Frenkel-Kontorova model

J. Frenkel, T. Kontorova. J Phys USSR 7 137, 1939 E. Allroth, H. Muller-Krumbhaar. Phys Rev A 27 1515, 1983 S. Stoyanov, H. Miiller-Krumbhaar. Resonance-induced cluster mobility Dynamics of a finite Frenkel-Kontorova model. Surf Sci 159 49, 1985. 

The comparison reveals that in addition to the activity of individual atoms, the next-neighbor interaction and collective atomic motion must play an important role in creating friction. This mechanism can be investigated more efficiently via the Frenkel-Kontorova model. 

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. 

Finally we have shown the possibility to build a thermal diode which exhibits a very significant rectifying effect in a very wide range of system parameters. Moreover, based on the phenomenon of negative differential thermal resistance observed in the thermal diode, we have built a theoretical model for a thermal transistor. The model displays two basic functions of a transistor switch and modulator/amplifier. Although at present it is just a model we believe that, sooner or later, it can be realized in a nanoscale system experiment. After all the Frenkel-Kontorova model used in our simulation is a very popular model in condensed matter physics(Braun and Kivshar, 1998). 

The essential properties of incommensurate modulated structures can be studied within a simple one-dimensional model, the well-known Frenkel-Kontorova model . The competing interactions between the substrate potential and the lateral adatom interactions are modeled by a chain of adatoms, coupled with harmonic springs of force constant K, placed in a cosine substrate potential of amplitude V and periodicity b (see Fig. 27). The microscopic energy of this model is ... 

Figure 7. Schematic representation of the one dimensional Frenkel Kontorova Tomlinson model, a and b denote the lattice constant of the upper sohd and the substrate, respectively. The substrate is considered rigid, and its center of mass is kept fixed. In the shder, each atom is coupled with a spring of lateral stiffness to its ideal lattice site and with a spring of stiffness 2 to its neighbor. The PT model is obtained for 2 0, while the Frenkel Kontorova model corresponds to k = 0. We will drop the subscripts for these two cases since a single spring is relevant.

There have been relatively few experimental tests of the Frenkel Kontorova model because of the difficulty in making sufficiently flat surfaces and of removing chemical contamination from surfaces. In one of the earliest experiments, Hirano et al. [126] examined the orientational dependence of the friction between atomically flat mica surfaces. They found as much as an order of magnitude decrease in fi iction when the mica was rotated to become incommensurate. Although this experiment was done in vacuum, the residual friction in the incommensurate case may have been due to surface contamination. When the surfaces were contaminated by exposure to air, there was no significant variation in friction with the orientation of the surfaces. 

Applications The HMM has been applied to the study of friction between two-dimensional atomically flat crystal surfaces, dislocation dynamics in the Frenkel-Kontorova model (i.e., considering a one-dimensional chain of atoms in a periodic potential, coupled by linear springs ), and crack propagation in an inhomogeneous medium. ... 

Brown, O. M., Kivshar, J. S. (2004). The Frenkel-Kontorova model. Concepts, Methods, and Applications Springer, 519 p. 

Most theories of structural superlubricity are based on the Prandtl-Tomlinson model or the more advanced Frenkel-Kontorova model [1043, 1044], in which the single atom/tip is replaced by a chain of atoms coupled by springs. However, Friedel and de Gennes [1045] noted recently that correct description of relative sliding of crystalline surfaces should include the motion and interaction of dislocations at the surfaces. This concept was taken up by Merkle and Marks [1045] and generalized using the well-established coincident site lattice theory and dislocation drag from solid-state physics. 

Braun, O.M. and Kivshar, Y.S. (2004) The Frenkel—Kontorova Model Concepts, Methods, Applications, Springer, Berlin. 

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