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A new potential model for the description of lattice stability in β-phase “alloys”
Scripta
METALLURGICA
Vol. 22, pp. I S 3 1 - 1 5 3 5 , 1988 P r i n t e d in the U.S.A.
Pergamon Press plc All r i g h t s r e s e r v e d
A NEW POTENTIAL MODEl, FOR THE DESCRIPTION OF LATTICE STABILITY IN ~-PHASE "ALLOYS" M. DE GRAEF, B. VERLINDEN & L. DELAEY Department of Metallurgy and Materials Engineering, Catholic University Leuven, de Croylaan 2, B-3030 Heverlee, Belgium ( R e c e i v e d J u n e 14, 1988) ( R e v i s e d J u l y 7, 1988) Introduction Over the past years it has become clear that the martensitic transformation in b.c.c.-based Hume-Rothery alloys can only be adequately described if both second and third order elastic constants (SOEC - TOEC) are taken into account (I)(2)(3). For a realistic computer simulation of the transformation mechanism, these constants should also be used in constructing the interatomic interaction potentials. Unfortunately, there are not many alloys for which both SOEC and TOEC are known (preferably as a function of temperature). Therefore, all Molecular Dynamics (MD) simulations so far have been carried out with an empirical potential fitted only to the SOEC; often this potential function is numerically manipulated in order to introduce certain stability relations between parent and product crystal structures. In (4) a Morse potential with an added oscillatory tail part (to mimic long range Friede] oscillations) has been used to study the effect of this tail part on the relative stability of b.c.c., 3R, pseudo-2H,... No detailed analysis of mechanical lattice stability was reported in this work. The same authors later used another empirical potential (5) to perform a two-dimensional MDsimulation of the nucleation of martensite: the potential was chosen in such. a way as to give a certain energy change in g o i n g from the initial to the final state. In (6) a phenomenological pair potential for pure Cu was used to calculate the minimum energy path for the Bain transformation from b.c.c, to f.c.c.; it was found that the angers-Burgers mechanism provided such a path. This potential, however, was not fitted to experimental elastic constants. Only one paper is known to the authors where a complete set of SOEC for a Cu-Zn alloy was used to construct a pair potential (7); the relative energies of different structures (9R, 2H, b.c.c.) were found to be almost identical and a study of the interface between the different structures showed that they can trnnsform easily from one to the other. It is clear that, in order to have a reliable potential function for e.g. MD-simulations, this potential must be able to describe (at least qualitatively) the softening of the lattice with decreasing temperature; experimentally, this softening is observed both in the decrease of the C' elastic constant and the anomalous behaviour of the [IIO]-TAa phonon branch. Especially, all results from linear elasticity theory concerning the changes of elastic constants under an applied deformation should be reproduced by this function. Recently, a full set of SOEC and TOEC as a function of temperature were reported for a Cu 20.8 at~ Zn - 12.7 at% A1 alloy (,Ms = 158 K) (8). Using a newly developed theoretical model potential (9) these constants can be used to compute a potential function that fully reproduces the elastic mechanical properties of the b.c.c.--lattice. The important features of this model as well as some results on the change of elastic constants as a function of deformation mode are explained in the next sections. The Potential Model As in the majority of potential models, the interaction between the atoms is assumed to be pairwise and central; for simplicity order is not taken into account (i.e. the alloy is regarded as an assembly of average atoms). The energy of the solid is then written as : E =
Eij { % ( r i j )
+ ~(rij). Q + B(rij).Q2
1531 0036-974R/88 $3.00
+ .00
+ 7(rij).Q3
)
(i)
1532
LATTICE
STABILITY
IN B - P H A S E
"ALLOYS"
Vol.
22,
No.
