Transient Gel Electrophoresis of a Spherical Colloidal Particle

The general theory is developed for the time-dependent transient electrophoresis of a weakly charged spherical colloidal particle with an electrical double layer of arbitrary thickness in an uncharged or charged polymer gel medium. The Laplace transform of the transient electrophoretic mobility of the particle with respect to time is derived by considering the long-range hydrodynamic interaction between the particle and the polymer gel medium on the basis of the Brinkman–Debye–Bueche model. According to the obtained Laplace transform of the particle’s transient electrophoretic mobility, the transient gel electrophoretic mobility approaches the steady gel electrophoretic mobility as time approaches infinity. The present theory of the transient gel electrophoresis also covers the transient free-solution electrophoresis as its limiting case. It is shown that the relaxation time for the transient gel electrophoretic mobility to reach its steady value is shorter than that of the transient free-solution electrophoretic mobility and becomes shorter as the Brinkman screening length decreases. Some limiting or approximate expressions are derived for the Laplace transform of the transient gel electrophoretic mobility.


Fundamental Electrokinetic Equations
Consider a charged spherical colloidal particle of radius a and relative permittivity ε p , carrying zeta potential ζ in a charged polymer gel medium containing an electrolyte solution of viscosity η and relative permittivity ε r . The Brinkman-Debye-Bueche continuum medium [44,45] is employed, in which polymer segments are considered to be resistance centers, exerting frictional forces on the liquid flowing through the gel medium. The gel medium is regarded as a uniform continuum medium, which contains fixed charges of density ρ fix , free mobile electrolyte ions of density ρ el (r) at position r, including added electrolyte ions and gel counterions. Let the electrolyte be composed of N ionic species of valence z i , bulk concentration (number density) n ∞ i and drag coefficient Λ i (i = 1, 2, . . . , N), and the gel counterions be of N + 1-th ionic species of valence z N+1 , bulk concentration (number density) n ∞ N+1 and drag coefficient Λ N+1 . The electroneutrality condition of the system is given by where e is the elementary electric charge.
We suppose that at time t = 0, a step electric field E(t) is suddenly applied to the particle, viz., where E o is a constant. The particle then starts to migrate with an electrophoretic velocity U(t) (U(t)cosθ, −U(t)sinθ, 0) in the direction parallel to E o , U(t) being the magnitude of U(t) (Figure 1).

