⇦Chinese Physics Letters, 2020, Vol. 37, No. 9, Article code 094701 Extended Nernst–Planck Equation Incorporating Partial Dehydration Effect Zhong Wang (王中)1†, Zhiyang Yuan (袁智扬)1‡, and Feng Liu (刘峰)1,2* Affiliations 1State Key Laboratory of Nuclear Physics and Technology, School of Physics, Peking University, Beijing 100871, China 2Center for Quantitative Biology, Peking University, Beijing 100871, China Received 30 April 2020; accepted 16 July 2020; published online 1 September 2020 Supported by the National Natural Science Foundation of China (Grant No. 11875076).
†Current address: Max Planck Institute of Microstructure Physics, Halle 06120, Germany.
‡Current address: Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA.
*Corresponding author. Email: liufeng-phy@pku.edu.cn Citation Text: Wang Z, Yuan Z Y and Liu F 2020 Chin. Phys. Lett. 37 094701    Abstract Novel ionic transporting phenomena emerge as nanostructures approach the molecular scale. At the sub-2 nm scale, widely used continuum equations, such as the Nernst–Planck equation, break down. Here, we extend the Nernst–Planck equation by adding a partial dehydration effect. Our model agrees with the reported ion fluxes through graphene oxide laminates with sub-2 nm interlayer spacing, outperforming previous models. We also predict that the selectivity sequences of alkali metal ions depend on the geometries of the nanostructures. Our model opens a new avenue for the investigation of the underlying mechanisms in nanofluidics at the sub-2 nm scale. DOI:10.1088/0256-307X/37/9/094701 PACS:47.61.-k, 66.10.Ed, 62.23.Kn, 81.05.ue © 2020 Chinese Physics Society Article Text Ion channels exhibit both high permeability and selectivity in ion transport in order to carry out their essential cell function. This high-performance ion transport often originates from a subnanometer selective filter.[1,2] Recently fabricated nanochannels also approach the subnanometer scale,[3–12] yielding novel properties, and paving the way for potential applications in relation to molecular sieving,[3] ion separation,[4,7,11] water desalination,[8,10] and biosensing.[13,14] Therefore, it is important to reveal the underlying mechanisms inherent in transport phenomena in subnanometer channels. Subnanometer ion transport has been studied using a variety of methods. On the one hand, the regularly used computational methods such as molecular dynamic (MD) simulations, Brownian simulations, ab initio methods[4,15–17] have certain limitations. Firstly, they computationally cost much more. Secondly, their results could be difficult to generalize, so people must perform specific simulations. Thirdly, their dynamic range is rather limited. For example, if the permeability of A is 1000 times larger than B, the MD simulation must run for a sufficient time for $\sim$$10^{4}$ cases of A permeation (so that the cases of B are $\sim $10) in order to verify the permeability ratio, which is usually unrealistic in practice. On the other hand, the traditional continuum equations relating to ion transport are computationally efficient and generalizable, but severely deviate from the experimental results in sub-2 nm nanochannels.[18–22] Some recent research has suggested that these equations can still be modified to account for the sub-continuum effect.[5,23–25] These effects include size exclusion,[26] partial dehydration,[17,27–29] adsorption,[4] anomalous dielectric constants,[30,31] and ionic concentration or species dependence on the electric double layer.[32,33] However, a general modification on traditional continuum equations is still required to match the accuracy of MD simulations. As a starting point, we focus on the partial dehydration effect. This effect is universal in all subnanometer channels, and is ion-dependent, therefore playing a vital role in ion selectivity.