Kosevich A.M. The crystal lattice (2ed., Wiley, 2005)(ISBN 3527405089)(342s)_PSa_

13.5 Dislocation Field in a Sample of Finite Dimensions 313

to the cylinder equilibrium equations should involve, apart from (13.3.3), additional stresses compensating the twisting moment (13.5.1), i. e., creating an average moment Mz = −Mz0. These stresses (and also the corresponding displacements) are easily obtained from a rod torsion theory. It is known that for a rod twisted under the action of the moment Mz, there arises a displacement vector component uϕ providing a torsion angle that is homogeneous along the rod length:

where C = (1/2)πGR4 is the torsional rigidity of a rod.

Taking this into account, it is easy to write an equilibrium distribution of additional displacements and stresses in a cylinder with a screw dislocation along its axis

The second term on the r.h.s. of (13.5.2) shows the role of finite dimensions of the specimen in generating the elastic stresses around the dislocation. With increasing

for L R may be essential. Torsional deformation is observed in long and thin thread-like crystals with a screw dislocation (whiskers).

Now consider a superstructure formed by a system of a large number of screw dislocations oriented along the axis of the cylinder of radius R and we will be interested in the macroscopic properties of such a superstructure. Then, in the main approximation the distribution of the dislocations can be assumed continuous, characterized by the density n0 = 1/S0, where S0 is the average area of the xy plane per dislocation.

The dislocation creates a stress (13.3.3) in an unbounded media (denote it as σ0). But because the stress field of the screw dislocations is similar to the electric field of linear charges, the stresses at a distance r from the axis of the cylinder are created by all the dislocations intersecting an area S = πr2 around the axis of the cylinder and are equal to the stresses around one dislocation lying along the axis of the cylinder (x = y = 0) and carrying the total “charge” (the Burgers vector bS/S0 ) of all those

These stresses, first, create a force acting on a dislocation lying a distance r from

13.6 Long-Range Order in a Dislocated Crystal 317

We note that the suggested definition of crystal long-range order does not give the information about the smallness of atomic displacements with respect to their equilibrium positions in the initial crystal lattice. Sometimes, the long-range order determined in this way is called topological order.

The vanishing of topological long-range order means the destruction of a crystal structure. When the absence of long-range order in a solid substance is not accompanied by the loss of shear elasticity, we assume that a solid is in an amorphous state. If the long-range order vanishes with increasing temperature simultaneously with shear elasticity (the substance starts leaking), we can speak of crystal melting. Unfortunately, there is no systematic theory of 3D crystal melting.

The situation is opposite in a 2D case. In a 2D crystal, a dislocation is a point defect, whose 2D displacement vector field u(ux, uy) is not different from that of a linear edge dislocation in a 3D crystal (13.3.5). The energy of this defect can be obtained by multiplying the linear dislocation energy per unit length by the lattice constant a:

Since the energy ε0 is comparable with the atomic energy of a 2D crystal and the dependence on crystal dimension (13.6.1) is very weak, a thermo-fluctuational initiation of the dislocation can be assumed in a 2D crystal.

With given temperature T, the thermodynamic equilibrium condition is determined by the minimum of the free energy of the solid F. We consider the change in F associated with the appearance of one dislocation and note that the dislocation contribution

Then

R

δF = E − TδS = (ε0 − 2T) ln a . (13.6.3)

It is clear that for T < (1/2)ε0 the appearance of a dislocation increases the free energy of the system and, thus, the dislocation initiated fluctuationally will inevitably escape from the crystal. In total thermodynamic equilibrium there are no isolated free dislocations.

For T > (1/2)ε0, the free energy of the system decreases with the appearance of a dislocation and thus the process of their initiation in a 2D crystal becomes advantageous thermodynamically. Hence, with increasing temperature at T = T0 = (1/2)ε0 there occurs a phase transition of a 2D crystal into a state with an arbitrary number of free dislocations (Berezinsky, 1970, Kosterlits, Taules, 1973). As a result, the initial long-range order in a 2D crystal is broken.

It is easy to follow what mechanism is responsible for breaking the 2D crystal longrange order. At low (T < T0) but finite temperatures a certain equilibrium density

318 13 Elastic Field of Dislocations in a Crystal

of dislocation dipoles exists in a crystal. This is the case because the energy of an opposite-sign dislocation pair, in contrast to (13.6.2) is independent of crystal dimen-

sion. Indeed, the dipole energy can be written as ED = 2E + Eint, where the interaction energy of a dislocation pair Eint is determined by (13.6.7) shown in solving

Problem 2. In this formula it is necessary to put b1 = b = −b2. Thus,

where the angle ϕ determines the dipole orientation and U0 is the dislocation core energy.

Since the energy (13.6.4) is bounded, there exists a finite density of dislocation dipoles of the given dimension r, proportional to the Boltzmann factor exp(−βED ), where β = (1/T). We calculate the mean square of the dislocation dipole dimension:

Since we are interested in the limit R → ∞ (R a) the mean square (13.6.5) will have different forms for 4 < 2βε0 when T < T0, and with 4 > 2βε0 when T > T0. In the first case T < T0, in the limit R → ∞ we have

Thus, the average dimension of equilibrium dislocation dipoles in an ordered 2D crystal is limited, but increases with increasing temperature and goes to infinity at the phase transition point T = T0.

In the second case (T > T0), in the limit R → ∞ we get r2 = ∞. Hence, it follows that when T > T0 the dislocation dipoles are broken down, on average, and a phase with free 2D dislocations is occurs. A more detailed description of the properties of the resulting phase requires taking into account the dislocation interaction and a knowledge of how to treat disordered systems.

