- Email: [email protected]
The [001] antiphase boundaries at the order-disorder transition of cu3au
Materials Science and Engineering B37 (1996) 177-180
The [OOl] antiphase boundaries at the order-disorder Cu,Au
transition of
H.M. Polatoglod Physics
Department,
Aristotle
University
of Thessaloniki,
GR-54006
Thessaloniki,
Greece
Abstract Atomistic simulations of the two possible [OOl] antiphase boundaries in Cu,Au are performed for temperatures below the order-disorder transition temperature. The method applied is the constant temperature, pressure and chemical potential difference Monte Carlo method. We used an N-body potential based on the tight-binding method in the second-moment approximation. One of the two possible antiphase boundaries is called conservative and the other non-conservative. For temperatures 20% below the transition temperature we observe that the ordering near the non-conservative boundary is less than in the bulk material. On the contrary, the ordering near the conservative boundary cannot be distinguished from the rest of the material. Therefore the simulations indicate that only the non-conservative boundary wets, in accordance with recent experimental results. Keywords:
Monte Carlo method; Order-disorder
transition;
Wetting;
1. Introduction The mechanical properties of materials are greatly influenced by the presence of defects, such as internal boundaries, and as a result a lot of work is devoted to their study [l-4]. Owing to their unique properties load-bearing intermetallics [S] have attracted considerable attention. At boundaries many interesting phenomena may occur, such as segregation [6], wetting phenomena [7-g]. A set of intermetallics, the so-called softly ordered, have an order-disorder transition below the melting point, and recent detailed observations produced evidence that some internal boundaries are wetted by the disordered phase below the order-disorder transition [lo]. The Cu,Au intermetallic is a model system since its structure is similar to those of many technologically important intermetallics, it has a relatively high orderdisorder transition and its constituents have a large difference in their atomic numbers. The last fact allows the observation of the different facets of the ordering effects near internal boundaries using high resolution electron microscopy. In fact it is observed that some antiphase boundaries are wetted by the disordered phase below the bulk order-disorder transition temper‘Fax: + $ (30) olymp.ccf.auth.gr.
31
206138.
e-mail:
polatoglou@
0921-5107/96/$15.00 0 1996 - Elsevier Science %A. All rights reserved
Segregation
ature [l]. From the high resolution images it is deduced that the conservative antiphase boundaries do not wet while the non-conservative wet. The aim of the present study is to investigate the effect of the presence of antiphase boundaries (conservative and non-conservative) in Cu,Au on the orderdisorder transition. We also examine wetting phenomena or the possible occurrence of segregation of one of the constituents to the boundary. The constant temperature, pressure and chemical potential difference Monte Carlo method is applied and a potential produced through the tight-binding method in the secondmoment approximation. The relaxation effects are included in a consistent wa’y, as in the case of the bulk order-disorder transition. Through the method we employ here we have obtained a unified description of the order-disorder transition in the bulk [12,13], the [OOl] surfaces [14] and the [loo] twist boundaries [9,15,16]. While the geometry of surfaces or of twist boundaries is quite different from that of the rest of the material, the geometry at antiphase boundaries is the same as in the bulk material. The only difference lies in the occupation of the different ordered sublattices. As a consequence the antiphase boundary energies are very small, of the order of some tenths of a millijoule per square metre, compared with some hundreds of millijoules per square metre for twist boundaries and of surfaces. Therefore these small energies indicate that the anti-
178
H.M.
Polatoglm
1 Materials
Science
phase boundaries may not be treated by semiempirical models and that the electronic structure of the materials should be included in a more detailed way. Previously the non-conservative boundaries in Cu,Au were studied using the cluster variation method, which is actually a mean field method, and pair potentials up to second neighbours [17,18].