[5,8] Experiments suggest that it is realistic to neglect all other effects in certain cases, particularly in the case of graphene oxide (GO) laminates with few charges.[4,8] Among the published models for the partial dehydration effect, we establish our model following the approach proposed in Zwolak's model,[28,29] as it is based on parameters obtained from MD simulations, and therefore combines the advantages of both analytical and numerical methods. Alternative models may lack sufficient accuracy, e.g., the Born-energy-based models of Eisenman[34] and Krauss et al.,[35] the models of Jain et al.[5] and Wu,[36] or it may be too difficult to obtain the necessary parameters, e.g., the model proposed by Laio et al.[27,36] Furthermore, all the models mentioned above were established for specific cases of interest and are not suitable for our aim, i.e., building a universal model for sub-2 nm ion transport. We propose a modified Nernst–Planck (N-P) equation, incorporating a partial dehydration effect for sub-2 nm ion transport. We apply this model to calculate the ion flux. Our model agrees with the reported ion fluxes through GO laminates with sub-2 nm interlayer spacing, for which the most concrete set of data is available. The predicted selectivity among ions is reasonably consistent with most experiments involving sub-2 nm channels. Theory. Ion transport in nanochannels is usually modeled via continuum equations, specifically the non-advective steady-state Poisson–Nernst–Planck (PNP) equations:[5,26] $$\begin{align} &\boldsymbol{J}_{\boldsymbol{i}}=-D_{i}\Big[\nabla n_{i}+\frac{z_{i}e}{kT}n_{i}\nabla \varphi \Big],~~ \tag {1} \end{align} $$ $$\begin{align} &-\epsilon_{\rm r}\epsilon_{0}\nabla ^{2}\varphi =\sum\limits_i {z_{i}en_{i}},~~ \tag {2} \end{align} $$ $$\begin{align} &\nabla \cdot \boldsymbol{J}_{\boldsymbol{i}}=0,~~ \tag {3} \end{align} $$ where $e$ is the electronic charge, $k$ is the Boltzmann constant, $T$ is the temperature, $\varphi$ is the electrical potential, $\epsilon_{0}$ is the dielectric constant of vacuum, and $\epsilon_{\rm r}$ is the relative dielectric constant of the solution. $\boldsymbol{J}_{i}$, $z_{i}$, $D_{i}$, and $n_{i}$ denote the flux density, the valence, diffusion coefficient, and number density of the $i^{\rm th}$ ion species, respectively. However, in sub-2 nm channels, it is questionable whether these continuum equations are still applicable. Firstly, the individual effects of atoms, ions, and molecules may dominate,[5] such that continuum equations may not be suitable, since they treat macroscopic quantities such as concentrations and flux rates, which are summed effects of large numbers of individual microscopic particles. Secondly, new interactions, such as partial dehydration, can arise in these channels, which are not included in traditional continuum equations. However, it is still reasonable to sum over the individual effects of microscopic particles when the channel is relatively long, and the correlation of ions is not significant. Moreover, these equations could be modified to take into account the confinement effects in sub-2 nm nanochannels. In this work, we modify these equations to take into account the partial dehydration effect. We alter the N-P equation for specific types of ion with a generalized chemical potential:[5,28,35] $$\begin{align} J={}&-D\cdot \frac{n}{kT}\nabla \mu,~~ \tag {4} \end{align} $$ $$\begin{align} \mu ={}& kT \ln n+ze\varphi +\mu_{\rm add},~~ \tag {5} \end{align} $$ where $\mu$ is the chemical potential consisting of electrochemical potential (the first two terms) and the additional chemical potential due to the confinement effect (the third term). Notably, without the third term, the extended N-P equation becomes equivalent to the traditional N-P equation [Eqs. (1)-(3)]. For a simplified case, we assume that the electrostatic interactions between the ions themselves, or between the ions and the nanopore wall are negligible. Hence, the Poisson equation is neglected here. Furthermore, we assume that the flux does not significantly vary over the cross section, so the steady-state N-P equation can be written in the one-dimensional form: $$\begin{alignat}{1} &J=-D \frac{n}{kT} \frac{d\mu }{dx}=-D\Big[\frac{dn}{dx}+\frac{n}{kT} \frac{d(ze\varphi +\mu_{\rm add})}{dx}\Big],~~ \tag {6} \end{alignat} $$ $$\begin{alignat}{1} &\frac{dJ}{dx}=0.