Half-space interaction approximation

PROBLEM L2.5 Derive approximation (L2.145) by expansion of Eq. (L2.144) and by differentiation of — [AHam/12jrZ2] for the interaction of half-spaces. 

We turn now to the other side of the colloidal particle interaction problem idealised to the case of two half spaces separated by salt water [3-10, 22-24]. Typically such particles will contain ionisable groups at their surfaces, so that the surfaces are charged. Imagine that the water, as before, retains its bulk properties up to the surface of the half spaces. The charged surfaces create an inhomogeneous profile of cationic and anionic density. For an isolated surface at the simplest level of approximation and schematically only, this distribution follows from the equation ... 

The description of the van der Waals interaction based on the Lifshitz approach is now sufficiently advanced to provide accurate predictions for the complete interaction energy. For the geometry of two half-spaces, the exact theory is available in a formulation suited for computational purposes. " In parallel with work on planar systems, there has been a focus on the interaction between spheres. " These developed theories have been used as the exact solutions in the validation of the approximate predictions using the Hamaker approach. The significant contribution of the continuum approach to our understanding of the van der Waals interaction lies in the reliable prediction of the Hamaker constant. The interaction energy for two half-spaces and two spheres is summarized below. 

This expression has the same form as the heuristic approximation discussed before and the interaction decays as 1 /L. When < m < Q or 2 < m < — i.e., when the dielectric constant of the film has a value in between those of the half-spaces, the interaction is repulsive and the film will tend to thicken. In all other cases, the interaction is attractive and the film will tend to thin. We also note that the interaction energy is proportional to T, which indicates that this is a purely entropic contribution coming from the classical part of the fluctuations of the electromagnetic field. 

The authors find very rapid convergence of (5.42). Model calculations indicated agreement with the exact formula to 2 percent. Many aerosols such as chain agglomerates and fibers may be viewed, to a first approximation, as cylinders. Interactions between parallel cylinders have been studied by LANGBEIN [5.96] who found interaction laws between those for interacting spheres and interacting half-spaces for nonretarded and presumably, therefore, for retarded cases. For separations respectively small and large with respect to ... 

Let us therefore try to find approximate dispersion functions which are valid at small separations of the interacting particles. The dispersion function between half-spaces is split up depending on the translational behavior of the interacting waves parallel to the surfaces. Each mode... 

Lateral tensions in soft membranes are essentially positive since these systems buckle so easily. Their action can only diminish the strength of out-of-plane fluctuations, thus weakening steric interaction. This may induce mutual adhesion where in the absence of stress the membranes would separate. In the following we deal with the effect of lateral tension on undulation forces and with the resulting equilibrium spacing. For simplicity the direct interaction is represented by the half-space approximation (1) of van der Waals attraction with H < H. In a first treatment, we will employ superposition although we are only on the borderline of its applicability. 

Our predictions concerning induced adhesion rest on the assumption that the direct interaction is satisfactorily described by van der Waals attraction in its half-space approximation (1). This restricts the range of mean spacings where they may be expected to be valid to, perhaps,... 

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