Retardation screening

The negative derivative of an energy with respect to distance is a force the force per area is pressure. When the spatial variation in Anam itself (which is due to retardation screening) is neglected, the energy per unit area -[AHam/(127r/2)] between half-spaces leads to a pressure that looks like — [AHam/67r/3)]. 

Mathematical form of dependence on material properties, 43 Mathematical form of the charge-fluctuation free energy, 45 Frequencies at which e s, A s, and Rn s are evaluated, 46 About the frequency spectrum, 51 Retardation screening from the finite velocity of the electromagnetic signal, 51 Effective power law of van der Waals interaction versus separation, 55 Van der Waals pressure, 57 Asymmetric systems, 58... 

Retardation screening from the finite velocity of the electromagnetic signal... 

Even at a separation of 100 A, 1/5 the principal absorption wavelength, there is damping. By a separation / = kabS = 500 A, practically no contribution occurs from the region of the absorption frequency. The effect of retardation screening can also be seen clearly in the changes of the density spectrum of contribution to the interaction energy at different frequencies (see Fig. LI. 14). 

The integrated consequence of this retardation screening shows up as a change in the contribution to the Hamaker coefficient AAm/Bm(0- This diminution in AAm/BmfZ) looks different when plotted versus log(Z) (see Fig. LI.16) or plotted versus Z by itself (see Fig. LI. 17). 

At separations greater than 20A, retardation screening begins progressively to snuff out the higher-frequency contributions. The terms in summation die out because both A(/ )2 and Rn(l ) go to zero over the same range of... 

The one-absorption example used for illustration here is appropriate to most van der Waals charge-fluctuation forces because of the important, usually dominant, UV-frequency range of fluctuations. Temperature is not usually a consideration the summation over sampling frequencies can often be smoothed into a continuous integration. However, retardation screening acts in any situation in which separations are more than a mere 20-30 A. Only for distances less than 20 A and, sometimes, for distances greater than 10,000 A, can the van der Waals interaction between parallel-planar surfaces be said to vary by the 1/Z2 power law that it demurely reveals in its simplest representation. 

Table LI.3. Typical Hamaker coefficients, symmetric systems, retardation screening neglected...

Without retardation screening, for separation w much less than layer thickness h, interaction is dominated by the first term, — (Ahw/hw/12ttw2), i.e., the same as if thickness h were infinite, as though substrate M were not even there, even if it had a very high polarizability compared with W and H. 

Even with the neglect of retardation screening, the apparent power of the van der Waals interaction varies with separation itself. The reason for this power-law variation is clear when we consider the interaction of two spheres of material 1 and 2 and of radius Ri and Ri with a center-to-center distance z = l + Ri + R2 in a medium m (see Fig. LI.35). 

This approximate form of Gss(z R1 R2) shows a general property of van der Waals interactions when formulated in the approximation (small differences in dielectric response, neglect of retardation) used here. The interaction is independent of length scale. If we were to change all the sizes and separations by any common factor, both the numerator RfR and the denominator z6 would change by the same factor to the sixth power. In reality, because retardation screening effectively cuts off interactions at distances of the order of nanometers, it makes sense to think of this inverse-sixth-power interaction only for particles that are the angstrom size of atoms or small molecules. 

This is smaller than the coefficient —3.2 x 10+3 nm6 kT already estimated for the zero-frequency contribution and therefore again indicates very weak interaction energy. Inclusion of the significant degree of retardation screening expected at these separations would have yielded still smaller energies. 

Even when not reduced by retardation screening, the van der Waals interaction between point particles will be small compared with thermal energy kT. This can be seen from a few examples. 

Actual (computed) retardation screening factor for small differences in dielectric response, energy between parallel flat surfaces. 

Table S.8. Point-particle interaction in vapor, like particles without retardation screening...

C.3.a. Two parallel cylinders, retardation screening neglected, solved by multiple reflection... 

C.3.b. Two parallel cylinders, pairwise-summation approximation, Hamaker-Lifshitz hybrid, retardation screening neglected C.3.b.l. All separations... 

Because of retardation screening, the integrals converge so rapidly that there is no chance for the material responses to vary, the response functions are taken to be effectively constant in frequency. 

In the limit of infinite separation at which /i 1, only the n = 0 term survives retardation screening ... 

This interaction is like the first n = 0 term for small-particle van der Waals forces but with ionic rather than retardation screening. In the limit of low salt concentration, Km - 0, it has the familiar l//6 form ... 

In the limit where there is no retardation screening, r = 0, only the first term in the integrand endures. In that case, the general interaction reduces to... 

When temperature is effectively zero, the discrete values of n = (2nkT/tyn merge into a continuum the interaction energy in the absence of ionic or retardation screening is... 

One, as frequency z approaches very high values, the relativistic retardation screening factor R(z) goes to zero. 

For good practice, it is probably a good idea to compute the Hamaker coefficient and the interaction free energy for several different film thicknesses to build an intuition about the magnitudes of forces and to see where retardation screening begins to be felt. 

The multiplication of the e s by /S s creates an effective dielectric response that includes ionic displacement. Double-layer screening of zero-frequency fluctuations is through the exponential e 2An . The formal resemblance to retardation screening comes clear here and in subsequent similar factors. 

The frequency of hypothyroidism has been studied in some specific circumstances in the United States. The chief among these has been congenital hypothyroidism, which has long been seen as one of the most common preventable causes of mental retardation. Screening for congenital hypothyroidism began in North America in 1972. Dussault et al. summarized the first miUion newborns screened at three sites in the United States and two in Canada. Primary hypothyroidism was identified in one in 4254 births secondary—tertiary hypothyroidism occurred in one in 100000 births. In addition, the prevalence of TBG deficiency was found to be one in 8913 births ishst et al., 1979). 

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