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Cascaded compression of size distribution of nanopores in monolayer graphene


Monolayer graphene with nanometre-scale pores, atomically thin thickness and remarkable mechanical properties provides wide-ranging opportunities for applications in ion and molecular separations1, energy storage2 and electronics3. Because the performance of these applications relies heavily on the size of the nanopores, it is desirable to design and engineer with precision a suitable nanopore size with narrow size distributions. However, conventional top-down processes often yield log-normal distributions with long tails, particularly at the sub-nanometre scale4. Moreover, the size distribution and density of the nanopores are often intrinsically intercorrelated, leading to a trade-off between the two that substantially limits their applications5,6,7,8,9. Here we report a cascaded compression approach to narrowing the size distribution of nanopores with left skewness and ultrasmall tail deviation, while keeping the density of nanopores increasing at each compression cycle. The formation of nanopores is split into many small steps, in each of which the size distribution of all the existing nanopores is compressed by a combination of shrinkage and expansion and, at the same time as expansion, a new batch of nanopores is created, leading to increased nanopore density by each cycle. As a result, high-density nanopores in monolayer graphene with a left-skewed, short-tail size distribution are obtained that show ultrafast and ångström-size-tunable selective transport of ions and molecules, breaking the limitation of the conventional log-normal size distribution9,10. This method allows for independent control of several metrics of the generated nanopores, including the density, mean diameter, standard deviation and skewness of the size distribution, which will lead to the next leap in nanotechnology.

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Fig. 1: Illustration of the compression cycles for nanopore creation in monolayer graphene.
Fig. 2: Characterization of nanopores and verification of the cascaded compression model.
Fig. 3: Solvent permeance and ultrafast nanofiltration by decoupling nanopore size distribution and density.
Fig. 4: Highly tunable and selective solute permeation beyond log-normal limitation.

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Data availability

All data needed to evaluate the conclusions herein are present in the article.

Code availability

The Python codes used in nanopore evolution simulation and Raman mapping are available at (ref. 55).


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J.W., J.-H.P. and J.K. acknowledge the financial support from the Air Force Office of Scientific Research (AFOSR) Multi-University Research Initiative FA9550-22-1-0166 and the US Army Research Office (ARO) under grant no. W911NF-18-1-0431. C.C. is the recipient and acknowledges the financial support from the Australian Research Council Australian Future Fellowship Award (FT210100364) funded by the Australian Government. A.-Y.L., J.-H.P. and J.K. acknowledge the US ARO through the Institute for Soldier Nanotechnologies at MIT, under cooperative agreement no. W911NF-18-2-0048. X.Z., T.Z. and J.K. acknowledge the support by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) under award DE-SC0020042, which provided STEM and other characterizations for the samples. Y.H., G.G. and B.S. acknowledge the support from Welch Foundation (C-2065-20210327). CAFM imaging was performed in part in the MIT.nano Characterization Facilities on the Cypher VRS enabled by the DURIP award (N000142012203). STEM imaging was conducted as part of a user project at the Center for Nanophase Materials Sciences (CNMS), which is a US DOE, Office of Science User Facility at Oak Ridge National Laboratory. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements of the US Government.

Author information

Authors and Affiliations



J.K. supervised the project. J.W. and J.K. conceived the experiments. J.W. conducted the theoretical analysis and simulations. J.W. established the LPCVD system with electrodes and prepared the nanoporous graphene film. C.C. and R.K. conceived and designed the nanofiltration and diffusion experiments and contributed to the design of the project. C.C. performed the nanofiltration and diffusion experiments. J.W. performed the Raman and CAFM characterization. X.Z., J.C.I., J.W., G.G., B.S. and Y.H. performed the STEM and HRTEM characterization. A.-Y.L. and J.W. wrote the Raman fitting program. B.G.S. and S.J.J. performed the STM characterization. J.-H.P., T.Z., H.W., X.J., L.J., L.-J.L. and R.K. participated in the characterization and discussion. J.W., C.C. and J.K. wrote the manuscript; all authors read and revised the manuscript.

Corresponding authors

Correspondence to Jiangtao Wang, Chi Cheng or Jing Kong.

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Competing interests

J.W., C.C. and J.K. are co-inventors on a patent application (provisional filling number 63/502,064) related to the research presented in this paper.

