Free energy of attraction

Afp free energy of attraction per unit area, due to depletion, between two plates... 

If we choose the energy at infinite separation as the energy zero, then the free energy of attraction between a pair of atoms or molecules at a separation d is... 

So the free energy of attraction decays very slowly, viz. reciprocally with the distance, and even slower than it does in the case of flat plates, where it goes reciprocally with the square of the distance, (eq. 48c of Part II). At larger separations of the two spheres the decay is of course faster, as for very great distances the attraction must die out as 1/if , but nevertheless the decay remains slower than 1/H until s exceeds 2.4, as may be read from a double logarithmic plot of eq. (85). 

The problem with high-energy adherend surfaces is that atmospheric contaminants are readily adsorbed on them, so reducing the surface free energy of attraction for the adhesive. Kinloch(2) suggests that the polar nature of structural adhesives will lead to displacement of the less polar, often hydrocarbon, contaminants. Polar water molecules, on the other hand, seem less likely to be readily displaced and, in view of their very small size in relation to an adhesive macromolecule, are able to adsorb in vast numbers on the surface. 

Mathematically, the simplest situation to analyze is that involving two hard, flat, effectively infinite surfaces separated by a distance, //, in a vacuum. The free energy of attraction per unit area in such a case is approximated by... 

A comparison of Equations (4.30) and (4.43) shows that the free energy of attraction between two surfaces falls off much more slowly than that between individual atoms or molecules. This extended range of bulk interactions plays an important role in determining the properties of systems involving surfaces and interfaces. A combination of the attractive and repulsive forces between surfaces leads to a curve such as in Figure 4.7a. 

Estimate the free energy of attraction between two spheres of radius 500 nm at separation distances of 1,10, and 100 nm for the following systems (a) water-air-water (b) pentane-water-pentane (c) hexadecane-water-hexadecane (d) quartz-air-quartz (e) quartz-air-hexadecane and (f) Teflon-water-Teflon. 

Employing the convention that attractive energy is negative, at infinite separation the energy of the system will be zero, so that the free energy of attraction at distance r will be... 

The fourth order perturbation energy (8.21) clearly equals the first term of a more general representation of the free energy of attraction according to Eqs. (3.48), (3.49). The respective dispersion function G( ) is split into four subdeterminants G( +, ) corresponding to the order of emission and absorption of photons q and r,... 

However, since the statistic weight of the excited states is affected by the electron-photon interaction, there might well arise a free energy of attraction from the energy terms of order zero and two. We have to check whether with decreasing separation there is a redistribution of occupied states, which gives rise to an energy of interaction of order four in U kiq) as well. 

Finally, we are interested in the free energy of the electron-photon system under investigation. The free energy of attraction of particles 1 and 2 is the difference between the free energies at separations rji and infinity. The free energy of any system of particles equals... 

To carry out the summation in Eq. (8.34) over all electron states, it is necessary to expand the photon terms with respect to — fto),. We obtain all terms up to order four in the interaction parameters U- (kiq) by expanding up to terms quadratic in —However, bearing in mind that all terms which do not depend on the separation of particles 1 and 2 cancel when the free energy of attraction is calculated, it is sufficient to indicate the terms quadratic in — ft m, by dots. Hence,... 

All electron states are on the average occupied according to the Fermi distribution (8.43). All photon states are on the average occupied according to the Bose distribution (3.6). On substituting Eq. (8.44) in Eqs. (8.37) and (8.32) we find for the free energy of attraction between particles 1 and 2... 

The free energy of attraction between particles 1 and 2 results by substituting the Fermi distribution and the Bose distribution into the fourth order energy expression (8.17). 

The final Eqs. (8.53)-(8.55) for the free energy of attraction between particles 1 and 2 confirm and refine the semiclassical results obtained in Chapter 3 6. There are contributions from the photons and the electrons to the free energy of attraction as well. The factor coth(< /2/cT) is not only caused by the Bose distribution, but is also required by the Fermi distribution. In accordance with the symmetric use of retarded and advanced susceptibilities in the case of damping in Section 3.6, we now distinguish between emission of a photon prior to absorption and absorption of a photon prior to emission. The result is a symmetric final integration over the dispersion function along the full imaginary frequency axis in both cases. 

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