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Cluster variation method for determining the energy of slip-induced anti-phase boundary in BCC alloys
Theoretical and Applied Fracture Mechanics 35 (2001) 243±254
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Cluster variation method for determining the energy of slip-induced anti-phase boundary in BCC alloys C.G. Sch on a,*, R. Kikuchi b a
Department of Metallurgical and Materials Engineering, Computational Materials Science Laboratory, Escola Polit ecnica da Universidade de S~ ao Paulo, CEP 05508-900, S~ ao Paulo, SP, Brazil b University of California at Berkeley, Berkeley, CA, USA
Abstract The cluster variation method formalism in the irregular tetrahedron approximation has been applied to the determination of the anti-phase boundary (APB) energies of anti-phases generated by the slip of multiple dislocations on a single plane in BCC alloys. The formalism has been applied to BCC Fe±Al alloys and compared with experimental results in the same system. The results shows that in the short-range ordered (SRO) state, most of the resistance to dislocation motion is felt by the ®rst moving dislocation, which characterises slip-plane softening. For the long-range ordered (LRO) state the calculation justi®es the formation of superdislocation con®guration as experimentally observed in the phases at room temperature deformation. The results have been discussed in connection with the experimentally observed deformation modes of BCC Fe±Al alloys. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Hardening mechanisms in metallic alloys are classi®ed depending on the crystal defect interacting with dislocations into work-hardening (interaction with other dislocations), solid±solution hardening (interaction with solute atoms and other point defects), Hall±Petch mechanism (interaction with grain boundaries) and particle hardening (also divided into dispersion and precipitation hardening, depending on the type of particle) [1]. Among these mechanisms, solid±solution hardening presents a simple and economic way to produce materials with improved mechanical properties, and hence the understanding of the basic principles behind this eect is essential. The typical interactions of solutes (and point defects) with dislocations in dilute alloys involve isolated atoms (elastic, modulus and electrostatic interactions) or ``atmospheres'' of atoms, condensed on dislocations or stacking-faults [1,2]. In concentrated alloys, however, another type of interaction appears, involving the interactions of the dislocation with the distribution or the local order, of alloying atoms. This corresponds to short-range order (SRO) 1 and long-range order (LRO) hardening.
*
Corresponding author. Tel.:+55-11-3818-5241; fax: +55-11-3818-5421. E-mail address: [email protected] (C.G. Schon). 1 Here understood in a broad sense, to include also systems with tendency to phase separation, i.e. with positive deviation of the ideal thermodynamic behaviour. 0167-8442/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 0 1 ) 0 0 0 4 8 - 9
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The procedure applied to modelling SRO and LRO hardening is based on the evaluation of the increase of internal energy of the system associated with the deformation process by counting the broken atomic bonds and subtracting from the atomic bonds which are restored by the glide of a model dislocation on the slip plane [3±7]. This procedure is referred to as ``bond-counting formalism''. In the case of LRO alloys, the main contribution to this increase of energy comes from the creation of a conservative anti-phase boundary (APB). The procedure is equivalent to the determination of the energy of this APB with anti-phase vector given by the Burgers vector of the model dislocation. In SRO alloys no APB can be de®ned in the strict sense, but the eect of the glide of a dislocation can be rationalised through the concept of ``diuse'' APBs [6,8], which can be de®ned as a plane of the crystal where the local correlation of atoms is less than that in the equilibrium bulk con®gurations. Right after the glide, we assume that short-range diusion is negligible at the temperature or at the time scale of the deformation process, so that local composition and order parameters of the crystal above and below the plane of the APB remain the same as in the bulk. The proper determination of APB energies is, therefore, the fundamental feature for the theories of SRO and LRO hardening. The calculation of APB energies is done by the use of various approximate solutions of the Ising-type model for the alloy statistical mechanics. Analytical or semi-analytical approximations can be obtained neglecting atom correlations up to a given limit. For example, in the Bragg±Willians±Gorskiy (BWG) approximation, no correlation is considered at all [3]. As a consequence, the diuse APB energies vanish in such calculations. Pair correlations have been introduced into the model by the use of Monte Carlo calculations [6,8] and in this case the diuse APB energies do not vanish. In a recent paper the present authors have extended the bond-counting formalism to the irregular tetrahedron cluster approximation of the cluster variation method (CVM) and applied it to the determination of a0 h1 0 0i 0 0 1, a0 h1 0 0i 0 1 1 and a0 =2h1 1 1i 0 1 1 APB energies [7]. 2 The ®rst one was derived for didactic purposes, but the two remaining APBs are experimentally observed as result of active slip systems in B2±FeAl and D03 ±Fe3 Al [9,10]. The purpose of the present paper is to address another important deformation con®guration experimentally observed in ordered BCC Fe±Al alloys, which was not considered in [7], namely the case of the glide of multiple a0 =2h1 1 1i dislocations in the same 0 1 1 slip plane.
2. Methods 2.1. Basic de®nitions and nomenclature The basic assumption of this work is that the energy of a BCC crystal can be written as a linear combination of energies contained in the basic BCC tetrahedron cluster. This treatment has often been used in phase diagram calculations and provided good results both for the numerical estimates of phase diagrams and for the description of other thermodynamic properties [11±13]. Fig. 1 presents the irregular tetrahedron cluster in the BCC lattice and the associated partition of this lattice into four sub-lattices. The distinction between lattice point and the species occupying this lattice point is essential in this work, and will be denoted by the use, respectively, of greek and roman indices. Using this sub-lattice partition, four dierent crystal structures can be built in a binary alloy (including the BCC basic lattice), corresponding to four superlattices and two BCC disordered solid solutions. The crystallographic conditions imposed to these structures are listed in Table 1, which also reproduces the traditional Strukturbericht notation for denomination of these superlattices.
2
The Miller indices and lattice parameter a0 refer to the disordered BCC alloy.
C.G. Schon, R. Kikuchi / Theoretical and Applied Fracture Mechanics 35 (2001) 243±254
245
Fig. 1. The irregular tetrahedron cluster in the BCC lattice. Table 1 Superlattices generated by the four sub-lattice partition of the BCC lattice, shown in Fig. 1, for the case of a binary A±B alloy [14] Stoichiometry
Strukturbericht notation
A A3 B AB AB AB3 B
A2 D03 B2 B32 D03 A2
Space group Im3m Fm3m Pm3m Fd 3m Fm3m Im3m
Condition a; b; c and d are equivalent a and b are equivalent a and b are equivalent, c and d are equivalent a and c are equivalent, b and d are equivalent a and b are equivalent a; b; c and d are equivalent
The APB energy calculation divides an in®nite crystal into two halves and moves one of them relative to the other by an anti-phase vector ~ b, the plane separating the half-crystals being the APB as well as the glide plane of the model dislocation with Burgers vector ~ b. The eect of introducing an APB in the system is to raise the internal energy to a value written as Eaft , after the glide. From the CVM equilibrium calculation we can obtain the internal energy of the crystal before the glide X abcd abcd ei;j;k;l qi;j;k;l ; Ebef Ueq 6N i;j;k;l
where the factor 6 is the number of tetrahedra per lattice point in a BCC crystal, N is the number of lattice points in a crystal plane parallel to the APB and qabcd i;j;k;l is the probability of a tetrahedron con®guration fi; j; k; lg (i.e., species i sitting on lattice point a, j on lattice point b and so on). The energy eabcd i;j;k;l per tetrahedron consists of three kinds of energies: those due (i) to four ®rst neighbours (each shared by three tetrahedra), (ii) to two second neighbours (each shared by four tetrahedra) and (iii) to four-body interactions within the tetrahedron [11]. The contribution of the APB to the increase of energy for the crystal can thus be obtained as: DU Eaft
Ebef :
1
The next task is to determine the expression for Eaft as a function of the equilibrium cluster probabilities. For this purpose, it is more convenient to introduce a dierent notation, explained in detail in a former publication [7]. The probability distribution before the glide of the dislocation qbef hr ws /t us is given by the equilibrium state calculated by the CVM. On the other hand the probability qaft hr ws /t us after the glide is the equilibrium probability distribution of an extended four-point cluster which changes to the (irregular) tetrahedron fabcdg after the shift. This variable is to be obtained using the concept of Kirkwood's superposition approximation [15].
