Point force models of primitive dislocation loops

Scripta Materialia, Vol. 39, Nos. 4/5, pp. 409 – 415, 1998 Elsevier Science Ltd Copyright © 1998 Acta Metallurgica Inc. Printed in the USA. All rights reserved. 1359-6462/98 $19.00 1 .00

Pergamon

PII S1359-6462(98)00215-2

POINT FORCE MODELS OF PRIMITIVE DISLOCATION LOOPS Craig S. Hartley Guest Researcher, Metallurgy Division, MSEL, National Institute for Standards and Technology and Professor, Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, FL 33431 (Received in final form May 15, 1998) Introduction Local continuum models of crystal defects suffer from the disadvantage of being unable to include information on the crystal structure of the medium. However, non-local models can be constructed by simulating the defect with an appropriate array of forces located on the defect’s neighboring atoms. The locations of the forces are determined by the positions of the nearest neighbor atoms to the defect and their magnitudes and directions are adjusted to produce the same long-range displacement field as the local continuum model of the defect. This model was first proposed by Kanzaki (1), and was later developed by Bullough and Hardy (2), who applied it successfully to obtain the atomic displacement fields around vacancies in copper and aluminum. Displacements and energies are calculated from the theory of an elastic continuum, where now the defect is not a point, but a non-local distribution of body force. The validity of this approach depends on a suitable choice for the forces which define the defect. It is also essential that the extent of the force array be as restricted as possible. Properties of defects obtained with non-local models can often be expressed in closed form, leading to analyses of the effects of elastic anisotropy and crystal structure which would be impossible with local continuum models. Similar information obtained with lattice models requires considerably more computation. The technique thus provides a bridge between local continuum models and lattice models, which can provide information on the role of crystal structure without the necessity for extensive numerical calculations. The displacement field of finite dislocations in an elastic medium can be constructed by the superposition of the displacement fields of point nuclei of strain (3, 4, 5, 6). In local models of such nuclei, the crystal structure of the medium only determines the Burgers vector and loop plane for the displacement dipole constituting the nucleus. However, a non-local model proposed by Hartley and Bullough (7) incorporates information on the crystal structure by defining a “primitive dislocation loop.” The primitive loop consists of an array of forces applied to atoms in the coordination cluster* which produce the displacement field of an infinitesimal dislocation loop located at the centroid of the array. The area of the loop is the area of the loop plane intersected by the coordination cluster. This construction does not correspond to an actual point defect in a crystal, but provides a concept which can

* The coordination cluster is defined as the array of nearest neighbor atoms surrounding a single atom taken as origin.

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be used to build extended defects by superposition in the same manner as unit cells are used to construct the crystal lattice. Models of dislocations constructed by superimposing primitive loops contain information on the non-local nature of the dislocation core, which is not available from the local continuum model. Since displacements are only defined for discrete atomic positions in the vicinity of the dislocation line, which does not pass through an atomic position, dislocations formed by superposition of primitive loops do not have a singularity in the stress and strain fields at the location of the dislocation line. Finally, the self energy of the dislocation can be obtained by summation of the self energies of the forces and their pairwise interaction energies. The interaction energy of the dislocation with other crystal defects constructed from non-local force models can be obtained by a similar summation of pairwise interaction energies. Formation of extended defects by superposition of primitive loops will not be treated in this work, but the process is illustrated for an isotropic, simple cubic lattice in reference 7. Primitive Dislocation Loops For the present purposes, we will consider forces applied only on nearest neighbors to the atom at the origin.† Directions and magnitudes are assigned to the forces so that the long-range displacement field of the array matches that of an appropriate infinitesimal dislocation loop, i.e. the dipole tensor of the force array is the same as that for the elastic dipole which constitutes the infinitesimal loop. The displacement, ui(rW), at a field point, Wr, due to an array of N point forces, f(,a)(rW(a)), is (8):

Of N

ui(rW) 5

(a) ,

(rW(a))Gi,(rW 2 Wr(a))

(1)

a51

where the ath force is located at Wr(a) from the origin and Gi,(rW 2 Wr9) is the Green’s Tensor function for elasticity. In equation (1) et seq. summation from 1 to 3 over repeated latin suffixes is implied. Expanding this expression about the origin yields

HO J HO J N

ui(rW) 5

N

~ a! ,

f

a51

f(,a)x~ka! Gi,,k(rW) 1 . . .

