Relatively Isotropic Crystals

Substances in this category include Krypton, sodium chloride, and diamond, as examples, and it is not surprising that differences in detail as to frictional behavior do occur. The softer solids tend to obey Amontons law with /i values in the normal range of 0.5-1.0, provided they are not too near their melting points. Ionic crystals, such as sodium chloride, tend to show irreversible surface damage, in the form of cracks, owing to their brittleness, but still tend to obey Amontons law. This suggests that the area of contact is mainly determined by plastic flow rather than by elastic deformation. 

Diamond behaves somewhat differently in that n is low in air, about 0.1. It is dependent, however, on which crystal face is involved, and rises severalfold in vacuum (after heating) [1,2,25]. The behavior of sapphire is similar [24]. Diamond surfaces, incidentally, can have an oxide layer. Naturally occurring ones may be hydrophilic or hydrophobic, depending on whether they are found in formations exposed to air and water. The relation between surface wettability and friction seems not to have been studied. 

A number of substances such as graphite, talc, and molybdenum disulfide have sheetlike crystal structures, and it might be supposed that the shear strength along such layers would be small and hence the coefficient of friction. It is true 

The structurally similar molybdenum disulfide also has a low coefficient of friction, but now not increased in vacuum [2,30]. The interlayer forces are, however, much weaker than for graphite, and the mechanism of friction may be different. With molecularly smooth mica surfaces, the coefficient of friction is very dependent on load and may rise to extremely high values at small loads [4] at normal loads and in the presence of air, n drops to a near normal level. 

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Whereas in many metals with relatively simple and isotropic crystal structures the parameter / has values between 0.5 and 1, it can have much more extreme values in materials in which the mobile species move through much less isotropic structures with 1-D or two-dimensional (2-D) channels, as is often the case with insertion reaction electrode materials. As a result, radiotracer experiments can provide misleading information about self-diffusion kinetics in such cases. 

There is a similar effect with diamond. In air we have //, = 0.1. In a vacuum, after heating to remove contamination and an oxide layer, //, rises approximately seven times. This can be explained again with an oxide coating. Many materials, in particular relatively isotropic, soft crystals such as krypton or sodium chloride, tie in the range 0.5-1.0. 

Note that AR = 1 for an isotropic crystal. The physical significance of this ratio is that it represents a measure of the relative response of cubic crystals to two different types of shear strain C44 is an indicator of shear resistance in the <100> directions on 100 planes, whereas the quantity (cn — ci2)/2 is an indicator of shear resistance in the <110> directions on 110 planes. Values of anisotropy ratios for various cubic crystals are listed in Table 3.1. Among cubic metals, W is essentially isotropic and A1 is very close to being so. Many materials listed in Table 3.1 have anisotropy ratios which depart significantly from unity. 

When a field is applied to an isotropic crystal, the relative impermeability relation reduces to... 

When highly branched polyethylene samples, either (dendritic) low density polyethylene or (comblike) very low density polyethylene, crystallize from oriented melts they do not form cylindrites because they contain insufficient linear chain segments to generate microfibrillar nuclei. In such cases the relatively slow crystallization kinetics and low crystallization temperature permit the molecules a relatively long time to relax prior to solidification. The lamellae that form under these circumstances are well separated from one another and do not share a common axis. The resultant semicrystalline morphology is similar to that of low density samples crystallized from an isotropic melt. 

This is the H.A. Lorentz-L. Lorentz formula. The right-hand side of this equation is a molar refraction R. It is fair for gases, nonpolar liquids and isotropic crystals (with cubic lattices). It is approximately appUcable to nonpolar liquids at relatively high frequencies when the orientation polarization is not exhibiting. 

Polarization effects are another feature of Raman spectroscopy that improves the assignment of bands and enables the determination of molecular orientation. Analysis of the polarized and non-polarized bands of isotropic phases enables determination of the symmetry of the respective vibrations. For aligned molecules in crystals or at surfaces it is possible to measure the dependence of up to six independent Raman spectra on the polarization and direction of propagation of incident and scattered light relative to the molecular or crystal axes. 

Wood and Blundy (2001) developed an electrostatic model to describe this process. In essence this is a continuum approach, analogous to the lattice strain model, wherein the crystal lattice is viewed as an isotropic dielectric medium. For a series of ions with the optimum ionic radius at site M, (A(m))> partitioning is then controlled by the charge on the substituent (Z ) relative to the optimum charge at the site of interest, (Fig. 10) ... 

Table 9.7 shows the results of the calculations of average parameters of PBU/P for isotropic DRP, fulfilled by Serra [134] and Meijering [152], Serra used VD-method while Meijering used the Johnson-Mehl s (JM) statistical model [150] of simultaneous growth of crystals until the total filling of the whole free space was accomplished. The parameter Nv in the table is the number of PBUs in a unit of system volume, thus Nv 1 is the mean volume of a single PBU, which is related to the relative density of the packing (1—e) with an interrelation... 

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