Sedimentation of a Charged Soft Sphere within a Charged Spherical Cavity

Abstract

The sedimentation of a soft particle composed of an uncharged hard sphere core and a charged porous surface layer inside a concentric charged spherical cavity full of a symmetric electrolyte solution is analyzed in a quasi-steady state. By using a regular perturbation method with small fixed charge densities of the soft sphere and cavity wall, a set of linearized electrokinetic equations relevant to the fluid velocity field, electrical potential profile, and ionic electrochemical potential energy distributions are solved. A closed-form formula for the sedimentation velocity of the soft sphere is obtained as a function of the ratios of core-to-particle radii, particle-to-cavity radii, particle radius-to-Debye screening length, and particle radius-to-porous layer permeation length. The existence of the surface charge on the cavity wall increases the settling velocity of the charged soft sphere, principally because of the electroosmotic enhancement of fluid recirculation within the cavity induced by the sedimentation potential gradient. When the porous layer space charge and cavity wall surface charge have the same sign, the particle velocity is generally enhanced by the presence of the cavity. When these fixed charges have opposite signs, the particle velocity will be enhanced/reduced by the presence of the cavity if the wall surface charge density is sufficiently large/small relative to the porous layer space charge density in magnitude. The effect of the wall surface charge on the sedimentation of the soft sphere increases with decreases in the ratios of core-to-particle radii, particle-to-cavity radii, and particle radius-to-porous layer permeation length but is not a monotonic function of the ratio of particle radius-to-Debye length.

Keywords: boundary effect; charged cavity; charged soft particle; electrokinetics; sedimentation velocity.

Figure 1

Geometric sketch for the sedimentation of a soft sphere inside a concentric spherical cavity.

Figure 2

The coefficient $H_1$ (accounting for the effect of the particle charge) in Equation (26) for the sedimentation of a soft particle in a cavity full of aqueous KCl solution: (a) versus the ratio of particle radius to Debye length $\kappa a$ with $r_0/a=0$ and $a/b=0.5$; (b) versus the particle-to-cavity radius ratio $a/b$ with $r_0/a=0$ and $\lambda a=1$; (c) versus the ratio of particle radius to porous layer permeation length $\lambda a$ with $\kappa a=1$ and $a/b=0.5$; (d) versus the core-to-particle radius ratio $r_0/a$ with $\kappa a=1$ and $\lambda a=1$.

Figure 3

The coefficient $H_2$ (accounting for the coupling effect of the fixed charges) in Equation (26) for the sedimentation of a soft particle in a cavity full of aqueous KCl solution: (a) versus the ratio of particle radius-to-Debye length $\kappa a$ with $r_0/a=0$ and $a/b=0.5$; (b) versus the particle-to-cavity radius ratio $a/b$ with $r_0/a=0$ and $\lambda a=1$; (c) versus the ratio of particle radius-to-porous-layer-permeation length $\lambda a$ with $\kappa a=1$ and $a/b=0.5$; (d) versus the core-to-particle radius ratio $r_0/a$ with $\kappa a=1$ and $\lambda a=1$.

Figure 4

The coefficient $H_3$ (accounting for the effect of the cavity charge) in Equation (26) for the sedimentation of a soft particle in a cavity full of aqueous KCl solution: (a) versus the ratio of particle radius-to-Debye length $\kappa a$ with $r_0/a=0$ and $a/b=0.5$; (b) versus the particle-to-cavity radius ratio $a/b$ with $r_0/a=0$ and $\lambda a=1$; (c) versus the ratio of particle radius-to-porous layer permeation length $\lambda a$ with $\kappa a=1$ and $a/b=0.5$; (d) versus the core-to-particle radius ratio $r_0/a$ with $\kappa a=1$ and $\lambda a=1$.

Figure 5

The normalized sedimentation velocity U/U₀₀ of a soft particle with the dimensionless space charge density $\overline{Q}=1$ in a cavity full of aqueous KCl solution versus the dimensionless surface charge density $\overline{\sigma }$: (a) for several values of the particle-to-cavity radius ratio $a/b$ with $r_0/a=0.5$, $\lambda a=1$, and $\kappa a=1$; (b) for several values of the ratio of particle radius to Debye length $\kappa a$ with $r_0/a=0.5$, $a/b=0.5$, and $\lambda a=1$; (c) for several values of the ratio of particle radius to porous layer permeation length $\lambda a$ with $r_0/a=0.5$, $a/b=0.5$, and $\kappa a=1$; (d) for several values of the core-to-particle radius ratio $r_0/a$ with $a/b=0.5$, $\lambda a=1$, and $\kappa a=1$.

Figure 5

The normalized sedimentation velocity U/U₀₀ of a soft particle with the dimensionless space charge density $\overline{Q}=1$ in a cavity full of aqueous KCl solution versus the dimensionless surface charge density $\overline{\sigma }$: (a) for several values of the particle-to-cavity radius ratio $a/b$ with $r_0/a=0.5$, $\lambda a=1$, and $\kappa a=1$; (b) for several values of the ratio of particle radius to Debye length $\kappa a$ with $r_0/a=0.5$, $a/b=0.5$, and $\lambda a=1$; (c) for several values of the ratio of particle radius to porous layer permeation length $\lambda a$ with $r_0/a=0.5$, $a/b=0.5$, and $\kappa a=1$; (d) for several values of the core-to-particle radius ratio $r_0/a$ with $a/b=0.5$, $\lambda a=1$, and $\kappa a=1$.