Thermally activated deformation in tantalum-base solid solutions

Journal

of the Less-Common

Metals,

THERMALLY ACTIVATED SOLID SOLUTIONS

L. A. GYPEN Departement (Received

86 (1982)

219

219

- 240

DEFORMATION

IN TANTALUM-BASE

and A. DERUYTTERE Metaalkunde,

March

Katholieke

Universiteit

Leuven,

B-3030

Heverlee

(Belgium)

15, 1982)

Summary

The thermally activated deformation behaviour of (i) binary Ta-Nb, Ta-W, Ta-Re, Ta-Mo and Ta-Hf alloys and (ii) ternary Ta-W-Nb, TaW-MO, Ta-W-Re and Ta-W-Hf alloys was investigated in the temperature range 77 - 573 K by means of tensile tests, compression tests, stress relaxation tests, stress dip tests and strain rate changes. In contrast with other published results, the internal stress in concentrated substitutional tantalum-base alloys remains constant with decreasing temperature. For all alloys investigated the activation volume u* and the effective stress CJ”are strain independent. The extrapolated effective stress a* of tantalum at 0 K is to a first approximation not influenced by the substitutional alloying elements niobium, tungsten, rhenium and molybdenum. In the temperature region 77 - 373 K a single thermally activated process dominates plastic deformation in concentrated substitutional tantalum-base alloys with hardnesses higher than 150 HV. The preexponential factor to of the rate equation is constant and is almost independent of the exact alloy composition. In addition, the power law model based on the Johnston-Gilman equation is valid for these hard alloys. The product m*T where m* is the dislocation velocity exponent is temperature independent and increases with increasing alloy content. The observed solid solution hardening both in the plateau region and below is explained in terms of interactions between solute atoms with a size misfit and polarity-reversing kinks and constrictions in (111) screw dislocations.

1. Introduction When recrystallized tantalum is tested at the ASTM standard strain rate, about half of its yield stress at room temperature is thermally activated [l] . Therefore in order to study the influence of alloying elements on the total flow stress u it is necessary to investigate the influence of these elements on both the athermal component ufi and the thermal component CJ*of the flow stress. The athermal substitutional solid solution hardening (SSH) in 0022-5088/82/0000-0000/$02.75

0 Elsevier

Sequoia/Printed

in The Netherlands

220

tantalum has recently been described [Z] . The influence of the substitutional alloying elements niobium, molybdenum, tungsten, hafnium and rhenium on the thermally activated deformation behaviour of polycrystalline tantalum in the temperature range 77 - 573 K ((0.02 - 0.2)2’,) is dealt with specifically in the present paper. Theories of thermally activated deformation are usually expressed in terms of the resolved shear stress r and the shear strain y. The following relations hold for polycrystalline materials:

CT=M7

y=Mc

and

(1)

where (Tis the normal true stress, E is the normal true strain and M is the Taylor factor [ 31. If the activation entropy can be neglected, the basic rate equation for thermally activated deformation is an Arrhenius equation of the form [4, 51 E =B,exp/-MkT*)/

(2)

where T is the absolute temperature and h is the Boltzmann constant. The activation enthalpy AH* is a function of the effective stress u* only. If the pre-exponential factor P,, is constant, the activation volume u* is given by c51

and the activation

enthalpy

AH” by [5] (4)

The thermally activated deformation behaviour of b.c.c. metals and alloys at low temperatures (T < 0.2T,) is currently described in terms of one of the following four basic models or a combination of them. (1) Intrinsic lattice hardening models based on the fact that the Peierls stress for the motion of screw dislocations in b.c.c. metals is very high; thermally activated deformation occurs by the nucleation and subsequent propagation of double kinks in screw dislocations (e.g. the Dorn-Rajnak model [6] ). (2) Impurity hardening models (e.g. ref. 7). (3) Phenomenological models such as the power law model based on the Johnston-Gilman equation [ 51. (4) Collective unpinning of edge dislocations from solute atoms in concentrated solid solutions [ 8 - lo] . In order to allow the experimental data to be compared with these theories, it is necessary first to outline some basic assumptions and characteristics of (i) the phenomenological power law model and (ii) the collective unpinning model of Butt and Feltham.

