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Solute hardening of close-packed solid solutions
SOLUTE
HARDENING
OF CLOSE-PACKED P. A.
SOLID
SOLUTIONS*
FLINN?
The contribution of local order, and of the segregation of solute atoms at extended dislocations, to the strength of close-packed solid solutions are investigated quantitatively. The two effects are roughly of the same order of magnitude, but either may predominate in a particular alloy. The theoretical predictions for the initial yield strength of silver-gold, copper-gold, and copper-zinc alloys are in good agreement with the experimental data.
DURCISSEMENT
“PAR
SOLUTION”
DES
SOLUTIONS
SOLIDES
A
RESEAU
COMPACT
L’auteur Qtudie quantitativement l’influence de I’ordre local et de la segregation d’atomes dissous aux dislocations sur la resistance des solutions solides a reseau compact. Les deux effets sent grossierement du m&me ordre de grandeur, mais l’un ou l’autre peut predominer dans un alliage paticulier. Les previsions theoriques pour la tension amorqant la deformation plaetique (initial yield strength) des alliages Ag-Au, &--Au et Cu-Zn sont en bon accord avec les resultats esperimentaux. LEGIERUNGSVERFESTIGUNG
DICHTEST
GEPACKTER
MISCHKRISTALLE
Der Beitrag der Nahordnung und der Anreicherung gel&tar Atome an ausgedehnten Versetzungen zur Streckgrenze dichtest gepackter Misohkristalle wurde quantitativ untersucht. Die beiden Einfliisse sind etwa van gleicher Griissenordnung; in einer bestimmten Legiertmg diirfte jedoch einer van beiden iibertiegen. Die theoretischen Voraussagen ftir die Streckgrenze van Silber-Gold-, Kupfer-Gold- und Kupfer--Zink-Legierungen sind in guter Ubereinstimmung mit den experimentellen Damn.
INTRODUCTION
Two mechanisms have been suggested to account for a major part of the hardening associated with substitutional solute atoms at large concentrations in close-packed solid solutions : the interaction between solute atoms and the faulted area of extended dislocations, proposed by Suzuki;(‘) and an effect associated with local order, proposed by Fisher.c2) Either mechanism, to a rough approximation, is of the correct order of magnitude to be in agreement with the limited amount of experimental data available. A more detailed investigation of the two mechanisms permits a decision as to the relative magnitudes of the two effects in various cases. The Fisher mechanism, of course, is not limited to close-packed structures; but the complications arising from interstitial interactions in body-centered cubic alloys preclude any simple interpretation of the strength of such alloys. Fisher estimated the contribution of local order to the yield st’rength of u-brass, and obtained a value several times the total observed strength. His analysis, however, was based on a degree of local order calculated from the thermodynamic functions of the system, rather than on local order directly measured by diffraction methods. It is known that no reliable * Received January 27, 1958. t Westinghouse Electric Corporation, Pibtsburgh 35, Pa. ACTA
METALLURGICA,
VOL.
6, OCTOBER
1958
correlation between actual local order and thermodynamic functions exists.@p4) Unfortunately, the only attempt at a direct measurement of local order in a-brasst5) was of rather low sensitivity. No such order was observed, although a small amount may have been present. For gold-silver and gold-copper alloys, however, direct measurements of local order by X-ray diffraction exist,(%‘) and the necessary data on the mechanical properties of alloy single crystals are available.(s,g) A quantitative calculation of the Fisher interaction in these systems leads to a predicted strengthening effect, in excellent agreement with experiment. ENERGY
ASSOCIATED
WITH
LOCAL
ORDER
The energy associated with the local ordering of nearest neighbors in a binary solid solution is given by:‘10,21’
E = NcmAm, vu where
N = c= ma,mB = v= cc =
(1)
number of atoms in the lattice, coordination number, mole fractions of A and B, interaction energy,: local order coefficient.
$ The interaction energy may be regarded as one-half the energy required to rearrange the lattice so as to form two A-B bonds from one A-A and one B-B bond. It cannot be calculated from the heat of solution of the alloy because the energy associated with forming the solution is not localized in nearest neighbor bonds.
