Transient Electrophoresis in Suspensions of Charged Porous Particles

Abstract

The start-up of electrophoretic motion in a suspension of uniformly charged, porous, spherical particles within an arbitrary electrolyte solution under a suddenly applied electric field is investigated. The unsteady Stokes/Brinkman equations, modified to include the electric body force, are solved for the fluid velocity field using a unit cell model to account for the particle-particle interactions. An explicit expression for the transient electrophoretic velocity of a porous particle in a unit cell is derived in the Laplace transform domain as a function of the key governing parameters. The transient electrophoretic velocity, when normalized by its steady-state counterpart, increases monotonically with both elapsed time and the ratio of particle radius to Debye length, with other parameters held constant. It generally increases with the ratio of particle radius to permeation length and with porosity, while decreasing monotonically with an increase in the particle-to-fluid density ratio. Similar to its steady-state value, the transient electrophoretic mobility of the suspension is typically a decreasing function of the particle volume fraction. However, under conditions of small elapsed time and large density ratio, the transient mobility may exhibit an initial increase with particle volume fraction.

1. Introduction

When an electric field is applied to a charged particle suspended in an electrolyte solution, the particle undergoes electrophoresis while the surrounding counterions and co-ions migrate, thereby inducing fluid motion. Electrophoresis represents one of the most widely utilized experimental techniques and operational methods for the manipulation, transport, assembly, separation, and characterization of colloidal particles. Analytical expressions for the steady electrophoretic mobility of charged particles have previously been derived for several idealized models, including the hard sphere (impermeable to solvent and ions) [1,2,3,4,5,6], the porous sphere (permeable) [7,8,9,10], and the soft sphere (with a rigid core and a porous surface layer) [11,12,13]. Although the original theoretical framework of electrophoresis focused on steady-state motion, transient effects are equally significant, since alternating or pulsed electric fields are frequently employed in measurements and applications involving colloidal electrophoresis [14,15,16,17,18,19,20,21]. In particular, the time-dependent behavior of particle mobility plays an essential role in high-speed colloidal processes within the sub-millisecond range, including electrophoretic separations in biochip devices, and the use of short-duration pulsed fields to separate particle migration from background electroosmotic flow or to suppress thermal broadening [22].

Assuming the fluid within the electric double layer surrounding a charged particle instantaneously reaches its electroosmotic terminal velocity when a uniform electric field is imposed, Morrison derived a closed-form solution for the starting electrophoretic velocity of a hard sphere under the thin double layer condition ($\kappa a\to \infty$, with $a$ being the particle radius and ${\kappa }^{-1}$ the Debye length) [23]. Subsequently, the transient start-up electrophoresis of a hard sphere [24] and a soft sphere [25] with a thin yet finite double layer ($\kappa a\ge 10$) was analyzed approximately by considering the dynamic relaxation of electroosmotic flow in the double layer under a suddenly applied electric field. On the other hand, theoretical investigations extended to transient electrophoresis of hard spheres [26,27], hard cylinders [28], and porous spheres (without consideration of particle porosity and particle volume fraction) [29] with arbitrary values of $\kappa a$ but small zeta potentials or fixed-charge densities, in response to a step change in the applied field.

In both steady and transient electrophoresis, concentrated suspensions of particles are frequently encountered. To evaluate particle–particle interactions on mean mobility, the unit cell model has been widely applied [30,31,32,33,34,35]. This model represents a homogeneous dispersion as an assembly of identical spherical cells, each containing one central particle and the surrounding concentric shell of fluid, thereby reducing the many-body suspension problem to a tractable single-cell formulation. Comparisons between measured mobilities in suspensions of charged particles and predictions of the cell model have shown good agreement across wide ranges of $\kappa a$ values and particle concentrations [36].

By adopting the unit cell model, the transient electrophoretic behavior of a suspension of identical hard spheres was investigated, taking into account dynamic electroosmosis in thin but finite double layers ($\kappa a\ge 10$) following the sudden application of an electric field [37]. More recently, the transient electrophoretic response of hard-sphere suspensions with arbitrary $\kappa a$ but low zeta potential was also analyzed (with no effects of particle porosity and particle permeability) [38,39]. In this study, we present an analysis of the start-up electrophoresis of suspensions composed of charged porous spheres with arbitrary $\kappa a$, particle porosity, particle permeability, particle volume fraction, and other relevant parameters under a suddenly imposed electric field. An explicit formula for the transient electrophoretic mobility of porous spheres is obtained in the Laplace transform domain.

