# Molecular streaming and its voltage control in ångström-scale channels

## Abstract

Over the past decade, the ability to reduce the dimensions of fluidic devices to the nanometre scale (by using nanotubes1,2,3,4,5 or nanopores6,7,8,9,10,11, for example) has led to the discovery of unexpected water- and ion-transport phenomena12,13,14. More recently, van der Waals assembly of two-dimensional materials15 has allowed the creation of artificial channels with ångström-scale precision16. Such channels push fluid confinement to the molecular scale, wherein the limits of continuum transport equations17 are challenged. Water films on this scale can rearrange into one or two layers with strongly suppressed dielectric permittivity18,19 or form a room-temperature ice phase20. Ionic motion in such confined channels21 is affected by direct interactions between the channel walls and the hydration shells of the ions, and water transport becomes strongly dependent on the channel wall material22. We explore how water and ionic transport are coupled in such confinement. Here we report measurements of ionic fluid transport through molecular-sized slit-like channels. The transport, driven by pressure and by an applied electric field, reveals a transistor-like electrohydrodynamic effect. An applied bias of a fraction of a volt increases the measured pressure-driven ionic transport (characterized by streaming mobilities) by up to 20 times. This gating effect is observed in both graphite and hexagonal boron nitride channels but exhibits marked material-dependent differences. We use a modified continuum framework accounting for the material-dependent frictional interaction of water molecules, ions and the confining surfaces to explain the differences observed between channels made of graphene and hexagonal boron nitride. This highly nonlinear gating of fluid transport under molecular-scale confinement may offer new routes to control molecular and ion transport, and to explore electromechanical couplings that may have a role in recently discovered mechanosensitive ionic channels23.

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## Data availability

The data that support the plots within this paper and other findings of this study are available in the main text and Extended Data Figures. Additional information is available from the authors upon reasonable request.

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## Acknowledgements

T.M. and L.B. acknowledge funding from ANR project Neptune. B.R. acknowledges a Royal Society Fellowship, a L’Oréal Fellowship for Women in Science, and EPSRC grant EP/R013063/1. A.S. acknowledges funding from the European Union’s Horizon 2020 (EU H2020) Framework Programme/ERC Starting Grant agreement number 637748—NanoSOFT. L.B. acknowledges funding from the EU H2020 Framework Programme/ERC Advanced Grant agreement number 785911—Shadoks. A.R.P. acknowledges funding from the EU H2020 Framework Programme/European Training Programme 674979—NanoTRANS. S.A.D. was funded by a scholarship from the University of Engineering and Technology, Lahore Pakistan. A.K., B.R. and A.K.G. were supported by Lloyd’s Register Foundation and European Research Council (ARTIMATTER). T.M. thanks S. Blin and H. Yoshida for assistance.

### Reviewer information

Nature thanks Rohit Karnik and the other anonymous reviewer(s) for their contribution to the peer review of this work.

## Author information

Authors

### Contributions

B.R., L.B. and A.S. designed and directed the project. A.K., B.R. and S.A.D. fabricated the devices. T.M. performed the measurements and their analysis. A.R.P., T.M. and L.B. provided theoretical support. T.M., L.B., B.R., A.K. and A.R.P. wrote the manuscript with inputs from A.K.G. All authors contributed to discussions.

### Corresponding authors

Correspondence to A. K. Geim or L. Bocquet or B. Radha.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Ångström-scale channel devices.

a, Optical image of a device with ångström channels. The square in light pink colour is the silicon nitride membrane which has a rectangular hole shown by the red dotted line. Covering the hole, the bottom graphite layer, spacer and top graphite layer are placed. Bottom and top graphite are visible in the image in light and bright yellow colours. b, Atomic force microscopy (AFM) image of the bilayer graphene spacer lines on the device. The histogram of the heights (below the AFM image) shows that the spacer is about 0.7 ± 0.1 nm thick.