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The examples calculated in [7] showed in detail how to apply the superposition approximation in the three cases described above. The steps shall not be repeated here, but since the results for the glide of a single dislocation, a0 =2h1 1 1i 0 1 1 APB, will be needed for comparison with the results to be obtained in this work, the evaluation of Eq. (1) for this case reported in [7] is reproduced below:
DU a0 =2h1 1 1i 0 1 1 (" abd # bc ac bd bd ac bc ad qacd qi;k;j qad qabc qacd qbcd qabd qabc qbcd N X abcd i;k;j ql;j l;j;i qj;k l;k;j qi;k k;i;l qj;l i;k;l qk;j l;j;k ql;i j;i;l qk;i i;l e qai qck qal qci 4 i;j;k;l i;j;k;l qdl qdj qbj qbk " ! avc abd bcd acd abc abc bcd acd X qabcd qabcd qabcd qabcd qabcd qabcd qabcd qabcd i;j;n;l qi;k;n i;j;k;n ql;j;n i;n;k;l qn;k;j n;j;k;l qn;i;l l;k;n;j qi;k;n l;k;i;n ql;j;n l;n;i;j qn;k;j n;k;i;j qn;i;l qac qac qad qad qbd qbc qbc qbd i;n n;i n;l n;l j;n n;j n n;k k;n
abcd qabcd i;j;n;l qi;k;n;l
qacd i;n;l abcd qabcd n;k;i;j qn;k;i;l
qabc n;k;i
!#
abcd qabcd i;j;k;n ql;j;k;n
qbcd j;k;n 24qabcd i;j;k;l
) :
abcd qabcd i;n;k;l qi;n;k;j
qabc i;n;k
abcd qabcd n;j;k;l qn;j;i;l
qabd n;j;l
abcd qabcd l;k;n;j qi;k;n;j
qbcd k;n;j
abcd qabcd l;k;i;n ql;j;i;n
qacd l;i;n
abcd qabcd l;n;i;j ql;n;k;j
qabd l;n;j 2
Consider a certain group of four atoms which form a tetrahedron after the glide. They are connected by the ®rst- and second-neighbour bonds. Before the glide, mutual distances among these four atoms were farther apart, forming an extended cluster. However, even considering that this extended cluster is not a tetrahedron, it still contains sub-clusters corresponding to ®rst-neighbour pairs, second-neighbour pairs or even isosceles triangles, which are also sub-clusters of the basic tetrahedron and hence, known from the equilibrium calculation. In evaluating the energy of the system before the glide, we need the probability of ®nding the four atoms in the spread out con®guration, which must be compatible with the known subcluster probabilities, and this is the foundation of the superposition approximation. As an example, let us consider the ®rst term inside the brackets in Eq. (2). The corresponding extended cluster is composed by an isosceles triangle 3 ai bk dj and a ®rst-neighbour pair ai dl , which share a common ai point, leading to the referred probability expression. The remaining terms are obtained in a similar way, as discussed in [7]. 2.2. Glide of multiple dislocations Eq. (2) is for a single glide of the dislocation. The case for glide of two dislocations is shown in Fig. 2, presenting the six con®gurations involved in the calculation of the 2 a0 =2h1 11i 0 1 1 APB energy. This ®gure shows a projection along the h0 1 1i axis of two atom layers of the BCC lattice. The APB is located in between and all clusters aected by the APB are contained in these atom layers. Small circles represent atoms on the ®xed half of the crystal and great circles are located on the shifted half of the crystal (the arrow indicates the anti-phase vector, i.e. the Burgers vector of the model dislocation). As simple inspection shows, the length of the arrows in Fig. 1 are all longer than the second-neighbour distance of the BCC lattice, and hence are not included in the basic tetrahedron cluster. In such cases, the application of the superposition approximation leads to the following two consequences: · The probability of the extended four points is a simple product of probabilities of the two separate subclusters (of the tetrahedron), one on the upper and other on the lower plane of the dislocation.
3 This isosceles triangle crosses the APB plane. Edge bd is located entirely on the shifted side of the crystal, as shown by the ``exchanged'' indices. The remaining edges have one point on the shifted side and another on the ®xed side.
C.G. Schon, R. Kikuchi / Theoretical and Applied Fracture Mechanics 35 (2001) 243±254
247
Fig. 2. (a)±(f) The six groups of tetrahedra for the 2 a0 =2h1 1 1i 0 1 1 APB; projection views along the h0 1 1i axis. Small circles: atoms in the ®xed layer. Great circles: atoms in the shifted layer. Double lines: second-neighbour bonds. Single lines: ®rst-neighbour bonds.
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· The probability expression does not change for the passage of three and more dislocations in the disordered phase. In the ordered phase, however, sub-lattices can make the energy expressions dierent for three, four and ®ve passages as will be shown below. For the case of Fig. 2(a), the probability expression obtained by the superposition approximation is: r u r s t u r s t u s t qaft qe qe qe : 3 hw/s hnfs hs nf When the ®xed and shifted planes are reversed, Fig. 2(b) reduces to Fig. 2(a) and the probability expression for Fig. 2(b) becomes the same as Fig. 2(a). Due to symmetry of the lattice, when we specify sub-lattice names into Eq. (3), both con®gurations result in the same probability expression. A similar situation also occurs for the cases shown in Figs. 2(c) and (d) as well as for Figs. 2(e) and (f). The ®nal energy expression can be thus simpli®ed by counting twice the twelve tetrahedra corresponding to Figs. 2(a), (c) and (e). The partition of the BCC lattice into four sub-lattices implies that only the ®ve ®rst passes of the model dislocation, i.e., up to the 5 a0 =2h1 1 1i 0 1 1 APB energy, are to be considered. The symmetry of the stable superlattice (see Table 1) imposes further degeneracy among the APB energy expressions, as will be discussed later. The probability expressions for the con®guration shown in Fig. 2(c) is given by r s t u r s t u r s u t qaft qe qe qe ; 4 hw/s hwfs hws f and for Fig. 2(e) we have r r s t u r s t u s t u qaft qe qe qe : hw/s hnvf nvf h
5
Table 2 shows the correspondence of the indices for the 12 tetrahedra which cross the APB plane for the 2 a0 =2h1 1 1i 0 1 1 APB. This table is employed in the evaluation of Eq. (1) for this case. Using these de®nitions, Eq. (1) can be written as N X abcd nh ad ad bc ac ac bd bd DU 2 a0 =2h1 1 1i 0 1 1 ei;j;k;l qi;l qj;k qbc j;k qi;l qi;k qj;l qj;l qi;k 2 i;j;k;l abc c bcd b acd a d qabd i;j;l qk qi;j;k ql qj;k;l qi qi;k;l qj i o b bcd c abc abcd d abd qai qacd q q q q q q 12q 6 j i;l;k j;l;k k j;i;l i;j;k;l : l j;i;k Table 2 Correspondence of the indices for the tetrahedra which cross the APB plane for the 2 a0 =2h1 1 1i 0 1 1 APB #