2. Analysis

The starting electrophoresis in a homogeneous suspension of identical charged porous spheres in a bounded solution of arbitrary electrolytes is considered. At the time $t=0$, a uniform electric field $E_{\infty }{e}_z$ is suddenly applied and carries on hereafter, where $e_z$ is the unit vector in the $z$ direction. Consequently, the particles undergo electrophoretic motion with a transient velocity equal to $U(t)e_z$ [knowing that $U(0)=0$] to be determined. As shown in Figure 1, we utilize the unit cell model in which each particle of radius $a$ is enclosed by a concentric shell of the electrolyte solution of outer radius $b$, and $\varphi ={(a/b)}^3$ represents the particle volume fraction of the suspension. The origin of the spherical coordinates $(r,\theta ,\phi )$ is set at the particle center, and the problem for a unit cell is axially symmetric about the $z$-axis (which coincides with $\theta =0$).

The fluid velocity $v(r,\theta )$ (satisfying the continuity equation $\nabla \cdot v=0$) and hydrodynamic pressure $p(r,\theta )$ of the electrokinetic flow is governed by the modified transient Stokes/Brinkman equations [40],

$$[1-h(r)(1-{\epsilon }_{\mathrm{p}})]\rho {\displaystyle \frac{\partial v}{\partial t}}-\eta {\nabla }^{2}v+h(r)f(v-U{e}_z)+\nabla p=\epsilon {\nabla }^2\psi \nabla \psi ,$$

(1)

where $\psi (r,\theta )$, $\eta$, $\rho$, and $\epsilon$ are the electric potential, viscosity, mass density, and dielectric permittivity, respectively, of the fluid, ${\epsilon }_{\mathrm{p}}$ and $f$ are the porosity and hydrodynamic friction coefficient per unit volume, respectively, of the particle, $h(r)$ is a step function equal to unity if $0\le r\le a$ and zero if $a<r\le b$, and Poisson’s equation is incorporated. Note that the effects of the porosity ${\epsilon }_{\mathrm{p}}$ in Equation (1) was not considered in a previous analysis [29] on the starting electrophoretic motion of an isolated porous sphere. The steady-state electrophoretic velocity $U(t\to \infty )=U_{\infty }$, which does not directly depend on the porosity and density of the particle, is available in the literature [34,41].

The electrical potential profile $\psi$ of the electrolyte solution in a unit cell can be expressed as the equilibrium potential profile ${\psi }_{\mathrm{eq}}(r)$ caused by the fixed charge within the porous sphere and ambient diffuse ions, which is assumed to be relatively small, added to the potential profile induced by the applied electric field $E_{\infty }{e}_z$,

$$\psi ={\psi }_{\mathrm{eq}}-E_{\infty }r\cos \theta .$$

(2)

The equilibrium potential satisfying the linearized Poisson–Boltzmann equation (under the Debye–Hückel approximation) and appropriate boundary conditions of the uniformly charged porous sphere in the cell is [34]

$${\psi }_{\mathrm{eq}}={\displaystyle \frac{Q}{\epsilon {\kappa }^2}}\{1-\left[\right(\kappa b-\kappa a)\cosh (\kappa b-\kappa a)+({\kappa }^{2}ab-1)\sinh (\kappa b-\kappa a\left)\right]{\displaystyle \frac{\sinh \left(\kappa r\right)}{\alpha \left(\kappa b\right)\kappa r}}\} \mathrm{if} 0\le r\le a,$$

(3)

$${\psi }_{\mathrm{eq}}={\displaystyle \frac{\alpha (\kappa a)Q}{\alpha \left(\kappa b\right)\epsilon {\kappa }^{3}r}}[\kappa b\cosh (\kappa b-\kappa r)-\sinh (\kappa b-\kappa r)] \mathrm{if} a\le r\le b,$$

(4)

where

$$\alpha \left(x\right)=x\cosh x-\sinh x,$$

(5)

$Q$ is the fixed-charge density within the porous sphere, which is assumed to be a constant, and $\kappa$ is the reciprocal of the Debye length.

