The interplay of cation/anion and monovalent/divalent selectivity in negatively charged nanopores: local charge inversion and anion leakage

Abstract

The anomalous mole fraction effect (AMFE) is widely regarded as a hallmark of calcium versus monovalent ion selectivity in negatively charged pores. While AMFE is well understood in highly cation-selective narrow ion channels, its microscopic origin in wide synthetic nanopores, where anions may also contribute to transport, remains less clear. Here, we use a reduced Nernst-Planck + Local Equilibrium Monte Carlo framework to study ionic transport in a negatively charged PET nanopore, with particular emphasis on how the modeling of surface carboxyl (COO$^{-}$) groups influences charge inversion, ionic currents, and AMFE. We systematically compare fixed point-charge models and explicit-particle representations of surface oxygens and identify two controlling parameters: the distance of closest approach (DCA) between ionic charges and pore charges and grid spacing that modulates localization (while keeping average surface charge constant). By fitting pore diffusion coefficients to three experimental conductance points, we reproduce the entire experimental AMFE curve as well as anion leakage in CaCl$_2$ seen in experiments and molecular dynamics simulations. Remarkably, vastly different microscopic models of the surface groups yield indistinguishable device-level conductance curves when the DCA is matched, despite substantial differences in local Ca$^{2+}$ concentration profiles. Our results demonstrate that AMFE in wide nanopores is governed by a delicate interplay between charge inversion, anion leakage, and ionic mobility, underlying that in wide pores monovalent vs.\ divalent cation selectivity is modulated by cations vs.\ anion selecivity.

Figures (13)

• Figure 1: (A) Scanning electron micrograph of a cross section of a biconical PET nanopore from the paper of Gillespie et al. gillespie_bj_2008_nanopore. The length of the pore (width of the PET membrane) is 12 $\mu$m. The large radii at the entrances are 79 nm, while the small radius at the center is $2.7$ nm. (B) The top panel shows the whole nanopore (axes aspect ratio is not to scale). The middle panel zooms on the blue rectangle of the top panel (axes aspect ratio is not to scale). The bottom panel zooms on the red rectangle of the middle panel (aspect ratio is correct). We explicitly simulate the central $H=12$ nm long portion (red rectangle), while the effect of the rest of the pore is taken into account by tuning the diffusion coefficients in the pore, $D_{i}^{\mathrm{p}}$. In our specific simulation cell, we have two baths on the two sides of the membrane (bottom panel) with their own diffusion coefficients.
• Figure 2: The four models for localizing the charges of the COO$^{-}$ groups (a) the $-e$ point charge is fixed on the surface of the pore ($r_{0}=0$), the model used in our previous work fabian_jml_2022; (b) the $-e$ point charge is behind the surface at $r_{0}$ distance from the surface; (c) the $-e$ point charge is located in the center of a hard sphere of radius $R_{\mathrm{f}}$ (this model is more appropriate for silanol groups of silica nanopores); (d) the $-e$ charge is distributed between two oxygen atoms in the form of $-e/2$ partial charges. For the latter two models, the oxygen atoms are confined to positions at distance $r_{0}$ behind the surface using the harmonic potential of Eq. \ref{['eq:harmonic']}. For more details, see Fig. \ref{['fig3']}.
• Figure 3: Variations of modeling the oxygen atoms considered in this work. The red oxygen atoms represented as charged hard spheres of radius $R_{\mathrm{f}}$ are localized by a harmonic potential (Eq. \ref{['eq:harmonic']}) with a reference position that is at distance $r_{0}$ from the wall (behind it). The radius of the confinement is $R_{\mathrm{loc}}$. From top to bottom: (i) the $R_{\mathrm{loc}}=0.3$ nm value allows larger mobility for the oxygen atoms hanging at the end of a relatively long side chain; (ii) the value $R_{\mathrm{loc}}=0.2$ nm allows smaller mobility for the oxygen atoms that are bound to the pore material more tightly; (iii) the value $r_{0}=0$ allows endgroups to protrude more deeply into the electrolyte; (iv) the group charge is located on a single oxygen atom.
• Figure 4: Conductances of ionic species (Ca$^{2+}$, K$^{+}$, and Cl$^{-}$) and total conductance as functions of Ca$^{2+}$ mole fraction, $\eta$, for different models of carboxyl groups. Lines refer to fixed-charge models with the point charge being "on the wall" ($r_0=0$) with $\Delta x$ grid spacing and reduced point charges $Q$ to maintain $\sigma=-1$$e$/nm$^{2}$. The arrows show the direction of increasing $\Delta x$. The small open triangles show the results obtained for modeling the carboxyl groups point charges at $r_0 =0.14$ nm distance "behind the wall" using $\Delta x=1$ nm spacing and $Q=-e$. The large gray circles denote experimental results gillespie_bj_2008_nanopore. Total concentration is fixed at $0.1$ M.
• Figure 5: Conductances of ionic species (Ca$^{2+}$, K$^{+}$, and Cl$^{-}$) and total conductance as functions of Ca$^{2+}$ mole fraction, $\eta$, for different models of carboxyl groups. Lines refer to fixed-charge models with the point charge being "behind the wall" at distance $r_0$ from the wall. The arrows show the direction of increasing $r_0$. The small open squares show the results obtained for modeling the carboxyl groups with two oxygen atoms (charged hard spheres with radius $R_\mathrm{f}=0.14$ nm) each carrying $-0.5e$ charge ($r_0 =0.14$ nm, $R_{\mathrm{loc}}=0.2$ nm). The large gray circles denote experimental results gillespie_bj_2008_nanopore. Total concentration is fixed at $0.1$ M.