thermally activated deformation of potassium

T . 11. BERNSTEIN and 5. C. N. LI: Thermally Activated Deformation of K

phys. stat. sol. 23, 539 (1967)

Subject classification: 10.1; 21.2

Ijenry Krumb School of Nines , Columbia University, Xew York City

Thermally Activated Deformation of Potassium BY

I. M. BERNSTEIN and J. C. M. L I ~ )

539

The thermally activated deformation of single crystal and polycrystalline potassium is examined. It is shown that the activation parameters describing low-temperature deformation can be correlated reasonably well with the b.c.c. transition metals on the basis of their respective shear moduli. This supports the suggestion that the same general mechanism controls low-temperature deformation for all b.c. c. materials. The possibility of this mechanism being either impurity or lattice controlled is examined and i t is shown that the Peierls-Nabarro lattice resistance model adequately explains the experimental behavior. The possible role of impurities a t low temperatures is briefly considered.

Die thermisch aktivierte Deformation von einkristallinem und polykristallinem Kalium mird untersucht. Es wird gezeigt, daD sich die Aktivierungsparameter, die die Deformation bei niedrigen Temperaturen beschreiben, ziemlich gut mit den k.r.z.-ubergangsmetallen auf der Basis ihrer betreffenden Schermoduli korrelieren lassen. Dies unterstutzt die Annahme, daB bei allen k.r.z .-Metallen der gleiche allgemeine Mechanismus die Deformation bei niedri- genTemperaturen steuert. Es wird untersucht, ob dieser Mechanismus durch Verunreinigun- gen oder durch das Gitter gesteuert wird und es wird gezeigt, daO das Gitterwiderstands- model1 von Peierls-Nabarro das experinientell gefundene Verhalten adequat beschreibt. Die mogliche Rolle der Verunreinigungen bei tiefen Temperaturen wird kurz diskutiert.

1. Introduction It has been demonstrated [I, 2, 31 that the parameters describing low-tem-

perature deformation of the b.c.c. transition metals can be correlated on the basis of the shear moduli of the respective metals. That is: 1. The extrapolated effective stress, T*, necessary to move mobile dislocations a t 0 “K is about 0.01 p, where p is the shear modulus. 2. The activation enthalpy a t very low stresses, AH,, is about 0.1 ,LA b3 where b is the Burgers vector. 3. The “activation volume” V* defined by kT (a In ilat), is about 20 b3 a t a shear stress of 10-3,u, where i is the applied strain rate.

These correlation have been cited as evidence [l, 2, 31 that the rate-controlling process for low-temperature deformation is the overcoming of the Peierls-Na- barro barrier, as opposed to the other reasonable alternative [4, 51 that it is controlled by the interaction of impurities with mobile dislocations. While the actual situation is not as clear cut as that, they can provide the basis for com- paring all metals and alloys of the b.c.c. structure. Partially for this reason, a study was undertaken of the b.c.c. alkali metal potassium. Its low melting point and low elastic moduli contrast it sharply from b.c.c. transition metals, yet its deformation behavior has been shown to be similar [6]. Furthermore, by the application of the third law of thermodynamics t o the thermally activated

l) The ant,hors are presently a t E. C. Bain Laboratory for Fundamental Research, U. S. Steel Corporation, Research Center, Monroeville, Pa. 15146.

540 I. M. BERNSTEIN and J. C. M. LI

processes for potassium, the parameters derivable from macroscopic strain-rate cycling techniques were shown to be internally consistent [7], thus allowing meaningful information to be obtained from such an experimental study.

2. Experimental Procedure Potassium was chosen as the alkali metal to be studied because cesium and

rubidium are too reactive to be handled easily and lithium and sodium undergo polymorphic transformations at low temperatures [S]. Owing to the high reac- tivity of potassium with water vapor and oxygen and its extreme softness a t room temperature, the preparation and testing of potassium required elaborate techniques. Since these have previously been recorded in considerable detail [6], only a brief description will be given here.

All specimen fabrication, preparation, and metallography were performed in two specially constructed dry boxes under a rigorously dried, oxygen-free, nitrogen atmosphere. Except for one series, polycrystalline and single crystal specimens were prepared from the highest purity potassium commercially available, which is rated to have < 125 ppm of impurities [6]. One test series was performed with commercially pure potassium which had a total impurity content of w 1 % [6].

