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Plastic anisotropy of doped alkali halide single crystals
Materials Science and Engineering, 32 (1978) 55 - 63
55
© Elsevier Sequoia S.A., Lausanne -- Printed in the Netherlands
Plastic Anisotropy of Doped Alkali Halide Single Crystals
W. SKROTZKI, R. STEINRRECH* and P. HAASEN
Institut fiir Metallphysik, Universita't G6ttingen, G6ttingen (FRG) (Received June 27, 1977)
SUMMARY
The critical resolved shear stress (CRSS) of various alkali halides doped with divalent ions was measured for slip on {110} and {100} planes in compression and shear. The results are explained in terms of the interaction of edge dislocations with dopant-vacancy dipoles. While {110} glide around room temperature has to overcome long-range elastic (Snoek-type) interaction, in general a Fleischer-type interaction with both elastic and electrostatic contributions is responsible for the glide resistance. The so-calculated plastic anisotropy at T = 0 is in qualitative agreement with that extrapolated from experiment. The observed correlation of the plastic anisotropy and solution hardening parameters with the polarizability ( ~ ) of the ions of the host lattice is explained by the dependence of the electrostatic interaction energy between dislocation and dipole on £ ~.
ZUSAMMENFASSUNG
Die kritische Schubspannung (CRSS) von verschiedenen Alkalihalogeniden, dotiert mit zweiwertigen Ionen, wurde im Druck- und Schubversuch ftir Gleitung auf {110} und {100} Ebenen gemessen. Die Ergebnisse werden durch die Wechselwirkung yon Stufenversetzungen mit Verunreinigungs-Leerstellen-Dipolen erkl~irt. W~ihrend fiir {110} Gleitung um Raumtemperatur die langreichweitige elastische (Snoek-) Wechselwirkung ausschlaggebend ist, wird im allgemeinen die Gleitung durch einen Fleischer-Mechanismus
*Present address: Lehrstuhl fiir Werkstoffwissenschaften A, Universit//t Dortmund, Dortmund, F.R.G.
bestimmt, der sowohl elastische als auch elektrostatische Wechselwirkungsanteile umfasst. Die aus der Theorie berechnete Anisotropie bei T = 0 stimmt qualitativ mit der aus dem Experiment extrapolierten iiberein. Der beobachtete Gang der plastischen Anisotropie und der Mischkristallh/irtungsparameter mit der Polarisierbarkeit (£~) der Matrixionen wird mit der Abh/ingigkeit der elektrostatischen Wechselwirkungsenergie zwischen Versetzung und Dipol von £ a erkl/irt.
1. INTRODUCTION
In ionic crystals of the NaCl-structure, slip is possible on several non-equivalent crystallographic planes. Depending on the type of plane, the critical resolved shear stress (CRSS, to) differs in magnitude and temperature sensitivity. This slip anisotropy has been investigated frequently in alkali halide crystals. In particular for {110} and {100} glide planes, ro was determined in single slip experiments [1 - 3]. They show that slip is more difficult on (100} than on (110}. The anisotropy decreases with increasing polarizability of the ions constituting the crystal. Moreover, some measurements on NaC1 + Ca 2. crystals [3] indicate an influence of divalent impurities on the plastic anisotropy. To explain the experimental situation, theories were developed which differentiate between intrinsic and extrinsic reasons for the slip anisotropy. Gutmanas [2] assumed that a high Peierls potential limits (100} slip. Gilman [4] argued that the particular ionic configuration in the dislocation core after half a [110](001) slip step causes the high ro {1°°}. With reference to the dependence of ro on the impurity content and ionic polarizability, Haasen [5, 6] sees in the anisotropy an
56
extrinsic effect. He explained the slip anisotropy by different electrostatic interactions of {110} and (100} edge dislocations with divalent impurity ions. It is the aim of the present work to produce single slip on (110} and {100} in different alkali halides and to check the above theories by comparison with more extensive results. A shear test, already applied to NaC1 [3], was improved and extended to KC1 and KBr. Additionally, the divalent impurity content was varied. Contrary to Franzbecker [3], it was difficult to prove that single {100} slip was actually obtained. Most of the {110} deformation tests were carried out in compression because ro{11°} did not differ from that measured in shear tests.
i
SA
bJ/~
;
:::
adhesive
Fig. 1. Schematic arrangement for the short " o b l i q u e " shear test (b: Burgers vector, GNP: glide plane normal, SA: specimen axis, SD: shear direction, S I : side 1, $2: side 2, A: grip distance).
2. E X P E R I M E N T A L
Shear tests were performed in a short "oblique" set-up [3] sketched in Fig. 1. The specimen was glued into grips with "uhu plus", a slowly hardening, two-component adhesive. The orientation of the specimen was chosen in such a way that the glide system we wanted to activate was favoured by the stress distribution of the "oblique" shear test [7], i.e., specimen axis (SA) (100) or (111), side 1 (S1) {100} or {110}, side 2 ($2) {100} or (112} for {110} or {100} slip, respectively. The samples were prepared from single crystal ingots of NaC1, KC1 and KBr. Sources of supply for crystals used, the purity of starting material, and the dopants are listed in Table 1. The divalent doping concentration was determined by atomic absorption analysis. Specimens for {110} slip were cleaved
from large ingots. (100} specimens had to be string-sawn. All samples were annealed for 24 h at a temperature 100 K below their melting points and afterwards furnace cooled (with AT/At <. 60 K/h). Standard techniques were used to polish and etch the samples on different crystallographic planes --NaC1 [8], KC1 [9], KBr [10]. After annealing the dislocation density was about No ~ 105/cm 2. The final dimensions were 3.5 X 3.5 X 20 m m 3 for shear and 4 X 4 X (16 - 18) m m 3 for compression tests. The shear tests were performed in an Instron machine using the shear apparatus described by Franzbecker [3]. The same apparatus could also be used in a home-made, evacuated, constant-strain-rate machine to make in-situ carbon replicas of the glide steps
TABLE 1 Crystal characteristics NaCl Crystal origin
KC1 I
II
III
I
I
p.a.
p.a.
p.a.
p.a.
p.a.
suprapur**
--
SrCl 2
--
--
SrBr 2
I
II
Purity o f starting material
pro analysi (p.a.)*
Dopant
CaC12
I: II: III: *" **-
KBr
SFB 126 GSttingen/Clausthal, Kristall-Labor. Physikalisches Institut, Universit~/t Frankfurt. Firma Leitz, Wetzlar. p.a.: max. 30 p p m divalent impurities } (supplier guarantee). suprapur: max. 3 p p m divalent impurities
57
[11]. Depending on the grip distance (1 mm < /x < 2 mm) the shear strain rate was about d ~ (4 - 8) • 10-4/s. For deformation tests at temperatures different from room temperature (RT) the shear apparatus could be immersed in various inert baths. The quality of the adhesive bonding, as well as horizontal stresses at higher and lower temperatures, respectively, limited the temperature range of the shear tests. Dynamic compression tests were performed at an average strain rate d ~ 10-4/s.