221

In the phenomenological power law model [5] the validity of the Johnston-Gilman and Orowan equations is assumed and therefore a power law is obtained between the strain rate 6 and the effective stress u* : * p=&)

WI*

(7

(511

i oo* 1

The dislocation the temperature that

velocity exponent rrz* is independent of < but depends on T, whereas u0 * is constant. From eqns. (2) and (5) it follows

AH* = m*kT In and from eqns. (3) and (6) * _ Mm*kT v --

(7)

U*

Equation

(5) can be written

in the form

(3

(3)

CY= l/m*T

(9)

where

If T = 0 K, then u* = uo* and AH* = 0 according to eqns. (8) and (6). It follows from thermodynamic considerations that the product m*T is constant and different from zero when T approaches 0 K [ 51. Experimentally it is often found that m*T is constant over an extended temperature range [ 51. Some of the above-mentioned equations of the power law model have been shown to be valid for a number of b.c.c. metals including tantalum [II, 121. The model of collective unpinning of edge dislocations from solute atoms by means of the double-kink mechanism proposed by Butt and Feltham [ 81 has recently [ 9, lo] been claimed to be valid for concentrated Nb-Mo and Nb-Re solid solutions. According to this model the stress dependence of the activation volume U* is given by the relation

(10) where u. is the yield stress at 0 K and u. * isThe extrapolated volume at 0 K.

2. Experimental

activation

procedure

Binary and ternary tantalum-base alloys with the alloying elements niobium, tungsten, molybdenum, rhenium and hafnium were melted in an

222

ultrahigh vacuum electron beam melting furnace. Recrystallized wires of diameter 1.5 mm with an ASTM grain size of 5 f 1 (pure tantalum excepted) were obtained as described elsewhere [ 13,141. These samples were tensile tested in an Instron temperature cabinet in the temperature range 200 573 K. Strain rate changes were incorporated. The strain rate ratios were 2X and 0.5X at 200 K and 295 K but 5X and 0.2X at 373,473 and 573 K. Relaxation measurements and dip tests were carried out as described elsewhere [1] , In addition, compression tests on cylindrical samples of diameter 2.9 mm and height 6 mm were carried out in liquid nitrogen (77 K). In order to avoid twinning, these samples had been predeformed by 3% at room temperature. The strain rate ratios at 77 K were 2X and 0.5X. For both the tension and compression tests uLyP or CJ~.~was taken as the yield stress depending on whether or not a Liiders elongation occurred. However, at 77 K the plastic strain rate iPl = 6, - &/EC where E, is the combined modulus of elasticity of the specimen and the machine was only a fraction of the applied strain rate I!, at E = 0.2%. Therefore the yield stress at 77 K was determined from an extrapolation of the elastic and steady state plastic deformation regions. The strain hardening due to the 3% plastic predeformation at room temperature was subtracted from this value. This correction was usually small (_+l% of the yield stress at 77 K).

3. Experimental

results

Tensile and compression tests at room temperature numerical value for the yield stress (Fig. 1).

yield the same

Fig. 1. Yield stress in tension us. yield stress in compression for tantalum and tantalumbase solid solutions at room temperature.

223

The temperature dependence of the yield stresses of Ta-Nb, Ta-W, Ta-Mo and Ta-Hf alloys is shown in Fig. 2. These alloy systems show large differences. Niobium has almost no influence (Fig. 2(a)). SSH is observed in Ta--Hf alloys at high temperatures, but solid solution softening (SSS) occurs at low temperatures (Fig. 2(d)). Tungsten and molybdenum increase the yield stress at both low and high temperatures (Figs. 2(b) and 2(c)). This is also the case for rhenium [ 21 .

Fig. 2. Temperature dependence of the yield stresses of (a) Ta-Nb, (b) Ta-W, (c) Ta-Mo and (d) Ta-Hf solid solutions.

224

Relaxation experiments (for experimental details see ref. 1) have yielded inconsistent results for the athermal stress of the hard alloys in contrast with the results for pure tantalum [l] and the soft alloys (see Section 4). For this reason all internal stresses determined in this investigation were obtained from dip tests (for experimental details see ref. 1). The internal stress at 3% true strain as obtained from dip tests at three different temperatures is almost constant (Fig. 3). The slight increase in athermal stress with increasing temperature for tantalum and Ta-l.ORe is due to a much higher strain hardening rate at the onset of plastic deformation at 473 and 373 K compared with that at room temperature. This difference in the initial strain hardening rate is negligible for the harder alloys. The results of dip tests below room temperature are not included because of poor temperature stability. Figure 4 shows the thermal stress u” uersus the absolute temperature for tantalum and Ta-Hf alloys and Fig. 5 shows log u* uersus temperature plots for the other alloy systems investigated. The data for tantalum given in Fig. 5(a) are omitted in Figs. 5(b) - 5(d) for clarity. The values of (T” in these figures were obtained as follows. The flow curve is almost horizontal at 77 and 200 K and u* is taken to be equal to the yield stress minus OHM,where uPM is the average value of the internal stresses u,, at 3% true strain at (i) 295 K, (ii) 373 K and (iii) 473 K. At the higher temperatures u* is obtained by subtracting oPM from the total stressat 3% true strain for all compositions except pure tantalum and the two binary Ta-Nb alloys. For these compositions the values of u, at each

01

1

I

I

300

350

‘00

TEMPERATURE IKI

I ‘50

J

m

0

1 0

w

m

m

‘00

500

TEMPERATURE IKI

Fig. 3. Athermal stress at 3% true elongation as determined by means of dip tests at three temperatures for tantalum and Ta-Re alloys. Fig. 4. Thermal stress vs. temperature for tantalum and Ta-Hf alloys.