Polycrystalline specimens 3/16 in. in diameter having a gauge length of w 314 in. were extruded at room temperature to an average grain size of w 0.05 in. for the high-purity material and w 0.03 in. for the commercial purity material. Single crystals of the same specimen dimensions were prepared from the high-purity potassium in oil-coated glass tubes, which were passed through a furnace held slightly above the melting point of potassium, All the crystals tested had an initial orientation near the center of the standard stereographic triangle. The high-purity polycrystals and single crystals are designated in the text as the M.S.A. and P series, respectively, and the commercial purity polycrystals are designated as the C.P. series.

The specimens were mounted into a jig which could be evacuated, removed from the dry box, and connected to a standard Instron testing machine. The whole unit was cooled to near 77 OK while still essentially under vacuum by using a small amount of pre-purified He gas as the heat transfer medium. This was done because the oxidation rate of potassium is negligible below w 195 OK

(91. Subsequent surface protection was afforded by a dynamic He gas flow. For tests below 77 "I< a flexible transfer tube, which moved with the Instron crosshead, was used to feed controlled amounts of liquid He into the specimen chamber. By careful flow control, temperatures between 77 OK and w 5 OK could be reached and maintained during the test period.

Temperature measurements were carried out using either a copper-constantan thermocouple above NN 30 OK or a calibrated carbon resistor for lower temperatures. In both cases the device was in intimate contact with the potassium and thus was capable of monitoring any temperature fluctuations of the specimen. The temperature control was +2O a t high temperatures and f l " below 20 OK. After testing, the jig was re-evacuated and returned to the dry box for specimen examination. Only limited surface contamination occurred during the testing period.

Strain-rate cycling experiments were carried out using a quick change gear box on the Instron which allowed crosshead speed changes of 2 : 1 to 100: 1 without relaxing the load on the specimen. For all tests reported here, the

Thermally Activated Deformation of K 541

crosshead speed was changed from 0.2 in./min (base strain rate) to 0.005 in./min corresponding to a strain-rate change of 40: 1. For uninterrupted stress-strain tests, the crosshead speed was always 0.2 in./min.

3. Experimental Results Fig. 1 and 2 summarize the stress-temperature behavior for the three series

of potassium tested. A detailed account of deformation and flow will be given in a future paper. Fig. 1 represents the temperature dependence of stress for M.S.A. potassium a t different true plastic strains. The yield stress was obtained from the deviation from linearity. This was usually well defined and could be determined to within 5% in most cases. For these materials no yield point phenomena were ever observed. The yield and flow stress are strongly temperature-dependent below T, M 25 OK. The yield stress shown a t 0 OK is not a data point, but is rather an extrapolated value whose validity has been previously substantiated [7]. The peak a t about 40 OK for the flow stress curves is a consequence of a peak in the work-hardening rate a t this temperature [6 ] . In order to compare the behavior of potassium with the higher melting b.c.c. transition metals the yield stress was plotted versus TIT,, where T , is the melting point (for potassium this is 63.7 "C [lo]). The results are shown in Fig. 2. The C.P. series with a much higher impurity content, exhibits a higher yield stress than M.S.A. potassium at all temperatures. However, the magnitude of this effect decreases a t very low temperatures. The reason why the P series single crystals also exhibit a higher yield stress than polycrystals of the same purity is not clear. However, the M.S.A. potassium is very coarse grained and if a texture exists, orientation is a possible cause. Note that the two series appear to have an approximately common yield stress a t 0 OK. Potassium dif- fers somewhat from the other b.c.c. metals in that its strong temperature- dependent region is initiated below M 0.1 T,, while for the transition series this occurs below 0.20 to 0.25 T , [ll].

The results of strain-rate cycling tests on potassium have been previously reported [7]. The parameters m* and t* derivable from such a study were

. \

'\

\ \ \ \

Fig. 1. Temperature dependence of the yield stress and the stress at 2 and 5 % strain for RI.S.6. potassium

0 a3 06 09 7xL Fig. 2. Yield stress YS. homologous temperature for the M.S.A., P, and C.P. potassium series


demonstrated to be consistent with the third law of thermodynamics, i.e. AS* = 0 (where AS+ is the activation entropy [12]). and as such are considered to be quantitatively meaningful. The quantity m* is the exponent of the Johnston-

(1) Gilman [13] relation

where V is the average velocity of a mobile dislocation and t* is the effective stress on a mobile dislocation in the absence of work hardening.

