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The interaction between extended dislocations and antiphase domain boundaries — I: superpartial separation and the yield stress
Intermetallics 9 (2001) 499–506 www.elsevier.com/locate/intermet
The interaction between extended dislocations and antiphase domain boundaries — I: superpartial separation and the yield stress T.S. Ronga,*, M. Aindowb, I.P. Jonesa a
IRC in Materials for High Performance Applications and School of Metallurgy and Materials, The University of Birmingham, Birmingham, B15 2TT, UK b Department of Metallurgy and Materials Engineering, Institute of Materials Science, Box U-3163, University of Connecticut, Storrs, CT 06269-3136, USA Received 24 January 2001; accepted 11 April 2001
Abstract The interaction between an extended superdislocation in an ordered structure and an antiphase domain boundary (APB) has been analysed. In contrast to previous treatments (Cottrell AH. Relation of properties to microstructure. ASM Monograph 1954. p. 131; Ardley GW. Acta Metall 1955;3:525) we include the effects of (i) the small strip of perfect crystal within an extended dislocation straddling an APDB and (ii) the ledge in an APDB which has been cut by a dislocation. We also revisit the concept of ‘APDB thickness’. The new analysis leads to a more marked variation of superpartial separation with domain size than previously calculated and also predicts a variation of yield stress with domain size which is in better agreement with experiment than are previous calculations. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: B. Yield stress; E. Mechanical properties, theory
1. Introduction Anti-phase boundaries (APBs) are planar defects which can arise in ordered superlattice structures which have simple disordered analogues such as fcc, bcc or hcp. These boundaries are characterised by displacement vectors, R, equal to exchange operations connecting sites of one sublattice in the ordered structure with those of the other. Since such operations correspond to lattice translation vectors in the appropriate disordered structure, the crystal lattice is largely unperturbed across the APB and it is only the positions of the sublattices which change. There are two main types of APB which can be identified on the basis of their morphology and the way in which they form, although we note that their topological properties are identical. Firstly, superlattices which form as the product of order/disorder transformations may contain closed domains which arise when different variants of the ordered transformation product grow to * Corresponding author. Tel.: +44-121-4146731; fax: +44-1214145232. E-mail address: [email protected] (T.S. Rong).
meet one another. We will refer to the APBs which separate such domains as anti-phase domain boundaries (APDBs). Since there will be an excess energy associated with the APDBs, one would expect the domains to grow and the APDBs to be eliminated as thermodynamic equilibrium is approached. In practice, however, APDBs are a ubiquitous feature of the defect microstructure in many ordered superlattices. Secondly, ribbons of APB are formed by the dissociation of superdislocations with large Burgers vectors equal to translation vectors in the superlattice structure. The dissociation products are usually pairs of superpartial dislocations, with Burgers vectors equal to lattice translations of the disordered structure, separated by a band of APB. The equilibrium separation of the superpartial dislocations is determined by the character of the dislocations and the APB energy. For the purposes of this paper we will consider only glide dissociations and will refer to these APBs as shear anti-phase boundaries (SAPBs), although we note that climb dissociations are also common at elevated temperatures. Cottrell [1] considered the strengthening effect of thermal APDBs. Assuming that a displacement of half the domain width represented complete disordering
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across the slip plane, he calculated an ‘averaged-out’ rise in strength c as t 1 c ¼ l l where is the APDB energy, l the domain size, t the APDB ‘thickness’ and a geometrical factor equal to about 6. This gives a maximum of c ¼ =4t at l ¼ 2t. Note that the existence of a maximum depends on a finite value for ‘t’. Ardley [2] reported strength vs domain size measurements which show maxima at small domain size. The measurements of domain size relied on electrical resistivity measurements cross referenced to earlier X-ray work. His expression c ¼
t 3 1 l l
again relies on a ‘work done to disorder’ approach and is an average value. The cube in the equation is from simple stereological considerations and the equation is similar to Cottrell’s for small t=l. Marcinkowski and Fisher [3] incorporated simple bonding considerations into the strength expression and also pointed out that finite t and small l will lead to an increase in the equilibrium separation of superpartials according to 1=ð1 t=lÞ3 . Our knowledge of the details of APDB structure is, of course, more detailed now than it was in 1954–1962. APDBs imaged via high resolution electron microscopy (HREM) appear to have no intrinsic thickness, except perhaps within a degree or so of Tc [4]. What they do have, and this appears to be universal, is a small additional dilatation perpendicular to the boundary. There remains the possibility of intrinsic or extrinsic chemical effects. These are now amenable to investigation by the new generation of field emission gun transmission electron microscopes (FEGTEMs), but few results have yet been reported. Neither of these effects is thought to have any significance for the analysis reported in this paper. In what follows we retain the concept of APDB ‘thickness’, primarily in order to compare our results with those of previous authors. In this paper we revisit the effect of l and t on strength and superpartial separation and in a second paper [5] consider the details of the disordering process.
2. Stresses on superpartial dislocations We now consider the interaction of an extended superdislocation with an APDB using the simple case depicted schematically in Fig. 1. In this situation a straight edge superdislocation is dissociated on its glide plane into two edge superpartial dislocations with equal
Burgers vectors b, separating a ribbon of SAPB with displacement vector RSAPB ¼ b. An APDB with displacement vector RAPDB ¼ RSAPB ,1 lies perpendicular to the glide plane and crosses it parallel to the line direction of the superpartial dislocations. Let us neglect lattice friction and designate the external shear stresses which must be applied to move the leading and trailing superpartial dislocations as L and T , respectively. Before the superpartial dislocations interact with the APDB [Fig. 1(a)], if we define shear stresses directed from the trailing superpartial dislocation towards the leading one as positive, L and T will be: d 1 ð1Þ L ¼ b r and T ¼
d 1 b r
ð2Þ
2
where d ¼ KðÞb 2 [6] is the equilibrium separation of coupled superpartial dislocations in a single domain and r is the actual separation. In the usual equilibrium situation r ¼ d and L ¼ T ¼ 0. The reason for distinguishing between r and d is that we will shortly consider a situation where the two superpartial dislocations are not in stress equilibrium. When the leading superpartial dislocation penetrates the APDB under the influence of the applied stress, a thin band of perfect crystal is formed on the glide plane with a width equal to b [Fig. 1(b)]. As shown in the inset to this figure, this narrow band is physically significant since it corresponds to a row of ‘‘correct’’ bonds across the glide plane in the middle of the SAPB. For the configuration shown in Fig. 1(b), where the trailing superpartial dislocation has not yet reached the position of the APDB, the forces on both of the superpartial dislocations will be virtually unchanged. To a first approximation, the applied shear stresses required to move them will be equal to those given in Eqs. (1) and (2).2 Once the trailing superpartial dislocation meets the APDB, however, further forward motion would require the creation of an additional area of SAPB as the dislocation crosses the band of perfect crystal. Thus, the stress required to move the trailing superpartial dislocation forward from this position changes from that given in Eq. (2) to: 1
Although RSAPB is determined explicitly by b, the indices of RAPDB are chosen from a set of equivalent exchange operations. Thus for simple ordered structures such as B2 for which there are only two sublattices with equal atomic densities, the condition RSAPB ¼ RAPDB will be valid in all situations. 2 Clearly if RSAPB 6¼ RAPDB then the junctions of APDBs with the SAPBs in Fig. 1(b) would have dislocation character which would introduce additional terms into the equations.