3. RESULTS
3.1 Stress-strain curves To determine the CRSS the stage I of the stress-strain curve is of particular interest. Characteristic curves for {110) and {100} slip of doped KBr crystals are shown in Fig. 2. By comparison with compression tests the (110} shear curves show a sharper transition from elastic to plastic behaviour followed by a plateau region. The (100} shear curves exhibit an upper yield stress, which becomes more pronounced with increasing doping. I
1ram
Fig. 3. Etched $1 surface of {110} shear specimen.
[N/mm 2]
K Br p.a.÷ 44 ppm Sr *÷ a (100) compression b {1101 shear c {100} shear
¢
a[%]
Fig. 2. Resolved stress-strain curves for shear and compression tests.
3. 2 Proof o f single slip The resolved stress-strain curves are calculated using formulae for single slip in compression [12] or for a shear test [13]. This presumes that slip actually occurs only in the specimen volume between the two grips only on one glide system. Figure 3 shows the etched $1 surface of a {110} shear specimen. Slip is seen mainly in the region between the grips. The grip distance differs by about 20% from the etchpitted region. Slip is also limited to the [101] ( 1 0 i ) glide system. Only a few isolated etch-pit traces show where the orthogonal [101] (101) system was also active. From the calculations of the stress distribution in shear tests, the appearance of orthogonal slip is understandable [7]. With increasing doping of the crystals the fraction of orthogonal glide decreased. Observation by replicas from $2 confirmed the etch pit results.
58 result from (001), (111) or (11:[) glide planes which have the same Burgers vector. Etch-pit experiments on $2 could not differentiate between them because no etch-pit traces appeared there. After deformation, the specimens were cleaved parallel (001) and etched. No slip bands parallel to the [ l f 0 ] Burgers vector could be observed. It is therefore concluded that both {110} and {100} shear tests essentially produced slip on the expected glide plane. 3.3 CRSS as a function o f doping concentration and temperature The CRSS in {110} compression and shear tests was taken as the intersection of the linearly extrapolated elastic region and stage I. In the case of {100} shear the lower yield stress was chosen. Figure 5 shows the dependence of To(n°} on the doping concentration (CM~2÷) at RT. For all three mate.~{~101r N/ram 2 3 5
0.Smm
/ 3
Fig. 4. Slip steps on $1 surface of {100} shear specimen.
o
2
On $2 replicas from {I00} specimens no individual slip lines could be observed. For a better resolution some specimens were bevelled on $2. This yielded two advantages: the Burgers vector of the main glide system now intersected $2 normally and the angle for heavy metal shadowingcould be made smaller than 10 degrees. Still the $2 surface showed only wavy slip lines. These observations are different from those reported by Franzbecker [3]. The wavy features on $2 could be caused by crossslip of the screw dislocations of the main glide system. It was more informativeto observe slip steps on the $1 surface (Fig. 4). Particularly near the round edges (a polishing effect), slip steps appeared parallel to the [110] sheardirection, even at low magnification. Etch-pit studies showed slip bands dominating in the same direction and, again, a few traces from oblique {I00} and {110} glide systems. The slip bands and slip steps could
/,//
/ J / 1~
5;
o I v KC~pa,+Sr;.+i(lOO)compression o KBr p.o.,, S~) • KBr p.o.+Sr°+ {110]shear
,;o
,;o
2~o
---
CMe++Eppm3
Fig. 5. CRSS as a function of divalent doping concentration; (I00) compression test. rials, To{11°} is a linear function of C M e 2 ÷ and the slopes of the straight lines are nearly the same (Aro/CMe2 ÷ ~ 2 • 104 N/mm2). Compression and shear values of To(11°} agree quite well, e.g., for KBr. It should be understood that ro:[llO}(CMe~*= 0) is n o t the CRSS of the pure materials. K B r crystals from "suprapur" starting material, grown and annealed in an argon atmosphere, show a still lower CRSS although they have the same initial dislocation density as standard crystals grown in air: To KBr{n°} (Cs~÷ = 0) ~ 0.5 N / m m 2,
59 "~oCN/mm23
1:o r N/ram23 7 Sr'"
6
\
,110} (I:0]i
KBr p.a.
O
K Br p.a. ÷ 71 ppm KBr p.a.+ 83 ppm
0
•
5
4
KBr p.a.+ Sr +* (100) compression
at 77 K
2
1t 2'o
4b
6b
8'o
,b0
" T[K]
Csr** [ p p m ]
Fig. 6. Concentration dependence of CRSS at 77 K; <100> compression test.
-~(o100}EN/mm2]
(100} shear at 296 K
2'o
;o
8'o = Csr. (ppm]
Fig. 8. Temperature dependence of the CRSS (open marks: <100) compression, filled marks: (100) shear).
at RT. %{loo} is higher than ro (n°} and increases parabolically with C M e :+ . The temperature dependence of ro {n°} and ro (l°°} for KC1 and KBr is drawn in Fig. 8. The parameter of the curves is the doping concentration, ro {11°} (T) can be divided into two regions with strong and weak temperature dependence, respectively. Normalizing %{11o} (T > 250 K) by the shear modulus (p) their ratio is independent of temperature (plateau region), ro{l°°}(T) could be determined only in a small temperature range (273 - 330 K) because of experimental problems, ro (1°°} (T) shows a strong temperature dependence, the stronger, the higher the concentration. It should be noted that in the measured temperature interval, no plateau behaviour is observed for KBr, contrary to the results of Franzbecker [3] for NaCl p.a.
