Solid-state nanopores made of two-dimensional materials such as molybdenum disulfide are of great interest thanks in part to promising applications such as ion filtration and biomolecule translocation. Controlled fabrication and tunability of nanoporous membranes require a better understanding of their ionic conductivity capabilities at the nanoscale. Here, we developed a model of ionic conductivity for a KCl electrolyte through sub 5-nm single-layer MoS_{2} nanopores using equilibrium all-atom molecular dynamics simulations. We investigate the dynamics of K^{+} and Cl^{−} ions inside the pores in terms of concentration and mobility. We report that, for pore dimensions below 2.0 nm, which are of particular interest for biomolecule translocation applications, the behaviors of the concentration and mobility of ions strongly deviate from bulk properties. Specifically, we show that the free-energy difference for insertion of an ion within the pore is proportional to the inverse surface area of the pore and that the inverse mobility scales linearly as the inverse diameter. Finally, we provide an improved analytical model taking into account the deviation of ion dynamics from bulk properties, suitable for direct comparison with experiments.

Solid-state nanopores (SSNs) made of two-dimensional (2D) materials such as MoS_{2} have emerged as versatile sensors for ion and biomolecule manipulation.^{1–10} One of the most promising applications of SSN is the sequencing of biological molecules.^{11} SSN sequencing experiments are based on the measurement of ionic current variations when a biomolecule immersed in an ionic solution is driven through the nanoporous membrane by applying a transverse electric field. During the process of translocation, the molecule occupies the pore volume, blocking the passage of ions. Consequently, an ultrafast monitoring of ionic flow during the passage of the biomolecule yields information about its structure and chemical properties, as experimentally demonstrated with sub-microsecond temporal resolution.^{12,13} However, understanding atomic and sub-nanometer sized pores in bare and functionalized 2-D membranes used for molecular and ionic selectivity are still a matter of study,^{14–19} since the fundamental principles behind electrical transport of ionic solution through those pores have not been explored in detail yet. Only two experimental studies of ionic conductance through MoS_{2} nanoporous membranes with diameters lower than 2.0 nm have been reported.^{3,8}

The analytical model used to predict the conductance *G*_{0} of nanoporous MoS_{2} membranes from the knowledge of their dimensions is a continuum model, which results from a combination of three resistors in series: one pore resistance *R _{pore}* modeled as a cylindrical resistor and two access resistances constituted by the mouth of the pore at each side of the membrane

^{20}

where *d** and *h*^{*} represent the effective pore diameter and membrane thickness, respectively. Effective pore diameter *d*^{*} and membrane thickness *h*^{*} correspond to the effective dimensions of the ionic conducting cylindrical channel of the nanoporous membrane experienced by solvent molecules and are extracted from the probability distributions of solvent molecules inside the pore. In Eq. (1), the only term related to the ionic properties is the bulk conductivity of the electrolyte, *σ _{bulk}*. In physical chemistry, the conductivity exhibited by an ionic solution is expressed as the product of the concentration

*c*of the ionic species, their charge

^{i}*q*=

^{i}*ez*, and their electrical mobility

^{i}*μ*

^{i}where *e* is the elementary (positive) charge, *z* is the charge number, and the index *i* represents the ionic species. For a neutral KCl ionic solution, Eq. (2) becomes $\sigma bulk=2ecbulk\u27e8\mu bulk\u27e9$, where *c _{bulk}* is the concentration of K

^{+}or Cl

^{−}ions and $\u27e8\mu bulk\u27e9=(\mu bulkK++\mu bulkCl\u2212)/2$. At the nanoscale, ions are confined in space whose dimensions are of similar sizes to that of the ionic radii. It follows that their concentration, mobilities, and hydration are different than their bulk counterparts, as already shown for graphene nanopores.