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Fig. 1. Schematic diagrams of the interaction of a straight extended edge superdislocation consisting of two edge superpartial dislocations with equal Burgers vectors b, with an APDB lying parallel to the line-direction of the dislocation and perpendicular to the slip plane; (a) before the leading superpartial dislocation meets the APDB; (b) after the leading superpartial dislocation has penetrated the APDB leaving a narrow band of perfect crystal; (c) after both the leading and trailing superpartial dislocations have penetrated the APDB leaving a step of width 2b in the boundary. An example of the atomic arrangements at the APDB are shown in the insets to these figures (drawn so that they correspond to {001} sections through a phase with B1 NaCl-type order).
KðÞb d ¼ þ1 T ¼ þ b 2r b r
ð3Þ
because the APB energy is now acting as a drag on the trailing superpartial dislocation, rather than pulling it forward. Thus, in this configuration, T > L , i.e. there is an additional retarding stress which will inhibit the penetration of the APDB by the trailing superpartial
dislocation and will lead to a separation r > d between the two superpartial dislocations. When the trailing superpartial dislocation does penetrate the APDB, the stress required to move it will revert to that given in Eq. (2). However, as noted by Cottrell [1], there will be a step in the APDB of width 2b on the glide plane [Fig. 1(c)] and this will give a net increase in the energy of the system due to the additional APDB area.
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3. The effect of domain size on superdislocation dissociation We now consider the effect of a distribution of APDBs crossing the glide plane on the equilibrium separation of coupled superpartial dislocations, following the approach of Marcinkowski and Fisher [3]. In their analysis a relationship was derived between the domain size, l, and the superpartial dislocation separation, r, which incorporated the effects of the APDB thickness, t. This analysis did not, however, include the effects of the thin band of perfect crystal that arises when the coupled superpartial dislocations straddle the APDB, and the additional boundary area of the domain once the whole superdislocation has passed the APDB. For simplicity we will consider the situation shown schematically in Fig. 2 which is a section passing through an extended edge superdislocation on the glide plane. The superdislocation is dissociated into two edge superpartial dislocations with equal Burgers vectors, b, and these intersect a distribution of APDBs which have RAPDB ¼ b and lie perpendicular to the glide plane. On the left of the diagram, we have the undisturbed APDBs ahead of the leading superpartial dislocation. Since APDBs will adopt arbitrary shapes of varying complexity it is not possible to obtain an accurate measure of the area of APDB on the glide plane, but following Marcinkowski and Fisher [3], we take the area fraction, f1 , as equal to the volume fraction occupied by the APDBs and so: t 3 : f1 ¼ 1 1 l
ð4Þ
In the centre of the diagram, we have the situation between the leading and trailing superpartial dislocations where there are thin bands of perfect crystal in the SAPB. These correspond to the positions where the leading superpartial dislocation has cut the APDBs. If we again assume that the distribution takes the form of Eq. (4), then the area fraction of SAPB between the superpartial dislocations, f2 , is:
tþb f2 ¼ 1 l
3 :
This expression will give an underestimate of the value for f2 , since it implies that the bands that do not correspond to SAPD in the glide plane have a uniform width of (t þ b). This is actually the maximum width of the bands and is only appropriate where the APDB cuts the glide plane perpendicular to b. Thus, for certain regular domain shapes, the value of f2 could be significantly higher than that given by Eq. (5). On the right of the diagram, we have the situation where both of the superpartial dislocations have cut through the APDBs. The areas of SAPB created by the leading superpartial dislocation have been removed by the trailing one, but bands of additional APDB area with a maximum width of 2b have been created. Using the same assumptions as before, the area fraction of APDB in the glide plane is: t þ 2b 3 f3 ¼ 1 1 ; l
ð6Þ
although, for the reasons noted above, this will overestimate f3 . In addition, it is worth adding that a numerical factor between 0 and 1, depending on geometry of antiphase domains, can be added to the distribution functions f1 , f2 and f3 . However, for simplicity and for comparing with the existing models which only include the effect of domain wall thickness, the numerical factor is set to 1. We can now obtain expressions for the stresses required to move the dislocations (neglecting lattice friction) by balancing the forces as before. Averaging the forces along the leading superpartial dislocation, the stress to move it is: d d ¼ f2 f1 : L ¼ f1 þ f2 b b br b r
Fig. 2. Schematic diagram of the interaction of a straight extended edge superdislocation consisting of two edge superpartial dislocations having equal Burgers vectors, with a distribution of APDBs lying perpendicular to the slip plane.