and Engineering
B37 (1996)
177-180
CusAu la0
D
m q
antiphase
LOOI] El’ei
B
g
0’0
q
0
boundary u;UA’J-OAUAO
A
A A A
0.8 2
--------------4--,-----_---__ A
4 A .dioe6
E¶ B
n
0
0
b
A 0
A A
0
0
0
0
OAOAo~OAOAo~
2. Results and discussion First we will explain the production of the conservative and the nonconservative antiphase boundaries. In Fig. 1 we show the unit cell of Cu,Au. The shaded spheres correspond to Au atoms and the open spheres to Cu atoms. The arrows show the possible exchanges of a gold atom with a copper atom. The exchange labelled with the arrows (a) produces a conservative antiphase boundary, while arrows (b) indicate a nonconservative boundary. In traversing the cyrstal in a direction perpendicular to the boundary plane a change in the occupation of a sublattice occurs, i.e. from the one side one sublattice is occupied by Au atoms while on the other side it is occupied by Cu atoms. Ordering in Cu,Au can be seen as a modulation of stoichiometry along some given direction. If we define the plane-averaged stoichiometry for planes parallel to the boundary plane, we can see that there is an alternation of planes containing Cu and Au in equal proportion (Cu-Au planes) and pure Cu planes. In the ordered Cu,Au the stacking of the planes along any of the cubic axis is * . *AaAajAaAa . . ., where A denotes a Cu-Au plane, a a Cu plane and j the boundary plane. The conservative antiphase boundary does not alter the above stacking, while the non-conservative has the following stacking: . . . aAaAlAaAa . . . .
0.2
t q oooo bulk CuSAu aaaAa non-concervative
I a
!
I
I
24
32
Fig. 2. Plane-averaged stoichiometry profile of the bulk Cu,Au and the non-conservative antiphase boundary for T/T, = 0.8.
Now we will consider the stoichiometry profile for the bulk Cu,Au material and the non-conservative antiphase boundary for T/T, = 0.8 (Fig. 2). We observe that for both cases the values alternate between values close to one-half and unity corresponding to the successive Cu-Au and Cu planes. For the non-conservative antiphase boundary for three layers on each side of the boundary departures from this pattern occur and the stoichiometry profile tends at this region to the average stoichiometry of the bulk material. This tendency shows that some Au atoms occupy the previously pure Cu planes and more Cu atoms occupy the previously mixed Cu-Au planes, indicating disorder. Within the accuracy of the calculations, with a statistical error of the order of 5%, no segregation can be noticed, CuoAu
1.0
00000 AAAAA
0.8
N -
0.6
antiphase
[OOl] bulk
boundary
CuaAu
non-conservativeT~~~~~~,~
B
1 AAAA
~“~8~0000000A00
I
-
A A
00
A
I
1 A A A
layer Fig. 1. Operations which produce the conservative and non-conservative [OOl] antiphase boundaries.
Fig. 3. Order parameter of each plane parallel to the interface of the non-conservative antiphase boundary for T/T, = 0.8.
H.&f. Polntoglou / Materials Science and Engineesing B37 (1996) 177-180
1.0 r
Cu3Au [OOl] antiphase
Cu3Au [OOl] antiphase
boundary
AAAdA non-conservative **A*A non-conservative
179
boundary
T/T,=0:6 T/T,=O.S AAAA
AAAAA
A
A A A
N- 0.6 c-
63 a
B
0
0
6
A 0
A 0
0
0
0
OAOAOAOAOAO~
% -0.4 m 00000 conservative AAAAA non-conservative l a*aa conservative AAAAA non-conservative
0.2
0.0
l&
0.0 oL
0
/
T/T,=02 T/T,=0.8 T/T,=0.9 T/T,=0.9
I
a
I
I
24
32
layer Fig. 4. Order parameter of the non-conservative antiphase boundary for T/T, = 0.8 and 0.9.
From
the order parameter,
Fig. 5. Plane-averaged stoichiometry profile of the [OOI] non-conservative and conservative antiphase boundaries in Cu,Au at T/T, = 0.8 and 0.9.
shown in Fig. 3 for
T/T= = 0.8, it can be seen that the region around the
boundary is disordered. We took the order parameter to be the square of the structure factor for the indicated wavevector [9]. Away from the boundary the order parameter has values close to the bulk values. Only for three layers around the boundary does the order parameter tend to zero, being zero at the boundary. The disordered region around the boundary becomes thicker as the temperature increases, as we can see in Fig. 4. This behaviour indicates the occurrence of wetting of the interface by the disordered phase [9,19]. The occurrence of the wetting phenomena about 100 K below the transition temperature is very impressive. Of course the simulations indicate wetting at the atomic level which is very small compared with the experimental observation. Experimentally the wetting is observed to occur some degrees below the transition temperature [ll], a region that is not easily accessible to the simulations because of the necessarily limited size of the simulation cell. We will now compare the behaviours of both antiphase boundaries by examining the stoichiometry profile, as shown in Fig. 5, for T/T= = 0.8 and 0.9. The conservative antiphase boundary behaves in a very bulk-like way, i.e. having a constant value throughout the simulation cell, and moving to the average value as the temperature approaches the transition temperature. Similar observations can be made for the order parameter (Fig. 6). The plane-averaged strain for both antiphase boundaries is presented in Fig. 7 for T/T, = 0.8 and 0.9. Positive or negative averaged strains indicate that the average positions of the atoms belonging to the given plane have moved to the right or to the left respectively.