Fundamental Electrokinetic Equations
Consider a charged spherical colloidal particle of radius a and relative permittivity εp, carrying zeta potential ζ in a charged polymer gel medium containing an electrolyte solution of viscosity η and relative permittivity εr. The Brinkman-Debye-Bueche continuum medium [44,45] is employed, in which polymer segments are considered to be resistance centers, exerting frictional forces on the liquid flowing through the gel medium. The gel medium is regarded as a uniform continuum medium, which contains fixed charges of density ρfix, free mobile electrolyte ions of density ρel(r) at position r, including added electrolyte ions and gel counterions. Let the electrolyte be composed of N ionic species of valence zi, bulk concentration (number density) and drag coefficient Λi (i = 1, 2, …, N), and the gel counterions be of N + 1-th ionic species of valence zN+1, bulk concentration (number density) and drag coefficient ΛN+1. The electroneutrality condition of the system is given by where e is the elementary electric charge.
We suppose that at time t = 0, a step electric field E(t) is suddenly applied to the particle, viz., where Eo is a constant. The particle then starts to migrate with an electrophoretic velocity U(t) (U(t)cosθ, −U(t)sinθ, 0) in the direction parallel to Eo, U(t) being the magnitude of U(t) (Figure 1). Our model uses a frame of reference fixed at the center of the particle. The origin of the coordinate system (r, θ, ϕ) is held fixed at the particle center, and the polar axis (θ = 0) is set parallel to E (t). The transient electrophoretic mobility µ(t) of the particle is defined by U(t) = µ(t)E(t) = µ(t)Eo. Our model treats the case in which the following conditions are fulfilled: (i) the liquid in the gel medium can be considered to be incompressible; (ii) the applied electric field E(t) is so weak that the particle velocity U(t) is proportional to E(t), and terms involving the square of the liquid velocity in the Navier-Stokes equation can be neglected in our model; (iii) the slipping plane, at which the liquid velocity u(r, t) relative to the particle is zero, is located on the particle surface (at r = a); (iv) electrolyte ions cannot penetrate the particle surface [46]; and (v) in equilibrium (in the absence of E(t)), Our model uses a frame of reference fixed at the center of the particle. The origin of the coordinate system (r, θ, φ) is held fixed at the particle center, and the polar axis (θ = 0) is set parallel to E (t). The transient electrophoretic mobility µ(t) of the particle is defined by Our model treats the case in which the following conditions are fulfilled: (i) the liquid in the gel medium can be considered to be incompressible; (ii) the applied electric field E(t) is so weak that the particle velocity U(t) is proportional to E(t), and terms involving the square of the liquid velocity in the Navier-Stokes equation can be neglected in our model; (iii) the slipping plane, at which the liquid velocity u(r, t) relative to the particle is zero, is located on the particle surface (at r = a); (iv) electrolyte ions cannot penetrate the particle surface [46]; and (v) in equilibrium (in the absence of E(t)), the ion distribution is assumed to be given by the Boltzmann distribution and the electric potential follows the Poisson-Boltzmann equation.
Under these conditions (i)-(v), the fundamental electrokinetic equations for the liquid flow velocity u(r, t) (u r (r, t), u θ (r, t), 0) at position r(r, θ, φ) and time t and the velocity v i (r, t) of i th ionic species are given by.
where k is the Boltzmann constant, T is the absolute temperature, ε o is the permittivity of a vacuum, p(r, t) is the pressure, ρ el (r, t) is the charge density and ψ(r, t) is the electric potential. Equation (3) is the Navier-Stokes equation, and Equation (4) is the equation of continuity for an incompressible flow (condition (i)). The term involving U (t) in Equation (3) arises from the fact that the particle has been chosen as the frame of reference for the coordinate system. Equation (5) means that the flow v i (r, t) of the i th ionic species is caused by u(r, t), and the gradient of the electrochemical potential µ i (r, t), given by Equation (8), in which µ o i is a constant term. Equation (6) is the continuity equation for the i th ionic species. Equation (9) is the Poisson equation. Note that in the absence of the particle, there exists a time-dependent transient electroosmotic flow, which is parallel to E(t). The transient electroosmotic flow velocity u EOF (t) = (u EOF (t)cosθ, −u EOF (t)sinθ, 0) obeys where u EOF (t) is the magnitude of u EOF (t).
The following initial condition and boundary conditions at the particle surface (at r = a) and far from the particle (r → ∞) must be satisfied: ψ(r, t) → −E(t)·r as r → ∞ wheren is the unit normal outward from the particle surface. Equation (12) is the no-slip boundary condition at the particle surface (condition (iii)). Equation (14) is derived from condition (iv). Equation (15) implies that ψ(r, t) tends to the potential of the applied electric field E(t) as r →∞. In addition, the particle velocity U(t) obeys the following equation of motion of the particle: where F H (t) and F E (t) are, respectively, the hydrodynamic and electric forces acting on the particle and are defined by Equation (16) serves as a boundary condition for u(r, t).

Weak Electric Field Approximation
For a weak electric field E(t), the deviations of n j (r, t), ψ(r, t) and µ j (r, t) from their equilibrium values due to E(t) are all small so that we may write where the quantities with superscript (0) refer to the equilibrium values and µ i (r) is assumed to be given by the Boltzmann distribution, and the equilibrium electric potential obeys the Poisson-Boltzmann equation (condition (v)), viz., The boundary conditions for ψ (0) (r) are given by By substituting Equations (19)-(21) into Equation (3) and neglecting the products of the small quantities, we finally obtain and form Equation (6) ∂ ∂t Further, from symmetry, we may write where E(t) is the magnitude of E(t), and h(r, t), φ i (r, t) and Y(r, t) are functions of r and t. (27) and (28), we obtain the following equations for h(r) and φ i (r), and Y(r):

By substituting Equations (29)-(31) into Equations
with y(r) = eψ (0) (r) kT (35) where the scaled equilibrium electric potential y(r) is introduced, λ is the reciprocal of the Brinkman screening length 1/λ, is a differential operator, and G(r, t) is defined by and is the kinematic viscosity.

