Capillary Condensation in Nanogaps: Nucleation or Film Coalescence?

Nucleation and film coalescence represent two fundamentally different pathways for capillary condensation. Yet, both have so far been proposed as the processes driving the condensation in nanometric confinements, leading to a long-standing and overlooked ambiguity. Here, we delineate the dichotomy between these mechanisms and test their validity using an experimental method capable of absolute distance measurement during capillary condensation. We show that the molecular content of the capillary meniscus given by the first nucleation theorem is far smaller than what the confinement geometry and the Kelvin equation require. In contrast, the analysis based on film coalescence reproduces the experimental observations and describes the final meniscus formation as a barrierless process, while allowing for an intermediate, first-order-like film-thickening transition prior to the meniscus formation.

💡 Research Summary

The paper tackles a long‑standing ambiguity in nanoconfined capillary condensation: does the liquid bridge form by stochastic nucleation, which requires overcoming a free‑energy barrier, or by deterministic film coalescence, which proceeds without a barrier? To answer this, the authors devised an experimental protocol that can measure the absolute tip‑substrate separation during bridge formation, using a scanning‑tunneling‑microscopy (STM)‑inspired setup. A Pt‑Ir tip approaches a highly doped Si(100) substrate (covered with a 2 nm native SiO₂ layer) in 0.4 Å steps while a small bias (≤ 70 mV) is applied. The onset of a water bridge is identified by a sudden increase of the measured current from the background (~0.3 fA) to tens of fA, a signal that is independent of the exact tip‑substrate distance and thus marks the formation of a conductive water meniscus. A second, distance‑sensitive current jump follows when the tip makes mechanical contact with the oxide film, confirming bridge formation.

To test the nucleation hypothesis, the authors applied the first nucleation theorem, which relates the survival probability of a bridge‑free state to the molecular content of the critical bridge. By performing 100 independent approach‑retract cycles at three relative humidities (ε = 0.80, 0.65, 0.50) they extracted the Poisson‑like survival probability P = exp(−k t) and obtained an upper bound for the number of water molecules in the critical bridge, Nₘ. The values are modest—on the order of 40–70 molecules—far below the number required to span the measured gap according to geometric considerations.

The authors then used the Kelvin equation together with a catenoid geometry to compute the molecular content needed for a stable meniscus at the same separations. The Kelvin‑based estimates are an order of magnitude larger (hundreds to thousands of molecules) and are consistent with the measured gap distances. This discrepancy shows that the nucleation picture cannot account for the observed bridge formation.

The alternative mechanism, film coalescence, is modeled following Chüraev et al. (2011). First, water adsorption isotherms on SiO₂ were fitted with a combined Frenkel‑Halsey‑Hill / Laaksonen model, yielding a disjoining pressure Π(h) = Π₀ exp(−h/λ) with Π₀ ≈ 1468 MPa and decay length λ ≈ 2.25 Å. When the two opposing films approach, an additional Hamaker term A/(6πh³) (A ≈ 4.8 × 10⁻²⁰ J) contributes to the total pressure, producing a free‑energy landscape ΔG(h) that exhibits a local minimum (stable thin film), a global maximum (unstable critical thickness), and an inflection point. As humidity decreases, these extrema shift toward the inflection point; at a critical film thickness (≈ 39 Å for ε ≈ 0.8) the first and second derivatives of ΔG vanish, defining a spinodal limit. Beyond this point, any fluctuation in film thickness grows spontaneously, leading to rapid film thickening and eventual coalescence that fills the gap. The experimentally observed current jumps coincide with the predicted transition from the metastable thin‑film state to the barrier‑free coalescence regime.

Overall, the study provides compelling evidence that capillary condensation in nanogaps proceeds via barrier‑free film coalescence rather than stochastic nucleation. The nucleation theorem underestimates the molecular content required by the Kelvin equation, while the film‑coalescence model quantitatively reproduces the measured critical distances, humidity dependence, and the nature of the transition. These findings call for a reassessment of models that rely on nucleation barriers in nanoscale fluidic systems and suggest that design strategies for nanofluidic devices, humidity‑controlled adhesion, and related technologies should incorporate the physics of thin‑film instability and coalescence.