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Solid solution hardening in iron doped MgO
Materials Science and Engineering, 22 (1976) 71-84
71
© Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands
Solid Solution Hardening in Iron Doped MgO*
B. REPPICH Institut fiir Werkstoffwissenschaften I, University Erlangen:Niirnberg, D-852 Erlangen (Germany)
(Received May 20, 1975; in revised form October 20, 1975)
SUMMARY The critical resolved shear stress To of quenched low iron doped MgO single crystals of different purity has been investigated between room temperature and 2000 K. It is shown that at low and elevated temperatures To is determined by the combined elastic and electrostatic interaction of the dislocations with single dipoles, composed of two trivalent iron ions and a charge-compensating cation vacancy. The elastic tetragonal hardening dominates. In the low-temperature decrease below 700 K ro is proportional to ~/CD (CD = dipole concentration), depends on strain rate, and is controlled by thermally activated passing of dislocations near dipoles located close to the slip plane (Fleischer interaction). In the temperature independent " p l a t e a u " above 700 K ro depends linearly on CD and can be explained by long-range dipole interaction via the induced Snoek effect. The dipole strength was estimated to be A~* = 0.42. The rapid high-temperature decrease of ro above 1300 K is attributed to the thermal dissociation of the dipoles.
1. INTRODUCTION The critical resolved shear stress (CRSS) of monovalent ionic crystals, e.g. NaC1 and LiF, is significantly increased by divalent impurity cations ([1, 2], refs. 1 - 6). After suitable thermal pretreatment the impurities can be present in the form of non-agglomerated pairs composed of an impurity cation and chargecompensating cation vacancy (hereinafter *Presented at the Friihjahrstagung des Arbeitskreises FestkSrperphysik der DPG, Freudenstadt, April 1974, Vortrag M 19.
called "dipoles"). The tetragonal strain field of the dipoles gives rise to a strong elastic interaction with the dislocations. Consequently, in the literature, the solid solution hardening of the alkali halides has been attributed to these elastic dipoles. The experimental results agree quantitatively [1, 2] with the theory of Frank [3, 4] which is an extension of the ideas of Fleischer [5, 6] and of Schoeck and Seeger [7]. Similar effects may be predicted for divalent ionic crystals doped with transition metal ions, e.g. Ti, V, Cr and Fe. On the other hand, the situation is more complex in these materials. For example, in iron
72
has yielded t o o small a contribution to the CRSS, while for the latter defect t y p e t o o high pair concentrations were obtained. According to Moon and Pratt [8] and Wicks and Lewis [9] the region of elevated temperatures, which is characterized b y a temperature and deformation rate independent CRSS, is attributed to the reorienting of tetragonal defects formed of Fe a÷ or other impurities in the elastic stress field of moving dislocations [ 7]. Miles et al. [10] perceive no correlations between the CRSS and Fe 3÷ vacancy dipoles and suppose that the hardening effect is associated with an impurity atmosphere pinning of the dislocations by Cottrell locking [14, 15]. Starting from this unsatisfactory situation in the present study the question is raised again -- which mechanisms control the solid solution hardening of typical ionic oxides doped with transition metal ions? Deformation experiments on low iron doped and suitably thermally pretreated MgO single crystals between r o o m temperature (RT) and 2000 K are described. The concept of dipole hardening, including electrostatic dipole interaction [ 1 6 ] , is applied on MgO and is examined experimentally.
2. T H E O R E T I C A L B A C K G R O U N D
2.1. Solid solution hardening Solid solution hardening is caused b y the interaction of dislocations with solute foreign atoms or other obstacles of atomic size. With regard to their theoretical approaches the existing theories of solid solution hardening can be classified into t w o groups [ 1 7 ] . In the first group the interaction of dislocations with individual obstacles is the elementary process (in the following called "localized" interaction). Theories of group t w o frequently consider statistical "fluctuations" of the interaction, averaged over a characteristic length, as the effective obstacles (in the following called interaction with "obstacle groups"). According to Labusch [17] all theories give the following relation between the critical resolved shear stress to, the maximum interaction force F, the range of interaction w b e t w e e n solute atoms and dislocations, the line tension t and the concentration of solute atdms c 4
1
FY w~- :__ ~o ~ , c~ . t~-
(1)
Labusch has further demonstrated that, depending on the choice of the given parameters, the calculations based on individual obstacles yield either eqn. (1) or the Fleischer - Friedel formula 3
F~- ± ro = , c: . t~-
(2)
Practically, eqn. (2) is limited to dilute solutions of "hard" single obstacles presented in concentrations of less than 10 -a [ 17 ]. But, in contrast with the f.c.c, metals, this condition is fulfilled in the present case*. Moreover, for the dipole hardening discussed here, the ratio F/t, which characterizes the strength of interaction, is larger than that for the combined parelastic/dielastic interaction in metallic alloys [14, 1 5 ] . In other words, it should be justifiable to describe the solid solution hardening in the present low iron d o p e d MgO on the basis of the localized interaction, eqn. (2) (see subsection 2.2)**. On the other hand, the argumentation is reversible; the experimental conditions are realized here for examining the theory of solid solution hardening for the special case of strong localized interaction. 2.2. Dipole hardening in Fe-doped MgO 2.2.1. Impurity - vacancy complexes in MgO The substitution of t w o trivalent iron ions Fe ~ * into the otherwise neutral MgO lattice requires formation of one cation vacancy (V). Owing to the Coulomb attraction between the effectively positively charged Fe 3÷ and the doubly negatively charged V, a cation vacancy pair forms [21 - 24] in analogy with the alkali halides [ 2 0 ] . Following Ilschner [25] thermodynamic and energetic points of view a complex of the form (Fe3÷V Fe 3÷) is suggested. From the possible crystallographic configurations the linear-symmetrical < 110> t y p e is the most favourable. 2.2.2. The critical resolved shear stress We assume that the (FeS*V Fe 3÷) complexes represent elastic dipoles [26] which have a *As shown in subsection 3.4 for the largest iron doping level the dipole concentration is <3 × 10 -3. **Localized interaction could be assumed with the same a r g u m e n t a t i o n for MgO containing the doped Fe 3÷ in f o r m o f dipole clusters [18] or precipitate magnesia ferrite particles [ 19 ].
73 clear t e n d e n c y for strong elastic i n t e r a c t i o n with dislocation lines due t o t h e i r t e t r a g o n a l strain field a c c o r d i n g t o N a b a r r o [ 2 7 ] , C o c h a r d t et al. [28] and Fleischer [ 5 ] . F r a n k [3, 4] has p r o p o s e d a m o d e l which describes t h e influence o f t e t r a g o n a l d e f e c t s in cubic crystals. In addition to the elastic i n t e r a c t i o n , the e l e c t r o s t a t i c dipole i n t e r a c t i o n m u s t be t a k e n into a c c o u n t . T h e c o n t r i b u t i o n o f the e l e c t r o s t a t i c i n t e r a c t i o n b e t w e e n charged screw dislocations and c a t i o n - v a c a n c y - d i p o l e s has been r e c e n t l y e s t i m a t e d b y F r a n k [ 1 6 ] . We a p p l y n o w the c o n c e p t o f the c o m b i n e d elastic/electrostatic dipole h a r d e n i n g o n MgO. T h e n f o r MgO o f sufficiently high p u r i t y containing the d o p e d Fe 3~ in the f o r m o f disperse d i s t r i b u t e d and n o n - a g g l o m e r a t e d dipoles, t h e following regions and m e c h a n i s m s m a y be e x p e c t e d (Fig. 1). (i) The low temperature decrease o f 7.o. At low t e m p e r a t u r e s ro decreases s t r o n g l y with increasing t e m p e r a t u r e s T a n d is given b y 7.o = 7.G + 7.~
(3a)
where rG = a Gb ~/p
(3b)
is the T - i n d e p e n d e n t c o n t r i b u t i o n t o 7.0 d u e to long range d i s l o c a t i o n / d i s l o c a t i o n i n t e r a c t i o n (~ is a c o n s t a n t ) . T h e t e m p e r a t u r e d e p e n d e n t c o m p o n e n t rs is d e t e r m i n e d b y s h o r t range dipole i n t e r a c t i o n . T h e t h e r m a l l y a c t i v a t e d dislocations " p a s s " t h e dipoles l o c a t e d close t o the slip plane resulting in an elastic contrib u t i o n to % (index " F " f o r Fleischer interaction) [ 3 ] : GAX* 7.sF _-- Z 1
~-~
~/CD (1 -- ~/01T)2;
(4)
Z 1 = 0 . 1 0 7 is a numerical factor, G t h e shear + m o d u l u s , b = 2.85 A t h e Burgers vector, p t h e dislocation density, d = 0.5 the distance o f the interacting dipoles f r o m the slip plane in units of b, c D the a t o m i c d i p o l e c o n c e n t r a t i o n . T h e function ¢1 = Z 2 - - 1 n GbaAX *
dipole dlssociehOn
dipole interection
f Fleischer sslng
I Induced Snoek I effect
elastic :}
[ \
electrostattc
,
-
0
RT
To2
To'
T'---'~ Fig. 1. Schematic representation of the contributions to the CRSS of low-iron-doped MgO at low and elevated temperatures due to combined elastic/electrostatic dipole interaction (temperature dependence of elastic constants neglected); TG is not drawn, see text. D e b y e f r e q u e n c y , Z2 = 10.82, a n u m e r i c a l factor. T h e " d i p o l e s t r e n g t h " Ak* is related to the free activation e n t h a l p y E 0 (at %r = 0), Z 3 = 0.092 is the n u m e r i c a l factor, Gb 3 Eo = Z3 - - A~*. d
(6)
T h e electrostatic c o n t r i b u t i o n to 7.s is [ 16] 7.sest = Z 4
qQ/b3KKo ~/C D ( 1 - - N / ~ ) 2 T ) 2 d2
(7)
with In , (8) qQ/~o w h e r e Z 4 = 0 . 1 4 2 and Z 5 = 8.18 are numerical factors, Q is the (positive) charge o f t h e dipoles, q t h e charge d e n s i t y per line length o f dislocation [30, 3 1 ] , K t h e static dielectric c o n s t a n t , Ko = 8.86 X 10 -14 ( c o u l o m b ) 2 Vsm. It is recognised t h a t eqns. (4) and (7), as well as eqns. (5) and (8), have the analogous f o r m ; the " m e c h a n i c a l " t e r m (Gb3AX *) corresponds t o the " e l e c t r o s t a t i c " t e r m (qQ/n ~o). N o t e t h a t b o t h % c o n t r i b u t i o n s have the same T and X/CD d e p e n d e n c e . (~2 = Z 5 -
(5)
contains t h e d e p e n d e n c e o n t h e d e f o r m a t i o n rate d. k is the B o l t z m a n n c o n s t a n t , vn the +The shear m o d u l u s is G = 1/2 (c11-- c12), w i t h t h e Voigt's elastic c o n s t a n t s a c c o r d i n g t o ref. 29.