ð5Þ
ð7Þ
The first and second terms correspond to the APDB removed by, and the SAPB created by the passage of the leading superpartial dislocation, respectively, whereas the third term comes from the interaction between the two superpartial dislocations. Similarly, the stress required to move the trailing superpartial dislocation is:
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d d ¼ f3 f2 þ : T ¼ f2 þ f3 þ b b br b r
ð8Þ
Moreover, the stress, P , required to move the whole extended superdislocation whose Burgers vector has a magnitude 2b is: P ¼
f1 þ f3 ¼ ðf3 f1 Þ 2b 2b
ð9Þ
The first term corresponds to the APDB removed by the leading superpartial dislocation and the second to that left by the trailing superpartial dislocation. The equilibrium separation for this configuration can be obtained by equating any pair of Eqs. (7), (8) or (9), i.e. by setting L ¼ T , L ¼ P , or T ¼ P . Using L ¼ T from Eqs. (7) and (8) and replacing f1 , f2 , and f3 by using Eqs. (4)–(6), we obtain: d tþb 3 1 ¼ 1 1 þ r l 2 ! h t i3 t þ 2b 3 1 þ 1 : l l
ð10Þ
This equation gives the ratio of d, the equilibrium separation of coupled superpartial dislocations in a single domain region, to r, the equilibrium separation in a multi-domain region. Thus, in a multi-domain region, the separation of coupled superpartial dislocations varies with the APDB thickness, domain size and the magnitude of
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the Burgers vector for the superpartial dislocations. For example, if we set t equal to 0.3 nm for the purposes of comparison with earlier work and take 0.28 nm as a reasonable value for b, then the variation of r/d with l expected from Eq. (10) is shown in the upper curve in Fig. 3. For comparison, the variation which one would expect if only the APDB thickness were considered [3] is shown in the lower curve. The difference between the values given by these two approaches increases with decreasing l. For example, when l=3 nm, r ¼ 1:4d when only the APDB thickness is considered. However, when Eq. (10) is used which includes the effects of the thin band of perfect crystal that arises when the coupled superpartial dislocations straddle the APDB, and the additional boundary area of the domain once the whole superdislocation has passed the APDB, r ¼ 14d, i.e. there is an order of magnitude difference in the values. Clearly, if superpartial dislocation separations are used to estimate for phases with fine domains without taking these effects into account, then very large errors could be introduced with misleadingly low values of being obtained in each case. We can calculate the separation of a pair of coupled pure edge superpartial 2dislocations as a function of and l if we use d ¼ KðÞb 2 in Eq. (10) and assume that the material is elastically isotropic so that the energy factor becomes [6]: cos2 KðÞ ¼ sin2 þ ; ð1 Þ
ð11Þ
Fig. 3. Separation of coupled superpartial dislocations (r) as a function of APD size (l) , for t ¼ 0:3 nm and b ¼ 0:28 nm. The lower curve is the variation expected if the effects of APDB thickness only are considered. The upper curve was obtained from Eq. (10) and includes also the effects of the narrow bands of perfect crystal and the steps on the APDBs as shown in Figs. 1 and 2. This curve is appropriate to, for example, B2. The curve in between is calculated from Eq. (13), and is appropriate, for example, to the L12 structure.
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where is the shear modulus, is Poisson’s ratio, and is the angle between the Burgers vector and the normal to the dislocation line. If we use b ¼ 0:28 nm and t ¼ 0:3 nm as before, assume ¼ 1=3 and take 50 GPa as a reasonable value for (for most intermetallic compounds, ¼ 20100 GPa [7]) then the variation of r with and l is as shown in Fig. 4. When both l and are small, the separation of coupled superpartial dislocations can exceed 1 mm. Experimental examples of such large spacings include Nb-based alloys with the B2 structure and small l , for which superpartial dislocation separations of a few hundred nanometres have been observed in the transmission electron microscope (TEM) by Hou [8]. Moreover, it can be shown that there is a critical grain size below which one would expect the formation of uncoupled superpartial dislocations. The details of this analysis are described in Part II [5]. In the above estimation of the separation of superpartial dislocations in a multiple domain situation, we assume that RSAPB ¼ RAPDB . This is true for some ordered structures, such as B1, B2 and L10, where a strip of perfect crystal is always created at APDBs when they are cut by a partial dislocation. However, if RSAPB 6¼ RAPDB , strips of perfect crystal may not be created at all APDBs when they are straddled by an extended superdislocation. Thus, to estimate the separation of superpartial dislocations in ordered structures where RSAPB 6¼ RAPDB , a numerical factor between 0 and 1, depending upon the specific ordering structure, would need to be introduced into the area fraction of SAPB between the superpartial dislocations, f2 , to compensate. For example, for the L12 structure, because the strips of perfect crystal between the superpartials are created at only about 1/3 of the APDBs that have been cut, f2 becomes
f2 ¼
t i3 1 t þ b 3 2h 1ð Þ þ 1 : 3 l 3 l
ð12Þ
Using the same logic, we can calculate the separation of superpartial dislocations. Eq. (10) can then be rewritten as d 1 t þ b 3 7 t 3 1 t þ 2b 3 ¼ 1 1 þ 1 þ 1 r 3 l 6 l 2 l ð13Þ Again, we set t ¼ 0:3 nm and b ¼ 0:28 nm. A curve calculated from the above equation showing r=d as a function of l for the L12 is included in Fig. 3. As expected, the curve for the L12 lattice is between the upper curve where the condition RSAPB ¼ RAPDB (e.g. B2) is satisfied and the lower curve where the effect of the narrow bands of perfect crystal is ignored. Obviously, RSAPB ¼ RAPDB gives the maximum separation. We should emphasise that the condition RSAPB ¼ RAPDB or RSAPB 6¼ RAPDB only affects the separation of the superpartial dislocations but not the stress required to move a dislocation. From Eq. (9) it is clear that the stress required to move a pair of superpartial dislocations is not a function of f2 . For whatever ordered structures, a ledge in an APDB will always be created after a complete dislocation cuts the APDB. Therefore, in the following discussion on the effect of domain size upon the yield stress, the condition of RSAPB ¼ RAPDB is ignored.