Z1- t * m * ,

3.1 Activation enfhalpy

A convenient and easily calculable relation for AH* is (121

Other forms of this relation are available [2, 141 with the major difference being

the substitution of the term for m*/ t * . Fig. 3 shows the temperature

dependence of AH+ for M.S.A. potassium a t yielding and for 3% plastic flow. The behavior of the P series was almost identical and therefore for clarity is not shown. Bernstein et al. (71 have demonstrated that for potassium m* T is constant below w 22 OK. It then directly follows from thermodynamics that AS+ is independent of stress [12], and since this is true for the whole range between 0' and M 22 OK the application of the third law of thermodynamics suggests that AS+ = 0 in the entire temperature range, and therefore the activation enthalpy equals the activation free energy AF* defined by

(3)

where 21, is the limiting velocity of a dislocation, i.e. when AF+ = 0 ; this is nor- mally considered to be the trans-

/' ' verse shear wave sound velocity for @&' +-7 // a dislocation moving on a particular

plane and in a particular direction. This velocity can be related to a critical strain rate through the fa- miliar relation

(4) where @ is a geometric factor and p is the mobile dislocation density. With the proviso that the mobile density is not a function of strain rate, equations (3) and (4) can be combined to give

I ' A F - M A . ' 006 -

-

i, = @ Q b vC , a04 -

A ~ - ~ ~ i .

i: AF+ = AH* = kT 1 n P . ( 5 ) E

30 Fig. 3. Variation of the activation enthalpy and energy with temperature and strain for M.S.A.

potassium ~/oxJ--, 0


With the applied strainrate E constant, equation ( 5 ) dictates that both AF* and AH+ should be linear with temperature in the region where AS* = 0 with the slope of the straight line equal to k In (&,/&). This is shown in Fig. 3 and the Ec are indicated. The values for &, agree within experimental error with those of Bern- stein e t al. [7] from a third law analysis at 0 OK. The difference in &, between yielding and 3% flow for M.S.A. potassium has been ascribed [7] to a difference in mobile dislocation density. Regrettably, the original density values calculated from equation (4) were all too large by a constant factor due to a nume- rical error in w,. From elastic constant data at 4.2 OK [15] a corrected wc of 1.15 x lo5 cm/s was obtained. Using this value, the following dislocation den- sities are calculated. For M.S.A. potassium, eys = 3 x lo9 @( 1 yo E ) = = 8 x lo8 and for the P series ~ ( 3 % F ) = 4x x lo7 Thus the original values quoted in [7] are in error by a factor of about 20. However, this in no way invalidates the conclusions reached in that paper.

At temperatures greater than 20 OK, A H + first increases above AF* and then decreases below it , indicating that in this temperature range AS* changes sign. It is quite likely that this points to the establishment of another rate-controlling mechanism somewhere in this transition temperature region.

~ ( 3 % E ) = 2 x 10s

3.2 Activation volume

The acbivation volume, V * , is defined in a strain-rate cycling test by

While equation (6) does have the dimensions of volume, this is an unfortunate definition since another activation volume exists which does define the volume change between the normal and activated state of a mobile dislocation, i.e.

where P is the hydrostatic pressure. To avoid confusion, it was recently suggested [16] that since the area a dis-

location sweeps out during thermal activation is a more realistic description of the event), the term activation area A* defined by V*/b should be adopted. This will be the position taken here. Fig. 4 shows a plot of A * vs. t* for potassium.

, l 4Ur - 1 1

1

v" 700 Z@ 3w 4 j O Fig. 4. The variation of the activation area with stress for the 31.S.A. and P series for yielding and 3% pla-

stic strain 7'"iP;i: -


Within experimental uncertainty, the results for both yielding and flow in M.X.A. potassium fall on a common curve. Therefore, the functional dependence between A * and t* is not sensitive to the state of strain. This implies that even in a strain-hardened material, the parameters describe the behavior of only mobile dislocations. The P series appears to exhibit a somewhat larger activation area.

4. Discussion

To compare t* at 0 OK among materials, it is important to use the same definition for t*. It is usually assumed that a t any point on the stress-strain curve z, = t * + z,, where t, is the applied stress and t, is the internal long range stress due to work hardening. However, t* is also considered to be made up of two components, a temperature-dependent stress, t * *, indicative of the thermally activated process and a component tf dependent on temperature only through the shear modulus. This stress may arise from the interaction of the mobile dislocation with the grown-in structure. It is common practice [l, 2, 31 to consider the essentially athermal high-temperature yield stress as tz and to substract it from the experimental yield stress to obtain t**. To make the comparisons valid this method will be used here also.

The extrapolated value of t:, a t 0 OK is 400psi for the M.S.A. polycrystals (making the assumption t,hat t rn lj2 a). From Fig. 1 a value of zf is estimated to be 30 psi; therefore t** = 370 psi.

