Sedimentation velocity and potential in a concentrated suspension of charged soft spheres

Sedimentation of a Charged Soft Sphere within a Charged Spherical Cavity

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.

Diffusiophoresis in Suspensions of Charged Soft Particles

The diffusiophoresis in a suspension of charged soft particles in electrolyte solution is analyzed. Each soft particle is composed of a hard core of radius r0 and surface charge density σ and an adsorbed fluid-penetrable porous shell of thickness a−r0 and fixed charge density Q. The effect of particle interactions is considered by using a unit cell model. The ionic concentration, electric potential, and fluid velocity distributions in a unit cell are solved as power expansions in σ and Q, and an explicit formula for the diffusiophoretic velocity of the soft particle is derived from a balance between the hydrodynamic and electrostatic forces exerted on it. This formula is correct to the second orders of σ and Q and valid for arbitrary values of κa, λa, r0/a, and the particle volume fraction of the suspension, where κ is the Debye screening parameter and λ is the reciprocal of a length featuring the flow penetration into the porous shell. The effects of the physical characteristics and particle interactions on the diffusiophoresis (including electrophoresis and chemiphoresis) in a suspension of charged soft particles, which become those of hard particles and porous particles in the limits r0=a and r0=0, respectively, are significant and complicated.

Sedimentation Velocity and Potential in Dilute Suspensions of Charge-Regulating Porous Spheres

The sedimentation of a charge-regulating porous sphere surrounded by an arbitrary electric double layer, which usually models a permeable polyelectrolyte coil or an aggregate of nanoparticles, is analyzed for the first time. The hydrodynamic frictional segments and ionogenic functional groups uniformly distribute in the porous sphere, and a regulation mechanism for the dissociation and association reactions occurring at these functional groups linearly relates the local electric potential to fixed charge density. The linearized electrokinetic equations governing the ionic concentration (or electrochemical potential energy), electric potential, and fluid velocity fields are solved for the case of a small basic fixed charge density by the regular perturbation method. Analytical formulae for the sedimentation velocity of a porous sphere and sedimentation potential of a dilute suspension of porous spheres are then obtained. The charge regulation tends to reduce the electrokinetic retardation to sedimentation velocity and the sedimentation potential (can be as much as 50 and 25%, respectively) compared to the case that the fixed charge density is a constant. Both the electrokinetic retardation to sedimentation velocity and the sedimentation potential vanish at the isoelectric point of the particles. The increase in the bulk concentration of the potential-determining ions crossing the isoelectric point changes signs of the fixed charges and thus causes a reversal in the direction of the sedimentation potential. The effects of charge regulation on the sedimentation of porous particles differ substantially from those of hard particles.

Sedimentation Velocity and Potential in Dilute Suspensions of Charged Porous Shells

The sedimentation of a charged spherical porous shell with arbitrary inner and outer radii, which can model a permeable microcapsule or vesicle, in a general electrolyte solution is analytically examined. The relaxation effect in the electric double layers of arbitrary thickness around the porous shell is considered. The differential equations governing the electric potential profile, ionic electrochemical potential energy (or concentration) distributions, and fluid velocity field are linearized by taking the system to be only slightly distorted from equilibrium. These linearized equations are solved using a perturbation method with the density of the fixed charge of the porous shell as the small perturbation parameter. Closed-form formulas for the sedimentation velocity of a porous shell and sedimentation potential in a suspension of porous shells are obtained from a force balance and a zero current requirement, respectively. Both the charge-induced sedimentation velocity retardation and sedimentation potential are monotonic increasing functions of the relative shell thickness, and these increases are substantial if the shell is thin. The sedimentation velocity and potential are complex functions of the electrokinetic radius and normalized flow penetration length of the porous shell. In the limit of the porous shells with zero inner radius, our formulas for the sedimentation velocity and potential reduce to the results obtained for the intact porous spheres.

Effect of induced electric field on migration of a charged porous particle

The effect of ambient fluid flow on a charged porous spherical particle suspended in an aqueous medium is analyzed. The porous particle is ion permeable and fluid penetrable. The induced electric field due to the polarization of the particle's electric double layer and counterion condensation leads to a hindrance effect on particle migration by producing an electric force. The influence of this retardation force on the hydrodynamics of the particle is studied through the Nernst-Planck equations, which are coupled with the Stokes-Brinkman equation. The interactions of the double-layer polarization, shielding effect, electroosmosis of unbalanced ions and fluid convection are analyzed. The settling velocity and fluid collection efficiency of the charged aggregate is determined. We have studied the electrokinetics for a wide range of fixed charge density and permeability of the particle with no assumption made on the thickness of the double layer relative to the dimension of the particle.

Sedimentation of a charged porous particle in a charged cavity

The sedimentation of a charged porous sphere at the center of a charged spherical cavity filled with an electrolyte solution is analyzed. The thickness of the electric double layers around the particle and cavity wall is arbitrary, and their relaxation effect is considered. Through the use of a set of linearized electrokinetic equations and a perturbation method, the ionic electrochemical potential energy, electric potential, and velocity fields in the fluid are solved with the fixed space charge density of the particle and surface charge density of the cavity as the small perturbation parameters, and an explicit formula for the sedimentation velocity is obtained. Due to the electroosmotic enhancement on the fluid recirculation in the cavity caused by the sedimentation-induced electric field, the presence of the surface charges on the cavity wall increases the sedimentation velocity of the porous particle. For the sedimentation of a porous particle in a cavity with their fixed charges of the same sign, the effect of electric interaction between the particle and cavity wall in general increases the sedimentation velocity. For the case of their fixed charges with opposite signs, the sedimentation velocity is increased/reduced if the magnitude of the fixed charge density of the cavity wall is relatively large/small. The effect of the surface charges at the cavity wall on the sedimentation of the porous particle increases with an increase in the permeability for fluid flow within the particle and with a decrease in the particle-to-cavity radius ratio (i.e., an increase in the surface area of the cavity wall relative to a given size of the particle, which enhances the fluid recirculation effect).