h
w
/
s
f
n
v
r
s
t
u
Fig. 2
1 2 3 4 5 6 7 8 9 10 11 12
a b c d a b c d a b c d
b a d c b a d c d c a b
c d b a c d b a c d b a
d c a b d c a b b a d c
d c a b ± ± ± ± c d b a
a b c d d c a b d c a b
± ± ± ± ± ± ± ± a b c d
i j k l i j k l i j k l
j i l k j i l k l k i j
k l j i k l j i k l j i
l k i j l k i j j i l k
(a) (a) (a) (a) (c) (c) (c) (c) (e) (e) (e) (e)
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249
As the reader will note, Eq. (6) can be understood in a simple way, if one considers that the cluster probabilities before the glide are composed of two parts: one corresponding to a sub-cluster on the ®xed side of the APB (where the index ``i'' corresponds to lattice point ``a'', ``j'' to lattice point ``b'' and so on) and another corresponding to a sub-cluster on the shifted side (where the lattice points have been ``exchanged'' by the anti-phase vector to a $ b and c $ d). As an example, we consider the ®rst term inside the brackets. The extended cluster is composed of two ®rst-neighbour pairs: ai dl on the ®xed side and aj kd on the shifted side (which would otherwise correspond to a bj kc pair in the bulk crystal). This allows us to derive the remaining APB energy expressions only considering the eect of the respective anti-phase vector on the distribution of sub-lattices relative to the frame of reference of the ®xed side of the crystal. For the case of the 3 a0 =2h1 1 1i 0 1 1 APB energy, for example, the eect of the anti-phase vector 3 a0 =2h1 1 1i can be thought as equivalent to the exchange of lattice points in the form a ! d, b ! c, c ! a and d ! b. The example of the ®rst term of the equation would involve thus the two pairs: ai dl on the ®xed side and akcj on the shifted side. Eq. (1) for this case will be written as: N X abcd nh ad ac bd ac bc bd ad ei;j;k;l qi;l qk;j qbc DU 3 a0 =2h1 1 1i 0 1 1 j;k ql;i qi;k ql;j qj;l qk;i 2 i;j;k;l abc b bcd d acd c a qabd i;j;l qk qi;j;k ql qj;k;l qi qi;k;l qj i o b abd c bcd d acd qai qabc 12qabcd 7 k;l;j qj qk;l;i qk ql;j;i ql qk;j;i i;j;k;l : After the fourth passage of the model dislocation, the sub-lattices are substituted by sub-lattice points of the same kind (e.g., an a point is substituted by another a point, and so on), which are always equivalent under the tetrahedron formalism. Eq. (1) will be given by N X abcd nh ad bc ac bd bd ac ad DU 4 a0 =2h1 1 1i 0 1 1 ei;j;k;l qi;l qj;k qbc q q q q q j;k i;l i;k j;l j;l i;k 2 i;j;k;l c abc d bcd a acd b qabd i;j;l qk qi;j;k ql qj;k;l qi qi;k;l qj i o b acd c abd d abc qai qbcd 12qabcd 8 j;k;l qj qi;k;l qk qi;j;l ql qi;j;k i;j;k;l : bc Looking again at the example of the ®rst term of the equation, we observe that the product qad i;l qj;k is abcd compared with the qi;j;k;l tetrahedron probability, which are essentially the same probability distributions, except that the ®rst neglects all correlations but that for the fadg and fbcg pairs. The 4 a0 =2h1 1 1i 0 1 1 APB will be diuse, both in the disordered and in the ordered B2 and D03 states. Finally, the ®fth pass of the dislocation produces an APB which is very similar to the case of the a0 =2h1 1 1i 0 1 1 APB (Section 2.1). The substitution scheme for this case is: a ! d, b ! c, c ! a and d ! b. Considering again the example of the ®rst term, it will involve the two pairs: ai dl on the ®xed side and bk dj on the shifted side. Comparing with the ®rst term of Eq. (2), we observe that the neither the triangle ai bk dj is involved nor the point ai is shared between the clusters anymore. In other words, the short-range correlations considered in the case of Eq. (2) are no more present in the 5 a0 =2h1 1 1i 0 1 1 APB case. The evaluation of Eq. (1) for this case is given by
DU 5 a0 =2h1 1 1i 0 1 1
N X abcd nh ad bd ac ac ad bd bc ei;j;k;l qi;l qk;j qbc q q q q q j;k l;i i;k l;j j;l k;i 2 i;j;k;l b abc a bcd d acd c qabd i;j;l qk qi;j;k ql qj;k;l qi qi;k;l qj i o b abc c acd d bcd qai qabd 12qabcd l;k;j qj ql;k;i qk ql;i;j ql qk;i;j i;j;k;l :
9
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3. Results As in [7], the application of the formalism developed above will be exempli®ed for the case of the ironrich part of the BCC Fe±Al system. The interaction parameters for this calculation are reproduced in Table 3. Details of the formalism and of the derivation of these parameters can be found elsewhere [12,16]. The cluster eigenenergies entering in Eqs. (2) and (6)±(9) can be obtained from the values listed in Table 3 as [16]: eabcd i;j;k;l
1 1 1 1 1 1 2 2 ~ i;j;k;l ; wi;k wi;l wj;k wj;l wi;j wk;l w 6 4
10
where the parameters not listed in Table 3 are obtained by application of the symmetry relations of the 1 1 lattice (e.g., wi;j wj;i ) or set to zero otherwise. This choice of parameters automatically sets the reference state for the calculation as the mechanical mixture of the components [16]. Fig. 3 presents the calculated BCC Fe±Al phase diagram, using the parameters of Table 3. The horizontal lines show the isotherms where APB energy calculations have been performed (T 1000 and 623 K, respectively). Three single-phase ®elds are observed in this phase diagram: A2, the ordered B2 and D03 superlattices. Dashed lines indicate second-order transformations, which are observed between the B2 and A2 and between the D03 and B2 superlattices (a tricritical point and the associated two-phase ®eld experimentally observed in the A2/B2 second-order line in the Fe±Al system cannot be reproduced by this calculation, as discussed elsewhere [16]). The 1000 K isotherm has been chosen since at this temperature there is optimal agreement between experimental and calculated values for the activity of aluminium for alloys up to xAl 0:5 [12]. The 623 K isotherm has been chosen since it is near the maximum temperature where normal temperature dependency Table 3 Interaction parameters for the CVM calculations in the BCC system Fe±Ala
a
~FeAlFeAl w
~FeAlAlAl w
35:1kB K
0
Units: 1 kB K 8:3145 (J/mol) 8:617 10
2
wFeAl 370kB K
5
1
wFeAl 840kB K
(eV/atom).
Fig. 3. The BCC Fe±Al phase diagram, calculated using the interaction parameters listed in Table 3 and the irregular tetrahedron approximation of the cluster variation method.
