Phase rule for capillary systems

Abstract

The phase rule is a classical topic in thermodynamics. For bulk phase systems, counting the number of degrees of freedom can be readily done by applying the well-known Gibbs phase rule. However, determining the number of degrees of freedom is not straightforward for capillary systems. This is essentially due to complications with the mechanical equilibrium constraints associated with curved liquid-fluid interfaces and curved three-phase contact lines. The only general rule to follow is that the number of degrees of freedom is obtained by subtracting the number of constraints from the number of variables. In this paper, first we reviewed the variables required to characterize the equilibrium states of simple single phase systems, including both moderately-curved and highly-curved surface phases and line phases. Then we discussed the mechanical equilibrium constraints and derived the phase rules for both moderately-curved and highly-curved capillary systems. Generally, the presence of curved interfaces in a capillary system results in fewer mechanical equilibrium constraints and hence more degrees of freedom, in comparison with composite bulk phase systems. There is not only a difference between the phase rule for bulk systems and the phase rule for capillary systems, but also a difference between the phase rule for moderately-curved capillary systems and the phase rule for highly-curved capillary systems. Some experimental confirmation of the number of degrees of freedom predicted by the phase rule for moderately-curved capillary systems was discussed. Effects of external interactions on the number of degrees of freedom of capillary systems were also considered. In addition, we studied the number of degrees of freedom for two special cases: solid-liquid-vapour capillary systems in the presence of a thin liquid film on the solid surface, and capillary systems with an elastic liquid-fluid interface. Overall, we concluded that the difference in the number of degrees of freedom stems from the difference in the thermodynamic descriptions of capillary systems, i.e. the thermodynamic fundamental equations. The fundamental equations for moderately-curved interfaces, highly-curved interfaces, interfaces with and without elasticity and thin liquid films are all different. The different fundamental equations require different numbers of variables and result in different numbers of equilibrium constraints, causing different numbers of degrees of freedom.