The Screw Dislocation

For many purposes, we will find that antiplane shear problems in which there is only one nonzero component of the displacement field are the most mathematically transparent. In the context of dislocations, this leads us to first undertake an analysis of the straight screw dislocation in which the slip direction is parallel to the dislocation line itself. In particular, we consider a dislocation along the X3-direction (i.e. = (001)) characterized by a displacement field Usixi, X2). The Burgers vector is of the form b = (0, 0, b). Our present aim is to deduce the equilibrium fields associated with such a dislocation which we seek by recourse to the Navier equations. For the situation of interest here, the Navier equations given in eqn (2.55) simplify to the Laplace equation (V ms = 0) in the unknown three-component of displacement. Our statement of equilibrium is supplemented by the boundary condition that for xi 0, the jump in the displacement field be equal to the Burgers vector (i.e. Usixi, O ) — M3(xi, 0 ) = b). Our notation usixi, 0+) means that the field M3 is to be evaluated just above the slip plane (i.e. X2 = e). 

This geometry of this problem inspires a description in terms of cylindrical coordinates, within which a plausible form for the displacements is the assumption that, like a helical ramp in a parking garage, the displacements increase linearly in the winding angle, 6, yielding the solution 

This displacement field may also be represented simply coordinates, resulting in the expression in terms of Cartesian 

The stress tensor corresponding to these strains may be written as 

Note that care must be taken with the choice of sign of the Burgers vector to actually use these equations in practice. 

---

Fig. VII-8. (a) Screw dislocation (from Ref. 115). (b) The slip that produces a screw-type dislocation. Unit slip has occurred over ABCD. The screw dislocation AD is parallel to the slip vector. (From W. T. Read, Jr., Dislocations in Crystals, McGraw-Hill, New York, 1953, p. 15.)...

The screw dislocation theory (27), often referred to as the BCE theory (after its formulators), shows that the dependence of growth rate on supersaturation can vary from a paraboHc relationship at low supersaturation to a linear relationship at high supersaturation. In the BCE theory, growth rate is given by... 

Fig. 9.10. The planking analogy of the screw dislocation. Imagine four planks resting side by side on a factory floor. It is much easier to slide them across the floor one at a time than all at the same time.

So far we have discussed the surface of a perfect crystal. But for an imperfect crystal there is another possibility to provide a step source. This is due to the screw dislocation. Assume that one cuts a crystal half-way from one side into the center, and slides the freshly created two faces against each other in... 

One can now immediately deduce the normal growth rate of a crystal due to the screw dislocation. Whenever a step edge passes by a fixed point on the crystal surface, this point gains the height of a lattice unit. The normal growth rate V of the crystal is then... 

The core structure of the (100) screw dislocation is planar and widely spread w = 2.66) on the 011 plane. In consequence, the screw dislocation only moves on the 011 glide plane and does so at a low Peierls stress of about 60 MPa. 

As a consequence edge and mixed (111) dislocations move with relative ease, whereas the Peierls barrier for screw dislocations is as high as 2 GPa. These results are in contrast to previous calculations [6], which have shown a splitting for the screw dislocations and also a much lower Peierls barrier. However, our results can perfectly explain most of the experimental results concerning (111) dislocations which will be discussed in the following section. 

The second type of line defect is the screw dislocation, which is rather less easy to visualise. Consider, however, a block of material, half of which is sheared one interatomic distance with respect to the other half, as shown in Fig. 20.306. The line cdthen constitutes a screw dislocation the arrangement of atoms around a screw dislocation is shown in Fig. 20.30c. 

The basic condition for experimental study of nucleation on an identical surface requires that this surface be a single crystal face without screw dislocations (page 306). Such a surface was obtained by Budevski et at. when silver was deposited in a narrow capillary. During subsequent deposition of silver layers the screw dislocations died out so that finally a surface of required properties was obtained. 

The electrocrystallization on an identical metal substrate is the slowest process of this type. Faster processes which are also much more frequent, are connected with ubiquitous defects in the crystal lattice, in particular with the screw dislocations (Fig. 5.25). As a result of the helical structure of the defect, a monoatomic step originates from the point where the new dislocation line intersects the surface of the crystal face. It can be seen in Fig. 5.48 that the wedge-shaped step gradually fills up during electrocrystallization after completion it slowly moves across the crystal face and winds up into a spiral. The resultant progressive spiral cannot disappear from the crystal surface and thus provides a sufficient number of growth... 

