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Solution hardening and softening in KCl-KBr single
crystals at low temperature
T. Kataoka, T. Yamada
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 7, Tome 41, Juillet 1980, page C6-150
Solution hardening and softening in KCl-KBr single crystals
at low temperature
T. Kataoka and T. Yamada
Department of Precision Engineering,
Faculty of Engineering, Osaka University, Suita, Osaka 565, Japan
Résumé. — Nous avons étudié la dépendance en température et concentration de la cission réduite critique des
KCl-KBr solutions solides dans un domaine de température de 4,2 K à 293 K. Il a été vérifié que la solution solide au-dessous de 10 mole % KBr et au-dessus de 90 mole % KBr a montré l'adoucissement de solution solide à basse température. La solution solide de l'autre concentration a été durcie à toutes les températures essayées. Le dépla-cement de dislocation n'a été empêché que par des ions de soluté aux hautes concentrations et par les barrières Peierls en plus des ions de soluté aux basses concentrations.
Nous avons déterminé théoriquement la cission réduite critique et le volume d'activation en fonction de la tem-pérature et de la concentration dans les solutions solides hautement concentrées. Les résultats théoriques et expé-rimentaux sont en accord.
Abstract. — The temperature and concentration dependences of critical resolved shear stress (CRSS) for
KC1-KBr solid solutions were studied from 4.2 K to 293 K. At lower temperatures the solid solutions less than 10 mol % KBr and more than 90 mol % KBr showed solution softening. The solid solutions of the other concen-trations were hardened at all the temperatures tested. The dislocation motion in the higher concentration region was impeded only by the solute ions, and in the lower concentration region by the Peierls barriers in addition to the solute ions.
CRSS and activation volume were theoretically determined as a function of temperature and concentration in highly concentrated solid solutions. The theoretical and experimental results show a good agreement.
1. Introduction. — Some theories of hardening for
highly concentrated solid solution have been propos-ed by taking into account of the overlapping of poten-tials due to solute ions. The dependence of critical resolved shear stress (CRSS) on solute concentration,
c, has been represented by the following formulas;
c5/3(ln cf given by Mottand Nabarro [1], c11/9(ln c)4 / 3
by Mott [2], [c(l - c)]2 / 3 by Riddhagni and
Asi-mow [3, 4], c2 / 3 by Labusch [5] and [c(l - c)]1 / 2
by Boser [6]. However, the CRSS of KCl-KBr, NaCl-NaBr and NaCl-NaBr-KBr shows the parabolic concentra-tion dependence of c(l — c) at room temperature [7]. This relation does not agree with any of above theo-ries.
The other interest concerning the deformation of alkali halide solid solutions at low temperature is the possibility of solution softening. In b.c.c. metals, the solution softening is no uncommon occurrence and the dislocation models based on the Peierls mechanism have been proposed [8, 9]. As the low temperature deformation of alkali halide pure crys-tals is controlled by the Peierls mechanism [10, 11], it is expected in their solid solutions that solution softening phenomena are observed.
The purpose of this paper is to show the outline of our theory on solution hardening [12] and experi-ments in KCl-KBr solid solutions [13] and,
further-more, to discuss the mechanism of solution softening which was observed at low temperatures.
2. A model of solution hardening. — If the straight edge dislocation is laid in the stress field made by solute ions and external force, it takes a wavy shape so as to decrease its energy. This configuration is assumed simply to be a rectangular double kink as shown in figure 1. The width and height of rectangle are determined by the condition that the energy difference between straight and kinked configurations takes a maximum value. This energy difference,
AU, can be statistically calculated by considering the
contributions from internal stress due to solute ions, line tension of dislocation and applied stress, T.
* . • . • • • • • •
• • * • •
p v • •••-> •
• • • • • •
• t •
Fig. 1. — Stable configuration (solid line) and activated configura-tion (broken line) of dislocaconfigura-tion.
