a computer simulation study of pipe diffusion in body centred cubic metals

Acta Mrrallurgico Vol. 29, pp. 1599 to 1606. 1981 Oool-6160/81iO91599-08602.00/O

Prmted m Great Bntam. All rights reserved Copyright 0 1981 Pergamon Press Ltd

A COMPUTER SIMULATION STUDY OF PIPE DIFFUSION IN BODY CENTRED CUBIC METALS

K. M. MILLER’, K. W. INGLE’t and A. G. CROCKER’

‘Central Electricity Generating Board, Berkeley Nuclear Laboratories, Berkeley, Gloucestershire, GL13 9PB and

*Department of Physics, University of Surrey, Guildford, Surrey, GU2 5XH, U.K.

(Received 19 January 1981; in revisedform 19 Fehruar), 1981)

Abstract-A real-space computer simulation study of vacancy migration along dislocations, known as pipe diffusion, has been carried out for models representing the b.c.c. metals iron and molybdenum. Migration along the cores of two straight edge dislocations with Burgers vector )[I 111 and lying in either the (170) or (ll?) slip planes has been investigated. It has been found that certain vacancy jumps along the dislocation cores require high migration energies. However in all cases the activation energy for pipe diffusion is smaller than for diffusion in the bulk. For both iron and molybdenum this energy is lower for the f-[ill] (112) than for the i[lll] (110) dislocation, being about 60% of the respective values for bulk diffusion.

Rbum&Nous avons effect& une simulation sur ordinateur dans l’espace direct de la migration des lacunes le long des dislocations (diffusion lintaire), pour des modtles reprtsentant les mCtaux C.C. fer et molybdirne. Nous avons ktudik cette migration le long du coeur de deux disloc$ions coil rectilignes de vecteur de Burgers +[ll 11, situbes respectivement dans un plan de glissement (110) ou (112). Nous avons montrk que certains sauts de la lacune le long du coeur de la dislocation nicessitaient une forte knergie de migration. Dans tous les cas toutefois, I’tnergie d’activation de la diffusion 1inCaire est plus faible que celle de la diffusion en volume. pans le fer et dans le molybdine, cettejnergie d’activation est plus petite pour la dislocation )[lll] (112) que pour la dislocation )[ll I] (110); elles valent environ 6OY, des valeurs correspondantes pour l’tnergie de la diffusion en volume.

Zusammenfassung-Die Wanderung von Leerstellen entlang von Versetzungen (pipe diffusion, ‘SchlauchdilTusion’) wurde mit Modellen fiir die k.r.z. Metalle Eisen und MolybdZn auf dem Rechner simuliert. Untersucht wurde die Wanderungentlang den Kernen zweier gerader Versetzungen mit Bur- gersvektor f[lll], die in der (liO)- oder der (llT)-Gleitebene liegen. Es wurde gefunden, da0 die Wanderungsenergie fiir gewisse Leerstellenspriinge hoch ist. In allen FIllen ist die Aktivierungsenergie fiir die_Schlauchdiffusion geringer alsfiir die Volumdiffusion. Diese Energie ist bei der Versetzung auf der (112)-Ebene kleiner als auf der (1 IO)-Ebene; sie betrlgt sowohl in Eisen als such in Molybdln nur etwa 60% der Energie fiir Volumdiffusion.

1. INTRODUCTION tals. For example, results have recently been

Measured rates of diffusion in crystalline materials presented on vacancies, interstitials and stacking

are often much greater than those deduced from faults [l], small vacancy clusters [2] and twist and tilt

mechanisms based on the migration of point defects boundaries [%5] in f.c.c. metals, and on twinning dis-

through the crystal lattice. It is usually assumed that locations [6] and interactions between vacancies and

these enhanced diffusion rates are associated with twin boundaries [7] and between vacancies and slip

extended defects, particularly dislocations and inter- dislocations [S] in b.c.c. metals.

