The Energy of Dislocations

For the present, consider only E (discussion on the contribution of the core follows later on). 

Tension applied to a rod of length, 1, and cross-section, A, extends the rod by dl and a strain energy, dE, is introduced. This may be expressed as  

Volume V = A1 remains constant during deformation and, by simple substitution, one can write  

This is obtained as follows Rewrite Eq. (3.26), substitute F from Eq. (3.27) then express A in terms of volume and express dl/1 by de from Eq. (3.28). This produces Eq. (3.30)  

In terms of shear stress and shear modulus, Eq. (3.32) should be written as  

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Next, let us compile some quantitative relations which concern the stress field and the energy of dislocations. Using elastic continuum theory and disregarding the dislocation core, the elastic energy, diS, of a screw dislocation per unit length for isotropic crystals is found to be... 

As explained in Sec. 6.1.7.4 of this chapter, the compressional elastic constant B vanishes at the SmA-SmA critical point so that the system is very close to a nematic. An interesting consequence is that the energy of dislocations becomes very weak and their proliferation may lead to a destruction of smectic order. Prost and Toner [109] have shown that depending on bare parameters of a particular system, either the nematic bubble or the critical point could be observed. 

The density of dislocations is usually stated in terms of the number of dislocation lines intersecting unit area in the crystal it ranges from 10 cm for good crystals to 10 cm" in cold-worked metals. Thus, dislocations are separated by 10 -10 A, or every crystal grain larger than about 100 A will have dislocations on its surface one surface atom in a thousand is apt to be near a dislocation. By elastic theory, the increased potential energy of the lattice near... 

However, it is not yet clear why the ener es of the SISF and the twin boundary increase with increasing A1 concentration. To find a clue to the problem, it would be needed to make out the effects of the short-range ordering of A1 atoms in excess of the stoichiometric composition of the HAl phase on the energies of planar faults and the stmcture of dislocation cores in the Al-rich HAl phase. 

Similarly, in studies of lamellar interfaces the calculations using the central-force potentials predict correctly the order of energies for different interfaces but their ratios cannot be determined since the energy of the ordered twin is unphysically low, similarly as that of the SISF. Notwithstcinding, the situation is more complex in the case of interfaces. It has been demonstrated that the atomic structure of an ordered twin with APB type displacement is not predicted correctly in the framework of central-forces and that it is the formation of strong Ti-Ti covalent bonds across the interface which dominates the structure. This character of bonding in TiAl is likely to be even more important in more complex interfaces and it cannot be excluded that it affects directly dislocation cores. 

The presence of dislocations is able to account for many features of crystal growth that cannot be explained if the growing crystal is assumed to be perfect. In these cases, the dislocation provides a low-energy site for the deposition of new material. 

The disruption to the crystal introduced by a dislocation is characterized by the Burgers vector, b (see Supplementary Material SI for information on directions in crystals). During dislocation motion individual atoms move in a direction parallel to b, and the dislocation itself moves in a direction perpendicular to the dislocation line. As the energy of a dislocation is proportional to b2, dislocations with small Burgers vectors form more readily. 

An elastic continuum model, which takes into account the energy of bending, the dislocation energy, and the surface energy, was used as a first approximation to describe the mechanical properties of multilayer cage structures (94). A first-order phase transition from an evenly curved (quasi-spherical) structure into a... 

The importance of dislocations becomes evident when we consider the strain on the microstructure of a simple crystal. The atoms or ions in a crystal are in symmetric energy wells and so vibrate around their lattice site. When we track across a crystal plane, the potential energy increases and decreases in a regular fashion with the minima at the lattice points... 

The surface layers of solids usually differ from the deeper zones of the same specimen in their chemical composition, their degree of lattice perfection (e.g., the frequency of dislocations), their state of stress, and so on. This renders unpalatable the notion of a surface tension in solids, but suggests the existence of a kind of surface energy, unknown in liquids, which it was proposed to designate as cuticular energy. 

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