Surface excess energy

This effect assumes importance only at very small radii, but it has some applications in the treatment of nucleation theory where the excess surface energy of small clusters is involved (see Section IX-2). An intrinsic difficulty with equations such as 111-20 is that the treatment, if not modelistic and hence partly empirical, assumes a continuous medium, yet the effect does not become important until curvature comparable to molecular dimensions is reached. Fisher and Israelachvili [24] measured the force due to the Laplace pressure for a pendular ring of liquid between crossed mica cylinders and concluded that for several organic liquids the effective surface tension remained unchanged... 

Second-order stress is difficult to observe and much less extensively studied. The causes of internal stress are still a matter for investigation. There are broad generalisations, e.g. frozen-in excess surface energy and a combination of edge dislocations of similar orientation , and more detailed mechanisms advanced to explain specific examples. 

In metal deposition, the primary products form adsorbates on the electrode surface rather than a supersaturated solution. Their excess chemical potential is directly related to polarization and given by nFAE. The total excess surface energy = 2 S,o,. Otherwise, all the results described above remain valid. 

Crystallization in general is a two-step process involving (1) nucleation and (2) growth of the nucleus to a macro size. Nucleation involves the activation of small, unstable particles with sufficient excess surface energy to form a new stable phase. This may occur in supersaturated solutions as a result of mechanical shock, the introduction of small crystals of the desired type, or the presence of certain impurities that can act as centers for growth. 

In situ precipitation also provides a method for preparing surfaces with uniform stoichiometry and purity, a small range of grain and particle diameters, and minimum excess surface energy. These properties should maximize catalyzed rates and minimize differences between experiments caused by nonuniformity of mixing and contact between solution and surface [112]. 

For the surface in Figure 2.6, it can be shown that the total excess surface energy is the sum of a contribution by the singular terraces and a contribution by the ledges, given by... 

A decrease in size of a condensed phase particle results in an increase in the chemical potential of the substance due to excess surface energy and the elevation of the Laplace pressure inside the particle (see Section 1.1). With a spherical particles of radius r, one can estimate the increment Aj of the chemical potential of component i of the continuous phase by the expression... 

Take the excess surface energy of compound K as equal to 1 J/m and the correlation coefficients between enthalpy of the elementary step and the potential barrier height for the transition state of elemen tary reactions 1 and 2 as equal to X = 0,5. 

Jjet us consider as an example the case of a saturated vapor which has been suddenly and adiabatically compressed to a vapor pressure P which is in excess of its equilibrium vapor pressure Po at the final temperature T. In order for liquid to form, it must grow by the growth of small droplets. If, however, we consider a very small droplet of the liquid phase present in the vapor, it will have an excess free energy, compared to bulk liquid, that is due to its extra surface. The magnitude of the excess surface energy is 4irrV, where surface tension and r is the radius of the drop. In order for the drop and vapor to be in equilibrium, the vapor pressure P must exceed the saturation vapor pressure Po by an amount which can be calculated from the Gibbs-Kelvin equation... 

Figure 2.16. Relationship between the excess surface energy and the latent heat of evaporation.

Fig. 7-11. The effect of decreasing particle size (represented by increasing surface area) on the bulk rate of dissolution. Three regions are described Region 1, particle dimensions are significantly greater than the distance between dislocation outcrops Region 2, particle dimensions are smaller than or equal to the distance between adjacent dislocations Region 3, particle dimensions are sufficiently small that excess surface energy increases the rate of release (reprinted with permission from Holdren and Speyer, 1985, Copyright 1985, Pergamon Press pic.)...

We have seen above that the kinetics of mineral dissolution is well explained by transition-state theory. The framework of this theory and kinetic data for minerals have shown that dissolution is initiated by the adsorption of reactants at active sites. Until now these active sites have been poorly characterized nevertheless, there is a general consensus that the most active sites consist of dislocations, edges, point defects, kinks, twin boundaries, and all positions characterized by an excess surface energy. Also these concepts have been strongly supported by the results of many SEM observations which have shown that the formation of crystallographically controlled etch pits is a ubiquitous feature of weathered silicates. 

The presence of excess surface energy in such gas-liquid disperse system predetermines its nonequilibrium. However, by virtue of the stabilizing effect of surfactants, foam possesses a metastable structure and has a certain lifetime [118], Its properties slowly relax under the action of external factors provided that the latter do not exceed some threshold values beyond which the foam structure is destroyed. 

This effect may be understood in terms of the concept of critical nucleus size for growth of crystals. Consider a small particle of material of diameter D immersed in another medium as shown in Fig. 10.23(a). If the particle is a droplet of liquid sitting in another liquid, such that an interfacial energy y exists between the two liquids, then the particle has an excess surface energy which causes a pressure excess 4y/D inside the particle according to Laplace s equation. 

FIGURE 2.4. The excess surface free energy of newly formed surface will depend on the nature of the new phase it contacts, (a) If the new surface contacts a vacuum, the excess free energy will be maximized, (b) If another phase is present (liquid or gas) the excess surface energy will be reduced by an amount depending on the new interactions. 

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