Cross plane p-,,v sli Zl~im~ry p plane
4.6 Defect behaviour in some real materials
4.6.1 Dislocation vector diagrams and the Thompson tetrahedron
The classification of defects into point, line, planar and volume is somewhat restrictive in presenting an overview of defect behaviour in materials, since it is clear, even from the discussion so far, that these defects are interrelated and interdependent. In the following
Defects in solids 105 sections these features will be brought out as well as those which relate to specific structures.
In dealing with dislocation interactions and defects in real material it is often convenient to work with a vector notation rather than use the more conven- tional Miller indices notation. This may be illustrated by reference to the fcc structure and the Thompson tetrahedron.
All the dislocations common to the fcc structure, discussed in the previous sections, can be represented conveniently by means of the Thompson reference tetrahedron (Figure 4.36a), formed by joining the three nearest face-centring atoms to the origin D. Here
w ~
ABCD is made up of four { 1 1 1 } planes (1 1 1 ), (1 1 1), (1 1 1) and (1 1 1) as shown by the stereogram given in Figure 4.36b, and the edges AB, BC, C A . . . corre- spond to the (1 1 0) directions in these planes. Then, if the mid-points of the faces are labelled c~,/~, y, 8, as shown in Figure 4.37a, all the dislocation Burgers vectors are represented. Thus, the edges (AB, B C . . . ) correspond to the normal slip vectors, a / 2 ( 1 10}. The half-dislocations, or Shockley partials, into which these are dissociated have Burgers vectors of the a/6(1 12) type and are represented by the Roman-Greek sym- bols Ay, By, Dy, A~, BS, etc, or Greek-Roman sym- bols yA, yB, yD, 8A, 8B, etc. The dissociation reaction given in the first reaction in Section 4.4.3.2 is then simply written
BC ~ B8 + 8C
and there are six such dissociation reactions in each of the four { 1 1 1} planes (see Figure 4.37). It is conven- tional to view the slip plane from outside the tetrahe- dron along the positive direction of the unit dislocation BC, and on dissociation to produce an intrinsic stack- ing fault arrangement; the Roman-Greek partial B8 is on the right and the Greek-Roman partial 8C on the left. A screw dislocation with Burgers vector BC which is normally dissociated in the 8-plane is capable of cross-slipping into the a-plane by first constricting B8 + 8C ~ BC and then redissociating in the a-plane BC ~ Ba + uC.
(a) ~ z (b)
I
! a _ . . [ t , o l
~ [t~ol
"
- - ' - I _ i " _ _ I
/ C
I [ , , o 1 x
Figure 4.36 (a) Construction and (b) orientation of the Thompson tetrahedron ABCD. The slip directions in a given {1 1 1 } plane may be obtained from the trace of that plane as shown for the (! 1 1 ) plane in (b).
106 Modern Physical Metallurgy and Materials Engineering
A- B
7
(a)
B
(b) D - <irO]' C [i'TO>
Figure 4.37 A Thompson tetrahedron (a) closed and (b) opened out. In (b) the notation [1 1 O) is used in place of the usual notation [1 1 O] to indicate the sense of the vector direction.
4.6.2 Dislocations and stacking faults in fcc s t r u c t u r e s
4.6.2.1 F r a n k loops
A powerful illustration of the use of the Thompson tetrahedron can be made if we look at simple Frank loops in fcc metals (see Figure 4.32a). The Frank par- tial dislocation has a Burgers vector perpendicular to the (1 1 1) plane on which it lies and is represented by Act, B/~, Cy, D3, ctA, etc. Such loops shown in the electron micrograph of Figure 4.38 have been pro- duced in aluminium by quenching from about 600~
Each loop arises from the clustering of vacancies into a disc-shaped cavity which then form a dislocation loop. To reduce their energy, the loops take up reg- ular crystallographic forms with their edges parallel to the (1 1 0) directions in the loop plane. Along a (1 1 0) direction it can reduce its energy by dissociating on an intersecting { 1 1 1} plane, forming a stair-rod at the junction of the two { 1 1 1 } planes, e.g. Act ~ A3 + 3ct
o
when the Frank dislocation lies along [ 1 0 1 ] common to both ct- and ~-planes.