~~ \tag {7} \end{alignat} $$ Given the electrostatic potential $\varphi (x)$ and additional chemical potential $\mu_{\rm add}(x)$, as well as the boundary conditions of $n(x=0)$ and $n(x=L)$, the steady-state ionic distribution $n(x)$ is determined, so one can derive the flux rate:[29,37] $$\begin{align} J={}&D \Big[n(x=0)\exp \Big(\frac{ze\varphi (x=0)+\mu_{\rm add}(x=0)}{kT} \Big)\\ &-n(x=L)\exp \Big(\frac{ze\varphi (x=L)+\mu_{\rm add}(x=L)}{kT} \Big)\Big]\\ &\cdot\Big[\int_0^L {\exp \Big(\frac{ze\varphi (x)+\mu_{\rm add}(x)}{kT}\Big)dx} \Big]^{-1},~~ \tag {8} \end{align} $$ where $x=0$ and $L$ denote the entrance and exit of the channel, respectively. Here, we consider a special case, in which the additional chemical potential has a discontinuous jump, $\mu_{1}$, at the entrance and exit of the channel: $$\begin{align} &\mu_{\rm add}(x)=\mu_{0}(x)+\mu_{1},~~~\left(0 < x < L \right)\\ &\mu_{\rm add}(x)=\mu_{0}(x),~~~ (x=0~{\rm or}~L) \end{align} $$ where $\mu_{0}(x)$ is a continuous function, and $\mu_{1}$ is a constant. Thus the total flux will depend exponentially on this discontinuous jump: $$\begin{align} J={}&D \Big[n(x=0)\exp \Big(\frac{ze\varphi (x=0)+\mu_{0}(x=0)}{kT} \Big)\\ &-n(x=L)\exp \Big(\frac{ze\varphi (x=L)+\mu_{0}(x=L)}{kT} \Big)\Big]\\ &\cdot\Big[\int_0^L {\exp \Big(\frac{ze\varphi (x)+\mu_{0}(x)}{kT} \Big)dx} \Big]^{-1}\cdot \exp \Big(-\frac{\mu_{1}}{kT} \Big)\\ ={}&J_{0}\exp \Big(-\frac{\mu_{1}}{kT} \Big).~~ \tag {9} \end{align} $$ Assuming that the partial dehydration effect is dominated by spatial confinement,[27,28] and the transition region at the entrance or exit of the channel is negligible, we can rewrite Eq. (9) in the specific form: $$ J=J_{0}\exp \Big(-\frac{{\Delta \mu }_{\rm deh}}{kT}\Big),~~ \tag {10} $$ where ${\Delta \mu }_{\rm deh}$ denotes the change of chemical potential due to the partial dehydration effect, and $J_{0}$ is the flux where there is no partial dehydration effect. Following the quantized ion conductance model,[28,29] we can model an ion in bulk solution, surrounded by several layers of hydration water molecules. These hydration layers are broken according to the channel geometry (see Fig. 1). Consequently, the partial dehydration energy can be written as follows: $$\begin{align} &\Delta U=\sum\limits_i {(f_{i}-1)U_{i}^{0}},~~ \tag {11} \end{align} $$ $$\begin{align} &f_{i}=1-\frac{\varOmega _{i,{\rm broken}}}{4\pi },~~ \tag {12} \end{align} $$ where $f_{i}$ is the fraction of the left part of the $i^{\rm th}$ layer (see Fig. 2), and the sum at Eq. (11) is taken over the part broken due to confinement. $U_{i}^{0}$ denotes the (total) hydration energy of the $i^{\rm th}$ layer. $\Delta U$ represents the energy change due to partial dehydration. $\varOmega _{i,{\rm broken}}$ denotes the spatial angle of the broken part of the $i^{\rm th}$ layer.
cpl-37-9-094701-fig1.png
Fig. 1. Partial dehydration process of ion permeation through nanopores. One fully hydrated ion (orange) with three layers of hydration shell (dashed lines mark the boundary) moves into the pore and exits at the other side under driving forces, e.g., electrical, osmotic, etc. If the pore is narrow enough, the hydration shells in the bulk must be partially broken so that the ion can transport through the pore.
cpl-37-9-094701-fig2.png
Fig. 2. Modeled hydration shell behavior under confinement. In a narrow slit, a certain fraction of a hydration shell, with spatial angle $\varOmega _{\rm broken}$, must be broken.