Gi,(rW) 1

(2)

a51

designating by a subscript comma partial differentiation with respect to the indicated spatial variable. Translational equilibrium requires that the first sum vanish and rotational equilibrium requires that the second sum be a symmetric, second rank tensor. We do not consider higher moments of the array in this work. Take the origin as the centroid of the array and assume that the vectors from the origin to the points of application of the forces are small compared with the distance to the field point. The displacement field of an infinitesimal dislocation loop located at the origin and having Burgers vector, bi, and area njdS is (3), um(rW) 5 [Cijk,binjdS]Gm,,k(rW)

(3)

where Cijk, is the elastic constant tensor. Now we can relate the second sum in equation (2) to the elastic constants and loop parameters by comparing equations (1) and (3):

† Although the concept is easily extended to second and higher neighbors, these extensions require additional assumptions about the relationships among forces on the various levels of neighboring atoms.

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411

Of

(4)

N

(a) (a) , k

x

5 Pk, 5 CijklbinjdS,

a51

where the vectors from the origin to the points of application of the forces connect the nearest neighbors to the atom at the origin and Pk, is the dipole tensor of the force array. In many crystal structures the atom at the center of the coordination cluster is located at a center of symmetry. Thus for each vector, Wr(a), there is a corresponding vector equal and opposite to it connecting a neighbor on the other side of the center of symmetry. Since the sum of all of the forces acting on nearest neighbor atoms to the defect must vanish, we can require that the forces occur in pairs, equal in magnitude and opposite in direction, applied on atoms located on either side of the center of symmetry. This reduces the number of independent forces by a factor of two and permits the sum to be taken over only half of the atoms in the cluster. Directions of the forces are chosen based on the symmetry of the defect itself, i.e. the Burgers vector and normal to the loop plane. No attempt is made in this work to prove uniqueness for the force arrays proposed. However, by St. Venant’s principle, any statically equivalent array will produce the same long-range displacement field and the same net force array characterizing an extended defect formed by superposition. The elastic constant tensor of an anisotropic cubic crystal can be expressed in terms of three independent elastic constants, C11, C12 and C44, using the conventional contracted notation for elastic constants referred to the cube axes (9). By equation (4) the dipole tensor for a primitive loop in such a material is:

S D

Pk, 5 C11[s1n1dkld,l 1 s2n2dk2d,2 1 s3n3dk3d,3] ubidSu 1 C12[s1n1(dk2d,2 1 dk3d,3) 1 s2n2(dk1d,1 1 dk3d,3) 1 s3n3(dkld,1 1 dk2d,2)] 1 C44[s1n3 1 s3n1)(dk1d,3 1 dk3d,1) 1 (s1n2 1 s2n1)(dk1d,2 1 dk2d,1) 1 (s2n3 1 s3n2)(dk2d,3 1 dk3d,2)]

(5)

As with any symmetrical, second rank tensor, isotropic and deviatoric components can be obtained, and eigenvectors can be found. In general, however, we find that it is most convenient to work with Equation (5) directly in determining the values of the forces on atoms surrounding the defect. Dislocations in BCC Crystals The coordination cluster for the body-centered cubic (BCC) lattice is the unit cell for the structure, shown in Figure 1. Taking the atom at the center of the cube as the origin, the atoms at the cube corners are numbered as shown. Choosing the forces to be in pairs as described in the previous section and noting that all vectors from the origin to nearest neighbors are of the form a/2,111. permits us to write the dipole tensor for the array as

Of 4

Pk, 5 a

(a) (a) , k

v

(6)

a51

where v(ka) are the direction numbers of the vectors from the atom at the origin to the ath near neighbor. Using the appropriate direction numbers for the nearest neighbors gives

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Figure 1. Coordination cluster for BCC. (2) (3) (4) P1, 5 a(f(1) , 2 f, 2 f, 1 f, ) (2) (3) (4) P2, 5 a(f(1) , 1 f, 2 f, 2 f, )

, 51 . . . 3

(7)

