On kinked screw dislocations in the b.c.c. lattice—II. Kink energies and double kinks

Acra metoN. Vol. 31. No. 10, Printedin ~rcat Britain

pp.

1759-1770.

ON KINKED LATTICE-II.

1983

OOOI-6160/83 53.00 + 0.00 Pergamon Press Ltd

SCREW DISLOCATIONS IN THE B.C.C. KINK ENERGIES AND DOUBLE KINKS

M. S. DUJZSBERY Division of Physics, National Research Council, Ottawa, Ontario, Canada KIA OR6 (Received 16 November 1982) Abstract-The self and interaction energies of kinks on screw dislocations in the b.c.c. lattice have been studied by computer simulation, using models representing both potassium and an iron-like material. Both the total energy and the spatial distribution of the energy are considered. The interaction energy is found to be as predicted by linear elasticity for separations of as low as half the width of a single kinlr; for still smaller separations the energy decreases linearly. The potassium results are compared with experiment and found to be reasonably consistent with a model in which the formation of double kinks is the rate-controlling mechanism in low temperature plastic deformation. R&um6-Nous avons Ctudit par simulation sur ordinateur les inergies propres et d’interaction des d6croclements sur des dislocations vis dans la structure CC., en utilisant des modbles reprisentant le potassium et les matbriaux semblables au fer. Nous avons consid& 1’Cnergietotale et la tipartition spatiale de Gnergie. L’Cnergied’interaction est cclle que pr6voit l’ilasticitC lin&ire pour dcs sCparations aussi petite-s que la moiti.5 de la largeur d’un d&crochementunique. Nous comparons les rCsultatspour le potassium avec I’expCrience;nous les trouvons en accord raisonnable avec un modhle dans lequel la formation de doubles dkrochements est le m6canisme qui contrdle la vitesse lors de la d&formation plastique g basse temp&ature. und Wechselwirkungsenergien von Kinken in Schraubenversetzungen des krz. ZU!MIlld mlbstGitters wurden mittels Computersimulation untersucht. Hiem wurden Modelle fur Kalxium und Eisen-Phnliche Materialien bcnutzt. Gesamtenergie und riiumliche Verteilung der Energie widen ermittelt. Die Wechselwirkungsenergie entspricht dejenigen. die aus der linearen Elastizitiitstheorie fir Abstiinde bis xur halben Weite e&r einxelnen Kinke folgt. Bei noch kleineren Abstfmden nimmt die Energie linear ab. Die Brgebnisse fir Kalzium werden mit experimentellen Werten verglichen. Sie sind him&&end vertriiglicb mit Modellen, bei denen die Bildung von Doppelkinken der geschwindigkeitsbcatimmende Mechanismus bei der plastischen Tieftemperaturverformung ist.

1. INTRODUCTION The low-temperature plastic behaviour of bodycentred cubic (b.c.c.) metals is believed to be controlled by the limited mobility of 4( 111) screw dislocations (for general reviews, see [l-3]). Studies using computer simulation [4] have suggested that the reason for this is a non-planar spreading out of the screw dislocation core; the core must be partially constricted in order for motion to occur and this requirement causes a strong lattice resistance (i.e. a large Peierls stress). However, these calculations at best predict the Peierls stress for a straight screw dislocation to be more than three times larger than the experimentally measured flow stresses extraplated to OK. Whether or not the calculated stresses would decrease in the observed manner if temperature were incorporated into the simulations is not known, but this seems unlikely. Rather, it is thought that motion of screw dislocations at finite temperatures takes place by the thermal activation and subsequent stress-driven migration of oppositely-signed pairs of kinks. In an earlier paper [5], the structure and Peierls stress of kinked screw dislocations in the b.c.c. lattice were studied by computer simulation, using both a

fundamental interatomic potential for potassium [6] and an empirical potential [7’j constructed for an iron-like material. It was shown that the Peierls stress at OK for an isolated kink is approximately 10m6G (where G is the { 110}(111) shear modulus) for potassium and about 7 x lo-’ G for iron (the second model will be referred to as iron for convenience, but it must be remembered that the potential [7j is empirical, chosen to match the elastic properties of iron), in both cases too small to play any significant part in determining the bulk low-temperature flow stress. Thus if the double kink model is to provide a valid description of the temperature dependence of the flow stress the controlling mechanism must lie in the activation, rather than the subsequent motion, of the kinks. A study of the activation process requires a calculation of the self and interaction energies of the kinks and it is to this purpose that the work reported herein is devoted. While, to the author’s knowledge, there has been no prior attempt to calculate kink interaction energies with an atomistic model, there exist two calculations of self-energies in the literature. Gehlen [8], using an iron-like empirical potential similar to [7] and model sizes up to 1900 atoms, finds I759

I760

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DISLOCATIONS IN THE B.C.C. LATTICE-II

a kink energy of 0.08 eV, while Wiithrich [9]. using the potential I?‘],and a block size of 600 atoms quotes self-energies ranging from 0.6 to 1.2 eV, depending on the type of kink. It is not clear whether the difference between the two calculations is due to the different-sized blocks or to some other factor, but the agreement is obviously rattler unsatisfactory. 2. A RECAPITULATION

Before discussing kink energies in detail, a brief review of the results reported in [S] is approp~ate. The screw dislocation core in the b.c.c. lattice can exist in two degenerate states which are distinguishable by the sense of displacement with respect to the elastic solution of the central three atom rows parallel to the dislocation line. The cores are labeled A and B depending on whether this displacement is parallel or anti-parallel to the Burgers vector. Based on these two states, two families of elementary kinks (i.e. those with the shortest possible line vector, 4(211>) can be established by inspection. The first and smaller family, termed homomo~ho~ kinks, encompasses those kinks in transit of which the dislocation core does not change its state. There are just two homomo~ho~ kinks, classified by the sequences ApA and AnA, in this terminology, the preceding and trailing letters refer to the core state on either side of the kink, and the letter p or n defines whether the kink line vector is a positive or negative f(Zl1) step. It follows from the symmetry of the lattice that ApA is degenerate with BnB and AnA with BpB, since the A- and B-kinks can be generated from each other by the operation of a lattice symmetry vector; the A- and B-states are not considered as being different. Atomistic relaxation reveals no new homomo~hous kink states. In the case of the potassium potential, the lattice coupling is so weak that there is no detectable preferred site for the kink, while for the Fe potential the preferred site is at a {111) atom plane, itself a diad-symmetric location. Thus only two homomorphous kinks need be considered. The members of the second family of kinks, termed heteromorphous, are each associated with a state-change of the dislocation core from type A to type B or vice-versa. Four kinks are evident from inspection. These can be grouped into two classes, depending on whether the relative orientation of the core states on either side of the kink corresponds to the “easy” or “hard” direction of motion of the screw dislocation [ltl]. The ApB and BnA kinks fall into the “easy” class. For each of these kinks, computer relaxation reveals two relaxed states, related to each other by a reIative translation of the (11 I) atom row passing through the centre of the kink. Moreover, none of these four states is diad-invariant, so that each has an associated degenerate state, which for present purposes will be ignored. The lattice sites are the same as for the homomorphous kinks. The second, “hard” class of heteromo~hous kinks con-

