Line defects

3.3 Line defects

3.3.1 Concept of a dislocation

All crystalline materials usually contain lines of structural discontinuities running throughout each crystal or grain. These line discontinuities are termed dislocations and there is usually about 10 10

to 10 12 m of dislocation line in a meter cube of material. 3 Dislocations enable materials to deform without destroying the basic crystal structure at stresses below that at which the material would break or fracture if they were not present.

A crystal changes its shape during deformation by the slipping of atomic layers over one another. The theoretical shear strength of perfect crystals was first calculated by Frenkel for the simple rectangular- type lattice shown in Figure 3.10 with spacing a between the planes. The shearing force required to move a plane of atoms over the plane below will be periodic, since for displacements x < b/2, where

b is the spacing of atoms in the shear direction, the lattice resists the applied stress, but for x > b/2 the lattice forces assist the applied stress. The simplest function with these properties is a sinusoidal relation of the form:

τ =τ m sin (2πx/b) ∼ =τ m 2πx/b, where τ m is the maximum shear stress at a displacement = b/4. For small displacements the elastic

shear strain given by x/a is equal to τ/µ from Hooke’s law, where µ is the shear modulus, so that τ m = (µ/2π)b/a

(3.7) and since b ≈ a, the theoretical strength of a perfect crystal is of the order of µ/10.

3 This is usually expressed as the density of dislocations ρ = 10 10 to 10 12 m −2 .

108 Physical Metallurgy and Advanced Materials

Unslipped b

Slipped

(a) Slip plane

Figure 3.11 Schematic representation of: (a) a dislocation loop, (b) edge dislocation and (c) screw dislocation.

This calculation shows that crystals should be rather strong and difficult to deform, but a striking experimental property of single crystals is their softness, which indicates that the critical shear stress to produce slip is very small (about 10 −5 µ or ≈ 50 gf mm −2 ). This discrepancy between the theoretical and experimental strength of crystals is accounted for if atomic planes do not slip over each other as rigid bodies but instead slip starts at a localized region in the structure and then spreads gradually over the remainder of the plane, somewhat like the disturbance when a pebble is dropped into a pond.

In general, therefore, the slip plane may be divided into two regions, one where slip has occurred and the other which remains unslipped. Between the slipped and unslipped regions the structure will be dislocated (Figure 3.11); this boundary is referred to as a dislocation line, or dislocation. Three simple properties of a dislocation are immediately apparent, namely: (1) it is a line discontinuity, (2) it forms

a closed loop in the interior of the crystal or emerges at the surface and (3) the difference in the amount of slip across the dislocation line is constant. The last property is probably the most important, since

a dislocation is characterized by the magnitude and direction of the slip movement associated with it. This is called the Burgers vector, b, which for any given dislocation line is the same all along its length.

3.3.2 Edge and screw dislocations

It is evident from Figure 3.11a that some sections of the dislocation line are perpendicular to b, others are parallel to b while the remainder lie at an angle to b. This variation in the orientation of the line with respect to the Burgers vector gives rise to a difference in the structure of the dislocation. When the dislocation line is normal to the slip direction it is called an edge dislocation. In contrast, when the line of the dislocations is parallel to the slip direction the dislocation line is known as a screw dislocation. From the diagram shown in Figure 3.11a it is evident that the dislocation line is rarely pure edge or pure screw, but it is convenient to think of these ideal dislocations since any dislocation can be resolved into edge and screw components. The atomic structures of simple edge and screw dislocations are shown in Figures 3.13 and 3.14.

3.3.3 The Burgers vector

It is evident from the previous sections that the Burgers vector b is an important dislocation parameter. In any deformation situation the Burgers vector is defined by constructing a Burgers circuit in the

Crystal defects 109

Figure 3.12 Burgers circuit round a dislocation A fails to close when repeated in a perfect lattice unless completed by a closure vector FS equal to the Burgers vector b.

dislocated crystal, as shown in Figure 3.12. A sequence of lattice vectors is taken to form a closed clockwise circuit around the dislocation. The same sequence of vectors is then taken in the perfect lattice when it is found that the circuit fails to close. The closure vector FS (finish-start) defines b for the dislocation. With this FS/RH (right-hand) convention it is necessary to choose one direction along the dislocation line as positive. If this direction is reversed the vector b is also reversed. The Burgers vector defines the atomic displacement produced as the dislocation moves across the slip plane. Its value is governed by the crystal structure because during slip it is necessary to retain an identical lattice structure both before and after the passage of the dislocation. This is assured if the dislocation has a Burgers vector equal to one lattice vector and, since the energy of a dislocation depends on the square of the Burgers vector (see Section 3.3.5.2), its Burgers vector is usually the shortest available lattice vector. This vector, by definition, is parallel to the direction of closest packing in the structure, which agrees with experimental observation of the slip direction.