22!i

,

I

I

MO

203

300

I

m

TEMPERATURE(K)

a0

m TEMPERATURE

TEMPERATURE

ml

(00

(K/

5m

(KI

Fig. 5. Log u* us. T for (a) Ta-Nb, (b) Ta--W, (c) Ta-Re U* is the thermally activated part of the yield stress.

and (d) Ta-Mo

solid solutions.

temperature instead of the arithmetic average uPM is subtracted from the total stress. The reason for this is given in the previous paragraph. The influence of plastic deformation on the total flow stress u, the athermal component u,, and the thermal component u* is shown in Fig. 6 for the system Ta-Nb at 473 K, in Fig. 7 for the system Ta-Re at 373 K and in Fig. 8 for the system Ta-Mo at 295 K. The other alloy systems provide

226

Fig. 6. (a) Flow stress u and (b) athermal stress u,, (large symbols) and thermal stress (7* (small symbols) us. true strain at 473 K for Ta and Ta-Nb alloys: 0, tantalum; A, Ta3.9Nb ; 0, Ta-7.9Nb.

TRUE STRAIN (a)

TRUE STRAIN

E

E

@I

Fig. 7. (a) Flow stress u and (b) athermal stress uU (large symbols) and thermal stress (small symbols) us. true strain at 373 K for Ta and Ta-Re alloys; 0, tantalum; A, Tal.ORe; 0, Ta-2.lRe; *, Ta-3.9Re.

U*

similar results at these temperatures [ 151. The data points at c = 1.34 in Fig. 8 were obtained on cold-drawn wires in the same way as described for pure tantalum [l]. A typical example of the influence of plastic deformation on the strain rate sensitivity (6a/6(ln6)}T and the activation volume u* obtained from strain rate changes at different temperatures is shown in Fig. 9. Both quantities are almost independent of the true strain e. The data points for the Liiders elongation zone are not plotted in Fig. 9. For all compositions the

227

i 01 0

005

ox)

015

TRUE STRAIN

020

03

13‘

E

(b)

Fig. 8. (a) Flow stress u and (b) athermal stress (Jo (large symbols) and thermal stress U* (small symbols) us. true strain at 295 K for tantalum and Ta-Mo alloys: 0, tantalum; A, Ta-0.6Mo; 0, Ta-1.3Mo; *, Ta-2.5Mo; 0, Ta-4.9Mo. The data at E = 1.34 were obtained from tensile tests on cold-drawn wires.

strain rate sensitivity in this zone was somewhat higher than that in the region where homogeneous plastic deformation occurred. Strain rate changes above and below the flow curve yield similar results for the activation volume u* except at low values of u” where a small difference is found (see T = 473 K and T = 573 K in Fig. 9). As suggested by Kocks [3] a value of 2.9 was used for the Taylor factor M. This is 5% less than the Taylor factor for crystallographic slip on (110) (M = 3.06) and 5% greater than the value for pencil glide (M = 2.75) in texture-free b.c.c. metals [ 31 . Log-log plots of v*/b3 uersus u* are shown in Fig. 10 for tantalum and the two Ta-Nb alloys and in Fig. 11 for the hardest alloys (harder than 150 HV). The value of u* for each composition and temperature in these figures is the average value of all activation volumes obtained in the

228

Fig. 9. Influence of the true strain E on the strain rate sensitivity &o/S(ln i) and the activation volume u* for the Ta-3.OW-0.2Hf alloy at different temperatures. The full and open symbols correspond to strain rate changes above and below the flow curve respectively. Fig. 10. Normalized activation volume u*/b3 us. thermal stress U* for tantalum and TaNb alloys. The measured hardness of these three compositions is between 72 and 76 HV. The straight line on this figure is taken from Fig. 11.