The strain is given in units of the lattice parameter. No strains occur within the accuracy of the calculations, which is calculated to be about 1%. Of the two antiphase boundaries the conservative exhibits on average a bulk-like behaviour in every detail. For the non-conservative antiphase boundary this is true only away from the boundary. Near the boundary a disordered region can be observed, which increases as the temperature increases. The last fact makes it evident that wetting may occur only for the non-conservative boundary. However, no noticeable segregation or strains, within the accuracy of the calculations, could be found. Cu3Au [OOl] antiphase
boundary
.OI 0.8 t; 2
6
Q
@
Q
2
o
0
o
o
’
’
‘A0 A
A
AAAA o 00
A
Fig. 6. Same as Fig. 5, but for the order parameter.
HA
180 CuaAu
[OOl]
Polatoglou / Materials Science and Engineerirjg B37 (1996) 177-180
antiphase
boundary
References
0’05 I
-0.03 I -0.05;
00000 conservative AAAAA non-conservative **oe* conservative *AAAA non-conservative I
,
a
I
T/T,=O.EI T/T,=O.El T/T,=0.9 T/T,=0.9
-
1
I
24
32
Fig. 7. Same as Fig. 5, but for the plane-averaged strains.
Acknowledgements This work has been partially supported by the NATO CRG 940267 and by a Greek Secretary of Research and Technology Grant.
[l] M.C. Payne, P.D. Bristowe and J.D. Joannopoulos, Phys. Rev. Lett., 58 (1987) 1348. [2] D.E. Luzzi, M. Yan, M. Sob and V. Vitek, Pl~>w. Rw Lctl., 67 (1991) 1894. [3] F. Ernst, M.W. Finnis, D. Hofmann, T. Maschik, U. Schonberger, U. Wolf and M. Methfessel, PII~s. Ret!. Le& 69 (1992) 620. [4] J.L. Putaux, J. Thibault, A. Jacques and A. George, Mater. Sci. Forum, 126-128 (1993) 13. [5] R.W. Cahn, MRS Bull., (May 1991) 18. [6] H.Y. Wang, R. Najafabadi, D.J. Srolovitz and R. LeSar, Interface Sci., Z(l993) 31. [7] G. Ciccotti, M. Guillope and V. Pontikis, P/~]~x Rec. B, 27 (1983) 5576. [S] S.R. Phillpot, S. Yip and D. Wolf, Coq~rf. P/I~x, 20 (November-December 1989). [9] H.M. Polatoglou, Comput. hhter. Sci., 3 (1994) 109. [lo] J.G. Antonopoulos, F.W. Schapink and F.D. Tichelaar, Philos. Mug. Lett., 61 (1990) 195. [ll] L. Potez and A. Loiseau, Interface Sci,, 2 (1994) 91. [12] H.M. Polatoglou and G.L. Bleris, Solid State Comm~m., 90 (1994) 425. 1131 H.M. Polatoglou and G.L. Bleris, Itrterfocc Sci., 2 (1994) 31. [14] H.M. Polatoglou, Proc. 6th Joint EPS-APS ht. Co& 0~1 Physics Computing, 1994, p, 605. [15] H.M. Polatoglou, J. Phyx Corrdcns. Matter, 6 (1994) 5621. [16] H.M. Polatoglou, Interfirce Sci., 2 (1994) 67. [17] A. Finel, V. Mazauric and F. Ducastelle, P/r))& Aeo. Lcrt., 65 (1990) 1016. [18] F. Ducastelle, C. Ricolleau and A, Loiseau, Proc. hr. Workshop on Reactive-DiJjirsive Tmrxformatioas, hsois, May 1993. [19] R. Lipowsky and W. Speth, P/IJX Ret?. B, 28 (1983) 3983.