(2) (3) (4) P3, 5 a(f(1) , 1 f, 1 f, 5 f, )

a set of nine equations in twelve unknowns, three components for each of the four forces. Equations (7) can be solved for the force components on any three atoms in terms of those on the fourth. Typical of these relationships are the equations: (1) f(2) , 5 2f, 1

(1) f(3) , 5 f, 1

(P2, 1 P3,) 2a

(P1, 1 P2,) 2a

(1) f(4) , 5 2f, 1

(8)

(P1, 1 P3,) 2a

Choosing appropriate combinations of Burgers vector and loop normal to substitute in equation (4) permits calculation of the components of the dipole tensor to substitute in equation (8). Solutions for the components, f(1) , , require additional assumptions about the directions of the forces, as illustrated in the following sections. Glide Loops On {110} Planes Consider a loop with Burgers vector (a/2)[111] lying on the (11# 0) plane. Then relevant parameters for the loop become sk 5

(dk1 1 dk2 1 dk3) ; Î3

nj 5

(dj1 2 dj2) ; Î2

a Î3 ubW u 5 ; 2

referred to the cube axes. By equation (5) the dipole tensor becomes

udSu 5 Î2 a2;

(9)

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3

C44 a 2

7

P5

3

2 A

0

4

1

0

2 21 2 A

1

21

413

0

(10)

in terms of the Anisotropy Ratio, A 5 2C44/(C11-C12). It is useful to note at this point that choosing other combinations of the same form for the Burgers vector and loop normal is equivalent to renumbering the atoms in the coordination cluster and results in the same terms in the dipole tensor with positions changed corresponding to the numbering system for the atoms. In order to determine the force components on atom 1 we make the following assumptions: 1. the force on 2 is parallel to the slip direction, [111]; and 2. the force on 4 is parallel to the Taylor axis, [112# ]. These conditions result in three independent equations which can be solved for the components of the force on atom 1 using equation (8). A self-consistent set of force components which satisfy these assumptions for the glide loop are: f(1) 1 5

(4 1 5A) ; 2A

(21A) f(1) ; 2 5 2 2A f(1) 3 5

(4 2 A) ; 2A

(2 1 A) f(2) ; 1 5 2 A

f(3) 1 5

(2 1 A) ; A

2(2 2 5A) ; 2A

f(4) 1 5

(4 2 A) f(3) ; 2 5 2 2A

f(2) 2 5 2

(2 1 A) f(2) ; 3 5 2 A

(4 2 A) f(3) ; 3 5 2 2A

f(4) 2 5

(1 2 A) A

(1 2 A) A

f(4) 3 5 22

(11)

(1 2 A) A

in units of C44a2/6. Note that it is possible for some of these force components to vanish for particular values of A. While this set of forces may not be unique for the primitive loop considered, it is easy to show that the components of the force on atom 1 cannot be chosen arbitrarily and satisfy equation (8) simultaneously. Dislocations having Burgers vectors of the form a,100. can also occur on {110} planes in BCC crystals. A primitive loop on the same {110} plane as above with a Burgers vector a[001] has the dipole tensor: 7

P 5 C44a

3

3

0

0

0

0 21

1 21

1 0

4

(12)

We now place the same requirements on the directions of the forces as before, with an appropriate change in the slip direction. The resulting force components, in units of C44a2/2, are f(1) 1 5 1;

f(2) 1 5 0;

f(3) 1 5 1;

f(4) 1 5 0

f(1) 2 5 21;

f(2) 2 5 0;

f(3) 2 5 21;

f(4) 2 5 0

f(1) 3 5 21;

f(2) 3 5 0;

f(3) 3 5 21;

f(4) 3 5 2

(13)

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Note that these forces do not depend on the anisotropy of the material, unlike those forming the previous loop. Prismatic Loops Prismatic loops are formed by the intersection of the loop plane with a cylindrical prism having the Burgers vector as its axis. In general the Burgers vector has components lying in the loop plane and normal to the loop plane. These loops have non-zero hydrostatic components to their dipole tensors, corresponding to the presence of a full or partial vacancy or interstitialcy associated with the defect. When the Burgers vector is normal to the loop plane, the loop is pure prismatic or edge in character; otherwise it is mixed. In BCC prismatic loops of both characters have been observed on {110}, {100} and {111} planes resulting from irradiation (10). In the following sections we construct primitive loops of each character on {110} and {100} planes. On {110} Planes Consider a dislocation loop with Burgers vector (a/2)[111] lying on the (110) plane. This is a mixed loop which can glide on the surface of a prism having the [111] direction as its axis. The loop parameters become: (dj1 1 dj2) ; nj 5 2 Î2

sk 5

(dk1 1 dk2 1 dk3)

