Hamaker constant retarded

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... 

Using Eqs. VI-30-VI-32 and data from the General References or handbooks, plot the retarded Hamaker constant for quartz interacting through water and through n-decane. Comment on the relative importance of the zero frequency contribution and that from the vuv peak. 

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). 

It is evident from Figure 10.7 that the measurements are consistent with both unretarded and retarded attractive forces at appropriate separation distances. It has also been possible to verify directly the functional dependence on radii for the attraction between dissimilar spheres (see Table 10.4), to determine the retardation of van der Waals forces (see Table 10.1), and to evaluate the Hamaker constant for several solids, including quartz. Values in the range of 6 10 20 to 7 10 20 J have been found for quartz by this method. This is remarkably close to the value listed in Table 10.5 for Si02. 

Here, n is the refractive index and is the ise the mean ionization frequency of the material. Typically this is ve k, 3x1015 Hz. If the absorption frequencies of all three materials are assumed to be the same, we obtain the following approximation for the non-retarded Hamaker constant ... 

In Table 6.3 non-retarded Hamaker constants are listed for different material combinations. Hamaker constants, calculated from spectroscopic data, are found in many publications [124-128], A review is given in Ref. [129],... 

In atomic force microscopy the tip shape is often approximated by a parabolic shape with a certain radius of curvature R at the end. Calculate the van der Waals force for a parabolic tip versus distance. We only consider non-retarded contributions. Assume that the Hamaker constant Ah is known. 

French, R.H., Cannon, R.M., DeNoyer, L.K. and Chiang, Y.-M., (1995), Full spectral calculation of non-retarded Hamaker constants for ceramic systems from interband transition strengths , Solid State Ionics, 75, 13-33. 

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. 

Van der Waals interactions between identical solid particles are always attractive [7]. However, if the Hamaker constant of the suspending fluid is intermediate between the Hamaker constants of two different particles, the van der Waals interactions will be repulsive [9]. Moreover, in view of the finite speed of propagation of electromagnetic radiation, the response of a molecule to perturbations in the electric field deriving from another nearby molecule is not instantaneous. Retardation effects are observable at separation distances as small as 1 or 2 nm, and they become prominent at larger distances (>10 nm) [50]. Gregory [50] has proposed a simple expression for describing retarded van der Waals interactions between flat plates ... 

Using this substitution and accounting for retardation effects, the effective Hamaker constant is given by [7]... 

A final interesting observation is the existence of a frequency scale, 3x10 see in Eq. (2-39). This is the frequency at which the electronic cloud around an atom fluctuates it is therefore the rate at which the spontaneous dipoles fluctuate. Since the electromagnetic field created by these dipoles propagates at the speed of light c = 3 x lO cm/sec, only a finite distance c/v 100 nm is traversed before the dipole has shifted. Since the dispersion interaction is only operative when these dipoles are correlated with each other, and this correlation is dismpted by the time lag between the fluctuation and the effect it produces a distance r away, the dispersion interaction actually falls off more steeply than r when molecules or surfaces become widely separated. This effect is called the retardation of the van der Waals force. The effective Hamaker constant is therefore distance dependent at separations greater than 5-10 nm or so. 

Figure 2,12 Van der Waals force F between two curved mica surfaces of radius /f 1 cm in water and electrolyte solutions. The line is the fitted van der Waals force with Hamaker constant Ah = 2.2 X 10 J. At distances D greater than 5 nm, the force is closer to zero than predicted because of retardation effects. (From Israelachvili and Adams 1978 and Israelachvili 1992, reprinted with permission from Academic Press.)...

The interaction energy between two identical particles depends on the potential and retarded Hamaker constant (cf. any handbook of colloid chemistry), and Fig. 3,83 shows that a ( potential of about (pins or minus) 40 mV assures an energy barrier that prevents fast coagulation even with relatively high Hamaker constant. When the absolute value of the ( potential is lower this barrier disappears (Fig. 3.82), and the stability ratio (Fig. 3.6) approaches 1. The theory of colloid stability is discussed in detail in handbooks of colloid chemistry. 

The asymptotic behavior of the dispersion interaction at large intermolecular separations does not obey Equation 5.166 instead, oc lf due to the electromagnetic retardation effect established by Casimir and Polder. Various expressions have been proposed to account for this effect in the Hamaker constant. ... 

The orientation and induction interactions are electrostatic effects, so they are not subjected to electromagnetic retardation. Instead, they are subject to Debye screening due to the presence of electrolyte ions in the liquid phases. Thus, for the interaction across an electrolyte solution the screened Hamaker constant is given by the expression " ... 

Figures 5.37 and 5.38 show the critical thicknesses of rupture, Rp for foam and emulsion films, respectively, plotted vs. the film radius." In both cases the film phase is the aqueous phase, which contains 4.3 x 10 M SDS + added NaCl. The emulsion film is formed between two toluene drops. Curve 1 is the prediction of a simpler theory, which identifies the critical thickness with the transitional one." Curve 2 is the theoretical prediction of Equations 5.270 to 5.272 (no adjustable parameters) in Equation 5.171 for the Hamaker constant the electromagnetic retardation effect has also been taken into account. In addition, Eigure 5.39 shows the experimental dependence of the critical thickness vs. the concentration of surfactant (dodecanol) for aniline films. Figures 5.37 to 5.39 demonstrate that when the film area increases and/or fhe electrolyte concentration decreases the critical film thickness becomes larger.

Table 7.1 Dielectric constant (e), refractive index (n), main absorption frequency in the UV region (ve) and non-retarded Hamaker constants (/ n) for two identical liquids, solids and polymers at 20°C interacting across vacuum (or air), calculated using Equation (567), where e3 = 1 and n3 = 1. From Equation (569)...

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... 

Note that I) 2 — Dl2 for sufficiently large separation r and D 2 decreases as the particle separation decreases. Figure 13.A.4 shows that the enhancement due to van der Waals forces is decreased when viscous forces are included in the calculations. For some values of the Hamaker constant there is an overall retardation of the coagulation rate due to the viscous forces. 

Coalescence frequency J depends on dimensionless parameters k, p, Sa, Sr, t, y, a. The parameter k characterizes relative sizes of interacting drops p is the viscosity ratio of drops and ambient liquid Sa and Sr are the forces of molecular attraction and electrostatic repulsion of drops r is the relative thickness of electric double layer, which depends, in particular, on concentration of electrolyte in ambient liquid y is the electromagnetic retardation of molecular interaction a is relative potential of surfaces of interacting drops. Let us estimate the values of these parameters. For hydrosols, the Hamaker constant is F 10 ° J. For viscosity and density of external liquid take m /s, 10 kg/m. ... 

For most substances, the retardation effect in the dispersion interaction energy becomes significant at distances greater than 100 A. The important point is that the Hamaker constant A defined in Eq. (167) is no longer constant but depends on the separation of interacting bodies. 

For D > 10 nm, retardation effects have to be considered, and the force decays with D instead of D. H is the so-called Hamaker constant that depends on the dielectric properties of the interacting materials and the medium between. Detailed calculations showed that Derjaguins approximation leads to errors that are usually less than 10% [161]. For large distances the precise shape of the probe has to be considered [19,162]. 

Van der Waals forces were calculated on the basis of Hamaker s theory with a retardation function introduced by Schenkel and Kitchener (5). The Hamaker constant of water-toluene-water was taken from Ref. 6. Both the... 

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