Introduction

The ion transport through two-dimensional (2D) nanochannels1,2,3,4 formed by graphene nanosheets5,6,7,8,9,10,11,12,13,14,15,16,17 and other 2D materials15,18,19,20,21,22,23 possesses broad application prospects in seawater desalination12,14,24,25,26,27, osmotic power generation18,28,29,30, nanodevices5,6,7,19, etc. since transport through 2D nanochannels shows numerous unexpected phenomena distinct from transport in bulk solution such as memristor effect5,6, transistor-like gating effect19, complete steric exclusion9,16. These exceptional phenomena root from the unique ion transport mechanism in confined nanochannels which is mainly governed by two characteristics, (1) water molecules form layered structure in nanochannels8,31,32 instead of uniform continuum of bulk water, (2) ion’s hydration shell (HS) becomes distorted or even partially dehydrated7,15,33. Very recently, these unique water structures have been explored to regulate the ion transport dramatically. For instance, the position of an ion relative to water layers can affect the transport of hydrated ions15; the spatial and temporal correlations between ions and water layers can lead to an ionic current with features of ionic rectifiers and logical gates34; the mobility of ions in nanochannels can depend on the polarization of water molecules11 or the hydration strength of the ion35. However, most of current work only provides qualitative descriptions of these phenomena, rarely digging into the quantitative mechanisms. Though a recent article reports the correlation that the relative ion mobility ~ ion core diameter15, the prediction error is ~75% and, therefore, not sufficient to yield more in-depth physical insights. Another crucial issue for these studies is that, the Nernst-Einstein relation (ion mobility μ \(\propto\) ion diffusivity D) is usually taken for granted, yet has seldom (if ever) been validated for nanochannels15,30. To validate such relation in 2D nanochannels is pivotal, as it was recently shown to breakdown in some 1D nanochannels36. Although how bulk continuum water influence the ion transport has recently been elucidated after a century of epics exploration by the science community37, the physical nature of how the water layers and distorted HSs in 2D nanochannels regulate the ion transport remains far from being resolved. Not to mention further developing a mechanism to quantitatively predict the transport behavior, such as diffusion and electromigration, for various ions in nanochannels.

Here, the transport behavior of monatomic ions (Li+, Na+, K+, Rb+, Cs+, Ca2+, Mg2+ and Cl-) in various 2D nanochannels formed by assembling the nanosheets of graphene, h-BN, MoS2, g-C3N4, involving ion self-diffusion and electromigration, are studied by molecular dynamics (MD) simulations. Several different force fields (FF)38 are employed, such as OPLS-AA39, Merz40,41, Netz42,43 and Williams FFs44, the latter 3 are named by the authors who developed them. To describe ion-graphene interaction, two versions of Lennard-Jones (LJ) interaction parameters between ions and channel wall atoms (\({\varepsilon }_{{{\rm{I}}}-{{\rm{W}}}}\)) were employed for each FF: one calculated by the Lorentz-Berthelot (LB) mixing rule (denoted as \({\varepsilon }_{{{\rm{I}}}-{{\rm{W}}}}^{{{\rm{LB}}}}\)) and the other recently optimized one which describes the ion-graphene interactions in solution accurately44,45 (denoted as \({\varepsilon }_{{{\rm{I}}}-{{\rm{W}}}}^{{{\rm{ion}}}-{{\rm{\pi }}}}\), see Method section and Supplementary Tables 1, 2 for details). These extensive simulations reveal a concise rule: The ratio of the ion self-diffusivity in 2D nanochannels to that in bulk water (Dchannel/Dbulk) for ions with small radius of the 1st HS (rHS) correlates linearly with the ion-water layer distance, while it keeps constant for the ions with large rHS. The mechanism of how water layers and HSs of ions regulate Dchannel is in-depth elucidated on the basis of fundamental physics such as ion-water friction, the residence time (τ) of water molecule in ion’s HS, and the free energy profiles of water molecules around the ion. Quantitative relationships among these physical quantities are elucidated with physical implications well explained. Furthermore, the above mechanism is also proved to be valid for ion electromigration in 2D nanochannels, i.e. the Nernst-Einstein relation is thoroughly validated for the first time in 2D nanochannels with both simulation data and theoretical derivations.