9
where ri~ is the distance between two atoms labelled i and j and O = I - V/Vo is the deviation from the equilibrium volume Vo. O is a small parameter so the effect of the higher order potential functions r,, ~ and ~ is small. For the present model 8 and ~ are taken to be constant. Relations between the elastic constants and the potential functions can be obtained by expanding the energy in powers of the deformation tensor and identifying the elastic moduli tensors of different orders with linear combinations of the potential derivatives. For details of this derivation we refer to (9); e.g. the second order elastic moduli tensor can be written as: C 876
= 2---~I Zj
{ 4~''r.~r~.r.Yr ~ .] 3] ] - 2~'[r'~r'~7~] ]
+ ~6~[~
+
2B --~]
_ K~7~=
+ rTr~6~B]j j
}
(2)
where primes denote di'fferentiation with respect to the distance squared, R is the atomic volume, 6 =~ is the Kronecker delta and Greek indices refer to Cartesian components; B is the constant function ~. The tensor K~s¥ ~ ensures the correct symmetry (with respect to the interchange of indices) and is given by:
KC~IBY6
= 2Gc~BsY6
- (~c~Y6~B
+ 6c¢6~Y~B)
(3)
This equation is valid for any crystal symmetry. For any set of experimental SOEC and TOEC the functions ~ and u and the constants B and C can now be determined, provided the equilibrium lattice positions are inserted into eqn. 2 and the corresponding equation for the TOEC (9); because only second and third order derivatives are involved, two integration constants must be supplied: the position of the minimum of the potential and the depth of the potential well. These can be derived from e.g. vacancy formation energy and/or stacking fault energy. Construction of a Potential Function In Table I the experimental elastic constants (from (8)) are listed for different temperatures (in units GPa). Both SOEC and TOEC were linearly approximated; for the TOEC this is probably not a very good fit hut insufficient data are available for e.g. a quadratic fit. The linearly fitted elastic constants are given in the last two columns of Table I. The potential functions ~ a n d = were written as fourth order spline functions: #(r)
= Z ~ = i A k ( r k - r) 4 H(r k-
r)
(4)
where Ak are fitting parameters and rk are knot points such that ri > ri+1 and rl = truncation radius (for the h.c.c, lattice the truncation distance was chosen at the fourth nearest neighbour distance). H(x) is the Reavyside stepfunction: H(x)=0 for x<0 and H(x)=l for x>0. For more details of the fitting procedure we refer to (9). The two integration constants (potential depth and position of the minimum) were chosen in such a way that equilibrium volume per atom for the f.c.c, phase and vacancy formation energy were well reproduced. Elastic Constants of a Strained BCC Lattice The change in free energy of a strained b.c.c, applied strain:
Fo
FK -
lattice can be written as a function of the
1 -- 5
r'ij Cij~it]j
+ 1
r.ijk Cijk
t]i~J ~]k
+
...
(5)
with ~i the Lagrangian strain parameters, Cij and Cijk the Brugger second and third order elastic constants and FK and F° the free energy per unit volume in rasp. the deformed and undeformed state. According to the Born-Milstein stability criteria (10) the b.c.c, lattice will be unstable as soon as one of the principal minors of the Jacobian determinant IFijKl is negative.
Vol.
22, No.
9
LATTICE
STABILITY
IN
B-PHASE
"ALLOYS"
1533
The resistance against a {110} type of shear in such a strained lattice can be determined by superimposing a small shear strain of the "bias" strain. The free energy change due to such a small shear strain can be expressed as: BF K F ~ = F K + r.i ~
1 ~2FK A~ i + ~ r-ij 8A~i%A~j
A~]iA~j
(6)
with Fg given by eqn. 5. The restoring force constant, ks, acting against this small shear strain is given by the second der ivat ire: k
~2
=
(FA
~A~ i
s
_ FK ) ~a~j
For a compression along the z-axis against a (110)[110] shear(Aql
(7)
(~3
= -~-, ~i
=A~/z,A92 =-AE/2)
ks(ll0)
1 = ~ (Fll
- F12)
= C'
= ~ 2 = ~/2 with ~.>0) the resistance or against a (i~0)[Ii0] shear is given by:
- --14 ( 3Cii 2 - 2C123
- Ciii)8
(8)
For the Cu-Zn-AI alloy described before (8) the restoring force constant is zero for a compression £ = 3.5 Z at 293 K and £ = 3.0 • at 183 K (dashed lines ,in Fig. i). (for this computation the smoothed elastic constants were used). This leads to the mechanical instability of the b.c.c, lattice. The critical values for E are the same as those predicted by the Born-Milstein criteria but the latter do not indicate the instability mode. The restoring force in the other {ii0} planes is given by: ks(101)
