Sphere-plane interactions

Evaluation of K Is possible using the forms for sphere-plane interactions (11) (a simplification necessitated by the otherwise complicated forms needed to account for pore wall curvature (35)). 

Interaction Energy Expressions. Previous papers (8,10,12,13) have used exact sphere-plane interaction energy expressions to approximate the sphere-cylinder Interaction. In this work, these exact expressions were replaced with recently published approximate expressions. For the double layer repulsion, this avoided the inconvenience and Inaccuracy of using tabular values ( ) while for the van der Waals attraction, using the approximate solution simplified the programing task. 

PROBLEM LI.I4 Obtain Eq. (LI.42) for a sphere-plane interaction from Eq. (LI.40) for a sphere-sphere interaction. 

Bell el al. (17) have found that, when wz > 5 and kx < 2, the error in using Derjaguin s approximation is less than 10%. This was based on the interaction of two spherical particles. For sphere-plane interactions, the error should be even less. To calculate the rates according to Eqs. [l] through [3], one needs accurate estimates of m. near 3 mOXf which is typically of the order of jc-1 hence... 

Lifshitz more recent continuum analysis of the dispersion interactions leads to approximately the same distance dependence. For the double-layer repulsive potential for the sphere-plane interaction energy we use the expression... 

Generally speaking, the interaction forces between two dielectric bodies with multipolar charge distributions are extremely difficult to compute. No attempt to review the literature is made here. Rather, a recent paper which addresses this problem in some generality for sphere-sphere and sphere-plane interactions among dielectrics is cited, and some results indicative of the importance of considering distributed, rather than point, charges and multipoles are described. 

The derivation of the expression for Gr involves numerical approximations that will not be presented here. The interested reader is referred to the original work for specifics. Values for sphere-sphere and sphere-plane interactions are presented in Table 4.6. The value x relates the distance of separation, r, to the particle radius R. 

In fact, from Equation 6.36, other cases such as sphere-plane interaction can be easily found. 

For the sphere-plane interactions, the exact results can be derived by substituting a, = a and letting 02 oo. In this way, for the nonretarded and retarded interactions one obtains... 

These two limiting forms show an equivalence between the closely approaching-sphere and the planar-block interactions. Ask, what area of the plane-plane interaction is equivalent to the sphere-sphere interaction Equate —[AHam/(12 /2)] L2 = [—(Anam/l2)] (R//) to see that two spheres look like two planes of area L2 = Ibr/. 

Conversely, if we compare the interaction free energy -(Alm/2m/12)(R/Z) of two equal-radius spheres with that of two circular parallel patches of radius R, area jrR2, on facing planes of the same materials in the same medium, we have [-( im/2m/12jr/2)]7rR2 = [-(Alm/2m/12)](R/Z)(R/Z). The plane-plane interaction per unit area is stronger by a variable factor (R/Z) 1. 

It is possible to understand how a potential or field effect on A5 could originate from the influence of the known field-dependent solvent orientation in the inner region of the double layer on the solvation-shell reorganization in the activation process through solvation co-sphere/solvent co-plane interaction. Quantitative formulation of this effect has not yet been made. 

At very small separation distances, the value of Gr becomes very small indicating that the velocity of approach of the moving particle to the second particle or surface becomes small as well. Such an effect may be expected to become significant in contexts such as particle flocculation, emulsion stabihty, and particle deposition onto surfaces. In summary, these hydrodynamic effects will be repulsive for approaching particles or surfaces and attractive for receding systems. It can be seen from Table 4.6 that the effect is much greater in the case of sphere-plane approach than that of sphere-sphere interaction. 

Nevertheless the main difficulty of the oscillating tip results concerns an accurate description of the viscoelastic response of the nanoprotuberance under the action of the attractive force between the sample and the tip. Many points remain open for the interpretation of the atypical variations of the amplitude. They are mostly linked to the multiple crude assumptions made with the model it is difncult to evaluate a local response when both the tip shape and effective force are unknown. One could discuss for example the sphere-plane surface interaction used, or ask about the local stiffness of the polymer, or question the unique relaxation time which is a to simple polymer viscoelastic response. Among our fit parameters, the sample properties should be independent of the drive amplitudes used. The variations reported in figure 15 indicate that a simple rheological model is unable unable to describe the whole growth process of the nanoprotuberances. 

For calculating the interactions for the sphere-plane geometry one can also use the Deqaguin model, expressed by Eqs. (70) and (72) by substituting Gjj = a. 

Even if Re, is small, the stress field around a sphere can interact with the wall, causing it to migrate inward. Ho and Leal (1974) have analyzed this problem extensively for both drag (Couette) and pressure-driven (plane Poiseuille) flow. They show that if the spheres are small enough and the flow rate is low. Brownian motion can keep the particles uniformly distributed. Their criteria for neglecting migration are... 

Following the derivation of the sphere-sphere interaction, we sketch that for a sphere interacting with a charged plane. Putting the dividing plane at Hq/2, where Hq is the closest sphere-plane distance, and with the charged plane situated at x = 0, we assume a potential of the form... 

The general picture of a tip interacting with a sample has been simplified to a paraboloidal object (e.g., the tip) of radius r interacting with an elastic, infinitely thick, infinitely planar film (e.g., the sample) (Scheme 8). Since the curvature can be much higher than the sample (especially when it is film like), this sphere -plane assumption is valid. 

The results obtained above can be directly applied to calculate the force between two different spheres. In the general case, the two interacting spheres can have different contact angles, 1 and 2, and different radii, l i and R2. The height coordinate h describes in this case the distance between the surface of sphere 1 and sphere 2. Rather than considering the shape of each sphere explicitly, we transform the geometry and consider the equivalent case of plane interacting with a sphere of... 

Hertz [27] solved the problem of the contact between two elastic elliptical bodies by modeling each body as an infinite half plane which is loaded over a contact area that is small in comparison to the body itself. The requirement of small areas of contact further allowed Hertz to use a parabola to represent the shape of the profile of the ellipses. In essence. Hertz modeled the interaction of elliptical asperities in contact. Fundamental in his solution is the assumption that, when two elliptical objects are compressed against one another, the shape of the deformed mating surface lies between the shape of the two undeformed surfaces but more closely resembles the shape of the surface with the higher elastic modulus. This means the deformed shape after two spheres are pressed against one another is a spherical shape. 

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