Electrophoretic Retardation of Colloidal Particles in Nonpolar Liquids

We have measured the electrophoretic mobility of single, optically trapped colloidal particles, while gradually depleting the co-ions and counterions in the liquid around the particle by applying a dc voltage. This is achieved in a nonpolar liquid, where charged reverse micelles act as co-ions and counterions. By increasing the dc voltage, the mobility first increases when the concentrations of co-ions and counterions near the particle start to decrease. At sufficiently high dc voltage (around 2 V), the mobility reaches a saturation value when the co-ions and counterions are fully separated. The increase in mobility is larger when the equilibrium ionic strength is higher. The dependence of the experimental data on the equilibrium ionic strength and on the applied voltage is in good agreement with the standard theory of electrophoretic retardation, assuming that the bare particle charge remains constant. This method is useful for studying the electrophoretic retardation effect and charging mechanisms for nonpolar colloids, and it sheds light on previously unexplained particle acceleration in electronic ink devices.

Popular Summary

Electrophoresis, the motion of charged particles in solution driven by an applied electric field, was discovered in 1807 and has, by now, been fairly well understood and broadly applied, as in DNA electrophoresis and electronic ink devices. It may seem that there is not much room left for new fundamental experiments in this rather mature field. In this paper, we present just such an experiment with optically trapped single latex spheres of micron size in a nonpolar solvent not typical in the traditional context of electrophoresis.

When a charged particle, such as the latex sphere we use, is dispersed in a solution, its charge attracts from the solution a cloud of ions of the opposite charge (what the experts call a “diffuse double layer”). When driven to move by an applied electric field, its motion is a result of three forces: the force coming from the electric field, a frictional force from the solution fluid, and a so-called electrophoretic retardation force that comes from the ion cloud that is driven to move by the same field, but in the opposite direction. The two centuries of development in electrophoresis notwithstanding, direct experimental measurements of the retardation force are very scarce. Measurements of this force through a continuous, controlled depletion of the ion cloud have not been done before. Our work fills this fundamental gap.

The central new idea underlying our experiment is that of stripping off the ion cloud in a continuous fashion to the limit of a “naked” charged latex particle by adding charged micelles formed by surfactant molecules to the solution. These micelles, which are nanometers in size, act as the counterions that form the ion cloud. Depleting them from the charged latex particles requires only the application of a very small electric field, because the charged micelles are not replaced as quickly by spontaneous generation as the atomic or molecular counterions present in traditional electrophoresis experiments. By optically trapping the latex particle and measuring its excursion in the trap at the application of an additional small ac field—an already established method—as the ion cloud is gradually reduced and finally fully stripped, we can then determine the electrophoretic retardation force and its dependence on the ion cloud.

While our experimental measurements do not change the theory of electrophoresis, they make an important addition to our fundamental understanding of electrophoresis. Already, they explain the previously unexplained particle acceleration seen in electronic ink devices. And our method should become useful for studying the electrophoretic retardation effect in colloidal systems.

Figure 1

(a) Schematic representation of a colloidal particle optically trapped at the midplane between two electrodes. The electrophoretic mobility is measured by applying an ac voltage $V_{\mathrm{ac}}$. In (b) the positively and negatively charged reverse micelles are completely or partially separated depending on the magnitude of the additional dc voltage $V_{\mathrm{dc}}$.

Figure 2

Surface plot of the normalized radial intensity profile of a spherical particle as a function of the particle position relative to the bottom substrate. The focal depth of the microscope is fixed at $z=28\mu \mathrm{m}$. By generating a $z$-stack fingerprint of each particle and using a least-squares-fitting procedure, the particle position can be determined with 20-nm precision over a range of tens of micrometers in any direction without needing to readjust the focus.

Figure 3

Electrophoretic mobility values of a single particle with radius $3.38\mu \mathrm{m}$ in dodecane with surfactant concentration ${\varphi }_m=0.001$ for different applied ac voltages $V_{\mathrm{ac}}$, normalized to the value ${\mu }_0$ at $V_{\mathrm{ac}}=5\mathrm{Vpp}$. The sinusoidal voltage $V_{\mathrm{ac}}$ at frequency $f=21\mathrm{Hz}$ is applied across the electrodes separated by $50\mu \mathrm{m}$ in the absence of a dc voltage. Below $V_{\mathrm{ac}}=10\mathrm{Vpp}$, the mobility is in the linear regime, while for voltages above 20 Vpp, the Stotz-Wien effect and other nonlinear effects start to play a role.