^{21}Consequently, the conductivity of the electrolyte in nanopores is expected to deviate from its bulk value and the conductance of open nanopores predicted by Eq. (1) is likely to be inaccurate for the smallest pores. We found previously that open pore conductance

*G*

_{0}predicted from Eq. (1) using

*σ*and the values obtained from experimental I-V curves

_{bulk}^{8}and molecular dynamics (MD) simulations

^{22}were overestimated for single-layer (SL) MoS

_{2}nanopores with diameters ranging from 1 to 3 nm. We showed that MD values of

*G*

_{0}for this system were better represented by a simple linear interpolation model

*G*

_{0}(

*d*

^{*}) =

*αd*

^{*}+

*β*where

*α*and

*β*are fitted parameters. In this simple linear relation, no current can be detected below a critical diameter $dmin*=\u2212\beta /\alpha \u223c0.7\u2009nm$.

^{22}Recent experimental measurements of ionic transport through sub-nanometer sized pores made of atomic vacancies fabricated in SL-MoS

_{2}show indeed that pores with diameters <0.6 nm display negligible conductance.

^{8}

Failure of Eq. (1) to reproduce experimental and MD data demands to reexamine the modeling of *G*_{0} at the atomic scale. Here, thanks to all-atom MD simulations for SL-MoS_{2} membranes with diameters ranging from 1 to 5 nm [Fig. 1(a)], we derive an analytical model of the electrolyte conductivity at room temperature in SL-MoS_{2} nanopores as a function of the pore diameter, and we named *σ _{pore}*(

*d*

^{*}). As shown in Fig. 1(b), the ion conductivity inside the pore deviates significantly from the bulk electrolyte conductivity for the range of diameters studied here, which corresponds to those used for biomolecule sensing. Replacing the bulk electrolyte conductivity

*σ*in Eq. (1) by

_{bulk}*σ*(

_{pore}*d*

^{*}) restores the validity of the continuum model, provided that the diameter dependence of the electrolyte conductivity is taken into account. The corrected continuum model derived here is given by Eq. (7) which can be used by experimentalists to extract the effective diameter of the SL-MoS

_{2}nanoporous membranes. We will now describe how the improved analytical model was derived from all-atom MD data at 300 K.

For each pore diameter, we performed a 10 ns all-atom MD simulation of the SL-MoS_{2} membrane with a 1 M KCl electrolyte as detailed in the supplementary material. We computed the ionic conductivity of each pore presented in Fig. 1(a), i.e., *σ _{pore}*, using Eq. (2). Ionic concentrations in nanopores were defined as the average number of ions inside each pore computed from the 10 ns MD run divided by the pore volume represented by a cylinder with effective diameter

*d*

^{*}and thickness

*h*

^{*}. Pore effective diameters (see Table SI) and thickness (0.96 nm for SL-MoS

_{2}) were defined from the water density profiles at the interface with the membrane (see Fig. S1). The bulk 1M KCl, i.e.,

*σ*, was computed from Eq. (2) for a 10 ns all-atom MD simulation of the bulk electrolyte without the MoS

_{bulk}_{2}membrane.

^{22}Ion mobilities were computed from the MD trajectories by applying the Einstein equation with the ion diffusion coefficients calculated from the mean-square displacements of K

^{+}and Cl

^{−}inside the pores. More details about this procedure can be found in the supplementary material. The exact same procedure was applied to extract the bulk ion mobilities from the MD run of the bulk electrolyte. All the values and their error bars extracted from MD runs are given in Table SI. Figure 1(b) presents the ratio between bulk and pore conductivities extracted from MD simulations for each pore diameter (gray filled circles). For diameters around 2 nm, the ion pore conductivity is about half the bulk value, which means that Eq. (1) overestimates the membrane conductance by a factor of 2. For diameters approaching 1 nm, the pore conductivity is only a third of the bulk value. To gain further insight into the origin of the deviations of the conductivity at the nanoscale, we reformulate the problem in terms of partition coefficients $\Phi i\u2261cporei/cbulk$ (concentration) and $\Gamma i\u2261\mu porei/\u27e8\mu bulk\u27e9$ (mobility). From Eq. (2), we have