4. The effect of domain size on the yield stress Cottrell’s equation for the increase in yield strength due to thermal APBD was [1] t c ¼ 1 ; ð14Þ l l and Ardley’s [2] was c ¼
t 3 1 : l l
ð15Þ
If we use Eq. (15) and differentiate with respect to l, then setting dc =dl ¼ 0, we obtain c ¼ cmax when l ¼ 4t. Substituting back into Eq. (15) gives: cmax ¼
Fig. 4. Separation of coupled superpartial dislocations as a function of APD size (l) and SAPB energy (), obtained from Eq. (10) assuming isotropic elasticity with t ¼ 0:3 nm, b=0.28 nm, =1/3 and =50 GPa.
t 3 27 1 0:105 : ¼ l l 256 t t
ð16Þ
The effect of l upon c for Cu3Au at room temperature was measured experimentally and fitted to Eq. (15) by Ardley [2]. Although very good agreement was claimed between the experimental measurements and
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the values predicted using Eq. (15), we note that values for of between 94 and 115 mJ/m2 were used. It has since been shown that these values are too high by a factor of 2: weak beam imaging in the TEM has been used to measure and values of 39 mJ/m2 [9] and 48 mJ/m2 [10] have been obtained. If we use ¼ 43:5 mJ/ m2, which is the average of these two measured values, and t ¼ 0:3 nm as before, then from Eq. (16) we would expect a maximum increase in the yield stress of about 15 MPa. However, the experimental data presented by Ardley [2] show a much larger increase than this. Our modified model also includes the effects of the thin band of perfect crystal that arises when the coupled superpartial dislocations straddle the APDB, and the additional boundary area of the domain once the whole superdislocation has passed the APDB. However, the stress in Eq. (9) varies only with the area fraction of APBs (including thermal APBs and SAPBs) before and after a dislocation cuts the APDBs. Although the displacement vector for the APBD and the Burgers vector of the dislocations may not be the same for different ordered structures, a ledge in an APBD that has been cut by a dislocation should always be created. Thus, we can still use Eq. (9) to relate yield stress to l for e.g. Cu3Au. Differentiating with respect to l, and setting d p =dl ¼ 0 as before, we obtain: " # dp 3t t 2 3ðt þ 2bÞ t þ 2b 2 ¼ 1 1 ¼ 0; 2b l2 l l dl l2 ð17Þ
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i.e. ðl tÞ2 t ¼ ðl t 2bÞ2 ðt þ 2bÞ:
ð18Þ
Rearranging in terms of l: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ¼ 2ðt þ bÞ þ t2 þ 2tb;
ð19Þ
or l ¼ 2ðt þ bÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t2 þ 2tb
ð20Þ
From the sign of d2 p =dl2 , we know that Eqs. (19) and (20) correspond to the maximum and minimum values of p respectively. Substituting Eq. (19) into Eq. (9), the maximum of p as a function of and t can be written as " 3 t max pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p ¼ 2b 2ðt þ bÞ þ t2 þ 2tb ð21Þ 3 # t þ 2b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2ðt þ bÞ þ t2 þ 2tb
For Cu3Au, a0 ¼ 0:3734 nm [11] and b ¼ a0 =2h110i (b 0.27 nm). Again using t ¼ 0:3 nm and substituting back into Eq. (21), we find pmax ¼ 34 MPa, which is more than twice as large as cmax . Three plots of yield stress against l for Cu3Au are presented in Fig. 5 corresponding to the experimental data from Ardley [2], to
Fig. 5. Yield stress as function of APD size (l) for Cu3Au. The curves obtained from Eqs. (9) and (15) using t ¼ 0:3 nm, b ¼ 0:27 nm and ¼ 43:5 mJ/m2, are compared with the experimental data of Ardley [2].
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Eq. (15) and to Eq. (9). Clearly, Eq. (9) gives a much better fit to the experimental data than does Eq. (15).
5. Discussion Our somewhat more detailed look at the interaction between extended superpartial dislocations and thermal APBD’s might be expected to improve the accuracy of the expressions for superpartial separation and strengthening and, indeed, we have presented some evidence above that this is so. The existence of a maximum in the strength c vs APBD size l graph still depends, however, as much for our approach as for the earlier ones, on a finite APBD thickness. And yet HREM micrographs of end on APDBs show no evidence of APDB thickness, unless at temperatures very close to the order-disorder transformation temperature. If we look elsewhere for possible reasons for a maximum in the strength, the ‘non-classical’ aspects of APBD’s which are either known or suspected are the small dilatation which seems to be a universal feature of all APDB’s and a change in the chemical composition. It is difficult to see how either effect would cause a maximum in the strength curve. Remembering that the origin of the strength maximum lies in the effective disordering that a finite APDB thickness brings with it, it seems more likely that the strength maximum lies in a decreasing degree of long range order within small domains. This might be allied to the physical processes used to achieve the small domain size, or might be a permanent feature of APDBs in the form of an atmosphere of disorder undetectable by normal HREM observation, but nevertheless corresponding effectively to a finite APDB thichness ‘t’. We favour the former possibility.
6. Conclusions In this paper we have considered the interactions of extended superdislocations and APDBs with particular emphasis on the effects of the thin band of perfect crystal that arises when the coupled superpartial disloca-
tions straddle the APDB, and the additional boundary area of the domain once the whole superdislocation has passed the APDB. It has been shown that if these effects are included, then, in an ordered phase containing multiple domains, we would expect the equilibrium separation of the coupled superpartial dislocations to vary more significantly with domain size than is predicted by previous models which only include the effects of APDB thickness. It has also been shown that these effects would also change the way in which yield stress varies with domain size and that the variation which one would expect gives a much better match to published experimental data for Cu3Au than previous models which only include the effects of APDB thickness.
Acknowledgements The authors would like to thank Professor D.G. Morris for helpful comments, Professors M.H. Loretto and I.R. Harris for the provision of laboratory facilities, and the Engineering and Physical Sciences Research Council for financial support under the IRC main grant.
References [1] Cottrell AH. Relation of properties to microstructure, ASM Monograph, 1954, p. 131. [2] Ardley GW. Acta Metall 1955;3:525. [3] Marcinkowski MJ, Fisher RM. J Appl Phys 1963;34:2135. [4] Finel A, Mazauric V, Ducastelle F. Phys Rev Lett 1990;65:1016. [5] Rong TS, Aindow M, Jones IP, Intermetallics 2001;9:507. [6] Hirth JP, Lothe J. Theory of Dislocations. 2nd ed. New York: John Wiley & Sons, 1982. [7] Westbrook JH, Fleischer RL. Intermetallic Compounds I. New York: John Wiley & Sons, 1995 (p. 891). [8] Hou DH. PhD thesis, The Ohio State University, USA, 1994. [9] Sastry SL, Ramaswami B. Phil Mag 1976;33:375. [10] Morris DG, Smallman RE. Acta Metall 1975;23:73 [11] Hudtgren R, Desai PD, Hawkins DT, Gleisser M, Kelley KK. Selected Values of the Thermodynamic Properties of Binary Alloys. American Society for Metals, 1973, p. 258.