Fig. 7. CRSS as a function of doping concentration; (100} shear test. 4. DISCUSSI DN
ro gsr{lz0} {"suprapur '') ~ 0.3 N / m m 2. Therefore ro{11°} (CMd ÷ = 0) is mainly determined by impurities (aliovalent cations and anions) already present in the starting material. Contrary to the room temperature results, the ro{ZZ°~(C) behaviour is not linear at lower temperatures (77 K), Fig. 6. Figure 7 shows to{ 1°°} as a function of doping concentration
The experimental results show that To{n°) as well as ro {x°°} are strongly influenced by aliovalent ions. Moreover, the investigation of KBr crystals from "suprapur" starting material grown under a protective atmosphere indicates that an impurity independent lowest limit of the CRSS has not reached so far. This
60 result contradicts the assumption that the measured ro (l°°} is determined by a high Peierls stress in the investigated temperature range. According to Gutmanas [2] no such strong concentration dependence is expected. Also screw and edge dislocations should be restrained in a similar way by a Peierls stress [14]. The easy cross slip of screw dislocations observed shows, however, that they are highly mobile on {100} planes. This is confirmed for NaC1 by HVEM [15]. An estimate of To{i°°} based on Gilman's shear model [4] yields values which are more than one order of magnitude higher than those measured for our purest crystals; i.e., from Gilman's equation e2
ro (1°°} = 0.034 Kb---~
(1)
(e = electron charge, K = static dielectric constant) one gets: ro {l°°} (NaC1) = 55 N / m m 2, ro {1°°} (KC1) = 43 N/mm 2, ro {1°°} (KBr) = 35 N/mm 2 -- to be compared with Fig. 7.
Dislocation-solute interaction Haasen's general idea of dislocation-dopant interaction [5, 6] is now extended into a form in which it is able to describe the measured plastic anisotropy qualitatively. In the investigated temperature range and for the chosen concentrations, it is assumed that a divalent ion and a vacancy associate to form a dipole. Both the concentration and the temperature dependence of To suggest that two different interaction mechanisms determine the CRSS. The one yielding the stronger interaction between dipole and dislocation should dominate: (a) For {110} glide, the dislocation induced Snoek effect (with the features: plateau in to(T), linearity of to(C)) is d o m i n a n t at RT. This effect is due to an elastic interaction. The short-range electrostatic forces of the dislocation are of no importance for the changes in the dipole orientation. The CRSS for the Snoek mechanism is given in ref. 16 as
ATo{II0} ----0 @ / A e C
(2)
Ca = 6.91, numerical factor for the stronger interacting edge component, p{110} = (Cll _ C12)/2 shear modulus, Ae = ell .... e22, principal strains of the tetragonal dipole distortion, C = concentration of dipoles). The slopes of the experimental ATo{11°}(C) curves
and the Ae values calculated from eqn. (2) are listed in Table 2. The latter appear reasonable. TABLE 2 Solution hardening parameters for Snoek interaction Matrix
NaC1
KCI
KBr
2÷ ~ o { 1 1 0 } / C M e 2÷ [104 N / m m 2]
Ca 2+
Sr 2.
Sr 2+
1.84
2.02
2.08
p{110}~RT) 2 [10 N/ram ]
1.80
1.70
1.45
Ae
0.15
0.17
0.21
(b) The second mechanism is based on a Fleischer-type interaction (judging from the temperature dependence of to(T) and the linearity in to(x/C)). Besides an elastic (elast) interaction of dislocations and dipoles which is stronger for edges than for screws, we consider for the edges also an electrostatic (elst) interaction. The total interaction energy results from the superposition of both contributions. In order to calculate the interaction energy, the position of the dipoles in the dislocation core has to be known. The dipole can be orientated with the divalent cation above (position 1) or below (position 2) the immaterial slip plane of the dislocation. The corresponding m a x i m u m interaction energies are max + ei elst max (i ---- 1,2). ei -- ei elast
The sign of the e max is positive or negative depending on the type of glide plane, position and size of the dipole. After Friedrichs and Haasen [ 17], the contribution to the CRSS by different obstacle types is given for Fleischer interaction as Aro
\ / ( % + e2) X/eL V~- • (1--(P(T))
(3)
((P(T) = 0 for T = 0, eL ~ ub2/2 line tension). It is assumed that the same number of cations are in positions 1 and 2. Setting ee~s~t p b 3 e l l ( ~ { l l O } = ( C l l _ C 1 2 ) / 2 ' $/~10o} = C44, ell = (2/3)Ae [18]) and using calculated values for e elst max [19], the anisotropy ratio
Aro{lOO} A
Aro{110}
can be determined. At T = 0 we obtain A = 2.3 for NaC1 + Ca 2÷ {see Tables 3 and 4).
61 TABLE 3 Elastic and electrostatic interaction energies of dipoles with edge dislocations Matrix
NaCl
KCl
KBr
Me 2+
Ca 2÷
Sr 2+
Sr 2+
b[10 -lOm] /~{-110} (T = 0) [104 N / m m 2 ] /~{100} (T = 0) [104 N / m m 2 ]
3.99 2.38 1.33
4.45 2.17 0.67 0.11 ± 1.31 ± 0.41
4.67 1.81 0.53 0.14 ÷ 1.61 +_0.47
ell max el,2elast ({110}) leVi elmax,2 elast ({100)) leVi elmaXelst({110}) [eV] max 62els~ ({110}) leVi elmaXelst({100}) leVi e2maXelst({100}) leVi
0.10 +- 0.94 -+ 0.53 0.13 0.75 0.77 --0.46
TABLE 4 Solution hardening parameters for Fleischer-type interaction and sum of the polarizabilities of the matrix ions NaCl
KCI
Ca 2+
ATokllOf/,~ "t
p.a.
KBr
Sr 2+
p.a.
Sr 2+
p.a.