(ii) The plateau. A b o v e t h e low t e m p e r a t u r e decrease o f 7.0, at elevated t e m p e r a t u r e s , a region is p r e d i c t e d w h e r e 7.0 is i n d e p e n d e n t o f T. In this region 7.o is c o m p o s e d o f 7"G and 7.p, and we have 7.o = 7.G + 7.p, w h e r e 7.p is c o n t r o l l e d b y long range d i p o l e i n t e r a c t i o n via t h e i n d u c e d
74
Snoek effect [ 3 3 ] , index " P " for plateau. The stress field of the moving dislocation produces a Snoek cloud around it, consisting of oriented (Fe s÷ V Fe a÷) dipoles [ 3 2 ] . In this way the dislocation "digs itself in" energetically and a local-order friction stress is needed to drive the dislocation through the crystal. For the elastic contribution to rp the theory yields [4] T~ = 1.48 (V A~*) CD
3. EXPERIMENTS AND RESULTS
3.1. Materials and methods Single crystals of MgO of different "basic p u r i t y " have been investigated. The pure MgO (4-Niner material, in the following called " 4 N " ) contains less than 100 p.p.m, impurities. The more impure MgO (3-Niner material, called " 3 N " ) contains roughly 1000 p.p.m, impurities. As shown in Table 1 both sorts differ mainly in the content of CaO and A1203. The thermal treatment was normally a solid solution annealing (1773 K, 1 day, in air) and subsequent quenching to RT with cooling rate of a b o u t 50 deg/min. In the (Mg/Fe)O solid solution the ratio Fea*/Fe 2÷ depends on the total iron content, temperature and oxygen partial pressure [21]. For a given iron level different Fe 3÷ concentrations can be established by variation of the oxygen partial pressure (via the composition of the gas atmosphere) and/or annealing temperature. Therefore, in addition to the above described normal treatment, samples were annealed in air at 1373 K, or in a controlled CO/CO2 gas mixture at 1773 K having oxygen partial pressure of 1.7 × 10 - s atm. The Fea÷/ F e 2+ ratio was estimated using a method by Brynestadt and Flood [21] which has demonstrated that in MgO the trivalent ions occur as (Fe a÷ V Fe a÷) complexes. Under this assumption the concentration of Fe a+ can be estimated from the total iron content*. The con-
(9)
while the electrostatic contribution, index "est", is given by [16] T ~ , S t = l . 7 ( K3q----~Q Kob
)C D.
(10)
In both formulas the characteristic linear dependence on the dipole concentration appears.
(iii) The high temperature region. The plateau is followed by a high temperature decrease of To. The reason may be: (a) At high temperatures the Snoek clouds can move with the moving dislocations, leading to a friction dragging effect. Formally, this can b e represented by inserting a T-dependent factor into eqns. (9) and (10) [4]. ~b) The (Fe a÷ V Fe a÷) dipoles dissociate thermally into free Fe a* ions and vacancies [9, 20, 2 5 ] . The resulting decrease in the dipole concentration CD and consequently in re, eqns. (9) and (10), is described b y the mass action law [20]. For the case that the doped trivalent iron ions are present in the form of free Fe a÷ ions, the theory of Frank is n o t appropriate. The CRSS may be then effected by electrostatic [8] and/or elastic interaction [2, 14, 34, 35] of the dislocations with the free Fe 3÷ ions. However, no exact quantitative formulations of these hardening mechanisms exist at the present time in the literature.
1
*Starting from the equation 2Fe 3+ + ¥ 0 2 = (FeS+V Fe 3÷) + 0 2 - and applying the mass action law one obtains eqn. (7) in ref. 21. Setting the n u m b e r of all cation sites to 1 this formula can be simplified for small total iron c o n c e n t r a t i o n s Fetota 1 to (Fe2÷) 2 +
(Fe2+)
(Fet°tal)
K(T)~/po: K(T)x/po:
o.
Insertion of the experimental values of the oxygen
TABLE 1 I m p u r i t y and iron d o p i n g c o n t e n t of the MgO single crystals used; figures in wt. p.p.m. a c c o r d i n g to t h e producer* SiO 2
CaO
A1203
4N
15
30
50
3N
20
800
100
TiO 2 -10
ZrO 2 0.5 10
MnO
Ni
B
Fe
--
--
--
Pure < 20 Doped <1.2%
5
2
5
Pure 100 Doped < 7 1 0
*W. and C. Spicer, England and Dr. R. L. Hansler, General Electric, Clev., Ohio, U.S.A.
75
--'T(K) 200
400 i
600
I
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800
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I
=--
1000 1200 14.00 1600 1300 2000 2200
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~
I
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I
I
r
l
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1
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i
100
t
To
60 M(A.m_.~)
6
40
20
-200
I
0
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I
200
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I
400
I
I
600
i
I
800
,
I
~
I
J
[
i
I
,
I
,
l
1000 1200 1400 1600 1800 2000
T(°C) Fig. 2. The CRSS of undoped MgO with different basic purity as a function of temperature. 3N MgO: different sorts (see Table 2); annealed and deformed in air. 4N MgO: full symbols: annealed and deformed in CO/CO 2 gas atmosphere. Open symbols: annealed and deformed in air. Half open symbols: annealed in CO/CO2, deformed in air. centration of dipoles is t hen given by c D Fea÷/2. The d e f o r m a t i o n o f the ~ 1 0 0 > oriented specimens having {100} outside faces a n d a size o f a b o u t 2 × 3 X 8 mm 3 was p e r f o r m e d in compression between RT and 2000 K in air or in CO/CO2 gas mixture in an Instron machine t y p e 1114 equipped with a high t e m p e r a t u r e jig with AI2Oa rams [ 36] and sapphire crystal spacers. Prior to the d e f o r m a t i o n tests the specimens were held at test t e m p e r a t u r e for 20 min in situ to establish the thermal equilibrium. The d e f o r m a t i o n rate was ~ = 6 X 10 - 4 s- 1 unless otherwise stated. The CRSS was obtained f r o m the measured shear stress (T) - shear strain (e) curves*; for the graphic det er m i nat i on o f the CRSS see Fig. 1 in ref. 2. The given data points represent average values of t w o or m ore tests u n d er the same conditions. ----
pressure PO2, a n n e a l i n g t e m p e r a t u r e T, Fetotal, a n d o f t h e e q u i l i b r i u m c o n s t a n t K ( T ) a c c o r d i n g t o In K(T) = AGo/kT, w i t h AG O = 25.3 k c a l / m o l . , yields the F e 2÷ c o n c e n t r a t i o n a n d c o n s e q u e n t l y the Fe 3÷ c o n c e n t r a t i o n . *By d e f o r m a t i o n of very p u r e u n d o p e d s p e c i m e n s at 1 4 7 0 K it c o u l d be d e m o n s t r a t e d t h a t also MgO single crystals of s u f f i c i e n t l y high p u r i t y e x h i b i t stress - strain curves w i t h t h r e e d i s t i n c t w o r k h a r d e n i n g stages well k n o w n f r o m m e t a l s a n d alkali halides.
3.2. The CRSS in dependence on temperature Undoped MgO Between RT and 2000 K two regions are observed for bot h 4N MgO and 3N MgO (Fig. 2): a low-temperature decrease of To (T < 800 K) as well as a t e m p e r a t u r e - i n d e p e n d e n t plateau (T > 800 K). For the more impure 3N MgO the plateau is superposed by a m a x i m u m with a To peak at about 1100 K. As d e m o n s t r a t e d in ref. 18 this peak is not effected by the Fe 3÷ present onl y in very low concent rat i ons in this material, but by clusters or by precipitations of ot her impurities, and will, therefore, n o t be discussed here. Fe-doped MgO There appear three distinct regions (Fig. 3): as with the u n d o p e d MgO the low t e m p e r a t u r e decrease (T < 700 K) and the plateau (T > 700 K) are present; but, between 1300 K and 1600 K a rapid high t e m p e r a t u r e fall of To is observed which begins at lower temperatures for higher Fe 3÷ concentrations. In the plateau the high doped MgO exhibits a special behavior. For 4N-MgO-8500 a sharp peak at 970 K exists with a To value higher than for RT. In contrast with the 3N MgO (Fig. 1) this peak is effected by precipitations o f Fe 3+ occurring during the d e f o r m a t i o n test procedure. As described in subsection 3.1 all specimens
76 r(K) 20~
200 '
I
400 '
(
600 '
I
800 '
I
'
r
1000 1200 1400 1600 1800 2000 2200 I ' I ' ] ' I ' I [ I
_ tSO
t60
gO
120
1
8
8O
64 N -23001620 4-
2
0
20
0
-2oo
[
o
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20o
I
~oo
~
i
[
aoo
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[
1ooo 12oo ~
[
I
1~o
L
,eoo 2ooo
T (oc) ------,..Fig. 3. T h e CRSS o f u n d o p e d a n d o f i r o n - d o p e d 4N MgO in d e p e n d e n c e o n t e m p e r a t u r e . T h e first n u m b e r specifies t h e t o t a l i r o n c o n c e n t r a t i o n in wt. p.p.m. T h e s e c o n d n u m b e r specifies the F e 3+ c o n c e n t r a t i o n in tool. p.p.m. O p e n s y m b o l s : e = 6 × 10 - 4 s - l ; full s y m b o l s : e = 1.2 × 10 - 2 s -1.
were held at test temperature before the deformation for 20 rain to establish t h e temperature equilibrium. In ref. 19 it could be demonsizated that in 4N MgO containing 8500 p.p.m, iron after 20 min isothermal ageing at 970 K, magnesia ferrite particles with a size of about 175 A precipitate. The hardening due to these particles superposes additively to the solid solution plateau in the form of a typical ageing peak. For more details see ref. 19. 3.3. The C R S S in dependence on deformation rate
The measured ~ dependence o f ro supports the above described classification into three
clearly distinguishable regions where different mechanisms may be operating. In the low temperature region the CRSS of u n d o p e d and of iron-doped MgO is increased markedly b y applying a deformation rate which is higher by a factor of 20 (Fig. 3). The g sensitivity at RT rises with increasing content b o t h of Fe s+ and of impurities (Fig, 4)*.