The Burgers vector is conveniently specified by its directional coordinates along the principal crystal axes. In the fcc lattice, the shortest lattice vector is associated with slip from a cube corner to a face center, and has components a/2, a/2, 0. This is usually written a/2[1 1 0], where a is the lattice parameter and [1 1 0] is the slip direction. The magnitude of the vector, or the strength of

2 2 2 2 the dislocation, is then given by √ {a (1 +1 +0 )/4 } = a/ 2. The corresponding slip vectors for the bcc and cph structures are b = a/2[1 1 1] and b = a/3[1 1 ¯2 0] respectively.

3.3.4 Mechanisms of slip and climb

The atomic structure of an edge dislocation is shown in Figure 3.13a. Here the extra half-plane of atoms is above the slip plane of the crystal, and consequently the dislocation is called a positive edge dislocation and is often denoted by the symbol ⊥. When the half-plane is below the slip plane it is termed a negative dislocation. If the resolved shear stress on the slip plane is τ and the Burgers vector of the dislocation b, the force on the dislocation, i.e. force per unit length of dislocation, is F = τb.

This can be seen by reference to Figure 3.13 if the crystal is of side L. The force on the top face (stress × area) is τ × L 2 . Thus, when the two halves of the crystal have slipped the relative amount

b, the work done by the applied stress (force × distance) is τL 2 b. On the other hand, the work done in moving the dislocation (total force on dislocation FL

× distance moved) is FL 2 , so that equating the work done gives F (force per unit length of dislocation) = τb. Figure 3.13 indicates how slip is

propagated by the movement of a dislocation under the action of such a force. The extra half-plane moves to the right until it produces the slip step shown at the surface of the crystal; the same shear will be produced by a negative dislocation moving from right to left. 4

4 An obvious analogy to the slip process is the movement of a caterpillar in the garden or the propagation of a ruck in a carpet to move the carpet into place. In both examples, the effort to move is much reduced by this propagation process.

110 Physical Metallurgy and Advanced Materials Extra half-plane

of atoms

b (a)

(d) Figure 3.13 Slip caused by the movement of an edge dislocation.

(b)

(c)

(c) Figure 3.14 Slip caused by the movement of a screw dislocation.

(a)

(b)

The slip process as a result of a screw dislocation is shown in Figure 3.14. It must be rec- ognized, however, that the dislocation is more usually a closed loop and slip occurs by the movement of all parts of the dislocation loop, i.e. edge, screw and mixed components, as shown in Figure 3.15.

A dislocation is able to glide in that slip plane which contains both the line of the dislocation and its Burgers vector. The edge dislocation is confined to glide in one plane only. An important difference between the motion of a screw dislocation and that of an edge dislocation arises from the fact that the screw dislocation is cylindrically symmetrical about its axis with its b parallel to this axis. To a screw dislocation all crystal planes passing through the axis look the same and therefore the motion of the screw dislocation is not restricted to a single slip plane, as is the case for a gliding edge dislocation. The process whereby a screw dislocation glides into another slip plane having a slip direction in common with the original slip plane, as shown in Figure 3.16, is called cross-slip. Usually, the cross-slip plane is also a close-packed plane, e.g. {1 1 1} in fcc crystals.

The mechanism of slip illustrated above shows that the slip or glide motion of an edge dislocation is restricted, since it can only glide in that slip plane which contains both the dislocation line and its Burgers vector. However, movement of the dislocation line in a direction normal to the slip plane can occur under certain circumstances; this is called dislocation climb. To move the extra half-plane either up or down, as is required for climb, requires mass transport by diffusion and is a non-conservative motion. For example, if vacancies diffuse to the dislocation line it climbs up and the extra half-plane will shorten. However, since the vacancies will not necessarily arrive at the dislocation at the same instant, or uniformly, the dislocation climbs one atom at a time and some sections will lie in one

Crystal defects 111

Slip plane Dislocation

line Pure screw

orientation

Pure edge

(a)

orientation

Unslipped Slipped

(b)

Slipped by one Burgers

vector displacement (c)

Figure 3.15 Process of slip by the expansion of a dislocation loop in the slip plane.