tains the AnB and BpA kinks. Relaxation reveals no new states, but the relaxed states are not diadinvariant, leading to consequent two-fold degeneracy which, as above, will be ignored; equilibrium lattice sites are the same as for the homomorphous kinks. The final type of defect considered in [5] is the flip, or core state change without kink; there are just two of these, to be called AxB and BxA, where x denotes the flip site and the symbols A and B have the same meaning as for the kinks discussed above. Using a block size of 30 x 30 atoms normal and 51 repeat distances parallel to the dislocation line, the Peierls stress was calculated for all kinks in the potassium model to be less than low6 G; in the iron model, only the ApB kink was studied in detail, proving to have a Peierls stress of about 5 x 10e4 G. 3. THE KINK SELF-ENERGY 3.1 The total energy For any system bound by the type of interatomic potential used here, the total energy V, can be expressed in the form

where r’, r’ are the position vectors of the atoms labeled f, 1’ and fJ is the total volume of the system. V( Ir I) is the central part of the potential, summed over all interacting atom pairs as indicated, while E(Q) is a contribution which depends formally only on the total volume of the system. For the potassium potential, E(Q) has its origin in the conduction electron gas; for the case of iron, E(Q) is simply a term which must be added in order to match the observed elastic constants. On the surface, calculation of the self-energy of a kink is a relatively straightforward matter, requiring the evaluation of (1) for two model crystal&s, containing respectively a kinked and a straight dislocation. The first term in equation (1) is easily evaluated, the energy difference is accumulated on an atom-by-atom basis, to avoid the numerical problems associated with the calculation of a small quantity from the difference between two large numbers. The second term in equation (1) proves to be a bit more tricky; some macroscopic change in vohnne is to be expected from the kinks, but with fixed boundary conditions there can formally be no volume change. However, the volume change that the system would undergo, if unconstrained, can be estimated by a method similar to that used by Hardy [l I], who reworked in lattice statics Eshelby’s 1121 continuum elastic calculation of the dilation produced by a source of point expansion. Writing w for the atomic volume, a for the lattice parameter and p = r/u for the dimensionless interionic distances, the energy (1) can be reexpressed as

DUESBERY: DISLOCATIONS IN THE B.C.C. LATTICE-R The perfect lattice must he in equilibrium with respect to volume change; thus by differentiation of (2) with respect to volume and equation of the result to zero, there results the well-known condition

1761

60, giving

where N is the number of atoms in the lattice. Applied to the kinked crystallite, equation (6) gives an estimate of the volume change due to the presence

of the kink and of the correction necessary to the energy; it may be noted that the energy correction, which is always negative, is second-order. In the practical calculation, there is a third corsmall volume change bw. The resulting energy change rection to be applied. The energy summation is most conveniently performed on subsets of the atoms, is rather than on the fuIl crystallite, in order to minimise rounding errors and to avoid the constrained boundaries. The sub-crystallite chosen is only partially constrained by the fixed boundaries, so that a volume (4) change can and does occur. In this event, the pair potential term in equation (1) can easily be caicuFor the imperfect solid the fhst term in equation lated, but the contribution of the volume term to the (4), linear in 60, will no longer be zero in genera&the energy must be added separately by computing the pair-potential term rdV/dr wiil change to reflect the volume change bo of the subcrystaL The corrective local environment, but the vofume force dE/dw is term, which can be inferred from equations (4) and unchanged. Provided that the distortions in the solid (S), is then just -p&, a first-order term. This is the are small enough for linear elasticity to hold in the energy adjustment for volume change described by case of the dislocated solid (this is true over most of Wiithrich [9]. The kink and flip energies and formation volumes the crystallite), the coefficient of So2/2a, in (4) is just have been calculated for a number of subsets of ‘the the bulk modulus, K. Thus (4) simplifies to model crystallite, each subset forming a cylinder of alJt=~ @‘-p&h (6”)” (5) atoms with axis coinciding with the [I 1I] atom row I i through the centre of the kink. The energy of the where p’ is the pressure at the site f due to the pair defects is confined in the main to a cylinder of radius potential and is de&d by analogy with (3); the about 3b; for cylinders of larger radii, up to 1lb (the quantity @ -PO) plays the same role in the lattice largest used), the energies change by no more than model as does the Green’s function which Eshelby f 10%. The calculated vaiues are shown in Table 1. [12] uses to &fine the strength of his point source. In this table, U,, is the energy contribution stemming The energy change can be minim&d with respect to from the pair potential. &U, and &Y, are the vohune In the derivation of this result, the outer summation in equation (2) can be removed &cause in the perfect Iattice the sites are identical. Consider now the application of (2) to an imperfect lattice and apply a

+$

I

Table 1. Kink energies for the potassium and iron models. Poiassium

APB BnA AIiB BPA APA AilA AXB BXA

0.028 0.052 0.043 0.113 0.037 0.108 0.052 0.013

APB BnA AnB BPA APA AnA AXB BXA

-0.342 0.599 -0.089 0.326 -0.206 0.399 0.632 0.100

0.010 -0.014

0.038 0.038

0.086 -0.105

4 4

] 0.076

0.070 -0.053

0.111 0.059

0.541 -0.338

2 2

] 0.170

0.044

-0.042

0.081

0.066

0.305 -0.274

22

] 0.147

- 0.034 0.035

0.018 0.048

-0.248 0.241

2 2

-0.257 0.498

0.114 -0.137

44

] 0.241

0.394

0.302 -0.075

0.429 -0.424

2 2F

1 0.227

0.218 -0.217

0.010 0.176

0.209 -0.256

22

] 0.186

- 0.223 0.202

0.408 0.300

-0.217 0.191

2 2

IrOn

-0.003 -0.013 -0.003 0 -0.002 -0.006 -0.001 -0.002

0.088 -0.088 -0.401

U,, is the nair-bindinn enerav. dU. and HJ. the volume term corrections. U, is the totai *’ener& and 6w isthe f~&sti& volume in atomic volumes. The columh g lists the degeneracy of each defect and UDKis the energy of the simplest double kinks. Ail

energies are given in eV.