631
632
ACTA
METALLURGICA,
A derivation of the general equation is given by J. M. Cowley.(21) We may divide up the total energy E among the NC/~ nearest neighbor bonds, so that the average energy per nearest neighbor bond due to local order is: E = 2EjNc = 2marn, vu (2) v may be calculated from an experimental cc by using the relation(lO) u P=m,m,[exp~-l]. (3) (1 - a)2 SLIP
IN A LATTICE
WITH
LOCAL
8 mamgvu a21/3
1/3
a2
’
(4)
Equating the work done in moving the dislocation, rb, to the energy increase y, we have for the shear stress 7:* ‘2 mgmgvu T= 16 3 a3 *
J
6, 1958
’ ’1 0 ---- - - - -
Data of Sachs and Weerts Pure Metal Term Suzuki Term Local Order Term Total Theoretical Strength
ORDER
Of the twelve nearest neighbors of an atom in a close-packed plane, (ill), of a f.c.c. lattice, six lie in this plane, three lie in the plane above, and three in the plane below. If we displace the plane above or below by one Burgers vector (b = l/2 [IlO]), two of the three neighbors in this plane are changed. Since, in most cases, next nearest neighbor positions are almost uncorrelated, the new neighbors will be randomly chosen. The energy of the system is thus increased by the amount of the energy associated with the local order destroyed: 2~ per atom in the slip plane. Dividing this by the area per atom in the slip plane, which is a2 2/3/4, we find y, the energy increase per unit area of slip, to be: &L_
VOL.
FIG. 1. Strength of Ag-Au single crystals.
Suzuki interaction may also be expected to make a small contribution (-0.1-0.2 kg/mm2), as discussed below. COMPOSITION ORDER
DEPENDENCE OF LOCAL STRENGTHENING
Experimental data are available for the strength of a number of Ag-Au alloys, so we may study the composition dependence of the effect in this system. Since the amount of local order is small, we can replace (3) by the approximate form:
We can now calculate the contribution of local order to the critical resolved shear stress for an alloy in which u is known, using equation (3) to calculate v and then equation (5) to calculate 7. For the alloy u = 2mAm, v/kT (6) CusAu in the disordered state, quenched from 4OO”C, and equation (5) then has the simple form: a = -0.15 and v = -0.028 eV.oO) From equation (4) we find y = 8.3 erg/cm2 and from (5) T = 3.2 2 (mAm17)2v2 7 = 32 kg/mm2. According to the data of Ardley,cg) the 3 a3kT * critical resolved shear stress for CusAu quenched For Ag-Au alloys from 420°C is about 5 kg/mm2. This agreement is v = -0.007 eV within the uncertainty due to the experimental error a = 4.08 A. in a. The mechanisms responsible for the strength of Assume T = 600°K (diffusion below this temperature pure metals will make a rather small contribution to range being too slow for any this strength, since the critical resolved shear stresses significant increase in local order are only about 0.1 kg/mm2 for pure Cu and Au. The under normal cooling conditions) * We assume that the temperature of the system is low so that: enough for entropy effects to be neglected, and the free 7 = 6(m,mB)2 (kg/mm2). (8) energy change is equal to the energy change.
J
FLINN:
SOLUTE
HARDENING
OF
CLOSE-PACKED
SOLID
633
SOLUTIONS
In Fig. 1 a plot of equation (8) is shown, along with the experimenta, strength data for Ag-Au alloys. It may be seen that the local order hardening can account for the major part of the strength of the alloys, except at low solute concentrations. The estimated contributions of the Suzuki interaction and the inherent strength are also shown. The sum of the three contributions is in re~~arkably good agreeu~ent with the experimental data. The “inherent strength” may not vary in the manner shown, nor be simply additive, but this term is probably small. THE
SUZUKI
INTERACTION SOLUTIONS
IN
REGULAR
Suzuki(l) has attributed virtually all of the solid solution hardening in such alloy systems as silvergold to a sort of chemical interaction between the stacking fault region of extended dislocations and the solute atoms. He tacitly assumes an ideal solution by neglecting the heat of solution, but his analysis is readily extended to regular solutions. For a regular solution. the molar free energy is given by:
+RT(mA
ln %A + mB ln mg)
(9)
where F,, F, are the molar free energies of the pure components il and B at temperature T. AH, = integral heat, of solution for mA = mg = l/2. We assume that the molar free energy of the faulted material, Ff, is given by a similar equation: F = m,lfPJf + mBfFBf + ~~_~f~~fAH~ + RT(mAf In mAf + mBf In mBf)
(10)
where AH, is the same as for the bulk material. With temperat,ure, pressure, and amount of faulted material constant 1 the equilibrium condition for the compositions of bulk and faulted materials is: 3F ...am,
i?Ff = __ . amAf
Combining this relation with (9) and (10) we find: (F,
-
FAf) -- (.FN -
FBf) + 4AH&m, -
mnf)
4AH, (mA --. m.f) + RT In zmf =: 0. (12) rnAtfmD
We now define & = (Fi, -
@,I) -
(Fn -
F,f) and
MA9 FIG.