Some of the loops shown in Figure 4.38 are not Frank sessile dislocations as expected, but prismatic dislocations, since no contrast of the type arising from stacking faults, can be seen within the defects. The fault will be removed by shear if it has a high stacking fault energy thereby changing the sessile Frank loop into a glissile prismatic loop according to the reaction
a/3[1 1 1] + a/6[1 1 2] ~ a/2[1 1 0]
Stressing the foil while it is under observation in the microscope allows the unfaulting process to be
Figure 4.38 Single-faulted, double-faulted (A ) and unfaulted (B) dislocation loops in quenched aluminium (after Edington and Smallman, 1965; courtesy of Taylor and Francis).
observed directly (see Figure 4.39). This reaction is more easily followed with the aid of the Thompson tetrahedron and rewritten as
D8 + ~C -+ DC
Physically, this means that the disc of vacancies aggre- gated on a (1 1 1) plane of a metal with high stack- ing fault energy, besides collapsing, also undergoes a shear movement. The dislocation loops shown in Figure 4.39b are therefore unit dislocations with their Burgers vector a/2[1 1 0] inclined at an angle to the original (1 1 1) plane. A prismatic dislocation loop lies on the surface of a cylinder, the cross-section of which is determined by the dislocation loop, and the axis of which is parallel to the [1 1 0] direction. Such a dislo- cation is not sessile, and under the action of a shear stress it is capable of movement by prismatic slip in the [1 1 0] direction.
Many of the large Frank loops in Figure 4.38 (for example, marked A) contain additional triangular- shaped loop contrast within the outer hexagonal loop.
Figure 4.39 Removal of the stacking fault from a Frank sessile dislocation by stress (after Goodhew and Smalhnan).
'--~ B -C ~ C ' ~ - - - ~ ~ " - ~ ' - 9
--~-~ A~~'~ "~""'~-~-A = -~.., A B ~ , ~ B ~ ~ ~ ' - " ~ ' ~ ~ "
a , . - ~ " A ~
,, i - . . . .
---~-c --'~" B ~_---~~_ "~'~"
Figure 4.40 The structure of a double dislocation loop in quenched aluminium (after Edington and Smallman, 1965;
courtesy of Taylor and Francis).
The stacking fault fringes within the triangle are usu- ally displaced relative to those between the triangle and the hexagon by half the fringe spacing, which is the contrast expected from overlapping intrinsic stacking faults. The structural arrangement of those double- faulted loops is shown schematically in Figure 4.40, from which it can be seen that two intrinsic faults on next neighbouring planes are equivalent to an extrinsic fault. The observation of double-faulted loops in aluminium indicates that it is energetically more favourable to nucleate a Frank sessile loop on an exist- ing intrinsic fault than randomly in the perfect lattice, and it therefore follows that the energy of a double or extrinsic fault is less than twice that of the intrinsic fault, i.e. ~ < 2Fl. The double loops marked B have the outer intrinsic fault removed by stress.
The addition of a third overlapping intrinsic fault would change the stacking sequence from the per- fect ABCABCABC to ABC ,1, B ~, A ,1, CABC, where the arrows indicate missing planes of atoms, and pro- duce a coherent twinned structure with two coherent twin boundaries. This structure would be energetically favourable to form, since Ytwi, < Yl < ~ . It is pos- sible, however, to reduce the energy of the crystal even further by aggregating the third layer of vacan- cies between the two previously-formed neighbouring intrinsic faults to change the structure from an extrin- sically faulted ABC $ B $ ABC to perfect ABC $ $,1, ABC structure. Such a triple-layer dislocation loop is shown in Figure 4.41.
4.6.2.2 Stair-rod dislocations
The stair-rod dislocation formed at the apex of a Lomer-Cottrell barrier can also be represented by the Thompson notation. As an example, let us take the interaction between dislocations on the 8- and c~- planes. Two unit dislocations BA and DB, respectively, are dissociated according to
BA ---, B& + &A (on the &-plane) and DB --+ Dc~ + otB (on the c~-plane)
and when the two Shockley partials orB and B& inter- act, a stair-rod dislocation c~& = a/6[ 1 01] is formed.
