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Solid solution hardening due to metal interstitials in the Pb-Ag system
SOLID SOLUTION HARDENING DUE TO METAL INTERSTITIALS IN THE Pb-Ag SYSTEM J. >I. GALLIG.L’i
and S. C. HU.kYG
Institute of Materials Science. University of Connecticut. Storrs. CT 06268. U.S.;\
Abstract-Solid solution alloys of PbAg. quenched from various solution temperatures. show a jutstantial hardening. d; dc c 3G. in agreement \\ith the hardening being a consequence of an asymmerncal defect. Furthermore. the hardening increases with increasing quenching temperature according to an .Arrhenius equation and this suggests that the solutes are partitioned between unlike Sites. Based on this result. the formation energy of the asymmetrical interstitial is calculated and is shown to decrease with increasing Ag concentration. These results are discussed. in consonance with a model of solid solutions in which an interstitial is composed of two silver atoms. R&urn&--Les solutions solides de plomb et d’argent. trempees depuis diverses temperatures. prisentsnt un durcissement d: dc I 3G notable. ce qui pourrait Ptre la consiquence d’un dCfaut as>mCtrique. De plus. Ie durcissement augmente lorsqu’on augmente la tempkrature de trempe selon I’kquation d’Arrh&ius. ce qui l&se penser que les constituants sont ripartis sur des sites diffirents. Grlce i ce rCsultat. on calcule I’inergie de formation d’un interstitiel asymitrique et on montre qu’elle dicroit lorsqu’on auzamente la concentration en argent. On discure ces r&ultats h la lumiere d’un mod2le de solution solids dans lequel un interstiticl ejt composi de deux atomes d’argent. Zusammenfassung-hIischkristall-Legierungen aus Pb-Ag, die van mehreren Lijsucgs-temperaturen abgeschreckt worden sind, zeigen eine betrichtliche Verfestigung. dr,dc z 3 G. in Ubereinstimmung mit der Verfestigung als Folge eines asymmetrixhen Defektes. Die Verfestigung steigt mit zunehmender Abschrecktemperatur entsprechend einer ArrheniusGleichung an. Das legt nahe. da!3 die gelasten Atome auf ungleiche PI&e aufgeteilt sind. Auseehend van diesem Ergebnis wird die Bildungsenergie des asymmetrischen Zwischengitterdefekts berechnsr; es wird gezeigt. da0 sie mit ansteigender .Ag-Konzentration abnimmt. Diese Ergebnisse werden im Einklang mit einem Modell. in dem ein Zwischengitterdefekt aus zwei Silberatomen besteht. diskutiert
where Di, and Xi, are constants, Tis the temperature and R the gas constant. Accordingly. the observed diffusion coefficient includes the energetics of the complex process as a whole-the interstitial formation energy and the interstitial diffusion energy. As such, measurements of diffusion coefficients in a leadnoble metal system are not capable of distinguishing between the formation of interstitials and the sub sequent migration of the interstitials. In order to study, in a distinct way. the partition of solutes between substitutional and interstitial sites, a measurement of some property which is sensitive to the solutes in one or the other types of sites must be made. Specifically. if the partition of solutes between the two types of sites occurs is as outlined above, then it is possible to rapidly quench a crystal from a high temperature and freeze in this partition. Such a quenched crystal would then have a partition of solutes between two types of sites and a property of a crystal which is very sensitive to such a partition of solutes is the yield stress of the crystal [3.-t]. This sensitivity applies, for example, if the solutes are partitioned between two types of sites. one of which is “symmetrical” and one of which is “asymmetrical”;
WTRODUCTION Noble metal solutes in lead crystals exhibit characteristically rapid solute diffusion rates, which are many orders of magnitude faster than the self-diffusion rates of the solvent atoms [l]. This rapid diffusion of the solutes has been interpreted as solutes, in part, occupying interstitial sites in the lattice and the solutes jumping between interstitial sites [2]. Such a partition of the solutes between interstitial sites and substitutional sites readily accounts for the rapid diffusion of the solute, since a solute, once in an interstitial site, needs not wait for a vacancy to diffuse around the crystal. As a result of this dissociative mechanism, the measured diffusion coeficient, D. can be related to the fraction of solutes in interstitial sites, X,, and the diffusion constant of solutes in interstitial sites. Di,
[?I
D = XiDi.