Figure 4

Measurement of the $z$ position of a particle with radius $3.38\mu \mathrm{m}$ in a micellar-ionic electrolyte containing surfactant OLOA 11000 with mass fraction ${\varphi }_m=0.0004$ in dodecane, sampled at 50 Hz. The particle is optically trapped in the $z$ direction with trapping constant $k_z=3\mathrm{pN}/\mu \mathrm{m}$, while a sinusoidal voltage with $f=21\mathrm{Hz}$ and $V=10\mathrm{Vpp}$ superposed with a stepwise increasing dc voltage $V_{\mathrm{dc}}$ is applied. For increasing dc voltage, the oscillation amplitude increases, and the average position increases as the shifting balance between gravitational, optical, and electrical forces pushes the particle upwards. The inset illustrates the fitting of a sinus to the data.

Figure 5

Mobility measurement of four particles of $3.38\mu \mathrm{m}$ radius optically trapped in the midplane between the electrodes, for different surfactant concentrations ${\varphi }_m$ and different applied voltages $V_{\mathrm{dc}}$. In the absence of surfactant (${\varphi }_m=0$), the mobility is independent of the increasing dc voltage. When the solution contains surfactant, the mobility increases for increasing values of $V_{\mathrm{dc}}$, because the micellar ions or charged reverse micelles are gradually separated, thereby reducing the electrophoretic retardation. The saturation value is reached for values of $V_{\mathrm{dc}}$ around 2 V.

Figure 6

The equilibrium mobility ${\mu }_0$ is the mobility measured in absence of a dc voltage. Without surfactant (${\varphi }_m=0$) the mobility is on average very small, but some particles (as in this case) get a larger charge after contact with a charged electrode. When surfactant is present, the mobility is about $-210\mu \mathrm{m}{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$ and is practically independent of the surfactant concentration.

Figure 7

Measurements (•) and simulations (▾) of the normalized mobility of $3.38\mathrm{\text{−}}\mu \mathrm{m}$ radius particles optically trapped in the midplane between the electrodes, as a function of the surfactant concentration ${\varphi }_m$ and of the applied dc voltage $V_{\mathrm{dc}}$. For increasing values of $V_{\mathrm{dc}}$, the micellar ions are increasingly separated, which reduces the electrophoretic retardation and increases the mobility. The saturation value is reached for values of $V_{\mathrm{dc}}$ around 2 V. The simulations are based on the standard theory of electrophoresis and on calculations of the local ionic strength, using the best-fitted relation $\overline{n}=2000\times {\varphi }_m\mu {\mathrm{m}}^{-3}$ for the equilibrium concentration of micellar ions. Similar to the experiments, the simulations are carried out with a 4-s delay between the dc voltage steps.

Figure 8

Measurement (•) of the saturated mobility ${\mu }_{\mathrm{sat}}$ at $V_{\mathrm{dc}}=2\mathrm{V}$ normalized to the equilibrium mobility ${\mu }_0$ at $V_{\mathrm{dc}}=0\mathrm{V}$, as a function of the surfactant concentration ${\varphi }_m$. The red line shows the expectancy from the standard theory, ${\mu }_{\mathrm{sat}}/{\mu }_0=[1+\kappa (\overline{n})R]/f[\kappa (\overline{n})R]$, using the relations $\overline{n}=2000\times {\varphi }_m\mu {\mathrm{m}}^{-3}$ and $R=3.38\mu \mathrm{m}$. For the green line, the retardation force is reduced with a correction factor $\xi =0.9$ while using the concentrations $\overline{n}=5300\times {\varphi }_m\mu {\mathrm{m}}^{-3}$ obtained from current measurements and $R=3.38\mu \mathrm{m}$.

Figure 9

Simulation of the concentration distribution of positively charged micelles (similar for negatively charged micelles) for $\overline{n}=1\mu {\mathrm{m}}^{-3}$ while the voltage $V_{\mathrm{dc}}$ is increased stepwise by 0.2 V every 4 s. The concentration profiles are shown at the end of each 4-s time interval. The electrodes are situated at $z=0$ and $50\mu \mathrm{m}$. For voltages below 1 V, the concentration profiles are not in the steady state. At voltages below 0.2 V, the bulk concentration remains approximately the same as $\overline{n}$. For voltages around 0.8 V, the concentrations of positively and negatively charged micelles in the middle become much lower. Above 1 V, the micellar ions are fully separated. In the inset, the concentration $n_{\mathrm{mid}}$ of positively and negatively charged micelles in the midplane between the electrodes (at $z=25\mu \mathrm{m}$) is shown as a function of $V_{\mathrm{dc}}$.