Results

Position-dependent relative ion diffusivity

Among all simulations employing different FFs (Supplementary Tables 1) in graphene nanochannels, when the ion distribution profile changes (Fig. 1a, Supplementary Fig. 1), so does its Dchannel/Dbulk ratio (Fig. 1b, Supplementary Tables 2, 3). This finding inspired us to calculate the average ion-wall distance (dion-wall) with Eq. (1) (See “Methods” section), which gives the following concise correlation: Dchannel/Dbulk linearly increases with increasing dion-wall (Fig. 1b). However, the bulky alkali metal ions with large rHS (≥ rHS of K+, see Supplementary Table 2 and Supplementary Fig. 2–5) usually give Dchannel/Dbulk lower than the linear correlation predicts for large dion-wall position (Fig. 1b). Moreover, various 2D nanochannels assembled by nanosheets of different materials, such as graphene, h-BN, MoS2, g-C3N4 were also simulated (Supplementary Fig. 6). In these simulations, LJ parameters between ion and channel wall atoms (\({\varepsilon }_{{{\rm{I}}}-{{\rm{W}}}}\)) were adjusted to gradually change dion-wall (Fig. 1c–e, Supplementary Fig. 7), i.e. with \({\varepsilon }_{{{\rm{I}}}-{{\rm{W}}}}\) increasing, the peaks of ion’s distribution profile move closer to the channel walls (Fig. 1c), which reduces dion-wall. All these simulations confirm that Dchannel/Dbulk varies linearly with dion-wall for ions with small rHS, while it is constant and independent on dion-wall for ions with large rHS, like K+ (Fig. 1d, e) in all kinds of studied 2D nanochannels, thereby we will focus on results of graphene 2D nanochannels hereafter. Furthermore, such Dchannel/Dbulk ~ dion-wall correlation (varying linearly for Li+ and Na+; constant for bulky K+, Rb+ and Cs+) holds true even when dion-wall of the studied ions in graphene 2D nanochannels is controlled by restraining their z coordinates with harmonic potential (Supplementary Fig. 8, note that all other simulations in this work were performed without any artificial restrains). The linear correlation agrees well with very recent experiments on graphene 2D nanochannels15, yet our results are more precise with a much smaller prediction error, e.g. 0.16 versus the reported ~0.7515. It can be attributed to that we simulated diluted solutions to avoid the interference of ion-ion interaction, and Dchannel/Dbulk correlates with dion-wall instead of the reported ion-core diameter or near-wall probability15. More importantly, we validated such rule for divalent ions, and for many other 2D nanochannels besides graphene. Quite a few new fundamental insights emerge from our work for the first time, e.g. Dchannel/Dbulk of bulky ions (K+, Rb+ and Cs+, see Fig. 1d, e and Supplementary Fig. 8) is independent on dion-wall; and Dchannel/Dbulk can be even above 1 for small ions with sufficiently large dion-wall (Na+, Ca2+, Mg2+ in Fig. 1b and Supplementary Fig. 8). In addition, though we define bulky and small ions based on rHS, discussions with ionic radii46 would yield similar results, due to the correlation between these radii (Supplementary Fig. 9). Note we usually discuss the Dchannel/Dbulk ratio instead of Dchannel (Supplementary Note 1 and Supplementary Fig. 10) to focus on a universal rule, i.e. how solvent structure (water layers) and ion motion in nanochannels differ from bulk solutions, instead of on specific details of individual ions or FFs.

Fig. 1: Ratio of ion self-diffusivity (Dchannel/Dbulk) increases linearly with increasing ion-wall distance (dion-wall) for hydrated ions with small rHS, while it keeps constant for ions with large rHS.
figure 1

a Simulation system of hydrated ions in 2D nanochannels, with distribution profiles of Na+ and water molecules in graphene 2D nanochannel. Distribution profiles were taken from simulations using Merz force field (FF)40,41. \({\varepsilon }_{{{\rm{I}}}-{{\rm{W}}}}^{{{\rm{LB}}}}\) and \({\varepsilon }_{{{\rm{I}}}-{{\rm{W}}}}^{{{\rm{ion}}}-{{\rm{\pi }}}}\) represent the original and optimized44,45 versions of the Lennard-Jones (LJ) parameters between ion and graphene, respectively (Supplementary Table 2). b Dchannel/Dbulk increases linearly with dion-wall for all ions and FFs under study (shadowed area represents the prediction error), except ions with large rHS (e.g. K+, Rb+, Cs+). Merz40,41, Netz42,43 and Williams44 FFs (named by the authors who developed them) and OPLS-AA39 FF were employed. The data for water molecules (black hollow diamond) is shown for comparison. c Distribution profiles of Na+ and K+ in graphene 2D nanochannel simulated with Williams FF. The \({\varepsilon }_{{{\rm{I}}}-{{\rm{W}}}}\) (LJ parameters between ion and wall atoms) of Na+ and K+ were adjusted smoothly from 0.001 kJ·mol-1 to corresponding \({\varepsilon }_{{{\rm{I}}}-{{\rm{W}}}}^{{{\rm{ion}}}-{{\rm{\pi }}}}\) values to gradually change their distribution profiles (c) and dion-wall (d). Each color in (c, d) represents an employed \({\varepsilon }_{{{\rm{I}}}-{{\rm{W}}}}\). d, e Dchannel/Dbulk increases linearly with dion-wall for Na+ with small rHS while keeps constant for K+ with large rHS. This finding holds true for 2D nanochannels constructed by graphene (d), h-BN, MoS2 or g-C3N4 nanosheets (e, \({\varepsilon }_{{{\rm{I}}}-{{\rm{W}}}}\) varies between 0.01 and 4 kJ·mol–1, using Williams FF), also true for other studied ions (see Supplementary Fig. 8 for Li+, Rb+, Cs+). The shadowed area in (b, d, e) are identical. Source data are provided as a Source Data file.

