Predicting tensorial electrophoretic effects in asymmetric colloids

We formulate a numerical method for predicting the tensorial linear response of a rigid, asymmetrically charged body to an applied electric field. This prediction requires calculating the response of the fluid to the Stokes drag forces on the moving body and on the countercharges near its surface. To determine the fluid's motion, we represent both the body and the countercharges using many point sources of drag known as Stokeslets. Finding the correct flow field amounts to finding the set of drag forces on the Stokeslets that is consistent with the relative velocities experienced by each Stokeslet. The method rigorously satisfies the condition that the object moves with no transfer of momentum to the fluid. We demonstrate that a sphere represented by 1999 well-separated Stokeslets on its surface produces flow and drag force like a solid sphere to 1% accuracy. We show that a uniformly charged sphere with 3998 body and countercharge Stokeslets obeys the Smoluchowski prediction [F. Morrison, J. Colloid Interface Sci. 34, 210 (1970)] for electrophoretic mobility when the countercharges lie close to the sphere. Spheres with dipolar and quadrupolar charge distributions rotate and translate as predicted analytically to 4% accuracy or better. We describe how the method can treat general asymmetric shapes and charge distributions. This method offers promise as a way to characterize and manipulate asymmetrically charged colloid-scale objects from biology (e.g., viruses) and technology (e.g., self-assembled clusters).

Figure 1

Asymmetrical, self-assembled $\mu \text{m}$-scale objects from nature and technology. (a) Spores of Tilletsiopsis fungi after Fig. IX of Ref. [22]. Long dimension is 8–10 $\mu \text{m}$. (b) Spores of Sporobolomyces odorus fungus after Fig. III of Ref. [22]. Long dimension is about 7 $\mu \text{m}$. (c) Human red blood cells, after [23]. Normal disk-shaped cells in center have a diameter of about 7 $\mu \text{m}$. Elongated cells are deformed by hemoglobin aggregates. (d) Most-probable eight-sphere colloidal cluster assembled from 1-$\mu \text{m}$ polystyrene spheres by depletion interaction, after Ref. [2], Fig. 2. (e) Collapsed colloidal shells after Ref. [3]. Scale bar is 3 $\mu \text{m}$.

Figure 2

(a) Depolarization charge, shown as pluses and minuses, on an insulating body in a conducting medium in the presence of an applied field ${\mathit{E}}_0$. The charge creates a field ${\mathit{E}}_{dp}$, which cancels the normal component of ${\mathit{E}}_0$ at the surface, as no current (and therefore no field) can penetrate the body. The resultant field ${\mathit{E}}_s$ is the total field at the object's surface. (b) Physical picture of electrophoresis. Counterions in the surrounding fluid form a screening cloud (upper blue gradient band) of thickness ${\lambda }_D$, which act to screen the surface charge (lower solid red band) of the object. (c) Stokeslet picture of electrophoresis. Both the body surface charges (lower layer of red dots) and screening charges (upper layer of blue dots) are represented as point sources of drag called Stokeslets, and exert respective drag forces ${\mathit{f}}^{\alpha }$ and ${\mathit{g}}^{\gamma }$ on the fluid. The two layers of Stokeslets are separated by a length ${\lambda }_D$.

Figure 3

Comparison of the velocity field from a sedimenting 1999-Stokeslet sphere (upper data set in red) and a uniformly charged electrophoretic sphere with ${\lambda }_D=3\%$ of the sphere radius (lower data set in blue), using the methods of Section 5 (sphere and Stokeslets are shown to scale). Vertical velocity of the fluid $u_z$, normalized by the calculated body speed $v_{\mathrm{obs}}$, is plotted versus horizontal distance $x$ from the center. For the sedimenting case, the fluid within the sphere is moving with the object to within a $1\%$ tolerance (horizontal red line) for $x<1$. For $x>1$ the velocity falls off as $1/r$ as shown in the log-log inset plot, and agrees well with theoretical predictions [32] (dashed line). The speed of the sphere implies a Stokes drag coefficient about $1\%$ smaller than the theoretical value for a sphere (upper red line). For electrophoresis, the fluid near the center moves somewhat faster than the sphere. External flow varies as $k/r^3$ as expected for mass dipole flow. The expected value of $k$ is 1/2 [17], indicated by the dashed line. The external velocity field matches that expected for a solid sphere moving at the observed velocity to within $1\%$. This observed velocity in turn agrees with the predicted velocity (lower blue line) to $11\%$, cf. Table 3.

Figure 4

Effect of number of Stokeslets on sedimentation flow, plotted as in Fig. 3. Inset indicates the number of Stokeslets for each data set. The lower data set in blue corresponds to 499 Stokeslets, while the upper data set in green corresponds to 1999. When more Stokeslets are used to represent a rigid body, better agreement with theory is obtained, as shown in Table 1. Furthermore, the transition from the interior to the exterior is sharper for 1999 Stokeslets than for 499, as the sphere blocks the flow better with more Stokeslets. The solid black line corresponds to the flow profile of a perfect hard sphere.

Figure 5

Effect of number of Stokeslets on the electrophoretic flow, plotted as in Fig. 4 (${\lambda }_D=6\%$ of $R$ in both cases). The data set in blue with a smooth transition at $x/R\approx 1$ corresponds to 499 Stokeslets, while the remaining curve in green corresponds to 1999. When more Stokeslets are used to represent a rigid body, better agreement with theory is obtained, as shown in Table 2. Furthermore, the transition from the interior to the exterior is sharper for 1999 Stokeslets than for 499, as the sphere blocks the flow better with more Stokeslets. The solid black line corresponds to the flow profile of a perfect hard sphere with zero electrostatic screening length.

Figure 6

Effect of Stokeslet size $a$ on the electrophoretic flow, plotted as in Fig. 4, for $N=1999$ and ${\lambda }_D=0.03$. The $a=0.0001$ data set is the light (green) data of smallest magnitude. The $a=0.001$ and $0.015$ sets are the successively higher downward sloping data on the left side of the plot. Finally, the $a=0.03$ and $a=0.1$ sets marked $b<2a$ are the upward sloping data. The more negative of these (black points) is the $a=0.1$ set. For very small $a$, the fluid inside the sphere moves much slower than the sphere itself. As $a$ increases, the interior flow matches better with the sphere velocity, so the sphere becomes more hydrodynamically opaque. However, when $a$ is greater than half the nearest-neighbor distance between Stokeslets $b$ or ${\lambda }_D$, Stokeslets begin to overlap, and the flow becomes unphysical.

Figure 7

Effect of electrostatic screening length ${\lambda }_D$ on the electrophoretic flow, plotted as in Fig. 4, for $N=1999$ and $a=0.015$. The top and bottom data sets on the left correspond to ${\lambda }_D=0.005$ and $0.01$, respectively. The (black) points with the smooth linear transition at $x/R\approx 1$ correspond to ${\lambda }_D=0.1$, while the remaining points correspond to ${\lambda }_D=0.03$. When $a<{\lambda }_D\ll R$, the interior flow matches well with the sphere velocity. However, when ${\lambda }_D<a$, the object and screening charge Stokeslet layers begin to overlap, and the flow becomes unphysical.

Figure 8

Streamlines for the quadrupolar sphere of Table 3 tilted ${45}^{\circ }$ to the right under an applied field in the $z$ direction. The sphere and its surface charge are shown at scale. The streamlines indicate only the direction of the flow, not its magnitude.