Fig. 11. Normalized activation volume with hardnesses greater than 150 HV.

u*/b3

us. effective

stress

U* ‘for tantalum

alloys

229

Fig, 12. {6o/6(in from Fig, 13.

i)}/T

Fig. 13. {&o/6(lni)}/T

us. -60/6T

for Ta-W alloys. The straight line in this figure is taken

us. -da/&T

for tantalum alloys harder than 150 HV

Fig. 14. Activation energy AH* us. temperature

T for tantalum alloys harder than 150 HV.

Fig. 15. AH* us. log U* for tantalum alloys harder than 150 HV.

homogeneous plastic deformation region (see Fig. 9). The numerical values of (T*in Figs. 10 and 11 are sfightly different from those in Figs. 4 and 5. They are average values of the true thermal companent of the flow stress determined by means of dip tests in the plastic deformation region between the Liiders elongation zone and the onset of necking (see Figs. 6 - 8). Most

230

curves of u*/b3 uersus u* for alloys with a hardness between 80 and 150 HV would lie between the two extremes shown in Fig. 10. Plots of {60/6(ln~)}T/Tuersus (&a/67’); are shown in Figs. 12 and 13 for different alloys. The values on the abscissae of both figures were obtained from the temperature dependence of the yield stress (see Figs. 2 and 5). The small influence of temperature on the modulus is neglected. Activation enthalpies were calculated using eqn. (4). For the hard alloys (harder than 150 HV) these values are plotted versus the absolute temperature in Fig. 14 and versus log (T* in Fig. 15. The values of CJ*in Fig. 15 were obtained in the same way as those of Fig. 5.

4. Discussion The strength differential effect in polycrystalline substitutional tantalum-base alloys at room temperature is negligible (Fig. 1). This is also the case for polycrystalline niobium at 173 K [ 161. Serrated yielding (Portevin-Le Chatelier effect) occurred at 573 K in most of the alloys investigated. The influence of substitutional alloying elements on this effect is discussed elsewhere [ 171. For the hard alloys (harder than 150 HV) a single thermally activated process dominates the plastic deformation in the temperature range 77 373 K as can be concluded from Figs. 13 and 14, taking into account eqns. (2) and (4). All the data fall on straight lines which extrapolate to the origin. In contrast, deviations from the fundamental equation for thermally activated deformation (eqn. (2)) occur for pure tantalum and for the soft alloys in the same temperature range. An example of this is shown in Fig. 12. The deviation for pure tantalum is in agreement with observations above 150 K made by Christian and Masters [IS] . This temperature is far below the temperature To at which the athermal plateau sets in (?400 K according to Fig. 2(a)). Because 77 K was the only temperature below 150 K at which experiments were carried out in the present investigation, no further attempts will be made to find a consistent activation analysis for pure tantalum and the soft alloys. As regards the hard alloys (harder than 150 HV), it follows from Figs. 13 and 14 that (i) a single process is rate determining in these alloys in the temperature range 77 - 373 K, (ii) the preexponential factor PO is constant (independent of temperature and strain rate), (iii) PO is independent of the alloying element and its concentration and (iv) the activation entropy AS* is constant or very small in agreement with the results of Arsenault [ 19, 201. Further, it follows from the present data that the predictions of the phenomenological power law model are valid in the case of the hard alloys; indeed (i) plots of log u* uersus T are linear (Fig. 5), (ii) a plot of Lw* uersus log u* is linear (Fig. 15) and (iii) the activation volume u* is inversely proportional to the effective stress (Fig. 11). The fact that the numerical values of the model parameters E’s, u,,* and m*T = l/a, as obtained from different rela-

231

tionships and graphs, are almost identical (Table 1) further supports the validity of the power law model for the hard alloys. In contrast, the data for the soft alloys show deviations from the predictions of the power law model (Figs. 5(a), 10 and 12). TABLE 1

.

Average values of ~0, GO*and m *T for tantalum alloys with a hardness value higher than 150 HV as determined from different relations Source

Relation

Equation

’

;:I, Fig. 13 Fig. 14 Fig. 11 Fig. 15 Table 2 --

{6@/6(lng)}/T us. 6a/6T &P vs. T v* us. l/u* AH* vs. log LT* log u* vs. T

(2), (4) (21 (7) (6) (8)