The [OOl] antiphase boundaries at the order-disorder Cu,Au
transition of
H.M. Polatoglod Physics
Department,
Aristotle
University
of Thessaloniki,
GR-54006
Thessaloniki,
Greece
Abstract Atomistic simulations of the two possible [OOl] antiphase boundaries in Cu,Au are performed for temperatures below the order-disorder transition temperature. The method applied is the constant temperature, pressure and chemical potential difference Monte Carlo method. We used an N-body potential based on the tight-binding method in the second-moment approximation. One of the two possible antiphase boundaries is called conservative and the other non-conservative. For temperatures 20% below the transition temperature we observe that the ordering near the non-conservative boundary is less than in the bulk material. On the contrary, the ordering near the conservative boundary cannot be distinguished from the rest of the material. Therefore the simulations indicate that only the non-conservative boundary wets, in accordance with recent experimental results. Keywords:
Monte Carlo method; Order-disorder
transition;
Wetting;
1. Introduction The mechanical properties of materials are greatly influenced by the presence of defects, such as internal boundaries, and as a result a lot of work is devoted to their study [l-4]. Owing to their unique properties load-bearing intermetallics [S] have attracted considerable attention. At boundaries many interesting phenomena may occur, such as segregation [6], wetting phenomena [7-g]. A set of intermetallics, the so-called softly ordered, have an order-disorder transition below the melting point, and recent detailed observations produced evidence that some internal boundaries are wetted by the disordered phase below the order-disorder transition [lo]. The Cu,Au intermetallic is a model system since its structure is similar to those of many technologically important intermetallics, it has a relatively high orderdisorder transition and its constituents have a large difference in their atomic numbers. The last fact allows the observation of the different facets of the ordering effects near internal boundaries using high resolution electron microscopy. In fact it is observed that some antiphase boundaries are wetted by the disordered phase below the bulk order-disorder transition temper‘Fax: + $ (30) olymp.ccf.auth.gr.
31
206138.
e-mail:
polatoglou@
0921-5107/96/$15.00 0 1996 - Elsevier Science %A. All rights reserved
Segregation
ature [l]. From the high resolution images it is deduced that the conservative antiphase boundaries do not wet while the non-conservative wet. The aim of the present study is to investigate the effect of the presence of antiphase boundaries (conservative and non-conservative) in Cu,Au on the orderdisorder transition. We also examine wetting phenomena or the possible occurrence of segregation of one of the constituents to the boundary. The constant temperature, pressure and chemical potential difference Monte Carlo method is applied and a potential produced through the tight-binding method in the secondmoment approximation. The relaxation effects are included in a consistent wa’y, as in the case of the bulk order-disorder transition. Through the method we employ here we have obtained a unified description of the order-disorder transition in the bulk [12,13], the [OOl] surfaces [14] and the [loo] twist boundaries [9,15,16]. While the geometry of surfaces or of twist boundaries is quite different from that of the rest of the material, the geometry at antiphase boundaries is the same as in the bulk material. The only difference lies in the occupation of the different ordered sublattices. As a consequence the antiphase boundary energies are very small, of the order of some tenths of a millijoule per square metre, compared with some hundreds of millijoules per square metre for twist boundaries and of surfaces. Therefore these small energies indicate that the anti-
178
H.M.
Polatoglm
1 Materials
Science
phase boundaries may not be treated by semiempirical models and that the electronic structure of the materials should be included in a more detailed way. Previously the non-conservative boundaries in Cu,Au were studied using the cluster variation method, which is actually a mean field method, and pair potentials up to second neighbours [17,18].
