Electroneutrality breakdown in nanopore arrays

The electrostatic screening of charge in one-dimensional confinement leads to long-range breakdown in electroneutrality within a nanopore. Through a series of continuum simulations, we demonstrate the principles of electroneutrality breakdown for electrolytes in one-dimensional confinement. We show how interacting pores in a membrane can counteract the phenomenon of electroneutrality breakdown, eventually returning to electroneutrality. Emphasis is placed on applying simplifying formulas to reduce the multidimensional partial differential equations into a single ordinary differential equation for the electrostatic potential. Dielectric mismatch between the electrolyte and membrane, pore aspect ratio, and confinement dimensionality are studied independently, outlining the relevance of electroneutrality breakdown in nanoporous membranes for selective ion transport and separations.

Figure 1

The simulation configuration in comsol (a) with the membrane, pore, and electrolyte (b) for an isolated pore that does not feel its periodic neighbors and (c) a periodic arrangement of pores on a square lattice. (d) A cross section of the system through the center of the cylindrical pore is shown to describe the domains and equations applied in the simulations. (e) A sample of continuum simulation results showing the progress from noninteracting to interacting channels for small charge densities. For all channels, the channel radius is 1 nm and the length is 100 nm. The membrane dielectric constant is ${\epsilon }_m=10{\epsilon }_0$ and the electrolyte dielectric constant is ${\epsilon }_w=80{\epsilon }_0$. The salt concentration is 1 mM. In order from left to right, the spacing $\ell$ between the channel centers is 129, 77, 46, 28, and 17 nm. Also in order, the amount of charge within the pore versus on the channel walls, $|Q_{\mathrm{in}}/Q_{\mathrm{out}}|$, is 12%, 13%, 17%, 26%, and 43%. As the channels get closer together and more interacting, the system returns closer to electroneutrality, which is evinced by the higher $\varphi$ values within the pore.

Figure 2

The results for an isolated pore with overlapping and nonoverlapping double layers within the pore. Plot of the potential profile for overlapping double layers at (a) $c_0=1$ mM (${\kappa }_{D}R\approx 0.1$) versus less overlapping double layers at (b) $c_0=100$ mM (${\kappa }_{D}R\approx 1$). The same parameters are used as in Fig. 1, except the center-to-center distance between pores is 500 nm such that the periodic channels are not interacting. For panel (a), 12% of the charge is contained within the pore, whereas for panel (b), 91% of the charge is contained within the pore. (c) The charge within the pore versus on the pore walls as a function of ${\kappa }_{D}R$ for the same channel as in panels (a) and (b). Electroneutrality breakdown occurs in the region of strong double-layer overlap ${\kappa }_{D}R\to 0$. For (c) the markers are the comsol simulations, whereas the line is the application of the approximate formula in Eq. (10). (d), (e) The integrated ionic charge as a function of the lateral position for $c_0=1$ and 100 mM. The charge is distributed over a wide area $O\left(L\right)$ extending beyond the pore mouth when electroneutrality is broken ($c_0=1$ mM) but is more localized when electroneutrality is maintained ($c_0=100$ mM). (e) Quantification of end effects for two different concentrations. The electrostatic potential is plotted as a function of the $z$ coordinate evaluated at the center axis of the channel. End effects are not significant for isolated channels.

Figure 3

Extent of electroneutrality breakdown for channels of different center-to-center separation distances on a lattice. (a) $|Q_{\mathrm{in}}/Q_{\mathrm{out}}|$ versus ${\kappa }_{D}R$ for $\ell /L=5$, $\ell /L=1$, $\ell /L=0.1$. (c) $|Q_{\mathrm{in}}/Q_{\mathrm{out}}|$ versus $\ell /L$ for $c_0=0.01$, 1, 100 mM for $\ell /L=0.02$ to $\ell /L=0.5$. For both panels (a) and (c), the markers are the comsol simulations, whereas the solid lines are the application of the approximate formula in Eq. (10) and the dotted lines are the application of the approximate formula in Eq. (14). (b) The integrated ionic charge as a function of the lateral position for $\ell /L=5$ and 0.05, with $c_0=1$ mM. The charge is distributed over a wide area when channels are isolated but is localized when the channels are closely spaced and strongly interacting. (d) Quantification of end effects for two different lattice spacings with $c_0=1$ mM. The electrostatic potential is plotted as a function of the $z$ coordinate evaluated at the center axis of the channel. End effects are significant when the channels are interacting.