Figure 2 illustrates the slit channel case as an example. In this case, the fraction part in Eqs. (11) and (12) can be written in the form: $$\begin{align} f_{i}={}&1-\varTheta (R_{i}-d_{\rm eff})\times \frac{2}{4\pi }\int\limits_0^{\arccos \frac{d_{\rm eff}}{R_{i}} } \int\limits_{0}^{2\pi } {{\sin}\theta d\varphi d\theta }\\ ={}&1-\mathit{\varTheta} (R_{i}-d_{\rm eff})\Big(1-\frac{d_{\rm eff}}{R_{i}}\Big),~~ \tag {13} \end{align} $$ where $R_{i}$ denotes the radius of the $i^{\rm th}$ layer, and $d_{\rm eff}$ is the half slit width. Here, $\varTheta (x)$ represents the Heaviside step function, noting that the hydration shell can be broken only when $R_{i}$ is larger than $d_{\rm eff}$. As an approximation, we assume that the sum in Eq. (11) can be written as $i=1$ to 3 by neglecting any fraction of the hydration shells outside of the 3$^{\rm rd}$ one. The experiments detailed in Refs. [4,8] indicate that the confinement in the region outer than the 3$^{\rm rd}$ shell does not make a difference in flux rates on different species of ions. For example, as the channel width is 0.98 nm, which is larger than the 3$^{\rm rd}$ shell but within the region of outer shells, its flux rates are nearly the same as those for the channel with a width of 1.3 nm. As some hydration shell parameters lack experimental data, we derive them from MD simulations, following Zwolak's approach.[28,29] In the radial distribution of water molecules around an ion, the radii of the hydration shells, $R_{i}$, are derived from peak locations, and the energy values, $U_{i}^{0}$, are calculated from the minima in the radial distribution: $$\begin{alignat}{1} &U_{i}^{0}=\frac{e^{2}}{8\pi \epsilon_{0}}\Big(1-\frac{1}{\epsilon_{w}}\Big)\Big(\frac{1}{R_{i}^{{\min}}} -\frac{1}{R_{i-1}^{{\min}}}\Big), ~~i=2,3,~~~~ \tag {14} \end{alignat} $$ $$\begin{alignat}{1} &U_{i=1}^{0}=\Delta G-\frac{e^{2}}{8\pi \epsilon_{0}}\Big(1-\frac{1}{\epsilon_{w}}\Big)\Big(0-\frac{1}{R_{1}^{{\min}}}\Big),~~ \tag {15} \end{alignat} $$ where $\epsilon_{w}=80$ is the dielectric constant of water, and $R_{i}^{{\min}}$ is the location of the $i^{\rm th}$ minimum. Here, $R_{i-1}^{{\min}}$ and $R_{i}^{{\min}}$ denote the inner and outer radii demarcating the $i^{\rm th}$ hydration layer. The simulated ion distribution agrees well with the available experimental data,[38] as summarized by Zwolak et al.[29] In relation to the first hydration shell (i.e., $i=1$), $R_{i-1}^{{\min}}$ is not well defined; we therefore calculate the hydration energy of the first shell by subtracting the calculated hydration energy of all outer shells [see Eq. (15)][39] from the experimental Gibbs free energy $\Delta G$.[39] Our model is significantly different from the models of Laio et al.[27] and Jain et al.[5] in that it takes into account not only the innermost water molecules, but also those in the 2$^{\rm nd}$ and 3$^{\rm rd}$ hydration shells. This model is thus possibly applicable to a wider range, where 2$^{\rm nd}$ and 3$^{\rm rd}$ broken shells are important. The parameters of Na$^{+}$, K$^{+}$, and Ca$^{2+}$ ions are derived from Ref. [29]. The energy parameters of Ca$^{2+}$ ions are modified to fit the experimental data, taking account of the potential deviation in the calculation of the hydration energy of alkali earth metal ions.[40,41] Those of Li$^{+}$ ions are derived from the MD simulation results taken from Ref. [11], $R^{\rm Li-H}\approx R^{\rm Li-O}+0.6$ Å, where $R^{\rm Li-H}$ is the Li–hydrogen distance, which is used for modeling each $R_{i}$ and $R_{i}^{{\min}}$, and where $R^{\rm Li-O}$ is the respective Li–oxygen distance derived from the MD results. All the parameters used in this study are listed in Table 1.