Screw dislocation

slip plane

Figure 3.16 Cross-slip of a screw dislocation in a crystal. plane and other sections in parallel neighboring planes. Where the dislocation deviates from one

plane to another it is known as a jog, and from the diagrams of Figure 3.17 it is evident that a jog in

a dislocation may be regarded as a short length of dislocation not lying in the same slip plane as the main dislocation but having the same Burgers vector.

112 Physical Metallurgy and Advanced Materials

A⬘ Edge dislocation

Figure 3.17 Climb of an edge dislocation in a crystal. Jogs may also form when a moving dislocation cuts through intersecting dislocations, i.e. forest 5

dislocations, during its glide motion. In the lower range of temperature this will be the major source of jogs. Two examples of jogs formed from the crossings of dislocations are shown in Figure 3.18. Figure 3.18a shows a crystal containing a screw dislocation running from top to bottom, which has the effect of ‘ramping’ all the planes in the crystal. If an edge dislocation moves through the crystal on

a horizontal plane then the screw dislocation becomes jogged as the top half of the crystal is sheared relative to the bottom. In addition, the screw dislocation becomes jogged since one part has to take the upper ramp and the other part the lower ramp. The result is shown schematically in Figure 3.18b. Figure 3.18c shows the situation for a moving screw cutting through the vertical screw; the jog formed in each dislocation is edge in character, since it is perpendicular to its Burgers vector, which lies along the screw axis.

A jog in an edge dislocation will not impede the motion of the dislocation in its slip plane because it can, in general, move with the main dislocation line by glide, not in the same slip plane (see Fig- ure 3.17b) but in an intersecting slip plane that does contain the line of the jog and the Burgers vector. In the case of a jog in a screw dislocation the situation is not so clear, since there are two ways in which the jog can move. Since the jog is merely a small piece of edge dislocation it may move sideways, i.e. con- servatively, along the screw dislocation and attach itself to an edge component of the dislocation line. Conversely, the jog may be dragged along with the screw dislocation. This latter process requires the jog to climb and, because it is a non-conservative process, must give rise to the creation of a row of point defects, i.e. either vacancies or interstitials depending on which way the jog is forced to climb. Clearly, such a movement is difficult but, nevertheless, may be necessary to give the dislocation sufficient

mobility. The ‘frictional’ drag of jogs will make a contribution to the work hardening 6 of the material. Apart from elementary jogs, or those having a height equal to one atomic plane spacing, it is possible to have multiple jogs where the jog height is several atomic plane spacings. Such jogs can

be produced, for example, by part of a screw dislocation cross-slipping from the primary plane to the cross-slip plane, as shown in Figure 3.19a. In this case, as the screw dislocation glides forward it trails the multiple jog behind, since it acts as a frictional drag. As a result, two parallel dislocations of opposite sign are created in the wake of the moving screw, as shown in Figure 3.19b; this arrangement is called a dislocation dipole. Dipoles formed as debris behind moving dislocations are frequently seen in electron micrographs taken from deformed crystals (see Chapter 6). As the dipole gets longer the screw dislocation will eventually jettison the debris by cross-slipping and pinching off the dipole to form a prismatic loop, as shown in Figure 3.20. The loop is capable of gliding on the surface of a prism, the cross-sectional area of which is that of the loop.

5 A number of dislocation lines may project from the slip plane like a forest, hence the term ‘forest dislocation’. 6 When material is deformed by straining or working the flow stress increases with increase in strain (i.e. it is harder to deform a material which has been strained already). This is called strain or work hardening.

Crystal defects 113

Figure 3.18 Dislocation intersections. (a, b) Screw–edge. (c) Screw–screw.

3.3.5 Strain energy associated with dislocations

3.3.5.1 Stress fields of screw and edge dislocations

The distortion around a dislocation line is evident from Figures 3.1 and 3.13. At the center of the dislocation the strains are too large to be treated by elasticity theory, but beyond a distance r 0 , equal to a few atom spacings, Hooke’s law can be applied. It is therefore necessary to define a core to the dislocation at a cut-off radius r 0 ( ≈b) inside which elasticity theory is no longer applicable. A screw dislocation can then be considered as a cylindrical shell of length l and radius r contained in an elastically isotropic medium (Figure 3.21). A discontinuity in displacement exists only in the z-direction, i.e. parallel to the dislocation, such that u = v = 0, w = b. The elastic strain thus has to accommodate a displacement w = b around a length 2πr. In an elastically isotropic crystal the accommodation must occur equally all round the shell and indicates the simple relation w = bθ/2π

114 Physical Metallurgy and Advanced Materials

Cross-slip plane

b Primary slip

plane (a)

Trailing dipole Multiple jog

Primary slip

planes (b)

Figure 3.19 (a) Formation of a multiple jog by cross-slip. (b) Motion of jog to produce a dipole.