faces. Unfortunately it is difficult to perform experi- Computer simulation studies have also provided in-

ments to examine this assumption and even more formation on the migration of point defects through

demanding to determine directly the relative diffusion the lattice. Thus, for example, the effect of applied

rates along extended defects and through the bulk stress on vacancy migration in b.c.c. metals has been

crystal. However, computer simulation methods investigated [9], quantitative values being obtained

based on empirical interatomic potentials are ideally for the increase in the migration energy caused by

suited to the investigation of atomic mechanisms of compressive stresses perpendicular to the vacancy

this kind. Most applications of this technique have jump. In addition vacancy migration near twin boun-

been aimed at the determination of the structures and daries in b.c.c. metals has been investigated [lo] and

self- and interaction-energies of defects in cubic crys- it has been demonstrated that migration energies parallel to the interface may be nearly 40% less than

Present address: Central Electricity Generating Board, in the bulk. The effect is smaller but still significant in

Generation Development and Construction Division, Bar- f.c.c. crystals [ 111. Larger decreases are anticipated for nett Way, Barnwood, Gloucester, GL4 7RS, U.K. vacancy migration along more general boundaries so

1599

1600 MILLER et al.: PIPE DIFFUSION IN B.C.C. METALS

that these results tend to confirm the important role of interfaces in diffusion processes. However, no re- liable computer simulation results have been published on vacancy diffusion, along dislocations, the mechanism popularly known as pipe diffusion, and the purpose of the present paper is to remedy this deficiency. Two straight edge dislocations in models representing the b.c.c. metals iron and molybdenum are considered. These dislocations have Burgers vector f[lll] and lie in either the (170) or the (112) slip plane. They are thus parallel to the [It?] and [liO] directions respectively. Four distinct groups of simu- lations are necessary for each dislocation in each metal. First, the vacancy formation and migration energies in an otherwise perfect single crystal are needed. Then the self energies and core structures of the dislocations are required. Next the binding energies of vacancies at different locations in the dislocation core must be determined. Finally, the migration energies of vacancies between different sites within the core have to be found. The computational methods used in carrying out the work are described in Section 2 of this paper, the results are then presented in Section 3 and discussed fully in Section 4. In particular the general implications of the results for other dislocations in other crystal structures are examined.

2. COMPUTATIONAL PROCEDURES

Real space computer simulation procedures were used to obtain all of the results in the present study. The defective model b.c.c. crystals were allowed to relax to their equilibrium configurations using lattice handling techniques known as DEVIL developed at AERE, Harwell, and described previously [12]. This relaxation procedure minimises the energy of the model with the atoms at rest so that the results obtained correspond to crystals at absolute zero temperature. The well-established Johnson J,, potential [13] and a new potential due to Miller [14] were used to represent iron and molybdenum respectively. Both of these empirical potentials which are com- pared in Fig. 1, hold the crystals in equilibrium at the correct lattice parameter, without the introduction of a volume dependent component. They are matched to the elastic constants and available data on defects.

The models used were in the form of rectangular blocks of atoms with (ill), (170) and (112) faces. Vacancy formation and migration energies were determined, as in previous studies [7-111, by removing an atom from the centre of the model and by displac- ing a neighbouring atom, by means of a sequence of small steps, into the vacant site. Following each of these steps the crystal was allowed to relax fully, except that the migrating atom had to be restrained from moving either back to its initial position or for- wards to its final position. The edge dislocations were introduced through the centre of the models by dis- placing each atom by an amount calculated from ani-

B

I I I I 06 0.8 1.0 1.2

?“a

Fig. 1. The empirical interatomic potentials I$, in electron volts, fol iron and molybdenum which were used in the calculations. The positions of the first and second nearest neighbours are indicated on the r/n axis, where a is the

edge of the unit cell.

sotropic elasticity theory. Periodic boundary conditions were then imposed on the pair of model faces perpendicular to the dislocation line and rigid boundary conditions on the other two pairs of faces. Thus for the i[lll] (170) and )[lll] (112) dislocations the (112) and (110) boundaries respectively were periodic, al1 other boundaries being fixed. Effectively the dislocations were then infinitely along. A mantle of atoms was placed outside the rigid boundaries so that sites near the edges of the computational cell had a full complement of neighbours.