I162

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energies due, respectively, to the “implied” volume change defined by equation (6) and to the subset volume change correction. The column headed Z/r lists the total energy and 60 is the formation volume in atomic volumes, computed as the sum of the volume changes leading to the terms 6UA and &J,. For reference, the column headed “g” enumerates the degeneracy of each kink and flip, including the diad degeneracies; this value can influence the activation statistics, since the rate of creation depends on the number of distinguishable states that can exist. In the final column of Table 1, the natural pairing of the kinks and the consequent double kink energy, UDK, is shown. The pairing shown is that which can occur without the presence of flip defects. Consider first the defect energies in the potassium model. The energies are quoted to 0.001 eV, with a precision of about 20.002 eV. This is a consequence simply of the fact thatthe kink energy is only a small perturbation on the dislocation energy, which is itself only a small fraction of the total binding energy. The accuracy of the energies, however, is probably much poorer. The interatomic potential is constructed specifically for the density of the perfect solid; the true potential in regions of enhanced or depleted density, such as occurs in particular for the flip defects (for the flips there is a localised volume change of f2% at the kink core [5]), may be Miciently different to change the calculated energy by rather more than the precision. An accurary of f5% for the kink and f 10% for the flips is a reasonable figure. The figures in Table 1 indicate that the lowestenergy defect is the AxB flip, with a self-energy of O.O18eV, while the single kink self-energies range from 0.038 eV (for the ApB and BnA kinks) to 0.11 eV, for the AnB kink. As anticipated, the second-order volume energy correction SU, is negligibly small; it is, however, included, because its contribution to the formation volume equation (6) is first-order and is not small. The double kink energies (strictly the energy of two single kinks at infinite separation and hence taken to be the sum of the self-energies of the constituent kinks) vary from a low of 0.076 eV for the ApBnA variety, through 0.147 eV, nearly twice as large, for the homomorphous ApAnA type, to 0.170 eV for the AnBpA pair. Because of the low energy of the flips, this simple spectrum is complicated by the possible existence of more complex double kinks. For example, the double kink BnAxBpa has an energy of 0.115 eV, thus lying between ApBnA and ApAnA in energy. The full spectrum of double kinks which can be constructed in this composite manner can be developed from the numbers given in Table 1. This will not be done explicitly, because of the primary purpose of the present work is to study kinks as they relate to plastic deformation. As is well-known, the mechanism of thermal activation discriminates dramatically between competing processes requiring different ener-

IN THE

B.C.C. LATTICE---II

gies; only processes with activation energies within IVkT of the minimum can contribute to any appreciable extent. Thus, for example, the relative contributions of the ApBnA double kink, with energy 0.076 eV, and the BnAxBpA complex, with energy of 0.1 IS eV, to a plastic deformation process are in the ratio of about exp (O.O39/kT), or at a temperature of 20 K, roughly ez3. For this reason, and to conserve space, only the ApB and BnA kinks will be considered in further detail. It may be noted in passing that the relative energies of the double kinks in potassium correlate strongly with the observed slip modes of the rigid screw dislocation. Consider the translation of an “A” screw dislocation through a “p” ( +$,a[f2m) vector. This translation takes place relatively easily if accompanied by a change in core structure to 9” type, which involves atomic displacements similar to those required for the ApB kink; the reverse BnA transition falls into the same class. The translation of an A core through p to an A core can occur, with suitable potentials [ 171or when subjected to specific non-glide stress constraints [18], but is a less preferred process: this process corresponds to the ApA and AnA kink configurations, which together form the double kink of intermediate energy. The final class of translation, that of the B core through a p vector with state change to A type has never, to the authors’ knowledge, been observed. This transition would require displacements similar to those observed for the highenergy BpAnB double kink, which exhibits a peculiar helical motion of one of the constituent fractional dislocations as the kink is traversed [5]. Whether the calculated minimum double kink energy of 0.076 eV is a reasonable value can be assessed by comparison with experiments. A common observation in the plastic deformation of b.c.c. metals at low temperatures is that, while the flow stress decreases rapidly, the activation energy, derived from the measurement of the temperature and strain rate dependences of the flow stress, increases approximately linearly with increasing temperature. At sufficiently high temperatures the decrease in flow stress slows and stops, while the activation energy becomes less dependent on temperature, finally becoming roughly constant. If the flow stress is indeed controlled by double kink activation, it seems reasonable to identify the transition between the two temperature regimes with the point at which thermal energy becomes sufficient to nucleate an independent pair of kinks; the activation energy would then be the sum of the self-energies, as given in Table 1. In potassium [13], the activation energy at low temperature is -20 kT, saturating at about 30 K. The energy at the saturation point is O.O5eV, about two-thirds of the calculated self-energy, 0.076 eV, of the lowest-energy ApBnA double kink. Next consider the results obtained with the Chang-Graham [7] model. It will be noticed immediately that several of the kinks have negative energies.