= ks(011)
= c'
1 + ~ ( 3CII 2 - 2 C 1 2 3
- C111)e
(9)
The restoring force constants computed with the potential model are drawn as solid lines in Fig. la and b; with eqn. 2 the complete second order tensor for a deformed lattice was computed at two temperatures. The (II0) force constant decreases to zero at £ = 3.33 Z (293 K) and £= 2.90 % (183 K); these values are in close agreement with the theoretical calculations. It should be emphasized that in the theoretical model the symmetry was always assumed to be cubic, even in the deformed state, whereas for the potential model the symmetry of the lattice was always the actual (deformed) symmetry. Therefore, it is no longer possible to define a C' elastic modulus just as a combination of Ci~ and Cl2; for the present calculations, averages were taken, e.g. on the (Ii0) plane the restoring force constant was taken as 0.25(C11+C22-C12C2t). As a result of this approach, the theoretical and computed curves are only tangent for small deformation. Nevertheless, the agreement for the "softening planes" is rather good. If a shear strain is applied to a b.c.c, lattice (e.g. (011)[0]I] with ~ z = - E/2 and ~3 = ~/2 and 6>0) the restoring force constants in the {II0} planes are (2): ks(011)
1 = ~ (F22
+ F33-
2F23)
=
C'
ks(101)
1 = ~ (Fll
+ F33-
2F13)
=
1 C' - ~ (3Cii 2- 2 C 1 2 3 -
Clll)e
ks(110)
= ~
+ F22-
2F[2)
=
C' +
Cl11)~
(Fll
(3Cll 2- 2 C 1 2 3 -
(io)
For the alloy under consideration an instability in the (I01) plane occurs for ~ : 6.9 • at 293 K and for ~ = 6.0 • at 183 K (dashed lines in Fig. 2.a and b). The numerical computations for this type of shear are also plotted as solid lines in Fig. 2. ; the agreement for the (I01) planes is again satisfactory. The instabilities occur for ~ = 6.14~ and E = 5.41~ at resp. 293 K and 183 K. It is clear that a more quantitative approach to the computation of the critical deformation should include the study of the complete second order tensor (the lattice is stable as long as this tensor is positive definite); an example of such a calculation will be presented elsewhere (ll).
1554
LATTICE STABILITY IN B-PHASE "ALLOYS"
Vol. 22, No. 9
Conclusion. The results from the potential modelconfirm those obtained from linear elasticity theory. In both cases it is clear that the resistance against a (iI0} type of shear is modified by an external strain which is important to explain the nucleation of martensite (1)(2)(12)(13). A more detailed analysis of the critical stresses needed to render the lattice mechanically unstable is also possible using the current potential model. This model might even provide a reliable interaction scheme for Molecular Dynmmics simulations, without the need to impose certai~ conditions on the potential function, as has been done in previous work (4)(5). Ack~nowle~ements. This research was carried out in the framework of the interuniversity networks for fundamental research financed by the Belgian Government (contract UIAP n°4). This work has been partially financed by the Belgian National Foundation for Science (FKFO). References
[I] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
P.C. Clapp, Phys. S t a t . Sol. ( b ) , 57, 561 (1973) G. Guenin and P.F. Gobin, Met. T r a n s . A, 13, 1127 (1982) T. S u z u k i , Met. T r a n s . A, 12, 709 (1981) P.C. Clapp and J . R i f k i n , P r o c . I n t . Conf. on S o l i d - S o l l d P h a s e T r a n s f o r m a t i o n s , eds. H . I . A a r o n s o n e t a l , 1165 (1982) J . R i f k i n and P.C. Clapp, J . de P h y s i q u e , 12, C4-157 (1982) P. Beauchemp, P h . D . T h e s i s , U n i v e r s l t e de P o i t i e r s (1981) G.N. B a r c e l o and A.G. C r o c k e r , J . de P h y s i q u e , 12, C4-179 (1982) B. V e r l i n d e n , T, S u z u k i , L. Delaey and G. Guenin, S c r i p t a M e t . , 18, 975 (1984) M. De G r a e f and L. Delaey, Phys. S t a t . Sol. ( b ) , 146, 427 (1988) F. M i l s t e i n , Phys. Rev. B, 3, 1130 (1971) M. De G r a e f , B. V e r l i n d e n and L. Delaey, t o b e p u b l i s h e d B. V e r l i n d e n and L. D e l a e y , J . Phys. F, 16, 1391 (1986) B. V e r l i n d e n and L. D e l a e y , Met. T r a n s . A, 19, 207 (1988)
TMS-AIME,
TABLE I Second and T h i r d O r d e r E l a s t i c C o n s t a n t s f o r a Cu - 2 0 . 8 at% Zn - 1 2 . 7 at% A1 A l l o y a t d i f f e r e n t T e m p e r a t u r e s ( t a k e n from ( 8 ) ) ; i n t h e l a s t two Columns t h e C o n s t a n t s a r e l i n e a r l y a p p r o x i m a t e d as C = ~T + b. A l l d a t a i n GPa.