where we neglect the effect of a slower diffusing solute in a substitutional site. Furthermore, Di involves an activation energy of interstitial diffusion, Ef, just as Xi includes the energy of formation of an interstitial. E/. so that 3-i
533
G.ALLIG.AN
ASiD
HL’ANG:
SOLID SOLUTIOK H.1RDENING
(Cl
Fig. 1. Three types of soiute configuration in a face-centered cubic crystal (a) symmetrical substitutional. (b) symmetric cubic interstitial and (c) asymmetric split-interstitial. solutes in substitutional and certain types of interstitial sites being symmetrical while solutes in certain interstitial sites being asymmetrical, Fig. 1. As shown by Fleischer [3], only an asymmetrical defect has a very large effect on the yield stress of a crystal: an asymmetrical defect being about 50 times more effective than a symmetrical defect. More specifically in the present case if we measure the change in mechanical properties of, say, a lead-silver crystal, which is quenched from different temperatures, then these changes in mechanical properties are related to solutes which are partitioned between symmetrical and asymmetrical sites. In this way the sensitivity of the mechanical properties to solutes in specific types of position in a lattice allows a unique assessment of the partition of solutes between different types of sites. We note in passing that the sample must be quenched to a low temperature, aell below room temperature, or the high temperature distribution may not be preserved [j]. The present experiments, then, are concerned with measuring the influence of the partition of silver solutes between symmetrical sites and asymmetrical sites on the mechanical properties of lead-silver ~rystals. In addition to the great technical interest in solid solution hardening of metal interstitiaIs themselves, such an experiment provides the first approach to the study of the interstitial formation process. especially the temperature dependence of the formation. EXI’ERIME%TAI.
tation. were grown by a modified Bridgemen technique; the orientation of the tensile axis as referred to a standard stereographic triangle is illustrated in Fig. 1. AI1 crystals were homogenized at 190°C for about 1 day to reduce solute segregation and provide a reference state for further experiments. The crystals then had gage sections milled in them with a laboratory scale chemical milling machine and they were then returned to the furnace for an additional twelve hours. now at the individua1 temperature from which each crystal was to be quenched. The crystals were then quenched from various temperatures into a bath and immediately of acetone held at -9j’C requenched into liquid nitrogen. This procedure has been previously shown to be successful in freezing silver atoms in solid solution without any precipitation[j.6]. The quenched samples were mounted into suitable grips for subsequent tensile testing while the specimen was held at liquid nitrogen temperature and transferred to a liquid nitrogen bath for testing, where all mechanical property measurements were carried out. After each tensile test the chemical composition of each specimen was ascertained by atomic absorption spectroscopy. In this method the limit of ~nsitivity of the technique is in the range of 5 ppm for silver, so that the composition determinations had an error limit, which ranged from It: 5:; for the Pb-100 ppm Ag crystals to about F 1% for the Pb-5OOppm Ag crystals. The mechanical property data are accordingly, related not to the nominal composition of the crystals but to the measured composition in each case.
PROCEDURE
The starting materials used in the present experiment were 99.9999 wt.*, pure kud and YY).YYY”, pure silver. Master alloys of these materials were thoroughly mixed in the liquid state and used in preparation of individual lots of single crystals. The silver contents of the alloys are 200-4GOppm (wt.), which represent a reasonable compromise between a large temperature range of solid solution and a high strengthening effect. Crystals, all of the same orien-
Fig. 2. The orientation of the tensile axis, as referred to a standard stereographic triangle, of ail of the specimen tested in the present experiment. Also shown are typical stress-strain curves of Pb-Ag crystals tested at 77 K: (1) 260 ppm (wt.) Ag, quenched from 560 K. (2) 210 ppm (wt.) Ag. quenched from 560 K and (3) 210 ppm (wt.) Ag. quench from 526 K.
Gr\LLIGAN
RESCLTS XXI
Ht’A4G:
.iSD
SOLID SOLL-TIOX 2.0
DISCCSSION
The yield stress. r. of quenched crl;staIs varies for a given solute concentration. c. directly with the quenching temperature. Fig. 3. Thus. as the quenching temperature is increased the yieid stress of these quenched crystals increases. This result bears directly on the question of the partition of solutes between various types of sites, since any interchange of solutes between like sites does not affect the mechanical properties of the solid. Accordingly-. the solutes must be partitioned between unlike sites. Furthermore. those solutes in one type of site must exhibit a larger strengthening effect than those in the other type of site and the former must increase in atom fraction with increasing quenching temperatures. The observed dependence of the yieid stress of quenched crystals upon quenching temperatures is. therefore. consistent with a partition of solutes between substitutional sites and interstitial sites. LVe rule out the possibihty that the observed dependence of the yield stress on quenching temperature is related to clustering or precipitation of the solutes on the basis that neither of those processes should depend on the solid solution temperature the crystals are quenched from. Indeed. if clustering is involved then the crystals should be softer when they are quenched from higher temperatures. while we expect that a precipitation reaction would be. roughly. independent of the quenching temperature. Quite clearly. then. the TEMPERATURE 32s
27s
300
(‘C
250
1 200
225
‘
1
-’
1
I.(1
.