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For a 2D nanochannel accommodating one hydrated ion and water layers, only ion-wall and ion-water layer interactions can affect ion diffusion. Further simulations indicate that hydrated ion-wall interaction hardly affect Dchannel/Dbulk, even when \({\varepsilon }_{{{\rm{I}}}-{{\rm{W}}}}\) parameters vary by two orders of magnitude (Supplementary Note 2 and Supplementary Fig. 11). Thus the above Dchannel/Dbulk ~ dion-wall rule can mainly be attributed to the hydrated ion-water friction force as follows. When ions approach water layers (small dion-wall) or possess large rHS like K+, they will contact closely with the water layers (Fig. 1 and Supplementary Fig. 8), i.e. the water density in the diffusion directions increases (Supplementary Fig. 12). Thus ions suffer from larger ion-water friction, and Dchannel decreases. Contrarily, ions possessing small rHS and large dion-wall suffer from small friction, and their Dchannel/Dbulk could be even above 1 (Ca2+ and Mg2+ in Fig. 1b). However, two questions remain open: (i) What is the physical nature of ion-water friction in the 2D nanochannel, and can it be explained from fundamental physics such as force, energy or even HS structure? (ii) Can a mechanism be quantitatively developed? For instance, why is Dchannel/Dbulk of hydrated K+ independent on dion-wall? The following discussions will focus on graphene 2D nanochannels to answer these questions in-depth, yet can also apply for other 2D nanochannels.

Correlations of physics quantities

The well-known Einstein equation relates the diffusivity to friction force37,47,48,49, \(D=\frac{{{{\rm{k}}}}_{{{\rm{B}}}}T}{\lambda }=\frac{{\gamma \left({{{\rm{k}}}}_{{{\rm{B}}}}T\right)}^{2}}{{I}_{{{\rm{FACF}}}}}\) (\(\lambda=\frac{{I}_{{{\rm{FACF}}}}}{{\gamma {{\rm{k}}}}_{{{\rm{B}}}}T}\) is the ion-water friction coefficient, \(\gamma\) is the dimensionality (3 for bulk solution, 2 for 2D nanochannel), \({I}_{{{\rm{FACF}}}}={\int }_{0}^{\infty }{{\rm{FACF}}}(t){dt}\), and FACF(t) refers to the Force Auto-Correlation Function, see “Methods” section). Thus, FACFs of the studied ions were calculated to analyze the ion-water friction force, as shown in Supplementary Figs. 13–18, where DFACF agree with DMSD (Supplementary Fig. 13, the superscript FACF and MSD indicate the diffusivity are calculated from FACF and MSD, respectively) and \({D}_{{{\rm{channel}}}}^{{{\rm{FACF}}}}\)/\({D}_{{{\rm{bulk}}}}^{{{\rm{FACF}}}}\) ~ dion-wall follows a similar rule as \({D}_{{{\rm{channel}}}}^{{{\rm{MSD}}}}\)/\({D}_{{{\rm{bulk}}}}^{{{\rm{MSD}}}}\) ~ dion-wall (Supplementary Fig. 14). Ions can be located in three different solvation environments or positions in this study, (i) bulk water, (ii) mid-position in nanochannels (the middle position inside the 2D nanochannel, dion-wall = 0.5 nm, Fig. 1a) with minimum water density, (iii) edge-position in nanochannels (the peak positions of ion distribution profiles which are close to water layers, dion-wall < 0.5 nm, Fig. 1a). As for a given ion, its \({I}_{{{\rm{FACF}}}}^{{{\rm{edge}}}}\) (IFACF for the ion locating at the edge-position) is usually ~20% larger than \({I}_{{{\rm{FACF}}}}^{{{\rm{bulk}}}}\) (IFACF for the ion in bulk water). However, \({I}_{{{\rm{FACF}}}}^{{{\rm{mid}}}}\) (IFACF at mid-position) can be larger than, similar to or even smaller than \({I}_{{{\rm{FACF}}}}^{{{\rm{bulk}}}}\) (Fig. 2a). The transport behavior of three representative ions is characterized as follows, (i) for Li+, 1/\({I}_{{{\rm{FACF}}}}^{{{\rm{edge}}}}\,\)< 1/\({I}_{{{\rm{FACF}}}}^{{{\rm{bulk}}}}\) ≤ 1/\({I}_{{{\rm{FACF}}}}^{{{\rm{mid}}}}\) with Dedge/Dbulk < 1 ≤ Dmid/Dbulk; (ii) for Na+, 1/\({I}_{{{\rm{FACF}}}}^{{{\rm{edge}}}}\) < 1/\({I}_{{{\rm{FACF}}}}^{{{\rm{mid}}}}\) ≈ 1/\({I}_{{{\rm{FACF}}}}^{{{\rm{bulk}}}}\) with Dedge/Dbulk <Dmid/Dbulk ≈ 1; (iii) for K+, 1/\({I}_{{{\rm{FACF}}}}^{{{\rm{edge}}}}\) ≈ 1/\({I}_{{{\rm{FACF}}}}^{{{\rm{mid}}}}\) < 1/\({I}_{{{\rm{FACF}}}}^{{{\rm{bulk}}}}\) with Dedge/Dbulk ≈ Dmid/Dbulk < 1 (showing a constant Dchannel/Dbulk, independent on dion-wall, Fig. 1d). The following discussion is focused on these three ions. Other ions’ IFACF and Dchannel/Dbulk just show features similar to either of them (Supplementary Tables 4-7). Further, the ions’ FACFs could be divided into the head part (violently oscillating for t <~ 0.3 ps) and the following tail part (smoothly decaying to 0, Fig. 2b), with corresponding integral as \({I}_{{{\rm{head}}}}={\int }_{0}^{{t}_{{{\rm{b}}}}}{{\rm{FACF}}}(t){{\rm{d}}}t\) and \({I}_{{{\rm{tail}}}}={\int }_{{t}_{{{\rm{b}}}}}^{\infty }\,{{\rm{FACF}}}(t){{\rm{d}}}t\), where tb is the boundary time separating these two parts (Supplementary Fig. 15–18). Note when a given ion is located in different solvation environments, Ihead changes very slightly, while the change of IFACF mainly comes from Itail (Fig. 2c, Supplementary Note 3).