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Extended data figures and tables

Extended Data Fig. 1 In situ deposition of Cu particles on the graphite electrode for in situ sputter.

a, Schematic of electrode reactions and enhanced copper evaporation in the CVD chamber. The surface electrochemical reaction changes Cu to Cu(OH)2, which is very easy to evaporate. Then Cu(OH)2 particles would be reduced to Cu on the graphite electrode in the atmosphere of H2. b, A scanning electron microscopy image of the graphite electrode surface after Cu particle deposition. c, The corresponding energy-dispersive spectroscopy mapping of Cu of the graphite electrode in b. The red dots indicate the deposited Cu particles. Scale bars, 5 μm. df, Illustration of whether sputter-etching happens or not with different types of electrode: graphite (d), graphite deposited with Cu particles (e) and flat Cu covered by graphene film (f). gi, The corresponding Raman spectrum of the graphene film after applying electric field with the anode materials shown in df. It proves that the in situ sputter results in the presence of exposed Cu particles at the anode. With more area of exposed Cu, the sputter effect would be stronger.

Extended Data Fig. 2 Optical images of the graphene film after in situ sputtering.

a, Optical image of the as-prepared nanoporous graphene on copper substrate. b, Optical image of the sample after being immersed in 80 °C water for 5 min. The contrast shows the wet-oxidation of different copper domains by water54, suggesting the presence of nanopores in the graphene film. c, Optical image of the sample after being oxidized in 150 °C ambient air for 2 min. The colour change indicates Cu oxidation owing to nanopores in graphene film. The arrow indicates a bilayer island that is not oxygen-permeable. Scale bars, 100 μm. d, Optical image of the whole sample after oxidation, in which only the indicated region was overlapped with the graphite electrode and sputtered.

Extended Data Fig. 3 Nanopore creation through in situ e-beam irradiation and STEM images of defects created by cascaded compression but with extended shrinking time in graphene film.

a, Random nucleation and linear expansion of two nanopores under in situ e-beam irradiation. The randomness of the size distribution comes from the fact that the nanopores are created at different starting times (that is, nucleation time), therefore, by the end of expansion time, the nanopores have different sizes. bg, The STEM images of the in situ creation and expansion of two nanopores at various times. Scale bar, 1 nm. Voltage: 100 kV, current: 38 pA, temperature: 700 °C. hk, STEM images of defects in graphene obtained by using extended regrowth time. h, A large topological defect with threefold symmetry. i, A divacancy defect. j, A linear 5–7 defect. k, A single-vacancy defect. The dotted circles indicate the area of the structural disorder. Scale bar, 0.5 nm.

Extended Data Fig. 4 Atomic characterization of nanopore density and size.

a, An STM image of 80 × 80 nm2. Corrugation with terrace width of about 20 nm was induced by the Cu substrate. Bright and dark dot features were observed in a large density compared with the conventional CVD graphene on Cu. b, Close-up STM image with atomic resolution for the area of the rectangle in a. Arrows indicated nanopore structures. c, Close-up STM image with atomic resolution for the area of the rectangle in b. The size of the nanopore is about 1 nm. The honeycomb structure of graphene is indicated with an inset of honeycomb cells. Nanopore density of 3.875 × 1012 cm−2 was counted in STM images, based on the characterization of the atomic resolution. dg, CAFM images of pristine graphene and nanoporous graphene samples with different nanopore diameters. The acquirements of the nanopore area are indicated by the white outlines. The diameter of the nanopore is calculated by using \(d=2\sqrt{A/\pi }\), in which A is the area of the nanopore. h,i, The experimental diameter distribution of samples #822 and #835, respectively. j,k, The simulated diameter distribution of samples #822 and #835, respectively.

Extended Data Fig. 5 Nanofiltration experimental setup and results analysis.

a, Methanol solutions of Rose bengal was used as feed to measure membrane rejection. The feed volume was 35 ml and had a solute concentration of 0.01 mM. To maintain an approximately constant solute concentration in the feed during each measurement, the total permeate that flowed out of the pressure cell was controlled to be less than 3 ml. Vigorous stirring was maintained on the feed side to minimize concentration-polarization effects. The stirrer was suspended above the membrane using a built-in mechanism inside the HP4750 High-pressure Stirred Cell (Sterlitech, Inc.). b, The benchmark of relative tail deviation and density. c, Rejection versus permeance plot of various membrane materials for organic solvent nanofiltration of Rose bengal in methanol. The innate permeance of nanoporous graphene is calculated using the measured membrane permeance normalized by the support porosity and subtracting the support flow resistance (see Extended Data Table 1).