*In o r d e r t o test t h e ~ d e p e n d e n c e o f t h e u n d o p e d 4N MgO (Figs. 3 a n d 4), s p e c i m e n s o b t a i n e d f r o m a larger single crystal b l o c k o f u n i q u e basic p u r i t y have b e e n used. One of t h e t w o p l o t t e d straight lines o f 4N MgO lying at higher values in Fig. 4 was o b t a i n e d f r o m a m o r e i m p u r e 4N crystal.
77
10
100
8 ~
-
- 80 I
-
To
6 _ ~ ~ ~ i ~
J-
60(_~2)
,/
2O (a) 0
I
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10-5
I
1 ] Illlll
10 ~
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10-3
I
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10-2
0
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30
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2,5
fMNl'i, tmq
k~m2)~
t5
1,5
(b) l
3.2 3.3
I
l
l
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I
I
1
I
~4
3.S
~6
&7
~8
&9
4.0
~,1
.....
I ~,2 ~,3
~
Fig, 4. The C R S ~ e n d e n c e on deformation rate d at RT: (a) represented as To x / 7 o - - "r G v s . x/In(co/e) according to eqns. (4) to (8).
In the plateau To does not depend on ~ (Fig. 3). Detailed results concerning the ~ dependence in the high-temperature decrease cannot yet be presented at this time, but preliminary results indicate a significant influence of ~ on the high temperature flow stress.
vs.
log c; (b) represented as
3.4. T h e C R S S in d e p e n d e n c e on the Fe 3÷ concentration L o w t e m p e r a t u r e region
In Fig. 5 the RT values of the CRSS are plotted versus x/Fe 3*. For high concentrations the data points follow strictly a straight line
78 I
!
I
l
I
200
20 E: = 6.
I
ld~s -t
,,°l
15
I
I
3N~ ~0
0
o
,o
20
30
~o
so
6o
70
so
0
FC'~'~(rnol.pprn) '/2 Fig. 5. The dependence of the CRSS on the Fe 3+ c o n c e n t r a t i o n and on the basic purity at RT.
through the origin independent o f the basic purity. For low concentrations the data points of 4N MgO or 3N MgO lie within two horizontal bands (hatched in Fig. 5). Their "highs" are given by the variation of the basic purity which could be demonstrated by the To values of specimens annealed in CO/COs gas atmosphere (see Table 2, subsection 3.5). From this it can be concluded that the CRSS of ironfree or very low-iron-doped MgO is determined by the basic purity only. Doping of Fe 3* has no effect on the CRSS as long as a minimum doping level is reached. These minimum values depend on the basic purity and are a b o u t 100 mol. p.p.m. Fe 3÷ for 4N MgO and 600 mol. p.p.m. Fe 3÷ for 3N MgO corresponding to dipole concentrations of 50 p.p.m, and 300 p.p.m, respectively. For higher doping levels the Fe 3÷ ions control the CRSS alone. Starting from this idea it is clear why the present lowiron-doped 3N MgO containing only 6 to 170 mol. p.p.m. Fe a÷ behaves as u n d o p e d impure MgO. Pla tea u The dependence of the CRSS on Fe 3÷ measured at 773 K is shown in Fig. 6a. Starting o from the values of the undoped MgO, To, a linear increase is found. Evidently, the contributions of the basic purity and of the Fe a÷ doping are additive in contrast with the low
temperature region (Fig. 5). Plotting the difference of (to -- r °) against C D = Fea÷/2, in order to separate the contribution o f the dipoles to the CRSS, one obtains a straight line through the origin for both 3N MgO and 4N MgO (Fig. 6b). High-temperature region Above 1600 K the remarkable result is observed that the CRSS decreases with increasing Fe 3÷ content (Fig. 3), which corresponds to solid solution softening. 3.5. The effect o f annealing temperature, gas atmosphere and Fe z+ content on the CRSS As pointed o u t in subsection 3.1, the Fe3÷/ Fe 2÷ ratio in MgO is strictly determined by the annealing temperature and the oxygen partial pressure of the gas mixture for given total iron content. Therefore, the influence of these two parameters on the CRSS is characterized uniquely by stating the Fe 3÷ concentration as described in subsections 3.2 - 3.4 and represented in Figs. 3 and 5. By annealing in the controlled CO/CO2 gas mixture, having an oxygen partial pressure of 1.7 × 10 - s atm., nearly all of the d o p e d iron is transferred into the divalent state (Table 2, last column). In this way t h e e f f e c t of the Fe 3÷ on the CRSS is eliminated and the role of the other impurities (subsection 3.4), and
79 I
I
,
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15 I
1,o
T=773K
./~
t,ooi
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ro (a)
I 1000
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I 2000
~ . I
i
I 3000
I
I 4000
0
Fe 3÷(rnoLpprn ) I
I
T=773K [] 3N-MgO 0 4N-MgO
I
150 l
~
100
J6
r -ro °
To-%° (b)
50[
t oI
o?
I
k
500
I
I
L
I
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1000 1500 CO(rnol.pprn)
2000
Fig. 6. The c o n c e n t r a t i o n dependence of the CRSS in the plateau region: (a) in a linear plot T O v s . Fe3*; (b) in a linear plot o f the same data (TO -- v ° ) v s . dipole c o n c e n t r a t i o n c D according to eqns. (9) and (10). TABLE 2 Effect of total iron c o n t e n t , gas a t m o s p h e r e or o x y g e n partial pressure and of annealing temperature on the Fe 3÷ c o n c e n t r a t i o n and on the CRSS at RT. MgO Total Fe type content (wt. p.p.m.)
4N
3N
Pure <20 310 2300 4300 7500 8500 Pure 70 140 300 710
Annealing conditions, Fe 3÷, To 1573 K / air
1773 K / air
1773 K / CO - CO 2
Fe 3+ (tool. p.p.m.)
To (kp m m - 2 )
Fe 3+ (tool. p.p.m.)
~l~ pmm -2)
Fe 3+ (mol. p.p.m.)
TO (kp m m 2)
"0" 47 885 1950 -4420
1 , 5 - 3,5 3 , 1 - 4,3 8,3 12,7 -18,1 - 18,9
"0" 24 622 1520 3150 3640
1 , 5 - 3,5 4,0 7,8 11,20 14,7 - 16,0 15,8 - 18,7
"0" "0" 0,5 1,7 5,3 6,7
1,5-3,5/ 2,0 2,4 - 2,9 ;~ 2,65 +_1,5 3,7 4,1 t 3,9 J
4,7 - 5,4 5,0 5,6 - 6,0 6,5-6,9
2,8 5,7 23,0 99,0
5,5 - 6,2 5,7 - 6,7 6,2 - 8,4 7,7-8,2
"0" "0" "0" "0"
5,7 - 6,4 t 5,0 5,5 ~ +_ 1,5 5,9 7,3 l- 6,5 4,9 5 , 1 )
\
6,2 11,8 42,7 170
80
also of the Fe e÷, can be studied. F r o m the comparison of the total iron level (second column in Table 2), which is equal to the Fe e* concentration after annealing in CO/COe, with the RT values of ro listed in the last column in Table 2, it follows that no correlation exists between the F e 2 ÷ concentration and the CRSS. The same result is obtained for the plateau. Hence, we conclude that for temperatures up to 1600 K the doped divalent iron ions have no influence on the strength of MgO.
4. A N A L Y S I S O F R E S U L T S
The CRSS of Fe-doped MgO exhibits those temperature regions (Fig. 3) which are expected for combined elastic/electrostatic dipole interaction. 4.1. T h e l o w t e m p e r a t u r e decrease o f ro ( T < 700 K )
According to eqns. (4) to (8) the elastic and the electrostatic contributions to the CRSS have the same dependence on temperature, deformation rate and dipole concentration. In order to obtain information on the tetragonality of the dipoles, Ak*, and on Eo we analyse the data points in the following way: The calculations of Frank [16] show that for 0 °K the electrostatic c o m p o n e n t of %, r~ 't, is only 20 - 28% of the elastic one. Therefore, above RT we can ignore for a m o m e n t the electrostatic contribution and ascribe the whole measured (ro -- rG) values to the elastic c o m p o n e n t r [ alone*. Then the data points can be represented after formula (4) to check the T and g dependence graphically. In the subsequent quantitative evaluation o f the obtained graphic representations the elastic comF ponent, %, and also AX* can be separated numerically.