Taking the curl of Equation (1) after the substitution of Equation (2), we obtain

$$[1-h(r)(1-{\epsilon }_{\mathrm{p}})]{\displaystyle \frac{1}{\nu }}{\displaystyle \frac{\partial }{\partial t}}\nabla \times v-[{\nabla }^2-h(r){\lambda }^2]\nabla \times v=-{\displaystyle \frac{\epsilon }{\eta }}{E}_{\infty }\nabla \times (e_z{\nabla }^2{\psi }_{\mathrm{eq}}),$$

(6)

where ${\lambda }^2=f/\eta$ is the reciprocal of the fluid permeability in the porous particle and $\nu =\eta /\rho$ is the kinematic viscosity of the fluid. The nonvanishing components of $v$ can be expressed in terms of the Stokes stream function $\Psi$, which immediately satisfies the continuity equation $\nabla \cdot v=0$,

$$v_r=-{\displaystyle \frac{1}{r^2\sin \theta }}{\displaystyle \frac{\partial \Psi }{\partial \theta }}, v_{\theta }={\displaystyle \frac{1}{r\sin \theta }}{\displaystyle \frac{\partial \Psi }{\partial r}}.$$

(7)

The application of Equation (7) to Equation (6) results in

$$E^2(E^2-{\lambda }^2-{\displaystyle \frac{{\epsilon }_{\mathrm{p}}}{\nu }}{\displaystyle \frac{\partial }{\partial t}}){\Psi }^{(\mathrm{i})}=-{\displaystyle \frac{\epsilon {\kappa }^2}{\eta }}{E}_{\infty }r{\displaystyle \frac{\mathrm{d}{\psi }_{\mathrm{eq}}}{\mathrm{d}r}}{\sin }^2\theta \mathrm{if} 0\le r\le a,$$

(8)

$$E^2(E^2-{\displaystyle \frac{1}{\nu }}{\displaystyle \frac{\partial }{\partial t}}){\Psi }^{(\mathrm{o})}=-{\displaystyle \frac{\epsilon {\kappa }^2}{\eta }}{E}_{\infty }r{\displaystyle \frac{\mathrm{d}{\psi }_{\mathrm{eq}}}{\mathrm{d}r}}{\sin }^2\theta \mathrm{if} a\le r\le b,$$

(9)

where the superscripts (i) and (o) represent the internal and external flows, respectively, and $E^2$ is the Stokes operator given by

$$E^2={\displaystyle \frac{{\partial }^2}{\partial r^2}}+{\displaystyle \frac{\sin \theta }{r^2}}{\displaystyle \frac{\partial }{\partial \theta }}({\displaystyle \frac{1}{\sin \theta }}{\displaystyle \frac{\partial }{\partial \theta }}).$$

(10)

The initial and boundary conditions of the flow field are

$$t=0: v^{(\mathrm{i})}=v^{(\mathrm{o})}=0,$$

(11)

$$r=a: v \mathrm{and} \tau -pI \mathrm{are} \mathrm{continuous},$$

(12)

$$r=b: v_r^{(\mathrm{o})}=0,$$

(13)

$${\tau }_{r\theta }^{(\mathrm{o})}=\eta \{r{\displaystyle \frac{\partial }{\partial r}}[{\displaystyle \frac{v_{\theta }^{(\mathrm{o})}}{r}}]+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial v_r^{(\mathrm{o})}}{\partial \theta }}\}=0 (\mathrm{for}\mathrm{the}\mathrm{Happel}\mathrm{model}),$$

(14)

$${[\nabla \times v^{(\mathrm{o})}]}_{\phi }={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}[r{v}_{\theta }^{(\mathrm{o})}]-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial v_r^{(\mathrm{o})}}{\partial \theta }}=0 (\mathrm{for}\mathrm{the}\mathrm{Kuwabara}\mathrm{model}),$$

(15)

where $\tau =\eta [\nabla v+{(\nabla v)}^{\mathrm{T}}]$ is the deviatoric stress tensor and $\mathbf{I}$ is the unit tensor.