(MPa)

m*T = l/cu (W

970 990

4800 4800 5050

(JO*

_

For the hard alloys (harder than 150 HV) it follows from Table 2 that the extrapolated thermal stress of each alloy at the absolute zero temperature is in the range 1000 MPa * 11%. The three softest compositions, i.e. tantalum and the two Ta-Nb alloys, extrapolate to a value close to 1000 MPa (Fig. 5(a)). It follows that to a first approximation the effective stress at T = 0 K is not influenced by substitutional alloying elements, In other words the thermal substitutions SSH Au* is equal to zero at T = 0 K. Recent flow stress measurements at temperatures close to and above 4 K on Nb-11Mo single crystals [ 21.1, Fe-Ni single crystals [ 22, 231 and Fe-Al single crystals [ 231 also suggest a thermal SSH which is almost zero at T = 0 K. From the foregoing it follows that the analytical expression o*(@, T) for the hard alloys (harder than 150 HV) is given by eqns. (8) and (9) in which uo* and ti, are to a first approximation independent of the alloy composition. In contrast, the value of m*T increases with increasing concentration (hardness) as can be seen in Fig. 16. The higher value of m*T for the hardest alloys results in a slower decrease of the effective stress with increasing temperature and thus also in an increase of the ~mperature To at which the athermal plateau sets in (e.g. Fig. 2(b)). Thus, for the hard alloys ACJ* is greater than zero at temperatures between T = 0 K and T = To,while Au* = OatT=OKandatT>T,. In terms of total stresses no true SSS was observed in the Ta-Nb, Ta-Re, Ta-W and Ta-Mo systems (Figs. Z(a) and 2(b) and ref. 2), in contradiction with the results of other ~vestigations (refs. 74 - 84 in the review by Pink and Arsenault [ 241). However, the present results are in agreement with the systematic experimental results of Gibala and Mitchell [ 251. They found no true SSS in outgassed ultrapure Ta-W, Ta-Re and TaMO single crystals, but significant alloy softening in impure binary alloys on

232 TABLE 2 Numerical values for (i) the power law model parameters (Yand uc* and (ii) the DornRajnak model parameters up* and T, for binary and ternary substitutional tantalum alloys with hardnesses greater than 150 HV Composition (at.%)

Hardness [I31 WV)

m*T = l/ff (Fig. 5) ( W

Ta-3.1W-l.ORe Ta-2.lRe Ta-6.3W Ta-4.9Mo Ta-3.5W-3.9Mo Ta-3.6W-4.3Mo Ta-10.3W Ta-3.9Re Ta-3.OW-2.7Re

157 161 166 171 181 193 217 226 234 -

5050 4600 4900 4900 5050 5050 5700 5200 5050

1020 910 1050 950 950 960 1100 1050

5050

990

Average

Fig. 16. Influence of solutes in concentrated l/a (calculated from Fig. 5).

890

T, (Fig. 2) W

Symbol in Fig. 18

520 473 500 520 500 500 573 573 513 -

0 v * V 0 0 * n n

tantalum-base alloys on the value of m *T =

the one hand and in ternary alloys of the substitutional-interstitial type on the other hand. When the present data are plotted on u* uersus T graphs [ 151, a little “pseudosoftening” (Au* < 0) is detected in the Ta-W, Ta-Re and Ta-Mo systems (not in Ta-Nb) at low concentrations. Taking the foregoing into account, this pseudosoftening is believed to be mainly due to the disturbing effect of residual interstitials. However, it should be mentioned that, in contrast with single crystals outgassed near the solidus line, pure tantalum-base polycrystalline alloys with fine grain size are very difficult to obtain because of the strong getter action of tantalum for interstitials at the recrystallization temperatures [ 141 of these alloys, as can be seen on the Ta-0 and Ta-N phase diagrams with isobars [26]. Because no true SSS was obtained in the Ta-Re, Ta-W and Ta-Mo systems in the present investigation, the perturbing effect of interstitials in the present investigation is assumed to be negligible at high solute concentrations (hardnesses greater