C11 C12 C4 4
ClII C11z Clzs C144
Clss C456
293 K
253 K
213 K
183 K
116.4 102.3 84.4
116.8 103.0 85.9
117.4 104.0 87.3
117.8 104.4 88.4
-2080 -1060 -920 -1020 -1020 -660
-1760 -790 -680 -790 -790 -680
-1740 -800 -720 -690 -700 -640
-1720 -680 -570 -720 -690 -470
a
•
b
-0.012948 -0.019609 -0.036232
120.1779 108.1191 95.0523
-3.048019 -3.066207 -2.717022 -2.786138 -2.753399 -1.567661
-II07.643 -ii0.613 -82.816 -149.063 -159.271 -243.468
Vol.
22,
No.
9
LATTICE
160
••i
128 Am
96
a.
~J
STABILITY
64
IN B - P H A S E
"ALLOYS"
1555
183 K
293 K
32
0
I
I
I
1
0
I
I
2
I
I
3
4
1
2
3
4
(%) FIG. I. Change of C' elastic constant for a z-compression at 293 K and 183 K; a = (110), b = (i01) and c = (011) restoring force constant. Note that all computed curves (full lines) are tangential to the theoretical curves (dashed lines) for small deformations.
160
183K, //
128
/
//'
/
/
A
m D= ¢.W
96 64 32
I
0
I
2
I
I
4
I
I
6
I
I
8
I
2
I
I
4
I
I
6
I
8
(%) FIG. 2. Analogous to Fig. I. for a (Ii0)[i~0] shear deformation; the (10l) ~estoring force constant tends to ze~-o with increasing de~'ormation.
METALLURGICA
Vol. 22, pp. I S 3 1 - 1 5 3 5 , 1988 P r i n t e d in the U.S.A.
Pergamon Press plc All r i g h t s r e s e r v e d
A NEW POTENTIAL MODEl, FOR THE DESCRIPTION OF LATTICE STABILITY IN ~-PHASE "ALLOYS" M. DE GRAEF, B. VERLINDEN & L. DELAEY Department of Metallurgy and Materials Engineering, Catholic University Leuven, de Croylaan 2, B-3030 Heverlee, Belgium ( R e c e i v e d J u n e 14, 1988) ( R e v i s e d J u l y 7, 1988) Introduction Over the past years it has become clear that the martensitic transformation in b.c.c.-based Hume-Rothery alloys can only be adequately described if both second and third order elastic constants (SOEC - TOEC) are taken into account (I)(2)(3). For a realistic computer simulation of the transformation mechanism, these constants should also be used in constructing the interatomic interaction potentials. Unfortunately, there are not many alloys for which both SOEC and TOEC are known (preferably as a function of temperature). Therefore, all Molecular Dynamics (MD) simulations so far have been carried out with an empirical potential fitted only to the SOEC; often this potential function is numerically manipulated in order to introduce certain stability relations between parent and product crystal structures. In (4) a Morse potential with an added oscillatory tail part (to mimic long range Friede] oscillations) has been used to study the effect of this tail part on the relative stability of b.c.c., 3R, pseudo-2H,... No detailed analysis of mechanical lattice stability was reported in this work. The same authors later used another empirical potential (5) to perform a two-dimensional MDsimulation of the nucleation of martensite: the potential was chosen in such. a way as to give a certain energy change in g o i n g from the initial to the final state. In (6) a phenomenological pair potential for pure Cu was used to calculate the minimum energy path for the Bain transformation from b.c.c, to f.c.c.; it was found that the angers-Burgers mechanism provided such a path. This potential, however, was not fitted to experimental elastic constants. Only one paper is known to the authors where a complete set of SOEC for a Cu-Zn alloy was used to construct a pair potential (7); the relative energies of different structures (9R, 2H, b.c.c.) were found to be almost identical and a study of the interface between the different structures showed that they can trnnsform easily from one to the other. It is clear that, in order to have a reliable potential function for e.g. MD-simulations, this potential must be able to describe (at least qualitatively) the softening of the lattice with decreasing temperature; experimentally, this softening is observed both in the decrease of the C' elastic constant and the anomalous behaviour of the [IIO]-TAa phonon branch. Especially, all results from linear elasticity theory concerning the changes of elastic constants under an applied deformation should be reproduced by this function. Recently, a full set of SOEC and TOEC as a function of temperature were reported for a Cu 20.8 at~ Zn - 12.7 at% A1 alloy (,Ms = 158 K) (8). Using a newly developed theoretical model potential (9) these constants can be used to compute a potential function that fully reproduces the elastic mechanical properties of the b.c.c.