Similar to *σ _{pore}*, the partition coefficients are diameter-dependent. The coefficient Φ

^{i}can be written as $\Phi i=Pporei/Pbulki$ with $Pporei$ and $Pbulki$ the probabilities to find an ion of species

*i*in the pore and in an equivalent volume in the bulk electrolyte, respectively. Therefore, Φ

^{i}is related to the difference between the free-energy of an ion of species

*i*in the pore and in the bulk electrolyte, named Δ

*G*, by the Boltzmann law: $\Phi i=Pporei/Pbulki=exp(\u2212\Delta Gi/RT)$, where

^{i}*R*is the perfect gas constant and

*T*the temperature. The free-energy difference is expected to be positive due to the loss of entropy and to the dehydration phenomenon

^{21,23,24}in the nanopore, both causes being dependent on the pore size. From the ion concentrations computed from MD data, we found that Δ

*G*is well represented by the following relation [see Fig. 2(a)] for the range of diameters studied (1 nm ≤

^{i}*d*

^{*}≤ 3 nm):

where $A*=\pi d*24$ is the nanopore effective area and *ϕ _{i}* is a positive fitted parameter. The variation of the free-energy difference Δ

*G*as a function of $1/d*2$ observed in MD data is significantly different from the one found in graphene nanopores

^{i}^{21}where Δ

*G*was fitted by the 1/

^{i}*d*

^{*}law. Finally, as shown in Fig. 2(b) for $\Phi i=exp(\u22124\phi i/\pi d*2)$, there is no large difference observed between K

^{+}and Cl

^{−}species. The values obtained from least-squares fitting are $\phi K+=0.832\u2009nm2$ and $\phi Cl\u2212=0.793\u2009nm2$.

Next, we study the mobility partition coefficient Γ^{i} inside MoS_{2} nanopores. By plotting the inverse mobility as a function of the inverse effective diameter [Fig. 3(a)], we find that $(1/\mu porei\u22121/\mu bulki)$ scales as 1/*d*^{*}. A similar behavior for mobilities was observed in graphene nanopores.^{21} This result leads to an expression of pore mobility $\mu porei$ for each ionic species *i*

where *γ ^{i}* are fitted parameters: $\gamma K+=4.27\xd710\u22123\u2009V\u2009s\u2009m\u22121$ and $\gamma Cl\u2212=4.61\xd710\u22123\u2009V\u2009s\u2009m\u22121$.

It is worth noting that there is a significant difference between K^{+} and Cl^{−} mobilities, K^{+} having a larger diffusion coefficient *D* than Cl^{−}, as it can be explained from Stoke's law. Indeed, from Stoke's law, $D(K+)/D(Cl\u2212)=R(Cl\u2212)/R(K+)\u22481.1$, where *R*(*K*^{+}) and *R*(*Cl*^{−}) are the ionic radii.^{25} Using Eq. (5), we can write an analytical expression for Γ and plot its evolution as a function of effective diameter *d*^{*} [Fig. 3(b)]

where $\delta i=\gamma i\u27e8\mu bulk\u27e9$ and $\u03f5i=\u27e8\mu bulk\u27e9/\mu bulki$, the corresponding values being $\delta K+=0.38\u2009nm$ and $\delta Cl\u2212=0.41\u2009nm,\u2009\u03f5K+=1.03$ and $\u03f5Cl\u2212=0.97$. As shown in Fig. 3(b), when the pore diameter is around 1.0 nm, mobility is reduced from the bulk value by about 40%. For the same diameter, the concentration was reduced by 70%. This means that for small diameters, the concentration of ions in the pore is the dominating factor. Finally, the analytical model for *σ _{pore}* developed in the present work is inserted in the continuum model of conductance [Eq. (1)] leading to the final model described by Eq. (7), which is compared to conductance values obtained from I-V curves extracted from non-equilibrium MD simulations with an external voltage performed in a previous work and to experimental data for sub 5-nm MoS

_{2}nanoporous membranes (Fig. 4)

For all the different *G*_{0} data reported in the literature, different experimental conditions were used leading to different values of ionic conductivity *σ _{bulk}* of KCl solutions depending on the temperature and the concentration of the electrolyte. In order to rationalize the data and as already done elsewhere for Si pores,

^{26}we decided to compute scaled conductance $G\u03030=(G0/\sigma bulk)given\xd7\sigma bulk1MKCl@RT$, where

*G*

_{0}and

*σ*were directly extracted from the literature and $\sigma bulk1MKCl@RT$ is the value of 11.18 S m

_{bulk}^{−1}measured experimentally recently.