2. Suzuki hardening for Ag-Au alloys
to an extra shear stress, rs, required to move an extended dislocation. We can calculate this extra shear stress by a method similar to that used for local order. We assume that the extended dislocation is moved a distance 6 by a stress TV,doing work per unit length of dislocation 611= &rJ. This must equal the increase in free energy of the system. One partial dislocation leaves behind a region of composition mRf, but unfaulted; the other enters material of composition m, and converts this into faulted material. Each of these regions is of volume (~~~3~~~ per unit length of dislocation (assuming the faulted region is two layers thick). The corresponding increase in energy is given by: w =
-F
mBf) z! LF(
f(mnf)]
V
2/3
where I’ is the volume per gram-atom. Using equations (10) and (II), we can simplify this: F(mBf) -
Ff(m,f)
= md[Fa -
m,[J',
-
+ Ff(m,) -
As shown by Suzuki, this composition difference leads
F(m,)
FA] -:- mBf[FLI -
F_4fl - m,[F,
-
Ts
=
2
2Q ---Am. 3 C’
J
Combining (13) and (16), we have
P,f]
F,l] = AmQ (15)
so that, equating W to (STY 6, we find:
mBf) = Am
(m, and expand the logarithm to obtain, to the first order in Am: m. m & --I AWL _ A N (13) RX
-
634
ACTA 1.6
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1 0
-
I 0
0.1
I 0.2
I
0’ 0
0
G&r and
o
Jomison
METALLURGICA,
VOL.
6, 1958
Since the Suzuki effect is of the looking type, that is, the dislocation need move only a short distance to escape the solute interaction, the possibility of thermally activated escape must be considered. Suzuki(l) has dealt with this possibility in some detail and found that the activation energy required for a significant reduction in yield strength is of the order of 1 eV per interatomic distance. At room temperature, thermal activation will be unimportant, since
I
kT w 0.026 eV.
Sachs and Sherill
Theoretic01 Suzuki Hardening
I
I
I
0.3
0.4
0.5
I
0.6
MZn
Fra. 3. Strength of cc-brasssingle crystals
Since even for alloys of quite similar metals, such as silver and gold, AH, is at least as large as RT, the term in brackets is quite important. For systems with continuous solid solubility, AH* must be negative unless it is very small. (A positive AH requires a solubility gap). A negative AH leads to a saturation of the effect at a relatively low solute concentration. In Fig. 2, the magnitude of the effect for Ag-Au alloys as given by Suzuki (ideal solution) is compared with that calculated from Equation (17) (regular solution), using AH, = -1070 cal/mole,(ll) T = 600°K and & = 131 cal/mole as assumed by Suzuki. The effect in C&Au alloys should be similar since AH+ M -1300 cal/mole,c12) and Q is not likely to be very different. For u-brass, we can make a direct estimate of Q, which is unusually large in this case. The energy of a twin boundary in copper as determined by Fullman(14) is about 25 erg/cc, so that we may assume the energy of a stacking fault to be roughly 50 erg/cc. At about 50% Zn, however, the energy of a stacking fault must be approximately zero, since the phase formed by low temperature transformation of B-brass has a mixed f.c.c. and h.c.p. character.(16) Assuming a linear variation of stacking fault energy with composition, we calculate Q to be about 500 cal/mole. The AH+ for u-brass is about - 2300 cal/mole.(13) Using these values, the T, given by equation (17) is plotted along with the experimental data(i6217) in Fig. (3). The agreement is quite satisfactory. It should be noted that the form of equation (17) gives a much better fit than (7), aside from the quantitative aspects of the matter.
