Transient Electroosmosis on a Soft Surface

Abstract

A general theory was developed for the time-dependent transient electroosmosis on a planar soft surface, i.e., a polyelectrolyte-coated solid surface in an electrolyte solution, when an electric field is suddenly applied. This serves as a simple model for the time-dependent electrokinetic phenomena occurring at biointerfaces. A closed-form approximate expression is derived for the electroosmotic velocity distribution within the polyelectrolyte layer as a function of both position and time. This analysis reveals that the temporal and spatial variations in the electroosmotic flow caused by the surface charges of the solid surface is confined to the region near the solid surface. In contrast, the variations due to the fixed charges within the polyelectrolyte layer extend over a wider region inside the polyelectrolyte layer.

1. Introduction

The measurement of the electrophoretic mobility of colloidal particles is a fundamental experimental technique for evaluating their zeta potential [1,2,3,4,5,6,7,8,9,10,11,12,13]. The zeta potential is a key parameter that governs the stability of colloidal dispersions and serves as a crucial factor in promoting dispersion stability. A thorough understanding of the zeta potential has significant implications for various applications, including pharmaceuticals, food science, and environmental systems. It provides insights into particle interactions, aggregation behavior, and the stability of colloidal dispersions under various conditions.

Traditionally, the focus of electrophoretic studies and their theoretical analyses has been on steady-state electrophoresis, which assumes the constant motion of colloidal particles in response to an applied static electric field. While this approach has yielded valuable information, it inherently overlooks dynamic processes that occur during the initial stages of electrophoresis, such as transient fluid flow. In contrast, transient electrophoresis, which captures the time-dependent behavior of colloidal particles in an electric field, provides a unique perspective. It reveals dynamic phenomena that cannot be observed under steady-state conditions, offering deeper insights into the interplay of electrostatic forces, hydrodynamic effects, and ion distributions. Numerous theoretical studies have been conducted on the transient electrokinetics of colloidal particles [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]. The theoretical groundwork for transient electrophoresis and electroosmosis (which is an electrokinetic phenomenon closely related to electrophoresis) was laid by Morison [14,15] and Ivory [16,17], with significant advancements later contributed by Keh and his collaborators [18,19,20,22,23,24,25,26]. A wide range of theoretical studies has examined transient electrophoresis, covering various types of particles such as rigid spheres [14,17,19,20,22,26,27,30], rigid cylinders [15,23,24,29], porous particles [25], permselective particles [21], and soft particles [28]. Beyond the transient free-solution electrophoresis theories outlined above, additional research has investigated transient gel electrophoresis, focusing on the behavior of colloidal particles undergoing electrophoresis within polymer gel matrices [31,32,33,34].

This paper investigates the problem of transient electroosmosis in systems consisting of planar solid surfaces coated with charged or neutral polymers. Electroosmosis refers to the flow of a liquid induced by an electric field near a charged surface, and its transient counterpart provides valuable information about the initial response of the system to electric stimuli. The study of transient electroosmosis is not only relevant for fundamental research but also holds potential for advancing technologies in microfluidics, biointerfaces, and colloidal assembly. Polyelectrolyte-coated surfaces considered in the present paper, often referred to as “soft surfaces,” serve as model systems for biological interfaces, including cell membranes and mucosal layers [35]. Understanding the behavior of soft surfaces under transient conditions is essential for elucidating the mechanisms governing biological colloids and their interactions with surrounding media.

By focusing on transient electroosmotic phenomena at soft surfaces, this study aims to bridge the gap between steady-state analyses and dynamic processes. The findings presented herein are expected to contribute to the broader understanding of electrokinetic phenomena in soft matter systems and their practical implications in various scientific and industrial domains.

2. Theory

Consider the system of a planar soft surface, that is, a planar solid surface covered with an ion-penetrable surface layer of polyelectrolytes of thickness D, immersed in an aqueous liquid with relative permittivity ε_r, mass density ρ₀, and viscosity η, which contains a general electrolyte consisting of N ionic species with valence z_i, and bulk concentration (number density) n_i (i = 1,…, N). From the electroneutrality condition for the bulk phase of the electrolyte solution, we obtain

$${\displaystyle \sum }_{i=1}^N{z}_{i}e{n}_i=0$$

(1)

We adopt a Cartesian coordinate system (x, y, z) with its origin O(0, 0, 0) located at the solid surface, where the x-axis is normal to the surface, and the z-axis is parallel to it, as shown in Figure 1.