--lattice. The important features of this model as well as some results on the change of elastic constants as a function of deformation mode are explained in the next sections. The Potential Model As in the majority of potential models, the interaction between the atoms is assumed to be pairwise and central; for simplicity order is not taken into account (i.e. the alloy is regarded as an assembly of average atoms). The energy of the solid is then written as : E =
Eij { % ( r i j )
+ ~(rij). Q + B(rij).Q2
1531 0036-974R/88 $3.00
+ .00
+ 7(rij).Q3
)
(i)
1532
LATTICE
STABILITY
IN B - P H A S E
"ALLOYS"
Vol.
22,
No.
9
where ri~ is the distance between two atoms labelled i and j and O = I - V/Vo is the deviation from the equilibrium volume Vo. O is a small parameter so the effect of the higher order potential functions r,, ~ and ~ is small. For the present model 8 and ~ are taken to be constant. Relations between the elastic constants and the potential functions can be obtained by expanding the energy in powers of the deformation tensor and identifying the elastic moduli tensors of different orders with linear combinations of the potential derivatives. For details of this derivation we refer to (9); e.g. the second order elastic moduli tensor can be written as: C 876
= 2---~I Zj
{ 4~''r.~r~.r.Yr ~ .] 3] ] - 2~'[r'~r'~7~] ]
+ ~6~[~
+
2B --~]
_ K~7~=
+ rTr~6~B]j j
}
(2)
where primes denote di'fferentiation with respect to the distance squared, R is the atomic volume, 6 =~ is the Kronecker delta and Greek indices refer to Cartesian components; B is the constant function ~. The tensor K~s¥ ~ ensures the correct symmetry (with respect to the interchange of indices) and is given by:
KC~IBY6
= 2Gc~BsY6
- (~c~Y6~B
+ 6c¢6~Y~B)
(3)
This equation is valid for any crystal symmetry. For any set of experimental SOEC and TOEC the functions ~ and u and the constants B and C can now be determined, provided the equilibrium lattice positions are inserted into eqn. 2 and the corresponding equation for the TOEC (9); because only second and third order derivatives are involved, two integration constants must be supplied: the position of the minimum of the potential and the depth of the potential well. These can be derived from e.g. vacancy formation energy and/or stacking fault energy. Construction of a Potential Function In Table I the experimental elastic constants (from (8)) are listed for different temperatures (in units GPa). Both SOEC and TOEC were linearly approximated; for the TOEC this is probably not a very good fit hut insufficient data are available for e.g. a quadratic fit. The linearly fitted elastic constants are given in the last two columns of Table I. The potential functions ~ a n d = were written as fourth order spline functions: #(r)
= Z ~ = i A k ( r k - r) 4 H(r k-
r)
(4)
where Ak are fitting parameters and rk are knot points such that ri > ri+1 and rl = truncation radius (for the h.c.c, lattice the truncation distance was chosen at the fourth nearest neighbour distance). H(x) is the Reavyside stepfunction: H(x)=0 for x<0 and H(x)=l for x>0. For more details of the fitting procedure we refer to (9). The two integration constants (potential depth and position of the minimum) were chosen in such a way that equilibrium volume per atom for the f.c.c, phase and vacancy formation energy were well reproduced. Elastic Constants of a Strained BCC Lattice The change in free energy of a strained b.c.c, applied strain:
Fo
FK -
lattice can be written as a function of the
1 -- 5
r'ij Cij~it]j
+ 1
r.ijk Cijk
t]i~J ~]k
+
...