### Extended Data Fig. 2 Gated pressure-driven current.

Streaming current per channel plotted as a function of ∆P/L with ∆V ranging between −100 mV and 100 mV (colour coded from blue to red with increasing voltage difference), KCl concentration of 100 mM and hBN channels of length L = 16 ± 0.1 µm.

### Extended Data Fig. 3 Control sample test.

a–c, Streaming current measured in a control sample without any channels as a function of the pressure. We varied the applied voltage from −100 to 100 mV (colour coded from blue to red). d–f, Same measurements as for a–c (coloured symbols) but compared with the streaming current measured with 200 graphite channels (black symbols). The streaming current is around 4 orders of magnitude larger, which confirms that channels remain mechanically stable and are not delaminated under pressure.

### Extended Data Fig. 4 Gated pressure-driven current and material dependency.

Streaming current per channel plotted as a function of ∆P/L for a KCl concentration varying from 1 mM to 300 mM and with ∆V ranging between −100 mV and 100 mV (colour coded from blue to red with increasing voltage difference). a–d, The channel length L for graphite is 5.7 ± 0.1 µm. e–h, For hBN, L = 16 ± 0.1 µm.

### Extended Data Fig. 5 Concentration dependence of the fit parameters of the gate-controlled mobility.

We report the fitting parameters of the voltage-gated streaming current. a, b, The quadratic dependence of the gated streaming current observed in graphite channels (Fig. 3d, main text) and described by equation (1): a, Vmin plotted as a function of the concentration; b, α as a function of the concentration. c, We report the fitting parameter β as a function of the concentration for hBN slits: β describes the linear dependence of the streaming current observed for hBN channels (Fig. 3e, main text) as given by equation (2). The dashed lines in b and c are linear fits.

### Extended Data Fig. 6 Geometry and effect of the asymmetry of the system.

A slit of uniform height $${h}_{0}=7$$ Å and length $$L=5$$ µm connects two asymmetric, divergent reservoirs of variable height $$h\left(x\right)$$. The asymmetry in the rate of divergence of the reservoir heights qualitatively mimics the asymmetry of the experimental geometry. A voltage $$\varphi ={\rm{\Delta }}V$$ and pressure $$P={\rm{\Delta }}P$$ are applied in the left reservoir (at $$x=-\infty$$); the voltage and pressure are held fixed at $$\varphi =0,P=0$$ in the right reservoir $$\left(x=+\infty \right)$$. The density in both reservoirs is held fixed at $$\rho ={\rho }_{{\rm{res}}}$$.

### Extended Data Fig. 7 Prediction of the streaming current from extended Poisson–Nernst–Planck modelling.

a, Mobility without applied voltage as a function of KCl concentration in linear–logarithmic coordinates for low water–wall friction and α+ > α. b, Streaming current per channel Istr for 300 mM as a function of the pressure gradient ∆P/L for ΔV varying from −75 mV (blue data) to +75 mV (red data). For each voltage, the dashed line corresponds to the linear fit of the data made to extract the mobility. c, Streaming mobility µ normalized by the K+ electrophoretic mobility μK+ and plotted as a function of the applied voltage for KCl concentration varying from 100 mM (blue data) to 1 M (red data). df, Same as in ac but with high water–wall friction and α+ = α. Parameters: ac, λ0/h0 = 1011 kg m−3 s−1, α+ = 1, α = 0.7; df, λ0/h0 = 1013 kg m−3 s−1, α+ = 0.01, α = 0.01. Dashed lines in a and d are guides to the eye corresponding to a constant value of µ and a linear variation with concentration, respectively.

### Extended Data Fig. 8 Total ionic concentration profiles from extended Poisson-Nernst-Planck modelling.

ad, Total ionic concentration profiles as a function of the normalized position x/L along the channel without (a, b) and with (c, d) applied pressure for c = 300 mM. The dashed vertical lines segregate the channel interior, x/L (−0.5, 0.5), from the left (x/L < −0.5) and right (x/L > 0.5) reservoirs. The curves are coloured according to the applied voltage from −50 mV (blue) to 50 mV (orange). a, The high-friction (hBN-like) configuration with ∆P/L = 0. b, The low-friction (graphite-like) behaviour with ∆P/L = 0. c, The high-friction (hBN-like) configuration with ∆P/L = 30 mbar µm−1. d, The low-friction (graphite-like) behaviour with ∆P/L = 30 mbar µm−1.