’
1.8 lOOO.‘T
*
’
1.9
’
’ 2.0
4
559
H.ARDENlSG
’
2.1
(*K“l
Fig. 3. Yield stress as a function of quenching temperature. for lead crwtals of various concentrations of Ag. The solubility limit is indicated by vertical lines for each composition.
0
I
8
, 100
8
200
I
300
3
b
400
r 5’
CONCEN~ATl~(Wt.p~)
Fig. 1. Yield stress vs Ag concentration. for samples quenched irom 300-C. Filled circles represent data taken from curves in Fig. 3. lvhile open circles arc data from individual tests. observed mechanical properties are unambiguously related to a solid solution hardening effect. In addition WF:expect on the basis of Fleischer’s calculation 131 that the solute which is important in changing the mechanical properties of quenched Pb-Ag crystals must reside in an asymmetrical site. since the change in yield stress per solute atom is so pronounced. Fig. 4.
Another variable which can be used to investigate the distribution of silver solutes in lead crystals involves the influence of solute concentration on the change in mechanical properties of such crystals, Fig. 4. This result can be compared to the observed hardening of other lead based crystals. such as that observed in lead-tin (61 cry-stals and lead-indium crystals [7], Fig. 5. Again. as shown in this figure, there is an extraordinary increase in the yield stress for the lead-silver system when compared to more conventional solid solution alloys which do not exhibit abnormally fast diffusion coefficients [S]. In addition the enormous solid solution hardening in the PbAg system can be appreciated by comparison with the M-C system in which the interstitial carbon occupies a symmetrical site in the f.c.c. lattice. A calculation using the present results shows that dridc c 3G. whereas measurements of Xi-C system shown a weak solid solution hardening with dr ‘dc 1 G/IO [9]. where G is the shear modulus. Furthermore. u’e note that the hardening associated with a silver solute of varying composition. now all referred to a common quenching temperature.
560
GALLIGAN
HUANG:
AND
”
_I
I .
/
HARDE?Z’iG
site. according to the equation
PRESENT DATA
‘
.
ln(As) = B -
0 PI-AS
lP
3
SOLID SOLUTION
Pb..‘”
A Pb-
sn
where B is a combination of constants. The energy involved in the formation of the pertinent interstitials is, therefore. equal to the product of 2R and the slope of a curve of the measured &Ar) vs 1 ?: The pertinent energy of formation. calculated in this way, is shown fo decrease with increasing solute con~tration. Fig. 6. so that E(c) = E, - EC,
0
0.1
0.2
0.4
0.3
C~CENTRATJO~
as
(at.%)
Fig. 5. A comparison of the present result with some previous results in Pb-In [7]. Pb-Sn [6] and Pb-Ag at higher concentrations [6]. All of the previous data were taken at 5.2 K, whereas the present data are at 77 K. All the results of Pb-Ag system in both of the experiments were referred to a quenching temperature of 29O’C.
depends on the solute concentration ner. Fig. 1.
in a linear man-
A.r = Bc(Ag), where Ar is the increase in the yield stress of pure Lead due to the presence of silver solute of concentration 4Ag) and fl is a constant. We expect. however, for a random array of strong obstacles that the change in yield stress should vary as [IO] Ar = rGbd I ” 9 where Y is a geometricaf factor, b is the Burgers vector of the dislocation cutting through the obstacIes of concentration cP This result can be readily explained in a manner which is consistent with the idea of an asymmetrical defect, by assuming that the dominant asymmetrical defect involves the product of the silver concentration. In that case the asymmetrical defect would consist of two silver solutes such that
(1)
where E, and E are constants. In contrast to this, we expect if the observed hardening involves a simple solid solution hardening then the most elementary calculation shows that the energy of solution should vary parabolically CL11. E(c) = E,c( 1 - c),
(2)
where E(c) is the energy of solution of the solute of concentration c and E, is a proportionality constant. This resuit can be explained by again invoking the idea that the pertinent defect is a split interstitial which forms according to the following sequence: First put the silver into substitutional solution. after which the silver solutes are partitioned into cubic interstitial sites according to the reaction A~(substitutiona1) -%Ag(cubic interstiti~) + vacancy. The silver interstitial being quite mobile wanders around to find a silver substitutional. forming a splitinterstitial according to the reaction Ag(cubic interstitial) + Ag(subs~tutiona1) ~A~(sptit-interstitiaI} The equilibrium
constant for the complete reaction
251
ci = const. fags, so that Ar = const. xGbc(Ag) as observed. A defect which is asymmetrical and involves two silver solutes is a split interstitial which, as shown below, is consistent with the measured dependence of Ahton quenching temperature. (iii) Formatior~ energy
of inrerstitials
As outlined above the observed change in yield stress in lead-sifver crystals is dependent upon the tem~rature the crystal is quenched from. Indeed, when ln(Ar) is plotted against l/T we can caicuiate the formation energy of the solute in the appropriate
0 OONCENTffATiON
f Wt.sgarf
Fig. 6. Formation energy of Ag spiit-interstitial vs Ag concentration in Pb crystals.