Fig. 2: Difference between ion-water friction in 2D nanochannels and bulk water: higher ratio τchannel/τbulk leads to larger ∆IFACF.
figure 2

a ∆IFACF (\({I}_{{{\rm{FACF}}}}^{{{\rm{channel}}}}-{I}_{{{\rm{FACF}}}}^{{{\rm{bulk}}}}\)) of Li+, Na+ and K+ at mid-position (blue area) or edge-position (red area). b FACFs of Na+ (simulated with Merz FF) in different solvation environments, bulk solution (green), mid- (red) or edge- (blue) position. These FACFs all oscillate violently in the head part, then smoothly decay to 0 in the tail part. See Supplementary Figs. 15–18 for FACFs of other ions. c Compared with bulk solution, Ihead changes slightly (∆Ihead = \({I}_{{{\rm{head}}}}^{{{\rm{channel}}}}-{I}_{{{\rm{head}}}}^{{{\rm{bulk}}}}\) ≈ 0), thus Itail dominates the change of IFACF (∆Itail = \({I}_{{{\rm{tail}}}}^{{{\rm{channel}}}}-{I}_{{{\rm{tail}}}}^{{{\rm{bulk}}}}\) ≈ ∆IFACF). d ∆λ (or ∆IFACF) correlates with τchannel/τbulk. ∆λ is the difference between ion-water friction coefficient (λ) in nanochannel and that in bulk solution, which equals to ∆IFACF multiplied by a constant, \(\frac{1}{\gamma {{{\rm{k}}}}_{{{\rm{B}}}}T}\) (see Eq. (4) in “Methods”). The black curve is exponential fitting result. The dash curve represents the quantitative correlation between ∆IFACF (unit: 1015 N2·mol-2 · s) and τchannel/τbulk: \(\Delta {I}_{{{\rm{FACF}}}}=-5.71\bullet \exp (-0.87\bullet \frac{{\tau }_{{{\rm{channel}}}}}{{\tau }_{{{\rm{bulk}}}}})+1.24\) and the shadowed area represents the prediction error. e Exchange of water molecules between the 1st HS of Na+ (simulated with Merz FF) and the surrounding water layers. 100 snapshots of a 1 ns simulation trajectory are superimposed. In each sub-figure, the left one is the side view, while the right is top view. For clarity, only the upper water layer is shown in the top view. When τ is small, water molecules exchange rapidly, while they exchange slowly when τ is large. Source data are provided as a Source Data file.

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It is known that IFACF is closely related to the speed of water molecules moving around the ion37, we further calculated the residence time (τ) of water molecules in the 1st HS of ions (abbreviated as HS). Figure 2d shows that ∆IFACF (\({I}_{{{\rm{FACF}}}}^{{{\rm{channel}}}}-{I}_{{{\rm{FACF}}}}^{{{\rm{bulk}}}}\)) correlates well with τchannel/τbulk. Specifically, when an ion moves from bulk water into the nanochannel, and if τ rises sharply (τchannel/τbulk > 3), IFACF also increases considerably (∆IFACF > 500×1012 N2·mol-2 · s), e.g. for Li+ and Na+ at edge-position, as well as K+ at both mid- & edge-positions (Fig. 2d). On the contrary, smaller τchannel/τbulk (≤2) leads to ∆IFACF ≈ 0 for Na+ at mid-position, or even below 0 for Li+ at mid-position (Fig. 2d). Shorter τ indicates more frequent exchanges of water molecules between HS and the surrounding solvent, and a more rapid change of water configurations around the ion (Fig. 2e). Consequently, the correlation between water configurations around the ion with large time interval (∆t > tb) becomes weaker, and FACF(∆t) becomes closer to 0. Thus FACF’s tail decays to 0 more quickly and Itail decreases, while Ihead hardly changes, leading to decreased IFACF consequently. Contrarily, longer τ indicates a slower change in the water configurations around the ion, and consequently a larger Itail and IFACF.