(T = 0)(exp)
[N/mm 2 ] Aro~ll%t/~/C (T = 0)(meor ) [ N / m m 2] (only elast.) [23 ]
1 563
1 231
872
783
648
Aro~lO0-~/~ C (T = 0)(exp) [N/mm 2 ] Aex p (T = O) Aex p (RT) Atheo r (T = 0) Y,0~ [10 _24 cm 3]
6 131
1 568 6.7
1.3 1.6
12.2 23.5
2.3 3.4
3.9 2.3 3.9
To calculate the experimental anisotropy ratio (Aexp), first the flow stress contribution of the dopant ions must be separated. Following Foreman and Makin [20] a quadratic superposition law is used 2 2 (AToMe2*)2 :- TO - - T o u
(rou = CRSS contribution from u n k n o w n impurities}. In the low temperature range the x/C-dependence of Aro {11°} is rather well established (Fig. 9). The same holds for Aro {1°°} at RT {Fig. 10). To get the CRSS at T = 0 the temperature dependence of ATo must be analysed. According to Ono [21] a r 1/2 v s . T 213 plot should give a straight line for nearly any type of obstacle profile deter-
4.5
5.6
mining the glide resistance. The experimental results, Fig. l l ( a ) and (b), are in good agreem e n t with this theoretical prediction. Of course, the extrapolation to T = 0 for {100} slip is connected with a larger error because the few measured points lie close together. The ATo/x/c and Aexp values taken from Fig. 11 for T = 0 are listed in Table 4. A comparison with Atheo r (T = 0) for NaC1 + Ca 2÷ shows an agreement in order of magnitude. This result is very satisfactory considering the crude theoretical estimates, which combine the atomistically calculated emaxi~l~twith an e maxi ~la~t, t h a t is calculated from linear elasticity. The significantly higher values for A~xp at RT result from the observed temperature
62 (110) A T o s r , . EN/rnm2]
.61/2 ~, 1/2, o(p, oj LI~I / m m J o (110} NaCl p.a. 1221 • 000) - [3] v (110} KCl p.a. c] (110) KBr p.a. • (100} •
i
(li01.77K
T 2/3 [ K 2/32
(a)
Ct/2 C(ppm)I/2.1 Sr*+
Fig. 9. Contribution of Sr2*-ions to the CRSS as a function of concentration; {110) glide, 77 K.
~-6~ . . . c N 1/2/ram3 o {110) NaCl÷ 16 pprn Ca*+r22] • (100~ NaCl÷18ppm Ca÷+E5] [] {110} K B r * ? $ ppm Sr *÷ {100) K B r * 8 3 ppm Sr÷*
A.~000} CNlmm 23 OSr*÷ 5 ~
1
1
2
KBr ÷ Sr ÷* i
{100}, 296K
J.
~
~
,'o
C 1/2 r(ppm) U2] Sr ÷¢
Fig. 10. As Fig. 9 for {100} glide, 296 K. dependences of To{11°) and to{l°°). In Fig. l l ( a ) and (b) nearly the same slopes are obtained for {100) and {110) slip. Furthermore, it must be concluded that the higher A~p values for p.a. material are due to different interaction energies of the u n k n o w n impurities. The relation between plastic anisotropy and electronic polarizability of the matrix ions can be deduced from the present model. With increasing Z ~ the electrostatic effect of impurity ions (and of dipoles) must decrease. In Table 4, ~ , A and for the investigated alkali halides show this clearly.
Aro/x/c
]
T2/3 E K 2/3. I
(b)
Fig. 11. Temperature dependence of the CRSS according to Ono [21 ]. (a) p.a. material; (b) contri• butlon of Me2 + .-Ions.
That theories based on purely elastic interactions are unable to explain is demonstrated by the experimental data. A calculation according to Mitchell and Heuer [23] yields Aro{ 1 1 0 = 0.35 M/~{1 1 0 }(Ae) 2 / 3 (Table 4) (M = anisotropic value for 1/(1 -- v), v = Poisson's ratio). The values increase from NaC1 + Ca2÷ to KBr + Sr 2+, contrary to observation.
Aro{ll°}/jc
}/x/c
63
How far this model is able to explain the experimental results quantitatively remains to be shown when further atomistic calculations are available. Especially calculations of the total interaction energy of edge dislocation with divalent ion-vacancy dipoles for different alkali halides and dopants would be helpful.
ACKNOWLEDGEMENT
The authors thank the GSttingen Akademie der Wissenschaften for support.
REFERENCES 1 J. J. Gilman, Acta Metall., 7 (1959) 608. 2 E. Yu. Gutmanas and E. M. Nadgornyi, Phys. Status Solidi, 38 (1970) 777. 3 W. Fra'nzbecker, Phys. Status Solidi (B), 57 (1973)545. 4 J. J. Gilman, J. Appl. Phys., 44 (1973) 982.
5 P. Haasen and W. Franzbecker, Nachr. GSttg. Akad. Wiss., (7) (1973) 135. 6 P. Haasen, in Dislocations et Processus de Transport, J. Phys. C, 34 (9) (1973) 205. 7 R. Steinbrech, P h . D . Dissertation, GSttingen, 1976. 8 R. Moran, J. Appl. Phys., 29 (1958} 1768. 9 S. V. Lubenets and N. F. Kostin, Kristallografiya, 7 (1962) 328. 10 W. Skrotzki, Diplomarbeit, GSttingen, 1977. 11 K.-H. Matucha, Phys. Status Solidi, 9 (1965) 209. 12 D. K. Bowen and J. W. Christian, Philos. Mag., 12 (1965) 369. 13 H. Scholl, Z. Metallkd., 44 (1953) 528. 14 P. Haasen, Scr. Metall., 9 (1975) 367. 15 H. Strunk, Phys. Status Solidi A, 28 (1975} 119. 16 W. Frank, Z. Naturforsch., 22a (1967) 377. 17 J. Friedrichs and P. Haasen, Philos. Mag., 31 (1975) 863. 18 D. M. Barnett and W. D. Nix, Acta Metall., 21 (1973) 1157. 19 H. H. Potstada, Diplomarbeit, Frankfurt, 1975. 20 A. J. E. Foreman and M. J. Makin, Can. J. Phys., 45 (1967) 511. 21 K. Ono, J. Appl. Phys., 39 (1968) 1803. 22 J. Hesse, in E. Rexer (ed.), Reinststoffprobleme, Bd. 3, Akademie-Verlag, Berlin, 1967. 23 T. E. Mitchell and A. H. Heuer, Mater. Sci. Eng., 28 (1977) 81.
