The theory of a one-dimensional dislocation model is developed. Besides acting as a pointer to developments of general dislocation theory, it has a variety of direct physical applications, particularly to monolayers on a crystalline substrate and to conditions in the edge row of a terrace of molecules in a growing crystal. Allowance is made in the theory for a difference in natural lattice-spacing between the surface layer or row and the substrate. The form and energy of single dislocations and of regular sequences of dislocations are calculated. Critical conditions for spontaneous generation (or escape) of dislocations are determined, and likewise the activation energies for such processes below the critical limits. Various physical applications of the model are discussed, and the physical parameters are evaluated with the aid of the Lennard-Jones force law for the above-mentioned principal applications.
A calculation of the interfacial energy between a crystalline film and a crystalline substrate of a different substance for the simple case in which the lattice parameters differ in one direction only, is presented. The results are expressed in terms of film thickness h, interfacial misfit η, interfacial bonding, and relative hardness of film and substrate. A parabolic interfacial potential has been used to investigate the effect of h showing that it is only a significant factor when either or both h and η are small. It is further shown that, in the minimum energy configuration of the system, the film is homogeneously strained. According to the calculations, a critical value of misfit ηc exists below which the film is strained to fit the substrate exactly and above which the required strain is an order of magnitude less than ηc. The misfit ηc is estimated to vary from as much as 13% for a ``soft'' monatomic layer which is tightly bound down to practically zero for thick films which are loosely bound. It is shown that the interfacial energy associated with an infinitely thick film as calculated with a Peierls-Nabarro type of interfacial force, is a useful approximation for many purposes. Approximate expressions for the strains in terms of the relevant parameters are deduced from this result.
This paper presents a calculation of the stresses and energies for interfaces between different crystals. Only simple cases corresponding to interfacial dislocations of pure screw or edge type are treated. The difference between the two crystals is expressed in terms of different elastic constants and variable interfacial bondings. Two kinds of interfacial forces appear: tangential forces with a periodic character and normal forces. The latter are induced by different normal displacements of the two contact surfaces of the crystals due to the equal and opposite tangential interfacial forces and are accounted for by assuming a linear relation between force and relative displacement. By using a periodic parabolic model to represent the periodic potential associated with the tangential forces, it is shown that the contribution of the normal force to the interfacial energies is negligible for the approximation used. When these normal forces are neglected, the Peierls-Nabarro representation of the interfacial forces yields simple results in terms of an interfacial rigidity modulus μ and an effective elastic constant λ+ defined by 1/λ+=1/λa+1/λb=(1−σa)μa+(1−σb)/μb, where σ is Poisson's ratio and the two crystals are designated by a and b. It is seen that ½λa≤λ+≤λa, where λa is the smaller of λa and λb. Apart from being proportional to μc, the interfacial energy E depends only on a parameter β=2π(c/p)(λ+/μ), where p is the dislocation spacing and the lattice constant c of the reference lattice is given by 2/c=1/a+1/b. The dependence on β is such that below a poorly defined value of the order of unity the variation of E is very rapid while beyond this it remains practically constant. It is further shown that the elastic energies stored in the two crystals are in the ratio λa/λb and that less than 2% of these energies is stored at distances from the interface greater than half the distance between dislocations. Similar results are shown to hold for simple twist and tilt boundaries.
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