M. BOEEK, P. G A T O C H V f L , and P. LuKAE: Effect of Impurities 1221

phys. stat. sol. 2, 1221 (1962)

Department of Solid State Physics, Charles University, Prague

The Effect of Impurities on the Critical Resolved Shear Stress of Zinc Single Crystals

BY M. BO~EK, P. KRATOCHV~L, and P. LuKA~:

The dependence of the critical resolved shear stress of metal single crystals on impurity concentration is explained on the basis of the existence of impurity microsegregation in zinc. The suggested model gives very good agreement between the theoretical and experi- mental values of the constants in the Y - e function.

Die Abhangigkeit der kritischen Schubspannung in Metalleinkristallen von der Stor- stellenkonzentration wird im Fall von Zn durch Mikroausscheidungen von Storstellen erklart. Das vorgeschlagene Model1 ergibt fiir die Konstanten in der Y - @-Funktion gute Uberein- stimmung mit dem Experiment.

Introduction In recent years progress has been made in the problem of the critical resolved

shear stress in single crystals. Many measurements have been performed to ex- plain in detail the effect of many parameters on the value of the critical resolved shear stress, the effect of impurities being of the greatest interest.

The authors have been dealing with this problem previously [l]. Here we want to give a more detailed discussion of this subject. We have supposed that du- ring crystal growth a microsegregation takes place which is, according to TILLER [2] and PFANN [3], connected with the existence of a dislocation network. From this point of view we have investigated the relation between the critical resolved shear stress of zinc single crystals and their purity. As there was no possibility to check the concentration difference causing the generation of dislocations, we used the expression2 COi Ari (C,i is i-th solute concentration, Art is the diffe- rence of atomic radii of the i-th solute and of the solvent atoms) for plotting the impurity - resolved shear stress curves. In this work we use more accurate ex- pression for A C 141.

Results For the evaluation the measurements made by many authors [5 to 151 on zinc

single crystals were collected. We take only those with solutes : 1. that are soluble in the used range, i.e. they form solid solutions. In this

case we only can use TILLER’S model [2] of the distortion of crystals lattice accord- ing to Vegard’s law,

2 . the equilibrium distribution coefficient of which is known. On this basis i t is possible to use the expression [4],

where C, is the bulk liquid concentration and w the factor summarizing the geo- metrical effect of the microsegregation formation. According to TILLER [4]

1222 M. B O ~ E K , P. KRATOCHV~L, and P. L u K ~ E

w M 0.1 for elongated cells, w w 0.2 for normal regular cells, w w 0.3 for well developed regular cells and w ~ 5 ! 0.4 for dendrites.

According to the above mentioned criterions copper and cadmium as solutes in zinc are favourable. Lead, iron and tin have not sufficient solubility [16]. But, i t is very difficult to exclude these impurities as i t is not clear wether the solubility of small amounts of solute near the melting point may not induce the appearing of microsegregation of the same type as copper and cadmium. One could think that after cooling down the samples, the dislocation network might persist in the crystal lattice though the dissolvent atoms could gather into some typc of precipitation nuclei [17].

We used the values of the equilibrium distribution coefficients according to [IS, 18, 191: 0.07 for cadmium in zinc and 3.0 for copper in zinc. It is pos- sible, according to metallographic examination of samples used in the work of one of the authors [5] , to divide the materials into four groups with respect to the value of w .

I n Table 1 the expression AC A r for Cd and Cu are shown. We suppose the mean size of the microsegregation substructure to be 30 microns. An additive effect of all impurities is presumed here. The theoretical value of the densitiy of dislocations

Reference

~ ~~

[53

2 AC Ar a r,"

Q = 3

Tab le 1

c u ~~

13.33 9.83 5.55

13.30 0.18 0.12 0.24 0.18 0.02 0.35 0.01 0.06 0.11 -

- -

- - 0.05 0.16

0.06 0.03

-

-

AC Ar at. yo em .

