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 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.
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.
It is found that a liquid drop of radius
a
and viscosity
η
d
with a slip length
Λ
in a liquid of viscosity
η
behaves like a spherical solid particle with an effective slip length
Λ
d
=
Λ
+
η
a
/
3
η
...d
. 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.
Graphical abstract
Electrophoretic mobility of a liquid drop with a slip surface
An approximate analytic expression for the electrophoretic mobility of an infinitely long cylindrical colloidal particle in a symmetrical electrolyte solution in a transverse electric field is ...obtained. This mobility expression, which is correct to the order of the third power of the zeta potential ζ of the particle, considerably improves Henry’s mobility formula correct to the order of the first power of ζ (Proc. R. Soc. London, Ser. A 1931, 133, 106). Comparison with the numerical calculations by Stigter (J. Phys. Chem. 1978, 82, 1417) shows that the obtained mobility formula is an excellent approximation for low-to-moderate zeta potential values at all values of κa (κ = Debye–Hückel parameter and a = cylinder radius).
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.
Graphical abstract
Electroosmotic velocity in an array of parallel cylindrical fibers with a slip surface
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.
Graphical abstract
Electrophoretic mobility of a sphere with a slip surface in a concentrated suspension
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.
Graphical abstract
Electrophoretic mobility of a cylinder with a slip surface in a transverse electric filed.
Theories of the electrostatic interaction between two soft particles (i.e., particles covered with an ion-penetrable surface layer of polyelectrolytes) in an electrolyte solution are reviewed. ...Interactions of soft particles after contact of their surface layers are particularly discussed. Interaction in a salt-free medium and the discrete-charge effect are also treated.
Display omitted
•Electrostatic interaction of soft particles before and after contact is discussed.•Electrostatic interaction of soft particles in a salt-free medium is discussed.•Discrete-charge effect on interaction of soft particles is discussed.