233

than 150 HV) as was for example also found with nitrogen in the Ta-Re-N system [ 271. In contrast with the Ta-W, Ta-Re, Ta-Nb and Ta-Mo systems, a true SSS is observed in the Ta-Hf system (Fig. 2(d)). This effect is even more clearly illustrated in Fig. 4. In all Ta--Hf alloys investigated thermal SSS (AcJ” < 0) is observed at 77 and 200 K and thermal SSH (Ao* > 0) at temperatures between 350 K and the onset of the athermal plateau. In addition, all Ta-Hf alloys investigated show athermal SSH (Fig. 2(d) and ref. 2). No systematic investigation on SSS in Ta-Hf alloys at low temperatures has been reported in the literature. However, it is believed that this effect is due to the high chemical affinity of hafnium for interstitials such as oxygen. According to thermodynamic data [ 281, hafnium has a much higher affinity for oxygen than tantalum, whereas molybdenum, tungsten and rhenium have a much lower affinity. Therefore a scavenging effect of a chemical nature is believed to occur in the Ta-Hf system, whereas in the Ta-Re, Ta-W and Ta-Mo systems a scavenging effect of a geometrical nature is possible. No SSH nor SSS is observed for the Ta-Nb system (Fig. 5(a)) in agreement with the facts that (i) niobium and tantalum have the same affinity for oxygen and (ii) the atomic radii of tantalum and niobium are almost identical. Reed-Hill and Donoso [ 51 suggest that the failure to obtain a successful activation analysis for the group Va metals vanadium, niobium and tantalum at temperatures higher than 150 - 250 K may be due to (dynamic) strain aging caused by hydrogen. It will be shown elsewhere [29] that hafnium increases the tendency of tantalum to absorb hydrogen, while tungsten, rhenium and molybdenum have the opposite effect. The suggestion of ReedHill and Donoso is further supported by the present observations that (i) the largest deviations from a single A~henius equation for thermally activated plastic deformation as well as large deviations from the power law model are observed in the Ta-Hf system and (ii) no such deviations are found in concentrated Ta-W, Ta-Mo and Ta-Re alloys. The different behaviour of Ta-Hf alloys is probably due to the presence of impurities such as oxygen, carbon, nitrogen and hydrogen. However, effects intrinsic to hafnium cannot be excluded completely. Indeed, the element hafnium differs from the elements tungsten, molybdenum and rhenium with respect to the following two points: (i) the lattice parameter change caused by hafnium in tantalum is positive while tungsten, rhenium and molybdenum additions cause a negative change and (ii) the number of s + d electrons in the outer shell is smaller in hafnium than in tantalum, while it is larger in tungsten, molybdenum and rhenium. The occurrence of SSS in substitutional molybdenum-base alloys has been claimed to depend on the number of s + d electrons [ 30, 311, while SSS in iron is linked with the size of the solute atoms [ 321. Experiments on ultrapure outgassed Ta-Hf or Nb-Hf single crystals should provide the correct explanation. Because thermally activated deformation in the hard alloys is in agreement with the power law model, the Johnston-Gilman equation is valid for the hard alloys. This equation is also valid for pure tantalum at room

234 temperature since both the relaxation method of Li [33] and the low temperature form of the plastic equation of state as defined by Hart [34] are valid for pure tantalum [l] . For the hard alloys inconsistent results were obtained for the athermal component of the flow stress when the relaxation method of Li was applied. The reason is that the dislocation velocity exponent m* increases with increasing alloy content (hardness), as shown in Fig. 16. From Table 2 it follows that for the hard alloys (hardness greater than 150 HV) the average value of m* at room temperature is equal to 17. Now, the relaxation method of Li yields reliable (sufficiently accurate) results for the four regression parameters only when m* is lower than 10, as is explained by White and Smith 1353 and Li [33] . In contrast, the dislocation velocity exponent for tantalum at room temperature is lower than 10 [I] and reliable results for the athermal component of the flow stress of pure tantalum were obtained which were in full agreement with results from other splitting methods such as the dip test, the saturation method and the extrapolation method of Seeger [l] . The internal stress is usually assumed to be temperature independent, except for the temperature dependence of the modulus @ [ 361. Spitzig and Leslie [37] found that the value of u,, extrapolated to zero strain ugo in iron and the alloys Fe-3Co, Fe-3Ni and Fe-3Si remains almost constant with decreasing temperature, In contrast, the relaxed flow stress obtained from dip tests has been reported to increase steeply with decreasing temperature in polycrystalline vanadium with high interstitial content [ 381. In a discussion on the scavenging mechanism, Pink et al. [39] expect the athermal stress to increase in Ta-Re alloys also. They suggest such an increase to be general in b,c.c. alloys. Our experimental results for Ta-Re alloys (Fig. 3) and for the other binary and ternary t~t~urn-bee alloys investigated [ 2, 151 prove that eELremains constant in the temperature range 473 - 295 K. Moreover, these results suggest that efi remains constant down to 77 K, since they agree with the activation analysis described above which is based on the assumption that up is constant. Relaxation experiments at very low temperatures should be interpreted with caution as has been explained in detail by Dotsenko ]36]. Because the phenomenological power law model is valid for the hard alloys, the increase of m* T with increasing concentration (Fig. 16) together with the fact that E. = MqO and us* = MT~* are independent of concentration implies the following (for 0 < T < ‘Z’s). (1) The work done by the external forces, given by the product T*U* according to refs. 36 and 40, increases with increasing concentration. Indeed, ?-*u* = km*T according to eqn. (7). (2) The energy provided by thermal activation, i.e. AH*, as well as the activation volume u* increase with increasing concentration at a given value of T* as follows from eqns. (6) and (7). (3) The ratio of the energy provided by thermal activation to the work done by external forces is concentration independent at a given value of T*. Indeed, AH*/u*r* = ln(Te*/T*) according to eqns. (6) and (7).