Table 1. Parameters of different types of ions. $R_{i}$ denotes the radius of the $i^{\rm th}$ hydration layer. $R_{i}^{{\min}}$ denotes the location of the $i^{\rm th}$ minimum. $\Delta G$ is the experimental hydration Gibbs free energy. $U_{i}^{0}$ is the hydration energy of the $i^{\rm th}$ layer. The last row shows the modified parameters of Ca$^{2+}$ ions.
Ion $R_{i}$ (Å) $R_{i}^{{\min}}$ (Å) $-\Delta G$ (eV) $-U_{i}^{0}$ (eV)
Li$^{+}$ 2.6, 5.0, 7.4$^{\rm a}$ 3.5, 6.2, 8.2$^{\rm a}$ 4.92$^{\rm b}$ 2.89, 0.87, 0.28$^{\rm a}$
Na$^{+}$ 2.9, 5.1, 7.5$^{\rm c}$ 3.8, 6.2, 8.4$^{\rm c}$ 3.80$^{\rm b}$ 1.51, 0.72, 0.30$^{\rm c}$
K$^{+}$ 3.3, 5.6, 7.8$^{\rm c}$ 4.2, 6.6, 8.8$^{\rm c}$ 3.07$^{\rm b}$ 1.15, 0.61, 0.27$^{\rm c}$
Ca$^{2+}$ (initial) 3.0, 5.1, 7.5$^{\rm c}$ 3.6, 6.1, 8.5$^{\rm c}$ 15.65$^{\rm b}$ 7.89, 3.23, 1.32$^{\rm c}$
Ca$^{2+}$ (modified) 3.0, 5.1, 7.5$^{\rm c}$ 15.65$^{\rm b}$ 10.9, 1.08, 0.47$^{\rm d}$
$^{\rm a}$Derived from Ref. [11] and including the method in the main text. $^{\rm b}$Experimental data.[39] $^{\rm c}$Derived from Ref. [29]. $^{\rm d}$Modified to fit the experimental flux rates.
Notably, as the entropy aspect ($T\Delta S$) of the aforementioned models is typically one or two orders of magnitude smaller than the energy aspect $\Delta U$,[27,28] we neglect the entropy part and adopt the approximation $\Delta \mu_{\rm dehydration}\approx \mathrm{\Delta }U$.
cpl-37-9-094701-fig3.png
Fig. 3. Calculated partial dehydration energy in slit structures, versus effective half spacing ${d}_{\rm{eff}}$ of the slit. For Li$^{+}$ (brown), Na$^{+}$ (red), K$^{+}$ (blue), Ca$^{2+}$ (olive) ions, compared with the results of this work (solid lines), the calculated partial dehydration energy, based on the extended Born energy model (dashed lines with diamonds) is higher, and the results based on the models of Laio et al. (dashed lines with down-triangles) and Jain et al. (dashed lines with up-triangles) are lower. Here, $d_{\rm eff}$ is the half spacing used for modeling, which may differ from the experimental values, which consider the sizes of wall atoms, ion-wall interactions, and wall atoms' relaxation.
Results and Discussions. We confirm that our model outperforms previously reported models in relation to the calculation of partial dehydration energy. The series of ionic permeation rates through the GO laminates reported in references[4,8] serve as the best benchmarks from which to examine our model. Although GO laminates usually carry negative charges,[42,43] particularly at the edges, the charge density in these GO laminates may be small, as the charge effect seems negligible. When the interlayer spacing is greater than 13 Å (corresponding to the pore size of $\sim $9–$10$ Å), it blocks all solutes with hydration diameters larger than 9 Å, i.e., the ion permeation rates depend solely on the size of the hydrated ions, rather than on the valence.[4] When the interlayer spacing is 6.4–9.8 Å, the ion permeation rates are dominated by the partial dehydration energy of ions.[8] In addition, water transport varies only slightly in these channels,[8] suggesting that water-water collisions and water-wall interactions are not significant when the slit width varies. Thus, partial dehydration is the only remained factor that can possibly account for variations in ion fluxes.[4,8] Consequently, we apply our model in the case of slit channels. Based on the hydration layer data listed in Table 1, we calculate the partial dehydration energy of Li$^{+}$, Na$^{+}$, K$^{+}$ and Ca$^{2+}$ ions via Eqs. (11)-(13), by tuning the half slit width $d_{\rm eff}$. The discontinuous points of the lines denote the boundary between the 1$^{\rm st}$ and 2$^{\rm nd}$ layers. We also extend the Born-energy model as a reference: $$ \Delta U=\left(1-\frac{1}{\epsilon_{w}} \right)\frac{z^{2}e^{2}}{8\pi \epsilon_{0}d_{\rm eff}} $$ in relation to the extended Born energy. Compared with the calculated partial dehydration energy based on our model, the results based on the extended Born energy model are greater, and the results based on the models of Laio et al. and Jain et al. (Fig. 3) are smaller. We exclude Wu's model[36] in our comparison, as it was designed for charged ion channels and is therefore not suitable for the sole discussion of the partial dehydration effect.