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251
Fig. 4. APB energies for the n a0 =2h1 1 1i 0 1 1 APBs for the BCC Fe±Al system at T 1000 K as a function of the aluminium content of the alloy. The vertical dashed line indicates the position of the second-order boundary for the B2/A2 equilibrium at this temperature.
of the critical resolved shear stress is observed (673 K) [9,10]. In this deformation stage, called stage A by Kr al, deformation is dominated by glide of superdislocations on f1 1 0g planes. These superdislocations usually correspond to four-fold con®gurations 4 a0 =2h1 1 1i, complete two-fold superdislocations of 2a0 h1 1 1i type, as well as imperfect 2 a0 =2h1 1 1i segments [9,10]. At stage A, thus, the formalism developed in this work is relevant. Fig. 4 presents the results of the APB energies for the T 1000 K isotherm. The curves are labelled using the number (n) of dislocation passes on the glide plane (i.e., for the n a0 =2h1 1 1i 0 1 1 APB). The symmetry of the B2 superlattice makes the a and b sub-lattices, and also the c and d sub-lattices equivalent (see Table 1), this means that the n 2 and n 4 curves are degenerate. The same is observed for the n 3 and n 5 curves. The otherwise existing degeneracy between that n 1 and n 3 curves is lifted due to the consideration of the short-range correlations in the energy expression for the n 1 case. In the actual plotting, however, that dierence is too small in the B2 ordered state and only two curves can be distinguished in the ®gure. In the disordered state the dierence due to the short-range correlations is somewhat bigger and the curve for the n 1 case can be distinguished from the remaining ones (which are all degenerate, since there is no sub-lattice partition in the disordered state). The n 2; 4 curve corresponds to a diuse APB and its functional form is very similar to the one observed for the a0 h1 0 0i 0 1 1 APB, obtained in a previous work [7]. The APB energy value is very small in the B2 stoichiometric composition xAl 0:5 and increases as the aluminium concentration decreases, reaching a maximum at the second-order boundary. The remaining curves present, on the contrary, a maximum at the B2 stoichiometric composition, which falls continuously as the aluminium content of the alloy decreases. The n 3; 5 and the n 2; 4 curves meet at the second-ordered boundary and present equal values in the disordered side of the phase diagram. Fig. 5 shows the results for the T 623 K isotherm. The D03 superlattice is less symmetric than the B2 superlattice and as a consequence, the n 2 and the n 4 curves are no more degenerate. In fact, inside the D03 phase ®eld, the n 2 APB is no more diuse and reaches a local maximum at the D03 stoichiometric composition xAl 0:25, while the n 4 diuse APB has a local minimum at this composition. Both curves meet at the second-order B2/D03 boundary, and are degenerate inside the B2 composition range. The ®rst-order equilibrium between the D03 superlattice and the A2 phase causes a discontinuous variation of the APB energies, but the trends are the same as observed in the T 1000 K isotherm. In particular, the values for the n 1 APB energy are slightly lower than for the remaining curves.
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1 1 APB for the BCC Fe±Al system at T 623 K as a function of the aluminium content Fig. 5. APB energies for the n a0 =2h1 1 1i 0 of the alloy. The vertical dashed line indicates the position of the second-order boundary of the B2/D03 equilibrium at this temperature.
4. Discussion Calculation of APB energies using the bond-counting formalism has often been applied to materials of technological interest, mainly on FCC-based superlattices (e.g., L12 phases [4,6]). These works develop the energy expression in function of experimentally observed pair correlations (Warren±Cowley order parameters) and the range of correlations has been extended to a rather high nearest neighbour distance (e.g., in [8] pairs up to the 14th neighbour have been considered). This formalism has the advantage to allow the use of diuse scattering experiments to derive the APB energies in the SRO state [5,6,8]. In the LRO state, SRO is usually neglected and the expressions are developed in terms of the LRO parameters by approximating the pair probabilities as a Bragg±Williams±Gorskyi-like product of point probabilities [5,9]. As a consequence, the diuse APB energies in the LRO state vanish, since no correlation is introduced in this case. The formalism of the present paper works with dierent premises: First of all, not only pair correlations are allowed in the APB energy expression, but also correlations corresponding to the isosceles triangles and to the irregular tetrahedron. This is expected to be more important near critical regions (in the neighbourhood of second-order phase transitions) and for the ®rst moving dislocations. Second, the present formalism is strongly based on statistical mechanics, which is related to the thermodynamic description of the system, and hence can be tested against a large source of experimental data, ranging from phase diagram determinations to calorimetric and chemical activity measurements. As observed in Figs. 4 and 5 the diuse APB energies do not vanish both in the SRO and in the LRO state. The calculation of APB energies using both formalisms (i.e., including long-range experimental pair correlations as in [5,10] and the present formalism), however, are usually in agreement. As an example, calculations were made [5] for the case of a fully ordered Fe0:72 Al0:28 alloy 85 5 mJ/m2 for an n 1 APB and 60 3 mJ/m2 for an n 2 APB. This can be compared with the values at T 623 K obtained in the present work, 4 respectively 108 and 58 mJ/m2 , and with the experimental value of 73 7 and 80 7 mJ/m2 , respectively. The eect of short-range order upon the deformation of solid solutions is mostly observed for the ®rst dislocation gliding on a given slip plane, the second dislocation feels a much lower resistance to motion and,
4
Calculated assuming 2a0 0:57923 nm (see [7]).
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253
according to our results, after the third dislocation no resistance is imposed to their motion. This absence of resistance is, of course, a consequence of using the tetrahedron interactions in the calculation. Calculations in [6], including long-range experimental pair correlations in FCC SRO alloys, showed that the diuse APB energy for the third and fourth dislocations oscillate around a constant value, which is eectively observed after the ®fth dislocation. Their results, however, agree with the conclusion that most of the resistance to motion is felt by the ®rst dislocation. As discussed in [8], in relation to FCC Ni±Mo alloys, this is one of the possible explanations for the formation of pile-up con®gurations of dislocations (i.e., planar slip) often observed in SRO alloys, since it is energetically more favourable for existing dislocation to move in glide planes where the correlations have been previously broken by a former dislocation. In the case of LRO alloys, the APB energy controls the separation distance of the components of the superdislocations, which controls the cross-slip ability of the dislocation. The increment of APB density due to the intersection of superdislocations and existing APBs, as suggested in [2], and the formation of locks and APB tubes, as observed in [9], seem to be the relevant factors on LRO hardening of BCC alloys (at least in Fe±Al alloys), leading to the positive dependency of the critical resolved shear stress. The APB energy should aect the moving dislocation in such cases in two ways: ®rst, the distance between the partials controls the surface area of the newly formed APB (both in the case of intersection with previously existing APBs and in the formation of the tube walls) and thus, it controls the energy increase of the crystal, and second, the formation of locks contribute to the immobilisation of the otherwise moving dislocation, contributing to the increase of the strain rate of the alloy.
Acknowledgements The ®nancial support of Cl audio G. Sch on by the S~ao Paulo State Research Foundation (FAPESP, Brazil) under grants No. 97/13761-5 and 99/07570-8 is gratefully acknowledged.
References [1] M.A. Meyers, K.K. Chawla, Mechanical Metallurgy, Principles and Applications, Prentice-Hall, Englewood Clis, NJ, 1984. [2] P.A. Flinn, Solid solution strengthening, in: Strengthening Mechanisms in Solids, American Society for Metals, Metals Park, OH, USA, 1962, pp. 17±50. [3] P.A. Flinn, Theory of deformation in superlattices, Trans. AIME 218 (1998) 145±154. [4] G. Inden, S. Bruns, H. Ackermann, Antiphase boundary energies in ordered FCC alloys, Philos. Mag. A 53 (1986) 87±100. [5] F. Kr al, P. Schwander, B. Sch onfeld, G. Kostorz, Calculation of antiphase boundary energies in intermetallics, Mater. Sci. Eng. A 234±236 (1997) 351±353. [6] B. Sch onfeld, P. Schwander, Slip induced con®gurational energy change in binary alloys, in: A. Gonis (Ed.), Stability of Materials, Plenum, New York, USA, 1996, pp. 657±662. [7] C.G. Sch on, R. Kikuchi, Antiphase boundary energy determination in superlattices based on the BCC lattice using the cluster variation method, Z. Metallk. 89 (1998) 868±878. [8] P. Schwander, Nahordnung und Versetzungen in einer Ni±Mo-Legierung, Ph.D. thesis, ETH-Z urich, Switzerland, 1990. [9] F. Kr al, Microstructure and mechanical properties of Fe3 Al with chromium, Ph.D. thesis, ETH-Z urich, Switzerland, 1996. [10] F. Kr al, P. Schwander, G. Kostorz, Superdislocations and antiphase boundary energies in deformed Fe3 Al single crystals with chromium, Acta Mater. 45 (1997) 675±682. [11] I. Ohnuma, C.G. Sch on, R. Kainuma, G. Inden, K. Ishida, Ordering and phase separation in the BCC phase of the Fe±Ti±Al system, Acta Mater. 46 (1998) 2083±2094. [12] C.G. Sch on, G. Inden, Concentration dependence of the excess speci®c heat capacity and of the thermodynamic factor for diusion in FCC and BCC ordering systems, Acta Mater. 46 (1998) 4219±4231. [13] C.G. Sch on, Thermodynamics of multicomponent systems with chemical and magnetic interactions, Ph.D. thesis, Universitat Dortmund, Germany, 1998.