- 1""- i

1.05 1.31 1.05 1.05 1.79 1.07 2.14 2.85 0.16 2.14 0.05 0.71 5.52 1.07

17.86 60.0

191.6 379.2

0.17 13.15 78.95

1.78 0.27 1.42

203 157 93

203 27 16 33 42 2.4

35 0.9

10.7 79.0 15

252 848

2708 5360

4.2 188.13

1116 25

20 4.2

252 242 238 220 167 157 134 72 3 1

145 18.4 65 80 57 94

274 825

1150 25 85

200 42 30 54

Effect of Impurities on the Critical Resolved Shear Stress 1223

is given in Table 1. The multiplication

factor cells (4 (-4) POI. 3 is the [2] and mean for value elongated for regular cells G- ‘ I : ; ! :/II According to SEEGER [12] the de- 8

pendence z = f (e ) (t is the critical 3 10‘

form resolved shear stress) should be of the 5

2 *-- ++-ti - t = a G b i e (3) ,03

10’2 5 70’2 5 10’ 2 5 where iy is a numerical constant de- pending on the geometry of the dis- location network, G the shear modulus, and b the Burgers vector. Our results given in Table 1 calculated by statistical methods give the t - Q function in the form

(4) This plot is shown in Fig. 1.

Now it is possible to prove the validity of the equation (4) comparing the value of the constant 2.6 to the expression a G b in (3). Using values a = 0.2, G = 4000 kg/mm2, b = 2.66 A, for zinc we obtain a G b = 2.08 glcm, which is in good agreement with (4).

These results : a) the square roote deFendence of T on e, b) the good value of the constant in (4), support the interpretation of the indirect effect of impurities on the critical resolved shear stress. The solute impurities present in the material induce the generation of dislocations in the manner proposed previously by TILLER [Z]. The ratio of these dislocations is involving the change of the value of the critical resolved shear stress according to SEEGER’S formula (3).

Further calculatior s have shown that some other metals behave in the same manner.

g ( c m ’1 - - Eig 1. Logarithmic plot of ther(p1 dependence

T = 2.6 (1;)’ 9 4 .

Authors wish to thank Dr. E. KLIER for many valuable comments on this work.

References [ l ] M. B O ~ E K , P. KRATOCHV~L, and P. LuKAE, Czech. J. Phys. B 11, 674 (1961). [2] W. A. TILLER, J. appl. Phys. 29, 611 (1958). [3] A. I. Goss, K. F. BENSON, and W. G. PFANN, Acta metall. 4, 332 (1956). [4] W. A. TILLER, Acta metall., (to be published). [5] M. BOEEK, Czech. J. Phys. B 10, 841 (1960). [6] D. C. JILLSON, J. Metals 2, 1129 (1950). [7] A. DERUYTTERE and G. B. GREENOUGH, J. Inst. Metals 84, 337 (1955/56). [8] W. FAHRENHORST and E. SCHMID, 2. Phys. 64, 845 (1930). [9] K. LUCKE et al., Z. Metallk. 46, 792 (1955).

[lo] P. ROSBAUD and E. SCHMIDT, Z. Phys. 38, 197 (1925). [ll] H. L. WAIN and A. H. COTTREL, Proc. Phys. SOC. B 67, 339 (1954). [12] J. J. GILMAN, J. Metals 8, 1326 (1956). [13] T. VREELAND e t al., J. Mech. Phys. Solids 6, 111 (1958). [14] M. J. DUMBLETON, Proc. Phys. SOC. B 67, 98 (1954). (151 A. SEECER, H. TRAUBLE, Z. Metallk. 51, 435 (1960). [16] M. HANSEN, K. ANDERKO, Constitution of Binary Alloys, Mc Graw-Hill, 1968.

1224 If. B O ~ E K , P. KRATOCHVfL, and P. LuKLE: Effect of Impurities

[I71 I. S. SERVI et al., Trans. A. I. M. E. 212, 361 (1958). [18] W. BOAS, Metallwirtschaft 11, 603 (1932). [19] P. KRATOCHV~L, P. L u K ~ ~ and M. VALOUCH, Czech. J. Fhys. B 10, 48 (1960). [20] M. BOEEK and P. KRATOCHVfL, Czech. J. Phys. 9,406 (1959). [21] A.SEEGER, Z.Naturf. 9a, 758 (1954).

(Received July 9, 1962)

Erratum In the paper by H. POLLAK:

Fe3+Ion Lifet ime in C O O Deduced from the Auger and Mossbauer Effects, phys. stat. sol. 2, 720 (1962),

all Li should read LI, and Liii (page 721, line 11) should read L,,,.