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This is a real phenomenon, due to the characteristics of the potential. If the density of pair bonds as a function of interatomic separation is calculated for the kinked state, it is found that a fair number of bonds are shifted into the region close to the first minimum of the potential (about 0.9~). For the potassium potential, a bond shift from 0.87~ (nearest neighbour) to the potential minimum results in a drop of 2.5%in the bond energy; for the iron model, in contrast, the analogous drop is 36%. It seems possible that it is this larger drop for the iron model which gives rise to the negative energies. A further problem which must be resolved is the discrepancy between the present results and those of Wiithrich [9], who used the same Chang-Graham potential but calculated very different energies-for example, Wfithrich quotes an energy of 0.8 eV for the ApB kink, shown in Table I to have an energy of -0.26 eV. Wiithrich’s study used a block size of 600 atoms, compared to the present study, which uses a size of 4.5 x IO” atoms. Since it has been shown [51 that boundary conditions can have a profound effect, the present calculations were repeated using the smaller block size, the result was essentially unchanged, in that the ApB kink energy proved to be negative. The solution to the problem lies not primarily in the boundary conditions, but in the initial displacement field assumptions used in [9]. As a starting solution to the ApB kink, [9] takes a displacement field linearly interpolated between the A and B fields over a precalculated width. If this procedure is followed with the small block size, an energy of 0.9 eV, close to that obtained in [9], is indeed real&d; if this solution is then relaxed with the large block size, however, the negative energy of Table I is obtained. Thus it can be concluded that the results [9] are artifacts of an unusual initial configuration, constrained by boundary forces. Despite these difficulties with the kink energies, the use of the iron model for comparison of many kink properties can remain justified. In a real material the energy of an isolated kink cannot be negative, since the creation of single kinks is possible during growth and by dislocation intersection. In the model crystallite however, the number of kinks is specified at the time of construction and cannot change thereafter except by the spontaneous generation of double kinks; as can be seen from Table I, the energy of the simple kink pairs is always positive, so that the model is stable relative to production of these. Some of the more exotic double kinks, for example BnAxBpA, can have negative energies, but since breakdown due to formation of these was never observed, it must be concluded that the model crystallite is metastable with respect to these states. Because of the negative energies, not much credence can be placed in the magnitudes of the double kink energies given in Table 1. A comparison with experiment, then, as was done for potassium, is of use only for the sake of completeness. In the deformation of non-alkaline b.c.c. metals, an activa-

IN THE

B.C.C. LATTICE-11

1763

tion energy of H u 27&T is commonly measured. With the additional information that the Peierls stress becomes negligible in iron at about 300 K, a double kink energy of 0.69 eV is obtained; this figure is not too far from twice the energy of some of the single kinks, but in each case the single kink is paired with a partner of very small or negative energy. It may be noted further from Table I that in the iron model the homomorphous double kink ApAnA apparently has the lowest energy, in contrast to the case for potassium. However, because the main reason for using the iron potential is comparison and because of the uncertainty about the magnitudes of the energies, further treatment of the iron model will be confined to the ApB and BnA kinks, as for potassium. 3.2. The energy distribution In the preceding section the total kink selfcnergy was calculated, in order to permit comparison with experiment and between kinks of different kinds. However, the total energy of a kink is a formless quantity: all the detailed information present in the model crystal has been submerged into this single number. It is the aim of this section to probe more deeply into the calculation and determine how the energy is distributed among the atoms which are perturbed by the presence of the kink. First, a word of caution is appropriate. Within the framework, defined in equation (l), of a total energy split into pairwise interactions and total volume terms, the term E(R) is formally independent of the detailed arrangement of the atoms and therefore is not included in the calculations reported below. To the same order of approximation, the pair potential V(r) changes slightly with total volume, but not with the local volume. Presumably a more accurate model of the problem may modify either of these statements; however, any corrections would necessarily be of higher order and hence expected to be small. The deviations would be expected to be larger where the local volume undergoes the greater change. Consider first the energy distribution normal to the dislocation line; this variation is shown in Fig. I. Each diagram in Fig. 1 shows a [l 1I] projection of the lattice in which the [TO11vector points vertically up and the [TXT] vector points horizontally to the right. The centre of each hexagon represents a [l I l] atomic row; the row at the centre of the kink is outlined boldly. The number within each hexagon is the kink energy contained within that atomic row, calculated as the difference between the row energy in the kinked crystallite less the mean of the energies of the corresponding rows in the dislocation arms on either side of the kink. For example, consider an ApB kink. Calculation of the energy of an atomic row, n, is a trivial matter. The quantity to be subtracted is the mean of the energies of the same row n, calculated in crystallites containing only a suitably sited dislocation firstly of type A and secondly of type B core.

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DISL~ATID~S

W.

W.

w.

Cd).

Fig. 1. The radial distribution of kink self-energy around the screw dislocation line. Each hexagon corresponds to a single [ 11l] atom row and contains a number which gives the kink energy stored in the row in units of meV for potassium, IOmeV for iron. The atom row at the centre of the kink is boldly outlined. Figure l(a)-(d) shows, respectively, the ApB kink in potassium, the BnA kink in potassium, the ApB kink in iron and the BnA kink in iron.

Figure i(a) and (b) shows the energy maps for the potassium kinks ApB and BnA, respectively: the row energies are in units of MeV. It can he seen from both maps that the energy falls off rapidly with distance from the kink centre, as mentioned in the previous section. In addition to the rows for which the energy is increased, there is a number for which the energy is substantially depleted. Consider the ApB kink [Fig. l(a)] for example. The energy stored in the centraf atom row is negligible compared with that contained in the shell of rows which are Fiat-nei~bo~ to the cc&e. In this shell, the two rows directly above the core contain 120 meV of excess energy while the remaining rows, beside and below the centre, are each lowered in energy, for a combined reduction of 37 meV. Comparing Fig. l(a) with the kink displacement gradient profile map of the same kink (Fig. 5(c) in reference [5J), it can be seen that the rows above the kink are in a state of compression (maximum volume change, 6V = - 1.2%) while those below centre are in tension (av = +0.6%). This is as far as the correlation between dilation and the sign of the stored energy goes; the central atom row is in tension (6V = +0.8x) and the rows on either side are compressed (SV = -0.2x), showing an opposite correlation with energy sign. It will also be noted that the map in Fig. l(a) shows approximate two foldsymmetry about a vertical axis through the centre, except very close to the central row, a symmetry not shown by the displacement profiles [S]. The analogous energy map for the BnA kink [Fig. l(b)] shows the same rough symmetry and pockets of enhanced and depleted energy, but is quite different in detail. The largest energy enhancement of 65 meV occurs right at the central atom row, for which the

