An asymptotic model of Poisson--Nernst--Planck--Stokes systems in narrow channels

Abstract

Ion transport through narrow channels is described by the coupled Poisson--Nernst--Planck--Stokes equations (PNPS) on a continuum scale. However, direct numerical simulations in two or three dimensions of boundary value problems for small aspect ratio geometries, a crucial characteristic of nanopores, can quickly become computationally intensive and thus limit the insights into the underlying mechanisms that control electrokinetic phenomena. Taking advantage of the small aspect ratio, we derive a systematic asymptotic reduction of the PNPS system. In contrast to existing one-dimensional reductions, which assume a Debye length much smaller than the channel radius, our analysis identifies a distinguished asymptotic regime in which the Debye length is allowed to be comparable to the channel width. Our approach has a significantly larger range of validity and contains existing approximations such as the Helmholtz--Smoluchowski approximation as limiting cases. The derived asymptotic model extends also to a generalized PNPS system, where finite-size constraints and solvation effects are taken into account and thus applies to other well-known models such as the Bikerman--Freise model. Using our asymptotic model we demonstrate that the ion current can undergo a number of different flow transitions and in particular predict that positively charged ions can be pushed against their electrostatic gradient. Furthermore, we show how finite-size effects can influence the ion current and enhance ion selectivity. Finally, we revisit case studies of protein-based channels from the literature to illustrate the predictive potential of our asymptotic model.

Figures (12)

• Figure 1: Sketch of an axially symmetric profile of an ion channel $\Omega$. The pore wall at $\tilde{r}= \tilde{R}(\tilde{z})$ is denoted with $S^\text{wall}$, the symmetry axis of the domain at $\tilde{r}=0$ is marked as $S^0$. At $\tilde{z}=0$ the pore has an outlet and at $\tilde{z}= \tilde{L}$ an inlet boundary.
• Figure 2: (a) Volumetric flow $\langle u^0\rangle$ plotted as a function of the bulk concentration $n^\text{bulk}$ for $a_\pm = 0$, $\Delta p = 0$ and different potentials $\Delta \varphi = -1$ (crosses), $\Delta \varphi = -5$ (plus signs) and $\Delta \varphi = -8$ (stars). The dashed line corresponds to the $\langle u^{\text{HS}}\rangle$ (Helmholtz-Smoluchowski) flow regime. The dotted line gives $\langle u^{\text{HS}}\rangle + \langle u^{\text{EDL}} \rangle$. (b) Volumetric flow scaled with $\Delta \varphi$.
• Figure 3: (a) Volumetric flow $\langle u^0\rangle$ for $n^\text{bulk} = 0.6$ ($a_\pm = 0$) plotted as a function of pressure $\Delta p$ for different potentials $\Delta \varphi = 0.2$ (crosses), $\Delta \varphi = 8$ (plus signs), $\Delta \varphi = -0.2$ (stars) and $\Delta \varphi = -8$ (circles). The dotted and dashed lines correspond to the $\langle u^{\text{PF}}\rangle$ (Poiseuille) and $\langle u^{\text{HS}}\rangle$ (Helmholtz-Smoluchowski) flow regimes, respectively. (b) Volumetric flow and pressure scaled with $C_u \Delta \varphi$.
• Figure 4: (a) Cation current $I_+$ plotted as a function of the bulk concentration $n^\text{bulk}$ for $a_\pm = 0$, $\Delta p = 0$ and different potentials $\Delta \varphi = -1$ (crosses), $\Delta \varphi = -5$ (plus signs) and $\Delta \varphi = -8$ (stars). The dashed line corresponds to the $I^\text{E}$ (potential-driven) regime. The dotted line gives $I^\text{E} + I^\text{C}$. (b) Cation current scaled with $C^\text{bulk} \Delta \varphi$.
• Figure 5: (a) Cation current $I_+$ for $n^\text{bulk} = 0.6$ ($a_\pm = 0$) plotted as a function of pressure $\Delta p$ for different potentials $\Delta \varphi = 1$ (crosses), $\Delta \varphi = 2$ (plus signs), $\Delta \varphi = -1$ (stars). The dotted and dashed lines correspond to the $I^{\text{P}}$ (pressure-driven) and $I^{\text{E}}$ (potential-driven) current regimes, respectively. (b) Current and pressure scaled with $C_+ \Delta \varphi$.