Let the surface charge density of the solid surface be σ, and the volume charge density of the fixed charges in the polyelectrolyte layer be ρ_fix.

We now apply a step electric field E(t) (0, 0, E(t)) in the z-direction at time t = 0, where E(t) is the z-component of E(t), viz.,

$$E\left(t\right)=\{\begin{array}{c}0, t=0 \\ E, t>0\end{array}$$

(2)

Here, E is constant, and the liquid starts to move with an electroosmotic velocity u(x, t) (0, 0, u(x, t)) in the direction parallel to E(t). Note that the liquid velocity u(x, t) and its z-component u(x, t) are functions of the normal distance x from the solid surface and time t. We treat the case where the following conditions are satisfied: (i) The thickness D of the polyelectrolyte layer is much larger than both the Debye length 1/κ and the Brinkmann screening length 1/λ, where κ and λ are the Debye–Hückel parameter and the Brinkmann parameter, respectively (as defined later in Equations (6) and (18)). Under this condition, the thickness D of the polyelectrolyte layer can effectively be considered infinite. (ii) The liquid can be regarded as incompressible. (iii) The applied electric field E(t) is weak so that the liquid velocity u(x, t) is proportional to E(t) and terms involving the square of the liquid velocity in the Navier–Stokes equation can be neglected. (iv) The slipping plane (at which the liquid velocity u(x, t) is zero) is located on the solid surface (at x = 0). (v) In the absence of E(t), the equilibrium ion distribution ψ(x) within the polyelectrolyte layer, which is a function of x, obeys the Boltzmann distribution, and the equilibrium electric potential is described by the Poisson–Boltzmann equation. (vi) The soft surface is weakly charged; that is, σ and ρ_fix are sufficiently low. (vii) We adopt the Brinkman–Debye–Bueche model [36,37], which assumes that polymer segments in the polyelectrolyte layer exert a frictional force on the liquid flow within the polyelectrolyte layer with a frictional coefficient ν_s.

Under the above conditions, the Poisson–Boltzmann equation for ψ(x) is given by

$${\displaystyle \frac{d^2\psi \left(x\right)}{d{x}^2}}=-{\displaystyle \frac{{\rho }_{\mathrm{el}}\left(x\right)}{{\epsilon }_{\mathrm{r}}{\epsilon }_0}}-{\displaystyle \frac{{\rho }_{\mathrm{fix}}}{{\epsilon }_{\mathrm{r}}{\epsilon }_0}}, 0<x<+\infty$$

(3)

with

$${\rho }_{\mathrm{el}}\left(x\right)={\displaystyle \sum }_{i=1}^N{z}_{i}e{n}_i\exp \left[-{\displaystyle \frac{z_{i}e\psi \left(x\right)}{kT}}\right]$$

(4)

where ρ_el(x) is the space charge density due to the electrolyte ions, e is the elementary electric charge, k is the Boltzmann constant, and T is the absolute temperature. Note here that the second term on the right-hand side of Equation (3) represents the contribution of the fixed charges with a density ρ_fix distributed within the polyelectrolyte layer.

For the case of a weakly charged soft surface, Equation (3) reduces to

$${\displaystyle \frac{d^2\psi \left(x\right)}{d{x}^2}}={\kappa }^2\psi \left(x\right)-{\displaystyle \frac{{\rho }_{\mathrm{fix}}}{{\epsilon }_{\mathrm{r}}{\epsilon }_0}}$$

(5)

with

$$\kappa =\sqrt{{\displaystyle \frac{{{\displaystyle \sum }}_{i=1}^N{z}_i^2{e}^2{n}_i^2}{{\epsilon }_{\mathrm{r}}{\epsilon }_{0}kT}}}$$

(6)

where κ is the Debye–Hückel parameter, ε₀ is the permittivity of a vacuum. Note that, since D >> 1/κ, the region 0 < x < D for ψ(x) has been approximately replaced by 0 < x < +∞ (condition (i)). The boundary condition for ψ(x) at x = 0 is

$${{\displaystyle \frac{d\psi \left(x\right)}{dx}}|}_{x=0^{+}}=-{\displaystyle \frac{\sigma }{{\epsilon }_{\mathrm{r}}{\epsilon }_0}}$$

(7)

