Helmholtz free energy, liquid interfaces

The most important property of a liquid-gas interface is its surface energy. Surface tension arises at the boundary because of the grossly unequal attractive forces of the liquid subphase for molecules at its surface relative to their attraction by the molecules of the gas phase. These forces tend to pull the surface molecules into the interior of the liquid phase and, as a consequence, cause liquids to minimize their surface area. If equilibrium thermodynamics apply, the surface tension 7 is the partial derivative of the Helmholtz free energy of the system with respect to the area of the interface—when all other conditions are held constant. For a phase surface, the corresponding relation of 7 to Gibbs free energy G and surface area A is shown in eq. [ 1 ]. 

Tarazona and Navascues have proposed a perturbation theory based upon the division of the pair potential given in Eq. (3.5.1). In addition, they make a further division of the reference potential into attractive and repulsive contributions in the manner of the WCA theory. The resulting perturbation theory for the interfacial properties of the reference system is constructed through adaptation of a method developed by Toxvaerd in his extension of the BH perturbation theory to the vapor-liquid interface. The Tarazona-Navascues theory generates results for the Helmholtz free energy and surface tension in addition to the density profile. Chacon et al. have shown how the perturbation theories based upon Eq. (3.5.1) may be developed by a series of approximations within the context of a general density-functional treatment. 

In order to establish the criteria of equilibrium when the interface between the phases is curved, let us consider the system of Fig. 2.9 at constant volume and constant temperature. The pressure inside the gas bubble is P and the surrounding liquid pressure is P". The volumes of the gas bubble and the surrounding liquid phase are V and V", respectively. The expressions for the differential Helmholtz free energy of the bubble and surrounding liquid for a single-component system are... 

In this section we shall consider some elementary thermodynamics relations involving interfaces [4]. Since molecules at an interface are in a different environment from molecules in the bulk, their energies and entropies are different. Molecules at a liquid-air interface, for example, have larger Helmholtz free energy than those in the Bulk. At constant V and T, since every system minimizes its Helmholtz free energy, the interfacial area shrinks to its minimum possible value, thus increasing the pressure in the liquid (Fig. 5.4). 

This point will be considered more closely below. Here it is asked first how O Eq. 6.20b relates to the balance of the excess energy for the interface and what the consequences for the wetting experiment are. These questions are important since the specific interactions in an interphase add their interaction energies to the balance, not the corresponding forces. For a liquid wetting a solid, the specific excess Helmholtz free energy of the SL-interface is given by (cf. O Sect. 4.2)... 

Other approaches for computing the surface tension start from the statistical mechanical expression for the Helmholtz free energy or for the pressure. The Kirkwood-Buff formula for the surface tension of a liquid/vapor interface of an atomic liquid described by the pair potential approximation is ... 

Before approaching the problem of dynamics of contact line, we shall briefly review the equilibrium properties of gas-liquid interfaces and their dependence on the proximity to solid surfaces. We shall consider the simplest one-component system a liquid in equilibrium with its vapor. Thermodynamic equilibrium in a two-phase system implies equilibrium of the interphase boundary, which tends to minimize its area. The thermodynamic quantity that expresses additional energy carried by the interface is surface tension, defined as the derivative of the Helmholtz or Gibbs free energy with respect to interfacial area E ... 

For simple case of the water-air interface (usually referred to as surface ), increasing the amphiphile concentration in the liquid phase results in an increasing number of surfactant molecules adsorbed at interface, and the water surface tension decreases accordingly. The surface tension y is an intensive thermodynamic function of state defined as the rate of increase with the area (A) of the interface of the Helmholtz (at constant volume) free energy (F) at constant temperature and composition ... 

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