Transient Electrophoresis of A Cylindrical Colloidal Particle

: We develop the theory of transient electrophoresis of a weakly charged, infinitely long cylindrical colloidal particle under an application of a transverse or tangential step electric field. Transient electrophoretic mobility approaches steady electrophoretic mobility with time. We derive closed-form expressions for the transient electrophoretic mobility of a cylinder without involving numerical inverse Laplace transformations and the corresponding time-dependent transient Henry functions. The transient electrophoretic mobility of an arbitrarily oriented cylinder is also derived. It is shown that in contrast to the case of steady electrophoresis, the transient Henry function of an arbitrarily oriented cylinder at a finite time is significantly smaller than that of a sphere with the same radius and mass density as the cylinder so that a cylinder requires a much longer time to reach its steady mobility than the corresponding sphere.

Li and Keh [14], in particular, derived the general expression for the Laplace transform of the transient electrophoretic mobility of a weakly charged infinitely long cylinder with arbitrary double-layer thickness in an applied transverse or tangential step electric field and calculated the transient electrophoretic mobility of the particle by using the numerical inverse Laplace transformation method. This method, however, requires tedious numerical calculation and it is not very convenient for practical purposes.
In a previous paper [18], we have shown that the fundamental electrokinetic equations describing the transient electrophoresis of a spherical colloidal particle are quite similar to those for the dynamic electrophoresis of the spherical particle in an applied oscillating electric field [20]. Indeed, it has been shown that there is a simple correspondence between the Laplace transform of the transient electrophoretic mobility and the dynamic electrophoretic mobility of a charged particle in an electrolyte solution [18]. As in the case of a spherical particle, it will be shown that there is the same correspondence relation between the Laplace transform of the transient electrophoretic mobility of a cylinder and its dynamic electrophoretic mobility [21].
The purpose of the present paper is to develop further the theory of transient electrophoresis of a weakly charged infinitely long cylinder in an applied transverse or tangential step electric field and derive closed-form expressions for the transient electrophoretic mobility of the cylinder without involving numerical inverse Laplace transformations.

Theory
Let us consider an infinitely long, cylindrical colloidal particle of mass density ρp, radius a, and zeta potential ζ in an aqueous electrolyte solution of mass density ρo, viscosity η, and relative permittivity εr. The electrolyte consists of N ionic species of valence zi, bulk concentration (number density) ni ∞ , and drag coefficient λi (i =1, 2, …, N). We suppose that a step electric field E(t) is suddenly applied transversely or tangentially to the cylinder at time t = 0, viz., where Eo is constant and the particle starts to move with an electrophoretic velocity U(t) in the direction parallel to Eo (Fig. 1). The transient electrophoretic mobility μ(t) of the particle is defined by U(t) = μ(t)E(t) = μ(t)Eo. The origin of the cylindrical coordinate system (r, θ, z) is held fixed at the center of the particle. We treat the case where (i) the liquid can be regarded as incompressible, (ii) the applied electric field E(t) is weak so that terms involving the square of the liquid velocity in the Navier-Stokes equation can be neglected and the particle velocity U(t) is proportional to E(t), and (iii) the relative permittivity of the particle εp is much smaller than that of the electrolyte solution εr (εp « εr).