Equation (5) subject to Equation (7) can be solved to yield

$$\psi \left(x\right)={\displaystyle \frac{\sigma }{{\epsilon }_{\mathrm{r}}{\epsilon }_0\kappa }}{e}^{-\kappa x}+{\displaystyle \frac{{\rho }_{\mathrm{fix}}}{{\epsilon }_{\mathrm{r}}{\epsilon }_0{\kappa }^2}}$$

(8)

and Equation (4) becomes

$${\rho }_{\mathrm{el}}\left(x\right)=-\kappa \sigma e^{-\kappa x}-{\rho }_{\mathrm{fix}}$$

(9)

Equation (8) shows that, in the bulk region within the polyelectrolyte layer (x » 1/κ), the electric potential ψ(x) is practically equal to the Donnan potential ψ_DON given by

$${\psi }_{\mathrm{DON}}={\displaystyle \frac{{\rho }_{\mathrm{fix}}}{{\epsilon }_{\mathrm{r}}{\epsilon }_0{\kappa }^2}}$$

(10)

The z-component u(x, t) of the liquid flow velocity u(x, t) at position x and time t is assumed to obey the following Navie–Stokes equation under conditions (ii) and(iii):

$${\rho }_0{\displaystyle \frac{\partial u\left(x, t\right)}{\partial t}}-\eta {\displaystyle \frac{{\partial }^{2}u\left(x, t\right)}{\partial x^2}}+{\nu }_{\mathrm{s}}u\left(x, t\right)-{\rho }_{\mathrm{el}}\left(x\right)E\left(t\right)=0, 0<x<+\infty$$

(11)

where the third term on the left-hand side of Equation (11) corresponds to the frictional force exerted on the liquid flow by the polymer segments, as described by the Brinkman–Debye–Bueche model (condition (vii)). The initial and boundary conditions for u(x, t) are

$$u\left(x, t\right)=0 \mathrm{for} t=0$$

(12)

$$u\left(x, t\right)=0 \mathrm{for} x=0$$

(13)

where Equation (13) is the no-slip condition on the solid surface at x = 0 (condition (iv)).

By using Equation (9), Equation (11) can be rewritten as

$${\displaystyle \frac{1}{\nu }}{\displaystyle \frac{\partial u\left(x, t\right)}{\partial t}}-{\displaystyle \frac{{\partial }^{2}u\left(x, t\right)}{\partial x^2}}+{\lambda }^{2}u\left(x, t\right)+{\displaystyle \frac{1}{\eta }}\left(\kappa \sigma e^{-\kappa x}+{\rho }_{\mathrm{fix}}\right)E\left(t\right)=0$$

(14)

with

$$\nu =\sqrt{{\displaystyle \frac{\eta }{{\rho }_0}}}$$

(15)

$$\lambda =\sqrt{{\displaystyle \frac{{\nu }_{\mathrm{s}}}{\eta }}}$$

(16)

Here, ν denotes the kinematic viscosity, and λ is the Brinkman parameter, where 1/λ represents the Brinkmann screening length.

Equation (14) is most easily solved by using the Laplace transformation with respect to time t. The Laplace transforms $\widehat{u}\left(x,s\right)$ of u(x, t) is given by

$$\widehat{u}\left(x,s\right)={{\displaystyle \int }}_0^{\infty }u\left(x,t\right)e^{-st}dt$$

(17)

Then, Equation (14) becomes

$${\displaystyle \frac{s}{\nu }}\widehat{u}\left(x,s\right)-{\displaystyle \frac{{\partial }^2\widehat{u}\left(x,s\right)}{\partial x^2}}+{\lambda }^2\widehat{u}\left(x,s\right)+{\displaystyle \frac{1}{\eta s}}\left(\kappa \sigma e^{-\kappa x}+{\rho }_{\mathrm{fix}}\right)=0$$

(18)

which can be solved to give

$$\widehat{u}\left(x,s\right)={\displaystyle \frac{\kappa \sigma E}{\eta s}}\left\{{\displaystyle \frac{e^{-\kappa x}-e^{-\sqrt{\frac{s}{\nu }+{\lambda }^2}x}}{{\kappa }^2-\left(\frac{s}{\nu }+{\lambda }^2\right)}}\right\}-{\displaystyle \frac{{\rho }_{\mathrm{fix}}E}{\eta \left(\frac{s}{\nu }+{\lambda }^2\right)s}}\left(1-e^{-\sqrt{{\displaystyle \frac{s}{\nu }}+{\lambda }^2}x}\right)$$

