Importance of core electrostatic properties on the electrophoresis of a soft particle

The impact of the volumetric charged density of the dielectric rigid core on the electrophoresis of a soft particle is analyzed numerically. The volume charge density of the inner core of a soft particle can arise for a dendrimer structure or bacteriophage MS2. We consider the electrokinetic model based on the conservation principles, thus no conditions for Debye length or applied electric field is imposed. The fluid flow equations are coupled with the ion transport equations and the equation for the electric field. The occurrence of the induced nonuniform surface charge density on the outer surface of the inner core leads to a situation different from the existing analysis of a soft particle electrophoresis. The impact of this induced surface charge density together with the double-layer polarization and relaxation due to ion convection and electromigration is analyzed. The dielectric permittivity and the charge density of the core have a significant impact on the particle electrophoresis when the Debye length is in the order of the particle size. We find that by varying the ionic concentration of the electrolyte, the particle can exhibit reversal in its electrophoretic velocity. The role of the polymer layer softness parameter is addressed in the present analysis.

Figure 1

Schematic description of the soft particle in electrophoresis under an applied electric field.

Figure 2

Variation of electrophoretic velocity at $\ell =3$ nm, $a=10.3$ nm, $b=13.6$ nm, $E_0={10}^4{\mathrm{Vm}}^{-1}$, with $\kappa b$ when (a) ${\rho }_c=5\times {10}^6{\mathrm{C}/\mathrm{m}}^3$ for different ${\rho }_f(=-5\times {10}^6,0,5\times {10}^6{\mathrm{C}/\mathrm{m}}^3$); (b) ${\rho }_f=5\times {10}^6{\mathrm{C}/\mathrm{m}}^3$ for different ${\rho }_c(=-5\times {10}^6,0,5\times {10}^6{\mathrm{C}/\mathrm{m}}^3$). Dashed lines are results of Gopmandal et al. [10]; (c) Potential distribution inside and around the particle when ${\rho }_f=5\times {10}^6{\mathrm{C}/\mathrm{m}}^3$ and ${\rho }_c=5\times {10}^6{\mathrm{C}/\mathrm{m}}^3, \kappa b=1$ and ${\varepsilon }_r=1$. Dashed thin lines are results of Phan et al. [9]; solid and dashed thick lines are the surface of the solid core and soft particle, respectively.

Figure 3

(a) Potential distribution on the surface of solid core $(r=\gamma ,\theta$ is the polar angle (in deg) from the positive $z\mathrm{axis})$ at ${\varepsilon }_r=1$ when $\kappa b=10, Q_f=50, Q_c=-50,0,50$. Symbols are results of Phan et al. [9]; variation of electrophoretic velocity with $\kappa b$ at $\mathrm{\Lambda }=1.16, \beta =3.3, e_p=0.25$ for different ${\varepsilon }_r(=0.1,1,10,100)$ when (b) $Q_f=15, Q_c=-15,0,15$. Inset figure shows the variation of $U_E$ with $\mathrm{\Lambda }$ when $\beta =3.3, Q_c=15, Q_f=15$ at different $\kappa b(=1,5,10)$ and ${\varepsilon }_r(=0.1,1,10,100)$; (c) $Q_c=-50,0,50, Q_f=50$. Circular symbols, computational result for ${\varepsilon }_r=0$; square symbol, the result of Ohshima [3] for ${\varepsilon }_r=0$; dashed lines are results of Gopmandal et al. [10].

Figure 4

Distribution of counterions at different ${\varepsilon }_r(=0.1,1,100)$ for $Q_f=15, Q_c=15, \kappa b=1, \beta =1, \mathrm{\Lambda }=1.16, e_p=0.25$. Dashed line represent the surface of the soft particle.

Figure 5

Variation of the (a) ratio of the drag forces on a charged and uncharged particle migrating at the same velocity with $Q_c$ when $Q_f=15, \kappa b=1,5,10, {\varepsilon }_r=0.1,1,10,100$; (b) electrophoretic velocity with $\beta$ when $Q_c=-15,0,15, Q_f=15, \kappa b=1,5,10$, and ${\varepsilon }_r=1$. Here $\mathrm{\Lambda }=1.16, e_p=0.25$.

Figure 6

Variation of the (a) electrophoretic velocity with $\kappa b$ when the net-fixed charge density of the soft particle is $Q_{\text{net}}=1.125$ (i.e., $Q_f=-9, Q_c=15$) or $Q_{\text{net}}=-1.125$ (i.e., $Q_f=9, Q_c=-15$) for different $\beta (=1,3,10)$. Symbols represent mobility of a porous sphere as obtained by Hermans and Fujita [38] at $\beta =10$; (b) critical value of charge density to achieve zero mobility of a soft particle at $\kappa b=1,5,10$ when $\beta =1,10$. Here $e_p=0.25, \mathrm{\Lambda }=1.16$, and ${\varepsilon }_r=1$.

Figure 7

Variation of the electrophoretic velocity $\left(U_E\right)$ for a net-zero-charged soft particle when ${\varepsilon }_r=100, \beta =1,3,10, a=75\mathrm{nm}, b=100\mathrm{nm}(e_p=0.25)$, and $\mathrm{\Lambda }=1.16$ with the (a) Debye length $\left(\kappa b\right)$ when ${\overline{Q}}_c=-{\overline{Q}}_f=\pm 5\times {10}^{-17}C$ and (b) with ${\overline{Q}}_f$ (or $-{\overline{Q}}_c$) at $\kappa b=50$; (c) variation of the electrostatic force with $\kappa b$ for a net-zero-charged soft particle, i.e., ${\overline{Q}}_c=-{\overline{Q}}_f=\pm {10}^{-16}C,\pm 2\times {10}^{-16}C,\pm 3\times {10}^{-16}C,\pm 5\times {10}^{-16}C$ at $\beta =3$. Symbols, result of bare polyelectrolyte as obtained by Hermans and Fujita [38] at $\beta =10$.