Up to lo4 atoms were used in the models, the detailed dimensions depending on the defects being examined. In order to minimise the approximations arising from the fixed boundaries it was necessary to have these as remote as possible from the dislocations. This was particul~ly true for boundaries perpendicular to the Burgers vector because of possible core-spreading in the slip plane. In addition any vacancy present in the computational cell was auto- matically converted into an infinite row of vacancies by the periodic boundary conditions. Thus, in order to eliminate interactions between these vacancies the dimension of the model parallel to the dislocation line had to be as large as possible. Experiments indicated that these requirements were met satisfactorily by models containing 36 (ill), 14 (110) and 18 (112) planes for the j-[ 11 l] (li0) dislocations and 50 (1 ll), 8

MILLER et al.: PIPE DIFFUSION IN B.C.C. METALS 1601

(110) and 36 (112) planes for the f[lll] (112) dislocations. No volume changes were allowed during the relaxation and vacancy migration processes. How- ever, the atom being displaced into a neighbouring vacant site was allowed to follow the lowest energy route using a method introduced earlier [ll]. This is particularly important when migration is occurring in a highly distorted structure, such as a dislocation core.

3. RESULTS

The vacancy formation energies EL and migration energies Er resulting from the present study are summarised in Table 1. When a vacancy is located in an otherwise perfect crystal P, EL (P) and Er(PP) are 1.37 eV and 0.68 eV respectively for iron and 2.42 eV and 1.39 eV for molybdenum. Thus the activation energy for bulk self diffusion Q(P) = EL (P) + Er (PP) is 2.05 eV for iron and 3.81 eV for molybdenum.

The relaxed structure of the i[lll] (170) edge dislocation in iron is shown in Fig. 2(a) projected on to the (112) plane which is perpendicular to the dislocation line. In this diagram no attempt has been made to distinguish between atomic sites in the six (112) planes which form the stacking sequence. Thus, although (111) is not a mirror plane, the twofold axis normal to (110) makes the projection appear symmetric about the vertical line through the centre of the dislocation. As the interplanar spacing of the (111) planes is one-third of the magnitude of the Burgers vector three extra half-planes are associated with the dislocation. The positions of these are indicated using conventional symbols in the figure and in particular the centre of the dislocation consists of a single extra half-plane above the slip plane. The vacancy site with the lowest formation energy lies at the edge of this half-plane and is labelled A. However, in order to

Table 1. Vacancy formation energies E{ and migration energies Er

EL E::

Fe MO Fe MO P 1.37 2.42 PP 0.68 1.39

A 0.67 1.25 AB 1.09 1.65 B 0.69 1.29 BC 1.09 1.60 C 0.75 1.29 CD 0.10 1.61 D 0.69 1.25 DA’ 1.07 0.25

; 0.15 1.08 0.23 2.02 Y 1.09 2.20

_%----&&____ploi ‘lY

1’ 0.15 0.23 YE’ 0.62 0.36

The energies are given in electron volts for sites in a perfect crystal (P), near a )[I1 l] (110) edge dislocation (A, B, C. D) and near a )[ll l] (112) edge dislocation (1./l. Y, a’) in iron and molybdenum. The locations A-D and a-a’ and the corresponding jumps are defined in Figs 225. The broken line separates results for the alternative a//j’ and a/y mechanisms for the )[l 1 l] (112) dislocation.

migrate by means of nearest neighbour )( 111) jumps along the dislocation line, which is parallel to [llz], this vacancy must leave the extra half-plane. For example it may first move to site B in an adjacent (111) plane and then proceed to C in the next plane. The following jump will be parallel to the Burgers vector to site D, which by symmetry is equivalent to B, and finally it will return to the extra half-plane at A’ which is equivalent to A. These four steps give a resultant migration of [l 121 along the dislocation line i.e.

+Clli] + t[iii] + )[iii] + +[iiil = ~112~ The (110) projection of this dislocation is shown in

Fig. 2(b), where different symbols are used to distinguish between atoms lying in the four (170) planes labelled + 1 and + 2 above the slip plane and - 1 and -2 below the slip plane in Fig. 2(a). The diagram clearly shows the displacement of atoms parallel to the axis of the dislocation, which arises because this direction is not perpendicular to the mirror plane. It also shows the effect of the two-fold [liO] symmetry axis normal to the slip plane. The main purpose of Fig. 2(b) however is to illustrate the four steps of the vacancy migration mechanism discussed above link-

t [Iid

l . . . . . . l ****... l .-_(A)+2

EDAOC

l .,... l **** . ...* ’ -co) +l

I I I ---c [I 111

. . . ..****.... l -(0)-l

. . . . . . l . l . . . . . -_(A)-2

(a)