SOLUTION HARDENING AND SOFTENING IN KC1-KBr SINGLE CRYSTALS C6-151
The result is given by the following equation ;
6
0
0
where the numerical constants A& and
& and the
function g(x) depend on the stress field on the slip plane made by one solute ion which is obtained from the modified solution of elasticity theory. The para- meter a is a numerical factor indicating the line ten- sion of dislocation, p1 and p2 are the elastic constants,b is the Burgers vector, c is the concentration of solute ions and Av is a misfit between solute and solvent ions which is defined using lattice constants al and
a , for both crystals, i.e. Av = (a: - a:)/4.
If the shear strain rate j is assumed to be described by an Arrhenius equation,
where j, is a frequency factor, k is Boltzmann's constant and T is temperature, we can derive the CRSS for every temperature, concentration and strain rate. At this time, the adjustable
-
parameters -contained in the theory are only a, AU,, 7, and j,,
in which A& and
;,
are not determined independ- ently.3. Results and discussion. - 3.1 LOWER SOLUTION HARDENING. -The CRSS is shown in figure 2 as a
function of temperature. The strain rate was
1.3 x s-l. The CRSS of KC1 pure crystal decreases rapidly with temperature up to approxi- mately 50 K, which has been explained by the Peierls
mechanism 1141. For the solid solutions over the
concentration range from 20 rnol
%
to 49 rnol%,
the CRSS decreases gradually and continuously from 4.2 K to 293 K. The CRSS of 4 rnol
%
solid solution decreases rapidly up to approximately 50 Kas well as that of pure KC1 and shows gradual temper- ature dependence above 50 K. On this curve, special attention should be paid to the fact that the CRSS below 20 K is actually lower than that for pure KCl.
The solution softening phenomenon was observed also on the alkali halide crystal at low temperature. The solid lines in the figure represent the theoretical curves in which the adjustable parameters
-
are chosen as follows; a = 0.035,AU,
= 0.18, Z, = 19 andj, = 9.4 x lo6 s-I. These values are used as para- meters hereafter. For solid solutions above 20 mol%,
all data points fall on the theoretical curve but for
4 rnol
%
solid solution, the disagreement is remarka- ble particularly in the low temperature region in which the deformation is affected' by the Peierls mechanism.The CRSS is shown in figure 3 as a function of
o KCI-49rnol0l0 KBr
0 100 200 300
TEMPERATURE T ( K )
Fig. 2. -Temperature dependence of critical resolved shear stress for pure KC1 and KCI-KBr solid solutions. The solid lines repre- sent the theoretical curves.
Fig. 3. - Concentration dependence of critical resolved shear stress at 4.2, 77 and 293 K. The solid lines represent the theoretical curves.
concentration at representative temperatures. The experimental values are consistent with theoretical predictions described by solid lines except for the region showing solution softening. However, the experiment shows somewhat: larger values than the theory for the higher concentration of KBr. The fit would become better, if one assumes that the value of a decreases with increasing concentration. The phenomenon of solution softening is most strikingly displayed on the curve at 4.2 K. Below about 10 rnol
%
and above about 90 rnol
%,
the solid solutions have a lower CRSS than the pure KC1 and KBr, respective- ly. The minimum points of CRSS appear at about5 and 95 rnol
%
solid solutions.C6- 152 T. KATAOKA AND T. YAMADA
V = kT(a In jldz),, and is shown in figure 4 as a function of CRSS. In KCl-20, -22, -26 and -49 mol
%
KBr solid solutions, the activation volume decreases monotonously with increasing CRSS. In KC1 pure crystal and KC1-4mol
%
KBr solid solution, how- ever, the activation volume has a minimum point and this abnormal property is more remarkable for KC1-4 moI%
KBr soIid solution. Theoretical curves represented by solid lines are calculated by partial derivative of equation (1) with respect to z,i.e. V = - (a AU/az),.
" 0 10 2 0 30 40 50 60
CRSS 7= (MPa)
Fig. 4. - Activation volume as a function of CRSS. The solid lines represent the theoretical curves.