The internal and external stream functions can be expressed as

$${\Psi }^{(\mathrm{i},\mathrm{o})}=g^{(\mathrm{i},\mathrm{o})}(r,t){\sin }^2\theta .$$

(16)

In terms of the Laplace transform (with a bar over the variable); Equations (8) and (9) with the substitution of Equation (16) can be expressed as

$$({\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{r}^2}}-{\displaystyle \frac{2}{r^2}})({\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{r}^2}}-{\displaystyle \frac{2}{r^2}}-{\lambda }^2-{\displaystyle \frac{{\epsilon }_{\mathrm{p}}s}{\nu }}){\overline{g}}^{(\mathrm{i})}(r,s)={\displaystyle \frac{r}{s}}G(r),$$

(17)

$$({\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{r}^2}}-{\displaystyle \frac{2}{r^2}})({\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{r}^2}}-{\displaystyle \frac{2}{r^2}}-{\displaystyle \frac{s}{\nu }}){\overline{g}}^{(\mathrm{o})}(r,s)={\displaystyle \frac{r}{s}}G(r),$$

(18)

where $s$ is the transform parameter and

$$G(r)=-{\displaystyle \frac{\epsilon }{\eta }}{\kappa }^2{E}_{\infty }{\displaystyle \frac{\mathrm{d}{\psi }_{\mathrm{eq}}}{\mathrm{d}r}}.$$

(19)

The general solution of Equations (8) and (9) or Equations (17) and (18) that satisfies the initial condition in Equation (11) is

$${\overline{\Psi }}^{(\mathrm{i})}={\left\{\right[\mathrm{C}}_1{({\displaystyle \frac{r}{a}})}^3+C_2+C_3\alpha (Br)+C_4\beta (Br)]{\displaystyle \frac{a}{r}}+{\overline{g}}_{\mathrm{p}}^{\left(\mathrm{i}\right)}(r,s)\}{\sin }^2\theta \mathrm{if} 0\le r\le a,$$

(20)

$${\overline{\Psi }}^{(\mathrm{o})}={\left\{\right[\mathrm{C}}_5{({\displaystyle \frac{r}{a}})}^3+C_6+C_7\gamma (Ar)+C_8\gamma (-Ar)]{\displaystyle \frac{a}{r}}+{\overline{g}}_{\mathrm{p}}^{\left(\mathrm{o}\right)}(r,s)\}{\sin }^2\theta \mathrm{if} a\le r\le b,$$

(21)

with the particular solution

$${\overline{g}}_{\mathrm{p}}^{\left(\mathrm{i}\right)}={\displaystyle \frac{1}{3B^{2}sr}}[J_3(r)-r^3{J}_0(r)]-{\displaystyle \frac{1}{B^{5}sr}}[\alpha (Br)I_{\beta }(r)-\beta (Br)I_{\alpha }(r)],$$

(22)

$${\overline{g}}_{\mathrm{p}}^{\left(\mathrm{o}\right)}={\displaystyle \frac{1}{3A^{2}sr}}[J_3(r)-r^3{J}_0(r)]-{\displaystyle \frac{1}{2A^{5}sr}}[\gamma (Ar)I_{\gamma -}(r)-\gamma (-Ar)I_{\gamma +}(r)],$$

(23)

where

$$\beta \left(x\right)=x\sinh x-\cosh x,$$

(24)

$$\gamma (x)=(1-x){\mathrm{e}}^x,$$

(25)

$$J_n(r)={\displaystyle {\int }_a^r{r}^{n}G(r)\mathrm{d}r,}$$

(26)

$$I_{\alpha }(r)={\displaystyle {\int }_a^r\alpha (Br)G(r)\mathrm{d}r,}$$

(27)

$$I_{\beta }(r)={\displaystyle {\int }_a^r\beta (Br)G(r)\mathrm{d}r,}$$

(28)

$$I_{\gamma \pm }(r)={\displaystyle {\int }_a^r\gamma (\pm Ar)G(r)\mathrm{d}r,}$$

(29)

$A=\sqrt{s/\nu }$, and $B=\sqrt{{\lambda }^2+{\epsilon }_{\mathrm{p}}s/\nu }$.