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[14] G. Inden, W. Pitsch, Atomic ordering, in: P. Haasen (Ed.), Phase Transformations in Materials, VCH, Weinheim, 1991, pp. 499± 552. [15] J.G. Kirkwood, Statistical mechanics of ¯uid mixtures, J. Chem. Phys 3 (1935) 300±313. [16] C.G. Sch on, G. Inden, CVM calculation of BCC phase diagrams using higher order cluster interactions, J. Chim. Phys. 94 (1997) 1143±1151.
www.elsevier.com/locate/tafmec
Cluster variation method for determining the energy of slip-induced anti-phase boundary in BCC alloys C.G. Sch on a,*, R. Kikuchi b a
Department of Metallurgical and Materials Engineering, Computational Materials Science Laboratory, Escola Polit ecnica da Universidade de S~ ao Paulo, CEP 05508-900, S~ ao Paulo, SP, Brazil b University of California at Berkeley, Berkeley, CA, USA
Abstract The cluster variation method formalism in the irregular tetrahedron approximation has been applied to the determination of the anti-phase boundary (APB) energies of anti-phases generated by the slip of multiple dislocations on a single plane in BCC alloys. The formalism has been applied to BCC Fe±Al alloys and compared with experimental results in the same system. The results shows that in the short-range ordered (SRO) state, most of the resistance to dislocation motion is felt by the ®rst moving dislocation, which characterises slip-plane softening. For the long-range ordered (LRO) state the calculation justi®es the formation of superdislocation con®guration as experimentally observed in the phases at room temperature deformation. The results have been discussed in connection with the experimentally observed deformation modes of BCC Fe±Al alloys. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Hardening mechanisms in metallic alloys are classi®ed depending on the crystal defect interacting with dislocations into work-hardening (interaction with other dislocations), solid±solution hardening (interaction with solute atoms and other point defects), Hall±Petch mechanism (interaction with grain boundaries) and particle hardening (also divided into dispersion and precipitation hardening, depending on the type of particle) [1]. Among these mechanisms, solid±solution hardening presents a simple and economic way to produce materials with improved mechanical properties, and hence the understanding of the basic principles behind this eect is essential. The typical interactions of solutes (and point defects) with dislocations in dilute alloys involve isolated atoms (elastic, modulus and electrostatic interactions) or ``atmospheres'' of atoms, condensed on dislocations or stacking-faults [1,2]. In concentrated alloys, however, another type of interaction appears, involving the interactions of the dislocation with the distribution or the local order, of alloying atoms. This corresponds to short-range order (SRO) 1 and long-range order (LRO) hardening.
*
Corresponding author. Tel.:+55-11-3818-5241; fax: +55-11-3818-5421. E-mail address: [email protected] (C.G. Schon). 1 Here understood in a broad sense, to include also systems with tendency to phase separation, i.e. with positive deviation of the ideal thermodynamic behaviour. 0167-8442/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 0 1 ) 0 0 0 4 8 - 9
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The procedure applied to modelling SRO and LRO hardening is based on the evaluation of the increase of internal energy of the system associated with the deformation process by counting the broken atomic bonds and subtracting from the atomic bonds which are restored by the glide of a model dislocation on the slip plane [3±7]. This procedure is referred to as ``bond-counting formalism''. In the case of LRO alloys, the main contribution to this increase of energy comes from the creation of a conservative anti-phase boundary (APB). The procedure is equivalent to the determination of the energy of this APB with anti-phase vector given by the Burgers vector of the model dislocation. In SRO alloys no APB can be de®ned in the strict sense, but the eect of the glide of a dislocation can be rationalised through the concept of ``diuse'' APBs [6,8], which can be de®ned as a plane of the crystal where the local correlation of atoms is less than that in the equilibrium bulk con®gurations. Right after the glide, we assume that short-range diusion is negligible at the temperature or at the time scale of the deformation process, so that local composition and order parameters of the crystal above and below the plane of the APB remain the same as in the bulk. The proper determination of APB energies is, therefore, the fundamental feature for the theories of SRO and LRO hardening. The calculation of APB energies is done by the use of various approximate solutions of the Ising-type model for the alloy statistical mechanics. Analytical or semi-analytical approximations can be obtained neglecting atom correlations up to a given limit. For example, in the Bragg±Willians±Gorskiy (BWG) approximation, no correlation is considered at all [3]. As a consequence, the diuse APB energies vanish in such calculations. Pair correlations have been introduced into the model by the use of Monte Carlo calculations [6,8] and in this case the diuse APB energies do not vanish. In a recent paper the present authors have extended the bond-counting formalism to the irregular tetrahedron cluster approximation of the cluster variation method (CVM) and applied it to the determination of a0 h1 0 0i 0 0 1, a0 h1 0 0i 0 1 1 and a0 =2h1 1 1i 0 1 1 APB energies [7]. 2 The ®rst one was derived for didactic purposes, but the two remaining APBs are experimentally observed as result of active slip systems in B2±FeAl and D03 ±Fe3 Al [9,10]. The purpose of the present paper is to address another important deformation con®guration experimentally observed in ordered BCC Fe±Al alloys, which was not considered in [7], namely the case of the glide of multiple a0 =2h1 1 1i dislocations in the same 0 1 1 slip plane.
2. Methods 2.1. Basic de®nitions and nomenclature The basic assumption of this work is that the energy of a BCC crystal can be written as a linear combination of energies contained in the basic BCC tetrahedron cluster. This treatment has often been used in phase diagram calculations and provided good results both for the numerical estimates of phase diagrams and for the description of other thermodynamic properties [11±13]. Fig. 1 presents the irregular tetrahedron cluster in the BCC lattice and the associated partition of this lattice into four sub-lattices. The distinction between lattice point and the species occupying this lattice point is essential in this work, and will be denoted by the use, respectively, of greek and roman indices. Using this sub-lattice partition, four dierent crystal structures can be built in a binary alloy (including the BCC basic lattice), corresponding to four superlattices and two BCC disordered solid solutions. The crystallographic conditions imposed to these structures are listed in Table 1, which also reproduces the traditional Strukturbericht notation for denomination of these superlattices.
2
The Miller indices and lattice parameter a0 refer to the disordered BCC alloy.