* S t a r t i n g f r o m F r a n k ' s result [16] t h a t for 0 °K r est relates to r sF as 1:4, o n e derives f r o m eqn. (4) the expression ( q Q / • • o ) = 0 . 1 8 8 ( G b 3 AX , ). S u b s t i t u t i n g i n t o f o r m u l a (5) or (8) yields for t h e t e m p e r a t u r e coefficients t h e c o n d i t i o n q52 = 4 @1. This m e a n s t h a t the e l e c t r o s t a t i c c o n t r i b u t i o n , Test decreases m u c h m o r e w i t h increasing t e m p e r a t u r e t h a n does t h e elastic comp o n e n t , r F. As an e x a m p l e , at RT r est a m o u n t s t o o n l y 5% o f the TF value a n d s o o n b e c o m e s zero at the characteristic t e m p e r a t u r e T 2, w h i c h is r o u g h l y a f o u r t h o f To1 for T e s t . T h e r e f o r e , the approximat.ive analysis des c r i b e d a b o v e is justified a b o v e RT.
According to eqn. (4) the plot of X/To -- rG versus x / T should give straight lines with inter-
section points on the ordinate A=
0.107 - ~
~/CD
(11)
and with slopes S=
(
1.155 dbk 2 X/CD In
(12)
Figure 7a shows the representation of the data points taken from Fig. 3. re was calculated according to formula (3) with ~ = 0.32 and p = 105 cm/cm a. The obtained straight lines have slopes which increase with increasing dipole concentration CD and decrease with higher g as required theoretically by eqn. (12). From the values of the ordinate intersections A in Fig. 6a, obtained by linear extrapolation to 0 °K, the dipole strength Ak* can be estimated by eqn. (11). Taking into account numerically the electrostatic contribution to %, as calculated by Frank, in the above described way, one obtains for the dipole strength A•* = 0.16 ... 0.62 if we chose d = 0.5 ... 1. In order to prove the g dependence, X/To -- rG is plotted as a function of [ln (~o/d)] v,, eo = P b2VD • Straight lines are expected from eqns. (4) and (5) with intersection points with the ordinate, A, given by eqn. (11) and with slopes $2 = A(10.82 k d T / G b a Ak*)1/'. Figure 4b illustrates the RT values, represented in this way taken from Fig. 4a, where 7G = 0.037 kp mm-2. The data points fit straight lines having slopes which increase with increasing dipole concentration as predicted theoretically. The extrapolation to 0 °K yields A values which agree with the A values in Fig. 7a. Using these A values and setting d = 0.5 ... 1 in eqn. (11) the dipole strength Ak* is estimated to be 0.17 ... 0.77. To test the dependence on the dipole concentration CD according to eqns. (4) and (7) ( T O - - TO)2 is plotted linearly versus CD in Fig. 8. Only data points of the pure 4N MgO from o Fig. 5 are used which are higher than the To values of the undoped material in order to eliminate the effect of other impurities. For comparison the same experimental data are represented in the Labusch plot, (to -- TG)a/2 versus CD, according to eqn. (1). Reasonable straight lines through the origin are obtained over the whole range of concentration, indicating that the assumed localized interaction
81
4
Ca)
23~
4N-8500]3640 4,N-4300/1520 4N-2300/6-20
l
I (b) 3N
4N
15 I
20 25 --Ill(K)'~2 I
I
30 I
Fig. 7. The temperature dependence of the CRSS of Fe-doped MgO (a) and of undoped MgO (b), represented in a x/To -- r G -- ~/T diagram after Frank, eqn. (4). The data points symbolized by • are shifted by 0.5 kp mm -2. Open symbols: ~ = 6 X 10-4 s-l; full symbols: ~ = 1.2 x 10 -2 s-1. is justified. T h e slope o f the linear Fleischer F r a n k p l o t in Fig. 8 includes n o w t h e electrostatic c o n t r i b u t i o n . If o n e separates this as described above, the t e t r a g o n a l i t y o f t h e defects is calculated b y use o f the slope, determ i n e d using eqn. (4), t o be 5X* = 0.43, setting d = 0.5. With this value t h e free a c t i v a t i o n e n t h a l p y is given b y eqn. (6) as E0 = 1.37 eV.
4.2. The plateau (T > 700 K) In t h e t e m p e r a t u r e - i n d e p e n d e n t r e g i o n elastic and e l e c t r o s t a t i c dipole i n t e r a c t i o n s result in a linear d e p e n d e n c e o f the CRSS on the dipole c o n c e n t r a t i o n (eqns. (9) and (10)). T h e e s t i m a t i o n o f F r a n k [16] shows t h a t the e l e c t r o s t a t i c c o n t r i b u t i o n t o the p l a t e a u stress a m o u n t s t o 12 - 18%. Using t h e l o w e r value, the expressions (9) a n d (10) can be c o m b i n e d t o re = (to - - r a ) = 1.48 (1.0 + 0 . 1 2 ) G AX*c D. T h e e x p e r i m e n t s d e m o n s t r a t e t h a t this relat i o n s h i p is fulfilled (Fig. 6b). As d e s c r i b e d in o w h e r e roo is t h e s u b s e c t i o n 3.4, re = ro -- To,
CRSS o f t h e u n d o p e d 4N MgO or 3N MgO, which already includes rG. Insertion o f the slope o f t h e p l o t o f Fig. 6b into t h e above form u l a gives AX* = 0.40, in g o o d a g r e e m e n t with AX* q u o t e d in the l o w - t e m p e r a t u r e region. If we p r e f e r the AX* values d e t e r m i n e d f r o m t h e dipole d e p e n d e n c e o f ro in the p l a t e a u and in t h e l o w - t e m p e r a t u r e region r a t h e r t h a n t h o s e AX* values o b t a i n e d f r o m the T and d e p e n d e n c e b y use o f the adjustable p a r a m e t e r d, t h e best value f o r the dipole s t r e n g t h is AX* (MgO : F e 3÷) = 0.42 in c o m p a r i s o n with previous estimates f o r alkali halides, AX* (NaC1 : Ca 2÷) = 0.57 [ 1 ] , and AX* (LiF : Mg 2÷) = 0.53 [ 2 ] .
4.3. The high-temperature region (T > 1300 K) The t h e o r e t i c a l l y p r e d i c t e d decrease o f the CRSS at high t e m p e r a t u r e s was f o u n d experim e n t a l l y (Fig. 3) and marks the end o f t h e plateau. If we c o m b i n e this result w i t h the
82 100
i
1
i
I
9O I 8O ~00
70 6O ~00
(kprnrr}2y/2
)2 -2.2
± 0 Davidge(1967) ~7 Moon&Pratt(1970) [ ] Srinivasan& Stoebe (1970) • present study
•
I
0
,
1000
~
I 2OOO
~
r00
I 300O
Co (rnol.ppm )
Fig. 8. T h e d e p e n d e n c e o f t h e C R S S o f F e d o p e d 4 N M g O at R T o n t h e d i p o l e c o n c e n t r a t i o n : b o t t o m : t h e F l e i s c h e r - F r a n k p l o t (I"o - - TG) 2 v s . CD, e q n . ( 2 ) o r e q n s . (4) a n d (7); t o p : t h e L a b u s c h p l o t o f t h e s a m e d a t a (T o - - TG) 312 vs. CD, e q n . (1). T h e c D values f o r d a t a p o i n t s c i t e d w e r e e s t i m a t e d f r o m t h e given t o t a l i r o n c o n t e n t as d e s c r i b e d in s u b s e c t i o n 3.1.
observation of Wicks and Lewis [9], namely a reduction of the number of Fe z÷ vacancy centres coupled by a simultaneous increase of the concentration of isolated F e 3÷ ions, the high-temperature decrease of ro could be interpreted informally by thermal dipole dissociation. But the fact, d o c u m e n t e d b y the To values above 1300 K in Fig. 3, that the dipole dissociation begins earlier for higher dipole concentrations, is not compatible with this explanation. Possibly the decrease of the plateau stress due to dipole dissociation is superposed b y other opposite effects [ 2 ] . Factors in the temperature region considered which can contribute to ro are: (a) Parelastic as well as dielastic interaction of dislocations with non-associated Fe z* ions by Cottrell locking [ 1 4 ] , which requires an extra stress for a dislocation to break loose from the Cottrell clouds formed in the stress field around the dislocation. At high temperatures and not t o o high strain rates the Fe 3÷ ions are sufficiently mobile so
that the Cottrell clouds can follow steadily the moving dislocations resulting in friction drag in analogy with the microcreep [14, 35]. Note the analogy with the friction stress in the plateau due to Snoek clouds consisting of oriented (Fe a÷ V Fe 3÷) dipoles. (b) Electrostatic interaction of charged dislocations or of dislocations having charged jogs with Fe a÷ ions [ 8 ] . On the other hand, all these mechanisms predict an increase of the CRSS with inc r e a s i n g F e 3+ concentration in contrast with the experimental results showing the inverse dependence. Eventually, a "doping-induced solid solution softening" plays an important role, as discussed for FeO [ 37, 3 8 ] .