cpl-37-9-094701-fig4.png
Fig. 4. Comparison between experimental and theoretical ion flux rates. Normalized ion flux rates $J/J_{0}$ as a function of experimental half spacing $d_{{\exp}}$ of the slit.[8] Experimental $J_{0}$ is obtained based on Ref. [4]. The upper and lower horizontal dashed lines indicate $J / J_{0}=1$ and the experimental detection limit, respectively. Experimental points below the limit line are put on the line. The theoretical results of this work (circles) are much closer to the experimental data (solid squares) than those based on the extended Born energy model (diamonds), the model proposed by Laio et al. (inverse triangles) or that proposed by Jain et al. (triangles).
We then substitute the calculated partial dehydration energy into Eq. (10) to derive the normalized flux $J/J_{0}$, and compare it with experimental data (see Fig. 4). Here, we consider the room temperature to be $T=298$ K. We define $d_{\rm eff}=d_{{\exp}}-0.9$ Å for theoretical calculations, to account for the effective radius of wall atoms. As the slit widths vary from 7.4 Å to 13.5 Å,[8] the ion fluxes in 13.5 Å GO are 3 orders of magnitude higher than the predictions for the standard diffusion equation, while in 9.8–7.4 Å GO they are 2–4 orders of magnitude lower. For a quantitative and systematic comparison, the experimental fluxes are normalized to unit cross section and unit driving force: $$ J_{i}=\frac{N_{\rm ions}}{S_{\rm eff}}\times \frac{F_{\rm reference}/L_{\rm reference}}{F/L_{\rm eff}},~~ \tag {16} $$ where $J_{i}=J$ or $J_{0}$ denote the flux, $S_{\rm eff}$ is the effective cross section, $L_{\rm eff}$ is the effective channel length, and $L_{\rm reference}$ is a reference length. Here, $F$ denotes the power of driving force, $F_{\rm reference}$ is the reference, and $N_{\rm ions}$ indicates the number of transported ions per unit of time. $J_{0}$ is derived from Ref. [4], and $J$ from Ref. [8]. In the case of the experiments considered here, $F=\Delta n$ is the concentration difference, whereas it can also represent the difference of electric potential, thermal difference, etc. in other cases. This normalization of driving force and channel length is based on the assumption that the flux rate is proportional to the driving force. The theoretical results of this work are much closer to the experimental data than those based on the other models (see Fig. 4). In addition, as $d_{\rm eff}=4.0$ and $3.05$ Å the calculated values of the partial dehydration energy for K$^{+}$ ions are 0.31 and 0.53 eV, respectively, which is comparable with the experimental data of 0.21 and 0.75 eV derived from the temperature dependence measurement.[8] This discrepancy could be attributed to the deviation between the realistic GO channel (e.g., rough edges) from the smooth surface assumption in the model. These results indicate that our model outperforms the previous models in calculating the partial dehydration effect on ion transport.
cpl-37-9-094701-fig5.png
Fig. 5. Predicted selectivity among alkali metal ions is reasonably consistent with experiments in slits[8,9,44,45] (left) and cylinders[11,46–48] (right). The predicted selectivities and their transition regions are shown as the red and light red areas, respectively. The selectivity measured in the experiments is shown by circles, triangles, squares, and diamonds, with half-filled symbols indicating intermediate selectivity (e.g, $\mathrm{K}^{+}\approx \mathrm{Na}^{+}>\mathrm{Li}^{+}$ is intermediate between $\mathrm{K}^{+}>\mathrm{Na}^{+}>\mathrm{Li}^{+}$ and $\mathrm{Na}^{+}>\mathrm{K}^{+}>\mathrm{Li}^{+}$).