The unknown constants $C_1\sim C_8$ in Equations (20) and (21) can be derived from the boundary conditions in Equations (12)–(15) and expressed as follows:

$$C_1=M_1{a}^2\overline{U}+M_2{a}^3,$$

(30)

$$C_2=-{\displaystyle \frac{J_3(0)}{3sa{B}^2},}$$

(31)

$$C_3=M_3{a}^2\overline{U}+M_4{a}^3,$$

(32)

$$C_4=-{\displaystyle \frac{I_{\alpha }(0)}{sa{B}^5,}}$$

(33)

$$C_5=M_5{a}^2\overline{U}+M_6{a}^3,$$

(34)

$$C_6=M_7{a}^2\overline{U}+M_8{a}^3,$$

(35)

$$C_7=M_9{a}^2\overline{U}+M_{10}{a}^3,$$

(36)

$$C_8=M_{11}{a}^2\overline{U}+M_{12}{a}^3,$$

(37)

where the dimensionless coefficients $M_1\sim M_{12}$ are lengthy functions of related parameters that can be provided on request.

The substitution of Equations (20) and (30)–(33) into Equation (7) yields the Laplace transform of the fluid velocity inside the porous sphere,

$${\overline{v}}_r^{(\mathrm{i})}=-F(r)\cos \theta ,$$

(38)

$${\overline{v}}_{\theta }^{(\mathrm{i})}={\displaystyle \frac{\partial }{2r\partial r}}[r^{2}F(r)]\sin \theta ,$$

(39)

where

$$\begin{gathered} F(r)={\displaystyle \frac{2a}{r^3}}[C_1{({\displaystyle \frac{r}{a}})}^3+C_2+C_3\alpha (Br)+C_4\beta (Br)] \\ +{\displaystyle \frac{2}{3B^2{r}^{3}s}}[J_3(r)-r^3{J}_0(r)]+{\displaystyle \frac{2}{B^5{r}^{3}s}}[\beta (Br)I_{\alpha }(r)-\alpha (Br)I_{\beta }(r)]. \end{gathered}$$

(40)

The electrostatic force acting on the porous sphere is

$$F_{\mathrm{e}}={\displaystyle \frac{4}{3}}\mathsf{\pi }{a}^{3}Q{E}_{\infty },$$

(41)

in the $z$ direction. The hydrodynamic force acting on the porous sphere is

$$F_{\mathrm{h}}=2\mathsf{\pi }{\displaystyle {\int }_0^{\mathsf{\pi }}{\displaystyle {\int }_0^a{e}_z\cdot f(v^{(\mathrm{i})}-U{e}_z)}}{r}^2\sin \theta \mathrm{d}r\mathrm{d}\theta ,$$

(42)

which is also in the $z$ direction, dependent on the friction coefficient $f=\eta {\lambda }^2$, and opposite to $F_{\mathrm{e}}$ in sign. Substituting Equations (38) and (39) into Equation (42), we obtain

$${\overline{F}}_{\mathrm{h}}=-{\displaystyle \frac{4}{3}}\mathsf{\pi }{a}^3\eta {\lambda }^2[F(a)+\overline{U}].$$

(43)

The summation of the electric force and hydrodynamic force is equal to the product of the mass and acceleration of the particle:

$$F_{\mathrm{e}}+F_{\mathrm{h}}={\displaystyle \frac{4}{3}}\mathsf{\pi }{a}^3(1-{\epsilon }_{\mathrm{p}}){\rho }_{\mathrm{p}}{\displaystyle \frac{\mathrm{d}U}{\mathrm{d}t,}}$$

(44)

where ${\rho }_{\mathrm{p}}$ is the mass density of the solid fragment of the porous particle. Taking the Laplace transform of Equation (44) and substituting Equations (41) and (43) into it, we obtain a transformed equation for the response of electrophoresis to a suddenly applied electric field,

$$\begin{gathered} \overline{U}={\displaystyle \frac{1}{3s{B}^5{a}^3}}{\{\eta {\lambda }^2[1+2M_1+2\alpha \left(Ba\right)M_3]+s(1-{\epsilon }_{\mathrm{p}}\left){\rho }_{\mathrm{P}}\right\}}^{-1} \\ \times [3B^5{a}^{3}Q{E}_{\infty }+2\eta {\lambda }^2\{B^3{J}_3\left(0\right)+3\beta \left(aB\right)I_{\alpha }(0)-3s{B}^5{a}^4[M_2+\alpha (aB)M_4\left]\right\}]. \end{gathered}$$

(45)

The particle’s transient velocity $U$ can be obtained by numerical inverse Laplace transform [42,43] of Equation (45).