(19)

From Equation (19), we obtain the following expression for the z-component u(x, t) of the transient electroosmotic velocity u(x, t), using the inverse Laplace transform:

$$u\left(x,t\right)=u_1\left(x,t\right)+u_2\left(x,t\right), t>0, 0<x<+\infty$$

(20)

with

$$u_1\left(x,t\right)={\displaystyle \frac{\sigma E}{\eta }}[{\displaystyle \frac{\kappa }{{\kappa }^2-{\lambda }^2}}{e}^{-\kappa x}\left\{1-e^{\left({\kappa }^2-{\lambda }^2\right)\nu t}\right\}-{\displaystyle \frac{1}{2}}{e}^{\lambda x}\mathrm{erfc}\left({\displaystyle \frac{x}{2\sqrt{\nu t}}}+\lambda \sqrt{\nu t}\right)-{\displaystyle \frac{1}{2}}{e}^{-\lambda x}\mathrm{erfc}\left({\displaystyle \frac{x}{2\sqrt{\nu t}}}-\lambda \sqrt{\nu t}\right)$$

$$+{\displaystyle \frac{1}{2}}{e}^{\left({\kappa }^2-{\lambda }^2\right)\nu t+\kappa x}\mathrm{erfc}\left({\displaystyle \frac{x}{2\sqrt{\nu t}}}+\kappa \sqrt{\nu t}\right)+{\displaystyle \frac{1}{2}}{e}^{\left({\kappa }^2-{\lambda }^2\right)\nu t-\kappa x}\mathrm{erfc}\left({\displaystyle \frac{x}{2\sqrt{\nu t}}}-\kappa \sqrt{\nu t}\right)]$$

(21)

and

$$u_2\left(x,t\right)=-{\displaystyle \frac{{\rho }_{\mathrm{fix}}E}{\eta {\lambda }^2}}\left[1-e^{-{\lambda }^2\nu t}\left\{1-\mathrm{erfc}\left({\displaystyle \frac{x}{2\sqrt{\nu t}}}\right)\right\}-{\displaystyle \frac{1}{2}}{e}^{\lambda x}\mathrm{erfc}\left({\displaystyle \frac{x}{2\sqrt{\nu t}}}+\lambda \sqrt{\nu t}\right)-{\displaystyle \frac{1}{2}}{e}^{-\lambda x}\mathrm{erfc}\left({\displaystyle \frac{x}{2\sqrt{\nu t}}}-\lambda \sqrt{\nu t}\right)\right]$$

(22)

where erfc(z) is the complementary error function, defined by

$$\mathrm{erfc}\left(z\right)={\displaystyle \frac{2}{\sqrt{\pi }}}{{\displaystyle \int }}_z^{\infty }{e}^{-x^2}dx$$

(23)

3. Results and Discussion

Equations (20)–(22) for the z-component u(x, t) of the electroosmotic flow velocity u(x, t) in the polyelectrolyte layer coating the solid surface are the principal results of the present paper. In Equation (20), u₁(x, t) represents the z-component of the electroosmotic flow velocity caused by the surface charges at x = 0, with a surface charge density σ, while u₂(x, t) corresponds to the flow velocity induced by the fixed charges distributed within the polyelectrolyte layer, characterized by a volume charge density ρ_fix. In this paper, we deal with the case of low potentials, where the potential distribution in the system is given by the sum of the potential due to the surface charge with a density σ and the potential arising from the fixed charges distributed within the polyelectrolyte layer with a density ρ_fix, as shown in Equation (8). Similarly, the velocity distribution in the system is given by the sum of the velocity due to the surface charge with a density σ and the velocity arising from the fixed charges distributed within the polyelectrolyte layer with a density ρ_fix, as shown in Equation (20).

