Incommensurate surfaces

An example of the velocity dependence of friction is given in Figure 5 for a boundary lubricant confined between two incommensurate surfaces.25 For the given choice of normal pressure and temperature, one finds four decades in sliding velocity for which Eq. [7] provides a reasonably accurate description. 

Simulations of incommensurate surfaces showed a similar dependence on Vi, with first-order instabilities occurring if Vi < Vj, where Vj is some positive, critical value that depends on the degree of mismatch between the lattice constants of the top and bottom surfaces. This process leads to nonvanishing Fk as l o goes to zero. In the case where Vi < V, the atoms are dragged with the wall that exerts the maximum lateral force. It, in turn, leads to friction that scales linearly with the sliding velocity. As a result, the friction force will go to zero with vq. 

Figure 22 Snapshots from the simulations leading to the friction coefficients shown in Figure 21. From left to right Atomically flat commensurate, atomically flat incommensurate, rough commensurate, and rough incommensurate geometries. Only the flat incommensurate surfaces remain undamaged, resulting in abnormally small friction coefficients. Reproduced with permission from Ref. 85.

Based on the discussion in earlier sections of this chapter, one may expect atomically flat incommensurate surfaces to be superlubric. Indeed the first suggestion that ultra-low friction may be possible was based on simulations of copper surfaces.6,7 Furthermore, the simulations of Ni(100)/(100) interfaces discussed in the previous section showed very low friction when the surfaces were atomically flat and misoriented (see the data for the atomically flat system between 30° and 60° in Figure 21). In general, however, it is reasonable to assume that bare metals are not good candidates for superlubric materials because they are vulnerable to a variety of energy dissipation mechanisms such as dislocation formation, plastic deformation, and wear. 

Fig. 5 Atomic configurations representing a commensurate surface (on the left) and incommensurate surface (on the right). Sliding friction is predicted and experimentally shown to be different between these cases.

For both models, it is possible to calculate the energy landscape generated by relative translation analytically. Both times it is found that (i) Tj = Fg/A is independent of A if the two periods of the two surfaces match, (ii) Tj decreases as A if the two surfaces are random, and (iii) Tj is zero if the surfaces are incommensurate. Contributions from the circumference of finite contacts between incommensurate surfaces yield contributions to that vanish with a higher power law than A (see also Fig. 6). 

Both the above simulations considered identical tips and substrates. Failure moved away from the interface for geometric reasons, and the orientation of the interface relative to easy slip planes was important. In the more general case of two different materials, the interfacial interactions may be stronger than those within one of the materials. If the tip is the weaker material, it will be likely to yield internally regardless of the crystallographic orientation. This behavior has been observed in experiments between clean metal surfaces where a thin tip is scraped across a flat substrate [31]. When the thin tip is softer than the substrate, failure is localized in the tip, and it leaves material behind as it advances. However, the simulations considered in this section treated the artificial case of a commensurate interface. It is not obvious that the shear strength of an interface between two incommensurate surfaces should be sufficient to cause such yield, nor is it obvious how the dislocation model of Hurtado and Kim applies to such surfaces. 

Sprensen et al. [63] also examined the effect of incommensurability. The tip was made incommensurate by rotating it about the axis perpendicular to the substrate by an angle 0. The amount of friction and wear depended sensitively on the size of the contact, the load, and 0. The friction between large slabs exhibited the behavior expected for incommensurate surfaces There was no wear, and the kinetic friction was zero within computational accuracy. The friction on small tips was also zero until a threshold load was exceeded. Then elastic instabilities were observed leading to a finite friction. Even larger loads lead to wear like that found for commensurate surfaces. 

Physisorbed molecules also provide a natural explanation for the logarithmic increase in kinetic friction with sliding velocity that is observed for many materials and represented by the coefficient A in the rate-state model of Eq. (5). Figure 16 shows calculated values of tq and a as a function of log for a sub monolayer of chain molecules between incommensurate surfaces [195]. The value of To becomes independent of v at low velocities. The value of a, which... 

As mentioned earlier, most studies of field interactions with liquid crystals are done using thin films with a well-defined initial state, usually a monodomain or a thin film with a simple distortion induced by incommensurate surface anchoring. These conditions simplify observation and theoretical analysis. However, most liquid crystal materials that are not specially prepared contain topological defects that are very important to their response to external fields. One class of defect commonly observed in nematics is the disclinalion line. At a disclination line the director field is ill defined. The director field turns around the disclination line a multiple of half-integer times. Several disclination lines are shown in Fig. 8. 

In the case quoted above, low friction between two incommensurable surfaces is explained by rotational anisotropy. A layer is shifted by a certain angle compared to the other layer, thus allowing an easier slip, with the atoms no longer coinciding (Figure 2.50(b)). The second case to be considered is the friction of two layers with different interatomic distances (Figure 2.50(c)). This condition allows an absence of coincidence between the atoms. In this section, friction experiments of IF mixtures are developed in order to study the possibility of the sliding of sheets with different interatomic distances. 

Figure 4.37 Incommensurate surface reconstruction of FeAI(l 10)(after [108]). The top layer of FeAl2 stoichiome-Xry with its unit mesh (white lines) is written as (J 4 ) in terms of the mesh (black line) of the unreconstructed second layer.

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