G.U_LIGAN to form a split-interstitial
XUD
HUAXG:
SOLID SOLUTIOY
K = Aexp[-(El;
CONCLCSIO~s
wilt be. in a reduced form.
where c,. csi and c, are the concentration of vacancy. spat-jnterstitia~ and substitutionai. respectively. The value of K can then be written as + El - IE{)iRT].
where .-I is a constant. 152 and E{ are formation energy of the solute in a split interstitial site and in a substitutional site. respectively. and Ei is formation energy of a vacancy. For the case where the energy of formation of the solid solution is determined by the elastic energy of adding the solute to the solution we know that. equation (2). E!’ = E,c(l - c) or for the small concentrations
we are dealing with
so that K is now of the form K = ,-l exp{ - [(E$ + EL) - &c]/R?J,
In summary. then. vve conclude on the btis of the experimental results that: (a) Relatively low concentrations of silver in lead impart a pronounced hardening to lead compared to other solid soiutions such as lead-tin or lead-indium. (b) The yield stress of lead-silver crystals of a given concentration. measured at liquid nitrogen temperature, varies with the temperature from which the crystal is quenched. This rules out the possibility that the solutes are all in interstitial positions or ah in subst~tLltiona1 positions. (cl The yield stress of quenched cr>stals. al1 quenched from the same temperature. varies vvith the first power of the concentration. (d) From the measured yield stress in quenched crystals, as a function of quenching temperature. it is shown that the energy of formation of the pertinent defect decreases with increasing c~ncent~~ion of silver solutes. (e) A11 of these results are consistent
EL = E,c
(3)
which is consistent with the observations. According to equation (3). the energy extrapolated to zero solute concentration is equal to (Es + E$). Taking E{ to be - 11 kcal:mole [12], E,/, wilI be - 16 kcal,‘mole. Given the uncertainties involved in this experiment, we take the value to be a qualitative estimate. Another important implication of the present results, Fig. 6 and equation (3) is that a high solute concentration, as well as a high temperature, favors the formation of the split interstitials. (iv) On the nature of the Defect Certainly the present quenching study is not a direct proof for the presence of the split-interstitial solutes in the Pb-Ag system. Nonetheless, an assessment of the data based on this type of solute can explain all of the present observations, as discussed above. This conclusion, furthermore, is consistent with the observations of other experiments such as the Mossbauer effect [ 131. internal friction measurements [14] and splat quenching [lj]. On the other hand, an interstitial in a cubic site [Z] should not produce such pronounced hardening as is observed, because of the symmetrical nature of the strain field associated with it [3]. Furthermore. a free interstitial, as well as a vacancy-interstitial pair [16], involves only one solute atom and its presence cannot explain the linear concentration dependence of yield stress, nor can it explain the concentration dependence of the energy of solution of the pertinent defect involved in the present quenching experiments.
561
H.4RDENING
with a model
in which the silver solutes occupy partially- asymmetrical positions, such as a split interstitial. was ,-tcitnorvledgmtenr-This work U.S.E.R.D..+. contract E(11-1) 2305.
supporied
by
. REFEREBCES 1. T. R. Anthony. in ~cutxies and ~tIre~sririu~s in ?iferais (edited by A. Seeger). Wiley, Nrw York (1970). 2. F. C. Frank and D. Turnbull, P&U. Rer. 104, 617 (1956). 3. R. L. Fleischer, in The Strengtheni,lg of Mer& (edited by D. Peckner). Reinhold, New York (196-U_ 4. I. Friedel. Dislocations. p. 382. Pergamon Pra Oxford (1963). 5. K. Tsumuraya. Y. Ohno and S. Xishikawa. Mat. Sci. Engng 17, 169 (1975). 6. J. H. Tregilgas and J. M. Galligan, Scriprir Mrr. 9. 1225 (1975). 7. C. S. Pang and J. hl. Galligan, unpublished data. 5. B. Okkerse, &ta Met. 2, 551 (1951). 9. P. A. Rinn. in Strengthening Mechanisms in Solids. ASM seminar. Cleveland (1962). IO. R. Labusch, Actu Mer. 20, 917 (197’). Il. J. D. Eshelby. in Solid Stare Physics (edited by F. Seitz and D. Turnbull). Vol. 3. Academic Press. liew York (1946).