As t→∞, u(x, t) tends to the steady-state electroosmotic velocity u(x, ∞), which is given by

$$u\left(x,\infty \right)=u_1\left(x,\infty \right)+u_2\left(x,\infty \right)$$

(24)

with

$$u_1\left(x,\infty \right)={\displaystyle \frac{\kappa \sigma E}{\eta }}{\displaystyle \frac{e^{-\kappa x}-e^{-\lambda x}}{{\kappa }^2-{\lambda }^2}}$$

(25)

$$u_2\left(x,\infty \right)=-{\displaystyle \frac{{\rho }_{\mathrm{fix}}E}{\eta {\lambda }^2}}\left(1-e^{-\lambda x}\right)$$

(26)

which can directly from Equation (19) by using the following relation:

$$u\left(x,\infty \right)=\underset{s\to 0}{\lim }\left[s\widehat{u}\left(x,s\right)\right]={\displaystyle \frac{\kappa \sigma E}{\eta }}{\displaystyle \frac{e^{-\kappa x}-e^{-\lambda x}}{{\kappa }^2-{\lambda }^2}}-{\displaystyle \frac{{\rho }_{\mathrm{fix}}E}{\eta {\lambda }^2}}\left(1-e^{-\lambda x}\right)$$

(27)

For the special case where λ = 0 and ρ_fix = 0, the soft surface reduces to a bare solid surface without a polyelectrolyte layer. Under these conditions, Equation (27) simplifies to

$$u\left(x,\infty \right)={\displaystyle \frac{\sigma E}{\eta \kappa }}\left(e^{-\kappa x}-1\right)$$

(28)

Far from the surface (x→∞), Equation (28) further reduces to

$$u\left(\infty ,\infty \right)=-{\displaystyle \frac{\sigma E}{\eta \kappa }}$$

(29)

This result is consistent with the electroosmotic flow velocity on a charged solid surface with a surface charge density σ.

3.1. Charged Solid Surface Covered with an Uncharged Polymer Layer (σ ≠ 0 and ρ_fix = 0)

First, consider the case where a charged solid surface with a surface charge density σ is covered with an uncharged polymer layer (ρ_fix = 0). In this case, Equation (8) for the equilibrium potential distribution ψ(x) becomes

$$\psi \left(x\right)={\displaystyle \frac{\sigma }{{\epsilon }_{\mathrm{r}}{\epsilon }_0\kappa }}{e}^{-\kappa x}$$

(30)

and u₂(x, t) = 0, that is, u(x, t) = u₁(x, t). Note that, in the special case of λ = 0, we find that

$$u\left(x,t\right)={\displaystyle \frac{\sigma E}{\eta \kappa }}\left\{e^{-\kappa x}\left(1-e^{{\kappa }^2\nu t}\right)-\mathrm{erfc}\left({\displaystyle \frac{x}{2\sqrt{\nu t}}}\right)+{\displaystyle \frac{1}{2}}{e}^{{\kappa }^2\nu t+\kappa x}\mathrm{erfc}\left({\displaystyle \frac{x}{2\sqrt{\nu t}}}+\kappa \sqrt{\nu t}\right)+{\displaystyle \frac{1}{2}}{e}^{{\kappa }^2\nu t-\kappa x}\mathrm{erfc}\left({\displaystyle \frac{x}{2\sqrt{\nu t}}}-\kappa \sqrt{\nu t}\right)\right\}$$

(31)

which agrees with the transient electroosmotic flow velocity on a solid surface immersed in a free electrolyte solution derived by Ivory [16].

Figure 2 shows a 3D plot of u₁(x, t) as a function of the scaled distance κx from the solid surface (x = 0) and the scaled time κ²νt, calculated using Equation (21) for the case of λ/κ = 0.5.

Figure 2 shows how u₁(x, t) approaches its steady-state value (Equation (25)) as t→∞. It reveals that the temporal and spatial variations in the electroosmotic flow caused by the surface charges on the solid surface is confined to the region near the solid surface.

3.2. Uncharged Solid Surface Covered with a Charged Polymer (Polyelectrolyte) Layer (σ = 0 and ρ_fix ≠ 0)

Next, consider the case where an uncharged solid surface (σ = 0) is covered with an uncharged polymer layer (ρ_fix = 0). In this case, Equation (8) becomes

$$\psi \left(x\right)={\displaystyle \frac{{\rho }_{\mathrm{fix}}}{{\epsilon }_{\mathrm{r}}{\epsilon }_0{\kappa }^2}}$$

(32)

and the z-component of the electroosmotic flow velocity u(x, t) is given by u₂(x, t) (Equation (22)). Figure 3 shows a 3D plot of u₂(x, t) as a function of the distance x from the solid surface (x = 0) and time t, calculated using Equation (22). It represents the temporal and spatial variations in the electroosmotic flow velocity u₂(x, t) caused by the fixed charges in the polyelectrolyte layer with a volume charge density ρ_fix. The 3D plot of the scaled electroosmotic velocity defined by u₂(x, t) = −u₂(x, t)/(ρ_fix./ηλ²) is given as a function of the reduced distance λx from the solid surface and the scaled time λ²νt. Here, ρ_fix./ηλ² is the bulk electroosmotic flow in the polyelectrolyte layer.

