Electrophoretic maneuvering of nonuniformly charged particles suspended in linear flows: Impact of the medium viscoelasticity

Electrophoresis in complex (non-Newtonian) fluidic media is becoming increasingly prominent owing to its applications in a wide variety of separation processes. Although most particles that undergo electrophoresis carry nonuniform surface charge, their motion in complex fluidic media has been addressed only recently, while many other separation processes also use externally imposed flows. Despite this, the impact of external flows on the electrophoretic motion of such particles in non-Newtonian fluids still remains largely unexplored. To address this, here we develop a semianalytical framework using a combination of matched asymptotic expansion and regular perturbation to examine the trajectories of nonuniformly charged particles suspended in a viscoelastic medium, subject to imposed shear flow and electric field. We assume the particle's surface charge to be weak but otherwise arbitrary, the electrical double layer to be thin, the suspending medium to obey the Oldroyd-B constitutive relation, and the overall flow to be weakly viscoelastic. Our results reveal that in the presence of imposed flows, any particle carrying a nonuniform surface charge will likely undergo cross-stream migration when the suspending medium is viscoelastic. It is shown that the very nature of a particle's trajectory and the extent of its migration will strongly depend on whether the imposed flow or the electrophoretic propulsion dominates particle motion. This is further supported by a reduced order model derived for Newtonian fluids, using which particles' velocities and their trajectories may be deduced without a detailed knowledge of the flow field. We demonstrate that in the limit of weak viscoelasticity, the general conclusions from this reduced order model may indeed be extended beyond the realm of Newtonian fluids. Our framework may be useful towards improving particle separation relevant to many biochemical applications.

Figure 1

Schematic of a spherical particle of radius $a$ carrying arbitrary surface charge density ${\sigma }^{\prime }(\overline{\theta },\overline{\varphi })$, suspended in a shear flow (${\mathbf{v}}_{\infty }=G{y}^{\prime }{\widehat{\mathbf{e}}}_z$) and subject to an electric field $E_0{\widehat{\mathbf{e}}}_z$. The particle translates with velocity ${\mathcal{U}}^{\prime }$ and rotates with angular velocity ${\mathit{\Omega }}^{\prime }$. The surrounding medium is viscoelastic with viscosity $\mu$, permittivity $\epsilon$, relaxation time ${\lambda }_1^{\prime }$, and retardation time ${\lambda }_2^{\prime }$ and obeys the Oldroyd-B constitutive relation. The laboratory-fixed, central, and body-fixed axes are shown. The central and the body-fixed axes are assumed to coincide at $t=0$.

Figure 2

Three-dimensional phase portraits of the body-fixed unit vector $\widehat{\mathit{k}}\left(t\right)$ for the flow-dominated scenario with $\alpha =0.4<{\alpha }_{\text{cr}}$ (a) and the electrophoresis-dominated scenario with $\alpha =1>{\alpha }_{\text{cr}}$ (b). We choose $\breve{\zeta }(\eta ,\overline{\varphi };t=0)=1-\sqrt{1-{\eta }^2}sin\overline{\varphi }$. The black lines indicate the integral paths of the unit vector $\widehat{\mathit{k}}$; the red dots are the fixed points corresponding to the two cases. All integral paths will lie on a unit sphere centered at the origin.

Figure 3

Plot of the translational velocity components $\left(\text{a}\right){\mathcal{U}}_x,\left(\text{b}\right){\mathcal{U}}_y,\left(\text{c}\right){\mathcal{U}}_z$, and the angular velocity components (d) ${\mathrm{\Omega }}_x$ and (e) ${\mathrm{\Omega }}_y$ for the same $\breve{\zeta }(\overline{\theta },\overline{\varphi };t=0)$ chosen in Eq. (43). The resulting particle trajectory is shown in (f). In all panels, the numerical simulations of Sec. 4b and the reduced order model of Sec. 5a1 have been compared. $\alpha =0.4$.