One other aspect of the Suzuki effect may be of some importance. Various anomalies have been observed in alloys as a consequence of prolonged annealing at low temperatures, e.g.: resistivity and lattice parameter changes in cc-brass,(ls)and resistivity and hardness changes in Ni3Cr.(lg) These effects have usually been ascribed to change in local or long-range order. In the case of Ni,Cr at least, Roberts and Swalin(20) have shown that no significant amount of order is present. These anomalies may be a consequence of the temperature dependence of the Suzuki effect (equation 13). Normal cooling will result in a segregation characteristic of a moderately high temperature. Prolonged lqw temperature annealing should increase segregation, increasing hardness and changing resistivity. Some effect on lattice parameters may also be expected. SUMMARY
Both the Fisher and Suzuki interactions may make significant contributions to the strength of solid solutions. We may expect the Suzuki effect to be dominant in dilute solutions, especially where the energy of a stacking fault varies rapidly with composition; and the Fisher effect to be dominant in concentrated solutions if any appreciable local order exists. The largest Fisher interaction is probably considerably more important than the largest Suzuki interaction. Since the submission of this manuscript, a discussion of the problem by Suzuki(l) and Seegerc2) has appeared. The contribution of local order to hardening derived by Suzuki (his equation 32) differs from that in this paper (equation 7) bv exactly a factor of 4, when proper conversion of units is made, although his analysis does not appear to differ in principle from that of this author. The origin of the discrepancy is not entirely clear, although one factor of 2 does seem to arise in his approximation for P,,. The result quoted by Seeger in his discussion (his equation l), after appropriate approximations are introduced, becomes identical with equation (7) of this paper.
FLINN:
SOLUTE
HARDENING
OF
(1) Hideki Suzuki, ~~~~t~o~ and ~ec~n~caZ Properties of Crystals p. 361. John Wiley, New York (1957). (2) A. Seeger, Ibid. p. 388.
CLOSE-PACKED 6.
7. 8. 9. 10. 11.
ACKNOWLEDGMENT
The author is grateful to several members of the department for enlightening discussion of this paper. REFERENCES 1. H. SUZUKI, Sci. Rep. Res. Insts To”hoku Univ. Ser A, 4, 455 (1952). 2. J. C. FISHER, Rcto iVet. 2, 9 (1954). 3. R. A. ORIANI, Rcta Met. 1.144 (1953). 4. B. L. AVERBACH, P. A. F&N and M. COREN, R&x Met. 2, 92 (1954). 5. D. T. KEATING, Aeta Met. 2, 885 (1954).
12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
SOLID
SOLUTIONS
635
NORMAX and B. E. WARREN, J. AppE. Phys. 22, 483 (1951). J. M. COWLEY,J. AppZ. P&s. 21, 24 (1950). G. SACHSand J. WEERTS, 2. Phyys. 62, 473 (1930). G. W. ABDLEY, Acta Met. 3, 525 (1955). P. A. FLINN, Phys. Rev. 104, 350 (1956). 0. KUBASCHEWSKIand J. A. CATTERALL,Thermoohemieal Data of Alloys, p. 64. Porgamon Press, London and New York (1956). 0. KUBAS~XEWSKIand J. A. CATTERALL,Ibid. p. 61. 0. KUBASCHEWSKIand J. A. CXIXXRALL,I&d. p. 66. R. L. FULLMAN,J. AppE. Phys. 22,448 (1951). T. B. IM~~~~~~~~ and C. S. BARRETT,Tram. Amer. Inst. Min. (Metall.) Eng7s. 209, 455 (1957). V. GOX,ERand G. SACHS,Z. Phys. 55, 581 (1929). R. E. JAMISONand F. A. SREBILL,Acta Met. 4,197 (1956). A. C. DAMASK,J. AppZ. Phys. 27, 610 (1956). R. NORDHEIMand N. J. GRANT, J. Inst. Met. 82, 440 (1953-54). B. W. ROBERTSand R. A, SW&IN, Trans. Amer. Ilzst. M&a. (M&all.) Engrs. ZSS, 845 (1957). J. M. COWLEY, Phys. Rev. 77, 669 (1950). N.