Further, the ratio of τchannel/τbulk for the studied ions correlates with the shape of their HSs: Ions with small τchannel/τbulk (e.g. Li+ or Na+ at mid-position in nano-channel) show spherical HSs similar to those in bulk water (Fig. 3a). Contrarily, ions with large τchannel/τbulk (e.g. Li+, Na+ at edge-position in channel and K+ at mid- or edge-position) show HSs distorted to rings & poles (Fig. 3a), due to the nanoconfinement effect overwhelming the z (channel height, or d-spacing direction) component of ion-water interaction, driving water molecules in the middle of HS toward the channel-walls for energetic reasons (Supplementary Note 4 and Supplementary Figs. 19–20). For instance, the ion-water interaction for K+ is so weak (insufficient to maintain a spherical HS) that its HS is distorted to two rings when locating at mid-position (left of Fig. 3a). In addition, HSs of ions at edge-position become too crowded to allow water molecules locate in regions between the ring and the pole (right of Fig. 3a, Supplementary Note 4). When comparing an ion at different solvation environments, the density of HS (ρHS) of the distorted HS usually increases, as its volume decreases yet the hydration number hardly changes (Supplementary Figs. 2–5). Thus, the free energy profile of water molecules around the ion was calculated from the density profile with Eq. (7) (Fig. 3b, Supplementary Fig. 21), and another correlation emerges: ln(τchannel/τbulk) decreases linearly with ∆Fchannel-∆Fbulk (Fig. 3c), where ∆F is the free energy difference between water in HS and water in the surrounding solvent (e.g. the water layers in nanochannel, or bulk water in the free solution), and the subscript (channel or bulk) indicates ions located in nanochannel or in bulk solution, respectively. Such correlation indicates that higher free energy of water molecules in HS (relative to surrounding solvent) helps water molecule to escape from the HS, which decreases τ.

Fig. 3: High free energy in the HS helps water molecules to escape, resulting in a small τchannel/τbulk.
figure 3

a Spatial distribution functions (SDFs) for water molecules in the 1st hydration shells (HS) of Li+, Na+ and K+ at the mid- or edge-position in a 2D nanochannel. At the mid-position, Li+ and Na+ possess sphere-like HSs while the K+ HS splits into two rings. At the edge-position, the HSs of all three ions are distorted to rings and poles. b Potential of mean force (PMF) profiles of water molecules around Li+, Na+ and K+ in bulk solution, mid- or edge-position of the 2D nanochannels, simulated with Merz FF. ∆F refers to the free energy difference between the ion HS and the surrounding (e.g. the water layers in nanochannel, or bulk water in the bulk solution), with subscripts ‘R’ and ‘P’ referring to the ring and pole part of HS, respectively. c ln(τchannel/τbulk) correlates linearly with ΔFchannel-ΔFbulk: \({\mathrm{ln}}\left(\frac{{\tau }_{{{\rm{channel}}}}}{{\tau }_{{{\rm{bulk}}}}}\right)=\frac{-0.00443({\Delta F}_{{{\rm{channel}}}}-{\Delta F}_{{{\rm{bulk}}}})}{{{\rm{R}}}T}+1.265\), where R is the ideal gas constant. Note at ∆Fchannel-∆Fbulk ≈ 0, τchannel/τbulk is yet above 1, which can be attributed to the nanoconfinement effect of 2D nanochannels, i.e. the channel walls prevent water molecules from leaving the HS along the channel height direction, thus hindering water molecules to exchange between HS and surrounding solvent in nanochannel52. Source data are provided as a Source Data file.

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Mechanism of the ion motion in 2D nanochannels

The physical nature of the Dchannel/Dbulk ~ dion-wall correlation and the ion-water friction can be explained as follows (Fig. 4a). When an ion approaches a water layer in 2D nanochannel (smaller dion-wal) or has a large rHS (like K+), the nanoconfinement effect significantly distorts its HS (Supplementary Note 4). This leads to a larger |∆F| between the HS and the water layer, which hinders water molecules to exchange between them, and τ significantly increases. Thus, the decay of FACF becomes slower with increasing IFACF, i.e. the ion-water friction rises, and thus Dchannel decreases consequently.

Fig. 4: Ion transport mechanism in 2D nanochannels and the involved quantitative relations.
figure 4

a Underlying mechanism how water layers and HSs affect ion transport in 2D nanochannels. The quantitative correlations between involving physical quantities are listed where applicable, also discussed in previous discussions on Figs. 2, 3 and Eq. (8) in “Methods” section. ∆F* refers to ∆Fchannel-∆Fbulk. As for some physical quantity such as τ, ∆F, the difference between this quantity in the nanochannel and the bulk water (τchannel/τbulk and ∆Fchannel-∆Fbulk) is more frequently discussed than the quantity itself, so as to focus on how water layers in 2D nanochannels affect these quantities. b Linear relation between ion diffusivity Dchannel and mobility μ under electric field E = 0.1 V·nm-1 in graphene 2D nanochannel (see Supplementary Figs. 22–23 for h-BN, g-C3N4 and MoS2 nanochannels). The black line is the linear fitting result (y = 0.0239·x, R2 = 0.9683), whose slope is consistent with the theoretical value of 0.0257 J · C–1 in the Nernst–Einstein relation. Error bars represent the standard error (n = 3 independent MD simulations), and the centers of error bars indicate the means. c Under E = 0.1 V·nm–1, μchannel/μbulk increases linearly with dion-wall for all ions and FFs under study (shadowed area represents the prediction error), except ions with large rHS (e.g. K+, Rb+, Cs+), which is similar to the Dchannel/Dbulk ~ dion-wall correlation in Fig. 1b. Source data are provided as a Source Data file.