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245
Fig. 1. The irregular tetrahedron cluster in the BCC lattice. Table 1 Superlattices generated by the four sub-lattice partition of the BCC lattice, shown in Fig. 1, for the case of a binary A±B alloy [14] Stoichiometry
Strukturbericht notation
A A3 B AB AB AB3 B
A2 D03 B2 B32 D03 A2
Space group Im3m Fm3m Pm3m Fd 3m Fm3m Im3m
Condition a; b; c and d are equivalent a and b are equivalent a and b are equivalent, c and d are equivalent a and c are equivalent, b and d are equivalent a and b are equivalent a; b; c and d are equivalent
The APB energy calculation divides an in®nite crystal into two halves and moves one of them relative to the other by an anti-phase vector ~ b, the plane separating the half-crystals being the APB as well as the glide plane of the model dislocation with Burgers vector ~ b. The eect of introducing an APB in the system is to raise the internal energy to a value written as Eaft , after the glide. From the CVM equilibrium calculation we can obtain the internal energy of the crystal before the glide X abcd abcd ei;j;k;l qi;j;k;l ; Ebef Ueq 6N i;j;k;l
where the factor 6 is the number of tetrahedra per lattice point in a BCC crystal, N is the number of lattice points in a crystal plane parallel to the APB and qabcd i;j;k;l is the probability of a tetrahedron con®guration fi; j; k; lg (i.e., species i sitting on lattice point a, j on lattice point b and so on). The energy eabcd i;j;k;l per tetrahedron consists of three kinds of energies: those due (i) to four ®rst neighbours (each shared by three tetrahedra), (ii) to two second neighbours (each shared by four tetrahedra) and (iii) to four-body interactions within the tetrahedron [11]. The contribution of the APB to the increase of energy for the crystal can thus be obtained as: DU Eaft
Ebef :
1
The next task is to determine the expression for Eaft as a function of the equilibrium cluster probabilities. For this purpose, it is more convenient to introduce a dierent notation, explained in detail in a former publication [7]. The probability distribution before the glide of the dislocation qbef hr ws /t us is given by the equilibrium state calculated by the CVM. On the other hand the probability qaft hr ws /t us after the glide is the equilibrium probability distribution of an extended four-point cluster which changes to the (irregular) tetrahedron fabcdg after the shift. This variable is to be obtained using the concept of Kirkwood's superposition approximation [15].
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The examples calculated in [7] showed in detail how to apply the superposition approximation in the three cases described above. The steps shall not be repeated here, but since the results for the glide of a single dislocation, a0 =2h1 1 1i 0 1 1 APB, will be needed for comparison with the results to be obtained in this work, the evaluation of Eq. (1) for this case reported in [7] is reproduced below:
DU a0 =2h1 1 1i 0 1 1 (" abd # bc ac bd bd ac bc ad qacd qi;k;j qad qabc qacd qbcd qabd qabc qbcd N X abcd i;k;j ql;j l;j;i qj;k l;k;j qi;k k;i;l qj;l i;k;l qk;j l;j;k ql;i j;i;l qk;i i;l e qai qck qal qci 4 i;j;k;l i;j;k;l qdl qdj qbj qbk " ! avc abd bcd acd abc abc bcd acd X qabcd qabcd qabcd qabcd qabcd qabcd qabcd qabcd i;j;n;l qi;k;n i;j;k;n ql;j;n i;n;k;l qn;k;j n;j;k;l qn;i;l l;k;n;j qi;k;n l;k;i;n ql;j;n l;n;i;j qn;k;j n;k;i;j qn;i;l qac qac qad qad qbd qbc qbc qbd i;n n;i n;l n;l j;n n;j n n;k k;n
abcd qabcd i;j;n;l qi;k;n;l
qacd i;n;l abcd qabcd n;k;i;j qn;k;i;l
qabc n;k;i
!#
abcd qabcd i;j;k;n ql;j;k;n
qbcd j;k;n 24qabcd i;j;k;l
) :
abcd qabcd i;n;k;l qi;n;k;j
qabc i;n;k
abcd qabcd n;j;k;l qn;j;i;l
qabd n;j;l
abcd qabcd l;k;n;j qi;k;n;j
qbcd k;n;j
abcd qabcd l;k;i;n ql;j;i;n
qacd l;i;n
abcd qabcd l;n;i;j ql;n;k;j
qabd l;n;j 2
Consider a certain group of four atoms which form a tetrahedron after the glide. They are connected by the ®rst- and second-neighbour bonds. Before the glide, mutual distances among these four atoms were farther apart, forming an extended cluster. However, even considering that this extended cluster is not a tetrahedron, it still contains sub-clusters corresponding to ®rst-neighbour pairs, second-neighbour pairs or even isosceles triangles, which are also sub-clusters of the basic tetrahedron and hence, known from the equilibrium calculation. In evaluating the energy of the system before the glide, we need the probability of ®nding the four atoms in the spread out con®guration, which must be compatible with the known subcluster probabilities, and this is the foundation of the superposition approximation. As an example, let us consider the ®rst term inside the brackets in Eq. (2). The corresponding extended cluster is composed by an isosceles triangle 3 ai bk dj and a ®rst-neighbour pair ai dl , which share a common ai point, leading to the referred probability expression. The remaining terms are obtained in a similar way, as discussed in [7]. 2.2. Glide of multiple dislocations Eq. (2) is for a single glide of the dislocation. The case for glide of two dislocations is shown in Fig. 2, presenting the six con®gurations involved in the calculation of the 2 a0 =2h1 11i 0 1 1 APB energy. This ®gure shows a projection along the h0 1 1i axis of two atom layers of the BCC lattice. The APB is located in between and all clusters aected by the APB are contained in these atom layers. Small circles represent atoms on the ®xed half of the crystal and great circles are located on the shifted half of the crystal (the arrow indicates the anti-phase vector, i.e. the Burgers vector of the model dislocation). As simple inspection shows, the length of the arrows in Fig. 1 are all longer than the second-neighbour distance of the BCC lattice, and hence are not included in the basic tetrahedron cluster. In such cases, the application of the superposition approximation leads to the following two consequences: · The probability of the extended four points is a simple product of probabilities of the two separate subclusters (of the tetrahedron), one on the upper and other on the lower plane of the dislocation.
3 This isosceles triangle crosses the APB plane. Edge bd is located entirely on the shifted side of the crystal, as shown by the ``exchanged'' indices. The remaining edges have one point on the shifted side and another on the ®xed side.
C.G. Schon, R. Kikuchi / Theoretical and Applied Fracture Mechanics 35 (2001) 243±254
247
Fig. 2. (a)±(f) The six groups of tetrahedra for the 2 a0 =2h1 1 1i 0 1 1 APB; projection views along the h0 1 1i axis. Small circles: atoms in the ®xed layer. Great circles: atoms in the shifted layer. Double lines: second-neighbour bonds. Single lines: ®rst-neighbour bonds.
