Solvents

One can match density and refractive index of PMMA particles with a well-tuned mixture of CHB and cis-decaline. Table5.2illustrates that density and refractive index of (bulk) PMMA lies between those of the two solvents. While decaline is a nonpolar solvent with a very low conductivity, CHB (C6H11Br) is a much more complex solvent. CHB is actually brominated cyclohexane that has some tendency to dissociate by producing H⁺and Br⁻ions [87]. They contribute to the unwanted screening of the particle charges. This means that any source enhancing dissociation should be excluded or set to a minimum.

Dissociation can be induced by sonication, ultraviolet light and also by heating, even a longer storage (∼ 10 months) will result in a higher ion density [87]. Therefore, samples were prepared with CHB that was purified (see below) not more than 2 months ago. The cis-decaline was used as received from the manufacturer (≥98%, Merck). Attempts with particularly pure anhydrous decaline (≥99%, Sigma Aldrich) showed no effect on the screening lengths.

solvent refractive density index [g∕cm³]

PMMA (bulk) 1.49 1.18

CHB 1.495 1.333

cis-decaline 1.481 0.895

good mix for RP45 n.a. 1.212

Table 5.2:Refractive index (manufacturer’s data + [87]) and density (PMMA: manufacturer’s data, rest:

measurements with an oscillating U-tube density meter (Anton Paar, DSA 5000 M) at𝑇 = 22^◦C) of the sample ingredients. The good mixture for sample RP45 is CHB/decaline80 %∕20 %by weight.

Purification of CHB

As delivered from the manufacturer, CHB (98%, Sigma-Aldrich) is not pure at all, sometimes one can even observe a yellowish colour of the solvent, which could possible be the result of some reaction in-volving dissociated H⁺and Br⁻ions. One approach is to wash the solvent with CaCl2(removes moisture) and to do a vacuum distillation at about 85^◦C [88]. This is not only time consuming but also does not decrease the ionic strength, since heating enhances the decomposition of CHB [87]. Therefore, within this work the solvent was purified with a method initially described by Pangborn et al. [89] and also mentioned by Leunissen [87]. Similar to water cleaning with activated carbon, a column of activated alumina is used (Al2O3activated, basic, Sigma-Aldrich). Alumina powder has a large surface area for the adsorption of dirt and any polar solvents like water. The alumina part of the column had a height of 20cm and a thickness of∼ 1.5cm. On the bottom of the column glass wool and a cellulose filter (Roth 13 A) with a retention of 5-6µm were used to impede the alumina powder from polluting the filtrated CHB. Before the solvent was poured through the column, the alumina was dried by heating it with a heat gun (∼ 200^◦C) for about 10 minutes, until no water droplets from the vaporization were left. After that it was necessary to wait 20 minutes for the column to cool down. In order to obtain 50ml of purified CHB, one has to wait almost one hour for the CHB to percolate through the column.

Conductivity of the solvents

Measuring the conductivity is a good way to check the purity of the solvents, especially if the goal is to have as few salt ions as possible to get the largest possible screening lengths. Unfortunately as most commercial conductivity meters are designed for water, the minimum conductivity they can measure is not much lower than0.05µS∕cm, which is the value for perfectly deionized water. To obtain at least a hint to the conductivity, a Keithley Model 6430 Source Meter was used, which is able to measure electric currents down to several picoamperes. The apparatus has its own voltage source for the measurement of such low currents. A standard platinum electrode was used with a cell constant of𝐾 = 0.1∕cm (WTW LTA 01). A moderate voltage of𝑈 = 10V was chosen for the measurements. The conductivity is given by

𝜎=𝐾 𝐼

𝑈, with standard units: 𝜎/ [ S cm

]

=𝐾∕[

cm⁻¹] 𝐼∕ [A]

𝑈∕ [V] , (5.3) where 𝐼 is the measured current. The Source Meter only allows the use of continuous current (DC), but after a few seconds a drop of the current was observed, which could be interpreted as the adsorption of ions on the electrode. With a single keypress it was possible to reverse the voltage and the same current values appeared again together with the drop after several seconds. Therefore, the high current values observed immediately after a voltage change were used for the estimation of the conductivity.