3.2 SOLID SOLUTION SOFTENING. - TWO quantita-
tive theories of solution softening have been proposed by Suzuki [8] and Sato and Meshii [9]. Suzuki has calculated the critical stress based on the modulus interaction of screw dislocation with solute atoms in the dislocation core and has shown that the softening and the hardening take place simultaneously at lower temperatures and only the hardening at higher temperatures. Sato and Meshii have simulated the screw dislocation motion under the stress field produc- ed by Peierls barriers and solute atoms. The results were able to explain the softening and hardening of b.c.c. solid solutions with comparatively lower con- centration. It is a characteristic common to both theories that, in b.c.c. crystals with high Peierls stress for screw dislocations, the solution softening takes place when the solute atoms assist the double
CONCENTRATION c (mol%)
Fig. 5. - Concentration dependence of CRSS at 4.2 K for some KC1 based solid solutions.
kink formation and the solution hardening takes place when the solute atoms prevent the sideward motion of the kink.
It has been suggested by Suzuki and Kim [lo, 111 that the low temperature deformation of alkali halide crystals is controlled by the Peierls mechanism. Solute ions can assist the double kink formation for the edge dislocation in alkali halides as well as for screw dislocation in b.c.c. metals. This is proved by performing the calculation similar to Sato and Meshii's one [9]. In this case, however, the interaction between solute ion and dislocation must be weak in order to cause solution softening because, if it is too strong, it acts as a pinning center for the disloca- tion motion. One proof of this idea is shown in figure 5, in which the concentration dependences of CRSS at 4.2 K are represented for some KC1 based solid solu- tions. The analysis of crystal composition has not been performed except for KCl-KBr and its values of data points mean a nominal content of solute ions mixed in a crucible when the crystal was grown from the melt. The arrows show that the specimen was fractured before the appearance of yield point. The values of parameter Au used in section 2 are shown in the figure. These values indicate the degree of interaction strength between solute ion and disloca- tion. The solid solutions of KCl-KBr and KCl- RbCl having weak interaction are softened but KCI-KI and KCl-NaCI having strong interaction are hardened.
DISCUSSION
SOLUTION HARDENING AND SOFTENING IN KCI-KBr SINGLE CRYSTALS C6-IS3
Reply. - T. KATAOKA. Question. - J . PHILIBERT.
An atomic absorption analysis indicated that the Is the softening you observe relative to the athermal CaZf content of our solid solution crystals was less stress or the thermally activated component of the than 1 mol ppm, but for the other impurities, we did stress ?
not make an analysis. The content of Ca2+ is largest
in the polyvalent impurities contained in our reagent- Reply. - T. KATAOKA.
grade powder of KC]. The stress which we measured means the total one.
References
[l] MOTT, N. F. and NABARRO, F. R. N., Report on Strength of
S o l i b (Phys. Soc., London) 1948, p. 1.
[2] MOTT, N. F., Imperfections in Nearly Perfect Crystals, eds. W. Shockley et al. (John Wiley and Sons, New York) 1952, p. 173.
[3] RIDDHAGNI, B. R. and ASIMOW, R. M., J. Appl. Phys. 39 (1968) 4144.
[4] RIDDHAGNI, B. R. and ASIMOW, R. M., J. Appl. Phys. 39 (1968) 5169.
[5] LABUSCH, R., Phys. Status Solidi 41 (1970) 659. [6] BOSER, O., Metall. Trans. 3 (1972) 843.
[7] KATAOKA, T. and YAMADA, T., Japan J. Appl. Phys. 16 (1977) 1119.
[8] SUZUKI, H., Nachr. Akad. Wiss. Gottingen 11. Math-Physik. Klasse (1971) 113.
[9] SATO, A. and MESHII, M., Acta Metall. 21 (1973) 758. [lo] SUZUKI, T. and KIM, H., J. Phys. Soc. Japan 39 (1975) 1566.
[ll] SUZUKI, T. and KIM, H., J. Phys. Soc. Japan 40 (1976) 1703. [12] KATAOKA, T. and YAMADA, T., Japan J. Appl. Phys. 18 (1979)
5s.
[13] KATAOKA, T., UEMATSU, T. and YAMADA, T., Japan J. Appl. Phys. 17 (1978) 271.
