Surface dispersion

There are numerous techniques which provide information related to the surface energy of solids. A large array of high-vacuum, destructive and non-destructive techniques is available, and most of them yield information on the atomic and chemical composition of the surface and layers just beneath it. These are reviewed elsewhere [83,84] and are beyond the scope of the present chapter. From the standpoint of their effect on wettability and adhesion, the property of greatest importance appears to be the Lifshitz-van der Waals ( dispersion) surface energy, ys. This may be measured by the simple but elegant technique of... 

Metal surface cleaning, by virtue of (limited) sequestration, dispersing, surface-acting, and detergent properties. 

The analysis demonstrates the elegant use of a very specific type of column packing. As a result, there is no sample preparation, so after the serum has been filtered or centrifuged, which is a precautionary measure to protect the apparatus, 10 p.1 of serum is injected directly on to the column. The separation obtained is shown in figure 13. The stationary phase, as described by Supelco, was a silica based material with a polymeric surface containing dispersive areas surrounded by a polar network. Small molecules can penetrate the polar network and interact with the dispersive areas and be retained, whereas the larger molecules, such as proteins, cannot reach the interactive surface and are thus rapidly eluted from the column. The chemical nature of the material is not clear, but it can be assumed that the dispersive surface where interaction with the small molecules can take place probably contains hydrocarbon chains like a reversed phase. 

OM Identification of compounding ingredients, particle size, dispersion, surface blooming, image analysis... 

Aid in the uniform dispersion of additives. Make powdered solids (e.g. particulate fillers with high energy and hydrophilic surface) more compatible with polymers by coating their surfaces with an adsorbed layer of surfactant in the form of a dispersant. Surface coating reduces the surface energy of fillers, reduces polymer/filler interaction and assists dispersion. Filler coatings increase compound cost. Fatty acids, metal soaps, waxes and fatty alcohols are used as dispersants commonly in concentrations from 2 to 5 wt %. 

The attractive force (F) is dependent on the Hamaker constant and the shortest distance between the particles, z. F may be decreased by decreasing A or increasing z. Theoretically, the Hamaker constant can be decreased by decreasing the densities of the two interacting particles. Since the separation distance plays a significant role in van der Waals attraction, any means to increase this distance will reduce the attractive force and increase the ease of dispersion. Surface roughening and the use of spacer particulates can increase interparticulate separation with the improved particle dispersion. 

We should first correct the wavevector inside the crystal for the mean refractive index, by multiplying the wavevectors by the mean refractive index (1 + IT). This expression is derived from classical dispersion theory. Equation (4. 18) shows us that is negative, so the wavevector inside the crystal is shorter than that in vacuum (by a few parts in 10 ), in contrast to the behaviom of electrons or optical light. The locus of wavevectors that have this corrected value of k lie on spheres centred on the origin of the reciprocal lattice and at the end of the vector h, as shown in Figure 4.11 (only the circular sections of the spheres are seen in two dimensions). The spheres are in effect the kinematic dispersion surface, and indeed are perfectly correct when the wavevectors are far from the Bragg condition, since if D 0 then the deviation parameter y, 0 from... 

For completeness we note that the dispersion surface is only strictly hyperbohc when the circles are accurately represented by straight lines. In three... 

Figure 4.11 The kinematic dispersion surface. The circles centred on the origin O and the relp H, with radius (1/ vacuum )(1+ 2) represent, in the plane shown, the allowable wavevectors in the crystal far from the diffracting condition. A section of the Ewald sphere is shown...

Figure 4.12 Magnification of the region near Lq in Figure 4.11. L is the Lorentz point (the Lane point corrected for the mean refractive index). A is one of the tie-points selected on the dispersion surface, and o and are the deviation parameters at that point (shown on branch 2)...

Any point on either branch of the dispersion surface is an equally good solution of the Maxwell equations. However, the only points that will be selected are... 

The above discussion has in effect been for materials with zero absorption, but this affects only the intensities. The construction of the dispersion surface and the wavevector matching are all performed on the real part of the wavevectors. When absorption is considered, the reflectivity in the Bragg case falls below 100% but it can still be over 99% for a low-absorption material such as silicon. 

Figure 4.14 The selection of tie-points on the dispersion surface for the transmission (Lane) case, using the construction of Figure 4.13 (branch 2 is on the left, branch 1 is on the right)...

Figure 4.17 The standing wavefields set up in symmetrical Laue-case reflections. The nodes of wavefields from branch 2 of the dispersion surface lie on the atomic planes and the wavefields experience low absorption. The antinodes of wavefields from branch 1 of the dispersion surface lie on the atomic planes and the wavefields experience high absorption...

Figure 4.19 The range of strong diffraction in (a) the Laue case, (b) the Bragg case. The heavy lining shows the region of the dispersion surface excited as the rocking curve is traversed...

In Fignre 4.19 the hyperbolic region is shown separately to clarify its dependence on the diameter of the dispersion surface. Since K o is large, we may write... 

Figure 4.20 The dependence of the range of strong diffraction on the diameter of the dispersion surface, (a) Symmetric reflection, (b) Asymmetric reflection...

For a perfect, uniform crystal, whether in bulk or as a thin layer, the Takagi-Taupin equations can be solved exactly as given in the next section. For the general case with multiple layers, however, it is necessary to integrate them numerically. The concepts of the dispersion surface are lost, and we cannot tell directly in which directions wavefields are propagating. They do give directly the intensities of the direct and diffracted beams emerging from the crystal, and all interference features are preserved. 

We see that very close to the Bragg condition, where the dispersion strrface is highly cttrved, R K and the crystal acts as a powerful angrtlar amplifier. A reaches 3.5xl0 in the centre of the dispersion surface for sihcon in the 220 reflection with MoK radiation. Far from the centre, the dispersion strrface becomes asymptotic to the spheres about the reciprocal lattice points and A approaches unity. Thus when the whole of the dispersion strrface is excited by a spherical wave, owing to the amplification close to the Bragg condition, the density of wavelields will be veiy low in the centre of the Borrmann fan and... 

Let us consider two points F and G between which the local reciprocal lattice varies from g to + rfg. If the deformation is small the shape of the dispersion surface does not change and only a displacement of the hyperbolae results. We can consider this as a rotation about the origin of reciprocal space,... 

Instead of considering the dispersion surface as a variable and the reciprocal lattice as invariant, it is usually easier to consider the reciprocal lattice as the variable. Then equation (8.30) determines the variation of the amplitude ratio of the reflected and transmitted components as the wavefield propagates through the crystal. The ratio R characterises a particular tie-point on the dispersion surface and if R varies the tie-point must migrate along the dispersion surface branch. This results in a change in the intensity of the transmitted and diffracted... 

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