Surprisingly, this brute-force method could reproduce Leunissen [87]’s value of1000µS∕cm for CHB as delivered from the manufacturer. They used a Scientifica model 627 conductivity meter which is no

5.1 Sample preparation more available on the market. For purified CHB (see the method above) a very low value of10-20µS∕cm was measured with this simple method, again in agreement with values from [87]. Pure cis-decaline (as delivered) did not show a measurable conductivity, the current fluctuated around50pA, with or without the electrode being in contact with the decaline, indicating𝜎 <0.5pS∕cm. The value for an80 %∕20 % mixture of CHB/decaline was comparable to purified CHB alone. This means that decaline should have no influence on the charges and their screening.

Sample preparation and density matching

In order to neutralize gravity the PMMA particles were put into a mixture of CHB and decaline with a mixing ratio of about80 %∕20 %w/w. This was done either by weighing the solvents and adding the mixture to the dried particles, or by adding CHB to particles that were dispersed in decaline. Just after putting them in the mixture they usually sediment during one night. But some days later, after the particles have swelled in the new solvent, the sedimentation is either gone, reverted or at least a lot slower.

After a waiting time of several days the density matching was refined using a centrifuge. Controlling the temperature during centrifugation is very important since the density of the solvents decreases faster with temperature than the density of the particles: Particles that are density matched at22^◦C will sediment at 30^◦C. The temperature was held constant at22^◦C, which is the common temperature in the laboratory.

Starting with500𝑔 the sample was centrifuged for10min and it was determined whether the particles went down to the bottom or rose up to the surface. In the first case more CHB was added to the sample, in the latter more decaline. This was repeated during several days. Note that a waiting time of at least one day is required for particles to adapt to the new solvent mixture by swelling or shrinking. If no sedimentation or buoyancy was observed, the acceleration and the centrifugation time were increased.

In the end a sample was stable under centrifugation at4000𝑔 for60min. In the last few steps of the process the addition of a very small drop (less than 1µl) to a vessel of3ml was often enough to change the direction of the particle movement.

The refinement via centrifugation is only required when particles from one synthesis process are density matched for the first time. For example, for the particle batch RP45 (rhodamine particles) the density of the good mixture was determined to be1.212g∕cm³. Later the density matching mixture could be reproduced by an iterative procedure of density measurements with the density meter (cf. Table5.2) and addition of either CHB or decaline. This made it possible to generate density matched samples without the need to centrifuge them. In order to determine the particle volume fractionΦit is better not to rely on weight or volume measurements of the suspension’s ingredients, due to the swelling of the particles in the solvent. Therefore, the particle detection software was used to obtain the number density𝑛= 𝑁∕𝑉 of the particles. With the knowledge of the particle radius𝑅, the volume fraction is given byΦ =𝑛⋅⁴

3𝜋𝑅³. Binary samples were either mixed from already density matched monodisperse samples or the density matching was done exclusively for the binary mixture, starting from the dry particle powders.

Density difference of small and big particles

Unfortunately it is not possible to use one perfect density matching mixture for all PMMA particles. It turned out that smaller particles require a somewhat higher density of the solvent than bigger particles. In mixtures it was even observed that big particles go up while small particles go down upon centrifugation.

An explanation is that smaller particles are indeed more dense. In the synthesis the particles are originally bubbles of monomer floating in a solvent. Since the pressure in bubbles is higher the smaller they are, the monomer density in small bubbles is higher. Therefore the resulting density of small particles is higher compared to bigger particles. However, the fact that one can only separate the particles at very high accelerations of about4000𝑔means that the density differenceΔ𝜌is very small. A spherical particle of

radius𝑅experiences the buoyant force𝐹 = Δ𝜌⁴

3𝜋𝑅³𝑔_𝑐when centrifuged at the acceleration𝑔_𝑐. From the balance of friction and driving force one obtains the equality:

Δ𝜌⁴

3𝜋𝑅³𝑔_𝑐 = 6𝜋𝜂𝑅𝑣 ⇒ Δ𝜌= 9 2

𝜂 𝑅²

𝑣 𝑔_𝑐 = 9

2

2.2mPa⋅s (1µm)²

1cm∕h

4000𝑔 ≈ 7 × 10⁻⁴ g

cm³ (5.4)

The calculation above uses an approximate viscosity of 𝜂 = 2.2mPa⋅s for the solvent mixture and a representative radius of1µm for the particles. So the density difference between the solvent and a typical particle that sediments with 𝑣 = 1cm∕h at 4000𝑔 is less than 0.1 %. Since big and small particles could only be separated at such high accelerations and only after about one hour of centrifuging, one can conclude that their relative density difference is at most in the range of0.1 %.