Fig. 2. Structure of the )[lll] (170) edge dislocation in a-iron projected on to (a) the (112) plane and (b) the (1iO) plane. Pipe diffusion of vacancies occurs along the closely related paths ABCDA’ and ABEDA’ which are marked by broken lines in (b) and are discussed in the text. The different symbols in (b) correspond to the Jour (110) planes labelled +2, f 1 in (a). where the six (112) planes comprising the stacking sequence are shown superimposed. The three extra half-planes associated with the dislocation are

indicated by conventional symbols in (a).


ing sites A, B, C, D and A’. The formation and migration energies for these sites and jumps are summarised in Table 1. The magnitudes of EL are all approximately 0.70 eV corresponding to vacancy-dislocation binding energies of about 0.67 eV. The migration energy E; is very small for the f [iii] step CD, parallel to the Burgers vector, but is approximat~iy 1.09 eV for the three $11 li] jumps. Thus it is com- paratively difficult for vacancies to migrate along the dislocation line. However, the net effect of the lower values of Et and the higher values of EF gives an activation energy for pipe diffusion Q(A, AB) = Et(AB) which is 1.76 eV, rather less than Q(P) = 2.05 eV.

The path ABCDA’ discussed above is not symmetric about the centre of the dislocation and therefore an alternative route ABEDA’ is possible where E as shown in Fig. 2 is equivalent to C. In this case the second and third jumps are effectively taken in the

reverse order and in opposite senses. Thus, although the detailed form of the jumps is changed, the overall migration energy is not. Also because of the symmetry imposed by the [l?O] two-fold rotation axis, vacancy migration along the dislocation in the opposite di- ection by means of routes A’DCBA and A’DEBA is equivalent to mechanisms ABEDA’ and ABCDA respectively.

Projections of the relaxed structure of the f[lll J (li0) edge dislocation in molybdenum are shown in

t [Iid . . . ..*.**....

l -_(r142

t Plzl . * A.*___“--c~D A l

ro LO I ,b A 0 ‘

:

. ‘3 yc * l

. A A.

A0 . 01’ . 0 . - [I 111

Fig. 3. Structure of the f[lll] (170) edge disloc$ion in molybdenum projected on to (a) (112) and (b) (110). Pipe diffusion of vacancies occurs along the path ABCDA marked by a broken line in (b). See the caption of Fig. 2 for

further particulars.

Fig. 3 and are very different from the corresponding projections for iron. In particular, as clearly indicated in Fig. 3(a), the dislocation core consists of two half- planes above the slip plane meeting one half-plane below. The corresponding vacancy migration path shown in Fig. 3(b) is therefore symmetric and unique being quite distinct from that of Fig. 2(b). In addition the lowest value of Ei; does not arise for the equivalent sites B and C at the edges of the two central half-planes but for the sites A and D, which are again equivalent, on the adjacent (111) planes. The migration path therefore consists of three parallel )[lli] steps AB, BC, CD followed by a +[iii] step DA parallel to the Burgers vector. The magnitudes of E{ and E: for these different sites and jumps are again given in Table 1. The two pairs of formation energies are in fact similar, being approximately 1.27 eV and thus corresponding to binding energies of about 1.15eV. The migration energy for the &iii] jump DA’ is again very smail but the remaining three jumps are appreciably higher being approximately 1.60 eV. Thus the concentration of vacancies along the dislocation is again expected to be higher than in the bulk but it is more difficult for these vacancies to move. The net effect however is that the activation energy for pipe diffusion Q(A,AB) = EL(A) + ET(AB) is 2.90eV substantially less than the bulk value of Q(P) = 3.81 eV.