CONCLUSION
The model of the combined elastic and electrostatic dipole hardening proposed by Frank for cubic crystals is applied on low-
83 i r o n - d o p e d MgO. T h e p r e s e n t c o n c e p t rests o n t h e a s s u m p t i o n t h a t t h e MgO o f s u f f i c i e n t l y high p u r i t y c o n t a i n s t h e d o p e d Fe 3÷ ions in t h e f o r m o f disperse d i s t r i b u t e d , n o n - a g g l o m e r a t e d s y m m e t r i c ( F e 3÷ V Fe a÷) dipoles, w h i c h h a v e a clear t e n d e n c y to act as s t r o n g "indiv i d u a l " o b s t a c l e s b u t n o t as o b s t a c l e groups. A c c o r d i n g l y , t h e critical resolved shear stress ro is c o n t r o l l e d at low t e m p e r a t u r e s b y therm a l l y a c t i v a t e d Fleischer i n t e r a c t i o n , while at elevated t e m p e r a t u r e s in t h e " p l a t e a u " ro is determined by the induced Snoek effect of o r i e n t e d dipoles in t h e stress field o f m o v i n g dislocations. In b o t h regions the elastic/ t e t r a g o n a l c o n t r i b u t i o n t o ro d o m i n a t e s (Fig. 1). T h e m e a s u r e d values o f ro w h i c h s h o w p r o n o u n c e d solid s o l u t i o n h a r d e n i n g e x h i b i t the theoretically expected dependencies upon t e m p e r a t u r e (Figs. 2, 3 a n d 7), d e f o r m a t i o n rate (Fig. 4) a n d dipole c o n c e n t r a t i o n (Figs. 5, 6 and 8). In b o t h regions q u a n t i t a t i v e agreem e n t w i t h t h e m o d e l is f o u n d . T h e d i p o l e s t r e n g t h was e s t i m a t e d at Ah* = 0.42 similar t o values o f d o p e d alkali halides. D i v a l e n t iron ions h a v e n o m e a s u r a b l e i n f l u e n c e o n to. T h e r a p i d h i g h - t e m p e r a t u r e fall o f ro is a t t r i b u t e d t o t h e t h e r m a l b r e a k i n g - u p o f the dipoles i n t o Fe 3÷ ions a n d c a t i o n vacancies. T h e solid s o l u t i o n s o f t e n i n g o b s e r v e d in this region c a n n o t be e x p l a i n e d .
p e r i m e n t s , a n d Dr. I. V a r n e r f o r r e a d i n g a n d c o r r e c t i n g t h e English m a n u s c r i p t . F i n a n c i a l support by the Deutsche Forschungsgemeins c h a f t is g r a t e f u l l y a c k n o w l e d g e d .
Note added in proof (December 22, 1975)
19
R e c e n t l y G i l m a n [ 3 9 ] has p r o p o s e d a n e w i n t e r e s t i n g m o d e l o f s h o r t r a n g e dipole i n t e r a c t i o n w h i c h is b a s e d o n c u t t i n g o f t h e dipoles b y gliding dislocations. A h l q u i s t [40] has applied this m o d e l t o d o p e d MgO a n d c o n c l u d e s t h a t also t h e C R S S o f MgO at R T is d e t e r m i n e d b y d i p o l e shearing a n d n o t b y t h e e l a s t i c / t e t r a g o n a l d i p o l e passing m o d e l o f Fleischer a n d F r a n k discussed here. But, it c a n be d e m o n s t r a t e d t h a t t h e G i l m a n - A h l q u i s t m e c h a n i s m explains 10% o f t h e m e a s u r e d values o n l y [41] a n d t h e r e f o r e c a n b e e x c l u d e d as t h e c o n t r o l l i n g p r o c e s s f o r t h e solid s o l u t i o n h a r d e n i n g in d o p e d MgO.
20
REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
21 22 23 24 25 26 27 28
ACKNOWLEDGEMENTS T h e a u t h o r wishes t o t h a n k Mr. B. K u m m e r f o r the p e r f o r m a n c e o f t h e d e f o r m a t i o n ex-
29
W. Frank, Phys. Status Solidi, 29 (1968) 391. B. Reppich, Acta Met., 20 (1972) 557. W. Frank, Z. Naturforsch., 22a (1966) 365. W. Frank, Z. Naturforsch., 22a (1966) 377. R. L. Fleischer, Acta Met., 10 (1962) 835. R. L. Fleischer, J. Appl. Phys., 33 (1962) 3504. G. Schoeck and A. Seeger, Acta Met., 7 (1959) 469. R. L. Moon and P. L. Pratt, Proc. Brit. Ceram. Soc., 15 (1970) 203. B. J. Wicks and M. E. Lewis, Phys. Status Solidi, (a), 6 (1970) 281. G. D. Miles, F. J. P. Clarke, B. Henderson and R. D. King, Proc. Brit. Ceram. Soc., 6 (1966) 325. R.W. Davidge, J. Mater. Sci., 2 (1967)339. G. W. Groves and M. E. Fine, J. Appl. Phys., 35 (1964) 3587. M. Srinivasan and T. G. Stoebe, J. Appl. Phys., 41 (1970) 3726. P. Haasen, in R. W. Cahn (ed.), Physical Metallurgy, North-Holland Publishing Co., Amsterdam, 1965, p. 821. R. L. Fleischer, in D. Peckner (ed.), The Strengthening of Metals, Reinhold, 1964, p. 93. W. Frank, personal communication. R. Labusch, Acta Met., 20 (1972) 917. B. Reppich, Mater. Sci. Eng., 19 (1975) 51; see also B. Reppich, Habil. Thesis, Univ. ErlangenNfirnberg, 1973. H. Knoch and B. Reppich, Acta Met., 23 (1975) ],055 and 1061. A. B. Lidiard, Handbuch der Physik, Band 20, Springer, 1957, p. 246. J. Brynestadt and H. Flood, Z. Elektrochem., (1958) 954. J. F. Wertz, J. W. Orton and P. Auzins, J. Appl. Phys., Suppl., 33 (1962) 322. A. L. Schawlow, J. Appl. Phys., Suppl., 33 (1962) 395. H. Wiedersich, Proc. Second Symp. on Low Energy, X- and Gamma-Sources and Applications, ORNL II C-10, Tennessee, U.S.A., 1967, p. 101. B. Ilschner, in O. Madelung (ed.), FestkSrperprobleme X, Advances in Solid State Physics, Pergamon, Vieweg, Braunschweig, 1970, p. 415. E. KrSner, Kontinuumstheorie der Versetzungen und Eigenspannungen, Springer, 1958. F. R. N. Nabarro, Rept. of Conf. on the Strength of Solids, Phys. Soc., London, 1958. A. Cochardt, G. Schoeck and H. Wiedersich, Acta Met., 3 (1955) 533. R. F. S. Hearmon, in K. H. Hellwege, LandoldBSrnstein, Zahlenwerte und Funktionen aus Naturwissenschaften und Technik, Bd. 1, Springer, 1966, p. 29.
84 30 R. Strumane and R. De Batist, Phys. Status Solidi, 6 (1958) 569. 31 R . W . Whithworth, Phil. Mag., 15 (1967) 305. 32 P. D. Southgate, Bull. Am. Phys. Soc., 7 (1962) 501. 33 J. Snoek, Physica, 8 (1941) 711. 34 A. Portevin and F. Le-Chatelier, Compt. Rend., Paris, 176 (1923) 507. 35 J. Weertman and J. R. Weertman, in R. W. Cahn (ed.), Physical Metallurgy, North-Holland Publishing Co., Amsterdam, 1965, p. 793. 36 B. Reppich, W. Blum and B. Ilschner, Ber. Deut.
Keram. Ges., 44 (1967) 41. 37 B. Ilschner, B. Reppich and E. Riecke, Discussions Faraday Soc., 38 (1964) 243. 38 B. Reppich, Phys. Status Solidi, 20 (1967) 69. 39 J. J. Gilman, J. Appl. Phys., 45 (1974) 508. 40 C. N. Ahlquist, in R.C. Bradt and R. E. Tressler (ed.), Proc. Symposium on Plastic Deformation of Ceramic Materials, held at Pennsylvania State Univ., July 1974, Plenum Press, New York, 1975, p. 233. 41 B. Reppich and A. Straub, in preparation.
71
© Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands
Solid Solution Hardening in Iron Doped MgO*
B. REPPICH Institut fiir Werkstoffwissenschaften I, University Erlangen:Niirnberg, D-852 Erlangen (Germany)
(Received May 20, 1975; in revised form October 20, 1975)
SUMMARY The critical resolved shear stress To of quenched low iron doped MgO single crystals of different purity has been investigated between room temperature and 2000 K. It is shown that at low and elevated temperatures To is determined by the combined elastic and electrostatic interaction of the dislocations with single dipoles, composed of two trivalent iron ions and a charge-compensating cation vacancy. The elastic tetragonal hardening dominates. In the low-temperature decrease below 700 K ro is proportional to ~/CD (CD = dipole concentration), depends on strain rate, and is controlled by thermally activated passing of dislocations near dipoles located close to the slip plane (Fleischer interaction). In the temperature independent " p l a t e a u " above 700 K ro depends linearly on CD and can be explained by long-range dipole interaction via the induced Snoek effect. The dipole strength was estimated to be A~* = 0.42. The rapid high-temperature decrease of ro above 1300 K is attributed to the thermal dissociation of the dipoles.
1. INTRODUCTION The critical resolved shear stress (CRSS) of monovalent ionic crystals, e.g. NaC1 and LiF, is significantly increased by divalent impurity cations ([1, 2], refs. 1 - 6). After suitable thermal pretreatment the impurities can be present in the form of non-agglomerated pairs composed of an impurity cation and chargecompensating cation vacancy (hereinafter *Presented at the Friihjahrstagung des Arbeitskreises FestkSrperphysik der DPG, Freudenstadt, April 1974, Vortrag M 19.
called "dipoles"). The tetragonal strain field of the dipoles gives rise to a strong elastic interaction with the dislocations. Consequently, in the literature, the solid solution hardening of the alkali halides has been attributed to these elastic dipoles. The experimental results agree quantitatively [1, 2] with the theory of Frank [3, 4] which is an extension of the ideas of Fleischer [5, 6] and of Schoeck and Seeger [7]. Similar effects may be predicted for divalent ionic crystals doped with transition metal ions, e.g. Ti, V, Cr and Fe. On the other hand, the situation is more complex in these materials. For example, in iron
72
has yielded t o o small a contribution to the CRSS, while for the latter defect t y p e t o o high pair concentrations were obtained. According to Moon and Pratt [8] and Wicks and Lewis [9] the region of elevated temperatures, which is characterized b y a temperature and deformation rate independent CRSS, is attributed to the reorienting of tetragonal defects formed of Fe a÷ or other impurities in the elastic stress field of moving dislocations [ 7]. Miles et al. [10] perceive no correlations between the CRSS and Fe 3÷ vacancy dipoles and suppose that the hardening effect is associated with an impurity atmosphere pinning of the dislocations by Cottrell locking [14, 15]. Starting from this unsatisfactory situation in the present study the question is raised again -- which mechanisms control the solid solution hardening of typical ionic oxides doped with transition metal ions? Deformation experiments on low iron doped and suitably thermally pretreated MgO single crystals between r o o m temperature (RT) and 2000 K are described. The concept of dipole hardening, including electrostatic dipole interaction [ 1 6 ] , is applied on MgO and is examined experimentally.