Using elastic constant data a t 4.2 OK and the averaging method suggested by Hill [17], an average polycrystalline shear modulus of 1 . 8 5 ~ lo5 psi was calculated. Thus to 06 = 2 x p. This can be compared with the values shown in Table 1 for other b.c.c. metals. While potassium falls on the lower end of the scale, i t does correlate reasonably with the other materials.

To estimate the value of AH: advantage is taken of the fact that AH+ w AF+ in the region where m* T is constant. Since AF+ is a function of stress only

( 8 )

Taking a (m* T),, value of 280 [7] and shear stress limits of 400 psi (at 0 OK)

and 30 psi (at rn 25 OK), which corresponds to the stress region of strong temperature dependence, AH? is calculated to be w 0.06 eV. For polycrystalline potassium a t T, w 25 OK, ,u b3 w 0.73 eV and therefore AH,+- rn 0 . 0 8 ~ b3 in good agreement with the b.c.c. transition metals (see Table 1). Alternatively AH$ can be estimatedfrom the value of AF+ a t T = T,. From Fig. 3 this value is found to be w 0.05 eV and therefore AH,+ w 0.07 p b3. The discrepancy is probably due both to the uncertainty in AF* and more importantly to the dif- ficulty in precisely defining T, (and its corresponding stress).

Finally the activation area a t p (i.e. 185 psi for potassium), is from Fig. 4, equal to about 30 b2 for polycrystalline potassium, again in good agreement with the other b.c.c. metals (Table 1) .

Correlations of this type should not be taken completely literally, for they do not directly take into account the detailed difference in electronic and dislocation core structures, particularly between the transition and alkali metals. Also results on iron [18] suggest that the Peierls stress may vary for different crystallographic planes. Nevertheless, the close agreement between metals in this crystal structure is taken as strong evidence that they all obey the same low-temperature deformation mechanism. It does not, however, allow the ad

AF: = AH: = 1 A* b d t * = (m* T),, k l d In t* .


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546 I. M. BERXSTEIN and J. C. M. Lr

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546 I. M. BERXSTEIN and J. C. M. Lr

hoc assumption to be made that this mechanism is overcoming the Peierls-Na- barro barrier. An interaction mechanism between impurities and dislocations could also be expected to have some form of a shear modulus correlation, although for all metals to obey the same rules would appear to be fortuitous. The most reasonable approach appears to be to examine tfhe various models that have been proposed and in the light of the experimental results for potassium, attempt to compare their relative merit. Bernstein [6] has examined these and concluded that the only possible rate controlling models for potassium are those based either on lattice resistance or solute-dislocation interactions. For this reason, only these two will be considered. The relation of the temperature dependence of stress to a dislocation-impurity interaction is due mainly to the solution hardening theory of Fleisher [ 5 ] . This model predicts certain functional dependencies which in principle should serve as tests of the theory’s applicability. In particular, a plot of l/p vs. Jr should yield a straight line and the activation area should be linear with l/w where to is the extrapolated yield stress a t 0 OK. For the case of potassium, these relations hold only if the ather- ma1 stress, t/.*, is included in the effective stress. If this is done the extrapolated zo is found to be nearly 600 psi. Since the experimental value of 400 psi was obtained with the guidance of the third law of thermodynamics [7], this value appears to be much too high. A more serious objection to this and in fact any other impurity based model is found if the results on M.S.A. and C.P. potassium are considered. As Fig. 2 shows, C.P. potassium by virtue of its larger (but un- specified) impurity content exhibits a higher yield stress than M.S.A. potassium at all stresses. But the curves are separated by a stress which is either independent of, or increasing with, temperature. I n other words there is a decreased sen- sitivity of stress with temperature for increasing impurity concentration. An impurity model would be expected to predict that the increase of stress due t.0 a concentration increase should be proportional to the stress itself.

Alternatively the results can be compared with a Peierls-Nabarro barrier model. Dorn and Rajnak [3] have analyzed in detail the situation where dislocation motion is controlled by the thermally assisted production of kinks and

Fig. 5a. A comparison of the temperature depen- Fig. 5b. A comparison of the stress dependence dence of the experimental yield stress Sor t,he of the experimental activation area for yielding H.S.A. and P series with the theoretical predic- in the M.S.A. and P series with the theoretical tion of Dorn and Rajnak [3] for a purely sinusoi- prediction of Dorn and Rajnak 131 for a purely