Figure 4

Particle trajectories (up to $t=30$ sec) for an initial surface charge distribution $\breve{\zeta }(\eta ,\overline{\varphi };t=0)=1-\sqrt{1-{\eta }^2}sin\overline{\varphi }$ are shown for various choices of $\text{De}=0$ (Newtonian), 0.05, 0.1, 0.2. $\alpha =0.4$ ($<{\alpha }_{\text{cr}}=0.667$), and ${\lambda }_2=0$. The light blue lines illustrate the imposed far-field linear flow (not to scale).

Figure 5

Plot of the translational velocity components $\left(\text{a}\right){\mathcal{U}}_x$, ($b$) ${\mathcal{U}}_y$, (c) ${\mathcal{U}}_z$, and the angular velocity components (d) ${\mathrm{\Omega }}_x$, (e) ${\mathrm{\Omega }}_y,$ and (f) ${\mathrm{\Omega }}_z$ of the particle as a function of time for $\text{De}=0,0.05,0.1,0.2$. All other parameters are identical to Fig. 4.

Figure 6

Particle trajectories (up to $t=30$ sec) for an initial surface charge distribution $\breve{\zeta }(\eta ,\overline{\varphi };t=0)=1+\eta +\frac{1}{2}\sqrt{1-{\eta }^2}(sin\overline{\varphi }+cos\overline{\varphi })(1+\eta )$ are shown for various choices of $\text{De}=0$ (Newtonian), 0.05, 0.1, 0.2. Here $\alpha =0.4<{\alpha }_{cr}$ ($=0.544$). Other parameters are the same as in Fig. 4.

Figure 7

Plot of the translational velocity components $\left(\text{a}\right){\mathcal{U}}_x$, (b) ${\mathcal{U}}_y$, (c) ${\mathcal{U}}_z$, and the angular velocity components (d) ${\mathrm{\Omega }}_x$, (e) ${\mathrm{\Omega }}_y$, and (f) ${\mathrm{\Omega }}_z$ of the particle as a function of time for $\text{De}=0,0.05,0.1,0.2$. All other parameters are identical to those in Fig. 6.

Figure 8

Particle trajectories (up to $t=30$ sec) for an initial surface charge distribution same as that of Fig. 4 are shown for various choices of $\alpha =0.4$ ($<{\alpha }_{\text{cr}}$), $\alpha =0.667={\alpha }_{\text{cr}}$, and $\alpha =1$ ($>{\alpha }_{\mathrm{cr}}$). We choose $\text{De}=0.15$, while all other entities are same as that of 4.

Figure 9

Plot of the translational velocity components $\left(\text{a}\right){\mathcal{U}}_x$, (b) ${\mathcal{U}}_y$, (c) ${\mathcal{U}}_z$, and the angular velocity components (d) ${\mathrm{\Omega }}_x$, (e) ${\mathrm{\Omega }}_y$, and (f) ${\mathrm{\Omega }}_z$ of the particle as a function of time for various values of $\alpha =0.4,0.667$, and 1. All other entities are identical to those in Fig. 8.

Figure 10

(a) Initial ($t=0$) $\breve{\zeta }(\eta ,\overline{\varphi })$ distribution of a Janus particle. (b) Variation in the $x$ coordinate of the particle vs $t$ till $t=40$ in Newtonian ($\text{De}=0$) and viscoelastic medium ($\text{De}=0.2$), for $\alpha =0.3<{\alpha }_{\text{cr}}=0.4454$ and $\alpha =0.9>{\alpha }_{\text{cr}}$. (c) ${\mathcal{U}}_x$ vs $t$ and (d) ${\mathrm{\Omega }}_x$ vs $t$ for the motion shown in panel (b). All other parameters are identical to Fig. 4.

Figure 11

Plot of the translational velocity components $\left(\text{a}\right){\mathcal{U}}_x$, (b) ${\mathcal{U}}_y$, (c) ${\mathcal{U}}_z$, and the angular velocity components (d) ${\mathrm{\Omega }}_x$, (e) ${\mathrm{\Omega }}_y$, and (f) ${\mathrm{\Omega }}_z$ of the particle as a function of time for two different time steps: $\mathrm{\Delta }t=0.01$ and $\mathrm{\Delta }t=0.005$. $\text{De}=0.1$. All other entities are identical to Fig. 4.