Tilt boundary

Figure 3.21 Low-angle grain boundary (a) two misaligned parts of the crystal represented as a series of steps and (b) a tilt boundary consisting of an array of edge dislocations. The angular orientation difference between the two parts is 0, and the Burgers vector of the dislocations is b.

Low-angle tilt boundaries are the most easily visualized. Two regions of crystal separated by a slight misorientation can be drawn as a set of interlocking steps (Fig. 3.21a)- The edge dislocations coincide with the steps. The separate parts can be linked to make the edge dislocation array clearer (Fig. 3.21b). In the situation in which the misorientation between the two parts of the crystal is 0, the distance between the steps A and C is given by twice the dislocation separation, 2d, where... 

The investigation ofboundary mobility was confined to <111> tilt boundaries with misorientation angles in the vicinity of the special misorientation X7 ((p = 38.2°). These... 

Fig. 4 Activation enthalpy H and preexponential factor A for <777 > tilt boundaries in pure Al of different origin (9 - Al I - Al II)...

Figure 5.8 Dislocation array in a low-angle tilt boundary.

Dislocations interact and tend to order if they can move. Consider the arrangement shown in Figure 3-6a. This is called an edge dislocation tilt boundary. It is seen that the number of lattice planes terminating at the boundary is n - (2/6)-sin 0/2, from which the (mean) spacing between the dislocations is found to be... 

Tilt boundaries occur if the axis of rotation between the two grains is located in the boundary (interface). In contrast, if the axis of rotation is perpendicular to the boundary, the boundary is called a twist boundary and consists of a collection of screw dislocations (Fig. 3-6b). An equation similar to Eqn. (3.14) holds for twist (and mixed) boundaries. Since dislocation theory is well understood, it is possible to quantitatively treat small-angle grain boundaries [J.P. Hirth, J. Lothe (1982)]. 

Figure 3-9. Grain boundary energies in aluminum. Rotation axis of the (symmetric) tilt boundary is <110) [after G. Hasson, C. Goux (1971)].

Figure 9.2 presents diffusivity data for a series of tilt boundaries as a function of the misorientation tilt angle. 

Figure 9.9 Calculated atom jumps in the core of a E5 symmetric (001) tilt boundary in b.c.c. Fe. A pair-potential-molecular-dynamics model was employed. For purposes of clarity, the scales used in the figure are [130] [310] [001] = 1 1 5. All jumps occurred in the fast-diffusing core region. Along the bottom, a vacancy was inserted at B, and subsequently executed the series of jumps shown. The trajectory was essentially parallel to the tilt axis. Near the center of the figure, an atom in a B site jumped into an interstitial site at I. At the top an atom jumped between B, I and B sites. From Balluffi et al. [14],...

The asymmetric small-angle tilt boundary in Fig. B.5a consists of an array of parallel edge dislocations running parallel to the tilt axis. During diffusion they will act as fast diffusion pipes. Show that fast self-diffusion along this boundary parallel to the tilt axis can be described by an overall boundary diffusivity,... 

Figure 13.1 Motion of small-angle symmetric tilt boundary by means of glissile motion...

Figure 13.8 Activation energy, EB, for the motion of 100 tilt boundaries in Al as a function of tilt angle. The arrows at the top indicate misorientations of singular boundaries. Data for Al of 99.99995%, 99.9992%, find 99.98% purity. From Fridman et al. 15).

Figure 13.9 Experimentally determined plot of In MB vs. 1/T for a (111) tilt boundary...

D.A. Molodov, C.V. Kopetskii, and L.S. Shvindlerman. Detachment of a special (E = 19, (111)) tilt boundary from an impurity in iron-doped aluminum bicrystals. 

Solution. Using Eq. 11.2, the force per unit length on each dislocation is /CT = symmetric tilt boundary is d = b/0 [7]. The pressure on the boundary due to all dislocations is then... 

Figure B.2 Symmetric large-angle (113)(TlO] tilt boundary in A1 viewed along the [110] tilt axis by high-resolution electron microscopy. The tilt angle is 50.48°. The inset shows a simulated image [4], Reprinted, by permission, from K.L. Merkle, L.J. Thompson, and F. Phillipp, Thermally activated step motion observed by high-resolution electron microscopy at a (113) symmetric tilt grain-boundary in aluminum," Philosophical Magazine Letters, vol. 82. pp. 589-597. Copyright (c) 2002 by Taylor and Francis Ltd., http //www.tandf.co.uk/journals.

Figure B.4 (a) Singular large-angle symmetrical tilt boundary in f.c.c. structure viewed...

Figure B.5 (a) Small-angle asymmetric tilt boundary in a primitive cubic lattice viewed...

Grain boundaries can also be classified as tilt boundaries, twist boundaries, and mixed boundaries. A tilt boundary s plane is parallel to the rotation axis used to define its crystal misorientation, as in Fig. B.4c. The crystals adjoining the boundary are related by a simple tilt around this axis. A twist boundary, as in Fig. B.56, is a boundary whose plane is perpendicular to the rotation axis. The two crystals adjoining the boundary are then related by a simple twist around this axis. All other types of boundaries are considered to be mixed. 

Figure 8.15 Symmetric small angle tilt boundary (drawn after [339]).

Calculate the average spacing between dislocations in a 1 /2° tilt boundary in aluminum. Look up any required data. 

A low-angle tilt boundary is composed of edge dislocations. 

Knowing that low-angle tilt boundaries are composed of edge dislocations, predict the most common planes for tilt boundaries in fee and in bcc metals. 

M.P. Dewald, W.A. Curtin Multiscale modelling of dislocation/grain boundary interactions. II. Screw dislocations impinging on tilt boundaries in Al. Phil. Mag. 87, 4615 1641 (2007)... 

Figure 1.7 View down the [001] direction of a tilt boundary between two crystals (A, B) with a misorientation angle of 36.9° about [001], The grain boundary is perpendicular to the plane of the page. Every fifth atom in the [010] direction in B is a coincidence point (shaded). The area enclosed by the CSL unit cell (bold lines) is five times that of the crystal unit cell, so 2 = 5. (After Lalena and Cleary, 2005. Copyright John Wiley Sons, Inc. Reproduced with permission.)...

For tilt boundaries, the value of E can also be calculated if the plane of the boundary is specified in the coordinate systems for both adjoining grains. This method is called the interface-plane scheme (Wolfe and Lutsko, 1989). In a crystal, lattice planes are imaginary sets of planes that intersect the unit cell edges. The tilt and twist boundaries can be defined in terms of the Miller indices for each of the adjoining lattices and the twist angle, , of both plane stacks normal to the boundary plane, as follows ... 

---