2. T H E O R E T I C A L B A C K G R O U N D
2.1. Solid solution hardening Solid solution hardening is caused b y the interaction of dislocations with solute foreign atoms or other obstacles of atomic size. With regard to their theoretical approaches the existing theories of solid solution hardening can be classified into t w o groups [ 1 7 ] . In the first group the interaction of dislocations with individual obstacles is the elementary process (in the following called "localized" interaction). Theories of group t w o frequently consider statistical "fluctuations" of the interaction, averaged over a characteristic length, as the effective obstacles (in the following called interaction with "obstacle groups"). According to Labusch [17] all theories give the following relation between the critical resolved shear stress to, the maximum interaction force F, the range of interaction w b e t w e e n solute atoms and dislocations, the line tension t and the concentration of solute atdms c 4
1
FY w~- :__ ~o ~ , c~ . t~-
(1)
Labusch has further demonstrated that, depending on the choice of the given parameters, the calculations based on individual obstacles yield either eqn. (1) or the Fleischer - Friedel formula 3
F~- ± ro = , c: . t~-
(2)
Practically, eqn. (2) is limited to dilute solutions of "hard" single obstacles presented in concentrations of less than 10 -a [ 17 ]. But, in contrast with the f.c.c, metals, this condition is fulfilled in the present case*. Moreover, for the dipole hardening discussed here, the ratio F/t, which characterizes the strength of interaction, is larger than that for the combined parelastic/dielastic interaction in metallic alloys [14, 1 5 ] . In other words, it should be justifiable to describe the solid solution hardening in the present low iron d o p e d MgO on the basis of the localized interaction, eqn. (2) (see subsection 2.2)**. On the other hand, the argumentation is reversible; the experimental conditions are realized here for examining the theory of solid solution hardening for the special case of strong localized interaction. 2.2. Dipole hardening in Fe-doped MgO 2.2.1. Impurity - vacancy complexes in MgO The substitution of t w o trivalent iron ions Fe ~ * into the otherwise neutral MgO lattice requires formation of one cation vacancy (V). Owing to the Coulomb attraction between the effectively positively charged Fe 3÷ and the doubly negatively charged V, a cation vacancy pair forms [21 - 24] in analogy with the alkali halides [ 2 0 ] . Following Ilschner [25] thermodynamic and energetic points of view a complex of the form (Fe3÷V Fe 3÷) is suggested. From the possible crystallographic configurations the linear-symmetrical < 110> t y p e is the most favourable. 2.2.2. The critical resolved shear stress We assume that the (FeS*V Fe 3÷) complexes represent elastic dipoles [26] which have a *As shown in subsection 3.4 for the largest iron doping level the dipole concentration is <3 × 10 -3. **Localized interaction could be assumed with the same a r g u m e n t a t i o n for MgO containing the doped Fe 3÷ in f o r m o f dipole clusters [18] or precipitate magnesia ferrite particles [ 19 ].
73 clear t e n d e n c y for strong elastic i n t e r a c t i o n with dislocation lines due t o t h e i r t e t r a g o n a l strain field a c c o r d i n g t o N a b a r r o [ 2 7 ] , C o c h a r d t et al. [28] and Fleischer [ 5 ] . F r a n k [3, 4] has p r o p o s e d a m o d e l which describes t h e influence o f t e t r a g o n a l d e f e c t s in cubic crystals. In addition to the elastic i n t e r a c t i o n , the e l e c t r o s t a t i c dipole i n t e r a c t i o n m u s t be t a k e n into a c c o u n t . T h e c o n t r i b u t i o n o f the e l e c t r o s t a t i c i n t e r a c t i o n b e t w e e n charged screw dislocations and c a t i o n - v a c a n c y - d i p o l e s has been r e c e n t l y e s t i m a t e d b y F r a n k [ 1 6 ] . We a p p l y n o w the c o n c e p t o f the c o m b i n e d elastic/electrostatic dipole h a r d e n i n g o n MgO. T h e n f o r MgO o f sufficiently high p u r i t y containing the d o p e d Fe 3~ in the f o r m o f disperse d i s t r i b u t e d and n o n - a g g l o m e r a t e d dipoles, t h e following regions and m e c h a n i s m s m a y be e x p e c t e d (Fig. 1). (i) The low temperature decrease o f 7.o. At low t e m p e r a t u r e s ro decreases s t r o n g l y with increasing t e m p e r a t u r e s T a n d is given b y 7.o = 7.G + 7.~
(3a)
where rG = a Gb ~/p
(3b)
is the T - i n d e p e n d e n t c o n t r i b u t i o n t o 7.0 d u e to long range d i s l o c a t i o n / d i s l o c a t i o n i n t e r a c t i o n (~ is a c o n s t a n t ) . T h e t e m p e r a t u r e d e p e n d e n t c o m p o n e n t rs is d e t e r m i n e d b y s h o r t range dipole i n t e r a c t i o n . T h e t h e r m a l l y a c t i v a t e d dislocations " p a s s " t h e dipoles l o c a t e d close t o the slip plane resulting in an elastic contrib u t i o n to % (index " F " f o r Fleischer interaction) [ 3 ] : GAX* 7.sF _-- Z 1
~-~
~/CD (1 -- ~/01T)2;
(4)
Z 1 = 0 . 1 0 7 is a numerical factor, G t h e shear + m o d u l u s , b = 2.85 A t h e Burgers vector, p t h e dislocation density, d = 0.5 the distance o f the interacting dipoles f r o m the slip plane in units of b, c D the a t o m i c d i p o l e c o n c e n t r a t i o n . T h e function ¢1 = Z 2 - - 1 n GbaAX *
dipole dlssociehOn
dipole interection
f Fleischer sslng
I Induced Snoek I effect
elastic :}
[ \
electrostattc
,
-
0
RT
To2
To'
T'---'~ Fig. 1. Schematic representation of the contributions to the CRSS of low-iron-doped MgO at low and elevated temperatures due to combined elastic/electrostatic dipole interaction (temperature dependence of elastic constants neglected); TG is not drawn, see text. D e b y e f r e q u e n c y , Z2 = 10.82, a n u m e r i c a l factor. T h e " d i p o l e s t r e n g t h " Ak* is related to the free activation e n t h a l p y E 0 (at %r = 0), Z 3 = 0.092 is the n u m e r i c a l factor, Gb 3 Eo = Z3 - - A~*. d
(6)
T h e electrostatic c o n t r i b u t i o n to 7.s is [ 16] 7.sest = Z 4
qQ/b3KKo ~/C D ( 1 - - N / ~ ) 2 T ) 2 d2
(7)
with In , (8) qQ/~o w h e r e Z 4 = 0 . 1 4 2 and Z 5 = 8.18 are numerical factors, Q is the (positive) charge o f t h e dipoles, q t h e charge d e n s i t y per line length o f dislocation [30, 3 1 ] , K t h e static dielectric c o n s t a n t , Ko = 8.86 X 10 -14 ( c o u l o m b ) 2 Vsm. It is recognised t h a t eqns. (4) and (7), as well as eqns. (5) and (8), have the analogous f o r m ; the " m e c h a n i c a l " t e r m (Gb3AX *) corresponds t o the " e l e c t r o s t a t i c " t e r m (qQ/n ~o). N o t e t h a t b o t h % c o n t r i b u t i o n s have the same T and X/CD d e p e n d e n c e . (~2 = Z 5 -
(5)
contains t h e d e p e n d e n c e o n t h e d e f o r m a t i o n rate d. k is the B o l t z m a n n c o n s t a n t , vn the +The shear m o d u l u s is G = 1/2 (c11-- c12), w i t h t h e Voigt's elastic c o n s t a n t s a c c o r d i n g t o ref. 29.
(ii) The plateau. A b o v e t h e low t e m p e r a t u r e decrease o f 7.0, at elevated t e m p e r a t u r e s , a region is p r e d i c t e d w h e r e 7.0 is i n d e p e n d e n t o f T. In this region 7.o is c o m p o s e d o f 7"G and 7.p, and we have 7.o = 7.G + 7.p, w h e r e 7.p is c o n t r o l l e d b y long range d i p o l e i n t e r a c t i o n via t h e i n d u c e d
74
Snoek effect [ 3 3 ] , index " P " for plateau. The stress field of the moving dislocation produces a Snoek cloud around it, consisting of oriented (Fe s÷ V Fe a÷) dipoles [ 3 2 ] . In this way the dislocation "digs itself in" energetically and a local-order friction stress is needed to drive the dislocation through the crystal. For the elastic contribution to rp the theory yields [4] T~ = 1.48 (V A~*) CD
3. EXPERIMENTS AND RESULTS
3.1. Materials and methods Single crystals of MgO of different "basic p u r i t y " have been investigated. The pure MgO (4-Niner material, in the following called " 4 N " ) contains less than 100 p.p.m, impurities. The more impure MgO (3-Niner material, called " 3 N " ) contains roughly 1000 p.p.m, impurities. As shown in Table 1 both sorts differ mainly in the content of CaO and A1203. The thermal treatment was normally a solid solution annealing (1773 K, 1 day, in air) and subsequent quenching to RT with cooling rate of a b o u t 50 deg/min. In the (Mg/Fe)O solid solution the ratio Fea*/Fe 2÷ depends on the total iron content, temperature and oxygen partial pressure [21]. For a given iron level different Fe 3÷ concentrations can be established by variation of the oxygen partial pressure (via the composition of the gas atmosphere) and/or annealing temperature. Therefore, in addition to the above described normal treatment, samples were annealed in air at 1373 K, or in a controlled CO/CO2 gas mixture at 1773 K having oxygen partial pressure of 1.7 × 10 - s atm. The Fea÷/ F e 2+ ratio was estimated using a method by Brynestadt and Flood [21] which has demonstrated that in MgO the trivalent ions occur as (Fe a÷ V Fe a÷) complexes. Under this assumption the concentration of Fe a+ can be estimated from the total iron content*. The con-
(9)
while the electrostatic contribution, index "est", is given by [16] T ~ , S t = l . 7 ( K3q----~Q Kob
)C D.