Cylinder in A Transverse Field
We first treat the case where E(t) is perpendicular to the cylinder axis ( Figure 1a). The ( , ) + ∇ • ( , ) ( , ) = 0 (5) where e is the elementary electric charge, k is the Boltzmann constant, T is the absolute temperature, εo is the permittivity of a vacuum, p(r, t) is the pressure, ρel(r, t) is the charge density, ψ(r, t) is the electric potential, FH(t) and FE(t) are, respectively, the hydrodynamic and electric forces acting on the cylinder. Equations. (2) and (3) are the Navier-Stokes equation and the equation of continuity for an incompressible flow (condition (i)). The term involving the particle velocity U (t) in Eq. (2) arises from the fact that the particle has been chosen as the frame of reference for the coordinate system. Equation (4) states that the flow v i (r, t) of the i th ionic species is caused by the liquid flow u(r, t) and the gradient of the electrochemical potential μ i (r, t). Equation  The initial and boundary conditions for u(r, t) and vi(r, t) are given by where is the unit normal outward from the particle surface. Equation (8) states that the slipping plane (at which u(r, t) = 0) is located on the particle surface. Equation (10) follows from the condition that electrolyte ions cannot penetrate the particle surface.
For a weak field E(t), the deviations of nj(r, t), ψ(r, t), and μj(r, t) from their equilibrium values (i.e., those in the absence of E(t)) due to the applied field E(t) are small so that we may write where the quantities with superscript (0) refer to those at equilibrium, the quantities, with δ referring to the deviations from the corresponding equilibrium values, and ( ) is a constant independent of r. It is assumed that the equilibrium concentration ( ) ( ) obeys the Boltzmann distribution and the equilibrium electric potential satisfies the Poisson-Boltzmann equation, viz., with = (17) where y(r) is the scaled equilibrium electric potential, κ is the Debye-Hückel parameter, and 1/κ is the Debye length. From symmetry, we may write where E(t) is the magnitude of E(t), h(r, t), and φi(r, t) are functions of r and t. By substituting Equations (11)-(13), (18), and (19) into Equations (2)- (5), we obtain the following equations for h(r): where and = is the kinematic viscosity. It follows from Equations (9) and (18) that the transverse transient electrophoretic mobility μ(t) can be obtained by We solve Equation (20) by introducing the Laplace transforms ℎ ( , ), ( , ), and ̂ ( ) of h(r, t), G(r, t), and μ⊥(r, t), respectively, which are given by and the Laplace transform of Equation (26) is The Laplace transform of Equation (20) By solving Equation (29) and using Equation (28), we obtain the following general expression for ̂ ( ): where Kn(z) is the n th order modified Bessel function of the second kind. Now consider the low ζ potential case. In this case, , it can be shown that (see Ref. [21]) and Equation (22) becomes The Laplace transform ( , ) of G(r, t) is given by where the equilibrium electric potential ψ (0) (r) for the low ζ potential case is given by which agrees with Li and Keh's result [14]. Li and Keh [14] obtained the transient electrophoretic mobility μ⊥(t) by using the numerical inverse Laplace transform of Eq. (35). This method, however, involves tedious numerical calculations and is not very convenient for practical uses. In order to avoid this difficulty, we employ the same approximation method as used for the static electrophoresis problem [22]. We first note that the integrand in Eq. (35) has a sharp maximum around r = a + δ/κ, δ being a factor of order unity. This is because the electrical double layer (of the thickness 1/κ ) around the cylinder is confined in the narrow region between r = a and r ≈ a +1/κ. Since the factor (1+a 2 /r 2 ) in the integrand of Eq. (35) varies slowly with r as compared with the other factors, one may approximately replace r in the factor (1+a 2 /r 2 ) by r = a + δ/κ and take it out before the integral sign. That is, we make the following approximate replacement of the difficult factor (1+a 2 /r 2 ) by an rindependent constant factor: We obtain μ⊥(t) from ̂ ( ) by using the inverse Laplace transformation, viz., where the integration is carried out along the vertical line Re(s)=γ in the complex plane, where γ is large so that all the singularities of ̂ ( ) lie to the left of the line (γ -i∞, γ +i∞) ( Figure 2). Since ̂ ( ) has a branch point at the origin s = 0, we convert this line integral into a contour integral over a large circle Γ with a cut along the negative part of the real axis Re(s). Since the integral over the large circle Γ vanishes as its radius R tends to infinity, the line integral is replaced by real infinite integrals along CD and EF together with the contribution from the small circle about the origin s = 0 [24].
We obtain from Equation (37) the following expression for μ⊥(t): where Jn(λ) and Yn(λ) are, respectively, the n th order Bessel functions of the first and second kinds. In the limit of t→∞, Equation (41) tends to the transverse steady electrophoretic mobility, viz., [22] (∞) = 2 which agrees with the following exact expression with negligible errors [22,23].
Equation (41) is the required approximate expression for the transverse transient electrophoretic mobility ( ) with negligible errors. In the limit of large κa (κa » 1), which agrees with the results of Morison [2] and Li and Keh [14]. For small κa (κa « 1),