Figure 3.20 Formation of prismatic dislocation loop from screw dislocation trailing a dipole.

Figure 3.21 Screw dislocation in an elastic continuum.

in polar (r, θ, z) coordinates. The corresponding shear strain γ θ z ( =γ zθ ) = b/2πr and shear stress τ θ z ( =τ zθ )

= μb/2πr, which acts on the end faces of the cylinder with σ 7 rr and τ rθ equal to zero. Alternatively, the stresses are given in cartesian coordinates (x, y, z):

τ xz ( zx )

2 =τ 2 = −μby/2π(x +y ) (3.8)

τ yz ( zy )

2 =τ 2 = −μbx/2π(x +y ),

7 This subscript notation zθ indicates that the stress is in the θ-direction on an element perpendicular to the z-direction. The stress with subscript rr is thus a normal stress and denoted by σ rr and the subscript rθ a shear stress and denoted

by τ rθ .

Crystal defects 115 with all other stresses equal to zero. The field of a screw dislocation is therefore purely one of shear,

having radial symmetry with no dependence on θ. This mathematical description is related to the structure of a screw which has no extra half-plane of atoms and cannot be identified with a particular slip plane.

An edge dislocation has a more complicated stress and strain field than a screw. The distortion associated with the edge dislocation is one of plane strain, since there are no displacements along the z-axis, i.e. w = 0. In plane deformation the only stresses to be determined are the normal stresses σ xx ,σ yy along the x- and y-axes respectively, and the shear stress τ xy which acts in the direction of the y-axis on planes perpendicular to the x-axis. The third normal stress σ zz = ν(σ xx +σ yy ) where ν is Poisson’s ratio, and the other shear stresses τ yz and τ zx are zero. In polar coordinates r, θ and z, the stresses are σ rr ,σ θθ , and τ rθ .

Even in the case of the edge dislocation the displacement b has to be accommodated round a ring of length 2πr, so that the strains and the stresses must contain a term in b/2πr. Moreover, because the atoms in the region 0 < θ < π are under compression and for π < θ < 2π in tension, the strain field must be of the form (b/2πr) f (θ), where f (θ) is a function such as sin θ which changes sign when θ changes from 0 to 2π. It can be shown that the stresses are given by

σ rr =σ θθ = −D sin θ/r; σ rθ = D cos θ/r;

y(3x 2 2 ) y(x 2 +y 2 −y ) σ xz = −D 2

+y 2 ) 2 ; (x σ yy =D (x 2 +y 2 ) 2 (3.9) x(x 2 2

σ −y )

xy =D (x 2 +y 2 2

) where D = μb/2π(1 − v). These equations show that the stresses around dislocations fall off as 1/r

and hence the stress field is long range in nature.

3.3.5.2 Strain energy of a dislocation

A dislocation is a line defect extending over large distances in the crystal and, since it has a strain energy per unit length (J m −1 ), it possesses a total strain energy. An estimate of the elastic strain energy of screw dislocation can be obtained by taking the strain energy (i.e. 1 2 × stress × strain per unit

volume) in an annular ring around the dislocation of radius r and thickness dr to be 1 2 × (μb/2πr) × (b/2πr) × 2πrdr. The total strain energy per unit length of dislocation is then obtained by integrating from r 0 (the core radius) to r (the outer radius of the strain field) and is:

μ b 2 r dr

(3.10) 4π r 0 r

ln

With an edge dislocation this energy is modified by the term (1 − v) and hence is about 50% greater

0 ]∼ so that the energy is approximately µb 2

= 0.25 nm, r ∼ = 2.5 µm and ln[r/r = 9.2, per unit length of dislocation, which for copper (taking µ = 40 GN m −2 ,b = 0.25 nm and 1 eV = 1.6 × 10 −19 J) is about 4 eV for every atom plane threaded

than a screw. For a unit dislocation in a typical crystal r 0 ∼

by the dislocation. 8 If the reader prefers to think in terms of 1 meter of dislocation line, then this length is associated with about 2 × 10 10 electron volts. We shall see later that heavily deformed metals contain approximately 10 16 mm −3 of dislocation line, which leads to a large amount of energy stored

8 The energy of the core must be added to this estimate. The core energy is about μb 2 /

10 or 0.5 eV per atom length.