The relaxed structure of the third dislocation examined, the )[lllf (112) edge in iron is illustrated in Fig. 4. The projection in Fig. 4(a) is on to the (li0) plane which is normal to the dislocation and reveals the lack of symmetry of the structure. In addition the locations of the three extra half-(1 11) planes are difficult to define in this case and therefore only one central dislocation symbol has been used. However the fact that the slip plane (llz) contains only one close- packed direction, the Burgers vector $ [ill], is appar- ent. Thus in order to migrate along the dislocation line any vacancy must move from one (112) plane to another. In particular in order to move from the most tightly bound position labelled IL in Fig. 4(a) a vacancy must either cross the slip plane to site fl or move away from the slip plane to site y. The structure of the dislocation appears to be much simpler in the (112) projection of Fig. 4@). This arises because the (170) plane is a mirror plane with a two-fold stacking sequence. Thus there are no atomic displacements parallel to the dislocation line and the possible migration paths alternate between either tl and /3 or tx and y sites as indicated. The energies EL and E: for this dislocation are again summarised in Table 1. The most striking feature of these results is that E{(z) has the very small value of 0.15 eV so that a high concentration of vacancies is expected at the tl sites. The two migration energies for jumps away from these sites are large, the preferred one being across the slip plane to the B sites. The activation energy for pipe diffusion Q(oz,a@) = Ed + EF(aj3) is then 1.26eV very much less than the bulk value of 2.05 eV.

MILLER et al.: PIPE DIFFUSION IN B.C.C. METALS 1603

t bll . . . . . .

.

.

.

.

.

.

.

.

.

.

. .I . . - (A)4

.

E . . - (O)+l

. -4 . .- (0) -1

. . . - (6) -2

. . . l - 8111

(a)

,’ ;I .O rd . 0 . 0

@I

Fig. 4. Structure of the fQ1 l] (112) edge dislocation in a-iron projected on to (a) (110) and (b) (112). Pipe diffusion of vacancies occurs along the path a/Xz or alternatively the path aya! marked by broken lines in @). The different symbols in (b) correspond to th_e four (112) planes labelled f 2, f 1 in (a), where the two (110) planes comprising the stacking sequence are shown superimposed. The centre of the

dislocation is indicated by a triple symbol.

t b4 . . 0’ . . .

. . . .I . -_(A)+2

. . Jli l = l --(*)+’

. . . l 8 .-(0)-l

. . . . - (A)-2

. . . . . - p111

(a)

0.

0.

@)

Fig. 5. Structure of the )[111] (11-i) edge dislocation in molybdenum. For details see the caption to Fig. 4.

Finally, the structure of the )[ll l] (112) edge dislocation in molybdenum is illustrated in Fig. 5. This

shows essentially the same features as the corresponding dislocation in iron except that, as in the case of the i[ 11 l] (110) molybdenum dislocation, the most tightly bound vacancy site does not occur at the

centre of the core region. The formation energy E{(cc) = 0.23 eV is again very small, and the preferred migration path is a/I. The self diffusion energy Q(u, c$) is 2.25 eV, much smaller than the perfect crystal value

of 3.81 eV.

4. DISCUSSION

Before examining the nature and significance of the results presented in Section 3, the reliability of the potentials and computing procedures which have been adopted will be discussed. The two potentials used [13,14] are similar in that they are empirical, equilibrium, spline-potentials terminating between second and third nearest neighbours. They hold the models in equilibrium at the correct lattice parameter and are matched to the elastic constants for iron and molybdenum. In addition the Johnson J,, potential for iron has been widely used in computer simulation studies and has enabled insight to be gained into the structures and energies of many crystal defects [&lo].

Although new, the molybdenum potential adopted here has been designed using the same general criteria as those for the well-established Jo potential [14]. It was thus anticipated that it would be equally well behaved during the present discrete calculations and indeed this was found to be the case. However, the detailed forms of the two potentials, as shown in Fig. 1, are very different from each other. It was therefore expected that they would give rise to quite distinct structures and energies and the results of Sec- tion 3 confirm this prediction. Whether or not these specific results represent characteristic properties of iron and molybdenum is, of course, uncertain but at least they suggest different effects that might arise in different b.c.c. metals.

The computing procedures that have been used are again well-established and as in previous applications

the relaxation routines have worked rapidly and re- liably. However, the defects examined have been complex and have imposed major demands on the models used. This is because the three-dimensional strain field of the migrating atom has been superimposed on the long-range two-dimensional strain field of the straight edge dislocation. It has been necessary therefore to use large models. The main approximations which have been introduced are: (a) the fixed boundaries based on the initial anisotropic elastic sol- ution for the displacement field of the dislocation and not on the relaxed structure, (b) the fixed size of the model, so that no volume changes accompany vacancy migration. In addition the relaxation procedure results in equilibrium structures with all atoms at rest so that the model is effectively at absolute zero

1604 MILLER et ~1.: PIPE DIFFUSION IN B.C.C. METALS

temperature rather than at high temperatures where vacancy diffusion occurs in practice. However, these limitations are not considered to be serious in the present application where it is the relative values of migration energies in the perfect crystal and along the dislocation cores which are required, rather than absolute energies. It is also emphasised that the migration paths have been freely determined by the relaxation procedure and are not subject to preconceived geometrical restrictions.