Equation (22) and Figure 3 exhibit that u₂(x, t) does not depend on the Debye length 1/κ, reflecting the fact that the electric potential ψ(x) within the polyelectrolyte layer is equal to the Donan potential ψ_DON (Equation (10)) everywhere in the polyelectrolyte layer. Figure 3 also shows that u₂₍x, t) approaches the bulk osmotic flow velocity of the magnitude ρ_fix/ηλ² at times of the order of 1/λ²ν, as also indicated in Equation (22). Similarly, u₂₍x, t) approaches ρ_fix/ηλ² at distances of the order of the Brinkmann screening length 1/λ, as shown in Figure 3 and also indicated in Equation (22). In contrast to Figure 2, the variations that, due to the fixed charges within the polyelectrolyte layer, extend over a wider region inside the polyelectrolyte layer.

3.3. Charged Solid Surface Covered with a Charged Polymer (i.e., Polyelectrolyte) Layer (σ ≠ 0 and ρ_fix ≠ 0)

Lastly, consider the case where the charged solid surface (σ ≠ 0) is covered with a charged polymer (i.e., polyelectrolyte) layer (ρ_fix ≠ 0). Figure 4 and Figure 5 are examples of the 3D plot of the z-component u(x, t) of the electroosmotic velocity u(x, t), calculated using Equations (20)–(22) for the case where λ/κ = 0.5 and κρ_fix/λ²σ = 0.4 (Figure 4) and =−0.4 (Figure 5). As described before, the linearity of the relationship for the liquid flow velocity holds due to the use of the low-potential approximation. The z-component of the electroosmotic velocity u(x, t) is thus given by the sum u(x, t) = u₁(x, t) + u₂(x, t) (Equations (20)–(22)). Accordingly, as shown in Figure 4 and Figure 5, the 3D plot of the electroosmotic flow velocity is the sum of the 3D plots of the respective flow velocities. Figure 4 represents the case where the charges on the solid surface and the fixed charges in the polyelectrolyte layer have the same sign, while Figure 5 illustrates the case where they have opposite signs. In both cases, it can be observed that the variation in the electroosmotic flow due to the solid surface charge with a density σ is confined to the vicinity of the solid surface, whereas the variation in the flow due to the fixed charges within the polyelectrolyte layer with a density ρ_fix extends throughout the bulk region of the polyelectrolyte layer.

Equations (21) and (22) and Figure 2, Figure 3, Figure 4 and Figure 5 show that the relaxation time for the z-component u(x, t) of the transient electroosmotic velocity u(x, t) to reach its steady-state value is characterized by 1/κ²ν and 1/λ²ν. The relaxation time is thus very short except when the electrolyte concentration is very low and/or the Brinkmann screening length is very large. In typical aqueous systems, the system reaches a steady state almost instantaneously. However, the findings obtained in this paper could potentially be applied to methods such as molecular dynamic simulations.

4. Conclusions

In this study, a general theory for time-dependent transient electroosmosis on a planar soft surface, specifically a polyelectrolyte-coated solid surface immersed in an electrolyte solution, was developed for the case where an electric field is suddenly applied. This theoretical framework provides a simple yet effective model for understanding time-dependent electrokinetic phenomena at biointerfaces. A closed-form approximate expression for the electroosmotic velocity distribution within the polyelectrolyte layer was derived, capturing its dependence on both spatial and temporal variables. The findings revealed that the temporal and spatial variations in the electroosmotic flow induced by the surface charges of the solid surface are confined to the vicinity of the solid surface. In contrast, variations driven by the fixed charges within the polyelectrolyte layer extend throughout a broader region inside the polyelectrolyte layer, highlighting the distinct contributions of these charge sources to the overall flow behavior.

This paper deals with the case where the thickness D of the polymer layer is sufficiently larger than the Debye length 1/κ and the Brinkmann screening length 1/λ, allowing it to be regarded as infinite. The focus is exclusively on examining the spatial and temporal variations in the liquid flow velocity distribution within the polymer layer. Accordingly, the flow velocity distribution in the electrolyte solution outside the polymer layer and at the boundary between the polymer layer and the electrolyte solution is not considered.