Surface tension and free energy

THL.6. R. Defay et I. Prigogine, Energie libre et Tension superficielle des surfaces non en equilibre (Free energy and surface tension of nonequilibrium surfaces), Bull. Cl. Sci. Acad. Roy. Belg. 32, 335-350 (1946). 

With regard to surface free energy and surface tension, one can simply try to fit the coefficients of an expansion in powers of of the Helmholtz free energy per atom, F N)/N, for several values of N ... 

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. 

A Surface free energy and surface tension of liquids... 

Table 2. Comparison of surface free energy and surface tensions of modified PDMS surfaces... 

Classical nucleation theory uses macroscopic properties characteristic of bulk phases, like free energies and surface tensions, for the description of small clusters These macroscopic concepts may lack physical significance for typical nucleus sizes of often a few atoms as found from experimental studies of heterogeneous nucleation. This has prompted the development of microscopic models of the kinetics of nucleation in terms of atomic interactions, attachment and detachment frequencies to clusters composed of a few atoms and with different structural configurations, as part of a general nucleation theory based on the steady state nucleation model [6]. The size of the critical nucleus follows straightforwardly in the atomistic description from the logarithmic relation between the steady state nucleation rate and the overpotential. It has been shown that at small supersaturations, the atomistic description corresponds to that of the classical theory of nucleation [7]. 

Before proceeding with the main discussions about wetting, we should review the concepts of surface free-energy and surface tension. 

The classical nucleation theory, based on Gibbs thermodynamics statements, uses the macroscopic properties characteristic of bulk phases, such as free energies and surface tensions, for the description of small clusters. Contradictory results arose in early studies of electrochemical nucleation [9], where the size of a critical mercury nucleus on a platinum substrate amounted to only a few atoms, with properties that could substantially differ... 

It was found that the deviation from graphite thermodynamics depends on the niekel partiele size as illustrated in Figure 5.12 and that the deviation eould be explained by the extra energy required by the higher surface energy, the elastie energy, and defect stracture of the carbon filaments. The eorrelation between free energy and surface tension is expressed by the Kelvin equation. In a simplified model [378] [381], the Kelvin equation beeomes ... 

An alternative expression for the free energy and surface tension in terms of the square of the density gradient can be made by starting wifi) the virial expression for the tension, (4.104), writing p(z2) of (7.1) as p(z,+z,2) and expanding h about p(z,). This route, together with the usual replacement of g(r,2, z,Z2) by g(r,2 p) leads to the following expression for the coefficient, ... 

In Chapter III, surface free energy and surface stress were treated as equivalent, and both were discussed in terms of the energy to form unit additional surface. It is now desirable to consider an independent, more mechanical definition of surface stress. If a surface is cut by a plane normal to it, then, in order that the atoms on either side of the cut remain in equilibrium, it will be necessary to apply some external force to them. The total such force per unit length is the surface stress, and half the sum of the two surface stresses along mutually perpendicular cuts is equal to the surface tension. (Similarly, one-third of the sum of the three principal stresses in the body of a liquid is equal to its hydrostatic pressure.) In the case of a liquid or isotropic solid the two surface stresses are equal, but for a nonisotropic solid or crystal, this will not be true. In such a case the partial surface stresses or stretching tensions may be denoted as Ti and T2-... 

Shuttleworth [26] (see also Ref. 27) gives a relation between surface free energy and stretching tension as follows. For an anisotropic solid, if the area is increased in two directions by dAi and dA2, as illustrated in Fig. VII-1, then the total increase in free energy is given by the reversible work against the surface stresses, that is. 

Surface free energy j surface tension [ 1.37,1.38]. Between the surface tension (y) and the surface free energy (a) the following relation exists y = a -I- A dojdA), for an area A. 

Definitions of surface and interfacial free energy, and surface and interfacial tension... 

Surface Tension, Surface Free Energy and Surface Stress of Solids... 

Any contamination, especially by surfactants, will lower surface tension and lower surface free energy. Some surface tension values of common liquids and solvents are shown in the following Tables 1.2 and 1.3. 

A solid is defined as a material that is rigid and resists stress. A solid surface may be characterized by its surface free energy and surface energy. The surface energy (tension) of a solid caimot be measured in a similar manner to that of a liquid, due to the difficulty caused by the reversible formation of its surface. The methods for the determination of surface energy of solids are described in this chapter. 

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