Figure 4.2 Quasi-hexagonal dislocation loop lying on the (111) glide plane of the diamond crystal structure. The <110> Burgers vector is indicated. A segment, displaced by one atomic plane, with a pair of kinks, is shown a the right-hand screw orientation of the loop. As the kinks move apart along the screw dislocation, more of it moves to the right.

Figure 6.9 Curvature of a screw dislocation near a free surface a. screw dislocation that has moved about halfway thru the specimen and is emerging from the surface, b. by becoming curved the screw dislocation reduces its length and hence its energy.

Dislocations are line defects that occur in crystals. There are many types of dislocation. The easiest to visualize are the edge dislocation, which consists of an extra half-plane of atoms inserted into a crystal and the screw dislocation that resembles... 

The reflections include a particular g in which the dislocation is invisible (i.e., g b = 0 when b is normal to the reflecting plane). With these criteria in diffraction contrast, one can determine the character of the defect, e.g., screw (where b is parallel to the screw dislocation line or axis), edge (with b normal to the line), or partial (incomplete) dislocations. The dislocations are termed screw or edge, because in the former the displacement vector forms a helix and in the latter the circuit around the dislocation exhibits its most characteristic feature, the half-plane edge. By definition, a partial dislocation has a stacking fault on one side of it, and the fault is terminated by the dislocation (23-25). The nature of dislocations is important in understanding how defects form and grow at a catalyst surface, as well as their critical role in catalysis (3,4). 

The second type of line defect, the screw dislocation, occurs when the Burger s vector is parallel to the dislocation line (OC in Figure 1.33). This type of defect is called a screw dislocation because the atomic structure that results is similar to a screw. The Burger s vector for a screw dislocation is constructed in the same fashion as with the edge dislocation. When a line defect has both an edge and screw dislocation... 

Figure 3.8. Explanation of dislocations in relation to glide. The solid arrow, b, corresponds to the Burgers vector of the dislocation. SV is the screw dislocation, WE is the edge dislocation, and VW is a mixed dislocation. The shaded area represents a glide plane.

Fig. 2.19 Elcctrocrystallization on a metal surface. Growth can proceed continuously in the -direction as the step rotates around the screw dislocation...

Dislocations are line defects. They bound slipped areas in a crystal and their motion produces plastic deformation. They are characterized by two geometrical parameters 1) the elementary slip displacement vector b (Burgers vector) and 2) the unit vector that defines the direction of the dislocation line at some point in the crystal, s. Figures 3-1 and 3-2 show the two limiting cases of a dislocation. If b is perpendicular to s, the dislocation is named an edge dislocation. The screw dislocation has b parallel to v. Often one Finds mixed dislocations. Dislocation lines close upon themselves or they end at inner or outer surfaces of a solid. 

These results indicate that in the present linear elastic model, the limiting velocity for the screw dislocation will be the speed of sound as propagated by a shear wave. Even though the linear model will break down as the speed of sound is approached, it is customary to consider c as the limiting velocity and to take the relativistic behavior as a useful indication of the behavior of the dislocation as v — c. It is noted that according to Eq. 11.20, relativistic effects become important only when v approaches c rather closely. 

By use of the proper experimental conditions and Ltting the four models described above, it may be possible to arrive at a reasonable mechanistic interpretation of the experimental data. As an example, the crystal growth kinetics of theophylline monohydrate was studied by Rodriguez-Hornedo and Wu (1991). Their conclusion was that the crystal growth of theophylline monohydrate is controlled by a surface reaction mechanism rather than by solute diffusion in the bulk. Further, they found that the data was described by the screw-dislocation model and by the parabolic law, and they concluded that a defect-mediated growth mechanism occurred rather than a surface nucleation mechanism. 

A dislocation is characterized by its Burgers vector. An atom-to-atom circuit that would close in a perfect crystal will fail to close if it is drawn around a dislocation. The closure failure is the Burgers vector of the dislocation. This is illustrated in Figure 5.6. The edge dislocation (middle) is perpendicular to its Burgers vector and the screw dislocation (right) is parallel to its Burgers vector. 

Fig. 23 a Transmission electron micrograph of permanganic-etched linear polyethylene. The micrograph shows the spiral development around a screw dislocation in a sample crystallised at 130 °C. Scale bar represents 10 pm. Courtesy of D.C. Bassett. From [105] with permission from Elsevier, UK. b Sketch showing c with respect to the fold surface of the crystal layers around the screw dislocation... 

---