(10)
In both formulas the characteristic linear dependence on the dipole concentration appears.
(iii) The high temperature region. The plateau is followed by a high temperature decrease of To. The reason may be: (a) At high temperatures the Snoek clouds can move with the moving dislocations, leading to a friction dragging effect. Formally, this can b e represented by inserting a T-dependent factor into eqns. (9) and (10) [4]. ~b) The (Fe a÷ V Fe a÷) dipoles dissociate thermally into free Fe a* ions and vacancies [9, 20, 2 5 ] . The resulting decrease in the dipole concentration CD and consequently in re, eqns. (9) and (10), is described b y the mass action law [20]. For the case that the doped trivalent iron ions are present in the form of free Fe a÷ ions, the theory of Frank is n o t appropriate. The CRSS may be then effected by electrostatic [8] and/or elastic interaction [2, 14, 34, 35] of the dislocations with the free Fe 3÷ ions. However, no exact quantitative formulations of these hardening mechanisms exist at the present time in the literature.
1
*Starting from the equation 2Fe 3+ + ¥ 0 2 = (FeS+V Fe 3÷) + 0 2 - and applying the mass action law one obtains eqn. (7) in ref. 21. Setting the n u m b e r of all cation sites to 1 this formula can be simplified for small total iron c o n c e n t r a t i o n s Fetota 1 to (Fe2÷) 2 +
(Fe2+)
(Fet°tal)
K(T)~/po: K(T)x/po:
o.
Insertion of the experimental values of the oxygen
TABLE 1 I m p u r i t y and iron d o p i n g c o n t e n t of the MgO single crystals used; figures in wt. p.p.m. a c c o r d i n g to t h e producer* SiO 2
CaO
A1203
4N
15
30
50
3N
20
800
100
TiO 2 -10
ZrO 2 0.5 10
MnO
Ni
B
Fe
--
--
--
Pure < 20 Doped <1.2%
5
2
5
Pure 100 Doped < 7 1 0
*W. and C. Spicer, England and Dr. R. L. Hansler, General Electric, Clev., Ohio, U.S.A.
75
--'T(K) 200
400 i
600
I
i
800
]
I
=--
1000 1200 14.00 1600 1300 2000 2200
I
~
I
i
I
I
I
r
l
I
I
I
1
I
i
100
t
To
60 M(A.m_.~)
6
40
20
-200
I
0
J
I
200
i
I
400
I
I
600
i
I
800
,
I
~
I
J
[
i
I
,
I
,
l
1000 1200 1400 1600 1800 2000
T(°C) Fig. 2. The CRSS of undoped MgO with different basic purity as a function of temperature. 3N MgO: different sorts (see Table 2); annealed and deformed in air. 4N MgO: full symbols: annealed and deformed in CO/CO 2 gas atmosphere. Open symbols: annealed and deformed in air. Half open symbols: annealed in CO/CO2, deformed in air. centration of dipoles is t hen given by c D Fea÷/2. The d e f o r m a t i o n o f the ~ 1 0 0 > oriented specimens having {100} outside faces a n d a size o f a b o u t 2 × 3 X 8 mm 3 was p e r f o r m e d in compression between RT and 2000 K in air or in CO/CO2 gas mixture in an Instron machine t y p e 1114 equipped with a high t e m p e r a t u r e jig with AI2Oa rams [ 36] and sapphire crystal spacers. Prior to the d e f o r m a t i o n tests the specimens were held at test t e m p e r a t u r e for 20 min in situ to establish the thermal equilibrium. The d e f o r m a t i o n rate was ~ = 6 X 10 - 4 s- 1 unless otherwise stated. The CRSS was obtained f r o m the measured shear stress (T) - shear strain (e) curves*; for the graphic det er m i nat i on o f the CRSS see Fig. 1 in ref. 2. The given data points represent average values of t w o or m ore tests u n d er the same conditions. ----
pressure PO2, a n n e a l i n g t e m p e r a t u r e T, Fetotal, a n d o f t h e e q u i l i b r i u m c o n s t a n t K ( T ) a c c o r d i n g t o In K(T) = AGo/kT, w i t h AG O = 25.3 k c a l / m o l . , yields the F e 2÷ c o n c e n t r a t i o n a n d c o n s e q u e n t l y the Fe 3÷ c o n c e n t r a t i o n . *By d e f o r m a t i o n of very p u r e u n d o p e d s p e c i m e n s at 1 4 7 0 K it c o u l d be d e m o n s t r a t e d t h a t also MgO single crystals of s u f f i c i e n t l y high p u r i t y e x h i b i t stress - strain curves w i t h t h r e e d i s t i n c t w o r k h a r d e n i n g stages well k n o w n f r o m m e t a l s a n d alkali halides.
3.2. The CRSS in dependence on temperature Undoped MgO Between RT and 2000 K two regions are observed for bot h 4N MgO and 3N MgO (Fig. 2): a low-temperature decrease of To (T < 800 K) as well as a t e m p e r a t u r e - i n d e p e n d e n t plateau (T > 800 K). For the more impure 3N MgO the plateau is superposed by a m a x i m u m with a To peak at about 1100 K. As d e m o n s t r a t e d in ref. 18 this peak is not effected by the Fe 3÷ present onl y in very low concent rat i ons in this material, but by clusters or by precipitations of ot her impurities, and will, therefore, n o t be discussed here. Fe-doped MgO There appear three distinct regions (Fig. 3): as with the u n d o p e d MgO the low t e m p e r a t u r e decrease (T < 700 K) and the plateau (T > 700 K) are present; but, between 1300 K and 1600 K a rapid high t e m p e r a t u r e fall of To is observed which begins at lower temperatures for higher Fe 3÷ concentrations. In the plateau the high doped MgO exhibits a special behavior. For 4N-MgO-8500 a sharp peak at 970 K exists with a To value higher than for RT. In contrast with the 3N MgO (Fig. 1) this peak is effected by precipitations o f Fe 3+ occurring during the d e f o r m a t i o n test procedure. As described in subsection 3.1 all specimens
76 r(K) 20~
200 '
I
400 '
(
600 '
I
800 '
I
'
r
1000 1200 1400 1600 1800 2000 2200 I ' I ' ] ' I ' I [ I
_ tSO
t60
gO
120
1
8
8O
64 N -23001620 4-
2
0
20
0
-2oo
[
o
[
20o
I
~oo
~
i
[
aoo
[
[
1ooo 12oo ~
[
I
1~o
L
,eoo 2ooo
T (oc) ------,..Fig. 3. T h e CRSS o f u n d o p e d a n d o f i r o n - d o p e d 4N MgO in d e p e n d e n c e o n t e m p e r a t u r e . T h e first n u m b e r specifies t h e t o t a l i r o n c o n c e n t r a t i o n in wt. p.p.m. T h e s e c o n d n u m b e r specifies the F e 3+ c o n c e n t r a t i o n in tool. p.p.m. O p e n s y m b o l s : e = 6 × 10 - 4 s - l ; full s y m b o l s : e = 1.2 × 10 - 2 s -1.
were held at test temperature before the deformation for 20 rain to establish t h e temperature equilibrium. In ref. 19 it could be demonsizated that in 4N MgO containing 8500 p.p.m, iron after 20 min isothermal ageing at 970 K, magnesia ferrite particles with a size of about 175 A precipitate. The hardening due to these particles superposes additively to the solid solution plateau in the form of a typical ageing peak. For more details see ref. 19. 3.3. The C R S S in dependence on deformation rate
The measured ~ dependence o f ro supports the above described classification into three
clearly distinguishable regions where different mechanisms may be operating. In the low temperature region the CRSS of u n d o p e d and of iron-doped MgO is increased markedly b y applying a deformation rate which is higher by a factor of 20 (Fig. 3). The g sensitivity at RT rises with increasing content b o t h of Fe s+ and of impurities (Fig, 4)*.
*In o r d e r t o test t h e ~ d e p e n d e n c e o f t h e u n d o p e d 4N MgO (Figs. 3 a n d 4), s p e c i m e n s o b t a i n e d f r o m a larger single crystal b l o c k o f u n i q u e basic p u r i t y have b e e n used. One of t h e t w o p l o t t e d straight lines o f 4N MgO lying at higher values in Fig. 4 was o b t a i n e d f r o m a m o r e i m p u r e 4N crystal.
77
10
100
8 ~
-
- 80 I
-
To
6 _ ~ ~ ~ i ~
J-
60(_~2)
,/
2O (a) 0
I
I
I IIIIll
10-5
I
1 ] Illlll
10 ~
t
I
I IIIII1
10-3
I
I
I IIIIll
10-2
0
104
30
25
2,5
fMNl'i, tmq
k~m2)~
t5
1,5
(b) l
3.2 3.3
I
l
l
I
I
I
1
I
~4
3.S
~6
&7
~8
&9
4.0
~,1
.....
I ~,2 ~,3
~
Fig, 4. The C R S ~ e n d e n c e on deformation rate d at RT: (a) represented as To x / 7 o - - "r G v s . x/In(co/e) according to eqns. (4) to (8).
In the plateau To does not depend on ~ (Fig. 3). Detailed results concerning the ~ dependence in the high-temperature decrease cannot yet be presented at this time, but preliminary results indicate a significant influence of ~ on the high temperature flow stress.
vs.
log c; (b) represented as
3.4. T h e C R S S in d e p e n d e n c e on the Fe 3÷ concentration L o w t e m p e r a t u r e region
In Fig. 5 the RT values of the CRSS are plotted versus x/Fe 3*. For high concentrations the data points follow strictly a straight line
78 I
!