The interaction between extended dislocations and antiphase domain boundaries — I: superpartial separation and the yield stress T.S. Ronga,*, M. Aindowb, I.P. Jonesa a
IRC in Materials for High Performance Applications and School of Metallurgy and Materials, The University of Birmingham, Birmingham, B15 2TT, UK b Department of Metallurgy and Materials Engineering, Institute of Materials Science, Box U-3163, University of Connecticut, Storrs, CT 06269-3136, USA Received 24 January 2001; accepted 11 April 2001
Abstract The interaction between an extended superdislocation in an ordered structure and an antiphase domain boundary (APB) has been analysed. In contrast to previous treatments (Cottrell AH. Relation of properties to microstructure. ASM Monograph 1954. p. 131; Ardley GW. Acta Metall 1955;3:525) we include the effects of (i) the small strip of perfect crystal within an extended dislocation straddling an APDB and (ii) the ledge in an APDB which has been cut by a dislocation. We also revisit the concept of ‘APDB thickness’. The new analysis leads to a more marked variation of superpartial separation with domain size than previously calculated and also predicts a variation of yield stress with domain size which is in better agreement with experiment than are previous calculations. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: B. Yield stress; E. Mechanical properties, theory
1. Introduction Anti-phase boundaries (APBs) are planar defects which can arise in ordered superlattice structures which have simple disordered analogues such as fcc, bcc or hcp. These boundaries are characterised by displacement vectors, R, equal to exchange operations connecting sites of one sublattice in the ordered structure with those of the other. Since such operations correspond to lattice translation vectors in the appropriate disordered structure, the crystal lattice is largely unperturbed across the APB and it is only the positions of the sublattices which change. There are two main types of APB which can be identified on the basis of their morphology and the way in which they form, although we note that their topological properties are identical. Firstly, superlattices which form as the product of order/disorder transformations may contain closed domains which arise when different variants of the ordered transformation product grow to * Corresponding author. Tel.: +44-121-4146731; fax: +44-1214145232. E-mail address: [email protected] (T.S. Rong).
meet one another. We will refer to the APBs which separate such domains as anti-phase domain boundaries (APDBs). Since there will be an excess energy associated with the APDBs, one would expect the domains to grow and the APDBs to be eliminated as thermodynamic equilibrium is approached. In practice, however, APDBs are a ubiquitous feature of the defect microstructure in many ordered superlattices. Secondly, ribbons of APB are formed by the dissociation of superdislocations with large Burgers vectors equal to translation vectors in the superlattice structure. The dissociation products are usually pairs of superpartial dislocations, with Burgers vectors equal to lattice translations of the disordered structure, separated by a band of APB. The equilibrium separation of the superpartial dislocations is determined by the character of the dislocations and the APB energy. For the purposes of this paper we will consider only glide dissociations and will refer to these APBs as shear anti-phase boundaries (SAPBs), although we note that climb dissociations are also common at elevated temperatures. Cottrell [1] considered the strengthening effect of thermal APDBs. Assuming that a displacement of half the domain width represented complete disordering
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across the slip plane, he calculated an ‘averaged-out’ rise in strength c as t 1 c ¼ l l where is the APDB energy, l the domain size, t the APDB ‘thickness’ and a geometrical factor equal to about 6. This gives a maximum of c ¼ =4t at l ¼ 2t. Note that the existence of a maximum depends on a finite value for ‘t’. Ardley [2] reported strength vs domain size measurements which show maxima at small domain size. The measurements of domain size relied on electrical resistivity measurements cross referenced to earlier X-ray work. His expression c ¼
t 3 1 l l
again relies on a ‘work done to disorder’ approach and is an average value. The cube in the equation is from simple stereological considerations and the equation is similar to Cottrell’s for small t=l. Marcinkowski and Fisher [3] incorporated simple bonding considerations into the strength expression and also pointed out that finite t and small l will lead to an increase in the equilibrium separation of superpartials according to 1=ð1 t=lÞ3 . Our knowledge of the details of APDB structure is, of course, more detailed now than it was in 1954–1962. APDBs imaged via high resolution electron microscopy (HREM) appear to have no intrinsic thickness, except perhaps within a degree or so of Tc [4]. What they do have, and this appears to be universal, is a small additional dilatation perpendicular to the boundary. There remains the possibility of intrinsic or extrinsic chemical effects. These are now amenable to investigation by the new generation of field emission gun transmission electron microscopes (FEGTEMs), but few results have yet been reported. Neither of these effects is thought to have any significance for the analysis reported in this paper. In what follows we retain the concept of APDB ‘thickness’, primarily in order to compare our results with those of previous authors. In this paper we revisit the effect of l and t on strength and superpartial separation and in a second paper [5] consider the details of the disordering process.
2. Stresses on superpartial dislocations We now consider the interaction of an extended superdislocation with an APDB using the simple case depicted schematically in Fig. 1. In this situation a straight edge superdislocation is dissociated on its glide plane into two edge superpartial dislocations with equal
Burgers vectors b, separating a ribbon of SAPB with displacement vector RSAPB ¼ b. An APDB with displacement vector RAPDB ¼ RSAPB ,1 lies perpendicular to the glide plane and crosses it parallel to the line direction of the superpartial dislocations. Let us neglect lattice friction and designate the external shear stresses which must be applied to move the leading and trailing superpartial dislocations as L and T , respectively. Before the superpartial dislocations interact with the APDB [Fig. 1(a)], if we define shear stresses directed from the trailing superpartial dislocation towards the leading one as positive, L and T will be: d 1 ð1Þ L ¼ b r and T ¼
d 1 b r
ð2Þ
2
where d ¼ KðÞb 2 [6] is the equilibrium separation of coupled superpartial dislocations in a single domain and r is the actual separation. In the usual equilibrium situation r ¼ d and L ¼ T ¼ 0. The reason for distinguishing between r and d is that we will shortly consider a situation where the two superpartial dislocations are not in stress equilibrium. When the leading superpartial dislocation penetrates the APDB under the influence of the applied stress, a thin band of perfect crystal is formed on the glide plane with a width equal to b [Fig. 1(b)]. As shown in the inset to this figure, this narrow band is physically significant since it corresponds to a row of ‘‘correct’’ bonds across the glide plane in the middle of the SAPB. For the configuration shown in Fig. 1(b), where the trailing superpartial dislocation has not yet reached the position of the APDB, the forces on both of the superpartial dislocations will be virtually unchanged. To a first approximation, the applied shear stresses required to move them will be equal to those given in Eqs. (1) and (2).2 Once the trailing superpartial dislocation meets the APDB, however, further forward motion would require the creation of an additional area of SAPB as the dislocation crosses the band of perfect crystal. Thus, the stress required to move the trailing superpartial dislocation forward from this position changes from that given in Eq. (2) to: 1
Although RSAPB is determined explicitly by b, the indices of RAPDB are chosen from a set of equivalent exchange operations. Thus for simple ordered structures such as B2 for which there are only two sublattices with equal atomic densities, the condition RSAPB ¼ RAPDB will be valid in all situations. 2 Clearly if RSAPB 6¼ RAPDB then the junctions of APDBs with the SAPBs in Fig. 1(b) would have dislocation character which would introduce additional terms into the equations.