and Engineering
B37 (1996)
177-180
CusAu la0
D
m q
antiphase
LOOI] El’ei
B
g
0’0
q
0
boundary u;UA’J-OAUAO
A
A A A
0.8 2
--------------4--,-----_---__ A
4 A .dioe6
E¶ B
n
0
0
b
A 0
A A
0
0
0
0
OAOAo~OAOAo~
2. Results and discussion First we will explain the production of the conservative and the nonconservative antiphase boundaries. In Fig. 1 we show the unit cell of Cu,Au. The shaded spheres correspond to Au atoms and the open spheres to Cu atoms. The arrows show the possible exchanges of a gold atom with a copper atom. The exchange labelled with the arrows (a) produces a conservative antiphase boundary, while arrows (b) indicate a nonconservative boundary. In traversing the cyrstal in a direction perpendicular to the boundary plane a change in the occupation of a sublattice occurs, i.e. from the one side one sublattice is occupied by Au atoms while on the other side it is occupied by Cu atoms. Ordering in Cu,Au can be seen as a modulation of stoichiometry along some given direction. If we define the plane-averaged stoichiometry for planes parallel to the boundary plane, we can see that there is an alternation of planes containing Cu and Au in equal proportion (Cu-Au planes) and pure Cu planes. In the ordered Cu,Au the stacking of the planes along any of the cubic axis is * . *AaAajAaAa . . ., where A denotes a Cu-Au plane, a a Cu plane and j the boundary plane. The conservative antiphase boundary does not alter the above stacking, while the non-conservative has the following stacking: . . . aAaAlAaAa . . . .
0.2
t q oooo bulk CuSAu aaaAa non-concervative
I a
!
I
I
24
32
Fig. 2. Plane-averaged stoichiometry profile of the bulk Cu,Au and the non-conservative antiphase boundary for T/T, = 0.8.
Now we will consider the stoichiometry profile for the bulk Cu,Au material and the non-conservative antiphase boundary for T/T, = 0.8 (Fig. 2). We observe that for both cases the values alternate between values close to one-half and unity corresponding to the successive Cu-Au and Cu planes. For the non-conservative antiphase boundary for three layers on each side of the boundary departures from this pattern occur and the stoichiometry profile tends at this region to the average stoichiometry of the bulk material. This tendency shows that some Au atoms occupy the previously pure Cu planes and more Cu atoms occupy the previously mixed Cu-Au planes, indicating disorder. Within the accuracy of the calculations, with a statistical error of the order of 5%, no segregation can be noticed, CuoAu
1.0
00000 AAAAA
0.8
N -
0.6
antiphase
[OOl] bulk
boundary
CuaAu
non-conservativeT~~~~~~,~
B
1 AAAA
~“~8~0000000A00
I
-
A A
00
A
I
1 A A A
layer Fig. 1. Operations which produce the conservative and non-conservative [OOl] antiphase boundaries.
Fig. 3. Order parameter of each plane parallel to the interface of the non-conservative antiphase boundary for T/T, = 0.8.
H.&f. Polntoglou / Materials Science and Engineesing B37 (1996) 177-180
1.0 r
Cu3Au [OOl] antiphase
Cu3Au [OOl] antiphase
boundary
AAAdA non-conservative **A*A non-conservative
179
boundary
T/T,=0:6 T/T,=O.S AAAA
AAAAA
A
A A A
N- 0.6 c-
63 a
B
0
0
6
A 0
A 0
0
0
0
OAOAOAOAOAO~
% -0.4 m 00000 conservative AAAAA non-conservative l a*aa conservative AAAAA non-conservative
0.2
0.0
l&
0.0 oL
0
/
T/T,=02 T/T,=0.8 T/T,=0.9 T/T,=0.9
I
a
I
I
24
32
layer Fig. 4. Order parameter of the non-conservative antiphase boundary for T/T, = 0.8 and 0.9.
From
the order parameter,
Fig. 5. Plane-averaged stoichiometry profile of the [OOI] non-conservative and conservative antiphase boundaries in Cu,Au at T/T, = 0.8 and 0.9.
shown in Fig. 3 for
T/T= = 0.8, it can be seen that the region around the
boundary is disordered. We took the order parameter to be the square of the structure factor for the indicated wavevector [9]. Away from the boundary the order parameter has values close to the bulk values. Only for three layers around the boundary does the order parameter tend to zero, being zero at the boundary. The disordered region around the boundary becomes thicker as the temperature increases, as we can see in Fig. 4. This behaviour indicates the occurrence of wetting of the interface by the disordered phase [9,19]. The occurrence of the wetting phenomena about 100 K below the transition temperature is very impressive. Of course the simulations indicate wetting at the atomic level which is very small compared with the experimental observation. Experimentally the wetting is observed to occur some degrees below the transition temperature [ll], a region that is not easily accessible to the simulations because of the necessarily limited size of the simulation cell. We will now compare the behaviours of both antiphase boundaries by examining the stoichiometry profile, as shown in Fig. 5, for T/T= = 0.8 and 0.9. The conservative antiphase boundary behaves in a very bulk-like way, i.e. having a constant value throughout the simulation cell, and moving to the average value as the temperature approaches the transition temperature. Similar observations can be made for the order parameter (Fig. 6). The plane-averaged strain for both antiphase boundaries is presented in Fig. 7 for T/T, = 0.8 and 0.9. Positive or negative averaged strains indicate that the average positions of the atoms belonging to the given plane have moved to the right or to the left respectively.