(5)
with ~i the Lagrangian strain parameters, Cij and Cijk the Brugger second and third order elastic constants and FK and F° the free energy per unit volume in rasp. the deformed and undeformed state. According to the Born-Milstein stability criteria (10) the b.c.c, lattice will be unstable as soon as one of the principal minors of the Jacobian determinant IFijKl is negative.
Vol.
22, No.
9
LATTICE
STABILITY
IN
B-PHASE
"ALLOYS"
1533
The resistance against a {110}
1 ~2FK A~ i + ~ r-ij 8A~i%A~j
A~]iA~j
(6)
with Fg given by eqn. 5. The restoring force constant, ks, acting against this small shear strain is given by the second der ivat ire: k
~2
=
(FA
~A~ i
s
_ FK ) ~a~j
For a compression along the z-axis against a (110)[110] shear(Aql
(7)
(~3
= -~-, ~i
=A~/z,A92 =-AE/2)
ks(ll0)
1 = ~ (Fll
- F12)
= C'
= ~ 2 = ~/2 with ~.>0) the resistance or against a (i~0)[Ii0] shear is given by:
- --14 ( 3Cii 2 - 2C123
- Ciii)8
(8)
For the Cu-Zn-AI alloy described before (8) the restoring force constant is zero for a compression £ = 3.5 Z at 293 K and £ = 3.0 • at 183 K (dashed lines ,in Fig. i). (for this computation the smoothed elastic constants were used). This leads to the mechanical instability of the b.c.c, lattice. The critical values for E are the same as those predicted by the Born-Milstein criteria but the latter do not indicate the instability mode. The restoring force in the other {ii0} planes is given by: ks(101)
= ks(011)
= c'
1 + ~ ( 3CII 2 - 2 C 1 2 3
- C111)e
(9)
The restoring force constants computed with the potential model are drawn as solid lines in Fig. la and b; with eqn. 2 the complete second order tensor for a deformed lattice was computed at two temperatures. The (II0) force constant decreases to zero at £ = 3.33 Z (293 K) and £= 2.90 % (183 K); these values are in close agreement with the theoretical calculations. It should be emphasized that in the theoretical model the symmetry was always assumed to be cubic, even in the deformed state, whereas for the potential model the symmetry of the lattice was always the actual (deformed) symmetry. Therefore, it is no longer possible to define a C' elastic modulus just as a combination of Ci~ and Cl2; for the present calculations, averages were taken, e.g. on the (Ii0) plane the restoring force constant was taken as 0.25(C11+C22-C12C2t). As a result of this approach, the theoretical and computed curves are only tangent for small deformation. Nevertheless, the agreement for the "softening planes" is rather good. If a shear strain is applied to a b.c.c, lattice (e.g. (011)[0]I] with ~ z = - E/2 and ~3 = ~/2 and 6>0) the restoring force constants in the {II0} planes are (2): ks(011)
1 = ~ (F22
+ F33-
2F23)
=
C'
ks(101)
1 = ~ (Fll
+ F33-
2F13)
=
1 C' - ~ (3Cii 2- 2 C 1 2 3 -
Clll)e
ks(110)
= ~
+ F22-
2F[2)
=
C' +
Cl11)~
(Fll
(3Cll 2- 2 C 1 2 3 -
(io)
For the alloy under consideration an instability in the (I01) plane occurs for ~ : 6.9 • at 293 K and for ~ = 6.0 • at 183 K (dashed lines in Fig. 2.a and b). The numerical computations for this type of shear are also plotted as solid lines in Fig. 2. ; the agreement for the (I01) planes is again satisfactory. The instabilities occur for ~ = 6.14~ and E = 5.41~ at resp. 293 K and 183 K. It is clear that a more quantitative approach to the computation of the critical deformation should include the study of the complete second order tensor (the lattice is stable as long as this tensor is positive definite); an example of such a calculation will be presented elsewhere (ll).