Extended Data Fig. 6 The uniformity comparison of the size and density of the nanoporous graphene grown on copper foil.

a,b, The mapping of D to G ratio and 2D to G ratio of sample #809 after transfer to SiO2/Si, respectively. The dashed blue line in a shows a copper grain boundary before the transfer. c,d, The mapping of D to G ratio and 2D to G ratio of sample #817, respectively. The preparation condition of sample #817 involves adding an extra 1-min shrinkage after the same conditions of sample #809. As a result, the D to G ratio of #sample 817 reflects the density of the nanopores of sample #809, and the D to G ratio difference between samples #809 and #817 indicates the variation of the nanopore size of sample #809.

Extended Data Fig. 7 Diametric evolution towards the designed asymptotic value for cascaded compression based on our modelling.

a,b, Nanopore density versus diameter distribution after different cycles of compression. The starting distributions (0 cycles) (\({d}_{{\rm{peak}}}=0.2\,{\rm{nm}},\,\widetilde{\sigma }=0.4\)) in our experiments are also log-normal distributions and are the same for a and b. After several compression cycles, such a log-normal distribution becomes a left-skewed distribution with total density increasing. The designed asymptotic values (d) are 0.8 nm and 0.6 nm, respectively, for a and b by tuning the compression factor from 0.750 to 0.667. c,d, The corresponding plots of nanopore density versus diameter of a and b in logarithmic coordinates. The high nanopore density with a slope of 0.1 or 0.05 nm dec−1 indicates high selectivity and permeance in these nanoporous graphene membranes.

Extended Data Fig. 8 Simulation of the cases beyond the ideal linear model.

a, The evolution of density, mean diameter and RSD based on the conditions of sample #825 and extra nine times higher nucleation rate. b,c, The corresponding final spatial (b) and diametric (c) distributions of a. d, The evolution of density, mean diameter and RSD based on the conditions of sample #825 and extra 90 cycles of compression. e,f, The corresponding final spatial (e) and diametric (f) distributions of d. Scale bars in b and e, 5 nm. With much higher density, the fusion is notable, but the diameter distribution is still very narrow. g, The evolution of density, mean diameter and RSD based on a nonlinear diametric transformation as shown in equations (29) and (30). h,i, The corresponding final spatial (h) and diametric (i) distributions of g. Red and black dots in b, e and h indicate the created nanopores and 5–7 defects (or single/double vacancies), respectively. Scale bars, 10 nm.

Extended Data Fig. 9 Numeric calculation of the nanopore shrinkage.

a,b, The linear shrinkage dynamics of the nanopores when the adsorption rate of carbon source on the copper surface is much faster than the rate of carbon atoms attaching to the graphene lattice. b, Corresponding logarithmic scale of a. c,d, The exponential compression of the nanopores when the adsorption rate of carbon source on the copper surface is much slower. d, Corresponding logarithmic scale of c.

Extended Data Fig. 10 Raman mappings and scattering plots of the nanoporous graphene film.

a, Processing stages of the samples with different shrinkage time. The fitted shrinkage rate is 0.078 s−1. bd, The Raman mapping and scattering plot of the sample after ten cycles of cascaded compression. eg, The Raman mapping and scattering plot of the sample with extra 30 s shrinkage. The samples are transferred onto SiO2/Si substrate.

Extended Data Table 1 Benchmark of membrane performance for organic solvent nanofiltration of Rose bengal in methanol

Supplementary information

Supplementary Information

This file includes Supplementary Notes 1–4, Supplementary Figs. 1–4 and Supplementary Tables 1 and 2.

Supplementary Video 1

The simulated time-resolved evolutions of the density, mean diameter and RSD of sample no. 825.

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Wang, J., Cheng, C., Zheng, X. et al. Cascaded compression of size distribution of nanopores in monolayer graphene. Nature 623, 956–963 (2023).

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