Half-space interaction

The advantage of the microscopic approach, used almost exclusively in the literature, is that analytical formulas can be derived for complicated geometries of the interacting particles, including the case of rough surfaces. In contrast, the more rigorous macroscopic approach can only be applied for half-space interactions. 

Similarly, the assumption that the contact area is small enough that the particle can be represented by an elastic half space allows the radii of the two contacting particles to be combined into a single effective radius that represents how the contacting shapes interact. 

Conjugated chains, 14, 46 Correlation diagrams, 44, 50 Cyclobutadiene, 171 Cyclobutane, 47, 222 orbital ordering, 26 through-space interactions, 26 Walsh orbitals, 27 Cyclobutene, 200 Cyclohexane, 278 Cyclohexene (half-boat), 274 Cyclopen tadiene, 225 Cvclopen tadienone, 269 Cyclopentadienyl anion, 237 Cyclopentane, 254 Cyclopen ten e, 241 Cyclopropane, 41, 47, 153 construction of orbitals, 19, 22 Walsh orbitals, 22, 36, 37 Cyclopropanone, 48, 197 bond lengths, 38 Cyclopropen e, 49, 132 reactivity, 40... 

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 ... 

For historical reasons, we write the interaction free energy G(Z) between the plane-parallel half-spaces of Fig. Ll.l in terms of a "Hamaker coefficient" AHam so as to put the interaction per unit area in a form... 

To see how these different e(i ) functions combine to create an interaction, consider the case of two hydrocarbon half-spaces A = B = H across water medium m = W. First plot eH(/ ) and ew(i ) as continuous functions [see Fig. LI.22(a)]. These are plotted at only the discrete sampling frequencies at which they are to be evaluated a log plot in frequency shows how compression of the arithmetically even spacing fit- = 0.159 n eV in index n works with the varying difference in eH(/ ) and ew(i ) [see Fig. LI. 22(b)],... 

Think now of the interaction between half-spaces, where e changes arbitrarily in each body (see Fig. LI.30). 

Conceptually these pairwise interactions emerge automatically from expressions for the interaction between half-spaces A and B across m. In this case, A is a suspension of particles a whose incremental contribution to eA is a (if) B, of particles b whose incremental contribution to eB is p(iii). The interaction, b(z) is what emerges when eA = em(i ) +Naa(i ), eB = em(i ) + Nbp(i ) so that... 

In the same spirit as that of the extraction of small-particle interactions, it is possible to specialize the general expression for the interaction of planar half-spaces in order to formulate interactions between point particles and substrates (see Fie. LI.44). 

To think efficiently, though formally, again use the simplest expression for two interacting half-spaces across a medium. Imagine that regions A and B are vapors that interact across a vacuum "medium" ... 

L2.1.B. Force and energy G GAmB(0 Free energy of interaction. Planar systems between half-spaces A and B across medium m of variable thickness Z. 

The previous result can be immediately specialized to the case of a layer of finite thickness interacting with a previously existing stack of N + 1 layers. Let half-space L, as well as all materials B, have the same dielectric properties as medium m. Let material A have the same properties as material B ... 

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) ... 

The general formula (Table R2.a.l) for the interaction of half-space A and half-space B coated with a layer of material Bi of thickness t> has the same outward form as the original Lifshitz formula for the interaction of two half-spaces. To recognize its inner possibilities, consider the single-layer interaction in terms of different variables of integration ... 

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... 

Hamaker summation for the case of a half-space A interacting with a finite slab of material B... 

For the interaction of planar slabs, the Hamaker approach entails integration over finite ranges of zA or zB. For the interaction between a half-space A and a parallel slab of B of finite thickness b, this procedure is equivalent to subtracting from E(l) = — (AHam/12 /2) an amount — [AHam/12 (Z +b)2] (see Fig. L2.10). This subtraction yields a form equivalent to the equation of Table P.2.b.3 (see Fig. L2.ll) ... 

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. 

Begin with the general form for interaction energy per unit area between half-spaces A and B ... 

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. 

Here we work out several cases for which there is an arbitrary continuously varying e = e(z) in a layer of fixed thickness between each outer half-space and the central medium of variable thickness 1. Although there is no general closed-form solution for arbitrary e (z), it is possible to derive a mathematical procedure for evaluation. For clarity, consider successively more difficult situations nonretarded interactions, symmetric and nonsymmetric geometry, retarded, nonsymmetric. 

As in the L m R geometry of the Lifshitz interaction between planar half-spaces, fluctuations in potential have the form of waves in the x,y directions parallel to the surfaces and an exponential f(z) that dies away from the surfaces.30 The general form is like that used in the derivation of the Lifshitz result. For each radial wave vector iu + jv, the potential [Pg.315]

Solution Feed Anam = 47 x 10 21 J, the Hamaker coefficient for tetradecane across vacuum, into the interaction energy per area — (AHam/l 2 r/2) of plane-parallel half-spaces of separation / = 3 x 1(T9 m ... 

Lei us first address some quantitative issues, using the more simple Hamaker approach. If one assumes pairwise additivity, the interaction between a sphere of kind (1) immersed in a fluid of kind (3) and a planar half-space of kind (2) is given by [27] ... 

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