55
© Elsevier Sequoia S.A., Lausanne -- Printed in the Netherlands
Plastic Anisotropy of Doped Alkali Halide Single Crystals
W. SKROTZKI, R. STEINRRECH* and P. HAASEN
Institut fiir Metallphysik, Universita't G6ttingen, G6ttingen (FRG) (Received June 27, 1977)
SUMMARY
The critical resolved shear stress (CRSS) of various alkali halides doped with divalent ions was measured for slip on {110} and {100} planes in compression and shear. The results are explained in terms of the interaction of edge dislocations with dopant-vacancy dipoles. While {110} glide around room temperature has to overcome long-range elastic (Snoek-type) interaction, in general a Fleischer-type interaction with both elastic and electrostatic contributions is responsible for the glide resistance. The so-calculated plastic anisotropy at T = 0 is in qualitative agreement with that extrapolated from experiment. The observed correlation of the plastic anisotropy and solution hardening parameters with the polarizability ( ~ ) of the ions of the host lattice is explained by the dependence of the electrostatic interaction energy between dislocation and dipole on £ ~.
ZUSAMMENFASSUNG
Die kritische Schubspannung (CRSS) von verschiedenen Alkalihalogeniden, dotiert mit zweiwertigen Ionen, wurde im Druck- und Schubversuch ftir Gleitung auf {110} und {100} Ebenen gemessen. Die Ergebnisse werden durch die Wechselwirkung yon Stufenversetzungen mit Verunreinigungs-Leerstellen-Dipolen erkl~irt. W~ihrend fiir {110} Gleitung um Raumtemperatur die langreichweitige elastische (Snoek-) Wechselwirkung ausschlaggebend ist, wird im allgemeinen die Gleitung durch einen Fleischer-Mechanismus
*Present address: Lehrstuhl fiir Werkstoffwissenschaften A, Universit//t Dortmund, Dortmund, F.R.G.
bestimmt, der sowohl elastische als auch elektrostatische Wechselwirkungsanteile umfasst. Die aus der Theorie berechnete Anisotropie bei T = 0 stimmt qualitativ mit der aus dem Experiment extrapolierten iiberein. Der beobachtete Gang der plastischen Anisotropie und der Mischkristallh/irtungsparameter mit der Polarisierbarkeit (£~) der Matrixionen wird mit der Abh/ingigkeit der elektrostatischen Wechselwirkungsenergie zwischen Versetzung und Dipol von £ a erkl/irt.
1. INTRODUCTION
In ionic crystals of the NaCl-structure, slip is possible on several non-equivalent crystallographic planes. Depending on the type of plane, the critical resolved shear stress (CRSS, to) differs in magnitude and temperature sensitivity. This slip anisotropy has been investigated frequently in alkali halide crystals. In particular for {110} and {100} glide planes, ro was determined in single slip experiments [1 - 3]. They show that slip is more difficult on (100} than on (110}. The anisotropy decreases with increasing polarizability of the ions constituting the crystal. Moreover, some measurements on NaC1 + Ca 2. crystals [3] indicate an influence of divalent impurities on the plastic anisotropy. To explain the experimental situation, theories were developed which differentiate between intrinsic and extrinsic reasons for the slip anisotropy. Gutmanas [2] assumed that a high Peierls potential limits (100} slip. Gilman [4] argued that the particular ionic configuration in the dislocation core after half a [110](001) slip step causes the high ro {1°°}. With reference to the dependence of ro on the impurity content and ionic polarizability, Haasen [5, 6] sees in the anisotropy an
56
extrinsic effect. He explained the slip anisotropy by different electrostatic interactions of {110} and (100} edge dislocations with divalent impurity ions. It is the aim of the present work to produce single slip on (110} and {100} in different alkali halides and to check the above theories by comparison with more extensive results. A shear test, already applied to NaC1 [3], was improved and extended to KC1 and KBr. Additionally, the divalent impurity content was varied. Contrary to Franzbecker [3], it was difficult to prove that single {100} slip was actually obtained. Most of the {110} deformation tests were carried out in compression because ro{11°} did not differ from that measured in shear tests.
i
SA
bJ/~
;
:::
adhesive
Fig. 1. Schematic arrangement for the short " o b l i q u e " shear test (b: Burgers vector, GNP: glide plane normal, SA: specimen axis, SD: shear direction, S I : side 1, $2: side 2, A: grip distance).
2. E X P E R I M E N T A L
Shear tests were performed in a short "oblique" set-up [3] sketched in Fig. 1. The specimen was glued into grips with "uhu plus", a slowly hardening, two-component adhesive. The orientation of the specimen was chosen in such a way that the glide system we wanted to activate was favoured by the stress distribution of the "oblique" shear test [7], i.e., specimen axis (SA) (100) or (111), side 1 (S1) {100} or {110}, side 2 ($2) {100} or (112} for {110} or {100} slip, respectively. The samples were prepared from single crystal ingots of NaC1, KC1 and KBr. Sources of supply for crystals used, the purity of starting material, and the dopants are listed in Table 1. The divalent doping concentration was determined by atomic absorption analysis. Specimens for {110} slip were cleaved
from large ingots. (100} specimens had to be string-sawn. All samples were annealed for 24 h at a temperature 100 K below their melting points and afterwards furnace cooled (with AT/At <. 60 K/h). Standard techniques were used to polish and etch the samples on different crystallographic planes --NaC1 [8], KC1 [9], KBr [10]. After annealing the dislocation density was about No ~ 105/cm 2. The final dimensions were 3.5 X 3.5 X 20 m m 3 for shear and 4 X 4 X (16 - 18) m m 3 for compression tests. The shear tests were performed in an Instron machine using the shear apparatus described by Franzbecker [3]. The same apparatus could also be used in a home-made, evacuated, constant-strain-rate machine to make in-situ carbon replicas of the glide steps
TABLE 1 Crystal characteristics NaCl Crystal origin
KC1 I
II
III
I
I
p.a.
p.a.
p.a.
p.a.
p.a.
suprapur**
--
SrCl 2
--
--
SrBr 2
I
II
Purity o f starting material
pro analysi (p.a.)*
Dopant
CaC12
I: II: III: *" **-
KBr
SFB 126 GSttingen/Clausthal, Kristall-Labor. Physikalisches Institut, Universit~/t Frankfurt. Firma Leitz, Wetzlar. p.a.: max. 30 p p m divalent impurities } (supplier guarantee). suprapur: max. 3 p p m divalent impurities
57
[11]. Depending on the grip distance (1 mm < /x < 2 mm) the shear strain rate was about d ~ (4 - 8) • 10-4/s. For deformation tests at temperatures different from room temperature (RT) the shear apparatus could be immersed in various inert baths. The quality of the adhesive bonding, as well as horizontal stresses at higher and lower temperatures, respectively, limited the temperature range of the shear tests. Dynamic compression tests were performed at an average strain rate d ~ 10-4/s.