IN THE

B.C.C. LATTICE-II

energy in the ApB kink was negligible. In the shell of nearest neighbour rows, the magnitudes drop to about half the central value, being positive below the core and negative above and beside it. A comparison of the radial dependence of the energies in Fig. l(a) and (b) suggests that BnA kink has a rather larger spatial extent than the ApB. Figure I(c) and (d) shows energy maps for the ApB and BnA kinks, respectively, for the iron model; the row energies are given in units of 10 meV. A number of differences from the potassium case are evident. In both iron kinks, the energy is much less con~ntrat~ in the centre than in the corresponding potassium kinks. The same approximate two-fold symmetry holds for the BnA kink in iron pig. l(d)], but there are substantial deviations from symmetry in the ApB [Fig. l(c)]; this may be a reflection of the instability which leads to negative total energy for the ApB kink (Table 1). Consider next the energy distribution along the dislocation line. In Fig. 2 are plotted the energies of successive (111) planes of atoms as a function of distance aIong the dislocation Enc. Figure 2(a) shows this variation for the ApB and BnA kinks in potassium. In both cases the kink centre lies between

I

~

.o

0.6

P

P bo’4 0.2

0.0

IO

I!5

20 25 30 (111 )Pkme index

35

40

35

40

(a). 1.0

rr

08

P

0.6

P P - 04 0.2 00 IO

15

20

25

30

(111) Plone index

(b). Fig. 2. The distribution of kink self-energy along the [I I I] dislocation line. The energy plotted is the kink energy stored in successive(I I I) planes normal to the dislocation line, normalized to a maximum value of unity. Figure 2(a) shows the energy distribution for the ApB _ and BnA kinks in . potassium and (b) shows the same fbr the iron model,

DUESBERY:

DISLOCATIONS IN THE B.C.C. LATTICE-II

planes 25 and 26 and to facilitate comparison the energies have been normalised in each case so that the (I 1I) plane of highest energy has unit energy. As the APB kink is approached from the A side [the lefthand edge of the diagram in Fig. Z(b)], the plane energy increases at a steadily faster rate which dies off slightly before increasing rapidly to a sharp maximum located within O.Sb of the A side of the kink centre. On proceeding through the maximum the energy drops to 65% of its maximum value in almost step-like fashion: this drop is essentially completed within 6 of the kink centre. The energy then decreases only very slightly for the next 2 - 36 before resuming a gradual drop which is similar to but not identical with the rise on the A side of the kink. The energy width of the kink, computed as twice the distance between the points at which the plane energy is 0.5, is 22.66, half as large again as the value of 15.36 suggested by displacement plots [5]. The BnA kink plane energy shows a similar behaviour when approached from the A side [the ~ght-hand edge of Fig. 2(a) in this case]. The same ~~ian-t~ rise is seen, to a sharp maximum followed by a step-like decrease, in this case to 50% of the maximum. The energy width of the BnA kink is only about lob, half that of the ApB (the displacement plots for the two suggested similar widths [q). Overall, the curves for both kinks resemble a fairly symmetric bell-like dependence with a sharp bite removed just on the B side of the kink centre. As can be seen from Fig. 2(b), which shows the same energy dependences for the ApB and BnA kinks in the iron model, it is the collapse of this “bite” region which leads to the negative energy of the ApB kink. Approac~ng the ApB kink from the A side, the energy increases as for potassium, but much more sharply, due to the relative narrowness of the iron kink. On reaching the maximum, however, the energy drops catastrophically [off-scale in Fig. 2(b)] to a value of -2.7, rejoining the overall “bell” of the curve nearly as sharply some 46 past the centre into the B side. The BnA kink in iron, which, as in potassium, has an energy width only half that of its ApB counterpart, does not show the same ‘drastic behaviour, although the “bite” drops the energy to little more than 10% of the maxims. Referring again to Table 1, it can be seen that the total energy of the BnA kink is positive. Thus it seems that an instability in the “bite” region is responsible for the negative energy of the iron ApB kink. An examination of the structure of the negative-energy atom planes revealed no localised feature which could be held responsible for the effect; rather, it seems just that the planes on the immediate B side of the centre have an exceptionally large number of pair-bonds with length close to the absolute minimum of the potential function. As explained earlier, this phenomenon would have a much greater effect with the iron potential than with the potassium potential.

4. THE KINK INTERACTION

1765 ENERGY

The kink self-energies, as discussed in the preceding section, are of prime importance in the high temperature limit of plastic deformation. To bridge the gap between this and the zero-Kelvin rigid screw dislocation regime, the interaction energy of the kinks in a double kink complex must be examined. At these inte~ediate temperatures, there is insuffi~ent thermal energy to create kink pairs with large separations. However, double kinks with smaller separations may be generated, because the interaction between opposite kinks is attractive, so that the total energy required for activation decreases as the separation decreases. Thus if the activation is to be successfully completed, there must exist an applied stress sufficiently large to overcome the kink interaction and force the activated complex to expand. The elastic theory of kinks was developed by Eshetby f14] and Seeger and Schiller 1151 and is reviewed in depth by Hirth and Lothe [14. For separations large compared to the width of an individual kink, the interaction energy should be inversely proportional to the separation [14]. However, the point at which this dependence breaks down and what happens at smaller separations is not known. It is just this kind of question which is amenable to the technique of computer modeling and this will be attempted below, by studying the behaviour of the ApBnA double kink in the potassium and iron models. First, a word is appropriate regarding the numerical method used. it was found that the block size of 30 x 30 x 51 atoms used in [S] was inad~uate for a proper treatment of the double kink, because of “end effects” from the (1 I I) boundaries at each end of the dislocation line. Accordingty, the length of the model crystallite was extended from 516 to 1016 parallel to the full [l II] dislocation tine, thus encompassing about 9 x lo” atoms. The use of fixed boundary conditions was retained. It was shown in [!il that flexible boundaries gave rise to unpredictable effects, while fixed boundary conditions produced influences for which allowance could be made. The technique used to detect the equilibrinm state of the stressed double kink also deserves some discussion. The ~~lib~um is an unstable one, since a small increase in kink separation decreases the interaction force, leading to uncontrolled expansion, while a small decrease in separation causes the interaction force to exceed the force due to the applied stress, resulting in mutual annihilation of the two members of the double kink. The procedure used consists of three steps. Initially, a state in stable equilibrium is created by constructing a double kink displacement field from the known single kink fields, with a specific kink separation and an appropriate applied stress. This is always possible, because of the harmonic force exerted by the fixed boundaries; it is a simple matter of trial and error to determine a stress and separation

II66

DUESBERY:

DISLOCATIONS

IN THE

which are in mutual equilibrium. In the second step, the applied stress is decreased, allowing the double kink complex to begin its collapse: this process is arrested at a desired kink separation. Lastly, the stress is increased incrementally until the unstable collapsing complex of step two reverses its motion and begins to expand again. This technique has been found to give quite reproducible results, provided that the separations are within + 3b of the initial separation; i.e. as long as the deviation from initial conditions remains within the limits of linear boundary force.