HARDENING
OF CLOSE-PACKED P. A.
SOLID
SOLUTIONS*
FLINN?
The contribution of local order, and of the segregation of solute atoms at extended dislocations, to the strength of close-packed solid solutions are investigated quantitatively. The two effects are roughly of the same order of magnitude, but either may predominate in a particular alloy. The theoretical predictions for the initial yield strength of silver-gold, copper-gold, and copper-zinc alloys are in good agreement with the experimental data.
DURCISSEMENT
“PAR
SOLUTION”
DES
SOLUTIONS
SOLIDES
A
RESEAU
COMPACT
L’auteur Qtudie quantitativement l’influence de I’ordre local et de la segregation d’atomes dissous aux dislocations sur la resistance des solutions solides a reseau compact. Les deux effets sent grossierement du m&me ordre de grandeur, mais l’un ou l’autre peut predominer dans un alliage paticulier. Les previsions theoriques pour la tension amorqant la deformation plaetique (initial yield strength) des alliages Ag-Au, &--Au et Cu-Zn sont en bon accord avec les resultats esperimentaux. LEGIERUNGSVERFESTIGUNG
DICHTEST
GEPACKTER
MISCHKRISTALLE
Der Beitrag der Nahordnung und der Anreicherung gel&tar Atome an ausgedehnten Versetzungen zur Streckgrenze dichtest gepackter Misohkristalle wurde quantitativ untersucht. Die beiden Einfliisse sind etwa van gleicher Griissenordnung; in einer bestimmten Legiertmg diirfte jedoch einer van beiden iibertiegen. Die theoretischen Voraussagen ftir die Streckgrenze van Silber-Gold-, Kupfer-Gold- und Kupfer--Zink-Legierungen sind in guter Ubereinstimmung mit den experimentellen Damn.
INTRODUCTION
Two mechanisms have been suggested to account for a major part of the hardening associated with substitutional solute atoms at large concentrations in close-packed solid solutions : the interaction between solute atoms and the faulted area of extended dislocations, proposed by Suzuki;(‘) and an effect associated with local order, proposed by Fisher.c2) Either mechanism, to a rough approximation, is of the correct order of magnitude to be in agreement with the limited amount of experimental data available. A more detailed investigation of the two mechanisms permits a decision as to the relative magnitudes of the two effects in various cases. The Fisher mechanism, of course, is not limited to close-packed structures; but the complications arising from interstitial interactions in body-centered cubic alloys preclude any simple interpretation of the strength of such alloys. Fisher estimated the contribution of local order to the yield st’rength of u-brass, and obtained a value several times the total observed strength. His analysis, however, was based on a degree of local order calculated from the thermodynamic functions of the system, rather than on local order directly measured by diffraction methods. It is known that no reliable * Received January 27, 1958. t Westinghouse Electric Corporation, Pibtsburgh 35, Pa. ACTA
METALLURGICA,
VOL.
6, OCTOBER
1958
correlation between actual local order and thermodynamic functions exists.@p4) Unfortunately, the only attempt at a direct measurement of local order in a-brasst5) was of rather low sensitivity. No such order was observed, although a small amount may have been present. For gold-silver and gold-copper alloys, however, direct measurements of local order by X-ray diffraction exist,(%‘) and the necessary data on the mechanical properties of alloy single crystals are available.(s,g) A quantitative calculation of the Fisher interaction in these systems leads to a predicted strengthening effect, in excellent agreement with experiment. ENERGY
ASSOCIATED
WITH
LOCAL
ORDER
The energy associated with the local ordering of nearest neighbors in a binary solid solution is given by:‘10,21’
E = NcmAm, vu where
N = c= ma,mB = v= cc =
(1)
number of atoms in the lattice, coordination number, mole fractions of A and B, interaction energy,: local order coefficient.
$ The interaction energy may be regarded as one-half the energy required to rearrange the lattice so as to form two A-B bonds from one A-A and one B-B bond. It cannot be calculated from the heat of solution of the alloy because the energy associated with forming the solution is not localized in nearest neighbor bonds.