116 Physical Metallurgy and Advanced Materials

Slip plane

h = b/⌰

Slip plane

(b) Figure 3.22 Interaction between dislocations not on the same slip plane: (a) unlike dislocations;

(a)

(b) like dislocations. The arrangement in ( b) constitutes a small-angle boundary. in the lattice (i.e. ≈4Jg −1 for Cu). Clearly, because of this high line energy a dislocation line will

always tend to shorten its length as much as possible, and from this point of view it may be considered to possess a line tension, T

2 ≈ αµb 1 , analogous to the surface energy of a soap film, where α ≈

3.3.5.3 Interaction of dislocations

The strain field around a dislocation, because of its long-range nature, is also important in influencing the behavior of other dislocations in the crystal. Thus, it is not difficult to imagine that a positive dislocation will attract a negative dislocation lying on the same slip plane in order that their respective strain fields should cancel. Moreover, as a general rule it can be said that the dislocations in a crystal will interact with each other to take up positions of minimum energy to reduce the total strain energy of the lattice.

Two dislocations of the same sign will repel each other, because the strain energy of two dislocations on moving apart would be 2

×b 2 , whereas if they combined to form one dislocation of Burgers vector 2b, the strain energy would then be (2b) 2 = 4b 2 ; a force of repulsion exists between them. The force

is, by definition, equal to the change of energy with position (dE/dr) and for screw dislocations is simply F = μb 2 / 2πr, where r is the distance between the two dislocations. Since the stress field around screw dislocations has cylindrical symmetry the force of interaction depends only on the distance apart, and the above expression for F applies equally well to parallel screw dislocations on neighboring slip planes. For parallel edge dislocations the force–distance relationship is less simple.

When the two edge dislocations lie in the same slip plane the relation is similar to that for two screws and has the form F = μb 2 / (1 − ν)2πr, but for edge dislocations with the same Burgers vector but not on the same slip plane the force also depends on the angle θ between the Burgers vector and the line joining the two dislocations (Figure 3.22a).

Edge dislocations of the same sign repel and opposite sign attract along the line between them, but the component of force in the direction of slip, which governs the motion of a dislocation, varies with

Crystal defects 117 the angle θ. With unlike dislocations an attractive force is experienced for θ > 45 ◦ but a repulsive

force for θ < 45 ◦ , and in equilibrium the dislocations remain at an angle of 45 ◦ to each other. For like dislocations the converse applies and the position θ = 45 ◦ is now one of unstable equilibrium. Thus, edge dislocations which have the same Burgers vector but which do not lie on the same slip plane will

be in equilibrium when θ = 90 ◦ , and consequently they will arrange themselves in a plane normal to the slip plane, one above the other a distance h apart. Such a wall of dislocations constitutes a small-angle grain boundary, as shown in Figure 3.22b, where the angle across the boundary is given by θ = b/h. This type of dislocation array is also called a sub-grain or low-angle boundary, and is important in the annealing of deformed metals.

By this arrangement the long-range stresses from the individual dislocations are cancelled out beyond a distance of the order of h from the boundary. It then follows that the energy of the crystal boundary will be given approximately by the sum of the individual energies, each equal

to {μb 2 / 4π(1 − ν)} ln(h/r 0 ) per unit length. There are 1/h or θ/b dislocations in a unit length, verti- cally, and hence, in terms of the misorientation across the boundary θ = b/h, the energy γ gb per unit area of boundary is

=E 0 θ [A − ln θ], where E 0 = μb/4π(1 − ν) and A = ln(b/r 0 ); this is known as the Read–Shockley formula. Values

from it give good agreement with experimental estimates, even up to relatively large angles. For θ ∼ 25 ◦ ,γ gb ∼ μb/25 or ∼ 0.4 J m −2 , which surprisingly is close to the value for the energy per unit area of a general large-angle grain boundary.

3.3.6 Dislocations in ionic structures

The slip system which operates in materials with NaCl structure is predominantly a/2 The closest packed plane {1 0 0} is not usually the preferred slip plane because of the strong elec-

trostatic interaction that would occur across the slip plane during slip; like ions are brought into neighboring positions across the slip plane for (1 0 0) but not for (1 1 0). Dislocations in these mate- rials are therefore simpler than fcc metals, but they may carry an electric charge (the edge dislocation on {1 1 0}, for example). Figure 3.23a has an extra ‘half-plane’ made up of a sheet of Na + ions and one of Cl − ions. The line as a whole can be charged up to a maximum of e/2 per atom length by acting as a source or sink for point defects. Figure 3.23b shows different jogs in the line which may either carry a charge or be uncharged. The jogs at B and C would be of charge +e/2 because the section BC has a net charge equal to e. The jog at D is uncharged.