The Total Surface Energy

If the variation of density and hence molar volume with temperature is small, it follows from Eqs. Ill-10 and III-8 that E will be nearly temperature-independent. In fact, Eq. Ill-11 with n = 1 may be written in the form 

Inspection of Table III-l shows that there is a wide range of surface tension and E values. It is more instructive, however, to compare E values calculated on an energy per mole basis. The area per mole of spherical molecules of molecular weight M and radius r is 

One may consider a molecule in the surface region as being in a state intermediate between that in the vapor phase and that in the liquid. Skapski [11] has made the following simplified analysis. Considering only nearest-neighbor interactions, if n, and denote the number of nearest neighbors in the interior of the liquid and the surface region, respectively, then, per molecule 

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The total surface energy generally is larger than the surface free energy. It is frequently the more informative of the two quantities, or at least it is more easily related to molecular models. 

The surface free energy can be regarded as the work of bringing a molecule from the interior of a liquid to the surface, and that this work arises from the fact that, although a molecule experiences no net forces while in the interior of the bulk phase, these forces become unbalanced as it moves toward the surface. As discussed in connection with Eq. Ill-IS and also in the next sections, a knowledge of the potential function for the interaction between molecules allows a calculation of the total surface energy if this can be written as a function of temperature, the surface free energy is also calculable. 

The illustrative data presented in Table VII-3 indicate that the total surface energy may amount to a few tenths of a calorie per gram for particles on the order of 1 /xm in size. When the solid interface is destroyed, as by dissolving, the surface energy appears as an extra heat of solution, and with accurate calorimetry it is possible to measure the small difference between the heat of solution of coarse and of finely crystalline material. 

These equations result from assuming that the total surface energy can be split into the sum of components associated with different types of bonding, for example dispersion plus polar yP (Eqs. 14 and 15), or Lifshitz-van der Waals... 

It is observed that the total surface energy increases with grafting (Table 5). The values become almost double for PEgDBM3. The increased surface energy of the grafted polymer comes mostly from the polar compo-... 

The total surface energy 7. together with 7s " (Liftshitz van der Waal s component), 7, (acidic component), 7s (basic component), and 7s (acidic and basic component) have been determined for the modified and control EPDMs. The results are plotted against radiation dose in Figure 31.6. 

More recently, the temperature dependence of the above vibrational component was studied13. Still for the 100 face of NaCl, the contribution of this component to the total surface energy, see Section II.3., was approximately —2 erg/cm2 and to the free surface energy near - 16 erg/cm2, both at 273 °K. No correction for the polarizability of the ions was employed. 

Several computations of the total surface energy (per unit area) Us start from the experimental value of the sublimation energy, Ls, of the crystal. When it is remembered that the intensity of interatomic forces frequently is derived from thisZ,S) the direct use of the experimental data does not appear to be a serious drawback. On the other hand, this approach does not differentiate between different crystal faces and can give only an averaged value for all of these. 

The total surface energy per unit area of liquids, U, is related to measurable surface tension or the specific free surface energy 7 by the thermodynamic equation... 

Treatment with fluorine/nitrogen mixtures (fluorination) increases the polar component of the surface energy initially, but longer treatment times result in a drop in both the polar component and the total surface energy (see Figure 16.3). At very long treatment times the surface energy ofthe treated surface can be lower than that of the polyolefin itself... 

At the critical temperature, Tc, and critical pressure, Pc, a liquid and its vapor are identical, and the surface tension, y, and total surface energy, as in the case of the energy of vaporization, must be zero (Birdi, 1997). At temperatures below the boiling point, which is 2/3 Tc, the total surface energy and the energy of evaporation are nearly constant. The variation in surface tension, y, with temperature is given in Figure A.l for different liquids. 

In the previous section we have noted that in the formation of an interfacial phase energy must be expended and that the free surface energy of an interface is related to the total surface energy by the Gibbs-Helmholtz equation... 

The relationship between surface tension and temperature leads to a series of interesting results bearing upon chemical constitution and the structure of the surface layer. As we have seen the surface tension is numerically equal to the free surface energy from which with the aid of the temperature coefficient the total surface energy may be obtained. 

Since is always negative for pure liquids the total surface energy... 

As examples of the magnitude of variation in the free surface energies and the approximate constancy of the total surface energies the following data for benzene (Whittaker, Proc. Roy. 8oc. A, Lxxxi. 21,1900), mercury and carbon tetrachloride (Harkins and Koberts, J.A.G.B. XLiv. 656,1922) may be cited (see tables, p. 20). 

A critical examination of the total surface energies of various liquids has been made by Langmuir and Harkins. It is found that the surface energies of the aliphatic hydrocarbons are mainly independent of the length of the hydrocarbon chain, whilst the surface... 

Debye s investigations on the energy content of substances at low temperatures be expressible in the form u — Uq aT , where a is determinable from the heat capacity of the surface film, and the temperature coefficient of the heat of wetting should decrease rapidly as we approach the absolute zero. Furthermore it is evident that at this temperature the free and total surface energies should be identical in value, the total surface energy sinking first slowly and then rapidly as the critical temperature is reached. Confirmation likewise of the assumption Lt = 0 or that the temperature... 

Starting from Gibbs concept [6] that the total surface energy times the surface area is at a minimum, the following concepts emerged ... 

The total surface energy per wavelength for the surface of revolution, Eq. 14.23, is... 

Figure 1.1. Derivation of Equation (1.12). When this equation is satisfied an infinitesimal change of position of the three-phase point, dx, will not change the total surface energy of the system and thus it is in equilibrium.

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