B.C.C. LATTICE--II

0.13

I

IO

20

30 ( ill

40

50

60

1 Plane index td.

4.1. The shape of the double kink (I.35f

The concept of a kink “shape”, familiar from the elastic string picture of a dislocation, is not so clear when dealing with the kink on an atomic scale, because except in cases of particular symmetry it is not possible to assign an unambiguous position to the dislocation. It is perhaps the best approach to generalise the meaning of “shape” and allow it to apply to any measurable quantity which undergoes change on passage through the kink. Thus the energy maps presented earlier can be thought of as defining a shape of the kink. The energy shape, however, is not the best analogue of the classical shape, because far from the kink in either direction along the dislocation line the energy shape drops to zero, whereas in the string model the position of the dislocation is different on either side of the kink. A shape connected with position therefore, is more desirable. As was mentioned in [S], the relative position of pairs of neigh~u~ng atomic rows undergoes permanent change on passage through the core. This can be seen easily by considering a pair of atom rows at a mean position vector r from the dislocation on the A-side of the ApB kink: on the B side of the kink the same pair of rows will be at t j, p from the dislocation, where p is the kink line vector. In general, the strains at these two positions ..wili be different and hence there will be a change in the relative positions of the two atomic rows. Far from the dislocation line this difference will be small, falling off as the inverse square of the distance, but close to the core the effect is signifi~nt and is made even larger by the different core structures of the A and B dislocation arms. The change in relative displacement of specific atoms rows was used in [5] to determine the position of the kink and this same property will be used here to define a shape for the kink. The choice of which pair of atomic rows to use is not pa~icularly critical, since quite similar shapes are observed. The pair actually chosen is that designated I and n in Fig. I of Ref. [5]; this choice has the advantage that the change in displacement is large, about b/6, so that the shape can be seen with greater resolution. In the case of a double kink, of course, the relative dispia~ement changes from one extreme to the other and then back, as long as the kinks are widely separated; when they

0.30 9 $

0.25 -

0.20 ~ 0.15 20

30 (111)

50

40 Plane

60

index

04.

3. The shapes of double kink loops with varying kink separation in equilibrium under an applied shear stress and the mutual attraction of the kinks. Figure 3(a) shows the ApBnA kink in potassium and (b) shows the same defect in iron. Fig.

become close enough to overlap, however, what will happen to the shape is not known. Figure 3(a), shows the calculated double kink shape in potassium for a variety of kink separations. The separation was determined by locating the centre of each of the constituent kinks by the method described in [5j; in concrete terms, this means that the separation is defined as the distance between the points at which the relative displacement curve intersects the line d = 0.25Ob in Fig. 3(a). This definition has some obvious drawbacks for small separations; for example, in the smallest double kink shown in Fig. 3(a), the peak of the curve does not even reach d = 0.250b. yet the bulged shape clearly exists. An alternative procedure might be to define the separation of the kinks as the distance between the points at which the displacement is half the maximum. If this is done, a curious property emerges; for kink separation of less than about 2Ob, calculated by the old method, the separation as determined by the new method remains constant at 21.5 f 0.36. This suggests that for separations less than 2Ob, or about 1.5 kink widths, the double kink behaves more like a single defect with a shape which can be written in parametric form as f(r) = constant, where r is the maximum displacement. This is in contrast with the

DUESBERY:

DISLOCATIONS IN THE B.C.C. LATTICE-11

case of large separations, for which the shape of each individual kink [S] is constant regardless of the separation. There is a considerable transition region between the two regimes, as can be seen from the curves in Fig. 3(a). The curve with the largest separation, of 41b or about three times the kink width, has a peak displacement of 0.3283, still ~~i~~ntly short of the value of 0.3336 which occurs for infinite separation. The smailest double kink in Fig. t corresponds to an applied stress of 8 MPa, or 6 x 1OA3 G, about 70% of the stress required to move a rigid screw dislocation. Figure 3(b) shows the double kink shapes calculated for the iron model. The effect of the smaller kink width for this model can be clearly seen. The largest double kink, with a separation of 28.514,or about seven kink widths, is pronouncedly square, with about lob of the peak lying at the asymptotic displacement value. The next smaller double kink, with a separation of 15.%, or about four widths, has not reached the displacement limit. The smallest double kink in Fig. 3(c), corresponding to an applied stress of 2 x tol MPa or 2.6 x 10m2G, has a remarkably high aspect ratio. ft is difficult to imagine the line tension of a dislocation allowing such a narrow bulge to develop and one wonders whether this shape and the narrowness of the iron kinks in genera1 is due to the artificiality of the potential used. 4.2. The energy of the double kink As has been shown above, the kink self-energiesare very small in comparison with the energy of the dislocation and the lattice. To attempt to calculate dire&y a quantity like the kink interaction energy, which at its maximum can only equal the self-energy would be an exercise in futility. Fortunately, there is a way around this problem. Since it is possible to construct equilibrium configurations of stressed double kinks, the interaction force between two kinks can be determined as a function of separation; the interaction energy is then just the integral of the force over the separation. The ~~~b~~ pairs of stress and separation for ~tassium are plotted in Fig. 4(a) (curve I). The stress required to maintain equilibrium has a limiting value as the separation approaches zero of 7,8MPa, or 5.9 x lo-‘G, rather smaller than the stress (9 x tOv3G) required to move the rigid screw. As the separation increases, the stress drops only slightly out to a separation of about 86, then rapidly to 1 MPa (7.6 x lo-‘G) for a separation of 22.56 (about 1.5 kink widths) and finally more slowly for larger separations. This curve can be integrated numerically to give the energy of the complex as a function of separation [curve II in Fig. 4(a)]. Bearing in mind that there is still an appreciable interaction between the kinks at the largest separation shown of 406, or nearly three kink widths, the estimated energy at this separation of 0.059 eV compares well with the calcu-