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· The probability expression does not change for the passage of three and more dislocations in the disordered phase. In the ordered phase, however, sub-lattices can make the energy expressions dierent for three, four and ®ve passages as will be shown below. For the case of Fig. 2(a), the probability expression obtained by the superposition approximation is: r u r s t u r s t u s t qaft qe qe qe : 3 hw/s hnfs hs nf When the ®xed and shifted planes are reversed, Fig. 2(b) reduces to Fig. 2(a) and the probability expression for Fig. 2(b) becomes the same as Fig. 2(a). Due to symmetry of the lattice, when we specify sub-lattice names into Eq. (3), both con®gurations result in the same probability expression. A similar situation also occurs for the cases shown in Figs. 2(c) and (d) as well as for Figs. 2(e) and (f). The ®nal energy expression can be thus simpli®ed by counting twice the twelve tetrahedra corresponding to Figs. 2(a), (c) and (e). The partition of the BCC lattice into four sub-lattices implies that only the ®ve ®rst passes of the model dislocation, i.e., up to the 5 a0 =2h1 1 1i 0 1 1 APB energy, are to be considered. The symmetry of the stable superlattice (see Table 1) imposes further degeneracy among the APB energy expressions, as will be discussed later. The probability expressions for the con®guration shown in Fig. 2(c) is given by r s t u r s t u r s u t qaft qe qe qe ; 4 hw/s hwfs hws f and for Fig. 2(e) we have r r s t u r s t u s t u qaft qe qe qe : hw/s hnvf nvf h
5
Table 2 shows the correspondence of the indices for the 12 tetrahedra which cross the APB plane for the 2 a0 =2h1 1 1i 0 1 1 APB. This table is employed in the evaluation of Eq. (1) for this case. Using these de®nitions, Eq. (1) can be written as N X abcd nh ad ad bc ac ac bd bd DU 2 a0 =2h1 1 1i 0 1 1 ei;j;k;l qi;l qj;k qbc j;k qi;l qi;k qj;l qj;l qi;k 2 i;j;k;l abc c bcd b acd a d qabd i;j;l qk qi;j;k ql qj;k;l qi qi;k;l qj i o b bcd c abc abcd d abd qai qacd q q q q q q 12q 6 j i;l;k j;l;k k j;i;l i;j;k;l : l j;i;k Table 2 Correspondence of the indices for the tetrahedra which cross the APB plane for the 2 a0 =2h1 1 1i 0 1 1 APB #
h
w
/
s
f
n
v
r
s
t
u
Fig. 2
1 2 3 4 5 6 7 8 9 10 11 12
a b c d a b c d a b c d
b a d c b a d c d c a b
c d b a c d b a c d b a
d c a b d c a b b a d c
d c a b ± ± ± ± c d b a
a b c d d c a b d c a b
± ± ± ± ± ± ± ± a b c d
i j k l i j k l i j k l
j i l k j i l k l k i j
k l j i k l j i k l j i
l k i j l k i j j i l k
(a) (a) (a) (a) (c) (c) (c) (c) (e) (e) (e) (e)
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249
As the reader will note, Eq. (6) can be understood in a simple way, if one considers that the cluster probabilities before the glide are composed of two parts: one corresponding to a sub-cluster on the ®xed side of the APB (where the index ``i'' corresponds to lattice point ``a'', ``j'' to lattice point ``b'' and so on) and another corresponding to a sub-cluster on the shifted side (where the lattice points have been ``exchanged'' by the anti-phase vector to a $ b and c $ d). As an example, we consider the ®rst term inside the brackets. The extended cluster is composed of two ®rst-neighbour pairs: ai dl on the ®xed side and aj kd on the shifted side (which would otherwise correspond to a bj kc pair in the bulk crystal). This allows us to derive the remaining APB energy expressions only considering the eect of the respective anti-phase vector on the distribution of sub-lattices relative to the frame of reference of the ®xed side of the crystal. For the case of the 3 a0 =2h1 1 1i 0 1 1 APB energy, for example, the eect of the anti-phase vector 3 a0 =2h1 1 1i can be thought as equivalent to the exchange of lattice points in the form a ! d, b ! c, c ! a and d ! b. The example of the ®rst term of the equation would involve thus the two pairs: ai dl on the ®xed side and akcj on the shifted side. Eq. (1) for this case will be written as: N X abcd nh ad ac bd ac bc bd ad ei;j;k;l qi;l qk;j qbc DU 3 a0 =2h1 1 1i 0 1 1 j;k ql;i qi;k ql;j qj;l qk;i 2 i;j;k;l abc b bcd d acd c a qabd i;j;l qk qi;j;k ql qj;k;l qi qi;k;l qj i o b abd c bcd d acd qai qabc 12qabcd 7 k;l;j qj qk;l;i qk ql;j;i ql qk;j;i i;j;k;l : After the fourth passage of the model dislocation, the sub-lattices are substituted by sub-lattice points of the same kind (e.g., an a point is substituted by another a point, and so on), which are always equivalent under the tetrahedron formalism. Eq. (1) will be given by N X abcd nh ad bc ac bd bd ac ad DU 4 a0 =2h1 1 1i 0 1 1 ei;j;k;l qi;l qj;k qbc q q q q q j;k i;l i;k j;l j;l i;k 2 i;j;k;l c abc d bcd a acd b qabd i;j;l qk qi;j;k ql qj;k;l qi qi;k;l qj i o b acd c abd d abc qai qbcd 12qabcd 8 j;k;l qj qi;k;l qk qi;j;l ql qi;j;k i;j;k;l : bc Looking again at the example of the ®rst term of the equation, we observe that the product qad i;l qj;k is abcd compared with the qi;j;k;l tetrahedron probability, which are essentially the same probability distributions, except that the ®rst neglects all correlations but that for the fadg and fbcg pairs. The 4 a0 =2h1 1 1i 0 1 1 APB will be diuse, both in the disordered and in the ordered B2 and D03 states. Finally, the ®fth pass of the dislocation produces an APB which is very similar to the case of the a0 =2h1 1 1i 0 1 1 APB (Section 2.1). The substitution scheme for this case is: a ! d, b ! c, c ! a and d ! b. Considering again the example of the ®rst term, it will involve the two pairs: ai dl on the ®xed side and bk dj on the shifted side. Comparing with the ®rst term of Eq. (2), we observe that the neither the triangle ai bk dj is involved nor the point ai is shared between the clusters anymore. In other words, the short-range correlations considered in the case of Eq. (2) are no more present in the 5 a0 =2h1 1 1i 0 1 1 APB case. The evaluation of Eq. (1) for this case is given by
DU 5 a0 =2h1 1 1i 0 1 1
N X abcd nh ad bd ac ac ad bd bc ei;j;k;l qi;l qk;j qbc q q q q q j;k l;i i;k l;j j;l k;i 2 i;j;k;l b abc a bcd d acd c qabd i;j;l qk qi;j;k ql qj;k;l qi qi;k;l qj i o b abc c acd d bcd qai qabd 12qabcd l;k;j qj ql;k;i qk ql;i;j ql qk;i;j i;j;k;l :
9
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3. Results As in [7], the application of the formalism developed above will be exempli®ed for the case of the ironrich part of the BCC Fe±Al system. The interaction parameters for this calculation are reproduced in Table 3. Details of the formalism and of the derivation of these parameters can be found elsewhere [12,16]. The cluster eigenenergies entering in Eqs. (2) and (6)±(9) can be obtained from the values listed in Table 3 as [16]: eabcd i;j;k;l
1 1 1 1 1 1 2 2 ~ i;j;k;l ; wi;k wi;l wj;k wj;l wi;j wk;l w 6 4
10
where the parameters not listed in Table 3 are obtained by application of the symmetry relations of the 1 1 lattice (e.g., wi;j wj;i ) or set to zero otherwise. This choice of parameters automatically sets the reference state for the calculation as the mechanical mixture of the components [16]. Fig. 3 presents the calculated BCC Fe±Al phase diagram, using the parameters of Table 3. The horizontal lines show the isotherms where APB energy calculations have been performed (T 1000 and 623 K, respectively). Three single-phase ®elds are observed in this phase diagram: A2, the ordered B2 and D03 superlattices. Dashed lines indicate second-order transformations, which are observed between the B2 and A2 and between the D03 and B2 superlattices (a tricritical point and the associated two-phase ®eld experimentally observed in the A2/B2 second-order line in the Fe±Al system cannot be reproduced by this calculation, as discussed elsewhere [16]). The 1000 K isotherm has been chosen since at this temperature there is optimal agreement between experimental and calculated values for the activity of aluminium for alloys up to xAl 0:5 [12]. The 623 K isotherm has been chosen since it is near the maximum temperature where normal temperature dependency Table 3 Interaction parameters for the CVM calculations in the BCC system Fe±Ala
a
~FeAlFeAl w
~FeAlAlAl w
35:1kB K
0
Units: 1 kB K 8:3145 (J/mol) 8:617 10
2
wFeAl 370kB K
5
1
wFeAl 840kB K
(eV/atom).