This low-energy dislocation is pure edge and there- fore sessile. If the other pair of partials interact then the resultant Burgers vector is (3A + Dot) = a/3[ 1 0 1 ]
Defects in solids 107
Figure 4.41 Triple-loop and Frank sessile loop in A1-0.65%
Mg (after Kritzinger, Smallman and Dobson, 1969; courtesy of Pergamon Press).
and of higher energy. This vector is written in Thomp- son's notation as 8D/Ac~ and is a vector equal to twice the length joining the midpoints of ~A and Dot.
4.6.2.3 Stacking-fault tetrahedra
In fcc metals and alloys, the vacancies may also clus- ter into a three-dimensional defect, forming a tetra- hedral arrangement of stacking faults on the four {1 1 1} planes with the six (1 10) edges of the tetrahe- dron, where the stacking faults bend from one { 1 1 1 } plane to another, consisting of stair-rod dislocations.
The crystal structure is perfect inside and outside the tetrahedron, and the three-dimensional array of faults exhibits characteristic projected shape and contrast when seen in transmission electron micrographs as shown in Figure 4.44. This defect was observed orig- inally in quenched gold but occurs in other materials with low stacking-fault energy. One mechanism for the formation of the defect tetrahedron by the dissociation of a Frank dislocation loop (see Figure 4.42) was first explained by Silcox and Hirsch. The Frank partial dis- location bounding a stacking fault has, because of its large Burgers vector, a high strain energy, and hence can lower its energy by dissociation according to a reaction of the type
a/3[ l l l ] ~ a / 6 [ 1 2 1 ] + a/6[ l O1]
(')
The figures underneath the reaction represent the ener- gies of the dislocations, since they are proportional to the squares of the Burgers vectors. This reaction is, therefore, energetically favourable. This reaction can be seen with the aid of the Thompson tetrahedron, which shows that the Frank partial dislocation Aot can dissociate into a Shockley partial dislocation (Aft, A&
or A y) and a low energy stair-rod dislocation (flot, &ot or ~ ) for example Aot ~ A y + yot.
108 Modern Physical Metallurgy and Materials Engineering
(a) (b) E3 (c) a
Figure 4.42 Formation of defect tetrahedron: (a) dissociation of Frank dislocations. (b) formation of new stair-rod dislocations, and (c) arrangement of the six stair-rod dislocations.
The formation of the defect tetrahedron of stacking faults may be envisaged as follows. The collapse of a vacancy disc will, in the first instance, lead to the formation of a Frank sessile loop bounding a stacking fault, with edges parallel to the (1 1 0) directions.
Each side of the loop then dissociates according to the above reaction into the appropriate stair-rod and partial dislocations, and, as shown in Figure 4.42a, the Shockley dislocations formed by dissociation will lie on intersecting {1 1 1} planes, above and below the plane of the hexagonal loop; the decrease in energy accompanying the dissociation will give rise to forces which tend to pull any rounded part of the loop into (1 10). Moreover, because the loop will not in general be a regular hexagon, the short sides will be eliminated by the preferential addition of vacancies at the constricted site, and a triangular-shaped loop will form (Figure 4.42b). The partials A~, A y and A~ bow out on their slip plane as they are repelled by the stair- rods. Taking into account the fact that adjacent ends of the bowing loops are of opposite sign, the partials attract each other in pairs to form stair-rod dislocations along DA, BA and CA, according to the reactions
yA + A ~ ---> yB, ~A + A y ---> ~y, ~A + A~ ---~ ~ In vector notation the reactions are of the type
a/6[1 1 2] + a/6[l 2 1] ---> a/6[0 1 1]
I 1
(the reader may deduce the appropriate indices from Figure 4.37), and from the addition of the squares of the Burgers vectors underneath it is clear that this reac- tion is also energetically favourable. The final defect will therefore be a tetrahedron made up from the inter- section of stacking faults on the four {1 1 1} planes, so that the (1 1 0) edges of the tetrahedron will consist of low-energy stair-rod dislocations (Figure 4.42c).