1’. R. Feder and A. S. Nowick, Phil. May. 15. SOS(1967). 13. C. F. Steen. D. G. Howard and R. H. Nussbaum. So/id State Comm. 9. 864 (1971). 11. T. J. Turner and C. H. Nielsen. BuU. Am phus. Sot. 18, 34’ (1973). 15. R. Ray. S. H. Hahn and B. G. Giessen. .-tcra Met. 20. t335 (197’). 16. J. W. Miller. Phyr. Rev. (B) 8. 241i (1973).
and S. C. HU.kYG
Institute of Materials Science. University of Connecticut. Storrs. CT 06268. U.S.;\
Abstract-Solid solution alloys of PbAg. quenched from various solution temperatures. show a jutstantial hardening. d; dc c 3G. in agreement \\ith the hardening being a consequence of an asymmerncal defect. Furthermore. the hardening increases with increasing quenching temperature according to an .Arrhenius equation and this suggests that the solutes are partitioned between unlike Sites. Based on this result. the formation energy of the asymmetrical interstitial is calculated and is shown to decrease with increasing Ag concentration. These results are discussed. in consonance with a model of solid solutions in which an interstitial is composed of two silver atoms. R&urn&--Les solutions solides de plomb et d’argent. trempees depuis diverses temperatures. prisentsnt un durcissement d: dc I 3G notable. ce qui pourrait Ptre la consiquence d’un dCfaut as>mCtrique. De plus. Ie durcissement augmente lorsqu’on augmente la tempkrature de trempe selon I’kquation d’Arrh&ius. ce qui l&se penser que les constituants sont ripartis sur des sites diffirents. Grlce i ce rCsultat. on calcule I’inergie de formation d’un interstitiel asymitrique et on montre qu’elle dicroit lorsqu’on auzamente la concentration en argent. On discure ces r&ultats h la lumiere d’un mod2le de solution solids dans lequel un interstiticl ejt composi de deux atomes d’argent. Zusammenfassung-hIischkristall-Legierungen aus Pb-Ag, die van mehreren Lijsucgs-temperaturen abgeschreckt worden sind, zeigen eine betrichtliche Verfestigung. dr,dc z 3 G. in Ubereinstimmung mit der Verfestigung als Folge eines asymmetrixhen Defektes. Die Verfestigung steigt mit zunehmender Abschrecktemperatur entsprechend einer ArrheniusGleichung an. Das legt nahe. da!3 die gelasten Atome auf ungleiche PI&e aufgeteilt sind. Auseehend van diesem Ergebnis wird die Bildungsenergie des asymmetrischen Zwischengitterdefekts berechnsr; es wird gezeigt. da0 sie mit ansteigender .Ag-Konzentration abnimmt. Diese Ergebnisse werden im Einklang mit einem Modell. in dem ein Zwischengitterdefekt aus zwei Silberatomen besteht. diskutiert
where Di, and Xi, are constants, Tis the temperature and R the gas constant. Accordingly. the observed diffusion coefficient includes the energetics of the complex process as a whole-the interstitial formation energy and the interstitial diffusion energy. As such, measurements of diffusion coefficients in a leadnoble metal system are not capable of distinguishing between the formation of interstitials and the sub sequent migration of the interstitials. In order to study, in a distinct way. the partition of solutes between substitutional and interstitial sites, a measurement of some property which is sensitive to the solutes in one or the other types of sites must be made. Specifically. if the partition of solutes between the two types of sites occurs is as outlined above, then it is possible to rapidly quench a crystal from a high temperature and freeze in this partition. Such a quenched crystal would then have a partition of solutes between two types of sites and a property of a crystal which is very sensitive to such a partition of solutes is the yield stress of the crystal [3.-t]. This sensitivity applies, for example, if the solutes are partitioned between two types of sites. one of which is “symmetrical” and one of which is “asymmetrical”;
WTRODUCTION Noble metal solutes in lead crystals exhibit characteristically rapid solute diffusion rates, which are many orders of magnitude faster than the self-diffusion rates of the solvent atoms [l]. This rapid diffusion of the solutes has been interpreted as solutes, in part, occupying interstitial sites in the lattice and the solutes jumping between interstitial sites [2]. Such a partition of the solutes between interstitial sites and substitutional sites readily accounts for the rapid diffusion of the solute, since a solute, once in an interstitial site, needs not wait for a vacancy to diffuse around the crystal. As a result of this dissociative mechanism, the measured diffusion coeficient, D. can be related to the fraction of solutes in interstitial sites, X,, and the diffusion constant of solutes in interstitial sites. Di,
[?I
D = XiDi.
