Capillary condensation between parallel walls of unequal length

TL;DR

The paper establishes a macroscopic framework for capillary condensation in slits with unequal wall lengths by introducing the edge contact angle and four distinct condensation states. It derives Kelvin-like onset relations for each state and maps comprehensive phase diagrams that highlight a geometric separatrix and a wedge-filling threshold at $\theta=\pi/4$, showing how confinement geometry governs condensation. The approach connects capillarity, wetting, and pinning phenomena, offering predictions for when and where condensation occurs in realistic finite slits and how states disappear at wedge-filling. These insights have potential implications for microfluidics, porous materials, and nanoscale devices where capillary forces are decisive, and they point toward future microscopic validation and extensions to heterogeneous or non-parallel geometries.

Abstract

We present a macroscopic theory of capillary condensation in slits formed by parallel walls of unequal length. Using the concept of an edge contact angle, we identify four distinct condensation states and derive Kelvin-like relations for their onset. The resulting phase diagrams, expressed in terms of wall geometry and contact angle, reveal two central organizing features: a geometric separatrix that divides distinct condensation regimes, and the wedge-filling threshold at $θ=π/4$, which separates a rich four-state scenario from a simpler two-state one. These results demonstrate how geometry dictates the onset and suppression of condensation in confined systems.

Figures (12)

• Figure 1: Schematic representations of possible condensed states: a) $1^+$ state, b $1^-$ state, c) $0$ state, and d) $2$ state.
• Figure 2: Schematic of a $1^+$ state for completely wet walls. The meniscus of radius $R$ is pinned at the edges of the top wall, while meeting the bottom wall tangentially. The edge contact angle at the top wall is $\theta_e$, and $\delta$ denotes the horizontal extension at which the meniscus meets the bottom wall.
• Figure 3: Schematic of a $1^-$ state for completely wet walls. The meniscus of radius $R$ is pinned at the edges of the bottom wall, while meeting the top wall tangentially. The edge contact angle at the bottom wall is $\theta_e$, and $\delta$ denotes the vertical extension at which the meniscus meets the top wall.
• Figure 4: Schematic of a $2$ state for completely wet walls. The meniscus of radius $R$ is simultaneously pinned at both walls, with edge contact angles $\theta_e^+$ (top) and $\theta_e^-$ (bottom).
• Figure 5: Schematic of a $0$ state for completely wet walls. The menisci of radius $R$ meet both walls tangentially, without being pinned at edges. The extensions are $\delta^+=R-L$ (top) and $\delta^-=R$ (bottom).