Agglomeration of oppositely charged particles in nonpolar liquids

We study the aggregation of insulating electrically charged spheres suspended in a nonpolar liquid. Regarding the van der Waals interaction as an irreversible sticking force, we are especially interested in the charge distribution after aggregation. Solving the special case of two oppositely charged particles exactly, it is shown that the surface charges either recombine or form a residual dipole, depending on the initial condition. The theoretical findings are compared to numerical results from Monte Carlo simulations.

Figure 1

Picture of two coated glass beads. The particle diameter is $\approx 6\mu \mathrm{m}$. (Picture taken by K.-E. Wirth, University of Erlangen-Nuremberg.)

Figure 2

(Color online) Schematic experimental setup of the coating process: Large and small particles are suspended in a nonpolar liquid. Charging them oppositely, the small particles are preferentially deposited at the surface of the large particles.

Figure 3

(Color online) Sketch of the coating process. Charges on the surface of particles are depicted by small spheres. Exact recombination of charges during agglomeration may be suppressed by translational or rotational Brownian motion.

Figure 4

(Color online) (a) Simplified model for electrostatically supported coating: A negatively charged pointlike particle is attracted by a positive charge located at the planar surface of an infinitely large spherical particle; (b) idealized setup to study the influence of rotational Brownian motion on charge recombination.

Figure 5

A thermally diffusing particle inserted randomly at finite distance $z=L$ is attracted by the target charge at the origin. Once the particle touches the target plane $z=0$, which represents the large particle to be coated, it sticks to its surface due to van der Waals forces. The single-particle problem can be recast as a quasi-stationary flow of a large number of particles inserted continously along the source plane that are removed whenever they touch the target plane.

Figure 6

Flux lines of the vector field $\vec{j}\left(\vec{r}\right)$ in the $x\text{−}z$ plane for the three-dimensional problem. The separatrix between flux lines terminating at the target charge and those terminating elsewhere at the wall is represented as a bold line. In the two-dimensional case, one obtains a qualitatively similar flow field.

Figure 7

Boundary conditions in the numerical simulation. The picture shows a two-dimensional cut through the three-dimensional simulation setup. The $x$ axis points perpendicular to the shown plane toward the reader.

Figure 8

(Color online) Influx of particles onto the wall within a circular-shaped region around the fixed charge. The straight line represents the analytical solution: $\pi ∕2+\pi R$. In both cases, the constant background is subtracted.

Figure 9

(Color online) Density profile of particle influx for one (a), two (b), three (c), and four (d) charges fixed on the $y$ axis. The bold (black) curves show data from our simulations, the light (red) curves are computed as described in the text.

Figure 10

Relative strength of the $\delta$ peaks for one (1), two (2), three ($3_{\mathit{I}}$ for the inner, $3_{\mathit{II}}$ for the outer peak) and four ($4_{\mathit{I}}$ for the inner, $4_{\mathit{II}}$ for the outer peak) charges.

Figure 11

Agglomeration of two spherical particles: Average dipole strength in units of particle radii $a$ as function of the Bjerrum length $l_B$ in units of $a$. For particle radii bigger than the Bjerrum length, the position of charges during agglomeration is randomized due to Brownian rotation. Hydrodynamic interactions between the particles lead to smaller remaining dipoles.