235

(4) This ratio decreases with increasing concentration at a given temperature. In other words increasing the concentration makes thermal activation more and more difficult at a given temperature. Although the data on the hard alloys are in full agreement with the power law model based on the Johnston-Gilman equation, this phenomenological model provides no physical explanation for the fact that both u* (Fig. 9) and cr* (Figs. 6 - 8) are independent of strain. Neither does it explain the specific influence of each alloying element and its concentration on the thermally activated deformation behaviour of these alloys. Therefore the agreement of the present data with the three other model groups mentioned in the introduction is discussed below. Classical impurity hardening models (e.g. ref. 7) do not provide an explanation for the fact that the thermal SSH Au* is zero at 0 K. The collective unpinning model of edge dislocations from solute atoms (Butt and Feltham [ 81) has been shown recently [ 9, lo] to describe well the temperature dependence of the total yield stress of concentrated Nb-Mo and Nb-Re alloys when values appropriate to each alloy are chosen for two model parameters. The same is possible with the data concerning the concentrated alloys given in Fig. 2 and in refs. 2 and 17. However, this model gives poor results as regards the strain rate dependence of the flow stress, as is illustrated in Fig. 17 for the two hardest binary tantalum alloys investigated, i.e. Ta-10.3W and Ta-3.9Re. According to eqn. (10) the two upper curves should be horizontal and equal to u,*/b3. The numerical values for CJ,,in eqn. (10) have been obtained by adding uU (Fig. 3 and ref. 2) to the numerical value of uO* given in Table 2. When compared with experimental data, the model of Butt and Feltham is always related to total stresses and not to thermal stresses u*. When their model is checked with the present

Fig. 17. Comparison with the double-kink curves) and effective

of the experimental data for the Ta-10.3W and Ta-3.9Re model of Butt and Feltham [ 81 applied to total stresses stresses (lower curves).

alloys (upper

236

data concerning the thermally activated part of the flow stress, deviations still occur (see the two lower curves in Fig. 17). The theory of Butt and Feltham is also unable to provide an explanation for the dip test results (e.g. Fig. 3), as pointed out elsewhere [ 21. These findings and the observation that the dislocation structure of Ta-9W single crystals deformed at 581 K and below consists predominantly of straight (111) screw dislocations [ 19, 411 support the view that the thermally activated deformation behaviour of concentrated tantalum-base alloys at low temperatures is still controlled by screw dislocations and not by edge dislocations. Intrinsic models based on the high Peierls barrier for (111) screw dislocations in b.c.c. metals can be divided into three groups [42] : (i) doublekink models, (ii) dissociation-recombination models and (iii) atomistic models. Dissociation-recombination models are not realistic because they assume the existence of stable single-layer stacking faults [42]. Atomistic models are more realistic, but they have not yet been developed sufficiently to provide quantitative relations between the activation parameters [43]. The most cited double-kink models are those of Seeger [44] and Dorn and Rajnak [6], The observed dependence u* = l/a* in Fig. 11 is in agreement with the double-kink model of Seeger. However, this model is only valid at low effective stress and does not predict an activation energy aH* = 0 at u* = as*, 1.e. ’ at T = 0 K. The latter boundary condition is valid for the hard alloys (Fig. 15 and Table 1). In Fig. 18 the present data for the hard alloys are compared with the Dorn-Rajnak double-kink model [ 61. The three curves indicated by cx = -1, o = 0 and CY= +l are the well-known universal u*/u,* us. T/T, Dorn-Rajnak curves for three different types of

Fig. 18. Comparison of the experimental results on tantalum alloys with hardness values higher than 150 HV with the double-kink model of Dorn and Rajnak [ 6 1. The alloy concentrations as well as the numerical values for the Peierls stress up* and the critical temperature Tc are given in Table 2.