1554
LATTICE STABILITY IN B-PHASE "ALLOYS"
Vol. 22, No. 9
Conclusion. The results from the potential modelconfirm those obtained from linear elasticity theory. In both cases it is clear that the resistance against a (iI0}
[I] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
P.C. Clapp, Phys. S t a t . Sol. ( b ) , 57, 561 (1973) G. Guenin and P.F. Gobin, Met. T r a n s . A, 13, 1127 (1982) T. S u z u k i , Met. T r a n s . A, 12, 709 (1981) P.C. Clapp and J . R i f k i n , P r o c . I n t . Conf. on S o l i d - S o l l d P h a s e T r a n s f o r m a t i o n s , eds. H . I . A a r o n s o n e t a l , 1165 (1982) J . R i f k i n and P.C. Clapp, J . de P h y s i q u e , 12, C4-157 (1982) P. Beauchemp, P h . D . T h e s i s , U n i v e r s l t e de P o i t i e r s (1981) G.N. B a r c e l o and A.G. C r o c k e r , J . de P h y s i q u e , 12, C4-179 (1982) B. V e r l i n d e n , T, S u z u k i , L. Delaey and G. Guenin, S c r i p t a M e t . , 18, 975 (1984) M. De G r a e f and L. Delaey, Phys. S t a t . Sol. ( b ) , 146, 427 (1988) F. M i l s t e i n , Phys. Rev. B, 3, 1130 (1971) M. De G r a e f , B. V e r l i n d e n and L. Delaey, t o b e p u b l i s h e d B. V e r l i n d e n and L. D e l a e y , J . Phys. F, 16, 1391 (1986) B. V e r l i n d e n and L. D e l a e y , Met. T r a n s . A, 19, 207 (1988)
TMS-AIME,
TABLE I Second and T h i r d O r d e r E l a s t i c C o n s t a n t s f o r a Cu - 2 0 . 8 at% Zn - 1 2 . 7 at% A1 A l l o y a t d i f f e r e n t T e m p e r a t u r e s ( t a k e n from ( 8 ) ) ; i n t h e l a s t two Columns t h e C o n s t a n t s a r e l i n e a r l y a p p r o x i m a t e d as C = ~T + b. A l l d a t a i n GPa.
C11 C12 C4 4
ClII C11z Clzs C144
Clss C456
293 K
253 K
213 K
183 K
116.4 102.3 84.4
116.8 103.0 85.9
117.4 104.0 87.3
117.8 104.4 88.4
-2080 -1060 -920 -1020 -1020 -660
-1760 -790 -680 -790 -790 -680
-1740 -800 -720 -690 -700 -640
-1720 -680 -570 -720 -690 -470
a
•
b
-0.012948 -0.019609 -0.036232
120.1779 108.1191 95.0523
-3.048019 -3.066207 -2.717022 -2.786138 -2.753399 -1.567661
-II07.643 -ii0.613 -82.816 -149.063 -159.271 -243.468
Vol.
22,
No.
9
LATTICE
160
••i
128 Am
96
a.
~J
STABILITY
64
IN B - P H A S E
"ALLOYS"
1555
183 K
293 K
32
0
I
I
I
1
0
I
I
2
I
I
3
4
1
2
3
4
(%) FIG. I. Change of C' elastic constant for a z-compression at 293 K and 183 K; a = (110), b = (i01) and c = (011) restoring force constant. Note that all computed curves (full lines) are tangential to the theoretical curves (dashed lines) for small deformations.
160
183K, //
128
/
//'
/
/
A
m D= ¢.W
96 64 32
I
0
I
2
I
I
4
I
I
6
I
I
8
I
2
I
I
4
I
I
6
I
8
(%) FIG. 2. Analogous to Fig. I. for a (Ii0)[i~0] shear deformation; the (10l) ~estoring force constant tends to ze~-o with increasing de~'ormation.