where we neglect the effect of a slower diffusing solute in a substitutional site. Furthermore, Di involves an activation energy of interstitial diffusion, Ef, just as Xi includes the energy of formation of an interstitial. E/. so that 3-i
533
G.ALLIG.AN
ASiD
HL’ANG:
SOLID SOLUTIOK H.1RDENING
(Cl
Fig. 1. Three types of soiute configuration in a face-centered cubic crystal (a) symmetrical substitutional. (b) symmetric cubic interstitial and (c) asymmetric split-interstitial. solutes in substitutional and certain types of interstitial sites being symmetrical while solutes in certain interstitial sites being asymmetrical, Fig. 1. As shown by Fleischer [3], only an asymmetrical defect has a very large effect on the yield stress of a crystal: an asymmetrical defect being about 50 times more effective than a symmetrical defect. More specifically in the present case if we measure the change in mechanical properties of, say, a lead-silver crystal, which is quenched from different temperatures, then these changes in mechanical properties are related to solutes which are partitioned between symmetrical and asymmetrical sites. In this way the sensitivity of the mechanical properties to solutes in specific types of position in a lattice allows a unique assessment of the partition of solutes between different types of sites. We note in passing that the sample must be quenched to a low temperature, aell below room temperature, or the high temperature distribution may not be preserved [j]. The present experiments, then, are concerned with measuring the influence of the partition of silver solutes between symmetrical sites and asymmetrical sites on the mechanical properties of lead-silver ~rystals. In addition to the great technical interest in solid solution hardening of metal interstitiaIs themselves, such an experiment provides the first approach to the study of the interstitial formation process. especially the temperature dependence of the formation. EXI’ERIME%TAI.
tation. were grown by a modified Bridgemen technique; the orientation of the tensile axis as referred to a standard stereographic triangle is illustrated in Fig. 1. AI1 crystals were homogenized at 190°C for about 1 day to reduce solute segregation and provide a reference state for further experiments. The crystals then had gage sections milled in them with a laboratory scale chemical milling machine and they were then returned to the furnace for an additional twelve hours. now at the individua1 temperature from which each crystal was to be quenched. The crystals were then quenched from various temperatures into a bath and immediately of acetone held at -9j’C requenched into liquid nitrogen. This procedure has been previously shown to be successful in freezing silver atoms in solid solution without any precipitation[j.6]. The quenched samples were mounted into suitable grips for subsequent tensile testing while the specimen was held at liquid nitrogen temperature and transferred to a liquid nitrogen bath for testing, where all mechanical property measurements were carried out. After each tensile test the chemical composition of each specimen was ascertained by atomic absorption spectroscopy. In this method the limit of ~nsitivity of the technique is in the range of 5 ppm for silver, so that the composition determinations had an error limit, which ranged from It: 5:; for the Pb-100 ppm Ag crystals to about F 1% for the Pb-5OOppm Ag crystals. The mechanical property data are accordingly, related not to the nominal composition of the crystals but to the measured composition in each case.
PROCEDURE
The starting materials used in the present experiment were 99.9999 wt.*, pure kud and YY).YYY”, pure silver. Master alloys of these materials were thoroughly mixed in the liquid state and used in preparation of individual lots of single crystals. The silver contents of the alloys are 200-4GOppm (wt.), which represent a reasonable compromise between a large temperature range of solid solution and a high strengthening effect. Crystals, all of the same orien-
Fig. 2. The orientation of the tensile axis, as referred to a standard stereographic triangle, of ail of the specimen tested in the present experiment. Also shown are typical stress-strain curves of Pb-Ag crystals tested at 77 K: (1) 260 ppm (wt.) Ag, quenched from 560 K. (2) 210 ppm (wt.) Ag. quenched from 560 K and (3) 210 ppm (wt.) Ag. quench from 526 K.
Gr\LLIGAN
RESCLTS XXI
Ht’A4G:
.iSD
SOLID SOLL-TIOX 2.0
DISCCSSION
The yield stress. r. of quenched crl;staIs varies for a given solute concentration. c. directly with the quenching temperature. Fig. 3. Thus. as the quenching temperature is increased the yieid stress of these quenched crystals increases. This result bears directly on the question of the partition of solutes between various types of sites, since any interchange of solutes between like sites does not affect the mechanical properties of the solid. Accordingly-. the solutes must be partitioned between unlike sites. Furthermore. those solutes in one type of site must exhibit a larger strengthening effect than those in the other type of site and the former must increase in atom fraction with increasing quenching temperatures. The observed dependence of the yieid stress of quenched crystals upon quenching temperatures is. therefore. consistent with a partition of solutes between substitutional sites and interstitial sites. LVe rule out the possibihty that the observed dependence of the yield stress on quenching temperature is related to clustering or precipitation of the solutes on the basis that neither of those processes should depend on the solid solution temperature the crystals are quenched from. Indeed. if clustering is involved then the crystals should be softer when they are quenched from higher temperatures. while we expect that a precipitation reaction would be. roughly. independent of the quenching temperature. Quite clearly. then. the TEMPERATURE 32s
27s
300
(‘C
250
1 200
225
‘
1
-’
1
I.(1
.