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Fig. 1. Schematic diagrams of the interaction of a straight extended edge superdislocation consisting of two edge superpartial dislocations with equal Burgers vectors b, with an APDB lying parallel to the line-direction of the dislocation and perpendicular to the slip plane; (a) before the leading superpartial dislocation meets the APDB; (b) after the leading superpartial dislocation has penetrated the APDB leaving a narrow band of perfect crystal; (c) after both the leading and trailing superpartial dislocations have penetrated the APDB leaving a step of width 2b in the boundary. An example of the atomic arrangements at the APDB are shown in the insets to these figures (drawn so that they correspond to {001} sections through a phase with B1 NaCl-type order).
KðÞb d ¼ þ1 T ¼ þ b 2r b r
ð3Þ
because the APB energy is now acting as a drag on the trailing superpartial dislocation, rather than pulling it forward. Thus, in this configuration, T > L , i.e. there is an additional retarding stress which will inhibit the penetration of the APDB by the trailing superpartial
dislocation and will lead to a separation r > d between the two superpartial dislocations. When the trailing superpartial dislocation does penetrate the APDB, the stress required to move it will revert to that given in Eq. (2). However, as noted by Cottrell [1], there will be a step in the APDB of width 2b on the glide plane [Fig. 1(c)] and this will give a net increase in the energy of the system due to the additional APDB area.
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3. The effect of domain size on superdislocation dissociation We now consider the effect of a distribution of APDBs crossing the glide plane on the equilibrium separation of coupled superpartial dislocations, following the approach of Marcinkowski and Fisher [3]. In their analysis a relationship was derived between the domain size, l, and the superpartial dislocation separation, r, which incorporated the effects of the APDB thickness, t. This analysis did not, however, include the effects of the thin band of perfect crystal that arises when the coupled superpartial dislocations straddle the APDB, and the additional boundary area of the domain once the whole superdislocation has passed the APDB. For simplicity we will consider the situation shown schematically in Fig. 2 which is a section passing through an extended edge superdislocation on the glide plane. The superdislocation is dissociated into two edge superpartial dislocations with equal Burgers vectors, b, and these intersect a distribution of APDBs which have RAPDB ¼ b and lie perpendicular to the glide plane. On the left of the diagram, we have the undisturbed APDBs ahead of the leading superpartial dislocation. Since APDBs will adopt arbitrary shapes of varying complexity it is not possible to obtain an accurate measure of the area of APDB on the glide plane, but following Marcinkowski and Fisher [3], we take the area fraction, f1 , as equal to the volume fraction occupied by the APDBs and so: t 3 : f1 ¼ 1 1 l
ð4Þ
In the centre of the diagram, we have the situation between the leading and trailing superpartial dislocations where there are thin bands of perfect crystal in the SAPB. These correspond to the positions where the leading superpartial dislocation has cut the APDBs. If we again assume that the distribution takes the form of Eq. (4), then the area fraction of SAPB between the superpartial dislocations, f2 , is:
tþb f2 ¼ 1 l
3 :
This expression will give an underestimate of the value for f2 , since it implies that the bands that do not correspond to SAPD in the glide plane have a uniform width of (t þ b). This is actually the maximum width of the bands and is only appropriate where the APDB cuts the glide plane perpendicular to b. Thus, for certain regular domain shapes, the value of f2 could be significantly higher than that given by Eq. (5). On the right of the diagram, we have the situation where both of the superpartial dislocations have cut through the APDBs. The areas of SAPB created by the leading superpartial dislocation have been removed by the trailing one, but bands of additional APDB area with a maximum width of 2b have been created. Using the same assumptions as before, the area fraction of APDB in the glide plane is: t þ 2b 3 f3 ¼ 1 1 ; l
ð6Þ
although, for the reasons noted above, this will overestimate f3 . In addition, it is worth adding that a numerical factor between 0 and 1, depending on geometry of antiphase domains, can be added to the distribution functions f1 , f2 and f3 . However, for simplicity and for comparing with the existing models which only include the effect of domain wall thickness, the numerical factor is set to 1. We can now obtain expressions for the stresses required to move the dislocations (neglecting lattice friction) by balancing the forces as before. Averaging the forces along the leading superpartial dislocation, the stress to move it is: d d ¼ f2 f1 : L ¼ f1 þ f2 b b br b r
Fig. 2. Schematic diagram of the interaction of a straight extended edge superdislocation consisting of two edge superpartial dislocations having equal Burgers vectors, with a distribution of APDBs lying perpendicular to the slip plane.