### Extended Data Fig. 9 Effect of the asymmetry of the system.

a, b, Plots show µ(∆V) versus ∆V as a function of asymmetry. a, Low-friction (graphite-like) behaviour. In this plot we take c = 100 mM, $${\alpha }_{+}=1$$, $${\alpha }_{-}=0.7$$, $${\mu }_{+}={\mu }_{+}^{{\rm{bulk}}}$$, $${\mu }_{-}=0.5{\mu }_{-}^{{\rm{bulk}}}$$ and $${\lambda }_{0}/{h}_{0}=1{0}^{11}$$ kg m−3 s−1, as in the main text, while varying the geometric parameters Γl and Γr, as indicated in the legend. b, High-friction (hBN-like) behaviour, c = 100 mM, $${\alpha }_{+}=0.01$$, $${\alpha }_{-}=0.01$$, $${\mu }_{+}={\mu }_{+}^{{\rm{bulk}}}$$, $${\mu }_{-}=0.5{\mu }_{-}^{{\rm{bulk}}}$$ and $${\lambda }_{0}/{h}_{0}=1{0}^{13}$$ kg m−3 s−1, as in the main text, while varying the geometric parameters Γl and Γr, as indicated in a.

### Extended Data Fig. 10 Influence of the friction parameters on the model predictions.

a–c, Plots show µ(∆V) versus ∆V for different concentrations (c = 100 mM, 300 mM and 1,000 mM) and frictional parameters. a, Low-friction (graphite-like) behaviour. In this plot, we take $${\alpha }_{+}=1$$, $${\alpha }_{-}=0.7$$, $${\mu }_{+}={\mu }_{+}^{{\rm{bulk}}}$$, $${\mu }_{-}=0.5{\mu }_{-}^{{\rm{bulk}}}$$ and $${\lambda }_{0}/{h}_{0}=1{0}^{11}$$ kg m−3 s−1. b, Intermediate-friction behaviour, $${\alpha }_{+}=0.02$$, $${\alpha }_{-}=0.01$$, $${\mu }_{+}={\mu }_{+}^{{\rm{bulk}}}$$, $${\mu }_{-}=0.5{\mu }_{-}^{{\rm{bulk}}}$$ and $${\lambda }_{0}/{h}_{0}=5.1{0}^{12}$$ kg m−3 s−1. c, High-friction (hBN-like) behaviour, $${\alpha }_{+}=0.01$$, $${\alpha }_{-}=0.01$$, $${\mu }_{+}={\mu }_{+}^{{\rm{bulk}}}$$, $${\mu }_{-}=0.5{\mu }_{-}^{{\rm{bulk}}}$$ and $${\lambda }_{0}/{h}_{0}=1{0}^{13}$$ kg m−3 s−1. d–f, Pressure-induced variation of the normalized electric potential ∆ϕ = ϕ(∆V, ∆P = 30 mbar µm−1) − ϕ(∆V, ∆P = 0) plotted as a function of the normalized channel coordinate x/L axis for ∆V = −50 mV, 0 mV and 50 mV. The dashed vertical lines segregate the channel interior, x/L (−0.5, 0.5), from the left (x/L < −0.5) and right (x/L > 0.5) reservoirs. The curves are coloured according to the applied voltage from −50 mV (blue) to 50 mV (orange). Panels df correspond to the parameters of ac, respectively. g, Table of the friction parameters corresponding to the data shown in ac. The table also shows the decomposition of λw(c) into its three components for the concentrations considered here.

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Mouterde, T., Keerthi, A., Poggioli, A.R. et al. Molecular streaming and its voltage control in ångström-scale channels. Nature 567, 87–90 (2019). https://doi.org/10.1038/s41586-019-0961-5

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