Interfaces estimating surface energies

The interface between a solid and its vapor (or an inert gas) is discussed in this chapter from an essentially phenomenological point of view. We are interested in surface energies and free energies and in how they may be measured or estimated theoretically. The study of solid surfaces at the molecular level, through the methods of spectroscopy and diffraction, is taken up in Chapter VIII. 

For the two-component mixed phases the surface and Coulomb contributions can be estimated along the lines of the analysis performed in Ref. [47] for the interface between nuclear and CFL matter. We find a gain in bulk energy of at most 6 MeV/fm3 which is already weight out by Coulomb and surface energy for relatively small values of the surface tension a 10 MeV/fm2. Note... 

Calculate the surface energies of each of these liquids and plot a graph of y for the CTAB solutions as a function of logio(conc.). Use your results and the Gibbs adsorption equation (see later) to estimate the minimum surface area per CTAB molecule adsorbed at the air-water interface. 

Atoms in the free surface of solids (with no neighbors) have a higher free energy than those in the interior and surface energy can be estimated from the number of surface bonds (Cottrell 1971). We have discussed non-stoichiometric ceramic oxides like titania, FeO and UO2 earlier where matter is transported by the vacancy mechanism. Segregation of impurities at surfaces or interfaces is also important, with equilibrium and non-equilibrium conditions deciding the type of defect complexes that can occur. Simple oxides like MgO can have simple anion or cation vacancies when surface and Mg + are removed from the surface,... 

The sequential addition method (SAM) recently developed in our studies makes it possible to estimate the solubility of oxides having different values of specific (molar) surface and, probably, of surface energy at the oxide—ionic melt interface boundary [333]. 

To calculate the surface energies at the oxide-ionic melt interface boundary by means of equation (3.6.57) it is necessary to know the radii of the oxide particles in contact with the saturated solution in the melt-solvent. In this connection there arises a question whether one can correctly estimate the values of the oxide particles radii from the data on the molar surface area obtained using any experimental method of estimating the specific surface area, e.g. the BET treatment. In order to answer this question we first consider the relationship between the volume and surface area of a spherical particle. 

The values of the surface energy at the oxide-chloride melt interface boundary were calculated in Ref. [350] to be within the range 30-40 J m-2. These values considerably exceed the corresponding parameters estimated for aqueous solutions of electrolytes, thus giving rise to essential changes of the solubility with the particle sizes (see Fig. 3.7.7). 

Over-estimation of the values of surface energies at the oxide-chloride melt interface boundary (if it actually takes place) can be caused by the action of factors other than surface effects, but it is not yet possible to find out these factors, even they exist at all. 

M is the molecular weight and p the density of the solid, d describes the particle size and a is a geometrical factor which depends on the shape of the particles. For approximately spherical particles with diameter d the geometry factor is a 6. According to Schindler [1967SCH] the mean free surface energy of the solid-liquid interface can be estimated by Eq. (VI1.44) ... 

Some theoretical models based on the thermodynamics of surfactant self-assembly have also been recently used to predict the critical surface aggregation concentration (the bulk concentration at which surfactants start to self-assemble at the solid-liquid interface), and the self-assembled surfactant structure at the solid-liquid interface (11). These models, although providing useful insight into the surfactant self-assembly process, require an estimate of the interaction energies, which are difficult to determine experimentally. Variations in the estimated interaction energies can lead to different self-assembled surfactant structures, depending on the values used for the calculations. 

Let us now return to the question of whether we can calculate the surface energies of polymers from first principles. The rough estimates in section 2.1 tell us correctly the order of magnitude of surface tensions and correctly draw attention to the intimate connection between surface energies and the cohesive forces in liquids, but they have a number of drawbacks. Firstly, temperature makes no appearance in these theories, despite the experimental fact that surface tensions depend quite strongly on temperature. Secondly, we have assumed that the density of the liquid near the surface is the same as the bulk density. These shortcomings are seen at their most extreme if we consider a liquid near the liquid-vapour critical point. Here the distinction between liquid and vapour vanishes completely the surface tension of the liquid approaches zero and the system becomes in effect all interface. An improved theory of surface tension must be able to accoxmt for these phenomena, at least qualitatively. 

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