Dispersion force retarded

In this section we consider electromagnetic dispersion forces between macroscopic objects. There are two approaches to this problem in the first, microscopic model, one assumes pairwise additivity of the dispersion attraction between molecules from Eq. VI-15. This is best for surfaces that are near one another. The macroscopic approach considers the objects as continuous media having a dielectric response to electromagnetic radiation that can be measured through spectroscopic evaluation of the material. In this analysis, the retardation of the electromagnetic response from surfaces that are not in close proximity can be addressed. A more detailed derivation of these expressions is given in references such as the treatise by Russel et al. [3] here we limit ourselves to a brief physical description of the phenomenon. 

The continuum treatment of dispersion forces due to Lifshitz [19,20] provides the appropriate analysis of retardation through quantum field theory. More recent analyses are more tractable and are described in some detail in several references [1,3,12,21,22],... 

A common approach to treating retardation in dispersion forces is to define an effective Hamaker constant that is not constant but depends on separation distance. Lxioking back at Eq. VI-22, this defines the effective Hamaker constant... 

A major advance in force measurement was the development by Tabor, Win-terton and Israelachvili of a surface force apparatus (SFA) involving crossed cylinders coated with molecularly smooth cleaved mica sheets [11, 28]. A current version of an apparatus is shown in Fig. VI-4 from Ref. 29. The separation between surfaces is measured interferometrically to a precision of 0.1 nm the surfaces are driven together with piezoelectric transducers. The combination of a stiff double-cantilever spring with one of a number of measuring leaf springs provides force resolution down to 10 dyn (10 N). Since its development, several groups have used the SFA to measure the retarded and unretarded dispersion forces, electrostatic repulsions in a variety of electrolytes, structural and solvation forces (see below), and numerous studies of polymeric and biological systems. 

McLachlan A D 1963 Retarded dispersion forces between molecules Proc. R. Soc. A 271 387... 

Hamaker constants can sometimes be calculated from refractive igdex data by the Lifshitz equations (8), but it now appears that Y values are closely related to refractive indices and are a direct measure of the Lifshitz attractions. In Equation 1 a correction factor f for "retardation" of dispersion forces is shown which can be determined from Figure 2, a graph of 1/f at various values of H and a as a function of Xj, the characteristic wavelength of the most energetic dispersion forces, calculable and tabulated in the literature (9). 

Figure 2. Retardation correction factor (f) for dispersion force attractions between spherical particles of radius (a) at separation distance (H), with dispersion force wavelength Xj. (10)...

The Lifshitz theory of dispersion forces, which does not imply pairwise additivity and takes into account retardation effects, shows that the Hamaker constant AH is actually a function of the separation distance. However, for the stability calculations that follow, only the values of the attraction potential at distances less than a few nanometers are relevant, and in this range one can consider that AH is constant. 

McLachlan AD (1963) Retarded dispersion forces between molecules. Proc Roy Soc (London) Ser A 271 387-401... 

Here the component Tlvw is inversely proportional to h4, similar to the Hamaker s formula which accounts for the electromagnetic retardation of dispersion forces. 

Equations of this type appear to fit Isotherms of type II (fig. 1.13) quite well, sometimes better than BET theory does. The exponent -1/3 stems from 11.5.51). In practice, values between -1/3 and -1/2 are usually found. From the viewpoint of dispersion forces this is difficult to account for. Retardation does not play a role and. even if it did, this would further reduce the exponent. Rather, the sum effect of all "hand-waving" approximations (including the assumption of surface homogeneity) leads to a semi-empirical Isotherm of the form j 1.5.54] in which the constants and exponent are, within certain limits, adjustable. Because of this, the equation is often written in the more general form... 

Casimir and Polder also showed that retardation effects weaken the dispersion force at separations of the order of the wavelength of the electronic absorption bands of the interacting molecules, which is typically 10 m. The retarded dispersion energy varies as R at large R and is determined by the static polarizabilities of the interacting molecules. At very large separations the forces between molecules are weak but for colloidal particles and macroscopic objects they may add and their effects are measurable. Fluctuations in particle position occur more slowly for nuclei than for electrons, so the intermolecular forces that are due to nuclear motion are effectively unretarded. A general theory of the interaction of macroscopic bodies in terms of the bulk static and dynamic dielectric properties... 

The attractive-force component between particles in a colloidal system is developed by summation of the London dispersion forces between all atom pairs in the particles. Neglecting retardation effects, the expression for the attractive energy Va between two particles of radius a at a distance of separation Hq, for a > Ho, is... 

Note first that in this older picture, for both the attractive (van der Waals) forces and for the repulsive double-layer forces, the water separating two surfaces is treated as a continuum (theme (i) again). Extensions of the theory within that restricted assumption are these van der Waals forces were presumed to be due solely to electronic correlations in the ultra-violet frequency range (dispersion forces). The later theory of Lifshitz [3-10] includes all frequencies, microwave, infra-red, ultra and far ultra-violet correlations accessible through dielectric data for the interacting materials. All many-body effects are included, as is the contribution of temperature-dependent forces (cooperative permanent dipole-dipole interactions) which are important or dominant in oil-water and biological systems. Further, the inclusion of so-called retardation effects, shows that different frequency responses lock in at different distances, already a clue to the specificity of interactions. The effects of different geometries of the particles, or multiple layered structures can all be taken care of in the complete theory [3-10]. 

Finally, it is worth remarking that retarded van der Waals dispersion potentials between molecules in ground and excited electronic states may also be calculated [12,51] using the fluctuating moment method. Because dispersion forces arise from the perturbation induced by the zero-point energy associated with the vacuum electromagnetic field, the expectation value of... 

Particle-Collector Interactions. Dispersion forces and double-layer forces were the two interaction forces considered between the particle and collector. The London dispersion forces can be expressed with the Hamaker constant H, the distance between the two particles 5, and a retardation factor... 

Equation 2.13 is vahd only for small values of h. For larger distances the propagation speed of the electromagnetic field has to be taken into account (so-called retardation effect), which gives weaker attractive forces. We avoid these complications here, also because the (approximate) summation procedures described in this section are now superseded by a macroscopic theory of dispersion forces in which these effects are treated in a natural way. 

In most cases e(electromagnetic spectrum. Fairly accurate interpolation formulas can be used for several systems. Numerical calculations have been made for soap films by Ninham and Parsegian. Their formula for e(w) was used to calculate the dispersion forces for our type of film. Results are given in Section VI. It is noteworthy that V h) is found to be not simply proportional to so that retardation effects cannot be neglected in our soap films. 

Casimir and Polder showed, in 1948, that London s theory must be modified if the molecules are some distance (in molecular terms) apart. This is because the electrostatic forces are not propagated instantaneously but take a finite, if small, time pass between two molecules. The net result of such retarded dispersion forces is that the energy of interaction between two particles with distance between them being greater than 20 nm is proportional to r instead of... 

McLachlan, A. D. 1963a. Retarded dispersion forces between molecules. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 271, no. 1346 387-101. doi 10.1098/rspa.l963.0025. 

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