Finally, we compare the predicted selectivity of alkali metal ions with the experimental results in sub-2 nm slits[8,9,44,45] and cylinders[11,46–48] (see Fig. 5). Notably, the flux data from Refs. [9,44,45] are difficult to normalize without proper $J_{0}$ as a reference, so we were unable to compare them quantitatively with the theoretical prediction, as we did for the data from Ref. [8]. The order of the selectivity, however, can still be used to test our model. We calculate ion transport in slits and cylinders separately, and test them with the corresponding experiments. Consistently with the experimental data, our model predicts that the fluxes of alkali metal ions are higher than those of higher-valence ions in all cases. The selectivity among alkali metal ions is quite interesting. The narrowest slits ($ < 2.4$ Å) and cylinders ($ < 2.5$ Å) favor $\mathrm{K}^{+}>\mathrm{Na}^{+}>\mathrm{Li}^{+}$ selectivity, whereas the widest ones (4.0–5.0 Å for slits and 4.7–5.0 Å for cylinders) favor the opposite $\mathrm{Li}^{+}>\mathrm{Na}^{+}>\mathrm{K}^{+}$ selectivity, and several changes happen in between. We note that this is mainly due to the radii of hydration layers of different ions. We can conclude that the predicted selectivity among alkali metal ions is consistent with most experimental data. The predicted selectivity which has seldom been observed in experiments, such as that for $\mathrm{Na}^{+}>\mathrm{K}^{+}>\mathrm{Li}^{+}$ in slits and $\mathrm{Li}^{+}>\mathrm{K}^{+}>\mathrm{Na}^{+}$ in cylinders, may pave the way to novel applications. Our model is significantly different from Eisenman's model of cation selective glass electrodes, which considers the electrostatic interaction between a charged electrode and a fully dehydrated ion.[34] The two models treat dehydration and electrostatic interactions very differently, thus the origins of the selectivity are different (see Table 2). However, the predicted selectivity in this work is somewhat comparable to the well-known Eisenman sequences. The sequence $\mathrm{K}^{+}>\mathrm{Na}^{+}>\mathrm{Li}^{+}$ happens in the largest electrodes in Eisenman's model, and in the narrowest channels in this work. The opposite sequence $\mathrm{Li}^{+}>\mathrm{Na}^{+}>\mathrm{K}^{+}$ appears in totally opposite situations. We note that for the largest electrodes, the electrostatic interaction is weakest and dehydration dominates, similarly to the case of the narrowest channels where dehydration is strongest in our model, and vice versa. On the other hand, in addition to the same sequences, such as $\mathrm{Li}^{+}>\mathrm{K}^{+}>\mathrm{Na}^{+}$ appearing in bot the models, our model also predicts new sequences, such as $\mathrm{Na}^{+}>\mathrm{K}^{+}>\mathrm{Li}^{+}$, $\mathrm{Na}^{+}>\mathrm{Li}^{+}>\mathrm{K}^{+}$ and $\mathrm{Li}^{+}>\mathrm{K}^{+}>\mathrm{Na}^{+}$.
Table 2. Comparison of our model to Eisenman's model.
