Direct Evidence for Curvature-Dependent Surface Tension in Capillary Condensation: Kelvin Equation at Molecular Scale

Abstract

Capillary condensation is the first-order vapor-to-liquid phase transition taking place in confined geometries. Such heterogeneous nucleation has been well described by thermodynamic laws such as the Kelvin equation, but the equation's applicability at the nanoscale is still unresolved. Here, we show that the Kelvin equation is valid down to approximately 0.5 nm radius of curvature when the curvature dependence of surface tension is taken into account. By the shear-mode atomic force microscopy, we have measured directly and accurately the critical tip-surface distance (d c) at which the water meniscus is capillary condensed in ambient condition; e.g., d c ≈ 1.2 nm at 10% relative humidity. In particular, we can determine the Tolman length, the unique characteristic of the curvature-dependent surface tension, as the single fitting parameter (δ ¼ 0.21 AE 0.05 nm). Our results that unify the validity of the Kelvin equation at molecular scale and the characterization of the curvature effect of surface tension may provide a better understanding of general nucleation phenomena in nature, including the role of nanometric aerosols in cloud formation.

Tolman lengths and rigidity constants from free-energy functionals—General expressions and comparison of theories

“…[2][3][4][5] The curvature dependence of the surface tension also has implications for other important examples such as the properties of biomembranes, 6 and wetting at the nanoscale. 7 The first quantitative description was proposed by Tolman. 8 By introducing the distance between the equimolar radius R e and the radius of the surface of tension R s , referred to as δ T (R s ), he proposed the expression…”

Effect of line tension on axisymmetric nanoscale capillary bridges at the liquid-vapor equilibrium

“…Since the line-tension force τ 2 /f (z 2 ) acting at the three-phase contact line always stays within the circular plane of the three-phase contact line (Fig. 2), the line tension does not contribute directly to the normal capillary force [16,23,24] parallel to the rotational axis (z−axis). Therefore, the normal capillary force can probe only the surface-tension force at the three phase contact line and the capillary pressure due to the pressure difference between the liquid and the vapor phase.…”

“…Recently, there appear the normal capillary force measurements of nanoscale bridges [16,24], which are interpreted by the size-or curvature-dependent liquid-vapor surface tension of nanoscale liquids using Tolman's formula [55,56]. As shown in Fig.…”

“…After the invention of various force spectrosopy techniques such as the surface force apparatus (SFA) and the atomic force microscopes (AFM), the problem of capillary bridge, in particular, the normal capillary force by the bridge have been attracted renewed interest recently [23,25,26]. Since the nanoscale liquid bridges can easily form even in the low relative humidity, it is * Permanent address Tokyo City University, Setagaya-ku, Tokyo 158-8557, Japan; iwamatm@tcu.ac.jp (corresponding author) also a convenient tool to study the surface properties such as the size-dependence of surface tension of the nanoscale liquid bridges [16,24]. It can also serve as a testing ground of various statistical mechanical theory of nanoscale liquids [27][28][29][30][31].…”

“…However, the classical capillary theory using the macroscopic concepts such as the surface tension [17,33] and the line tension [33,34] is still useful [29,35,36]. In particular, the analytical or semi-analytical results obtained from the classical capillary theory have been useful to understand the physics of capillary phenomena and to analyze experimental results directly [16,24].…”