3. RESULTS
3.1 Stress-strain curves To determine the CRSS the stage I of the stress-strain curve is of particular interest. Characteristic curves for {110) and {100} slip of doped KBr crystals are shown in Fig. 2. By comparison with compression tests the (110} shear curves show a sharper transition from elastic to plastic behaviour followed by a plateau region. The (100} shear curves exhibit an upper yield stress, which becomes more pronounced with increasing doping. I
1ram
Fig. 3. Etched $1 surface of {110} shear specimen.
[N/mm 2]
K Br p.a.÷ 44 ppm Sr *÷ a (100) compression b {1101 shear c {100} shear
¢
a[%]
Fig. 2. Resolved stress-strain curves for shear and compression tests.
3. 2 Proof o f single slip The resolved stress-strain curves are calculated using formulae for single slip in compression [12] or for a shear test [13]. This presumes that slip actually occurs only in the specimen volume between the two grips only on one glide system. Figure 3 shows the etched $1 surface of a {110} shear specimen. Slip is seen mainly in the region between the grips. The grip distance differs by about 20% from the etchpitted region. Slip is also limited to the [101] ( 1 0 i ) glide system. Only a few isolated etch-pit traces show where the orthogonal [101] (101) system was also active. From the calculations of the stress distribution in shear tests, the appearance of orthogonal slip is understandable [7]. With increasing doping of the crystals the fraction of orthogonal glide decreased. Observation by replicas from $2 confirmed the etch pit results.
58 result from (001), (111) or (11:[) glide planes which have the same Burgers vector. Etch-pit experiments on $2 could not differentiate between them because no etch-pit traces appeared there. After deformation, the specimens were cleaved parallel (001) and etched. No slip bands parallel to the [ l f 0 ] Burgers vector could be observed. It is therefore concluded that both {110} and {100} shear tests essentially produced slip on the expected glide plane. 3.3 CRSS as a function o f doping concentration and temperature The CRSS in {110} compression and shear tests was taken as the intersection of the linearly extrapolated elastic region and stage I. In the case of {100} shear the lower yield stress was chosen. Figure 5 shows the dependence of To(n°} on the doping concentration (CM~2÷) at RT. For all three mate.~{~101r N/ram 2 3 5
0.Smm
/ 3
Fig. 4. Slip steps on $1 surface of {100} shear specimen.
o
2
On $2 replicas from {I00} specimens no individual slip lines could be observed. For a better resolution some specimens were bevelled on $2. This yielded two advantages: the Burgers vector of the main glide system now intersected $2 normally and the angle for heavy metal shadowingcould be made smaller than 10 degrees. Still the $2 surface showed only wavy slip lines. These observations are different from those reported by Franzbecker [3]. The wavy features on $2 could be caused by crossslip of the screw dislocations of the main glide system. It was more informativeto observe slip steps on the $1 surface (Fig. 4). Particularly near the round edges (a polishing effect), slip steps appeared parallel to the [110] sheardirection, even at low magnification. Etch-pit studies showed slip bands dominating in the same direction and, again, a few traces from oblique {I00} and {110} glide systems. The slip bands and slip steps could
/,//
/ J / 1~
5;
o I v KC~pa,+Sr;.+i(lOO)compression o KBr p.o.,, S~) • KBr p.o.+Sr°+ {110]shear
,;o
,;o
2~o
---
CMe++Eppm3
Fig. 5. CRSS as a function of divalent doping concentration; (I00) compression test. rials, To{11°} is a linear function of C M e 2 ÷ and the slopes of the straight lines are nearly the same (Aro/CMe2 ÷ ~ 2 • 104 N/mm2). Compression and shear values of To(11°} agree quite well, e.g., for KBr. It should be understood that ro:[llO}(CMe~*= 0) is n o t the CRSS of the pure materials. K B r crystals from "suprapur" starting material, grown and annealed in an argon atmosphere, show a still lower CRSS although they have the same initial dislocation density as standard crystals grown in air: To KBr{n°} (Cs~÷ = 0) ~ 0.5 N / m m 2,
59 "~oCN/mm23
1:o r N/ram23 7 Sr'"
6
\
,110} (I:0]i
KBr p.a.
O
K Br p.a. ÷ 71 ppm KBr p.a.+ 83 ppm
0
•
5
4
KBr p.a.+ Sr +* (100) compression
at 77 K
2
1t 2'o
4b
6b
8'o
,b0
" T[K]
Csr** [ p p m ]
Fig. 6. Concentration dependence of CRSS at 77 K; <100> compression test.
-~(o100}EN/mm2]
(100} shear at 296 K
2'o
;o
8'o = Csr. (ppm]
Fig. 8. Temperature dependence of the CRSS (open marks: <100) compression, filled marks: (100) shear).
at RT. %{loo} is higher than ro (n°} and increases parabolically with C M e :+ . The temperature dependence of ro {n°} and ro (l°°} for KC1 and KBr is drawn in Fig. 8. The parameter of the curves is the doping concentration, ro {11°} (T) can be divided into two regions with strong and weak temperature dependence, respectively. Normalizing %{11o} (T > 250 K) by the shear modulus (p) their ratio is independent of temperature (plateau region), ro{l°°}(T) could be determined only in a small temperature range (273 - 330 K) because of experimental problems, ro (1°°} (T) shows a strong temperature dependence, the stronger, the higher the concentration. It should be noted that in the measured temperature interval, no plateau behaviour is observed for KBr, contrary to the results of Franzbecker [3] for NaCl p.a.