631
632
ACTA
METALLURGICA,
A derivation of the general equation is given by J. M. Cowley.(21) We may divide up the total energy E among the NC/~ nearest neighbor bonds, so that the average energy per nearest neighbor bond due to local order is: E = 2EjNc = 2marn, vu (2) v may be calculated from an experimental cc by using the relation(lO) u P=m,m,[exp~-l]. (3) (1 - a)2 SLIP
IN A LATTICE
WITH
LOCAL
8 mamgvu a21/3
1/3
a2
’
(4)
Equating the work done in moving the dislocation, rb, to the energy increase y, we have for the shear stress 7:* ‘2 mgmgvu T= 16 3 a3 *
J
6, 1958
’ ’1 0 ---- - - - -
Data of Sachs and Weerts Pure Metal Term Suzuki Term Local Order Term Total Theoretical Strength
ORDER
Of the twelve nearest neighbors of an atom in a close-packed plane, (ill), of a f.c.c. lattice, six lie in this plane, three lie in the plane above, and three in the plane below. If we displace the plane above or below by one Burgers vector (b = l/2 [IlO]), two of the three neighbors in this plane are changed. Since, in most cases, next nearest neighbor positions are almost uncorrelated, the new neighbors will be randomly chosen. The energy of the system is thus increased by the amount of the energy associated with the local order destroyed: 2~ per atom in the slip plane. Dividing this by the area per atom in the slip plane, which is a2 2/3/4, we find y, the energy increase per unit area of slip, to be: &L_
VOL.
FIG. 1. Strength of Ag-Au single crystals.
Suzuki interaction may also be expected to make a small contribution (-0.1-0.2 kg/mm2), as discussed below. COMPOSITION ORDER
DEPENDENCE OF LOCAL STRENGTHENING
Experimental data are available for the strength of a number of Ag-Au alloys, so we may study the composition dependence of the effect in this system. Since the amount of local order is small, we can replace (3) by the approximate form:
We can now calculate the contribution of local order to the critical resolved shear stress for an alloy in which u is known, using equation (3) to calculate v and then equation (5) to calculate 7. For the alloy u = 2mAm, v/kT (6) CusAu in the disordered state, quenched from 4OO”C, and equation (5) then has the simple form: a = -0.15 and v = -0.028 eV.oO) From equation (4) we find y = 8.3 erg/cm2 and from (5) T = 3.2 2 (mAm17)2v2 7 = 32 kg/mm2. According to the data of Ardley,cg) the 3 a3kT * critical resolved shear stress for CusAu quenched For Ag-Au alloys from 420°C is about 5 kg/mm2. This agreement is v = -0.007 eV within the uncertainty due to the experimental error a = 4.08 A. in a. The mechanisms responsible for the strength of Assume T = 600°K (diffusion below this temperature pure metals will make a rather small contribution to range being too slow for any this strength, since the critical resolved shear stresses significant increase in local order are only about 0.1 kg/mm2 for pure Cu and Au. The under normal cooling conditions) * We assume that the temperature of the system is low so that: enough for entropy effects to be neglected, and the free 7 = 6(m,mB)2 (kg/mm2). (8) energy change is equal to the energy change.
J
FLINN:
SOLUTE
HARDENING
OF
CLOSE-PACKED
SOLID
633
SOLUTIONS
In Fig. 1 a plot of equation (8) is shown, along with the experimenta, strength data for Ag-Au alloys. It may be seen that the local order hardening can account for the major part of the strength of the alloys, except at low solute concentrations. The estimated contributions of the Suzuki interaction and the inherent strength are also shown. The sum of the three contributions is in re~~arkably good agreeu~ent with the experimental data. The “inherent strength” may not vary in the manner shown, nor be simply additive, but this term is probably small. THE
SUZUKI
INTERACTION SOLUTIONS
IN
REGULAR
Suzuki(l) has attributed virtually all of the solid solution hardening in such alloy systems as silvergold to a sort of chemical interaction between the stacking fault region of extended dislocations and the solute atoms. He tacitly assumes an ideal solution by neglecting the heat of solution, but his analysis is readily extended to regular solutions. For a regular solution. the molar free energy is given by:
+RT(mA
ln %A + mB ln mg)
(9)
where F,, F, are the molar free energies of the pure components il and B at temperature T. AH, = integral heat, of solution for mA = mg = l/2. We assume that the molar free energy of the faulted material, Ff, is given by a similar equation: F = m,lfPJf + mBfFBf + ~~_~f~~fAH~ + RT(mAf In mAf + mBf In mBf)
(10)
where AH, is the same as for the bulk material. With temperat,ure, pressure, and amount of faulted material constant 1 the equilibrium condition for the compositions of bulk and faulted materials is: 3F ...am,
i?Ff = __ . amAf
Combining this relation with (9) and (10) we find: (F,
-
FAf) -- (.FN -
FBf) + 4AH&m, -
mnf)
4AH, (mA --. m.f) + RT In zmf =: 0. (12) rnAtfmD
We now define & = (Fi, -
@,I) -
(Fn -
F,f) and
MA9 FIG.