Ignoring relaxation effects, the formation energy of a vacancy in a b.c.c. metal is -4&r,) - 34(r,), where 4(r) is the potential and r1 and r2 are first and second nearest neighbour distances. Thus, to a first approximation, the vacancy formation energy reflects the depth of the potential well. Similarly the migration energy is controlled by the close approach of the moving atom to its near neighbours as it jumps into a vacant site. Thus, again to a first approximation, the magnitude of the migration en’ergy reflects the slope of the potential curve at atomic separations less than nearest neighbour. Reference to Fig. 1 and Table 1 demonstrates that all the results of the present project satisfy this general rule, the formation and migration energies for molybdenum being very roughly twice as large as the corresponding values for iron.

The most striking features of the dislocation structures which have been obtained, arise from crystallo- graphic factors rather than the potentials adopted. For example edge dislocations with f ( 111) Burgers vectors in b.c.c. crystals have three extra half-planes and not one. In the case of the )[lll] (110) dislocation, as shown in Fig. 2(a) and Fig. 3(a), these three half-planes can be clearly distinguished in symmetric cores extended in the slip plane. However, the structure of the *[ll l] (112) dislocation is asymmetric and the locations of the extra half-planes are ill-defined. The (112) plane is, of course, the twinning plane and the dislocations in Figs 4 and 5 are, in fact, tending to dissociate into a twinning disl~ation with b = l/6 [ 1111 on the left and a complementary twinning dislocation with b = l/3 [ 11 l] on the right. The more subtle differences between the iron and molybdenum dislocations which arise from the potentials, involve the detailed st_ructure of the core regions. Thus for the $[l 1 l] (110) dislocation a single extra half-plane occurs at the centre of the core region in iron, whereas in molybdenum two planes above the slip plane meet one below. Also in iron the core of the % [ 1111 (112) dislocation is more widely spread, the twinning dislocation being dissociated [6] into two twinning partials with Burgers vector &[lll]. This difference is indicated by the isosceles triangle formed by three projected atomic sites at the extreme left of the slip plane in Fig. 5(a). This triangle is the basic interfacial unit of the isosceles twin boundary structure [12] and for the corresponding dislocation in iron arises beyond the limits of the section of the model shown in Fig. 4(a). These results are generally consistent with the well-established effect that core

spreading increases as potentials become shallower and hence interatomic forces become weaker [6].

The detailed structures of the dislocation cores con- trol the magnitudes of the vacancy formation energies given in Table 1. The favoured sites are those at which creation of a vacancy eliminates as many highly compressed nearest neighbour bonds as possible. These sites are expected to lie at the edge of an extra ham-pl~e of the dislocation, but more important are the shortest bonds which tend to be amongst those which cross the slip plane. In particular in the case of the $[ll l] (112) dislocations creation of a vacancy at site c1 in Fig. 4 or Fig. 5 elimmates a pair of very short bonds linking c( and p sites. The resulting vacancy formation energy is therefore extremely low for both iron and molybdenum. The effect is less marked for the +[lll] (170) dislocations as the (110) planes are more widely spaced, resulting in the bonds crossing the slip plane being less compressed.

Vacancy migration energies in b.c.c. metals are controlled by the atom which moves having to squeeze through two equilateral triangles of neighbouring atoms before reaching the vacant site [9]. For a perfect crystal, this gives rise to a characteristic potential barrier with two symmetric maxima, as shown in Fig. 6. In the core region of a dislocation the two equilateral triangles are highly distorted or even de- stroyed and the initial and final sites of the migrating atom are not in general equivalent. The barrier therefore becomes asymmetric and typically has only one maximum. On the compression side of the dislocation, where vacancies are most tightly bound, the triangles perpendicular to the dislocation line will effectively become smaller than in the perfect crystal so that migration will be consequently more difficult. An example of this behaviour is the step c$ for iron, the potential barrier for which is illustrated in Fig. 6. The result is a significant increase in the migration

I 8 t f

0.0 I I I I I I 0. 2 0. 4 d 0.6 0. 8 1.0

Fig. 6. Comparison of the variation of potential energy E in electron volts, with distance d in units of the nearest neighbour separation, for vacancy migration through a perfect crystal PP and along the core step ab of the )[l 1 l]

(112) edge dislocation in iron defined in Fig. 4.

