Retarded van der Waals constant

Quantity of electricity, electric charge Q Retarded van der Waals constant B,P... 

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 ... 

Figure 3.4 Non-retarded van der Waals interaction free energy w between two spheres, lu = -A/ 60) R Rif (Jii + 2)) where R and R2 are their radii and D is the distance between their surfaces. A = ir Cpi p2 is the Hamaker constant with Pi and p2 the number of atoms per unit volume of each sphere and C is given by eq. 3.18.

Figure 3,5 Non-retarded van der Waals interaction free enei w between a sphere and a flat surface, w = — (A/6D) R with R the radius of the sphere, D the distance between the two surfaces and A the Hamaker constant (fig. 3.4).

A rough approximation to understand qualitatively the transition from the non-retarded van der Waals force to the retarded regime can be derived using Eq. (2.45). Note that only the London dispersion contribution is affected by retardation. When retardation comes into play, the Hamaker constant A123 becomes, in fact, a Hamaker function that will depend on separation x. Under the simplifying assump-... 

At this point we should darify that the Casimir force is not a really new type of force. It is simply another term for a special case of the van der Waals forces, namely, the retrarded van der Waals force between metallic surfaces. While the terms retarded van der Waals force or retarded London dispersion force are prevalent in the physical chemistry and colloid community, the term Casimir force or Casimir-Polder force has become popular in the physics community. This means that in principle the lifshitz theory is apphcable to describe the Casimir forces. The problem with using Lifshitz theory for ideal metals is the fact that for these the didectric constant diverges (e oo) and therefore the Lifshitz theory breaks down. However, for real metals, the use of the Lifshitz theory is possible with corresponding dielectric models of the metals. 

The repulsive force depends on the double layer potential and thickness, the particle radius and the dielectric constant of the medium, whereas the attractive force arises from retarded London/van der Waals forces. 

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. 

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. 

With respect to the molecular interactions the simplest asymmetric films are these from saturated hydrocarbons on a water surface. Electrostatic interaction is absent in them (or is negligible). Hence, of all possible interactions only the van der Waals molecular attraction forces (molecular component of disjoining pressure) can be considered in the explanation of the stability of these films. For films of thickness less than 15-20 nm, the retardation effect can be neglected and the disjoining pressure can be expressed with Eq. (3.76) where n = 3. When Hamaker s constants are negative the condition of stability is fulfilled within the whole range of thicknesses. 

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.)...

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. 

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