Energy of a dislocation

Exactly the same procedure can be followed to define the free energy of a dislocation core. It should be surrounded by a box, the terminating planes of which can be dealt with exactly as above. Special attention has to be given to the atoms at the comers of the box, but this presents no particular problems their weights are simply a oroduct of the weights generated by the planar terminations which they share. 

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. 

A dislocation line may only terminate al the surface of the crystal. The energy of a dislocation is largely stored as strain in the surrounding lattice. The important property of a dislocation is its ability to move quite easily... 

We advanced the idea that the energy of a dislocation line could be written in a local approximation as... 

It is important to note that, in these examples, we have considered unit strength dislocations only. Since the energy of a dislocation is... 

A perfect dislocation has a Burgers vector equal to an atom-to-atom vector in the crystal. As the energy of a dislocation is proportional to b, the most common dislocations in metals have small Burgers vectors. 

The important result is that the strain energy of a dislocation is proportional to the square of the Burgers vector b and the shear modulus of the material. 

The strain energy (or self energy) of a dislocation actually depends on the character of the dislocation, but setting E = aGb is a good estimate, where a is -0.5. 

Because such a large number of dislocations exist in a crystal, dislocations often intersect, and the interaction between dislocations results in various physical phenomena occiuring in crystals. The energy of a dislocation is proportional to the square of its Burger s vector b. Two dislocations can coalescence spontaneously if the total energy of the coalesced dislocation is smaller than the sum of the energies of the two... 

In order to examine the mechanics of dislocation formation in a compliant film-substrate system, the energy of a dislocation at an arbitrary position in this structure in the absence of any other stress field is required. This energy is equivalent to the configurational force on the dislocation as a function of position through the thickness of the composite layer. In order to determine an approximate form for the critical thickness condition in Section 6.7.1, an ad hoc assumption on the variation of this force was made in (6.66). While the assumed variation of the force is asymptotically correct near either free surface and it has the obvious virtue of simplicity, the quality of the approximation is not evident. Thus, in this section, the variation of this force with position is examined in greater detail. 

Dislocations are the most elementar defects of a SmA phase. Since they cannot end in the smectic material, they must form closed loops (at least in the thermodynamic limit of an infinite sample). The elastic energy of a dislocation line is proportional to its length [31]. It is then characterized by a line tension Yo- The total free energy per unit length is ... 

The elastic stress field around a dislocation affects the elastic energy of a dislocation and the interaction between parallel dislocations. The elastic energy per unit length of dislocation between two cylindrical surfaces of radius Cq and R is given by ... 

Let us consider, as an example, a heterogeneous nucleation caused by a dust particle whose shape is close to a circular bicone of height 21 and base radius R (Fig. 19). It creates deformations which are relaxed by a set of dislocations loops. The energy of a dislocation... 

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