dal PcierLs barrier sinusoidal Peierls barrier


t'heir subsequent motion in an initially st,raight dislocation line. The barrier is considered to be sinusoidal in shape, but the analysis was somewhat generalized by allowing for some distortion in the shape. Their model predicts universal relations between both zp b A*/2 E,and TIT, (w A H + / 2 Ek) vs. c**/tp where tp is the Peierls stress, 2 E, the energy to form a double kink and T , is the temperature above which the thermal component of the stress, described by this model, is negligible. By the use of these universal curves, it is possible to compare theoretical t** vs. T or A* vs. t** plots with the experimental resultas. The theoretical curves for polycrystalline potassium for the particular case of a purely sinusoidal energy barrier are shown in Fig. 5 a and 5 b with the experimental results for yielding in both polycrystalline and single crystal potassium. It is of note that by subtracting the respective t: from the M.S.A. and P series results, the A* values for the two series are now quite compatible. While the reasonable agreement between theory and experiment really only demonstrates the applicability of the force-distance curve used in the analysis, it does provide further justification, when considered in the light of the other experimental results on potassium, that the Peierls-Nabarro model adequately explains low-temperature deformation in this material. The impurities do play a modifying role but the exact nature of this is not yet understood. They may only affect the athermal stress, T:, which in turn can partially restrict kink motion. Li [16] has shown that if the long-range stress field is periodic, the effective internal stress produced by it may be insensitive to temperature, particularly for large m*. Or possibly the impurities have the effect of directly modifying the Peierls barrier. In any event their role is not considered to be rate-controlling. Then, on the basis of the correlations established, it might be expected that the low- temperature behavior of the b.c.c. transition metals is also controlled by the overcoming of a Peierls-Nabarro barrier.

6. Summary

The temperature dependence of the yield and flow stress of the b.c.c. alkali metal potassium was examined, with the following results : 1. Potassium exhibits the same strong temperature dependence of yield and flow stress as the other b.c.c. metals. 2 . The low-temperature parameters which describe thermally activated deformation in potassium correlate quite well with the high melting b.c.c. transition metals, if normalized on the basis of their respective shear moduli. This suggests that the same general mechanism controls low-temperature deformation for this crystal structure. 3. The Dorn-Rajnak model based on the thermally activated nucleation of double kinks over the Peierls barrier adequately explains the experimental result's for potassium and therefore would appear to be the rate controlling mechanism for all b .c.c. metals. Impurities only supply a relatively athermal contribution to the yield and flow stress.

Acknowledgements

The authors gratefully acknowledge the advice and interest of Prof. M. Gen- samer. The research was supported by t,heAir Force Office of Scientific Research, under Contract AF 49(638)408, and the Office of Naval Research under Con- tract No. 4643(00).

548 I. M. BERNSTEIN and J. C. M. L1: Thermally Activated Deformation of K

References [l] H. CONRAD and W. HAYES, Trans. ASM 56, 249 (1963). [2] J. W. CHRISTIAN and B. C. MASTERS, Proc. Royal SOC. 281A, 240 (1964). [3] J. E. DORN and S. RAJNAK, Trans. AINE 230, 1052 (1964). [4] D. F. STEIN, J. R. LOW, and A. U. SEYBOLT, Acta metall. 11, 1253 (1963). [5] R. L. FLEISHER, J. appl. Phys. 33, 3504 (1962). [GI I. M. BERNSTEIN, Ph.D. Thesis, Columbia Univ., New York 1965. [7] I. M. BERNSTEIN, J. C. M. Lr, and M. GENSAMER, Acta metall. 15, 801 (1967). [a] C. S. B-ARELETT, Acta cryst. 9, 671 (1956). 191 J. V. CATRCART and G. P. SMITH, J. Electrochem. Soc. 107(27), 141 (1960).

1101 S. L. WALTERS and R. R. MILLER, Ind. Engng. Chem., Analyt. Ed. 18, 469 (1946). [ll] H. CONRAD and W. HAYES, Trans. ASM 56, 125 (1963). [12] J. C. M. LI, Trans. AIME 233, 219 (1965). [13] W. G. JOHNSTON and J. J. GILMAN, J. appl. Phys. 30, 129 (1959). [14] Z. S. BASINSKI, Acta metall. 5, 684 (1957). [15] W. R. MARQUARDT and J. TRIVISONNO, J. Phys. Chem. Solids 26, 273 (1965). [I61 J. C. M. LI, Rattelle Conference on Dislocation Dynamics, to be published 1967. 1171 R. HILL, Proc. Phys. Soc. A65, 349 (1952). [18] A. S. KEH and Y. NAKADA, Canad. J. Phys. 45(2), 1101 (1967). [19] D. HULL and H. M. ROSENBERG, Proc. X. Internal Congr. Refrigeration, Copenhagen

1959 (p. 58).

(Received June 26, 1967)

thermally activated deformation of potassium

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