Electro-osmotic flow through nanopores in thin and ultrathin membranes

We theoretically study how the electro-osmotic fluid velocity in a charged cylindrical nanopore in a thin solid state membrane depends on the pore's geometry, membrane charge, and electrolyte concentration. We find that when the pore's length is comparable to its diameter, the velocity profile develops a concave shape with a minimum along the pore axis unlike the situation in very long nanopores with a maximum velocity along the central pore axis. This effect is attributed to the induced pressure along the nanopore axis due to the fluid flow expansion and contraction near the exit or entrance to the pore and to the reduction of electric field inside the nanopore. The induced pressure is maximal when the pore's length is about equal to its diameter while decreasing for both longer and shorter nanopores. A model for the fluid velocity incorporating these effects is developed and shown to be in a good agreement with numerically computed results.

Figure 1

A schematic diagram of our modeled nanopore structure with the electric potential in the background. (For this plot, the membrane charge is $\rho =0.4e/{\mathrm{nm}}^3$ and the bulk electrolyte concentration is $C=0.1$ M).

Figure 2

Contour plots of the computed EOF velocity with streamlines for a nanopore with (a) $R_p=5$ nm and (b) 10 nm.

Figure 3

Fluid velocity profile in $x$ direction in the center of the pore for different nanopore lengths $L_p$ and membrane charge densities: (a) $R_p=5$ nm and (b) $R_p=10$ nm. The solid curves are the results of the numerical simulations while the dashed (dotted) curves are the results of Eq. (6) with $\mathrm{\Delta }\mathrm{\Phi }$ $\left({\mathrm{\Phi }}_0\right)$. For both dashed and dotted curves, $E_z=V_e/(L_p+\alpha R_p)$ where values of parameter $\alpha$ are given in text (see Sec. 3 for details).

Figure 4

$z$ component of the electric field $E_z$ along the central pore axis for a nanopore with (a) $R_p=5$ nm and (b) $R_p=10$ nm and different pore lengths. The horizontal dashed lines represent the values of the electric field as computed by $E_z=V_e/(L_p+\alpha R_p)$ with $\alpha$ given in the text, and the vertical dot-dashed lines show the location of the inlet $\left(z=0\right)$ and outlet $(z=15,z=50$, and $z=100$ nm) for each nanopore.

Figure 5

Induced pressure $P$ along the central pore axis for nanopores of different lengths and (a) $R_p=5$ nm, (b) $R_p=10$ nm $\left(\rho =1.2e/{\mathrm{nm}}^3\right)$. The total pressure drop $\mathrm{\Delta }P$ is defined as difference between the maximum and minimum pressure values. The vertical dot-dashed lines show the location of the inlet $\left(z=0\right)$ and outlet $(z=15,z=50$, and $z=100$ nm) for each nanopore. The inset in (a) shows distribution of the pressure in the fluid.

Figure 6

Induced pressure drop $\mathrm{\Delta }P$ vs aspect ratio of the pore $\mathrm{\Pi }=L_p/\left(2R_p\right)$ for $\rho =1.2e/{\mathrm{nm}}^3$. Dots connected by dashed lines are the results of calculations, solid curves are the result of Eq. (13) with $\beta =1.75$. Note that the same $\beta$ is used for both $R_p=5$ and 10 nm.

Figure 7

Same as in Fig. 3 but with dashed lines from Eqs. (12) and (13) with $\beta =1.75$ and dotted lines with $\beta =3$.

Figure 8

The fluid velocity profile in $x$ direction at the center of the nanopore of length $L_p=25$ nm for different bulk electrolyte concentrations $C$: (a) $R_p=5$ nm and (b) 10 nm. The dashed lines are the results of Eq. (12).

Figure 9

The fluid velocity profile in $x$ direction at the center of the nanopores with $R_p=5$ nm and varying length $L_p$ (solid curves). The dashed lines are the results of Eq. (6) with $E_z=V_e/L_p$.