237

Peierls barrier for the case that kink nucleation in screw dislocations is rate determining. The Peierls stress up* was set equal to uO* and the critical temperature T, was estimated from p1ot.s of o uersus T and cr* versus T (Table 2). Figure 18 shows that the experimental results for the hard alloys deviate considerably from the predicted universal curves. It can be shown that for pure tantalum and the soft alloys there are similar but much smaller deviations. The increase of the activation volume at low effective stresses (373 K) (Fig. 10) is explained by the fact that dynamic strain aging in tantalum and the Ta-Nb alloys starts at 473 K in contrast with the other alloys and that negative strain rate sensitivities are observed at 573 K (which is a typical characteristic of the Portevin-Le Chatelier effect [36]). After abstracting the data at 373 K, all other data in Fig. 10 for tantalum and Ta-Nb alloys on the one hand and all data for the hard alloys on the other hand are qualitatively in agreement with the theory of Sato and Meshi [45] . According to this theory solute atoms with a size misfit promote kink nucleation at low temperatures but impede lateral motion at higher temperatures. However, according to this model SSS should occur at low temperatures in pure Ta-Re, Ta-W and Ta-Mo alloys; this was not found in the present investigation nor by Gibala and Mitchell [ 251. The latter data do not even suggest pseudosoftening [46, 471. Several other theories which describe the influence of solute atoms on the double-kink mechanism have been proposed (for a review see ref. 24). Although many of them are able to explain some of the data mentioned above, none of them provides good fits to the complete set of the present data and observed relations. However, it should be mentioned that (i) all these theories start from a rather arbitrary and empirical approximation of the Peierls barrier and (ii) the polarity and the threefold symmetry of (111) screw dislocations in b.c.c. metals are not taken into account. Notwithstanding the shortcomings of the present theoretical doublekink models, we believe that the double-kink mechanism which is now generally accepted to be the rate-controlling mechanism at low temperatures in pure b.c.c. metals [43] does still control the thermally activated deformation in the hard alloys (hardnesses greater than 150) because (i) the dislocation structure of Ta-SW single crystals at 581 K and below consists predominantly of straight (111) screw dislocations [ 19, 411, (ii) the experimental curve u*/up* versus T/T, (Fig. 18) for the hard alloys is indeed independent of the exact composition of the tantalum alloys as predicted by the DornRajnak model, (iii) the experimentally observed activation volumes ( lob3 100b3 according to Fig. 11) are typical of the Peierls mechanism [4] and (iv) both the activation volume u* and the effective stress u* are independent of strain (Figs. 6 - 9). It is our opinion that the polarity of (111) screw dislocations, which is a scalar quantity as high as O.lb [ 481, should be taken into account in refining the present double-kink models for dilute and concentrated solid solutions. In particular, hydrostatic stress fields associated with polarity-reversing kinks of the K++ and the K_Y type [49] as well as with constrictions [49] in (1111)

238

screw dislocations are expected to interact with long-range hydrostatic stress fields caused by variations in the concentration of solute atoms with a size misfit. Such variations are expected from both statistical and thermodynamic considerations. At high temperatures these interactions lead to athermal SSH dominated by the atomic size misfit effect [ 131, as has been observed in the present tantalum-base alloys. Indeed, the binary athermal SSH rate Aa,/AC correlates well with the atomic size misfit parameter e, = (l/a)(Aa/Ac), being the highest for rhenium followed by molybdenum and tungsten while niobium (ea = 0) had no effect at all [ 21. In addition, at lower temperatures or higher strain rates, thermal activation may no longer be sufficient to overcome fast enough short-range interactions between hydrostatic stress fields around polarity-reversing kinks and constrictions on the one hand and small-scale hydrostatic stress fields around single atoms on the other hand. A typical value for u* is 50b3 (Fig. 11). Thus in concentrated alloys a successful activation step has to overcome several solute atoms together, which leads to thermally activated SSH still dominated by the size misfit effect. From Fig. 2 it can be seen that the order of effectiveness of the alloying elements in increasing m*T and therefore also u* (eqn. (8)) is identical with the one just mentioned. In addition, niobium does not influence the temperature dependence of the yield stress at all in the whole temperature range investigated (Fig. 2(a)), in agreement with the SSH behaviour proposed above.

5. Conclusions (1) The internal stresss in concentrated substitutional tantalum-base alloys with hardnesses greater than 150 HV remains constant with decreasing temperature. (2) In the temperature range 77 - 373 K a single thermally activated process dominates the plastic deformation in these alloys. The preexponential factor go of the rate equation is constant and independent of the exact composition. (3) The data for the hard alloys are in agreement with the power law model based on the Johnston-Gilman equation. The product m*T where m* is the dislocation velocity exponent is temperature independent and increases with increasing solute concentration. (4) Thermal substitutional solid solution hardening in both hard and soft alloys extrapolates to almost zero at T = 0 K. (5) The activation volume U* and the effective stress u* are almost independent of strain. This and other arguments suggest that plastic deformation in concentrated substitutional tantalum-base alloys is still controlled by the double-kink mechanism for (111) screw dislocations. (6) The specific effect of the alloying elements both in the plateau region and below is explained in terms of interactions between on the one hand solute atoms with a size misfit and on the other hand polarity-reversing kinks and constrictions in (111) screw dislocations.

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41 42 43 44 45 46

Rep. COOOH). Rep. COOOH).