The tetrahedron of stacking faults formed by the above sequence of events is essentially symmetrical, and the same configuration would have been obtained if collapse had taken place originally on any other (1 1 1) plane. The energy of the system of stair-rod dislocations in the final configuration is proportional
_ _ 1 I ~ .
to 6 x ~8 5, compared with 3 x ~ 1 for the orig- inal stacking fault triangle bounded by Frank partials.
Considering the dislocation energies alone, the disso- ciation leads to a lowering of energy to one-third of the original value. However, three additional stacking fault areas, with energies of y per unit area, have been newly created and if there is to be no net rise in energy these areas will impose an upper limit on the size of the tetrahedron formed. The student may wish to ver- ify that a calculation of this maximum size shows the side of the tetrahedron should be around 50 nm.
De Jong and Koehler have proposed that the tetra- hedra may also form by the nucleation and growth of a three-dimensional vacancy cluster. The smallest cluster that is able to collapse to a tetrahedron and subsequently grow by the absorption of vacancies is a hexa-vacancy cluster. Growth would then occur by the nucleation and propagation of jog lines across the faces of the tetrahedron, as shown in Figure 4.43. The hexa- vacancy cluster may form by clustering di-vacancies and is aided by impurities which have excess positive change relative to the matrix (e.g. Mg, Cd or A1 in Au). Hydrogen in solution is also a potent nucleating agent because the di-vacancy/proton complex is mobile and attracted to 'free' di-vacancies. Figure 4.44 shows the increase in tetrahedra nucleation after preannealing gold in hydrogen.
4.6.3 Dislocations and stacking faults in cph structures
In a cph structure with axial ratio c/a, the most closely packed plane of atoms is the basal plane ( 0 0 0 1 )
Figure 4.43 Jog line forming a ledge on the face of a tetrahedron.
Defects in solids 109
S
A
T
m
and the most closely packed directions (1 120). The smallest unit lattice vector is a, but to indicate the direction of the vector (u, v, w) in Miller-Bravais indices it is written as a/3(1 1 20) where the mag- nitude of the vector in terms of the lattice param- eters is given by a[3(u 2 + uv + v 2) + (c/a)2w2] 1/2.
The usual slip dislocation therefore has a Burgers vector a/3(I 120) and glides in the ( 0 0 0 1) plane.
This slip vector is (a/3, a/3, 2(a/3), 0) and has no
Figure 4.45 Burgers vectors in the cph lattice (after Berghezan, Fourdeux and Amelinckx, 1961" courtesy of Pergamon Press).
component along the c-axis and so can be writ- ten without difficulty as a / 3 ( 1 1 2 0 ) . However, when the vector has a component along the c-axis, as for example (a/3, a / 3 , 2 ( a / 3 ) , 3c), difficulty arises and to avoid confusion the vectors are referred to unit distances (a, a,a, c) along the respective axes (e.g.
1 / 3 ( 1 1 2 0 ) and 1 / 3 ( 1 1 2 3 ) ) . Other dislocations can be represented in a notation similar to that for the fcc structure, but using a double-tetrahedron or bipyramid instead of the single tetrahedron previously adopted, as shown in Figure 4.45. An examination leads to the following simple types of dislocation:
1. Six perfect dislocations with Burgers vectors in the basal plane along the sides of the triangular base ABC. They are AB, BC, CA, BA, CB and AC and
m
are denoted by a or 1 / 3 ( 1 1 2 0 ) .
2. Six partial dislocations with Burgers vectors in the basal plane represented by the vectors Aty, Bor, Ca and their negatives. These dislocations arise from dissociation reactions of the type
AB -~ At~ + oB
l (10]-0).
and may also be written as p or
3. Two perfect dislocations perpendicular to the basal plane represented by the vectors ST and TS of magnitude equal to the cell height c or ( 0 0 0 1 ) . 4. Partial dislocations perpendicular to the basal plane
represented by the vectors ~S, t~T, Scr, Tcr of (0001).
magnitude c/2 or