Fig. 3. The BCC Fe±Al phase diagram, calculated using the interaction parameters listed in Table 3 and the irregular tetrahedron approximation of the cluster variation method.
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251
Fig. 4. APB energies for the n a0 =2h1 1 1i 0 1 1 APBs for the BCC Fe±Al system at T 1000 K as a function of the aluminium content of the alloy. The vertical dashed line indicates the position of the second-order boundary for the B2/A2 equilibrium at this temperature.
of the critical resolved shear stress is observed (673 K) [9,10]. In this deformation stage, called stage A by Kr al, deformation is dominated by glide of superdislocations on f1 1 0g planes. These superdislocations usually correspond to four-fold con®gurations 4 a0 =2h1 1 1i, complete two-fold superdislocations of 2a0 h1 1 1i type, as well as imperfect 2 a0 =2h1 1 1i segments [9,10]. At stage A, thus, the formalism developed in this work is relevant. Fig. 4 presents the results of the APB energies for the T 1000 K isotherm. The curves are labelled using the number (n) of dislocation passes on the glide plane (i.e., for the n a0 =2h1 1 1i 0 1 1 APB). The symmetry of the B2 superlattice makes the a and b sub-lattices, and also the c and d sub-lattices equivalent (see Table 1), this means that the n 2 and n 4 curves are degenerate. The same is observed for the n 3 and n 5 curves. The otherwise existing degeneracy between that n 1 and n 3 curves is lifted due to the consideration of the short-range correlations in the energy expression for the n 1 case. In the actual plotting, however, that dierence is too small in the B2 ordered state and only two curves can be distinguished in the ®gure. In the disordered state the dierence due to the short-range correlations is somewhat bigger and the curve for the n 1 case can be distinguished from the remaining ones (which are all degenerate, since there is no sub-lattice partition in the disordered state). The n 2; 4 curve corresponds to a diuse APB and its functional form is very similar to the one observed for the a0 h1 0 0i 0 1 1 APB, obtained in a previous work [7]. The APB energy value is very small in the B2 stoichiometric composition xAl 0:5 and increases as the aluminium concentration decreases, reaching a maximum at the second-order boundary. The remaining curves present, on the contrary, a maximum at the B2 stoichiometric composition, which falls continuously as the aluminium content of the alloy decreases. The n 3; 5 and the n 2; 4 curves meet at the second-ordered boundary and present equal values in the disordered side of the phase diagram. Fig. 5 shows the results for the T 623 K isotherm. The D03 superlattice is less symmetric than the B2 superlattice and as a consequence, the n 2 and the n 4 curves are no more degenerate. In fact, inside the D03 phase ®eld, the n 2 APB is no more diuse and reaches a local maximum at the D03 stoichiometric composition xAl 0:25, while the n 4 diuse APB has a local minimum at this composition. Both curves meet at the second-order B2/D03 boundary, and are degenerate inside the B2 composition range. The ®rst-order equilibrium between the D03 superlattice and the A2 phase causes a discontinuous variation of the APB energies, but the trends are the same as observed in the T 1000 K isotherm. In particular, the values for the n 1 APB energy are slightly lower than for the remaining curves.
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1 1 APB for the BCC Fe±Al system at T 623 K as a function of the aluminium content Fig. 5. APB energies for the n a0 =2h1 1 1i 0 of the alloy. The vertical dashed line indicates the position of the second-order boundary of the B2/D03 equilibrium at this temperature.
4. Discussion Calculation of APB energies using the bond-counting formalism has often been applied to materials of technological interest, mainly on FCC-based superlattices (e.g., L12 phases [4,6]). These works develop the energy expression in function of experimentally observed pair correlations (Warren±Cowley order parameters) and the range of correlations has been extended to a rather high nearest neighbour distance (e.g., in [8] pairs up to the 14th neighbour have been considered). This formalism has the advantage to allow the use of diuse scattering experiments to derive the APB energies in the SRO state [5,6,8]. In the LRO state, SRO is usually neglected and the expressions are developed in terms of the LRO parameters by approximating the pair probabilities as a Bragg±Williams±Gorskyi-like product of point probabilities [5,9]. As a consequence, the diuse APB energies in the LRO state vanish, since no correlation is introduced in this case. The formalism of the present paper works with dierent premises: First of all, not only pair correlations are allowed in the APB energy expression, but also correlations corresponding to the isosceles triangles and to the irregular tetrahedron. This is expected to be more important near critical regions (in the neighbourhood of second-order phase transitions) and for the ®rst moving dislocations. Second, the present formalism is strongly based on statistical mechanics, which is related to the thermodynamic description of the system, and hence can be tested against a large source of experimental data, ranging from phase diagram determinations to calorimetric and chemical activity measurements. As observed in Figs. 4 and 5 the diuse APB energies do not vanish both in the SRO and in the LRO state. The calculation of APB energies using both formalisms (i.e., including long-range experimental pair correlations as in [5,10] and the present formalism), however, are usually in agreement. As an example, calculations were made [5] for the case of a fully ordered Fe0:72 Al0:28 alloy 85 5 mJ/m2 for an n 1 APB and 60 3 mJ/m2 for an n 2 APB. This can be compared with the values at T 623 K obtained in the present work, 4 respectively 108 and 58 mJ/m2 , and with the experimental value of 73 7 and 80 7 mJ/m2 , respectively. The eect of short-range order upon the deformation of solid solutions is mostly observed for the ®rst dislocation gliding on a given slip plane, the second dislocation feels a much lower resistance to motion and,
4
Calculated assuming 2a0 0:57923 nm (see [7]).
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according to our results, after the third dislocation no resistance is imposed to their motion. This absence of resistance is, of course, a consequence of using the tetrahedron interactions in the calculation. Calculations in [6], including long-range experimental pair correlations in FCC SRO alloys, showed that the diuse APB energy for the third and fourth dislocations oscillate around a constant value, which is eectively observed after the ®fth dislocation. Their results, however, agree with the conclusion that most of the resistance to motion is felt by the ®rst dislocation. As discussed in [8], in relation to FCC Ni±Mo alloys, this is one of the possible explanations for the formation of pile-up con®gurations of dislocations (i.e., planar slip) often observed in SRO alloys, since it is energetically more favourable for existing dislocation to move in glide planes where the correlations have been previously broken by a former dislocation. In the case of LRO alloys, the APB energy controls the separation distance of the components of the superdislocations, which controls the cross-slip ability of the dislocation. The increment of APB density due to the intersection of superdislocations and existing APBs, as suggested in [2], and the formation of locks and APB tubes, as observed in [9], seem to be the relevant factors on LRO hardening of BCC alloys (at least in Fe±Al alloys), leading to the positive dependency of the critical resolved shear stress. The APB energy should aect the moving dislocation in such cases in two ways: ®rst, the distance between the partials controls the surface area of the newly formed APB (both in the case of intersection with previously existing APBs and in the formation of the tube walls) and thus, it controls the energy increase of the crystal, and second, the formation of locks contribute to the immobilisation of the otherwise moving dislocation, contributing to the increase of the strain rate of the alloy.
Acknowledgements The ®nancial support of Cl audio G. Sch on by the S~ao Paulo State Research Foundation (FAPESP, Brazil) under grants No. 97/13761-5 and 99/07570-8 is gratefully acknowledged.
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