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Such mechanism well explains all above findings. For instance, when Na+ (or other small ions) moves from mid- to edge-position, its sphere-like HS becomes distorted to a ring and a pole (Fig. 3a), with ρHS increasing from 17.3 g·cm–3 to 44.2 g·cm–3 (see Supplementary Note 4 for explanation) and |∆F| increasing by ~2.5 kJ·mol–1 (Fig. 3b). This leads to an increasing ratio τchannel/τbulk, and ∆IFACF increases up to ~1000 × 1012 N2·mol–2 · s (Fig. 2d), which results in a slower ion diffusion due to rised friction. Therefore, Dchannel/Dbulk decreases with decreasing dion-wall (Fig. 1b). Contrarily, when K+ moves from mid-position to edge-position, the ρHS and ∆F change only very slightly (Fig. 3b), which gives a constant Dchannel/Dbulk (Fig. 1b). Such mechanism also explains why ions with large rHS usually show a negative deviation (Fig. 1b), particularly for location with large dion-wall. That is, large rHS leads to distorted HS, larger |∆F | , increased τ and IFACF, and consequently, decreased Dchannel/Dbulk (Fig. 4a). Therefore, when dion-wall is large, Dchannel/Dbulk of large-rHS ions (independent on dion-wall) may be quite smaller than the prediction of Dchannel/Dbulk ~ dion-wall correlation, whose slope is positive (Fig. 1b).

Besides the self-diffusion of hydrated ions, the ion migration behavior under external electric field in the confined 2D nanochannels are also investigated. We simulated the ion electromigration in 2D nanochannels constructed by graphene, h-BN, g-C3N4 or MoS2 under electric field between 0.1 and 0.5 V·nm–1 along the x direction (Fig. 4b, Supplementary Figs. 22, 23), and found that the ion mobility μ (independent on electric field, see Supplementary Fig. 24) is proportional to their self-diffusivity Dchannel, which agrees well with the Nernst–Einstein relation: D = kBTμ/q, where kB is Boltzmann constant, T is temperature and q is ion’s charge. Combining simulations and theoretical derivations (Supplementary Note 5), the validity of Nernst–Einstein relation in 2D nanochannel is proved for the first time. Such validity could be attributed to that, the electric field hardly changes the structure of water layers (Supplementary Fig. 25), in which the ions diffuse (without external field) or drift (driven by the electric field). Furthermore, the relationship between μchannel/μbulk and dion-wall is similar to the ratio of the self-diffusivities without external field (Fig. 4c), indicating that the above mechanism (Fig. 4a) applies for both ion self-diffusion (Fig. 1b) and electromigration (Fig. 4c) in 2D nanochannels. Please refer to Supplementary Table 8 for all studied systems and resulting conclusions in this work.

Discussion

Our research provides a complete theoretical framework for the transport of hydrated monatomic ions in various 2D nanochannels assembled by graphene, h-BN, g-C3N4, MoS2, including self-diffusion and electromigration. Through in-depth MD simulations employing multiple force fields, we unveil that the diffusivity of ions in a 2D nanochannel Dchannel depends on their HS radius (rHS) and the position in the nanochannel, differing from their constant bulk diffusivity Dbulk. In detail, the ratio of ion diffusivity Dchannel/Dbulk shows for ions with small rHS a linear correlation with dion-wall, and for ions with large dion-wall, Dchannel/Dbulk > 1 is found. In contrast, for ions with large rHS (e.g., K+, Rb+, Cs+) Dchannel/Dbulk is constant, independent on dion-wall. Importantly, the underlying physical mechanism has been revealed with quantitative correlations, which bridges fundamental physical quantities such as the free energy profile of water molecules around the ion, the residence time of water molecule in HS and the water-ion friction force. Moreover, besides the ions’ self-diffusivity, their mobility under an electric field also follows the above rule, and the well-known Nernst–Einstein relation is proved to be valid for electrolytes in 2D nanochannels. To our knowledge, this work provides a universal and general approach to quantitatively study the ion transport in confined 2D nanochannels tracing back to the physical nature. The position-dependent diffusivity/mobility of ions confined in the studied 2D nanochannels is to the sharp contrary of bulk solution, where ions’ diffusivity/mobility are constant. This indicates ions’ transportation in 2D nanochannels could be adjusted by controlling their position by some external fields4, e.g. electric- or magnetic-field, which may be exploited for various nano-devices5,6,7,19. Furthermore, considering the variation range of diffusivity (or mobility) shown in Figs. 1 and 4, more attention should be put on the entrance effect17 or specific ion-wall attraction50 when employing 2D nanochannels for ion-sieving. In brief, all the findings in our work pave the way to thoroughly understand the ion transport phenomena in nanochannels, also help design and construction of high-performance 2D nanochannels for a wide range of application fields, such as ion-sieving, nano-device, osmotic energy conversion, etc.

Methods

MD simulations

MD simulations were performed for the 2D nanochannel systems shown in Fig. 1a: salt solution containing one ion (Li+, Na+, K+, Rb+, Cs+, Ca2+, Mg2+ or Cl-) and a certain number of water molecules (see later explanation and Supplementary Table 8) is confined in a 2D nanochannel whose wall material is graphene, h-BN, MoS2 or g-C3N4 (Supplementary Fig. 6). The x-y size of the simulation box equals to that of different wall materials (see Supplementary Table 8). In an 2D nanochannel, the distance between the channel walls is 1 nm (Supplementary Fig. 6). Note in MoS2 nanochannel, it refers to the distance between the 2 inner S atom planes. The atoms of channel walls were fixed, which have yielded accurate simulation results as plenty of recent publications51,52 did.