’
1.8 lOOO.‘T
*
’
1.9
’
’ 2.0
4
559
H.ARDENlSG
’
2.1
(*K“l
Fig. 3. Yield stress as a function of quenching temperature. for lead crwtals of various concentrations of Ag. The solubility limit is indicated by vertical lines for each composition.
0
I
8
, 100
8
200
I
300
3
b
400
r 5’
CONCEN~ATl~(Wt.p~)
Fig. 1. Yield stress vs Ag concentration. for samples quenched irom 300-C. Filled circles represent data taken from curves in Fig. 3. lvhile open circles arc data from individual tests. observed mechanical properties are unambiguously related to a solid solution hardening effect. In addition WF:expect on the basis of Fleischer’s calculation 131 that the solute which is important in changing the mechanical properties of quenched Pb-Ag crystals must reside in an asymmetrical site. since the change in yield stress per solute atom is so pronounced. Fig. 4.
Another variable which can be used to investigate the distribution of silver solutes in lead crystals involves the influence of solute concentration on the change in mechanical properties of such crystals, Fig. 4. This result can be compared to the observed hardening of other lead based crystals. such as that observed in lead-tin (61 cry-stals and lead-indium crystals [7], Fig. 5. Again. as shown in this figure, there is an extraordinary increase in the yield stress for the lead-silver system when compared to more conventional solid solution alloys which do not exhibit abnormally fast diffusion coefficients [S]. In addition the enormous solid solution hardening in the PbAg system can be appreciated by comparison with the M-C system in which the interstitial carbon occupies a symmetrical site in the f.c.c. lattice. A calculation using the present results shows that dridc c 3G. whereas measurements of Xi-C system shown a weak solid solution hardening with dr ‘dc 1 G/IO [9]. where G is the shear modulus. Furthermore. u’e note that the hardening associated with a silver solute of varying composition. now all referred to a common quenching temperature.
560
GALLIGAN
HUANG:
AND
”
_I
I .
/
HARDE?Z’iG
site. according to the equation
PRESENT DATA
‘
.
ln(As) = B -
0 PI-AS
lP
3
SOLID SOLUTION
Pb..‘”
A Pb-
sn
where B is a combination of constants. The energy involved in the formation of the pertinent interstitials is, therefore. equal to the product of 2R and the slope of a curve of the measured &Ar) vs 1 ?: The pertinent energy of formation. calculated in this way, is shown fo decrease with increasing solute con~tration. Fig. 6. so that E(c) = E, - EC,
0
0.1
0.2
0.4
0.3
C~CENTRATJO~
as
(at.%)
Fig. 5. A comparison of the present result with some previous results in Pb-In [7]. Pb-Sn [6] and Pb-Ag at higher concentrations [6]. All of the previous data were taken at 5.2 K, whereas the present data are at 77 K. All the results of Pb-Ag system in both of the experiments were referred to a quenching temperature of 29O’C.
depends on the solute concentration ner. Fig. 1.
