Inter-lamellar shear

Although a lamellar crystal structure is favourable for solid lubrication, the inter-lamellar spacing and the nature of the inter-lamellar bonding are of major importance in determining the resistance to inter-lamellar shear, and therefore the sliding friction, of lamellar compounds. 

Strong electronic (covalent or electrovalent) bonding is desirable within a crystal lamella, to provide structural strength and to ensure that when shear forces are applied to the crystal, shear takes place between lamellae, and not within them. Conversely, strong bonding between lamellae is undesirable, as it leads to high inter-lamellar shear resistance and high friction. Ideally, the inter-lamellar forces are limited to weak van der Waals forces, and the inter-lamellar space is often called the "van... 

It has been known since the middle of the nineteenth century that atoms or molecules can be introduced between the carbon layers in graphite, and more recently it has been recognised that many other lamellar crystalline materials will behave in the same way. This phenomenon of "intercalation" results in modification of many of the properties of the basic crystalline material. In particular it leads to an expansion of the inter-lamellar gap, but the effect on inter-lamellar shear resistance will also depend on any effects on bonding which may arise. 

In the case of the equivalent niobium compounds, the same electronic effects are not present. He postulated that in pure stoichiometric niobium disulphide this results in poor lubrication. When good lubrication behaviour is observed, it is probably caused by additional niobium atoms intercalated between the lamellae, which contribute non-bonding electrons. On the basis of this theory, non-bonded atoms intercalated between the lamellae can increase the inter-lamellar spacing, whereas bonded intercalated atoms increase the resistance to inter-lamellar shear, and therefore the friction. However, an alternative interpretation is that certain intercalated atoms alter the interaction between the niobium atoms, allowing rearrangement to the 2H structure of molybdenum disulphide, and it is the favourable structure which provides good lubrication performance. 

Jamison found that low concentrations of intercalated copper or silver in niobium disulphides and diselenides promoted good lubricating performance. Higher concentrations increased the resistance to inter-lamellar shear, and therefore the friction, but improved high temperature performance due to the reduced intracrystalline shear and some sacrificial oxidation of the intercalated metals. 

Gupta and Ward found modulus crossover points in drawn and annealed sheets of low density po yethylene and also in the b-c and it-b sheets similar to that observed by Takayanagi in drawn and annealed high density polyethylene. The crossover points were attributed to inter-lamellar shear. When the tensile stress is applied along the draw direction ( measurements in a drawn and annealed sheet) or along either the c or a direction in the special structure sheets ( , and Ea measure-... 

The cross-over point in high density polyethylene occurs above the a-relaxation, which it is proposed is an inter-lamellar shear relaxation. This highlights a major difference between high density polyethylene and low density polyethylene, where the a-relaxation is attributed to the c-shear process which gives rise to the anomalous mechanical anisotropy. 

The success of the model for the loss anisotropy led Owen and Ward to use equivalent assumptions to calculate the modulus anisotropy. The loss anisotropy calculations assume simple shear between the lamellae only, which for parallel lamellae sheet would imply that inter-lamellar shear is not activated when the tensile stress is applied along the initial draw direction i.e. parallel to the lamellar plane normals). A very appreciable fall in tensile modulus was. however, observed in this case, although as e.xpected by comparison with the corresponding loss factor in... 

The most compliant directions, and thus those in which tan is a maximum, occur when the maximum resolved shear stress is parallel with the lamellar planes. An inter-lamellar shear mechanism is suggested, the mechanical anisotropy of the j relaxation being determined by the configuration of the lamellae, and not by the molecular orientation within the lamellae, which differs between annealed, a-c and b-c sheets. Differences in the magnitudes of the relaxation peaks for different types of specimen will depend on the spread of deviations from the mean angle of the lamellae, and on the lamellae in the annealed sheet being arranged in cones around the stretch direction rather than in flat layers. 

Support for the inter-lamellar shear hypothesis is provided by the isotropy of the dielectric / process, which shows that the mechanical anisotropy cannot be a consequence of an ordered inter-lamellar material. 

The original analysis of the relaxation mechanism by Stachurski and Ward was qualitative, and based on simplifying assumptions such as all lamellae at 45° to the draw direction. More recently Davies et have presented a quantitative theory for mechanical isotropy of cylindrically symmetric specimens based on an inter-lamellar shear mechanism. As will be discussed later the a relaxation in high density polyethylene shows a similar anisotropy to the /) relaxation in low density polyethylene, and is therefore proposed as being a consequence of inter-lamellar shear. 

McCrum and Morris also compared torsion data for samples originally with an oriented surface layer and then with the layer removed. They did resolve the a peak, which they found diminished by removal of the oriented layer. Interpretation was made in terms of a two phase model of Iwayanagi involving lamellar boundary slip. It was concluded that the relaxation was caused by an inter-lamellar shear process. 

The relaxation associated with interlamellar shear has been further investigated by Stachurski and Ward, who confirmed and extended Takayanagi s earlier measurements. The anisotropy of the a peak in annealed samples, with tan 5q > tan 45 > tan 9o, was similar to that of the P relaxation in low density polyethylene, being attributed to an inter-lamellar shear mechanism. Evidence of this process was present also in cold drawn samples, but was less clearly defined. 

The authors commented that their results contrasted with those for low density polyethylene, in which both inter-lamellar shear and slip between folded chains within lamellae were found to be reversible, but Owen and Ward have shown that in low density polyethylene which has been annealed the inter-lamellar slip contribution is enhanced. 

For both cold and hot drawn specimens Eo > 4 5 > 90 over the complete temperature range. This suggested that an inter-lamellar shear process was not taking place, presumably because inter-lamellar ties were restraining the non-crystalline regions. 

Samples annealed at 145°C showed an increase of Eoo relative to Eo and 45, giving Eo > Ego > 45 above 20°C and for specimens annealed at 158°C 90 > 0 > 45 above 40°C. The low value of 45 implied that a shearing process was taking place, and evidence favoured an inter-lamellar shear mechanism, which could occur presumably because of relaxation of intercrystalline tie molecules. 

Driscoll, T.P., Nakasone, R.H., Szczesny, S.E., Elliott, D.M., and Mauck, R.L. (2013) Biaxial mechanics and inter-lamellar shearing of stem-cell seeded electrospun angle-ply laminates for annulus fibrosus tissue engineering. / Orthop. Res., 31 (6), 864-870. 

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