1767

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(b) Fig. 4. The kink-kink interaction force and energy for the ApBnA double kink in potassium. Figure 4(a) shows a linear plot of the dependence of force and energy on kink separation. Figure 4(b) shows a Iowa plot of the interaction force: the sotid straight fine indicates an inverse square dependence.

lated energy for infinite separation of 0.076 eV (Table 1). Experimental work on potassium [i3] indicates that for the orientation of applied shear stress used here, the flow stress extrapolated to OK is about 3 x 10q3G, which would correspond in Fig. 4(a) to a double kink separation of - 146 and an energy of O.O46eV,the energy provided by the work done by the applied stress can be estimated roughly as 14pb3, or 0.025 eV, leaving 0.021 eV to be supplied in the form of thermal energy. At 3OK, by which tanperature the flow stress has dropped to 5 7 x IO-’ G and is no longer strongty temperature dependent, the kink separation suggested by Fig. 4(a) is 256 and the double kink energy is 0.055 eV. The work done by the applied stress, estimated as above, is 25pb’ or 0.011 eV, leaving in this case 0.044 eV as the thermal contribution. The value of 0.021 eV at absolute zero is, of course, too high, but at 30 K the available thermal energy can easily give the required plastic flow rate with a process requiring 0.044 eV per activation. In view of the crudity of the estimates, the overafl picture is reasonably satisfactory. The functional dependence of the force on the kink separation can be seen moreclearly by piotting the points on a logarithmic scale, the result of which is

1768

DUESBERY: DISLOCATIONS IN THE B.C.C. LATTICE--II

shown in Fig. 4(b). As can be seen, for kink separations 286, the force is given reasonably by an inverse square law [the straight line in Fig. 4(b)]; for ~86, the force is essentially constant, as noted above. For large separations L, elasticity theory predicts that the interaction force F between two opposite kinks of length a is, from [ 161 F=-

- azb2 2L

(7)

in which K(j?) is the energy factor of the kink and B is the “character angle” [ 161of the kink (i.e. the angle between the kink line and the Burgers vector). For kinks on a screw dislocation with sufficiently large separation, /3 can be taken to be n/2 regardless of the shape of the kink. In the present case, the attractive force equation (7) is balanced by the force Fp due to the applied glide shear stress p, which is simply Fp -pba. Combining this with equation (7), the relation between applied stress and kink separation is

4

2

6

Kink

IO

6

separation,

12

L/b

(a)

x

(8) If this expression is fitted to the data of Fig. 4(b), a value for (K + a’K/@?2) of 1.33 x IO’ MPa is obtained. The value of K(n/2) for potassium, calculated using anisotropic elasticity theory, is 2.30 x 103MPa; the second derivative of K(x/2), estimated by numerical differentiation, is - 1.24 f 0.20 x 10’ MPa. Thus the elastic value of the bracketed term in equation (8) is 1.06 f 0.20 x 10s MPa, in fair agreement with the observed value. This agreement between modeling experiments and continuum theory is not surprising. The continuum expressions are rigorous for suIIlciently large kink separations, so that agreement with these is simply a confirmation of the validity of the model. Of greater interest is the observation, from Fig. 4(b), that the inverse square law is valid for separations as small as 86, or half the width of a single kink. This statement deserves some amplification in view of the earlier comments on measurement of kink separations. In this section, the separations have been computed using the constant relative displacement criterion, with a critical displacement value of 0.250b. As indicated earlier, however, this criterion is dubious for separations less than 2Ob,in which circumstances the double kink acts more like a single defect, characterised chiefly by the extent of advance of the kink peak. Perhaps the force would be better plotted against this variable for very narrow kinks. However, because of the constancy of the loop shape relative to the advance of the kink peak, this latter is directly proportional to the width as measured by any constant relative displacement criterion. Thus the diagrams in Fig. 4 would be unchanged, except for replacement of the horizontal coordinate by the kink peak height. It seems quite remarkable that the inverse square law, with a modulus reasonably close to the elastic value, should hold in this regime, since

’

-I -2

I

I

I

I

J

-1

0

I

2

3

In (separation,

L/b 1

(b) Fig. 5. The kink-kink interaction force and energy for the ApBnA double kink in iron. Figure S(a) shows a linear plot of the dependence of force and energy on kink separation. Figure S(b) shows a logarithmic plot of the interaction force: the solid straight line indicates an inverse square depcndence. the step length of the kink varies by more than a factor of two. It is possible, however, that the step length measured on the shape diagrams in Fig. 3 is not an exact analogue of the classical step length and that the latter, if a way were found to measure it, would prove to be more constant. The force-separation and energy-separation diagrams for the double kink in the iron model are shown in Fig. 5(a). There is an overall similarity to the potassium behaviour [Fig. 4(a)], but with changes of scale due to the higher stresses and smaller kink width in iron. The stress required for equilibrium drops slowly at first, then faster as the separation exceeds about 26; the rate of decrease of the stress then gradually drops as the separation increases further. The energy of the double kink increases steadily, as for potassium, but to a value at 106 separation of 0.74 eV, which is three times the energy (0.24 eV) given in Table 1. This latter value represents the sum of a positive-energy BnA kink (0.498 eV) and a negative-energy ApB kink (-0.257 eV). Thus it seems that the catastrophe which leads to the negative energy of the isolated ApB kink does not take place when the kink is bound in close proximity to its BnA partner. The resulting double-kink energy is a

DuEsRERY: DISLOCATF~NS IN THE B.C.C. L.4ITICE-4 much more reasonable value for an iron-like material. Presumably if the separation of the kinks is increased, there comes a point when the ApB kink will collapse to negative energy again; however, this was not checked. In Fig. 5(b) are shown the force-separation data for iron plotted logarithmically. The inverse-square region is not as clear as for potassium but is nevertheless present, operating over a range of separations from about 3b to 86. The solid line drawn through the points corresponds to a value of 2.79 x IO4MPa for (K + azK/d#?‘). As was the case for potassium, this value is close to but rather larger than the result of 2.62 2 0.20 x IO4MPa obtained from elasticity theory [for iron, K(lc/2) = 1.262 x 1O’MPa and (J”K(n/2)/~?/3~) = 1.36 k 0.20 x IO’ MPa]. Deviations from the inverse square dependence can he seen in Fig. S(b) for large kink separations. These are due to the influence of the kink Peierls force, which has a magnitude of about 50 MPa. In principle one could exploit the ~mul~neous presence of elastic and lattice forces to study the energy barrier associated with the kink Peierls force, but this was considered outside the scope of the current work.