Eisenman's model This work
Physical process Static binding Dynamic transport
Dehydration Full Partial
Electrostatic interactions Dominate Not considered
Origin of changes of selectivity Electrostatic interactions Partial dehydration
In summary, we are able to incorporate the partial dehydration effect to the steady-state N-P equation for the nanostructures with a scale below 2 nm. Our model is consistent with the experiments reported, and outperforms previous models. This work sheds light on the mechanisms of sub-2 nm transport, and may facilitate the design of artificial nanostructures. Dealing with the partial dehydration effect can be seen as a first step in terms of research on this topic. Our model may be extended for other sub-2 nm effects, including size exclusion, adsorption, and electrostatic interactions. The one-dimensional N-P equation may be further extended to a three-dimensional form for greater accuracy. References Instantaneous ion configurations in the K + ion channel selectivity filter revealed by 2D IR spectroscopyMicroED structure of the NaK ion channel reveals a Na+ partition process into the selectivity filterUnimpeded Permeation of Water Through Helium-Leak-Tight Graphene-Based MembranesPrecise and Ultrafast Molecular Sieving Through Graphene Oxide MembranesHeterogeneous sub-continuum ionic transport in statistically isolated graphene nanoporesMolecular transport through capillaries made with atomic-scale precisionHighly Selective Ionic Transport through Subnanometer Pores in Polymer FilmsTunable sieving of ions using graphene oxide membranesSize effect in ion transport through angstrom-scale slitsUltrafast ion sieving using nanoporous polymeric membranesUltrafast selective transport of alkali metal ions in metal organic frameworks with subnanometer poresUltrafast selective ionic transport through heat-treated polyethylene terephthalate track membranes with sub-nanometer poresNanopore-based Fourth-generation DNA Sequencing TechnologyFundamental transport mechanisms, fabrication and potential applications of nanoporous atomically thin membranesUnderstanding Water Permeation in Graphene Oxide MembranesDehydration as a Universal Mechanism for Ion Selectivity in Graphene and Other Atomically Thin PoresA coupled effect of dehydration and electrostatic interactions on selective ion transport through charged nanochannelsTests of Continuum Theories as Models of Ion Channels. I. Poisson−Boltzmann Theory versus Brownian DynamicsTests of Continuum Theories as Models of Ion Channels. II. Poisson–Nernst–Planck Theory versus Brownian DynamicsTesting the Applicability of Nernst-Planck Theory in Ion Channels: Comparisons with Brownian Dynamics SimulationsCollective diffusion in carbon nanotubes: Crossover between one dimension and three dimensionsCharge transport in confined concentrated solutions: A minireviewIon transport in sub-5-nm graphene nanoporesMolecular streaming and its voltage control in ångström-scale channelsImprovements in continuum modeling for biomolecular systemsSelecting Ions by Size in a Calcium Channel: The Ryanodine Receptor Case StudyPhysical Origin of Selectivity in Ionic Channels of Biological MembranesQuantized Ionic Conductance in NanoporesDehydration and ionic conductance quantization in nanoporesWater Dielectric Effects in Planar ConfinementAnomalously low dielectric constant of confined waterSingle-Walled Carbon Nanotubes: Mimics of Biological Ion ChannelsEffect of Interfacial Ion Structuring on Range and Magnitude of Electric Double Layer, Hydration, and Adhesive Interactions between Mica Surfaces in 0.05–3 M Li + and Cs + Electrolyte SolutionsCation Selective Glass Electrodes and their Mode of OperationSelectivity sequences in a model calcium channel: role of electrostatic field strengthMicroscopic model for selective permeation in ion channelsSelf-consistent analytic solution for the current and the access resistance in open ion channelsStructure and dynamics of hydrated ionsThermodynamics of solvation of ions. Part 5.—Gibbs free energy of hydration at 298.15 KMultisite Ion Model in Concentrated Solutions of Divalent Cations (MgCl 2 and CaCl 2 ): Osmotic Pressure CalculationsImproved model of hydrated calcium ion for molecular dynamics simulations using classical biomolecular force fieldsScalable Graphene-Based Membranes for Ionic Sieving with Ultrahigh Charge SelectivityVertically Transported Graphene Oxide for High‐Performance Osmotic Energy ConversionSelf-Assembly: A Facile Way of Forming Ultrathin, High-Performance Graphene Oxide Membranes for Water PurificationCharge- and Size-Selective Ion Sieving Through Ti 3 C 2 T x MXene MembranesObservation of ionic Coulomb blockade in nanoporesPolystyrene Sulfonate Threaded through a Metal-Organic Framework Membrane for Fast and Selective Lithium-Ion SeparationWater desalination using nanoporous single-layer graphene
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