Î3

;

udSu 5 Î2 a2;

ubW u 5

a Î3 ; 2

(14)

and the dipole tensor is:

3

C44a 7 P52 2

3

2(1 1 rA) A

2

2

2(1 1 rA) A

1

1

1

2r

1

4

;

r5

C12 . C44

(15)

To determine the components of the force on atom 1 we use equation (8) with the additional conditions: 1. The force on atom 1 is parallel to the Burgers vector, i.e. to [111]; 2. The forces on atoms 2 and 4 are normal to the Burgers vector. These assumptions yield the following array of force components (2 1 4rA 1 5A) (2 1 4rA 2 4A) (4 1 2rA 1 A) f(1) ; f(2) ; f(3) ; 1 5 2 1 5 1 5 A A A

(4 1 2rA 2 2A) f(4) 1 52 A

(2 1 4rA 1 5A) f(1) ; 2 52 A

f(4) 2 5

(4 1 2rA 2 2A) (3) (4 1 2rA 1 A) f(2) ; f2 5 2 ; 2 52 A A

(2 1 4rA 2 4A) A

(2 1 4rA 1 5A) (2 2 2rA 1 2A) (2 1 4rA 2 A) (2 2 2rA 1 2A) f(1) ; f(2) ; f(3) ; f(4) 3 5 2 3 5 3 5 2 3 5 A A A A (16)

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in units of C44 a2/4. Note that the assumptions on the forces on atoms 1, 2 and 4 lead to the result that the force on atom 3 lies in the (11# 0) plane. Discussion The primitive loops described in the preceding sections can be employed to create extended dislocations by superposition as for the local continuum model of dislocations. Displacements due to these force arrays must be computed using the appropriate form of the Green’s Function for anisotropic materials (8). Self- and interaction energies of the extended defects can be computed by the methods described by Hartley and Bullough (7). It is also possible to construct such models of kinks and jogs using the procedures described by Hartley and Georges (11). Finally, the effects of crystal structure on the interaction energies with point defects constructed using non-local models can be calculated (12). This approach provides a rapid and useful method of assessing the effects of elastic anisotropy and crystal structure on properties of crystal defects, forming a bridge between the local continuum model and the more computation-intensive lattice models. Acknowledgments This research was conducted during a sabbatical leave from Florida Atlantic University with research facilities provided by the Metallurgy Division, National Institute of Standards and Technology. The helpful comments of R. deWit, R. Thomson and other colleagues at NIST are gratefully acknowledged. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

H. Kanzaki, Phys. Chem. Solids. 2, 24 (1957). R. Bullough and J. R. Hardy, Philos. Mag., 17, 833 (1968). E. Kro¨ner, Kontinuumstheorie der Versetzungen und Eigenspannungen, Springer Verlag, Berlin (1958). F. Kroupa, in Theory of Crystal Defects, ed. B. Gruber, p. 210, Academia, Prague (1966). J. S. Koehler, J. Appl. Phys., 37, 4351 (1966). P. P. Groves and D. J. Bacon, J. Appl. Phys., 40, 4207 (1969). C. S. Hartley and R. Bullough, J. Appl. Phys., 48, 4557 (1977). C. Teodosiu, Elastic Models of Crystal Defects, Springer-Verlag, New York (1982). J. F. Nye, Physical Properties of Crystals, p. 138, Clarendon Press, Oxford (1986). R. Bullough and R. C. Perrin, Proc. Roy. Soc., A 305, 541 (1968). J.-P. Jacques Georges and C. S. Hartley, in The Structure and Properties of Crystal Defects, ed. V. Paidar and L. Lejcek, p. 384, Elsevier, New York, NY (1984). C. S. Hartley, in Dislocations in Solids, ed. H. Suzuki, T. Ninomiya, K. Sumino and S. Takeuchi, p. 37, University of Tokyo Press, Tokyo (1985).