Cylinder in a tangential field
We next treat the case where the applied electric field E(t) = (0, 0, E(t)) is parallel to the cylinder axis (Figure 1b). The liquid velocity u(r, t) can be expressed as u = u(0, 0, uz(r, t)). The Navier-Stokes equation for uz(r, t)) is given by As in the case of ( ), by using the inverse Laplace transform ̂ ( ), i.e., We obtain the following expression for ∥ ( ): In the limit of t→∞, Equation (53) tends to the tangential steady electrophoretic mobility [22,23], viz., In the limit of large κa, Eq. (53) tends to while for small κa, Equation (53) tends to It should be noticed that as in the case of a sphere [18], there is a simple correspondence between the Laplace transform of the transient mobility of a cylinder and its dynamic mobility. That is, ̂ ( ) and ̂∥( ) of the transient electrophoretic mobilities ( ) and ∥ ( ), respectively, can be obtained from the dynamic electrophoretic mobility ( ) and ∥( ) of a cylinder under an oscillating electric field of frequency ω by replacing -iω with s and G(r) by G(r)/s.

Results and Discussion
The principal results of the present paper are Equations (41) and (53) for the transverse and tangential transient electrophoretic mobilities, respectively. We define the timedependent transient Henry function as We thus obtain and As t →∞, the transverse transient Henry functions ( , ) and ∥ ( , ) given by Eqs. (60) and (61), respectively, tend to the following steady Henry functions [22]: Note that Equations (41)  In the present paper, we treat an infinitely long cylinder, neglecting the end effects. Sherwood [25] demonstrated that the end effects can be neglected under the condition that the cylinder length is much longer than the double-layer thickness 1/κ, Under this condition, it can also be assumed that there is no interaction between cylinders when we consider a dilute suspension of infinitely long cylinders.
Finally, let us consider a cylindrical particle oriented at an arbitrary angle between its axis and the applied electric field. In the present paper we have treated the two types of fields, that is, transverse and tangential electric fields. When an electric field is applied at an arbitrary angle relative to the cylinder axis, the electrophoretic mobility is given by the weighted average of ( , )and ∥ ( , ). Thus -the transient electrophoretic mobility fav(κa, t) averaged over a random distribution of orientation is given by [26]:  It is seen from Fig. 4 that the average transient Henry function fav(κa, t) of a cylinder at a finite time is considerably lower than the transient Henry function fsp(κa, t) of a sphere with the same radius a and mass density ρp so that a cylinder requires a much longer time to reach its steady mobility than the corresponding sphere, in contrast to the case of steady electrophoresis, where fav(κa, t) is quite similar to fsp(κa, t).
The shape and size dependence of the steady Henry function decreases as its size relative to the Debye length (1/κ) increases and vanishes in the thin double-layer limit (i.e., in the limit of κa →∞ for a sphere and a cylinder, each with radius a) so that a sphere and a cylinder exhibit the same mobility value as that of a particle with a planar surface. On the other hand, even in this limit, the transient Henry function always depends on the particle shape and size.
The present theory can be extended to other types of applied electric fields. It can be shown that in the case where the applied field is an oscillating electric field with frequency ω (i.e., the applied electric field is proportional to e -iωt ), the inverse Laplace transforms

Conclusions
We developed the theory of transient electrophoresis of a weakly charged, infinitely long cylindrical colloidal particle under an application of a transverse or tangential step electric field. We derived closed-form expressions for the transient electrophoretic mobilities ( , ) and ∥ ( , ) of a cylinder (Equations. (41) and (53)) without involving numerical inverse Laplace transformations and the corresponding time-dependent transient Henry functions ( , ) and ∥ ( , ) (Eqs. (60) and (61)). The transient Henry function fav(κa, t) of an arbitrarily oriented cylinder is also derived (Eq. (64)). It is shown that in contrast to the case of steady electrophoresis, the transient Henry function fav(κa, t) of an arbitrarily oriented cylinder at a finite time is significantly smaller than the transient Henry function fsp(κa, t) of a sphere with the same radius a and mass density ρp as the cylinder so that a cylinder requires a much longer time to reach its steady mobility than the corresponding sphere. It is also shown that, unlike the steady Henry function, the transient Henry function for a cylinder differs from that of a sphere even in the limit of large κa.