The number of water molecules (Nwater) in 2D nanochannels shown in Fig. 1a was determined by simulating water-soaked 2D nanochannel system51: a 2D nanochannel (identical with that in Fig. 1a) placed in the center of a simulation box containing 15,723 water molecules (box size: about 10 nm × 10 nm × 5 nm). Nwater was calculated as \({N}_{{{\rm{water}}}}=\left\langle {N}_{{{\rm{c}}}}\right\rangle \cdot A/{A}_{{{\rm{c}}}}\), where A is the x-y area of 2D nanochannel (e.g., 4.920 × 5.124 nm2 for graphene 2D nanochannel. See Supplementary Table 8); Ac = 3 × 3 nm2 is the x-y area of center region of the 2D nanochannel in water-soaked 2D nanochannel system; \(\left\langle {N}_{{{\rm{c}}}}\right\rangle\) is the ensemble-averaged number of water molecules in the center region of water-soaked 2D nanochannel (whose x-y area is Ac).

MD simulations for an ion in bulk solution were also performed using a cubic simulation box (size: 2.55 nm × 2.55 nm × 2.55 nm). The simulation system consists of 1 ion and 553 water molecules (corresponding to the bulk water density of 0.998 g/cm3, excluding the mass of ion). The simulation systems in our work usually contain 1 ion, if not otherwise specified, to avoid the interference of ion-ion correlations, so that we could focus on the ion-water interactions.

The intermolecular interactions include coulombic and van der Waals interactions, where van der Waals interactions were described as the Lennard-Jones (LJ) potential. If not otherwise specified, LJ parameters were derived from Lorentz-Berthelot (LB) mixing rule. SPC/E water model53 was employed in this work; the force field (FF) parameters for graphene carbon atoms (σC = 0.3214 nm, εC = 0.4900 kJ·mol–1) were taken from the literature54. Four sets FF parameters for ions from Netz et al.42,43, Merz et al.40,41, Williams et al.44 (referred to as Netz FF, Merz FF, and Williams FF, respectively) and OPLS-AA39 FF were employed in this work (Supplementary Table 1), which well reproduced the experimental diffusivity of ions in bulk solutions (Supplementary Fig. 26). Our simulations also reproduced the experimental ion mobility (Supplementary Fig. 27), both in bulk solutions and 2D nanochannels15. For the LJ parameters between ions and graphene, two versions of FFs (Supplementary Table 2) were employed: (i) original FFs: calculated with the LB mixing rule (denoted as \({\varepsilon }_{{{\rm{I}}}-{{\rm{W}}}}^{{{\rm{LB}}}}\)); (ii) optimized FFs: the recently optimized LJ parameters44,45 (denoted as \({\varepsilon }_{{{\rm{I}}}-{{\rm{W}}}}^{{{\rm{ion}}}-{{\rm{\pi }}}}\)) which describe the ion-π interaction between ion and graphene accurately (LJ parameters, \({\sigma }_{{{\rm{I}}}-{{\rm{W}}}}\), were all calculated with the LB mixing rule). The LJ parameters and atomic charges for wall atoms of h-BN, MoS2 and g-C3N4 were taken from refs. 55,56,57. (Supplementary Table 1).

The simulation trajectories were integrated via the leapfrog algorithm with a time step of 2 fs with periodic boundary conditions (PBCs) applied to x and y directions (the channel wall plane). For simulations of water-soaked 2D nanochannel system and bulk solution system, PBCs were applied to x, y and z directions. If not otherwise specified, the simulation setups for simulations of water-soaked 2D nanochannel system and bulk solution system were identical to the simulations of 2D nanochannels shown in Fig. 1a. The intermolecular LJ interactions were computed with a cutoff of 1.2 nm with tail correction. The long-range electrostatic interactions were calculated by the particle mesh Ewald (PME) method58, with a real space cutoff of 1.2 nm. The temperature was controlled at 298.15 K by Nose-Hoover thermostat59,60. The length of bonds involving H atoms was constrained by SETTLE algorithm61. In a typical MD simulation procedure, an energy minimization for 10,000 steps with a steepest-descent minimization algorithm was performed, followed by a 2 ns NVT MD simulation increasing the temperature to 298.15 K; subsequently, a 55 ns production simulation was performed in NVT ensemble and the last 50 ns data were used for analysis. For water-soaked 2D nanochannel system, a 10 ns production simulation was performed in NPT ensemble (the pressure was controlled at 1 bar by Parrinello-Rahman barostat) and the last 5 ns data were used for calculating Nwater. All MD simulations were performed by GROMACS 5.1.5 simulation package62 and VMD software63 was utilized for visualization.

