The maximum value of κmagagg that can be measured with an OMC experiment depends on the actuation time.
When the particle aggregation is such high that all magnetic dimers become chemical dimers, the ratio in equation 3.3 becomes one. In this case, the maximum rate that can be measured with an actuation time of 20 seconds is κmagagg = 0.1 s−1. The lowest rate that can be measured, due to the noise in the signal, is κmagagg = 0.005 s−1. This gives a dynamic range for measuring the rate of approximately a factor 20.
3.4 Ideal particles
In the actuation phase of Fig. 3.4, the Fourier amplitude increases linearly in time, which is interpreted as a linear increase of the number of dimers in the solution. This is based on several assumptions. For example, the scattering cross-section at every single dimer is equal, the dimer formation rate is constant, and only dimers contribute to the scattering signal.
The light scattering at a dimer depends on the size and the effective refractive index of the particles. For an equal scattering cross-section of dimers, all dimers should contain particles of the same size and effective refractive index. Clusters made of three particles (triplets) have a different scattering cross-section, compared to dimers. So ideally, the formation of triplets (or larger clusters) should be prevented during the actuation phase, this can be realized by using only short actuation phases (∼ 20s)[16]. The last assumption, that is related to scattering, is that all the dimers in the solution have the same orientation. This can be realized when every dimer has the same magnetic and viscous torque. So the size and magnetic properties of each dimer should be the same and the magnetic field strength should be homogeneous. In summary, all particles in the solution should be monodisperse.
Not only the scattering at each dimer should be the same to get a linear increase of the Fourier amplitude with concentration, but also the dimer formation rate should be constant. The number of dimers that are formed per unit of time depends on the monomer concentration [m] and the magnetically induced encounter rate κmagenc , as described by equation 3.4. This equation only holds in the limit that the dimer concentration is low, ([Nmag] [m]), where the formation rate of a triplet is negligible.
∂[Nmag]
∂t = κmagenc [m]2, (3.4)
Note that there is no loss term in the equation, because magnetic dimers will not break up and it is assumed that no triplets are formed. The monomer concentration is decreasing in time and depends on the number of dimers as follows
[m] = [m0] − 2[Nmag], (3.5)
where [m0]is the initial monomer concentration.
Filling in equation 3.5 in equation 3.4 and solving the obtained differential equation results in [Nmag] = κmagenc [m0]2t
1 + 2κmagenc [m0]t. (3.6)
The number of magnetic dimers according to equation 3.6 is plotted versus time t in Fig. 3.5a with m0 = 1 pM and κmagenc = 6 × 10−9M−1s−1. At low values for t (t <20 s) the number of dimers increases approximately linearly in time, the error with a linear increase is less than 10 % below the 20 seconds. At higher values the dimer formation rate decreases, due to depletion of the monomers. Fig. 3.5b shows the results of a real measurement of magnetic particles. The|A4f| peak is plotted against the actuation time. This curve shows the same depletion effect as predicted in Fig. 3.5a. In order to stay in the linear regime of the actuation (the marked area in the graphs), the actuation phase is limited to 20 seconds.
In summary, the OMCE is most accurate when the actuation phase is shorter than the typical time at which depletion occurs, and a linear signal can be realized when the particles are monodisperse in size, in
refract-3.4. IDEAL PARTICLES CHAPTER 3. THE OPTOMAGNETIC CLUSTER EXPERIMENT
(a) (b)
Fig. 3.5: The depletion effect. (a) Depletion effect as described by equation 3.15, with κenc = 6 × 109s−1 and p0 = 2 × 10−12M. (b) Measured depletion effect of magnetic particles at an initial particle concentration of 2 × 10−12M, a magnetic field strength of 4 mT and a rotation frequency of 5 Hz. The marked area is the linear regime of the actuation. The actuation should not be too long to stay in this regime.
ive index, and in magnetic properties. In the next chapter the most ideal particle is selected for the OMC experiments of this project.
.
Chapter 4
Particle selection for the OMC experiment
In this chapter, particles that can meet the requirements for OMC experiments are selected. In previous work from Ranzoni et al.[32], and Romijn et al.[10] the Ademtech Masterbeads were used. A disadvantage of the Ademtech particles is their large size dispersion (CV = 25 %). Two alternatives for the Ademtech particles are the polystyrene superparamagnetic microparticles and the silica superparamagnetic microparticles, from the manufacturer MicroParticles GmbH. The manufacturer of these particles claims a smaller size distribution (CV
< 5 %). However, apart from the size dispersion also the magnetic and chemical properties determine how suited the particles are for the OMC experiment. In this chapter the Ademtech Masterbeads, the silica particle from microparticles GmbH, and the polystyrene particles from microparticle GmbH are shortly called the Ademtech, Silica, and Polystyrene particles respectively. At the end of this chapter, the most suited particle type, for use in the OMC experiment, is selected.
4.1 Non-magnetic properties
The three different particle types considered here have a mean diameter of about 500 nm. A SEM image of the three different particle types is shown in Fig. 4.1. Also the diameter, the density, and the zeta potential of the particles are given in this figure. The three particle types are composed of iron-oxide (a mixture of magnetite: Fe3O4, and hematite: Fe2O3) grains that are randomly dispersed in a non-magnetic matrix. The
(a) (b) (c)
Fig. 4.1: SEM images of Ademtech particles (a), polystyrene particles (b) and silica particles (c). The white bar corresponds to 500 nm and the scale is the same on all three images.