2.2.4. Retardation Effect in van der Waals Interaction

The derived van der Waals interaction potential equations for the geometrical shapes described earlier are unretarded (without retardation effect) interaction equations for sphere, cylinder, and rod-to-solid flat surfaces and are derived based on an implicit assumption that the speed of light is infinite. “Retardation effect” is the correction factor that must be made because of the reduction in the van der Waals interaction potential due to the finite speed of light [26,36]. Owing to the electromagnetic nature of the interaction, the actual van der Waals potential between two interacting bodies is reduced at large separation distances because the time it takes for the electric field to propagate from one body to another and back is such that the fluctuating electric moments become slightly out of phase. Such an effect may become very pronounced for macroscopic bodies at separation distances larger than about 5 nm [24]. For a large particle with diameter of 1 µm or greater, omission of the retardation has proved to lead to a serious overestimate of the van der Waals interaction[19]. In principle, the Hamaker approach can be modified to account for this retardation by multiplying the unretarded vdW interaction potential by a correction factor:

Where the correction factor, f(p), depends on the reduced distance, p= 2πrλ, and λ is the “London characteristic wavelength” of the interaction. λ is assumed to be about 100 nm for most materials and the retardation only becomes significant when the separation distance between particles is of the same order as the characteristic wavelength [19] as shown in Figure 6. Since the rod particles are so much smaller than the flat plate, the existing correction factor for the sphere-plate interaction potential may be most suitable to account for the retardation effect on the rod-flat plate interaction system. The correction factor given by [37] as reported by [19] is as follows:

where, λ is a retardation parameter and “s” is a constant (s = 11.116). Multiplying the unretarded van der Waals interaction potentials by f (D, λ) yields the approximate retarded van der Waals interaction potential. When using a bacterium as the macroscopic body, there is only a small difference (up to 10–30J) between retarded and non-retarded vdW forces, and therefore will be ignored in this study.

The van der Waals forces with non-retarded and retarded regimes [23].