ð5Þ
ð7Þ
The first and second terms correspond to the APDB removed by, and the SAPB created by the passage of the leading superpartial dislocation, respectively, whereas the third term comes from the interaction between the two superpartial dislocations. Similarly, the stress required to move the trailing superpartial dislocation is:
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d d ¼ f3 f2 þ : T ¼ f2 þ f3 þ b b br b r
ð8Þ
Moreover, the stress, P , required to move the whole extended superdislocation whose Burgers vector has a magnitude 2b is: P ¼
f1 þ f3 ¼ ðf3 f1 Þ 2b 2b
ð9Þ
The first term corresponds to the APDB removed by the leading superpartial dislocation and the second to that left by the trailing superpartial dislocation. The equilibrium separation for this configuration can be obtained by equating any pair of Eqs. (7), (8) or (9), i.e. by setting L ¼ T , L ¼ P , or T ¼ P . Using L ¼ T from Eqs. (7) and (8) and replacing f1 , f2 , and f3 by using Eqs. (4)–(6), we obtain: d tþb 3 1 ¼ 1 1 þ r l 2 ! h t i3 t þ 2b 3 1 þ 1 : l l
ð10Þ
This equation gives the ratio of d, the equilibrium separation of coupled superpartial dislocations in a single domain region, to r, the equilibrium separation in a multi-domain region. Thus, in a multi-domain region, the separation of coupled superpartial dislocations varies with the APDB thickness, domain size and the magnitude of
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the Burgers vector for the superpartial dislocations. For example, if we set t equal to 0.3 nm for the purposes of comparison with earlier work and take 0.28 nm as a reasonable value for b, then the variation of r/d with l expected from Eq. (10) is shown in the upper curve in Fig. 3. For comparison, the variation which one would expect if only the APDB thickness were considered [3] is shown in the lower curve. The difference between the values given by these two approaches increases with decreasing l. For example, when l=3 nm, r ¼ 1:4d when only the APDB thickness is considered. However, when Eq. (10) is used which includes the effects of the thin band of perfect crystal that arises when the coupled superpartial dislocations straddle the APDB, and the additional boundary area of the domain once the whole superdislocation has passed the APDB, r ¼ 14d, i.e. there is an order of magnitude difference in the values. Clearly, if superpartial dislocation separations are used to estimate for phases with fine domains without taking these effects into account, then very large errors could be introduced with misleadingly low values of being obtained in each case. We can calculate the separation of a pair of coupled pure edge superpartial 2dislocations as a function of and l if we use d ¼ KðÞb 2 in Eq. (10) and assume that the material is elastically isotropic so that the energy factor becomes [6]: cos2 KðÞ ¼ sin2 þ ; ð1 Þ
ð11Þ
Fig. 3. Separation of coupled superpartial dislocations (r) as a function of APD size (l) , for t ¼ 0:3 nm and b ¼ 0:28 nm. The lower curve is the variation expected if the effects of APDB thickness only are considered. The upper curve was obtained from Eq. (10) and includes also the effects of the narrow bands of perfect crystal and the steps on the APDBs as shown in Figs. 1 and 2. This curve is appropriate to, for example, B2. The curve in between is calculated from Eq. (13), and is appropriate, for example, to the L12 structure.
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where is the shear modulus, is Poisson’s ratio, and is the angle between the Burgers vector and the normal to the dislocation line. If we use b ¼ 0:28 nm and t ¼ 0:3 nm as before, assume ¼ 1=3 and take 50 GPa as a reasonable value for (for most intermetallic compounds, ¼ 20100 GPa [7]) then the variation of r with and l is as shown in Fig. 4. When both l and are small, the separation of coupled superpartial dislocations can exceed 1 mm. Experimental examples of such large spacings include Nb-based alloys with the B2 structure and small l , for which superpartial dislocation separations of a few hundred nanometres have been observed in the transmission electron microscope (TEM) by Hou [8]. Moreover, it can be shown that there is a critical grain size below which one would expect the formation of uncoupled superpartial dislocations. The details of this analysis are described in Part II [5]. In the above estimation of the separation of superpartial dislocations in a multiple domain situation, we assume that RSAPB ¼ RAPDB . This is true for some ordered structures, such as B1, B2 and L10, where a strip of perfect crystal is always created at APDBs when they are cut by a partial dislocation. However, if RSAPB 6¼ RAPDB , strips of perfect crystal may not be created at all APDBs when they are straddled by an extended superdislocation. Thus, to estimate the separation of superpartial dislocations in ordered structures where RSAPB 6¼ RAPDB , a numerical factor between 0 and 1, depending upon the specific ordering structure, would need to be introduced into the area fraction of SAPB between the superpartial dislocations, f2 , to compensate. For example, for the L12 structure, because the strips of perfect crystal between the superpartials are created at only about 1/3 of the APDBs that have been cut, f2 becomes
f2 ¼
t i3 1 t þ b 3 2h 1ð Þ þ 1 : 3 l 3 l
ð12Þ
Using the same logic, we can calculate the separation of superpartial dislocations. Eq. (10) can then be rewritten as d 1 t þ b 3 7 t 3 1 t þ 2b 3 ¼ 1 1 þ 1 þ 1 r 3 l 6 l 2 l ð13Þ Again, we set t ¼ 0:3 nm and b ¼ 0:28 nm. A curve calculated from the above equation showing r=d as a function of l for the L12 is included in Fig. 3. As expected, the curve for the L12 lattice is between the upper curve where the condition RSAPB ¼ RAPDB (e.g. B2) is satisfied and the lower curve where the effect of the narrow bands of perfect crystal is ignored. Obviously, RSAPB ¼ RAPDB gives the maximum separation. We should emphasise that the condition RSAPB ¼ RAPDB or RSAPB 6¼ RAPDB only affects the separation of the superpartial dislocations but not the stress required to move a dislocation. From Eq. (9) it is clear that the stress required to move a pair of superpartial dislocations is not a function of f2 . For whatever ordered structures, a ledge in an APDB will always be created after a complete dislocation cuts the APDB. Therefore, in the following discussion on the effect of domain size upon the yield stress, the condition of RSAPB ¼ RAPDB is ignored.