5.SUMMARY

AND DISCUSSION

An attempt has been made to provide a reasonably comprehensive study of the self and interaction energies of kinks on screw dislocations in the b.c.c. lattice. Computer models have been constructed both for

potassium, using a f~damen~l interatomic force law and for an iron-like material, using an empirical potential law. Although different in scale and detail, the two models present broadly similar pictures, reinforcing the view that many of the plastic properties of b.c.c. materials which arc absent in other structures are determined by the geometry of the b.c.c. lattice, rather than by some coincidental quirk of the binding forces or by some extrinsic agent, such as impurity atoms. As shown in the first part of this manuscript (51, many different kinks can exist on the b.c.c. screw dislocation, due to the abnormal core structure of the latter. By themselves, the kinks can be paired to form double kinks in only a small number of ways, but if the existence of flip defects 114 is allowed, the list of possible kink complexes becomes almost endIess. Fortunately, except for some symmetry-based degeneracies, the energies of the various complexes are relatively widely separated. Thus in internal friction spectra, peaks corresponding to different double kinks should be easily resolved in most cases. In the case of plastic deformation, the difference in double kink energies is large enough to rule out all but the lowest-energy defect as an effective contributor. The dist~bution of kink energy in the lattice is confined mostly to a cyhndrical region concentric 4M31110-u

1769

with the dislocation line and with radius -36. Due to the relative softness of the lattice parallel to the dislocation line, both displacement and energy distributions are elongated along this axis, the energy field having a rather larger spread. The energy distribution along the dislocation line is asymmetric, with distinct features breaking up an otherwise continuous, bellshaped variation. The most obvious such feature is a discontinuous drop in energy on one side of such kink. extending for a few atomic distances before rejoining the envelope. In the iron model, probably because of the assumed form of the potential, this discontinuous drop is greatly exaggerated, leading to a negative kink energy in some cases. The magnitude of the lowest kink energies in potassium is reasonably consistent with the observed temperatures (30 K) above which the Peierls stress is negligible [13]. A similar comparison cannot be made for the iron model, because of the negative energies encountered. The formation and equilibrium shape of double kinks has been studied by balancing the attracitve elastic force between the kinks with an externally applied shear stress. It is found that the tip of the double kink (i.e. the point of maximum advance of the defeat normal to the dislocation line) does not stabilise in the next equivalent lattice site until the separation of the kinks is greater than about five times the width of individual kinks (i.e. about 75b in potassium or 206 in iron). For separations greater than this value the kinks behave as two fully independent (though interacting) defects. For very small separations, less than about 2Ob or 1.5 kink widths in potassium, the double kink behaves more like a single defect with a shape which is a function only of the extent of advance of the kink tip. For kink separations of 1.5 to 5 kink widths the behaviour is transitional between the two extremes. For both models, the interaction force between two opposite kinks is found to be attractive, with a magnitude which remains roughly constant for separations of less than about half a kink width. For larger separations the force varies as the inverse square of the separation, with a magnitude close to that predicted by anisotropic continuum theory [I 61. The double kink energy, obtained by integrating the fork-~paration curve, therefore increases linearly for small separations and inversely with the separation for larger values, finally approaching a limit consistant (in the case of potassium) with the sum of the self energies of the two constituent kinks. Assuming a model in which the formation of a double kink is the rate controlling process, the interaction force and energy in potassium have magnitudes in reasonable agreement with the observed low-temperature flow-stress.

author wishes to acknowledge the technical assistance of H. G. Champion and C. Lacelle in performing the above research. Ack~~w~e~~~meffts-The

1770

DUESBERY:

DISLOCATIONS

REFERENCES I. P. B. Hirsch, Trans. Jupun Inst. Met&, Suppl. XXX (1968). 2. J. W. Christian, Proc. Btd Inr. Con/. on Strength of Mefals and Alloys, Asilomar, p. 31. Am. Sot. Metals.

(1970). 3. B. Sestak, Proc. 3rd. Int..Symposium “Reinststoffe in Wissenschati und Technik. Dresden. D. _ 221. Akademie, Berlin. (1970). 4. V. Vitek, Proc. R. Sot. Lond. A332, 85 (1976). 5. M. S. Duesbery, Acla metall. (1983). 6. L. Dagens, M. Rasolt and R. Taylor, Phys. Reu. Bll, 2726 (1975). 7. R. Chang and L. F. Graham, Physica status solidi 18, 99 (1966). 8. P. C. Gehlen Interatomic Potentials and Simulation of Lattice Defects (edited by P. C. Gehlen, J. R. Beeler and

IN THE B.C.C. LATTICE--II R. 1. Jaffee), p. 475. Plenum Press, New York (1972). 9. C. Wiithrich, Phil. Mag. 35, 325; ibid. 35, 337 (1977). 10. Z. S. Basinski, M. S. Due&xv and R. Taylor, Can. J. Phys. 49, 2160 (1971). I I. J. k. Hardy, J. phys: them. Solids 29, 2009 (1968). 12. J. D. Eshelbv. SolidState Physics (edited bv F. Seitz and D. Turnbullj; Vol. 3, p. 79. Academic P&s, New York (1956). 13. Z. S. Basinski, M. S. Duesbery and G. S. Murty, Acta metall. 29, 801 (1981). 14. J. D. Eshelby, Proc. R. Sot. Lond. 266A, 222 (1962). 15. A. Seeger and P. Schiller, Acta metall. 10, 348 (1962). 16. J. P. Hirth and J. Lothe, Theory of Dislocations, 2nd edn. Wiley. New York (1982). 17. M. S. Duesbery, V. Vitek and D. K. Bowen, Proc. R. Sot. Land. A332, 85 (1973). 18. M. S. Duesbery, Proc. R. Sot. Lond. (1983).