Equilibrium of multi-phase systems in gravitational fields

Area of Science:

• Thermodynamics
• Physical Chemistry
• Surface Science

Background:

• Previous work by Ward and Sasges (1998) established four equilibrium conditions for solid-liquid-vapor systems using entropy maximization, assuming constant surface tension with elevation.
• Gibbs (1876) previously derived the Young equation separately and obtained a more general Laplace equation that included elevation-dependent surface tension, a detail often overlooked.

Purpose of the Study:

• To derive the equilibrium conditions for isolated solid-liquid-vapor systems in a gravitational field, specifically allowing for liquid-vapor surface tension to vary with elevation.
• To unify the derivation of these conditions within a single framework, incorporating Gibbs's generalized Laplace equation.

Main Methods:

• Employed an entropy maximization approach, similar to Ward and Sasges (1998).
• Derived equilibrium conditions for systems where liquid-vapor surface tension is a function of elevation.
• Applied the methodology to two distinct geometries: a sessile drop and a conical capillary tube.

Main Results:

• Successfully derived the four standard equilibrium conditions: thermal equilibrium, the Young equation, and modified Laplace and chemical potential conditions.
• Incorporated Gibbs's generalized Laplace equation, which accounts for the variation of surface tension with elevation, into the unified framework.
• Obtained consistent equilibrium conditions for both the sessile drop and conical capillary tube geometries.

Conclusions:

• The inclusion of elevation-dependent surface tension, via Gibbs's generalized Laplace equation, refines the understanding of thermodynamic equilibrium in solid-liquid-vapor systems.
• The unified approach provides a more comprehensive set of equilibrium conditions applicable to systems with varying surface tension.
• The findings are validated across different system geometries, suggesting broad applicability.