2. Suzuki hardening for Ag-Au alloys
to an extra shear stress, rs, required to move an extended dislocation. We can calculate this extra shear stress by a method similar to that used for local order. We assume that the extended dislocation is moved a distance 6 by a stress TV,doing work per unit length of dislocation 611= &rJ. This must equal the increase in free energy of the system. One partial dislocation leaves behind a region of composition mRf, but unfaulted; the other enters material of composition m, and converts this into faulted material. Each of these regions is of volume (~~~3~~~ per unit length of dislocation (assuming the faulted region is two layers thick). The corresponding increase in energy is given by: w =
-F
mBf) z! LF(
f(mnf)]
V
2/3
where I’ is the volume per gram-atom. Using equations (10) and (II), we can simplify this: F(mBf) -
Ff(m,f)
= md[Fa -
m,[J',
-
+ Ff(m,) -
As shown by Suzuki, this composition difference leads
F(m,)
FA] -:- mBf[FLI -
F_4fl - m,[F,
-
Ts
=
2
2Q ---Am. 3 C’
J
Combining (13) and (16), we have
P,f]
F,l] = AmQ (15)
so that, equating W to (STY 6, we find:
mBf) = Am
(m, and expand the logarithm to obtain, to the first order in Am: m. m & --I AWL _ A N (13) RX
-
634
ACTA 1.6
I
1 0
-
I 0
0.1
I 0.2
I
0’ 0
0
G&r and
o
Jomison
METALLURGICA,
VOL.
6, 1958
Since the Suzuki effect is of the looking type, that is, the dislocation need move only a short distance to escape the solute interaction, the possibility of thermally activated escape must be considered. Suzuki(l) has dealt with this possibility in some detail and found that the activation energy required for a significant reduction in yield strength is of the order of 1 eV per interatomic distance. At room temperature, thermal activation will be unimportant, since
I
kT w 0.026 eV.
Sachs and Sherill
Theoretic01 Suzuki Hardening
I
I
I
0.3
0.4
0.5
I
0.6
MZn
Fra. 3. Strength of cc-brasssingle crystals
Since even for alloys of quite similar metals, such as silver and gold, AH, is at least as large as RT, the term in brackets is quite important. For systems with continuous solid solubility, AH* must be negative unless it is very small. (A positive AH requires a solubility gap). A negative AH leads to a saturation of the effect at a relatively low solute concentration. In Fig. 2, the magnitude of the effect for Ag-Au alloys as given by Suzuki (ideal solution) is compared with that calculated from Equation (17) (regular solution), using AH, = -1070 cal/mole,(ll) T = 600°K and & = 131 cal/mole as assumed by Suzuki. The effect in C&Au alloys should be similar since AH+ M -1300 cal/mole,c12) and Q is not likely to be very different. For u-brass, we can make a direct estimate of Q, which is unusually large in this case. The energy of a twin boundary in copper as determined by Fullman(14) is about 25 erg/cc, so that we may assume the energy of a stacking fault to be roughly 50 erg/cc. At about 50% Zn, however, the energy of a stacking fault must be approximately zero, since the phase formed by low temperature transformation of B-brass has a mixed f.c.c. and h.c.p. character.(16) Assuming a linear variation of stacking fault energy with composition, we calculate Q to be about 500 cal/mole. The AH+ for u-brass is about - 2300 cal/mole.(13) Using these values, the T, given by equation (17) is plotted along with the experimental data(i6217) in Fig. (3). The agreement is quite satisfactory. It should be noted that the form of equation (17) gives a much better fit than (7), aside from the quantitative aspects of the matter.
