Electroosmotic flow in soft microchannels at high grafting densities

This paper reports the results of a theoretical modeling of the fully developed electroosmotic flow in a rectangular microchannel with a high-density polyelectrolyte layer (PEL) attached to the walls. At these conditions, the ions are partitioned between the PEL and the fluid outside the PEL owing to the difference between the permittivities of the two media. It is taken into account that the dynamic viscosity is higher within the PEL because of hydration effects. Solutions are obtained for the electric potential and velocity distributions as well as the mean velocity by making use of a variational approach, applied to the linearized form of the governing equations, which treats the whole area under consideration as a single domain with variable physical properties. The resulting equations are solved using a spectral method. Closed-form analytical expressions are obtained for a slit geometry, representing the case of high aspect ratios. The solutions obtained are validated by comparing to finite-element simulations of the full nonlinear equations. It is shown that the electric potential drop inside the channel increases due to the depletion of the counterions within the PEL caused by the ion partitioning effect. This effect, surprisingly, magnifies the electroosmotic flow rate because of the increase of the space charge outside the PEL. As expected, the hydration effects reduce the flow rate, especially for thick PELs.

Figure 1

Schematic of the soft microchannel under consideration including the dimensions and the coordinate system. The polyelectrolyte layer (PEL) is the region between the rigid wall and the dashed lines. It contains fixed ions, shown as large circles, whereas the electrolyte ions, represented by small circles, are mobile. Different physical properties are considered inside and outside the PEL.

Figure 2

Profiles of ${\psi }^{*}$ at the vertical centerline for different values of ${\sigma }^{*}$, while keeping $W^{*}=1$, $t^{*}=0.1$, $K=5$, ${\eta }_{\lambda }=2$, $\mathrm{\Xi }=1.078$, and ${\eta }_{\varepsilon }=0.8$. The results of the spectral method obtained utilizing $l_{\max }=m_{\max }=100$ are compared with the predictions of the FEM-based numerical solution of Eq. (12). The inset shows the results for ${\sigma }^{*}=5$.

Figure 3

Heightwise variation of ${\psi }^{*}$ at the vertical centerline for different values of $t^{*}$ and ${\eta }_{\varepsilon }$, while keeping $W^{*}=1$ and $K=5$. The results of the spectral method are compared with the predictions of the full numerical solution of Eq. (11).

Figure 4

Cross-sectional distributions of $c_{-}^{*}$ and $c_{+}^{*}$ for $W^{*}=1$, $t^{*}=0.1$, $K=4$, and ${\eta }_{\varepsilon }=0.8$.

Figure 5

Profiles of the dimensionless velocity at two different values of $\alpha$, while keeping $W^{*}=1$, $t^{*}=0.1$, $K=4$, ${\eta }_{\varepsilon }=0.8$, and ${\eta }_{\mu }=1.5$.

Figure 6

Heightwise variation of $u^{*}$ at the vertical centerline for different values of $t^{*}$, $K$ (and ${\eta }_{\lambda }$), ${\eta }_{\varepsilon }$, and ${\eta }_{\mu }$. The default parameters are $W^{*}=1$, $t^{*}=0.1$, $K=10$, ${\eta }_{\varepsilon }=0.8$, ${\eta }_{\mu }=1.5$, and $\alpha =1$. The results of the spectral method are compared with the predictions of the full numerical solution of Eqs. (11) and (25).

Figure 7

Heightwise variation of $c_{-}^{*}-c_{+}^{*}$ at the vertical centerline for different values of ${\eta }_{\varepsilon }$ and $W^{*}=1$, $t^{*}=0.1$, $K=5$.

Figure 8

Dependence of the average velocity on $W^{*}$ for different values of $t^{*}$, while keeping $K=5$, ${\eta }_{\varepsilon }=0.8$, ${\eta }_{\mu }=1.5$, and $\alpha =1$. The results of the general solution are compared with the predictions of the special solution for $1\ll W^{*}$.

Figure 9

Dependence of the average velocity on ${\eta }_{\varepsilon }$, ${\eta }_{\mu }$, and $\alpha$ at different values of $t^{*}$. The default parameters are $W^{*}=1$, $K=10$, ${\eta }_{\varepsilon }=0.8$, ${\eta }_{\mu }=1.5$, and $\alpha =1$. The results of the spectral method are compared with the predictions of the full numerical solution of Eqs. (11) and (25).