MILLER et al.: PIPE DIFFUSION IN B.C.C. METALS 160.5

energy for an atom moving in the sense @ and a corresponding decrease for the opposite sense. How- ever, in this particular case, and indeed in the other cases examined, the net effect of a lower formation energy and a higher migration energy is a smaller potential barrier. Thus the results show that the activation energy for self-diffusion along edge dislocations in both iron and molybdenum is smaller than through the bulk. The effect is more marked for the $[ll l] (112) dislocations where the pipe diffusion energy is about 60% of the bulk value in both metals. For the f[lll] (110) dislocation the corresponding values are 86 and 767; for iron and molybdenum respectively.

It is anticipated that the formation energies of vacancies at screw and mixed dislocations will be greater than the values deduced in Section 3 for edge dislocations and that the associated migration energies will be smaller. The resulting activation energies for self-diffusion are thus expected to be similar to those given here. Thus self-diffusion by means of vacancy migration along a network of dislocation lines of various orientations is likely to be perhaps 2q< easier than through the bulk crystal. In any case the difference is considerably less than the factor of 2 that is often quoted. More or less the same conclusion is reached for iron and molybdenum and the result is probably valid for all b.c.c. metals. Indeed as the effect arises from vacancies being bound to the compressed region of edge dislocations and then having difficulty in moving, the result should be generally valid for all metals, regardless of crystal structure. In addition the same general conclusions should hold for diffusion of solutes along dislocations and through the bulk as the controlling mechanism is vacancy migration. How- ever, the case of pipe-diffusion of interstitials needs to be considered separately as it is expected that these defects are most tightly bound to the tension side of edge dislocations where they should migrate more easily. Thus both the formation and migration energies are likely to be lower than in the perfect crystal so that pipe-diffusion should be particularly favoured.

The only ather computer simulation study of pipe diffusion which has been undertaken appears to be that of Fidel’man and Zhuravlev [15,16]. They used pseudopotentials to obtain results for [lOO] (010) t[lll] (liO)and~[lll] (112)edge dislocations in the b.c.c. metals iron, molybdenum and tungsten and also for perfect and dissociated edge dislocations in the f.c.c. metals aluminium, silver, gold, copper and nickel. This was clearly an ambitious project but unfortunately the results obtained are not comprehen- sive and they are presented and discussed in a rather casual manner. In particular, differences in the structures of the dislocations in the various metals are not considered and hence the details of the vacancy jump sequences cannot be described. The vacancy formation energies were determined for a very limited number of locations in the dislocation cores and un-

fortunately it has not been possible to fully interpret the diagrams which are used to define these locations and also the vacancy jumps. In addition, ranges of migration energies are presented rather than specific values for well-defined jumps. It is therefore difficult to compare the relevant results for iron and molybdenum with those of the present paper. However, it appears that some of the vacancy sites and migration steps which are critical in the present analysis were not considered. In particular, the very low formation energies for the 0: sites of Figs 4 and 5 were not dis- covered. Also the crucial result of the present paper, that the vacancy migration energies in the dislocation cores are considerably greater than those in the perfect crystal was not found. In the case of the $[lll] (112) dislocation these two effects fortuitously com- pensate to give similar percentage reductions in the activation energies for self-diffusion as in the present work. However, for the i[ll l] (lib) dislocations the two studies give very different results demonstrating that great care has to be exercised in performing and interpreting these calculations.