The strain is given in units of the lattice parameter. No strains occur within the accuracy of the calculations, which is calculated to be about 1%. Of the two antiphase boundaries the conservative exhibits on average a bulk-like behaviour in every detail. For the non-conservative antiphase boundary this is true only away from the boundary. Near the boundary a disordered region can be observed, which increases as the temperature increases. The last fact makes it evident that wetting may occur only for the non-conservative boundary. However, no noticeable segregation or strains, within the accuracy of the calculations, could be found. Cu3Au [OOl] antiphase
boundary
.OI 0.8 t; 2
6
Q
@
Q
2
o
0
o
o
’
’
‘A0 A
A
AAAA o 00
A
Fig. 6. Same as Fig. 5, but for the order parameter.
HA
180 CuaAu
[OOl]
Polatoglou / Materials Science and Engineerirjg B37 (1996) 177-180
antiphase
boundary
References
0’05 I
-0.03 I -0.05;
00000 conservative AAAAA non-conservative **oe* conservative *AAAA non-conservative I
,
a
I
T/T,=O.EI T/T,=O.El T/T,=0.9 T/T,=0.9
-
1
I
24
32
Fig. 7. Same as Fig. 5, but for the plane-averaged strains.
Acknowledgements This work has been partially supported by the NATO CRG 940267 and by a Greek Secretary of Research and Technology Grant.
[l] M.C. Payne, P.D. Bristowe and J.D. Joannopoulos, Phys. Rev. Lett., 58 (1987) 1348. [2] D.E. Luzzi, M. Yan, M. Sob and V. Vitek, Pl~>w. Rw Lctl., 67 (1991) 1894. [3] F. Ernst, M.W. Finnis, D. Hofmann, T. Maschik, U. Schonberger, U. Wolf and M. Methfessel, PII~s. Ret!. Le& 69 (1992) 620. [4] J.L. Putaux, J. Thibault, A. Jacques and A. George, Mater. Sci. Forum, 126-128 (1993) 13. [5] R.W. Cahn, MRS Bull., (May 1991) 18. [6] H.Y. Wang, R. Najafabadi, D.J. Srolovitz and R. LeSar, Interface Sci., Z(l993) 31. [7] G. Ciccotti, M. Guillope and V. Pontikis, P/~]~x Rec. B, 27 (1983) 5576. [S] S.R. Phillpot, S. Yip and D. Wolf, Coq~rf. P/I~x, 20 (November-December 1989). [9] H.M. Polatoglou, Comput. hhter. Sci., 3 (1994) 109. [lo] J.G. Antonopoulos, F.W. Schapink and F.D. Tichelaar, Philos. Mug. Lett., 61 (1990) 195. [ll] L. Potez and A. Loiseau, Interface Sci,, 2 (1994) 91. [12] H.M. Polatoglou and G.L. Bleris, Solid State Comm~m., 90 (1994) 425. 1131 H.M. Polatoglou and G.L. Bleris, Itrterfocc Sci., 2 (1994) 31. [14] H.M. Polatoglou, Proc. 6th Joint EPS-APS ht. Co& 0~1 Physics Computing, 1994, p, 605. [15] H.M. Polatoglou, J. Phyx Corrdcns. Matter, 6 (1994) 5621. [16] H.M. Polatoglou, Interfirce Sci., 2 (1994) 67. [17] A. Finel, V. Mazauric and F. Ducastelle, P/r))& Aeo. Lcrt., 65 (1990) 1016. [18] F. Ducastelle, C. Ricolleau and A, Loiseau, Proc. hr. Workshop on Reactive-DiJjirsive Tmrxformatioas, hsois, May 1993. [19] R. Lipowsky and W. Speth, P/IJX Ret?. B, 28 (1983) 3983.