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The general expression is derived for the Laplace transform of the time-dependent transient electrophoretic mobility (with respect to time) of a spherical colloidal particle when a step electric field is applied. The transient electrophoretic mobility can be obtained by the numerical inverse Laplace transformation method. The obtained expression is applicable for arbitrary particle zeta potential and arbitrary thickness of the electrical double layer around the particle. For the low potential case, this expression gives the result obtained by Huang and Keh. On the basis of the obtained general expression for the Laplace transform of the transient electrophoretic mobility, we present an approximation method to avoid the numerical inverse Laplace transformation and derive a simple approximate analytic mobility expression for a weakly charged particle without involving numerical inverse Laplace transformations. The transient electrophoretic mobility can be obtained directly from this approximate mobility expression without recourse to the numerical inverse Laplace transformation. The results are found to be in excellent agreement with the exact numerical results obtained by Huang and Keh.

The general expression is obtained for the diffusiophoretic mobility of a mercury drop in an electrolyte concentration gradient. On the basis of the obtained general mobility expression, an approximate analytic mobility expression which is correct to the second order of the drop zeta potential is derived.

The general expression is derived for the diffusiophoretic velocity of a spherical soft particle (that is, a spherical hard particle consisting of the particle core covered with an ion-penetrable surface layer of polyelectrolytes) in an electrolyte concentration gradient. For a weakly charged soft particle, the obtained general expression for the diffusiophoretic velocity is shown to reproduce the results derived by Huang and Keh (J Phys Chem B (2012) 116: 7575–7589). A simple approximate analytic expression is obtained for the diffusiophoretic velocity applicable for the case where the particle core radius and the thickness of the polyelectrolyte layer are much larger than the Debye length and the Brinkman screening length.

The general expression is derived for the diffusiophoretic velocity of a large spherical colloidal particle of radius a in a concentration gradient of general electrolytes of Debye-Hückel parameter κ. On the basis of this expression, simple approximate analytic expressions for the diffusiophoretic velocity correct to the order of (1/κa)0 are derived, which can be applied for large particles with κa ≥ 50 at arbitrary values of the particle zeta potential with negligible errors.

It is found that a liquid drop of radius a and viscosity ηd with a slip length Λ in a liquid of viscosity η behaves like a spherical solid particle with an effective slip length Λd=Λ+ηa/3ηd. On the basis of this correspondence relation, equations for various electrokinetic quantities of slip drops can easily be derived from those for a spherical solid particle with a slip surface. In particular, expressions are derived for the electrophoretic mobility of a weakly charged liquid drop with a slip surface in an aqueous electrolyte solution. The obtained mobility expression covers previous results for the mobility of a spherical solid particle with a slip surface and that of a liquid drop with a no-slip surface.Electrophoretic mobility of a liquid drop with a slip surface

A general theory is presented to analyze the time-dependent, transient diffusiophoresis of a charged spherical colloidal particle in an uncharged gel medium containing a symmetrical electrolyte when an electrolyte concentration gradient is suddenly applied. We derive the inverse Laplace transform of an approximate expression for the relaxation function R(t), which describes the time-course of the ratio of the diffusiophoretic mobility of a weakly charged spherical colloidal particle, possessing a thin electrical double layer, to its steady-state diffusiophoretic mobility. The relaxation function depends on the mass density ratio of the particle to the electrolyte solution, the particle radius, the Brinkman screening length, and the kinematic viscosity. However, it does not depend on the type of electrolyte (e.g., KCl or NaCl), which affects only the steady-state gel diffusiophoretic mobility. It is also found that the expression for the relaxation function in transient gel diffusiophoresis of a weakly charged spherical colloidal particle with a thin electrical double layer takes the same form as that for its transient gel electrophoresis.

A theory of electroosmosis in an array of parallel cylindrical fibers of Kozak and Davis (J Colloid Interface Sci 112:403–411, 1986) is extended to cover the case where the hydrodynamic slip occurs on the fiber surface. An analytic formula for the electroosmotic velocity for low zeta potentials is obtained, and its simple approximate expression without involving numerical integration is also derived.

A theory is developed of the electrophoresis of a spherical colloidal particle with a slip surface in a concentrated suspension on the basis of Kuwabara’s cell model. We introduce the slipping length on the particle surface, which is the measure of the particle surface hydrophobicity. We derive the general expression of the particle electrophoretic mobility and its approximate analytic expressions for a particle carrying a low zeta potential. Expressions for other electrokinetics, that is, electrical conductivity, sedimentation velocity, and potential in concentrated suspensions, are also derived. Furthermore, it is shown that as in the case of a dilute suspension, a similarity is found between the electrokinetics of charged spherical solid particles with a slip surface in a concentrated suspension and that for liquid drops.

A theory is developed of the electrophoretic mobility of a charged spherical colloidal particle in a gel medium due to the long-range hydrodynamic particle-gel interaction by using the Brinkman-Debye-Bueche model and the general mobility expression is derived. On the basis of the obtained general mobility expression, a simple analytic mobility expression for particles with low zeta potential is derived, which is found to be in excellent agreement with the exact mobility equations expressed in terms of exponential integrals derived by Allison et al. for an uncharged gel (J Colloid Interface Sci 313: 328, 2007) and by Li et al. for a charged gel (J Colloid Interface Sci 423: 129, 2014). The relative error is less than 1.6%. An approximate mobility expression for a particle with arbitrary zeta potential in an uncharged gel medium applicable for large κa > 30, where a is the particle radius and κ is the Debye-Hückel parameter, is also derived.

General expressions of the electrophoretic mobility-zeta potential relationship for a cylindrical colloidal particle with a hydrodynamically slipping surface in an aqueous electrolyte solution under a transverse or tangential electric field are obtained on the basis of the Navier boundary condition. Approximate expressions for the electrophoretic mobility of cylindrical particles carrying a low zeta potential are derived. As in the case of a sphere, the electrophoretic mobility of a cylinder increases with increasing slip length, which characterizes the hydrophobicity of the particle surface.