3. Results and Discussion

For the case of a single charged porous sphere [$\phi ={(a/b)}^3=0$] at maximum porosity (${\epsilon }_{\mathrm{p}}=1$), the transient electrophoretic velocity $U$, normalized by the terminal electrophoretic velocity $U_{\infty }$ [7] was obtained using the numerical inverse transform of a simplified form of Equation (45) for various values of the dimensionless elapsed time $\nu t/a^2$, the ratio of particle radius to Debye length $\kappa a$, the ratio of particle radius to permeation length $\lambda a$, and the particle-to-fluid density ratio ${\rho }_{\mathrm{p}}/\rho$ [29]. The normalized electrophoretic velocity $U/U_{\infty }$ was found to increase monotonically with $\nu t/a^2$ (developing continuously from zero at $t=0$ to unity as $t\to \infty$) and with $\kappa a$, but to decrease monotonically with ${\rho }_{\mathrm{p}}/\rho$ (developing more slowly for heavier particles and vanishing in the limit ${\rho }_{\mathrm{p}}/\rho \to \infty$), keeping all other parameters fixed. Moreover, $U/U_{\infty }$ generally rises with increasing $\lambda a$ (or decreasing permeability of the porous sphere) for given values of $\nu t/a^2$, $\kappa a$, and ${\rho }_{\mathrm{p}}/\rho$, though it may decline again once $\lambda a$ becomes relatively large. The results of the transient electrophoretic velocity obtained in the present work under the same conditions ($\phi =0$ and ${\epsilon }_{\mathrm{p}}=1$) agree very well with these previous results [29] of a single charged porous sphere at maximum porosity.

The normalized electrophoretic velocity $U/U_{\infty }$ of a single charged porous sphere with various values of the porosity ${\epsilon }_{\mathrm{p}}$ is shown in Figure 2, Figure 3, Figure 4 and Figure 5 as a function of the parameters $\nu t/a^2$, $\lambda a$, $\kappa a$, and ${\rho }_{\mathrm{p}}/\rho$, respectively. These results reveal that $U/U_{\infty }$ typically increases with ${\epsilon }_{\mathrm{p}}$ from ${\epsilon }_{\mathrm{p}}\to 0$ (the particle is essentially impermeable) to ${\epsilon }_{\mathrm{p}}\to 1$, indicating that particles of lower porosity exhibit longer relaxation times and lag behind particles of higher porosity in the buildup of electrophoretic velocity. On the other hand, the normalized sedimentation velocity of the porous sphere may slightly decrease with an increase in ${\epsilon }_{\mathrm{p}}$ when the value of ${\rho }_{\mathrm{p}}/\rho$ is relatively small [40]. For finite ${\epsilon }_{\mathrm{p}}$, Figure 2, Figure 3, Figure 4 and Figure 5 further demonstrate that $U/U_{\infty }$ develops steadily with time from zero at $t=0$ to unity as $t\to \infty$, generally increases with $\lambda a$ to a maximum before falling with further increases in $\lambda a$, increases monotonically with $\kappa a$, and decreases monotonically with ${\rho }_{\mathrm{p}}/\rho$. However, the normalized sedimentation velocity decreases monotonically with an increase in $\lambda a$ [40].

The results for the dimensionless starting electrophoretic velocity $\eta U/a^{2}Q{E}_{\infty }$ of porous spheres with constant fixed-charge density $Q$ in a suspension, obtained from numerical inverse transforms of Equation (45) for both Happel and Kuwabara cell models, are plotted against the dimensionless time $\nu t/a^2$ in Figure 6, Figure 7 and Figure 8 for various values of the particle volume fraction $\phi$, the ratio of particle radius to permeation length $\lambda a$, and the porosity ${\epsilon }_{\mathrm{p}}$, respectively. In contrast, this dimensionless velocity is plotted against the particle volume fraction $\phi$ in Figure 9 and Figure 10 for various values of the particle-to-fluid density ratio ${\rho }_{\mathrm{p}}/\rho$ and the ratio of particle radius to Debye length $\kappa a$, respectively. The data are shown up to $\phi =0.74$, which represents the maximum achievable packing for a collection of identical spheres [30]. The steady electrophoretic velocity $\eta U_{\infty }/a^{2}Q{E}_{\infty }$ is independent of ${\epsilon }_{\mathrm{p}}$ and ${\rho }_{\mathrm{p}}/\rho$, and is generally a decreasing function of $\phi$, consistent with published findings [34,41]. The effect of $\phi$ and other relevant parameters on the time evolution of $\eta U/a^{2}Q{E}_{\infty }$ are significant.

