Capillary condensation of water is ubiquitous in nature and technology. It routinely occurs in granular and porous media, can strongly alter such properties as adhesion, lubrication, friction and corrosion, and is important in many processes used by microelectronics, pharmaceutical, food and other industries1,2,3,4. The century-old Kelvin equation5 is frequently used to describe condensation phenomena and has been shown to hold well for liquid menisci with diameters as small as several nanometres1,2,3,4,6,7,8,9,10,11,12,13,14. For even smaller capillaries that are involved in condensation under ambient humidity and so of particular practical interest, the Kelvin equation is expected to break down because the required confinement becomes comparable to the size of water molecules1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22. Here we use van der Waals assembly of two-dimensional crystals to create atomic-scale capillaries and study condensation within them. Our smallest capillaries are less than four ångströms in height and can accommodate just a monolayer of water. Surprisingly, even at this scale, we find that the macroscopic Kelvin equation using the characteristics of bulk water describes the condensation transition accurately in strongly hydrophilic (mica) capillaries and remains qualitatively valid for weakly hydrophilic (graphite) ones. We show that this agreement is fortuitous and can be attributed to elastic deformation of capillary walls23,24,25, which suppresses the giant oscillatory behaviour expected from the commensurability between the atomic-scale capillaries and water molecules20,21. Our work provides a basis for an improved understanding of capillary effects at the smallest scale possible, which is important in many realistic situations.
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All the mentioned data to support this study and its conclusions are available upon request from Q.Y.
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This work was funded by Lloyd’s Register Foundation, the European Research Council, Graphene Flagship and the Royal Society. Q.Y. acknowledges support from the Leverhulme Early Career Fellowship, and F.C.W. from the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB22040402) and the CAS Youth Innovation Promotion Association.
The authors declare no competing interests.
Peer review information Nature thanks Patrick Huber and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
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Extended data figures and tables
a, Simplified flow chart for our fabrication procedures. (1) Graphene spacers and the bottom crystal of either mica or graphite (shown in yellow) were assembled on top of an oxidized Si wafer. (2) A suspended SiN membrane with a rectangular hole was prepared separately. (3) The two-layer assembly was transferred from the Si oxide wafer onto the SiN membrane. The opening was extended through the assembly by RIE. (4) The top crystal of mica or graphite was placed on top of graphene spacers. b, AFM micrograph of graphene spacers with N = 5. The colour scale is given by the height profile (blue curve). c, Optical image of a final mica device used in our experiments. The bottom mica crystal shows up in purple on top of the square SiN membrane. Graphene spacers (N = 3) and the top mica layer are outlined in blue and yellow, respectively. d, Cross-sectional scanning transmission electron microscopy image of a graphite channel with N = 2. The blue ticks mark the channel’s edges.
a, Our AFM set-up. Humidified nitrogen gas flows through the bottom chamber made from an aluminium alloy. A silicon wafer of 15 × 15 mm2 in size is seen to cover the chamber, flush with its top surface. The white rubber gasket was lowered during AFM measurements to seal the space above the Si wafer. Inset, cross-sectional schematic showing how capillary devices were mounted during AFM measurements. b, Schematic of a water plug inside our capillaries. For brevity, the layered structure of water is ignored in this sketch. When the top chamber is at low RH, the meniscus slightly retracts inside the capillary to create a vapour pressure gradient. The RH gradient stabilizes two menisci with the same curvature at both exit and entrance. The distance from the exit meniscus to the opening is expected to be short because, in our atomically flat capillaries, water moves much faster as liquid than vapour26.
a–c, Images of a graphite capillary with N = 3 at RH of 55%, 70% and 95% (a, b and c, respectively). The upper part of each image shows sagging of the top graphite crystal (H ≈ 80 nm) into the 2D channel. The lower part shows the area immediately outside the channel, which is not covered by the top graphite. The black dotted lines mark a border between the two regions (edge of the top crystal). The colour scales for the lower and upper parts of the AFM images are given by the green and black curves, respectively. The profiles are averaged over ∼100 nm along the y direction, and the curves in the upper parts of all the panels are provided on the same scale given by the black arrows in panel a. A small number of horizontal scanning lines (x direction) around the black-dot dividing lines were removed for clarity because they contained numerous instabilities caused by the AFM tip moving along the edge of the top crystal and jumping up and down. Such instabilities are typical for AFM scanning close to edges.
a, Sagging profiles for a graphite capillary (N = 4) with increasing and decreasing RH between 75% and 80%. Black curve, initial dry-state profile. Red curve, RH was increased to 80%. Then, RH was returned to 75% and maintained at this humidity. AFM profiles were taken after 4 h, 9 h and 16 h (colour coded). b, The N = 6 graphite capillary was brought from the dry state (black curve) into the state filled with water and kept for an hour at 95% RH (red). The humidity was then decreased to ∼30%, well below the condensation transition observed at 62.5 ± 2.5% for this device. The colour-coded curves show the time evolution towards the original dry state. Note that the sagging depths δ for such hysteretic loops were highly reproducible but details of sagging profiles could differ in different RH cycles. For example, the top crystal’s adhesion to the right wall was different in the original and final dry states, as seen in a (compare black and purple curves). This hysteresis is attributed to irreproducible vdW attachments of top crystals to channel sidewalls.
a, b, Sagging profiles for two N = 5 graphite capillaries with different δ0. RH was increased in 5% steps (colour coded). The water condensation transition occurred between 80% and 85% RH in a and between 70% and 75% in b. The difference in RHC for the same N is attributed to different h in the two cases.
a, Schematic of top crystal sagging. b, Typical behaviour observed for the sagging depth δ as a function of RH, after the condensation transition occurred at RH < 60%. Symbols: Measurements for two different mica capillaries with N = 8. The solid curves are best fits using equations (3) and (4) (colour-coded). The grey symbol with error bars indicates the experimental accuracy.
a, Its density profiles at different distances h between two rigid capillary walls with the contact angle θ ≈ 11°. b, Same calculations but for contact angle 85°. The orange dashed lines mark positions of the surfaces that defined the 2D channels. Water exhibits a pronounced layered structure near each surface, and the structures start to overlap for h < 15 Å. Top insets, cross-sectional profiles for water droplets placed on the surfaces with the given θ.
Calculated Δγ(h) for several characteristic θ. The arrows indicate the number of molecular layers of water that fit inside the 2D channels.
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Yang, Q., Sun, P.Z., Fumagalli, L. et al. Capillary condensation under atomic-scale confinement. Nature 588, 250–253 (2020). https://doi.org/10.1038/s41586-020-2978-1