in a linear man-
A.r = Bc(Ag), where Ar is the increase in the yield stress of pure Lead due to the presence of silver solute of concentration 4Ag) and fl is a constant. We expect. however, for a random array of strong obstacles that the change in yield stress should vary as [IO] Ar = rGbd I ” 9 where Y is a geometricaf factor, b is the Burgers vector of the dislocation cutting through the obstacIes of concentration cP This result can be readily explained in a manner which is consistent with the idea of an asymmetrical defect, by assuming that the dominant asymmetrical defect involves the product of the silver concentration. In that case the asymmetrical defect would consist of two silver solutes such that
(1)
where E, and E are constants. In contrast to this, we expect if the observed hardening involves a simple solid solution hardening then the most elementary calculation shows that the energy of solution should vary parabolically CL11. E(c) = E,c( 1 - c),
(2)
where E(c) is the energy of solution of the solute of concentration c and E, is a proportionality constant. This resuit can be explained by again invoking the idea that the pertinent defect is a split interstitial which forms according to the following sequence: First put the silver into substitutional solution. after which the silver solutes are partitioned into cubic interstitial sites according to the reaction A~(substitutiona1) -%Ag(cubic interstiti~) + vacancy. The silver interstitial being quite mobile wanders around to find a silver substitutional. forming a splitinterstitial according to the reaction Ag(cubic interstitial) + Ag(subs~tutiona1) ~A~(sptit-interstitiaI} The equilibrium
constant for the complete reaction
251
ci = const. fags, so that Ar = const. xGbc(Ag) as observed. A defect which is asymmetrical and involves two silver solutes is a split interstitial which, as shown below, is consistent with the measured dependence of Ahton quenching temperature. (iii) Formatior~ energy
of inrerstitials
As outlined above the observed change in yield stress in lead-sifver crystals is dependent upon the tem~rature the crystal is quenched from. Indeed, when ln(Ar) is plotted against l/T we can caicuiate the formation energy of the solute in the appropriate
0 OONCENTffATiON
f Wt.sgarf
Fig. 6. Formation energy of Ag spiit-interstitial vs Ag concentration in Pb crystals.
G.U_LIGAN to form a split-interstitial
XUD
HUAXG:
SOLID SOLUTIOY
K = Aexp[-(El;
CONCLCSIO~s
wilt be. in a reduced form.
where c,. csi and c, are the concentration of vacancy. spat-jnterstitia~ and substitutionai. respectively. The value of K can then be written as + El - IE{)iRT].
where .-I is a constant. 152 and E{ are formation energy of the solute in a split interstitial site and in a substitutional site. respectively. and Ei is formation energy of a vacancy. For the case where the energy of formation of the solid solution is determined by the elastic energy of adding the solute to the solution we know that. equation (2). E!’ = E,c(l - c) or for the small concentrations
we are dealing with
so that K is now of the form K = ,-l exp{ - [(E$ + EL) - &c]/R?J,
In summary. then. vve conclude on the btis of the experimental results that: (a) Relatively low concentrations of silver in lead impart a pronounced hardening to lead compared to other solid soiutions such as lead-tin or lead-indium. (b) The yield stress of lead-silver crystals of a given concentration. measured at liquid nitrogen temperature, varies with the temperature from which the crystal is quenched. This rules out the possibility that the solutes are all in interstitial positions or ah in subst~tLltiona1 positions. (cl The yield stress of quenched cr>stals. al1 quenched from the same temperature. varies vvith the first power of the concentration. (d) From the measured yield stress in quenched crystals, as a function of quenching temperature. it is shown that the energy of formation of the pertinent defect decreases with increasing c~ncent~~ion of silver solutes. (e) A11 of these results are consistent
EL = E,c
(3)
which is consistent with the observations. According to equation (3). the energy extrapolated to zero solute concentration is equal to (Es + E$). Taking E{ to be - 11 kcal:mole [12], E,/, wilI be - 16 kcal,‘mole. Given the uncertainties involved in this experiment, we take the value to be a qualitative estimate. Another important implication of the present results, Fig. 6 and equation (3) is that a high solute concentration, as well as a high temperature, favors the formation of the split interstitials. (iv) On the nature of the Defect Certainly the present quenching study is not a direct proof for the presence of the split-interstitial solutes in the Pb-Ag system. Nonetheless, an assessment of the data based on this type of solute can explain all of the present observations, as discussed above. This conclusion, furthermore, is consistent with the observations of other experiments such as the Mossbauer effect [ 131. internal friction measurements [14] and splat quenching [lj]. On the other hand, an interstitial in a cubic site [Z] should not produce such pronounced hardening as is observed, because of the symmetrical nature of the strain field associated with it [3]. Furthermore. a free interstitial, as well as a vacancy-interstitial pair [16], involves only one solute atom and its presence cannot explain the linear concentration dependence of yield stress, nor can it explain the concentration dependence of the energy of solution of the pertinent defect involved in the present quenching experiments.
561
H.4RDENING
with a model
in which the silver solutes occupy partially- asymmetrical positions, such as a split interstitial. was ,-tcitnorvledgmtenr-This work U.S.E.R.D..+. contract E(11-1) 2305.
supporied
by
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1’. R. Feder and A. S. Nowick, Phil. May. 15. SOS(1967). 13. C. F. Steen. D. G. Howard and R. H. Nussbaum. So/id State Comm. 9. 864 (1971). 11. T. J. Turner and C. H. Nielsen. BuU. Am phus. Sot. 18, 34’ (1973). 15. R. Ray. S. H. Hahn and B. G. Giessen. .-tcra Met. 20. t335 (197’). 16. J. W. Miller. Phyr. Rev. (B) 8. 241i (1973).