Half-space interaction limit

There are two major sources of the deformation in contact-mode SFM the elasticity of the cantilever and the adhesion between the tip and sample surface. For purely elastic deformation, a variety of models have been developed to calculate the contact area and sample indentation. The lower limit for the contact diameter and sample indentation can be determined based on the Hertz model without taking into account the surface interactions [79]. For two bodies, i.e. a spherical tip and an elastic half-space, pressed together by an external force F the contact radius a and the indentation depth 8 are given by the following equations ... 

Examining the full expression for the interaction between half-spaces to see which features are revealed in its specialized limiting forms (Section L2.3.A) ... 

When modern theory is restricted to the limits at which all relativistic retardation is neglected and differences in the dielectric susceptibilities are small, the interaction between half-spaces (omitting magnetic terms) goes as... 

This is for the interaction between a half-space L and an infinitely layered half-space R. For the large limiting value of N used here, the right-hand half-space R disappears from the formulation. 

The method considered here for determination of these forces is the modified Lifshitz macroscopic approach, considering the microscopic approach results. This approach uses the optical properties of interacting macroscopic bodies to calculate the van der Waals attraction from the imaginary part of the complex dielectric constants. The other possible approach—the microscopic theory—uses the interactions between individual atoms and molecules postulating their additive property. The microscopic approach, which is limited to only a few pairs of atoms or molecules, has problems with the condensation to solids and ignores the charge-carrier motion. The macroscopic approach has great mathematical difficulties, so the interaction between two half-spaces is the only one to be calculated (Krupp, 1967). 

When charged colloidal particles in a dispersion approach each other such that the double layers begin to overlap (when particle separation becomes less than twice the double layer extension), then repulsion will occur. The individual double layers can no longer develop unrestrictedly, as the limited space does not allow complete potential decay [10, 11]. The potential v j2 half-way between the plates is no longer zero (as would be the case for isolated particles at 00). For two spherical particles of radius R and surface potential and condition x i <3 (where k is the reciprocal Debye length), the expression for the electrical double layer repulsive interaction is given by Deryaguin and Landau [10] and Verwey and Overbeek [11],... 

Other methods to construct a crystal-liquid interface are possible. One example applies to systems under triple-point (three-phase) conditions where a block of crystal is sandwiched in the z direction between regions of empty space. Keeping one half of the crystal region fixed and heating the other so that it melts, a three-phase system can be constructed. If this process is done carefully enough the system should come to equilibrium so that the densities of the various phases adjust to the proper triple-point values. One advantage of this procedure is that the coexistence conditions need not be determined beforehand, but are by-products of the calculation. The method is extremely limited, however, since only one point along the crystal-liquid phase coexistence line can be studied. Also, the method is not applicable to purely repulsive potentials, such as hard spheres or inverse power interactions, which have only one fluid phase. 

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