Commensurate surfaces

It may come as a surprise to some that two commensurate surfaces withstand finite shear forces even if they are separated by a fluid.31 But one has to keep in mind that breaking translational invariance automatically induces a potential of mean force T. From the symmetry breaking, commensurate walls can be pinned even by an ideal gas embedded between them.32 The reason is that T scales linearly with the area of contact. In the thermodynamic limit, the energy barrier for the slider to move by one lattice constant becomes infinitely high so that the motion cannot be thermally activated, and hence, static friction becomes finite. No such argument applies when the surfaces do not share a common period. 

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.

On the other hand, there is also a qualitatively new feature, since now the experimentally observed, so-called (a/6 x a/6) pseudo-commensurate surface oxide structure displayed in Fig.5.11 is not only a kinetic precursor but... 

Further results of the overlap model [61] are as follows (iv) The prefactor that determines the strength of the exponential repulsion has no effect on Fj at fixed normal load L, (v) the lateral force scales linearly with L for any fixed lateral displacement between slider and substrate, (vi) allowing for moderate elastic interactions within the bulk does not necessarily increase Fj, because the roughness may decrease as the surfaces become more compliant, and (vii) the prefactor of F for nonidentical commensurate surfaces decreases exponentially with the length of the common period 5 . This last result had already been found by Lee and Rice [62] for a yet different model system. We note that the derivation of properties (iv) and (v) relied strongly on the assumption of exponential repulsion or hard disk interactions. Therefore one must expect charged objects to behave differently concerning these two points. 

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. 

Fig. 7a - d. Models of commensurate structures observed for I adsorption on W(110). The partially transparent circles represent the I atoms and the black quadrilaterals highlight the unit cells. The solid lines represent the commensurate surface unit cell while the dashed lines the adsorbate unit cell, (a) The structure observed at 0.25 ML. Compressing this structure uniaxially as indicated by the arrows creates incommensurate structures with coverages up to 0.33 ML. (b) The (3x2) structure observed at 0.5 ML. The arrows show how expanding this structure leads to lower coverage structures, (c) The (2x1) structure also observed at 0.5 ML. Compression of the (2x1) unit cell as indicated by the arrows ultimately leads to the c(2x6) structure shown in (d) observed at the saturation coverage of 0.58 ML. 

The balance between these different types of bonds has a strong bearing on the resulting ordering or disordering of the surface. For adsorbates, the relative strength of adsorbate-substrate and adsorbate-adsorbate interactions is particularly important. Wlien adsorbate-substrate interactions dominate, well ordered overlayer structures are induced that are arranged in a superlattice, i.e. a periodicity which is closely related to that of the substrate lattice one then speaks of commensurate overlayers. This results from the tendency for each adsorbate to seek out the same type of adsorption site on the surface, which means that all adsorbates attempt to bond in the same maimer to substrate atoms. 

A superlattice is temied commensurate when all matrix elements uij j are integers. If at least one matrix element uij j is an irrational number (not a ratio of integers), then the superlattice is temied incommensurate. A superlattice can be inconnnensiirate in one surface dimension, while commensurate in the other surface dimension, or it could be mconmiensurate in both surface dimensions. 

Flame spraying is no longer the most widely used melt-spraying process. In the power-feed method, powders of relatively uniform size (<44 fim (325 mesh)) are fed at a controlled rate into the flame. The torch, which can be held by hand, is aimed a few cm from the surface. The particles remain in the flame envelope until impingement. Particle velocity is typically 46 m/s, and the particles become at least partially molten. Upon impingement, the particles cool rapidly and soHdify to form a relatively porous, but coherent, polycrystalline layer. In the rod-feed system, the flame impinges on the tip of a rod made of the material to be sprayed. As the rod becomes molten, droplets of material leave the rod with the flame. The rod is fed into the flame at a rate commensurate with melt removal. The torch is held at a distance of ca 8 cm from the object to be coated particle velocities are ca 185 m/s. 

The adsorption of alkali metals on single crystal surfaces can result in the formation of ordered structures (commensurate or incommensurate super-... 

Studies based on the Frenkel-Kontorova model reveal that static friction depends on the strength of interactions and structural commensurability between the surfaces in contact. For surfaces in incommensurate contact, there is a critical strength, b, below which the depinning force becomes zero and static friction disappears, i.e., the chain starts to slide if an infinitely small force F is applied (cf. Section 3). This is understandable from the energetic point of view that the interfacial atoms in an incommensurate system can hardly settle in any potential minimum, or the energy barrier, which prevents the object from moving, can be almost zero. 

Solid contacts are incommensurate in most cases, except for two crystals with the same lattice constant in perfect alignment. That is to say, a commensurate contact will become incommensurate if one of the objects is turned by a certain angle. This is illustrated in Fig. 30, where open and solid circles represent the top-layer atoms at the upper and lower solids, respectively. The left sector shows two surfaces in commensurate contact while the right one shows the same solids in contact but with the upper surface turned by 90 degrees. Since the lattice period on the two surfaces, when measured in the x direction, are 5 3 A and 5 A, respectively, which gives a ratio of irrational value, the contact becomes incommensurate. 

In reality, static friction is always observed regardless of whether the surfaces in contact are commensurate or not. This raises a new question as to why the model illustrated in Fig. 29 fails to provide a satisfactory explanation for the origin of static friction. 

Fig. 30—Change in commensurability of two surface in contact, on the left is a commensurate contact, on the right the contact becomes incommensurate when the upper body being turned by 90 degrees, the open and solid circles denote the surface atoms of the upper and...

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