^{8}First, as shown in Fig. 4, the difference between the pore and bulk conductivities has a significant effect on the value of the predicted conductance for pore with diameters lower than 2.0 nm. For such diameters, the original model [Eq. (1)] overestimates conductance by a factor of 2. This overestimation becomes a factor 5 for the diameter around 1.0 nm. In addition, Eq. (7) developed here using equilibrium MD simulations is in very good agreement with conductances computed from I-V curves extracted from non-equilibrium MD simulations using an external electric field. Compared to the linear empirical model proposed elsewhere,

^{22}the conductance for the SL-MoS

_{2}nanoporous membrane becomes negligible (10

^{−1}nS) for diameters below 0.6 nm. Moreover, from experimental conductance data for SL-MoS

_{2}nanopores with diameters around 2.0 nm,

^{3,4}according to the present model [Eq. (7)], the effective diameter of the pore would be closer to 3.0 nm than to 2.0 nm. Finally, experimental data

^{1}for diameters larger than 2.0 nm are very close to the present model [Eq. (7)] within the error bar when available. We also added into the conductance graphs presented in Fig. 4 conductance values extracted from measurements of WS

_{2}nanopores.

^{27}As shown in Fig. 4, for nanopores of diameters between 2 and 5 nm, conductance values are similar within the error bars. Therefore, the present model may be used also for other TMDs such as WS

_{2}. Finally, for few-Angstrom size defect pores (diameters lower than the limit of 0.6 nm), according to our model, conductance values are often overestimated. In a recent work,

^{8}we showed using MD simulations in the presence of an applied voltage that defect pores characterized by effective diameter d

^{*}< 0.6 nm do not conduct ions, characterized by negligible conductance below 20 pS. For those particularly tiny pores, more experimental measurements are needed to test the validity of the model since the passage of an ion across the membrane is a rare event. However, for pores with diameters around 1.0 nm, our model is in good agreement with experiments.

In summary, we developed a model of ionic conductance for sub 5-nm SL-MoS_{2} nanoporous membranes using MD simulations. Our model, which takes into account the concentration and the mobility of ions in the nanopores, shows that the behavior of the KCl electrolyte deviates by 50% from bulk properties for diameters below 2.0 nm. Moreover, our model is in very good agreement with simulation and experimental data of conductances in MoS_{2} nanoporous membranes. This model is essential for understanding the behavior of 2-D nanopores in this range of diameters to design and fabricate sensors for DNA or protein sequencing applications.

See supplementary material for nanopore modeling and numerical calculations, the ion concentrations and mobilities (Table SI), radial distribution of water molecules (supplementary Fig. S1), number of ions in the nanopore (supplementary Fig. S2), ion mean square displacements (supplementary Fig. S3), ion residence time (supplementary Fig. S4), and trajectories of ions crossing the nanopore (supplementary Fig. S5).

The simulations were performed using HPC resources from DSI-CCuB (Université de Bourgogne). This work was supported by a grant from the Air Force Office of Scientific Research (AFOSR), as part of a joint program with the Directorate for Engineering of the National Science Foundation (NSF), Emerging Frontiers and Multidisciplinary Office Grant No. FA9550-17-1-0047, and the NSF Grant No. EFRI 2-DARE (EFRI-1542707). Part of the work was funded by the Conseil Régional de Bourgogne Franche-Comté (Grant Nos. PARI NANO2BIO and ANER NANOSEQ).