One other aspect of the Suzuki effect may be of some importance. Various anomalies have been observed in alloys as a consequence of prolonged annealing at low temperatures, e.g.: resistivity and lattice parameter changes in cc-brass,(ls)and resistivity and hardness changes in Ni3Cr.(lg) These effects have usually been ascribed to change in local or long-range order. In the case of Ni,Cr at least, Roberts and Swalin(20) have shown that no significant amount of order is present. These anomalies may be a consequence of the temperature dependence of the Suzuki effect (equation 13). Normal cooling will result in a segregation characteristic of a moderately high temperature. Prolonged lqw temperature annealing should increase segregation, increasing hardness and changing resistivity. Some effect on lattice parameters may also be expected. SUMMARY
Both the Fisher and Suzuki interactions may make significant contributions to the strength of solid solutions. We may expect the Suzuki effect to be dominant in dilute solutions, especially where the energy of a stacking fault varies rapidly with composition; and the Fisher effect to be dominant in concentrated solutions if any appreciable local order exists. The largest Fisher interaction is probably considerably more important than the largest Suzuki interaction. Since the submission of this manuscript, a discussion of the problem by Suzuki(l) and Seegerc2) has appeared. The contribution of local order to hardening derived by Suzuki (his equation 32) differs from that in this paper (equation 7) bv exactly a factor of 4, when proper conversion of units is made, although his analysis does not appear to differ in principle from that of this author. The origin of the discrepancy is not entirely clear, although one factor of 2 does seem to arise in his approximation for P,,. The result quoted by Seeger in his discussion (his equation l), after appropriate approximations are introduced, becomes identical with equation (7) of this paper.
FLINN:
SOLUTE
HARDENING
OF
(1) Hideki Suzuki, ~~~~t~o~ and ~ec~n~caZ Properties of Crystals p. 361. John Wiley, New York (1957). (2) A. Seeger, Ibid. p. 388.
CLOSE-PACKED 6.
7. 8. 9. 10. 11.
ACKNOWLEDGMENT
The author is grateful to several members of the department for enlightening discussion of this paper. REFERENCES 1. H. SUZUKI, Sci. Rep. Res. Insts To”hoku Univ. Ser A, 4, 455 (1952). 2. J. C. FISHER, Rcto iVet. 2, 9 (1954). 3. R. A. ORIANI, Rcta Met. 1.144 (1953). 4. B. L. AVERBACH, P. A. F&N and M. COREN, R&x Met. 2, 92 (1954). 5. D. T. KEATING, Aeta Met. 2, 885 (1954).
12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
SOLID
SOLUTIONS
635
NORMAX and B. E. WARREN, J. AppE. Phys. 22, 483 (1951). J. M. COWLEY,J. AppZ. P&s. 21, 24 (1950). G. SACHSand J. WEERTS, 2. Phyys. 62, 473 (1930). G. W. ABDLEY, Acta Met. 3, 525 (1955). P. A. FLINN, Phys. Rev. 104, 350 (1956). 0. KUBASCHEWSKIand J. A. CATTERALL,Thermoohemieal Data of Alloys, p. 64. Porgamon Press, London and New York (1956). 0. KUBAS~XEWSKIand J. A. CATTERALL,Ibid. p. 61. 0. KUBASCHEWSKIand J. A. CXIXXRALL,I&d. p. 66. R. L. FULLMAN,J. AppE. Phys. 22,448 (1951). T. B. IM~~~~~~~~ and C. S. BARRETT,Tram. Amer. Inst. Min. (Metall.) Eng7s. 209, 455 (1957). V. GOX,ERand G. SACHS,Z. Phys. 55, 581 (1929). R. E. JAMISONand F. A. SREBILL,Acta Met. 4,197 (1956). A. C. DAMASK,J. AppZ. Phys. 27, 610 (1956). R. NORDHEIMand N. J. GRANT, J. Inst. Met. 82, 440 (1953-54). B. W. ROBERTSand R. A, SW&IN, Trans. Amer. Ilzst. M&a. (M&all.) Engrs. ZSS, 845 (1957). J. M. COWLEY, Phys. Rev. 77, 669 (1950). N.