Direct experimental evidence for enhanced diffusion along an isolated dislocation in terms of the transfer of matter by either a vacancy or an interstitial mechanism is extremely limited. Volin er al. [17], using transmission electron microscopy, have measured the shrinkage of voids in aluminium on annealing due to diffusion of vacancies along a single dislocation to the surface. They found that the activation energy for the process was approximately 0.6 times the activation energy for self-diffusion in the perfect crystal. By far the largest body of evidence, however, comes indirectly from measurements of grain boundary diffusion rates. In general, it is found that the grain boundaries which consist of relatively dense planar arrays of discrete dislocations show enhanced diffusion rates. attributed to the low diffus- ivity paths of the component dislocations. Measure- ments of self-diffusion rates along both tilt and twist boundaries have been reported for several metals [ 181 and the results interpreted within the model of a high- di~usivity grain boundary slab representing the planar dislocation array. Assuming an effective dislocation pipe area of 25 x 1O-2o m2, the estimated activation energies for diffusion along an isolated dislocation are between 0.3 and 0.7 times that corresponding to diffusion in the bulk. In particular, the result for b.c.c. iron of James and Leak [19] gives a ratio of grain boundary to host lattice diffusion activation energy of 0.68 which is in good agreement with the results of the present work. Further indirect evidence for pipe diffusion comes from measurements of inter- nal friction [2OJ and of change in creep rate with temperature [2i J. All the experimental evidence reported is consistent with a significant reduction in diffusion activation energy from that in the bulk by a factor of between 0.3 and 0.7. However. the estimated cross sectional areas of the dislocation pipes vary over several orders of magnitude, and are all larger than


the value of approximately 15 x 10-20m2, found in the present work. This area corresponds to pipes containing sites A-E in Figs 2 and 3 and sites u and /? in Figs 4 and 5.

In a recent review of the experimental evidence for point defect diffusion along dislocations [22], Balluffi and Granato argue that the results are more consistent with a vacancy than an interstitial mechanism. The computer simulation results presented here confirm this. However, Balluffi and Granato [22] also predict that vacancy migration is easier and interstitial migration more difficult in the core region than in the perfect crystal. The present results conflict with this view, showing an increase in migration energy for the vacancy, and suggesting that interstitial migration may take place more easily in the core. This will be the subject of further computer simulation investiga- tions.

Acknowledgements-This work was supported in part by the Science Research Council. The paper is published by permission of the Central Electricity Generating Board.

REFERENCES 1. K. M. Miller, Physica status solidi (b) 98, 387 (1980). 2. A. G. Cracker. M. Donerrhan and K. W. Ingle, PhiI.

Mag. A 41, 21’(1980). - _

3. P. D. Bristowe and A. G. Cracker, Phil. Mag. A 38,487 (1978).

4. K. W. Ingle and A. G. Cracker, Phil. Mag. A 41, 713 (1980).

5. A. G. Cracker and B. A. Faridi, Acta metall. 28, 549 (1980).

6. P. D. Bristowe and A. G. Cracker, Acta metall. 25, 1363 (1977).

7. K. W. Ingle, P. D. Bristowe and A. G. Cracker, Phil. Msg. A 33,663 (1976).

8. K. W. Ingle and A. G. Cracker, Acta metall. 26, 1461 (1978).

9. K. W. Ingle and A. G. Cracker, J. nucl. Mater. 69/70, 667 (1978).

10. K. W. Ingle and A. G. Cracker, Phil. Mag. A 37, 297 (1978).

11. B. A. Faridi and A. G. Cracker, Phil. Mag. A 41, 137 (1980).

12. P. D. Bristowe and A. G. Cracker, Phil. Mag. 31, 503 (1975).

13. R. A. Johnson, Phys. Rev. A 134, 1329 (1964). 14. K. M. Miller, J. Phys. F. 11, 1175 (1981). 15. V. A. Zhuravlev and V. R. Fidel’man, Fizika metal.

metalloved. 43, 222 (1977). 16. V. R. Fidel’man and V. A. Zhuravlev, Fizika metal.

metalloved. 46, 106 (1978). 17. T. E. Volin. K. H. Lie and R. W. Balluffi. Acta metall.

19, 263 (19jl). 18. R. W. Balluffi, Physica status solidi 42, 11 (1970). 19. D. W. James and G. M. Leak, Phil. Mag. 12, 491

(1965). 20. H. Rahmatalla and C. F. Burdett, J. Mater. Sci. 12,

2396 (1977). 21. H. E. Evans and G. Knowles, in Proc. 5th International

Conf on Strength of Metals and Alloys, p 301, Aachen (1979).

22. R. W. Balluffi and A. V. Granato, in Dislocations in Solids, 4, Dislocations in Metallurgy (edited by F. R. N. Nabarro), North Holland, Amsterdam (1979).

a computer simulation study of pipe diffusion in body centred cubic metals

Documents