Capillary condensation follows classical law even at the nanoscale
When water vapour spontaneously condenses inside capillaries just 1 nm thick, it behaves according to the 150-year-old Kelvin equation – defying predictions that the theory breaks down at the atomic scale. Indeed, researchers at the University of Manchester have showed that the equation is valid even for capillaries that accommodate only a single layer of water molecules (Nature 588 250).
Condensation inside capillaries is ubiquitous and many physical processes – including friction, stiction, lubrication and corrosion – are affected by it. The Kelvin equation relates the surface tension of water to its temperature and the diameter of its meniscus. It predicts that if the ambient humidity is between 30–50%, then flat capillaries less than 1.5 nm thick will spontaneously fill with water that condenses from the air.
Real world capillaries can be even smaller, but for them it is impossible to define the curvature of a liquid’s meniscus so the Kelvin equation should no longer hold. However, because such tight confinement is difficult to achieve in the laboratory, this had yet to be tested.
Manchester’s Andre Geim, Qian Yang and colleagues created tiny capillaries by sandwiching strips of graphene between atomically flat crystals of mica or graphite using a process called Van der Waals assembly. The graphene strips act as spacers and their thicknesses can be varied, allowing for capillaries of varying heights. Some are just one atom high, allowing only a single layer of water molecules to pass through.
Using atomic force microscopy, the team imaged the capillaries as they filled with water – finding that capillary condensation follows the Kelvin equation even in these tiny structures. “The result came as big surprise,” Yang says. “We expected a complete breakdown of the equation since the properties of water change at this scale, with its structure becoming distinctly discrete and layered.”
At the atomic scale, Yang explains that researchers rewrite the Kelvin equation in terms of how water molecules in both gas and liquid phases interact with solid surfaces. In this form, macroscopic quantities such as the contact angle of water with the capillary wall, the surface tension of water and its meniscus curvature all disappear from the equation. This remains valid if the energy of the water-surface interactions does not notably change.
“In practice, however, the condition breaks at about four to five layers of confined water (which are less than 2 nm thick in total),” she says. “Under stronger confinement still, the water structure strongly changes, and the interaction energies (primarily the liquid water-surface energy) inevitably change.” In this regime, the Kelvin equation should fail – mainly because huge oscillations in the relative humidity at which condensation occurs are expected because of layered structure of water.
However, the Manchester team found that these oscillations are strongly suppressed by the elasticity of the capillary walls. Although these walls adjust their position by less than 0.1 nm in response to the high pressures present during capillary condensation under ambient humidity, this is enough to snugly accommodate only an integer number of water-molecule layers. As a result, she concludes, “the Kelvin equation remains valid down to a monolayer of confined water”.