Fig. 7. CRSS as a function of doping concentration; (100} shear test. 4. DISCUSSI DN
ro gsr{lz0} {"suprapur '') ~ 0.3 N / m m 2. Therefore ro{11°} (CMd ÷ = 0) is mainly determined by impurities (aliovalent cations and anions) already present in the starting material. Contrary to the room temperature results, the ro{ZZ°~(C) behaviour is not linear at lower temperatures (77 K), Fig. 6. Figure 7 shows to{ 1°°} as a function of doping concentration
The experimental results show that To{n°) as well as ro {x°°} are strongly influenced by aliovalent ions. Moreover, the investigation of KBr crystals from "suprapur" starting material grown under a protective atmosphere indicates that an impurity independent lowest limit of the CRSS has not reached so far. This
60 result contradicts the assumption that the measured ro (l°°} is determined by a high Peierls stress in the investigated temperature range. According to Gutmanas [2] no such strong concentration dependence is expected. Also screw and edge dislocations should be restrained in a similar way by a Peierls stress [14]. The easy cross slip of screw dislocations observed shows, however, that they are highly mobile on {100} planes. This is confirmed for NaC1 by HVEM [15]. An estimate of To{i°°} based on Gilman's shear model [4] yields values which are more than one order of magnitude higher than those measured for our purest crystals; i.e., from Gilman's equation e2
ro (1°°} = 0.034 Kb---~
(1)
(e = electron charge, K = static dielectric constant) one gets: ro {l°°} (NaC1) = 55 N / m m 2, ro {1°°} (KC1) = 43 N/mm 2, ro {1°°} (KBr) = 35 N/mm 2 -- to be compared with Fig. 7.
Dislocation-solute interaction Haasen's general idea of dislocation-dopant interaction [5, 6] is now extended into a form in which it is able to describe the measured plastic anisotropy qualitatively. In the investigated temperature range and for the chosen concentrations, it is assumed that a divalent ion and a vacancy associate to form a dipole. Both the concentration and the temperature dependence of To suggest that two different interaction mechanisms determine the CRSS. The one yielding the stronger interaction between dipole and dislocation should dominate: (a) For {110} glide, the dislocation induced Snoek effect (with the features: plateau in to(T), linearity of to(C)) is d o m i n a n t at RT. This effect is due to an elastic interaction. The short-range electrostatic forces of the dislocation are of no importance for the changes in the dipole orientation. The CRSS for the Snoek mechanism is given in ref. 16 as
ATo{II0} ----0 @ / A e C
(2)
Ca = 6.91, numerical factor for the stronger interacting edge component, p{110} = (Cll _ C12)/2 shear modulus, Ae = ell .... e22, principal strains of the tetragonal dipole distortion, C = concentration of dipoles). The slopes of the experimental ATo{11°}(C) curves
and the Ae values calculated from eqn. (2) are listed in Table 2. The latter appear reasonable. TABLE 2 Solution hardening parameters for Snoek interaction Matrix
NaC1
KCI
KBr
2÷ ~ o { 1 1 0 } / C M e 2÷ [104 N / m m 2]
Ca 2+
Sr 2.
Sr 2+
1.84
2.02
2.08
p{110}~RT) 2 [10 N/ram ]
1.80
1.70
1.45
Ae
0.15
0.17
0.21
(b) The second mechanism is based on a Fleischer-type interaction (judging from the temperature dependence of to(T) and the linearity in to(x/C)). Besides an elastic (elast) interaction of dislocations and dipoles which is stronger for edges than for screws, we consider for the edges also an electrostatic (elst) interaction. The total interaction energy results from the superposition of both contributions. In order to calculate the interaction energy, the position of the dipoles in the dislocation core has to be known. The dipole can be orientated with the divalent cation above (position 1) or below (position 2) the immaterial slip plane of the dislocation. The corresponding m a x i m u m interaction energies are max + ei elst max (i ---- 1,2). ei -- ei elast
The sign of the e max is positive or negative depending on the type of glide plane, position and size of the dipole. After Friedrichs and Haasen [ 17], the contribution to the CRSS by different obstacle types is given for Fleischer interaction as Aro
\ / ( % + e2) X/eL V~- • (1--(P(T))
(3)
((P(T) = 0 for T = 0, eL ~ ub2/2 line tension). It is assumed that the same number of cations are in positions 1 and 2. Setting ee~s~t p b 3 e l l ( ~ { l l O } = ( C l l _ C 1 2 ) / 2 ' $/~10o} = C44, ell = (2/3)Ae [18]) and using calculated values for e elst max [19], the anisotropy ratio
Aro{lOO} A
Aro{110}
can be determined. At T = 0 we obtain A = 2.3 for NaC1 + Ca 2÷ {see Tables 3 and 4).
61 TABLE 3 Elastic and electrostatic interaction energies of dipoles with edge dislocations Matrix
NaCl
KCl
KBr
Me 2+
Ca 2÷
Sr 2+
Sr 2+
b[10 -lOm] /~{-110} (T = 0) [104 N / m m 2 ] /~{100} (T = 0) [104 N / m m 2 ]
3.99 2.38 1.33
4.45 2.17 0.67 0.11 ± 1.31 ± 0.41
4.67 1.81 0.53 0.14 ÷ 1.61 +_0.47
ell max el,2elast ({110}) leVi elmax,2 elast ({100)) leVi elmaXelst({110}) [eV] max 62els~ ({110}) leVi elmaXelst({100}) leVi e2maXelst({100}) leVi
0.10 +- 0.94 -+ 0.53 0.13 0.75 0.77 --0.46
TABLE 4 Solution hardening parameters for Fleischer-type interaction and sum of the polarizabilities of the matrix ions NaCl
KCI
Ca 2+
ATokllOf/,~ "t
p.a.
KBr
Sr 2+
p.a.
Sr 2+
p.a.