13.6 Long-Range Order in a Dislocated Crystal 319

13.6.1

Problems

1. Calculate the shear stresses generated by an infinitely extended dislocation wall (Fig. 10.8c) at large distances.

Hint. Write down the total field of all dislocations

and make use of the Poisson summation formula (9.4.3).

2. Find the interaction energy of two linear edge dislocations lying in the parallel glide planes.

Hint. Make use of the fact that the desired energy is equal to the work done to remove one dislocation in the glide plane from infinity (from crystal surface) to the line of its location in the field of a fixed second dislocation. Thus, the interaction energy per unit length of the dislocation equals

3. Describe the distribution of straight dislocations near the obstacle under the action of a homogeneous compressive stress.

Hint. Make use of (13.3.13), setting σxye = σ0 =constant, a2 = 0, and find a1 = −L

14

Dislocation Dynamics

14.1

Elastic Field of Moving Dislocations

Let us now elucidate what form the total system of equations of elasticity theory takes that determines strains and stresses in a crystal when dislocations perform a given motion.

Equation (13.1.6) is independent of whether the dislocations are at rest or in motion. However, in the dynamic case the distortion tensor should change with time and this will be determined by the character of dislocation motion.

If during the deformation of a medium the dislocations remain stationary the following equality is valid

where v = v(r, t) is the velocity of displacement of a medium element with a coordinate r at time t.

If the dislocations move and their density changes with time, (14.1.1) is incompatible with (13.1.6). Thus, we replace it with

where the tensor jik should be chosen so that these equations are compatible. The compatibility condition of (13.1.6), (13.1.2) has the form

To associate the tensor jik with dislocation motion, we note that the condition (14.1.3) can be regarded as the differential form of the conservation law of the Burgers vector in a medium. Indeed, we integrate both sides of (14.1.3) over the surface

The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices, Second Edition. Arnold M. Kosevich Copyright c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ISBN: 3-527-40508-9

322 14 Dislocation Dynamics

spanned on a certain closed line L and introduce the total Burgers vector b of the dislocations enveloped by the contour L. Using Stokes’s theorem, we obtain

It is clear from this equality that the integral on its r.h.s. determines the Burgers vector “flowing” per unit time through the contour L, i. e., carried over by dislocations that cross the line L. Thus, it is natural to consider jik as the dislocation flux density tensor, and (14.1.3) as an equation for the continuity of dislocation flow.

In particular, it is obvious that in the case of an isolated dislocation loop, the tensor jik has the form

where V is the dislocation line velocity at a given point. The vector of the flux jik dli through an element dl of the contour is then proportional to dl [τV ] = V [dlτ], i. e., to the projection velocity V onto the direction perpendicular both to dl and τ. It follows from geometrical considerations that only this projection results in the dislocation crossing the element dl.

If the dislocation distribution is described by the continuous functions αsik (14.1.4)

is generalized by

jik = eilm ∑αsikVms , s

where the index s denotes the densities of dislocations of various types (for instance, dislocations of different sign in the case of a system of parallel edge dislocations) and the vector V s is equal to the average velocity of dislocations of the type s at a given point in the crystal.

The tensor jik has an independent meaning and is the principal characteristic of dislocation motion.

Finally, we obtain a total system of differential equations describing the elastic fields in a crystal with moving dislocations. This system consists of (13.1.6), (14.1.2) and the equations of motion of a continuous medium:

In these equations the tensors αik and jik are the given functions of the coordinates (and time) characterizing the dislocation distribution and motion.

The compatibility conditions for (14.1.5) are the conservation laws

14.1 Elastic Field of Moving Dislocations 323

Using the definition for the tensor of the dislocation flow density and the system of (14.1.5), the dynamics of dislocations in an elastic medium can be developed (Kosevich, 1962; Mura, 1963).

The relation between the trace of the tensor jik (j0 ≡ jkk) and the equation for continuity of a continuous medium is of special interest. The convolution j0 is involved in the equation obtained from (14.1.5):

It is easy to explain the physical meaning of (14.1.7). Indeed, the convolution ε kk is a relative elastic change in the medium element volume obviously related with a corresponding relative change in its density (ρ):

Substituting (14.1.8) into (14.1.7) and using the linearity of the theory, we get the

If, with a moving dislocation, the medium elements move without breaking continuity, the l.h.s. of (14.1.9) vanishes due to the continuity equation and (14.1.10)

For linear dislocations (14.1.10) has a simple interpretation. Indeed, in the case of an isolated linear dislocation, the convolution j0 is proportional to [bτ] V , i. e., proportional to the projection of the dislocation velocity onto the directional perpendicular to the vectors τ and b, or in other words, onto the direction perpendicular to the dislocation glide plane. Thus, (14.1.10) implies that with medium continuity preserved the vector of the dislocation velocity V lies in the glide plane, so that the mechanical motion of a dislocation can only take place in this plane.

If dislocation motion is accompanied by the formation of some discontinuities, for instance, by a macroscopic cluster of vacancies along some part of the dislocation line, the l.h.s. of (14.1.9) is nonzero and equals the velocity of a relative inelastic increase or decrease of the mass of some elementary volume of the medium.

The action of this mechanism amounts to a macroscopic clustering of vacancies or interstitial atoms along the dislocation line. We denote by q(r) a relative increase of the specific volume of a medium at the point r per unit time. According to the formulae (14.1.4), (14.1.9), the motion of an isolated dislocation should then generate the following value of q:

when ξ is a 2D radius vector measured from the dislocation axis.