4. The effect of domain size on the yield stress Cottrell’s equation for the increase in yield strength due to thermal APBD was [1] t c ¼ 1 ; ð14Þ l l and Ardley’s [2] was c ¼
t 3 1 : l l
ð15Þ
If we use Eq. (15) and differentiate with respect to l, then setting dc =dl ¼ 0, we obtain c ¼ cmax when l ¼ 4t. Substituting back into Eq. (15) gives: cmax ¼
Fig. 4. Separation of coupled superpartial dislocations as a function of APD size (l) and SAPB energy (), obtained from Eq. (10) assuming isotropic elasticity with t ¼ 0:3 nm, b=0.28 nm, =1/3 and =50 GPa.
t 3 27 1 0:105 : ¼ l l 256 t t
ð16Þ
The effect of l upon c for Cu3Au at room temperature was measured experimentally and fitted to Eq. (15) by Ardley [2]. Although very good agreement was claimed between the experimental measurements and
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the values predicted using Eq. (15), we note that values for of between 94 and 115 mJ/m2 were used. It has since been shown that these values are too high by a factor of 2: weak beam imaging in the TEM has been used to measure and values of 39 mJ/m2 [9] and 48 mJ/m2 [10] have been obtained. If we use ¼ 43:5 mJ/ m2, which is the average of these two measured values, and t ¼ 0:3 nm as before, then from Eq. (16) we would expect a maximum increase in the yield stress of about 15 MPa. However, the experimental data presented by Ardley [2] show a much larger increase than this. Our modified model also includes the effects of the thin band of perfect crystal that arises when the coupled superpartial dislocations straddle the APDB, and the additional boundary area of the domain once the whole superdislocation has passed the APDB. However, the stress in Eq. (9) varies only with the area fraction of APBs (including thermal APBs and SAPBs) before and after a dislocation cuts the APDBs. Although the displacement vector for the APBD and the Burgers vector of the dislocations may not be the same for different ordered structures, a ledge in an APBD that has been cut by a dislocation should always be created. Thus, we can still use Eq. (9) to relate yield stress to l for e.g. Cu3Au. Differentiating with respect to l, and setting d p =dl ¼ 0 as before, we obtain: " # dp 3t t 2 3ðt þ 2bÞ t þ 2b 2 ¼ 1 1 ¼ 0; 2b l2 l l dl l2 ð17Þ
505
i.e. ðl tÞ2 t ¼ ðl t 2bÞ2 ðt þ 2bÞ:
ð18Þ
Rearranging in terms of l: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ¼ 2ðt þ bÞ þ t2 þ 2tb;
ð19Þ
or l ¼ 2ðt þ bÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t2 þ 2tb
ð20Þ
From the sign of d2 p =dl2 , we know that Eqs. (19) and (20) correspond to the maximum and minimum values of p respectively. Substituting Eq. (19) into Eq. (9), the maximum of p as a function of and t can be written as " 3 t max pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p ¼ 2b 2ðt þ bÞ þ t2 þ 2tb ð21Þ 3 # t þ 2b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2ðt þ bÞ þ t2 þ 2tb
For Cu3Au, a0 ¼ 0:3734 nm [11] and b ¼ a0 =2h110i (b 0.27 nm). Again using t ¼ 0:3 nm and substituting back into Eq. (21), we find pmax ¼ 34 MPa, which is more than twice as large as cmax . Three plots of yield stress against l for Cu3Au are presented in Fig. 5 corresponding to the experimental data from Ardley [2], to
Fig. 5. Yield stress as function of APD size (l) for Cu3Au. The curves obtained from Eqs. (9) and (15) using t ¼ 0:3 nm, b ¼ 0:27 nm and ¼ 43:5 mJ/m2, are compared with the experimental data of Ardley [2].
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Eq. (15) and to Eq. (9). Clearly, Eq. (9) gives a much better fit to the experimental data than does Eq. (15).
5. Discussion Our somewhat more detailed look at the interaction between extended superpartial dislocations and thermal APBD’s might be expected to improve the accuracy of the expressions for superpartial separation and strengthening and, indeed, we have presented some evidence above that this is so. The existence of a maximum in the strength c vs APBD size l graph still depends, however, as much for our approach as for the earlier ones, on a finite APBD thickness. And yet HREM micrographs of end on APDBs show no evidence of APDB thickness, unless at temperatures very close to the order-disorder transformation temperature. If we look elsewhere for possible reasons for a maximum in the strength, the ‘non-classical’ aspects of APBD’s which are either known or suspected are the small dilatation which seems to be a universal feature of all APDB’s and a change in the chemical composition. It is difficult to see how either effect would cause a maximum in the strength curve. Remembering that the origin of the strength maximum lies in the effective disordering that a finite APDB thickness brings with it, it seems more likely that the strength maximum lies in a decreasing degree of long range order within small domains. This might be allied to the physical processes used to achieve the small domain size, or might be a permanent feature of APDBs in the form of an atmosphere of disorder undetectable by normal HREM observation, but nevertheless corresponding effectively to a finite APDB thichness ‘t’. We favour the former possibility.
6. Conclusions In this paper we have considered the interactions of extended superdislocations and APDBs with particular emphasis on the effects of the thin band of perfect crystal that arises when the coupled superpartial disloca-
tions straddle the APDB, and the additional boundary area of the domain once the whole superdislocation has passed the APDB. It has been shown that if these effects are included, then, in an ordered phase containing multiple domains, we would expect the equilibrium separation of the coupled superpartial dislocations to vary more significantly with domain size than is predicted by previous models which only include the effects of APDB thickness. It has also been shown that these effects would also change the way in which yield stress varies with domain size and that the variation which one would expect gives a much better match to published experimental data for Cu3Au than previous models which only include the effects of APDB thickness.
Acknowledgements The authors would like to thank Professor D.G. Morris for helpful comments, Professors M.H. Loretto and I.R. Harris for the provision of laboratory facilities, and the Engineering and Physical Sciences Research Council for financial support under the IRC main grant.
References [1] Cottrell AH. Relation of properties to microstructure, ASM Monograph, 1954, p. 131. [2] Ardley GW. Acta Metall 1955;3:525. [3] Marcinkowski MJ, Fisher RM. J Appl Phys 1963;34:2135. [4] Finel A, Mazauric V, Ducastelle F. Phys Rev Lett 1990;65:1016. [5] Rong TS, Aindow M, Jones IP, Intermetallics 2001;9:507. [6] Hirth JP, Lothe J. Theory of Dislocations. 2nd ed. New York: John Wiley & Sons, 1982. [7] Westbrook JH, Fleischer RL. Intermetallic Compounds I. New York: John Wiley & Sons, 1995 (p. 891). [8] Hou DH. PhD thesis, The Ohio State University, USA, 1994. [9] Sastry SL, Ramaswami B. Phil Mag 1976;33:375. [10] Morris DG, Smallman RE. Acta Metall 1975;23:73 [11] Hudtgren R, Desai PD, Hawkins DT, Gleisser M, Kelley KK. Selected Values of the Thermodynamic Properties of Binary Alloys. American Society for Metals, 1973, p. 258.