General Expression for the Laplace Transform of the Transient Gel Electrophoretic Mobility
The transient electrophoretic mobility µ(t) can be obtained from Equation (13), viz., Here h(r, t) is the solution to Equation (32), which can be most easily solved by using the Laplace transformation with respect to time t. The Laplace transformsĥ(r, s),û EOF (s), G(r, s) andμ(s) of h(r, t), u EOF (t), G(r, t) and µ(t), respectively, are defined bŷ Thus, the Laplace transform of Equation (32) yields which is solved to giveĥ with where C 1 -C 3 are integration constants to be determined. From the Laplace transform of Equation (10), we obtain By determining the integration constants C 1 -C 3 in Equation (46) to satisfy the boundary conditions (Equations (11)- (16)) and using Equation (40), we finally obtain the following expression for the Laplace transformμ(s) of the transient gel electrophoretic mobility µ(t) of a sphere: with

Results and Discussion
Equation (50) is the required general expression forμ(s), which is applicable for arbitrary values of the particle zeta potential ζ and κa. The transient electrophoretic mobility µ(t) can be obtained numerically from Equation (50) by the inverse transform method.
Consider the following two limiting cases. In the limit of t → ∞, µ(t) tends to the steady gel electrophoretic mobility µ(∞) = µ s , which can be obtained fromμ(s) by using the following formula: The result is with Equation (53) agrees with the general expression for the steady electrophoretic mobility m(t) of a sphere in a polymer gel medium [35]. Next, in the limit of ρ fix = 0 and λ = 0, i.e., β = √ s/ν), Equation (50) reduces tô with which agrees with the general expression for the Laplace transformμ(s) of the transient electrophoretic mobility µ(t) of a sphere in a free solution [14]. It is thus found that in the above two limiting cases, Equation (50) reduces to the correct limiting forms. Now consider the case where the particle ζ potential is low, and the relative permittivity of ε p of the particle is much smaller than that of the electrolyte solution ε r (ε p « ε r ) so that ε p is practically equal to zero. In this case, Equations (33) and (34) give and Equation (38) becomes where κ is the Debye-Hückel parameter (1/κ is the Debye length). The Laplace transform G(r, s) of G(r, t) is thus given bŷ where the equilibrium electric potential ψ (0) (r) for low ζ potential is given by which is obtained from the linearized Poisson-Boltzmann equation ∆ψ (0) (r) = κ 2 ψ (0) (r) (see Equation (23)). By substituting Equation (58) into Equation (55), we obtain Equation (62) can be rewritten in terms of exponential integrals aŝ where E n (κa) is the exponential integral of order n and is defined by Equations (62) and (63) are the generalization of the result of Saad and Faltas [40] and are applicable for low zeta potentials and arbitrary values of κa.
Equations (62) and (63) involve integration or exponential integrals, so they are not very convenient for practical use. To avoid this inconvenience, we approximately replace r in the factor (1 + a 3 /2r 3 ) by r = a + δ/, viz., In the steady gel electrophoresis [35], we have found that the best approximation can be achieved if δ is chosen to be δ = (2.33κ + 1.52λ)/(κ + λ), and the maximum relative error becomes less than 1.6%. We use this choice of δ in the transient gel electrophoresis problem. By using this approximation, the integration in Equation (62) can be carried out analytically to giveμ We next consider the following two limiting cases.
(i) In the limit of κa → ∞ (Smoluchowski limit), Equation (66) becomeŝ (ii) In the limit of κa → 0 (Hückel limit), Equation (66) becomeŝ   Figure 2 shows that the relaxation time required for µ(t) to reach its steady value µ s becomes shorter as λa increases. An approximate expression for the relaxation time T for large λa can be derived as follows. For large λa, β in Equation (60) Here T can be regarded as the relaxation time. The relaxation time T f for the transient free-solution electrophoresis is given by so that T T f = 1 Ω s = 1 1 + λa + λ 2 a 2 /9 (75) which shows that the relaxation time T for the transient gel electrophoresis is shorter than the relaxation time T f for the transient free-solution electrophoresis by a factor Ω s and becomes shorter as λa decreases. This is because the steady gel electrophoretic mobility µ s itself becomes smaller as λa increases [36], and the time required to reach the steady value becomes smaller as λa increases. The dotted curves (λa = 10, and 100) are the results calculated via Equation (72) for the large λa approximate gel electrophoretic mobility.

Conclusions
We have derived an approximate expression (Equation (63)) and its approximate form with negligible errors (Equation (66)) for the Laplace transformμ(s) of the transient gel electrophoretic mobility µ(t) of a sphere in a polymer gel medium. Equations (63) and (66) are the generalization of the result of Saad and Faltas [40] and are applicable for low zeta potentials and arbitrary values of κa. Equation (66), in particular, which does not involve exponential integrals, is convenient for practical use. It is shown that the relaxation time T for the transient gel electrophoretic mobility µ(t) to reach its steady value µ s is shorter than that for the transient free-solution electrophoretic mobility, and T becomes shorter as λa increases.