For given $\nu t/a^2$, ${\rho }_{\mathrm{p}}/\rho$, ${\epsilon }_{\mathrm{p}}$, and $\phi$, as in Figure 7 and Figure 10, the transient electrophoretic velocity $\eta U/a^{2}Q{E}_{\infty }$ (non-dimensionlized with the chosen length scale $a$) decreases with increasing $\lambda a$ and $\kappa a$, in the same manner as the steady velocity $\eta U_{\infty }/a^{2}Q{E}_{\infty }$ [41]. Furthermore, as in Figure 8, $\eta U/a^{2}Q{E}_{\infty }$ increases with increasing ${\epsilon }_{\mathrm{p}}$ from ${\epsilon }_{\mathrm{p}}\to 0$ to ${\epsilon }_{\mathrm{p}}\to 1$, holding other parameters constant. For given $\nu t/a^2$, ${\rho }_{\mathrm{p}}/\rho$, $\kappa a$, $\lambda a$, and ${\epsilon }_{\mathrm{p}}$, the transient velocity $\eta U/a^{2}Q{E}_{\infty }$ generally decreases with $\phi$, as seen in Figure 6 and at steady state, though it may increase with $\phi$ when $\nu t/a^2$ is small, ${\rho }_{\mathrm{p}}/\rho$ is large, and $\phi$ is small, as in Figure 9 and Figure 10. For all combinations of $\nu t/a^2$, ${\rho }_{\mathrm{p}}/\rho$, $\kappa a$, $\lambda a$, ${\epsilon }_{\mathrm{p}}$, and $\phi$, as illustrated in Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10, the Kuwabara cell model predicts smaller values of $\eta U/a^{2}Q{E}_{\infty }$ than the Happel model, because the zero-vorticity condition in the Kuwabara model induces greater energy dissipation in the cell than particle drag alone due to the added stress at the outer cell boundary [44], but the discrepancies are generally negligible. The dependence of the dimensionless transient electrophoretic velocity $\eta U/a^{2}Q{E}_{\infty }$ on the parameters $\nu t/a^2$, ${\rho }_{\mathrm{p}}/\rho$, $\kappa a$, and $\phi$ shown in the present work is consistent with that of the dimensionless transient electrophoretic velocity $\eta U/a\sigma E_{\infty }$ on these parameters [38] for a suspension of hard spheres with constant surface charge density $\sigma$.

4. Conclusions

In this work, the starting electrophoretic motion in a suspension of uniformly charged, porous, spherical particles within an arbitrary electrolyte solution caused by the sudden application of an electric field is analyzed. The unsteady Stokes/Brinkman equations, modified to include the electric force, are solved using a unit cell model. An explicit formula for the transient electrophoretic velocity of the porous particle in a unit cell is obtained in the Laplace transform domain as a function of related parameters. The transient electrophoretic velocity normalized by its steady-state value increases monotonically with increases in the scaled elapsed time $\nu t/a^2$ and the ratio of particle radius to Debye length $\kappa a$, generally increases with increases in the ratio of particle radius to permeation length $\lambda a$ and the porosity ${\epsilon }_{\mathrm{p}}$, but decreases monotonically with an increase in the particle-to-fluid density ratio ${\rho }_{\mathrm{p}}/\rho$, with other parameters held constant. Similar to its steady-state value, the transient electrophoretic mobility is generally a decreasing function of the particle volume fraction $\phi$ of the suspension, but it may exhibit an initial increase with $\phi$ when $\nu t/a^2$ is small and ${\rho }_{\mathrm{p}}/\rho$ is large.