(T = 0)(exp)
[N/mm 2 ] Aro~ll%t/~/C (T = 0)(meor ) [ N / m m 2] (only elast.) [23 ]
1 563
1 231
872
783
648
Aro~lO0-~/~ C (T = 0)(exp) [N/mm 2 ] Aex p (T = O) Aex p (RT) Atheo r (T = 0) Y,0~ [10 _24 cm 3]
6 131
1 568 6.7
1.3 1.6
12.2 23.5
2.3 3.4
3.9 2.3 3.9
To calculate the experimental anisotropy ratio (Aexp), first the flow stress contribution of the dopant ions must be separated. Following Foreman and Makin [20] a quadratic superposition law is used 2 2 (AToMe2*)2 :- TO - - T o u
(rou = CRSS contribution from u n k n o w n impurities}. In the low temperature range the x/C-dependence of Aro {11°} is rather well established (Fig. 9). The same holds for Aro {1°°} at RT {Fig. 10). To get the CRSS at T = 0 the temperature dependence of ATo must be analysed. According to Ono [21] a r 1/2 v s . T 213 plot should give a straight line for nearly any type of obstacle profile deter-
4.5
5.6
mining the glide resistance. The experimental results, Fig. l l ( a ) and (b), are in good agreem e n t with this theoretical prediction. Of course, the extrapolation to T = 0 for {100} slip is connected with a larger error because the few measured points lie close together. The ATo/x/c and Aexp values taken from Fig. 11 for T = 0 are listed in Table 4. A comparison with Atheo r (T = 0) for NaC1 + Ca 2÷ shows an agreement in order of magnitude. This result is very satisfactory considering the crude theoretical estimates, which combine the atomistically calculated emaxi~l~twith an e maxi ~la~t, t h a t is calculated from linear elasticity. The significantly higher values for A~xp at RT result from the observed temperature
62 (110) A T o s r , . EN/rnm2]
.61/2 ~, 1/2, o(p, oj LI~I / m m J o (110} NaCl p.a. 1221 • 000) - [3] v (110} KCl p.a. c] (110) KBr p.a. • (100} •
i
(li01.77K
T 2/3 [ K 2/32
(a)
Ct/2 C(ppm)I/2.1 Sr*+
Fig. 9. Contribution of Sr2*-ions to the CRSS as a function of concentration; {110) glide, 77 K.
~-6~ . . . c N 1/2/ram3 o {110) NaCl÷ 16 pprn Ca*+r22] • (100~ NaCl÷18ppm Ca÷+E5] [] {110} K B r * ? $ ppm Sr *÷ {100) K B r * 8 3 ppm Sr÷*
A.~000} CNlmm 23 OSr*÷ 5 ~
1
1
2
KBr ÷ Sr ÷* i
{100}, 296K
J.
~
~
,'o
C 1/2 r(ppm) U2] Sr ÷¢
Fig. 10. As Fig. 9 for {100} glide, 296 K. dependences of To{11°) and to{l°°). In Fig. l l ( a ) and (b) nearly the same slopes are obtained for {100) and {110) slip. Furthermore, it must be concluded that the higher A~p values for p.a. material are due to different interaction energies of the u n k n o w n impurities. The relation between plastic anisotropy and electronic polarizability of the matrix ions can be deduced from the present model. With increasing Z ~ the electrostatic effect of impurity ions (and of dipoles) must decrease. In Table 4, ~ , A and for the investigated alkali halides show this clearly.
Aro/x/c
]
T2/3 E K 2/3. I
(b)
Fig. 11. Temperature dependence of the CRSS according to Ono [21 ]. (a) p.a. material; (b) contri• butlon of Me2 + .-Ions.
That theories based on purely elastic interactions are unable to explain is demonstrated by the experimental data. A calculation according to Mitchell and Heuer [23] yields Aro{ 1 1 0 = 0.35 M/~{1 1 0 }(Ae) 2 / 3 (Table 4) (M = anisotropic value for 1/(1 -- v), v = Poisson's ratio). The values increase from NaC1 + Ca2÷ to KBr + Sr 2+, contrary to observation.
Aro{ll°}/jc
}/x/c
63
How far this model is able to explain the experimental results quantitatively remains to be shown when further atomistic calculations are available. Especially calculations of the total interaction energy of edge dislocation with divalent ion-vacancy dipoles for different alkali halides and dopants would be helpful.
ACKNOWLEDGEMENT
The authors thank the GSttingen Akademie der Wissenschaften for support.
REFERENCES 1 J. J. Gilman, Acta Metall., 7 (1959) 608. 2 E. Yu. Gutmanas and E. M. Nadgornyi, Phys. Status Solidi, 38 (1970) 777. 3 W. Fra'nzbecker, Phys. Status Solidi (B), 57 (1973)545. 4 J. J. Gilman, J. Appl. Phys., 44 (1973) 982.
5 P. Haasen and W. Franzbecker, Nachr. GSttg. Akad. Wiss., (7) (1973) 135. 6 P. Haasen, in Dislocations et Processus de Transport, J. Phys. C, 34 (9) (1973) 205. 7 R. Steinbrech, P h . D . Dissertation, GSttingen, 1976. 8 R. Moran, J. Appl. Phys., 29 (1958} 1768. 9 S. V. Lubenets and N. F. Kostin, Kristallografiya, 7 (1962) 328. 10 W. Skrotzki, Diplomarbeit, GSttingen, 1977. 11 K.-H. Matucha, Phys. Status Solidi, 9 (1965) 209. 12 D. K. Bowen and J. W. Christian, Philos. Mag., 12 (1965) 369. 13 H. Scholl, Z. Metallkd., 44 (1953) 528. 14 P. Haasen, Scr. Metall., 9 (1975) 367. 15 H. Strunk, Phys. Status Solidi A, 28 (1975} 119. 16 W. Frank, Z. Naturforsch., 22a (1967) 377. 17 J. Friedrichs and P. Haasen, Philos. Mag., 31 (1975) 863. 18 D. M. Barnett and W. D. Nix, Acta Metall., 21 (1973) 1157. 19 H. H. Potstada, Diplomarbeit, Frankfurt, 1975. 20 A. J. E. Foreman and M. J. Makin, Can. J. Phys., 45 (1967) 511. 21 K. Ono, J. Appl. Phys., 39 (1968) 1803. 22 J. Hesse, in E. Rexer (ed.), Reinststoffprobleme, Bd. 3, Akademie-Verlag, Berlin, 1967. 23 T. E. Mitchell and A. H. Heuer, Mater. Sci. Eng., 28 (1977) 81.