Figure 4

Role of dielectric mismatch on extent of electroneutrality breakdown. (a) Results for isolated pore with same properties as in Fig. 1, but with ${\epsilon }_m={\epsilon }_0$, ${\epsilon }_m=10{\epsilon }_0$, and ${\epsilon }_m=100{\epsilon }_0$. (b) $|Q_{\mathrm{in}}/Q_{\mathrm{out}}|$ versus ${\kappa }_{D}R$ for varying ${\epsilon }_m$. (c) $|Q_{\mathrm{in}}/Q_{\mathrm{out}}|$ versus $\ell /L$ with $c_0=1$ mM for varying ${\epsilon }_m$. For both panels (b) and (c), the markers are the comsol simulations, whereas the solid lines are the application of the approximate formula in Eq. (10) and the dotted lines are the application of the approximate formula in Eq. (14).

Figure 5

Role of aspect ratio on extent of electroneutrality breakdown. (a) Results for isolated pore with same properties as in Fig. 1 but with $L=10$ nm, $L=100$ nm, and $L=1000$ nm, also with $c_0=1$ mM. (b) $|Q_{\mathrm{in}}/Q_{\mathrm{out}}|$ versus ${\kappa }_{D}R$ for varying $L$, with $\ell /L=5$. (b) $|Q_{\mathrm{in}}/Q_{\mathrm{out}}|$ versus $\ell /L$ with $c_0=1$ mM for varying $L$. For both panels (b) and (c), the markers are the comsol simulations, whereas the solid lines are the application of the approximate formula in Eq. (10) and the dotted lines are the application of the approximate formula in Eq. (14).

Figure 6

Role of dimensionality of confinement by inspecting a slit pore geometry. (a) Results for isolated pore with the same properties as in Fig. 1 but for a slit pore. (b) $|Q_{\mathrm{in}}/Q_{\mathrm{out}}|$ versus ${\kappa }_{D}R$ for varying $\ell /L$. (c) $|Q_{\mathrm{in}}/Q_{\mathrm{out}}|$ versus $\ell /L$ for varying $c_0$. (d), (e) The same as panels (b) and (c) but with $L=1000$ nm. For panels (b)–(e), the markers are the comsol simulations, whereas the solid lines are the application of the approximate formula in Eq. (18) and the dotted lines are Eq. (14) with constants given by Eqs. (20) and (21).

Figure 7

Role of nonlinearity on extent of electroneutrality breakdown. (a) Results for isolated pore with same properties as in Fig. 1, but with $q_s\to 0$, $q_s=0.001\mathrm{C}/{\mathrm{m}}^2$, and $q_s=0.01\mathrm{C}/{\mathrm{m}}^2$ with $c_0=1$ mM. (b) $|Q_{\mathrm{in}}/Q_{\mathrm{out}}|$ versus ${\kappa }_{D}R$ for varying $q_s$. (c) $|Q_{\mathrm{in}}/Q_{\mathrm{out}}|$ versus $\ell /L$ with $c_0=1$ mM for varying $q_s$. For both panels (b) and (c), the markers are the comsol simulations, whereas the solid line is the application of the approximate formula in Eq. (10) and the dotted line is the application of the approximate formula in Eq. (14).

Figure 8

(a) The dimensionless conductance and (b) cation transference number through a negatively charged nanochannel for varying channel separation distances. The solid lines are the predictions using Eq. (10) and the dotted lines are the predictions using Eq. (14). The dotted lines are good predictors of the extent of electroneutrality breakdown at small $\ell /L$, but fail at large $\ell /L$, e.g., the blue dotted lines. The plateau in conductance at low concentration is only apparent when $|Q_{\mathrm{in}}/Q_{\mathrm{out}}|=1$. As the ratio $\ell /L$ decreases, the system moves closer to electroneutrality. When electroneutrality is broken, the cation transference number does not saturate to 1 at low concentration.

Figure 9

(a) The configuration of multiple channels in a membrane is reduced to (b) a single-pore problem. The partial differential equation and boundary conditions are listed for the given single-pore problem.

Figure 10

Electric-field magnitude normalized by the surface charge as a function of position within a cross section of the membrane-channel system, for the same conditions as Fig. 1. As the unit cell becomes smaller and channels are closer together, the electric-field lines are strongest at the entrance regions of the channel. End effects can be important in this regime.

Figure 11

The extent of electroneutrality breakdown for two different unit-cell shapes: square and hexagonal. The center-to-center distance between channels, $\ell$ is maintained the same between the channels. The conditions are identical to Fig. 3. As $\ell$ is varied, the hexagonal and square unit cells appear to have similar extents of electroneutrality breakdown.

Figure 12

The capacitance (a) normalized by the pore surface area (b) normalized by the membrane surface area per unit cell and (c) normalized by the membrane-pore volume. In each plot, the separation distance between channels is varied, changing the extent of electroneutrality breakdown. The solid lines are the predictions using Eq. (10) and the dotted lines are the predictions using Eq. (14). The dotted lines are good predictors of the extent of electroneutrality breakdown at small $\ell /L$, but fail at large $\ell /L$, e.g., the blue dotted lines.