Simulation data analyses

Average ion-wall distance (d ion-wall)

We have calculated dion-wall as follow:

$${d}_{{{\rm{ion}}}-{{\rm{wall}}}}=\int _{-0.5}^{0.5}{\rho }_{i}\left(z\right)\left(0.5-{{\rm{|z|}}}\right){{\rm{d}}}z$$
(1)

where z values of +/- 0.5 and 0 corresponds to the position of the bottom/upper channel wall (the inner S planes for MoS2 channel wall), the middle of the nanochannel, respectively, ρi(z) is ion’s distribution probability along z direction (Supplementary Figs. 1, 7), 0.5-|z| is ion’s distance from the nearest channel wall. The average (water’s) oxygen-wall distance was calculated similarly, show in Fig. 1b.

Diffusivity calculated with the mean square displacement (MSD) method

Ion’s diffusivity (D) could be obtained by computing the slope of its MSD curve51:

$$D={{\mathrm{lim}}}_{t\to \infty }\frac{\left\langle {\left|{{\bf{r}}}\left(t\right)-{{\bf{r}}}(0)\right|}^{2}\right\rangle }{\beta \cdot t}$$
(2)

where \({{\bf{r}}}(t)\) stands for ion’s position at time t, \(\left\langle {\left|{{\bf{r}}}\left(t\right)-{{\bf{r}}}(0)\right|}^{2}\right\rangle\) is the ensemble average of ion’s MSD for a given simulation time t, the coefficient β is 4 for ion in 2D nanochannel or 6 for ion in bulk water. For D calculation, we performed 5 independent 55-ns production simulations, with averaged results and stand error reported. Diffusivities in this work were usually calculated with the MSD method if not otherwise specified.

Diffusivity calculated with the force auto-correlation function (FACF) method

We also obtained D using the FACF method64:

$${{\rm{FACF}}}(t)=\left\langle F(t)F(0)\right\rangle$$
(3)
$$D=\frac{{{{\rm{k}}}}_{{{\rm{B}}}}T}{\lambda }=\frac{\gamma {{{{\rm{k}}}}_{{{\rm{B}}}}}^{2}{T}^{2}}{{\int }_{0}^{\infty }{{\rm{FACF}}}(t){{\rm{d}}}t}$$
(4)

where \(\left\langle F(t)F(0)\right\rangle\) is the Force Auto-Correlation Function (FACF); F(t) is the total force that water molecules exert on the ion at time t, respectively; <…> represents the ensemble average; \(\lambda=\frac{1}{{\gamma {{\rm{k}}}}_{{{\rm{B}}}}T}{\int }_{0}^{\infty }{{\rm{FACF}}}(t){{\rm{d}}}t\) is the ion-water friction coefficient (the coefficient γ is 2 for ion in 2D nanochannel or 3 for ion in bulk solution); kB and T are Boltzmann constant and temperature, respectively. In FACF simulations, the ion was fixed so that the ‘infinite mass’ approximation could apply65. Note as for simulations elsewhere in this work, the ion was free to move.

Residence time of water molecules in ions’ 1st hydration shell

The residence time (τ) for water molecules staying in ion’s 1st HS was calculated by the residence auto-correlation function R(t)66:

$$\tau=\int _{0}^{\infty }R(t){{\rm{d}}}t$$
(5)
$$R\left(t\right)=\frac{1}{{N}_{{{\rm{h}}}}}{\sum }_{i=1}^{{N}_{{{\rm{h}}}}}\left\langle {\theta }_{i}(t){\theta }_{i}(0)\right\rangle$$
(6)

where θi (t) is the Heaviside function for the ith water molecule in ion’s 1st HS, which is 1 if the water molecule’s Ow atom stays in the 1st HS at time t and 0 otherwise.

Potential of mean force (PMF)

The PMF profiles for water molecules around the ion were calculated as follow67,68:

$${{\rm{PMF}}}\left(r,z\right)=-{{{\rm{k}}}}_{{{\rm{B}}}}T{\mathrm{ln}}\frac{{\rho }_{(r,z)}}{{\rho }_{0}}$$
(7)

where r and z are the radial distance and z-direction distance from the ion, respectively; kB and T are Boltzmann constant and temperature, respectively; ρ(r,z) is the local water density in (r, z) position; ρ0 represents bulk water density (1 g·cm–3) for the ion in bulk water or the peak density of water layers (3.27 g·cm–3) for the ion in nanochannel. The free energy difference (∆F) between HS of an ion and the surrounding solvent was calculated as:

$$\Delta F=-{{{\rm{k}}}}_{{{\rm{B}}}}T{\mathrm{ln}}\frac{{\rho }_{{{\rm{HS}}}}}{{\rho }_{0}}$$
(8)

where ρHS is the peak water density (maximum ρ(r,z)) in the HS of the ion.

Ion mobility

Ion’s mobility (μ) can be calculated as:

$$\mu=\frac{v}{E}=\frac{1}{E}\cdot {{\mathrm{lim}}}_{t\to {{\infty }}}\frac{\left\langle {{\bf{|r}}}\left(t\right)-{{\bf{r}}}\left(0\right){{|}}\right\rangle }{t}$$
(9)

where v is ion’s speed, E is the strength of electric field (along the x direction for ions in 2D nanochannel); \({{\bf{r}}}(t)\) stands for ion’s position at time t and \(\left\langle {{\bf{|r}}}\left(t\right)-{{\bf{r}}}\left(0\right)|\right\rangle\) is the ensemble average of ion’s displacement for a given simulation time t.