Theoretical predictions

Stimulated by the above-presented results, Holger Stark and Michael Reichert from the University of Konstanz developed a theoretical prediction for our experiments. As a first step, Stark solved the Smoluchowski equation (see eq. 2.18 of chapter 4) for a single, driven particle in a sawtooth potential. He described the driving force and the sawtooth potential by the three constant forces k, k_s1, k_s2, as shown in figure 3.15. Stark found an average velocity of the particle hvi normalized by the particle velocity without the sawtooth potential v0:

hvi v0

=

1 + q²

(1−q)[1−(1−q)δ] ×

δ− 1 C

1−e^{−(1−q)δC} (1−q)[1−(1−q)δ]

⁻¹

(3.3)

v₀ = v(q= 0) = k

γ velocity without sawtooth potential γ = friction coefficient of particle

q = k_s2

k measure for amplitude of sawtooth δ = L2

L measure for asymmetry of sawtooth C = k L

k_BT dissipated energy per sawtooth in units of kBT

Since the potential depth of the sawtooth in the experiment is not known, we have to introduce a fitting parameter Pf it to compare his theoretical prediction with our experimental data. The breaking force k_s2 of the sawtooth is experimentally adjusted by the laser intensity variation

5.3. ACCELERATED PARTICLE CLUSTER MOTION IN ASYMMETRIC

POTENTIAL 93

∆I from top to bottom of a sawtooth. According to Ashkin [Ash70, AD87, Ash92], the force on a particle due to this intensity variation should depend linearly on the intensity change, i.e.

k_s2 =P_{f it}∆I. Therefore, the experimental amplitudes ∆Iwe used as x-coordinates in figures 3.13, 3.14 are correlated to the normalized sawtooth potential depth q of equation 3.3 by q= ^{Pf it}_k^·∆I.

In figure 3.16 we compare equation 3.3 with our one-particle measurements for different saw-tooth potential depths. P_{f it} is the only fitting parameter. Although, for δ = ^L_L² we had to make an assumption, because the real sawtooth in the experiment will, related to the final size of the laser focus, deviate from the theoretical sawtooth (δ ≈ 0). Since our focus is very small (about 0.8µm), the deviation will be small (q.v. section 4.26). δ≈0.1 should be a good approximation.

All the other parameters (L, v_o, k, C) are known from the experiment.

C is the energy dissipated on the length L relative to the thermal energy k_BT. It is linked to the conventional Peclet number P e = k a/kBT = aC/L (see section 1.2.2). Since C 1 in our case, one might naively expect a purely deterministic motion. In general, however, the type of motion depends on the value q = k_S2/k which serves as a measure for the amplitude of the sawtooth potential. For q < 1 the motion is indeed purely deterministic and determined by the first term in equation (3.3). At q = 1, it becomes stochastic since the particle experiences no force (k+k_S2 = 0) in the second segment of the potential and therefore just diffuses around until it enters the drift motion on the first segment. At q > 1 it even experiences a potential barrier which gives rise to a stick-slip motion.

We find in figure 3.16 an astonishingly good agreement between theory and experiment. This emphasizes that the simple model, which led to equation 3.3, is a valid description of our exper-imental situation. This is surprising, because we will have, as already mentioned, a smeared out potential (q.v. section 2.3) in our experiments, and, we will not have a completely constant driving force k, since we drive the particles with the same laser beam that is modulated to construct the sawtooth potential. In section 4.26 we will calculate this driving force and measure the sawtooth potential in our experiment. As we will see there, it is indeed a good approximation to consider the driving force constant along the circle in our experiments. From the sawtooth potential based on our data, we can extract the functional form of a realistic potential for our experiments. Com-parison of computer simulations for the ’perfect’ sawtooth potential used in equation 3.3 and for the experimental gained potential are also in qualitatively good agreement (see figure 3.17).

Stark also calculated numerically, in the same model, the solution of the Smoluchowski equation for the 2-particle cluster on the circle. The hydrodynamic effects were taken into account on a Rotne-Prager level (see eq.2.26 in chapter 4). The result is shown in figure 3.17 (blue-dotted and blue-broken line). The different starting points for these two curves at q = 0 are due to the different initial distances of the two particles (3 and 4 particle radii center-to-center distance respectively). Since Stark did not consider any Brownian motion, the motion of the particles in his model is deterministic, i.e. the initial equilibrium distance does not change over time without the sawtooth potential (q = 0). For smaller initial particle distances, the hydrodynamic draft and, therefore, the 2-particle cluster velocity increases. For q > 1 the mean velocity is governed by the potential wells of the sawteeth; the motion gets independent of the initial conditions, i.e.

both curves merge into one in figure 3.17. The theoretical predictions are in the range of the experimental data points.

Michael Reichert produced additional computer simulations which take also into account the Brownian motion (Temperature T = 300K), hydrodynamic effects (Rotne-Prager), and the elec-trostatic Yukawa-potential (Debye-length=300nm, 8000 charges per particle) between the parti-cles. The simulations were done for the ’perfect’ sawtooth potential (δ = L2/L= 0.1) of Starks’

analytical theory and for the fitted potential from our measurements (for details see section 4.26).

The results are shown as dark grey curves with open symbols in figure 3.17. The simulations for the sharp angle potential of figure 3.15 and the experimental potentials, like that of figure 4.25, are equal up to q = 1. For q > 1, the 2-particle cluster can still move in the smeared-out

ex-94 CHAPTER 5. NON-EQUILIBRIUM SYSTEMS

Figure 3.16. Comparison between the experimental mean velocity of a single particle (black squares) for different potential depths and the equation 3.3 (blue line). Parameters used: v0 = 7.24µm/sec, k = (γ ·v0) = 245f N, δ = (L2/L) = 0.108, L2 = 1µm, L = 10.3µm, C = (k·L/k_BT) = 610.5. ks2 was fitted byP_{f it}·∆I. ∆I is here the intensity variation in the sawtooth potential, according to the function generator reading of the sawtooth potential depth in mV (see figure 3.13).

perimental potentials, because it is shallower, while its motion in the ’perfect’ sawtooth potential would already have been stopped. This explains why the experimentally determined 2-particle cluster velocities remain still high for q >1.1.

5.3. ACCELERATED PARTICLE CLUSTER MOTION IN ASYMMETRIC

POTENTIAL 95

Figure 3.17. Experimental data for single particle (black squares, green triangles) and 2-particle cluster (red circles) velocities of figure 3.13. Blue curves: Theoretical predictions of single particle (solid line) and 2-particle cluster (dotted and broken line) motion by H. Stark. The dotted and broken line belong to different initial 2-particle distances (3 and 4 radii). Dark grey curves: Computer simulations of [LRSB05]. The open squares are simulation of 2-particle cluster in the sawtooth potential of figure 3.15.

The open circles are results for the 2-particle cluster in the experimentally determined potential of our measurements (for details see section 4.26). The triangles simulate the motion of a single particle in the experimental potential.

96 CHAPTER 5. NON-EQUILIBRIUM SYSTEMS