To avoid the hydrodynamic friction between the particles and the substrate, we needed to elevate the particles about 40µm above the substrate in the experiments of chapter 5. To get the necessary three-dimensional tweezer, we had to make some small changes to the setup of figure 2.4.
So far, we used a low aperture objective above the sample cell to focus the laser to a beam waist of 3 µm close to the lower bottom plate of our particle cell. The focus of this beam was strong enough to confine the particle laterally on a circular channel, but not sufficient to move the particle against the radiation pressure in the vertical direction. The radiation pressure pushed the particles against the lower substrate. In some experiments we changed the optical path to trap the particles from below with a high aperture oil-immersion objective (PL Fluotar M=100× A=1.32 (Leitz-Wetzlar)). This has the advantage that the light pressure gets balanced by gravitation and a strong vertical intensity gradient. Indeed, the particles stay in the focus and can be elevated over the substrate. The focus of our laser beam has a diameter of about 0.6 µm, which rigidly confines the particle radial motion (see figure 3.9, right). The radial potential from figure 3.9 (right) is calculated from the Boltzmann distribution of the radial fluctuations of a particle along the channel from a single sphere measurement.
To improve the force balance of the particles further, we employ silica spheres (Bangs Inc.) with a diameter of 3µm, dispersed in pure ethanol. The silica particles in ethanol display the same electrostatic properties as in water [RSLV+01]. This system favors three-dimensional trapping due to its similar refraction indices (nSpheres = 1.39, nEthanol = 1.37). This seems on the first glance counterintuitive, since the lateral forces should be best for a high refraction index difference. But
2.3. THREE-DIMENSIONAL TRAPPING AND DRIVING 33
lowering the gap between the refraction indices minimizes also the radiation pressure which tends to push the particles upwards [TMBZ98].
To get deep enough into the particle cell, we also work with a new cell design. As bottom substrate, we use silica microscope cover glasses (22×22mm, Pr¨azisionsGlas & Optik GmbH or Menzel GmbH) with a thickness of 100 or 160 µm. On this substrate we put microscope slides, which have 600-800µm deep and 15-18 mm wide spherical coves etched into them (Paul Marienfeld KG). The cove is filled with the ethanol-silica particle suspension. Then, the microscope slides and the cover glasses are sealed with ultraviolet curing (Norland Optical Adhesive 68 ,Norland Products Inc.). Both glass parts of this new cell were plasma-cleaned before assembling.
For the measurement of chapter 5, we operate our circular trap in the regime II of section 2.1. That is, the particle motion exhibits a constant driving force due to the repeated kicks of the passing optical trap. We apply a mirror frequency of 76 Hz with a laser intensity of about 200mW in the focus. Figure 3.9 (left) is an example of the percentage deviation of the single particle velocity along circle circumference for our measurements. The curve is averaged over 50 particle revolutions. The velocity deviates at the most 20% from its average value. According to equations 1.2, and 1.3 this would equal an intensity variation along the circle of ±10%. This intensity variation can be ascribed to a misalignment between the cell, the immersion objective, and the laser beam. Nevertheless, the average particle velocityvP in the measurement of figure 3.9 (left) can still be determined very precisely to 8.46±0.02µm/sec, which is the decisive quantity for most of our measurements in chapter 5. As we will see, the error in the intensity is far smaller than the hydrodynamic effects we study.
The video data for the experiments with the driven particles were taken via a video compressor (ADVC-100, Canopus), which allowed us to follow the dynamics of the particle motion with a frame rate of 25 pictures per second. This frame rate is high enough to follow the average transport of the particles we are interested in. It does not resolve the single kicks of the rotating tweezer on the particle (a particle gets pushed by the tweezer about 3 times between each picture), which gives films of our experiments likewise the impression of a constant driving force (see videos on the enclosed CD).
Chapter 3
Correlations in one-dimensional systems
To study the characteristics of one-dimensional systems in chapter 2, we built a microscopic, experimental model system consisting of a circular laser trap for microscopic, strongly repelling particles in a solvent of low ion concentration (see figure 0.1). In this chapter, we will examine the one-dimensional system’s structural behavior for different particle densities. We will show that an exact analytical theory provides us the possibility of both predicting the pair-distribution function in our system, and, of extracting the pair-interaction potential from our data without the use of theoretical approximation. Besides the interaction potential, this theory will allow us to extract the complete equation of state for our system.
These experiments are of fundamental importance because, up to now, it has been impossible to determine the exact pair-interaction potential in higher-dimensional, experimental systems.
The common Ornstein-Zernike approach (see section 1.2.1) leads to ambiguous or even wrong results due to its approximations, especially for higher particle densities [BBS+02].
3.1 Experimental density profile
Figure 1.2 presents the pair density profile determined from four of our measurements like those of figure 0.1. The line density ρL for the four measurements was ρL = 0.202, 0.183, 0.168, 0.155 [1/µm]. In each of the four experiments we had the same particle number N = 45 in the
Figure 0.1. Top view of 45 electrostatically-stabilized, polystyrene spheres of diameter σ = 2.9µm.
Particle motion is limited to a circular line by a ring-shaped optical trap.
36 CHAPTER 3. CORRELATIONS IN ONE-DIMENSIONAL SYSTEMS
Figure 1.2. Pair density profiles from four measurements along the circle. The mean density, i.e. the line density ρL for the four measurements was ρL = 0.202 [1/µm] (top-left, ), ρL = 0.183 [1/µm]
(bottom-left,◦ ),ρL= 0.168 [1/µm] (top-right, 4),ρL= 0.155 [1/µm] (bottom-right, D).
circular trap, under the same ambient conditions. By changing slightly the circle diameter the density can be decreased and, as shown in figure 1.2, a significant decay in the particle correlations demonstrated. To obtain the pair-interaction potential from our density profiles, we can now use a mathematically exact theory as outlined in the following section.