discovered and explained a dynamic effect which diminishes high potential barriers and induces a caterpillar-like motion of paired particles. This effect could be significant to explain recent in vivo experiments on biological motors. Their results demonstrate that the speed of coupled motor proteins is increased compared to the speed of a single motor [KKS+05].
Our experiments confirmed the value of colloidal suspensions as model systems, not only for structural analysis, but especially for the investigation of diffusive or driven particle dynamics.
Outlook
Future experimental work on single-file systems will go in the direction of a more precise imitation of biological and atomic channels. New theoretical results [TM05] propose that the single-file diffusion remains still proportional to the square-root of time √
tfor (sinusoidal) corrugated chan-nels, but with new dependencies of the single-file mobilities on temperature, density, viscosity. In case of a driven single-file system, these new results predict a maximum in the mobility related to the driving force.
We used only smooth, static walls for the single-file diffusion experiments in this work. In atomic systems, the wall particles move due to thermal lattice vibrations. Hence, one can think of creating channels not exclusively by light potentials, but also by walls consisting of real particles.
This gives the opportunity to study the possible influence of thermal lattice vibrations (phonons) on the particle transport [KC01]. Measurements aimed at uncovering the contribution of lattice vibrations are carried out at the moment in our research group. In these experiments, the con-finements for a one-dimensional colloidal system consist of rows of particles, each trapped in a laser focus. A single laser beam is deflected with an acousto-optical deflector (AOD) to form these lines of optical traps. The deflection angle of the laser can be changed much faster with an AOD than with the galvanometric mirrors used in this study, thus permitting the creation of larger, well-defined structures. Moreover, the depth of trap potentials is chosen to still allow the particles fluctuations due to the Brownian motion. The distance between the wall particles is so small that the particles can feel the vibrations of their neighbors and particle density fluctuations can propagate along the walls. Initial experiments in such systems have already provided promising results.
A new research field, which could be studied extremely well with colloidal particles, is the inquiry of the relations between microscopic dynamics and the macroscopic results of equilibrium thermodynamics [Sek98].
Danksagung
Ich danke Herrn Prof. Paul Leiderer daf¨ur, daß ich den ersten Teil meiner Dissertation an seinem Lehrstuhl durchf¨uhren durfte.
Besonders danke ich Herrn Prof. Clemens Bechinger f¨ur die M¨oglichkeit bei ihm zu promovieren, f¨ur hilfreiche Diskussionen und sein Verst¨andnis f¨ur meine k¨unstlerischen Ambitionen. Mit seiner Unterst¨utzung und durch seine Mo-tivierung konnte ich – trotz einiger R¨uckschlage, die wohl jeder Wissenschaftler zu verzeichnen hat – schließlich meine Messungen erfolgreich abschließen.
Dar¨uberhinaus danke ich Roland Roth, Hendrik Hansen-Goos f¨ur die theo-retische Beschreibung der gemessenen Korrelationsfunktionen. Ebenso Holger Stark und Michael Reichert f¨ur ihre Berechnungen und Simulationen zur quan-titativen Beschreibung der getriebenen Systeme.
Mein Dank f¨ur die Verbesserungen hinsichtlich der Abfassung der Dissertation in Englisch gilt vor allem Prof. David Marr, der trotz seiner bevorstehenden R¨uckreise in die USA die Zeit gefunden hat, einen Großteil dieser Arbeit zu korrigieren.
Des Weiteren danke ich allen Kollegen und Mitarbeitern in Konstanz und Stuttgart, die mir bei meinen Experimenten geholfen haben. Insbesondere m¨ochte ich Louis Kukk und Uwe Rau danken, die meine technischen Ideen praktisch umgesetzt haben. Ralf Bubeck danke ich f¨ur die Einf¨uhrung in die Visual C Programmierung. Carmen Schmitt, Valentin Blickle, J¨org Baum-gartl, Stefan Bleil, Denis Kobasevic und Jules Mikhael danke ich f¨ur ihre Hilfsbereitschaft – auch ¨uber das Berufliche hinaus – und f¨ur sch¨one Stun-den in Stuttgart nach der Arbeit.
Meiner Familie, vor allem Gerlint und Simone, verdanke ich moralische und finanzielle Unterst¨utzung, ohne die diese Arbeit nicht m¨oglich gewesen w¨are.
Der gr¨oßte Dank aber gilt meiner Freundin Marilena Santuccio f¨ur ihre best¨andige Hilfe, die sie mir trotz des Zeitverlustes f¨ur ihre eigene Dissertation gew¨ahrt hat, und f¨ur ihre Geduld, da ich ihr wegen meiner Dissertation nur wenig Zeit widmen konnte. Tu dai bellezza a tutti i giorni.
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