Coincidence-site lattice theory

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. 

The grain boundary energy 7gb should be proportional to . For small values of high coincidence occurs and the number of broken bonds can be minimized. = 1 corresponds to complete coincidence of the ideal crystal. Experimentally it was found that the correlation between 7Gb and is not that simple due to volume expansions or translations at the grain boundaries. A principal problem of the coincident site lattice model is that, even arbitrarily small variations of the lattice orientation lead mathematically to a complete loss of coincidence. This is physically not reasonable because an arbitrarily small deviation should have a small effect. This problem was solved by the O-lattice theory [343], For a comprehensive treatment of solid-solid interfaces and grain boundaries, see Refs. [344,345],... 

The simple CSL model is directly applicable to the cubic crystal class. The lower symmetry of the other crystal classes necessitates the more sophisticated formalism known as the constrained coincidence site lattice, or CCSL (Chen and King, 1988). In this book we treat only cubic systems. Interestingly, whenever an even value is obtained for E in a cubic system, it will always be found that an additional lattice point lies in the center of the CSL unit cell. The true area ratio is then half the apparent value. This operation can always be applied in succession until an odd value is obtained thus, E is always odd in the cubic system. A rigorous mathematical proof of this would require that we invoke what is known as O-lattice theory (Bollman, 1967). The O-lattice takes into account all equivalence points between two neighboring crystal lattices. It includes as a subset not only coinciding lattice points (the CSL) but also all nonlattice sites of identical internal coordinates. However, expanding on that topic would take us well beyond the scope of this book. The interested reader is referred to Bhadeshia (1987) or Bollman (1970). 

There are two well-known models of GBs that were developed primarily from studies of metals by considering the relative misorientation of the adjoining grains. These are the coincidence-site lattice (CSL) theory and the dis-placement-shift-complete lattice (DSCL). We first define two special quantities E and E Imagine two infinite arrays of lattice points (one array for each crystal) they both run throughout space and have a common origin. For certain orientations, a fraction of the points in each lattice will be common to both lattices. 

Eqs. (31) and (32) becomes more complicated. The dimensionality of the set of equations, however, coincides with that of the system in the QCA. A more exact description is obtained with the correlators of greater dimensionality m>2 (see, e.g., Refs. [90,91]). Of special interest are the one-dimensional systems with s — 2. Exact solutions for the migrating adspecies on the one-dimensional lattice have been obtained for a small number of sites [92]. In Refs. [93,94] the procedure of numerical analysis of the hierarchical system of equations has been elaborated, which is applicable not only to the one-dimensional [94,95] but also to the two-dimensional lattices [95,96], as well, the interaction with the second neighbors being taken into account (d — 1) [97]. Also, it should be noted that the expansions (virial or diagrammatic) [98] similar to the common expansion in the equilibrium theory of condensed systems [77] are used for closing the kinetic equations. 

---