Linear Electro-Osmosis of Charged and Uncharged Polymer Solutions

Abstract

We investigate electro-osmosis in aqueous solutions of polyelectrolytes using Poisson-Boltzmann and hydrodynamic equations. A solution of positively charged polyelectrolytes is confined between two negatively charged planar surfaces, and an electric field is applied parallel to the surfaces. When electrostatic attraction between the polymer and the surface is strong, the polymers adhere to the surface, forming a highly viscous adsorption layer that greatly suppresses the electro-osmosis. Conversely, electro-osmosis is enhanced by depleting the polymers from the surfaces. We also found that the electro-osmotic flow is invertible when the electrostatic potential decays to its bulk value with the opposite sign. These behaviors are well explained by a simple mathematical form of the electro-osmotic mobility.

Appendix 2

2.1.1 2.A Local Equilibrium Conditions for the Components

Because we apply an external field E along the x direction (see Fig. 2.1), the total electrostatic potential is not $\psi (z)$ in Eq. (2.4), but instead is $\psi _\mathrm {total}(x,z)=\psi (z)-Ex$. Assuming the local equilibrium condition, the chemical potential of the cation and anion is given by Eqs. (1.44) and (1.44) replaced $\psi _\mathrm {total}$ instead of $\psi$. In the geometry of the investigated system, the diffusion flux of the ion, given by

$$\begin{aligned} {\varvec{j}}^i=-\frac{c^i}{\zeta ^i}\nabla \mu ^i, \end{aligned}$$

(2.30)

is divided into two components:

$$\begin{aligned} {\varvec{j}}^i= & {} j^i_x \hat{{\varvec{x}}}+j^i_z\hat{{\varvec{z}}}, \end{aligned}$$

(2.31)

$$\begin{aligned} j^i_x= & {} \pm \frac{c^i}{\zeta ^i}E,\end{aligned}$$

(2.32)

$$\begin{aligned} j^i_z= & {} -\frac{c^i}{\zeta ^i}\frac{\mathrm{{d}}}{\mathrm{{d}}z}\left[ k_\mathrm{B}T\ln (c^ia_i^3)\pm e\psi (z)\right] . \end{aligned}$$

(2.33)

Here $\hat{{\varvec{x}}}$ and $\hat{{\varvec{z}}}$ are the unit vectors along the x and z axes, respectively. Because the system is confined by the walls at $z=0$ and 2L, the diffusion flux along the z direction vanishes at steady state. Thus, we obtain the Boltzmann distribution along the z axis. On the other hand, the diffusion flux remains along the x axis. Because the applied electric field is sufficiently weak and orthogonal to $-\nabla \psi$, it influences neither the concentration fields nor the polymer conformation.

2.1.2 2.B Scaling Behaviors in Polyelectrolyte Solutions

The scaling behaviors of polyelectrolyte solutions are known to widely differ from those of uncharged polymer solutions. At the overlap concentration $c^*$ in a polyelectrolyte solution, the monomer density inside the coil equals the overall monomer density in the solution [10]. In our notation, the overlap concentration in a theta solvent is determined by $c^*(1+2c_\mathrm{b}^+/c^*f)^{-3/2}\approx N^{-2}a^{-2}\ell _\mathrm{B}^{-1}f^{-2}$.

In the low-salt or salt-free regime, the overlap concentration becomes $c^*\approx (a^2\ell _\mathrm{B}Nf)^{-1}$. Conversely, it approaches $c^*\cong \{8(c_\mathrm{b}^{+})^3a^{-4}\ell _\mathrm{B}^{-2}f^{-7}N^{-4}\}^{1/5}$ in the high-salt regime. Between these two extremes, the overlap concentration decreases as f increases. Given the same polymer length N, polyelectrolyte chains expand more than their uncharged counterparts.

The viscosity of polyelectrolyte solutions also obeys scaling behaviors, which depend on the solvent quality and the polymer concentration regime. For example, the viscosity of a semidilute solution in a theta solvent is given by $\eta \approx \eta _0 N a \ell _\mathrm {B}^{1/2}fc^{1/2}(1+2c_\mathrm {b}^+/fc)^{-3/4}$. If the salt is not dissolved or is insufficiently dilute, this expression approaches $\eta \approx \eta _0 N a\ell _\mathrm {B}^{1/2}fc^{1/2}$; that is, the viscosity is proportional to $c^{1/2}$ (Fuoss law). On the other hand, in highly saline conditions the viscosity behaves as $\eta \approx \eta _0 N a\ell _\mathrm {B}^{1/2} (c_\mathrm {b}^+)^{-3/4}f^{7/4}c^{5/4}$. The viscosity depends on the polymer concentration as $c^{5/4}$, identical to that of an uncharged polymer solution in a theta solvent, namely $\eta \approx \eta _0 N(ca^3)^{1/(3\nu -1)}$ with $\nu =3/5$. Physically, this result implies that electrostatic interactions in a polyelectrolyte solution are well screened by the salt.