I
l
I
200
20 E: = 6.
I
ld~s -t
,,°l
15
I
I
3N~ ~0
0
o
,o
20
30
~o
so
6o
70
so
0
FC'~'~(rnol.pprn) '/2 Fig. 5. The dependence of the CRSS on the Fe 3+ c o n c e n t r a t i o n and on the basic purity at RT.
through the origin independent o f the basic purity. For low concentrations the data points of 4N MgO or 3N MgO lie within two horizontal bands (hatched in Fig. 5). Their "highs" are given by the variation of the basic purity which could be demonstrated by the To values of specimens annealed in CO/COs gas atmosphere (see Table 2, subsection 3.5). From this it can be concluded that the CRSS of ironfree or very low-iron-doped MgO is determined by the basic purity only. Doping of Fe 3* has no effect on the CRSS as long as a minimum doping level is reached. These minimum values depend on the basic purity and are a b o u t 100 mol. p.p.m. Fe 3÷ for 4N MgO and 600 mol. p.p.m. Fe 3÷ for 3N MgO corresponding to dipole concentrations of 50 p.p.m, and 300 p.p.m, respectively. For higher doping levels the Fe 3÷ ions control the CRSS alone. Starting from this idea it is clear why the present lowiron-doped 3N MgO containing only 6 to 170 mol. p.p.m. Fe a÷ behaves as u n d o p e d impure MgO. Pla tea u The dependence of the CRSS on Fe 3÷ measured at 773 K is shown in Fig. 6a. Starting o from the values of the undoped MgO, To, a linear increase is found. Evidently, the contributions of the basic purity and of the Fe a÷ doping are additive in contrast with the low
temperature region (Fig. 5). Plotting the difference of (to -- r °) against C D = Fea÷/2, in order to separate the contribution o f the dipoles to the CRSS, one obtains a straight line through the origin for both 3N MgO and 4N MgO (Fig. 6b). High-temperature region Above 1600 K the remarkable result is observed that the CRSS decreases with increasing Fe 3÷ content (Fig. 3), which corresponds to solid solution softening. 3.5. The effect o f annealing temperature, gas atmosphere and Fe z+ content on the CRSS As pointed o u t in subsection 3.1, the Fe3÷/ Fe 2÷ ratio in MgO is strictly determined by the annealing temperature and the oxygen partial pressure of the gas mixture for given total iron content. Therefore, the influence of these two parameters on the CRSS is characterized uniquely by stating the Fe 3÷ concentration as described in subsections 3.2 - 3.4 and represented in Figs. 3 and 5. By annealing in the controlled CO/CO2 gas mixture, having an oxygen partial pressure of 1.7 × 10 - s atm., nearly all of the d o p e d iron is transferred into the divalent state (Table 2, last column). In this way t h e e f f e c t of the Fe 3÷ on the CRSS is eliminated and the role of the other impurities (subsection 3.4), and
79 I
I
,
I
15 I
1,o
T=773K
./~
t,ooi
10
ro (a)
I 1000
I
I 2000
~ . I
i
I 3000
I
I 4000
0
Fe 3÷(rnoLpprn ) I
I
T=773K [] 3N-MgO 0 4N-MgO
I
150 l
~
100
J6
r -ro °
To-%° (b)
50[
t oI
o?
I
k
500
I
I
L
I
I
1000 1500 CO(rnol.pprn)
2000
Fig. 6. The c o n c e n t r a t i o n dependence of the CRSS in the plateau region: (a) in a linear plot T O v s . Fe3*; (b) in a linear plot o f the same data (TO -- v ° ) v s . dipole c o n c e n t r a t i o n c D according to eqns. (9) and (10). TABLE 2 Effect of total iron c o n t e n t , gas a t m o s p h e r e or o x y g e n partial pressure and of annealing temperature on the Fe 3÷ c o n c e n t r a t i o n and on the CRSS at RT. MgO Total Fe type content (wt. p.p.m.)
4N
3N
Pure <20 310 2300 4300 7500 8500 Pure 70 140 300 710
Annealing conditions, Fe 3÷, To 1573 K / air
1773 K / air
1773 K / CO - CO 2
Fe 3+ (tool. p.p.m.)
To (kp m m - 2 )
Fe 3+ (tool. p.p.m.)
~l~ pmm -2)
Fe 3+ (mol. p.p.m.)
TO (kp m m 2)
"0" 47 885 1950 -4420
1 , 5 - 3,5 3 , 1 - 4,3 8,3 12,7 -18,1 - 18,9
"0" 24 622 1520 3150 3640
1 , 5 - 3,5 4,0 7,8 11,20 14,7 - 16,0 15,8 - 18,7
"0" "0" 0,5 1,7 5,3 6,7
1,5-3,5/ 2,0 2,4 - 2,9 ;~ 2,65 +_1,5 3,7 4,1 t 3,9 J
4,7 - 5,4 5,0 5,6 - 6,0 6,5-6,9
2,8 5,7 23,0 99,0
5,5 - 6,2 5,7 - 6,7 6,2 - 8,4 7,7-8,2
"0" "0" "0" "0"
5,7 - 6,4 t 5,0 5,5 ~ +_ 1,5 5,9 7,3 l- 6,5 4,9 5 , 1 )
\
6,2 11,8 42,7 170
80
also of the Fe e÷, can be studied. F r o m the comparison of the total iron level (second column in Table 2), which is equal to the Fe e* concentration after annealing in CO/COe, with the RT values of ro listed in the last column in Table 2, it follows that no correlation exists between the F e 2 ÷ concentration and the CRSS. The same result is obtained for the plateau. Hence, we conclude that for temperatures up to 1600 K the doped divalent iron ions have no influence on the strength of MgO.
4. A N A L Y S I S O F R E S U L T S
The CRSS of Fe-doped MgO exhibits those temperature regions (Fig. 3) which are expected for combined elastic/electrostatic dipole interaction. 4.1. T h e l o w t e m p e r a t u r e decrease o f ro ( T < 700 K )
According to eqns. (4) to (8) the elastic and the electrostatic contributions to the CRSS have the same dependence on temperature, deformation rate and dipole concentration. In order to obtain information on the tetragonality of the dipoles, Ak*, and on Eo we analyse the data points in the following way: The calculations of Frank [16] show that for 0 °K the electrostatic c o m p o n e n t of %, r~ 't, is only 20 - 28% of the elastic one. Therefore, above RT we can ignore for a m o m e n t the electrostatic contribution and ascribe the whole measured (ro -- rG) values to the elastic c o m p o n e n t r [ alone*. Then the data points can be represented after formula (4) to check the T and g dependence graphically. In the subsequent quantitative evaluation o f the obtained graphic representations the elastic comF ponent, %, and also AX* can be separated numerically.
* S t a r t i n g f r o m F r a n k ' s result [16] t h a t for 0 °K r est relates to r sF as 1:4, o n e derives f r o m eqn. (4) the expression ( q Q / • • o ) = 0 . 1 8 8 ( G b 3 AX , ). S u b s t i t u t i n g i n t o f o r m u l a (5) or (8) yields for t h e t e m p e r a t u r e coefficients t h e c o n d i t i o n q52 = 4 @1. This m e a n s t h a t the e l e c t r o s t a t i c c o n t r i b u t i o n , Test decreases m u c h m o r e w i t h increasing t e m p e r a t u r e t h a n does t h e elastic comp o n e n t , r F. As an e x a m p l e , at RT r est a m o u n t s t o o n l y 5% o f the TF value a n d s o o n b e c o m e s zero at the characteristic t e m p e r a t u r e T 2, w h i c h is r o u g h l y a f o u r t h o f To1 for T e s t . T h e r e f o r e , the approximat.ive analysis des c r i b e d a b o v e is justified a b o v e RT.
According to eqn. (4) the plot of X/To -- rG versus x / T should give straight lines with inter-
section points on the ordinate A=
0.107 - ~
~/CD
(11)
and with slopes S=
(
1.155 dbk 2 X/CD In
(12)
Figure 7a shows the representation of the data points taken from Fig. 3. re was calculated according to formula (3) with ~ = 0.32 and p = 105 cm/cm a. The obtained straight lines have slopes which increase with increasing dipole concentration CD and decrease with higher g as required theoretically by eqn. (12). From the values of the ordinate intersections A in Fig. 6a, obtained by linear extrapolation to 0 °K, the dipole strength Ak* can be estimated by eqn. (11). Taking into account numerically the electrostatic contribution to %, as calculated by Frank, in the above described way, one obtains for the dipole strength A•* = 0.16 ... 0.62 if we chose d = 0.5 ... 1. In order to prove the g dependence, X/To -- rG is plotted as a function of [ln (~o/d)] v,, eo = P b2VD • Straight lines are expected from eqns. (4) and (5) with intersection points with the ordinate, A, given by eqn. (11) and with slopes $2 = A(10.82 k d T / G b a Ak*)1/'. Figure 4b illustrates the RT values, represented in this way taken from Fig. 4a, where 7G = 0.037 kp mm-2. The data points fit straight lines having slopes which increase with increasing dipole concentration as predicted theoretically. The extrapolation to 0 °K yields A values which agree with the A values in Fig. 7a. Using these A values and setting d = 0.5 ... 1 in eqn. (11) the dipole strength Ak* is estimated to be 0.17 ... 0.77. To test the dependence on the dipole concentration CD according to eqns. (4) and (7) ( T O - - TO)2 is plotted linearly versus CD in Fig. 8. Only data points of the pure 4N MgO from o Fig. 5 are used which are higher than the To values of the undoped material in order to eliminate the effect of other impurities. For comparison the same experimental data are represented in the Labusch plot, (to -- TG)a/2 versus CD, according to eqn. (1). Reasonable straight lines through the origin are obtained over the whole range of concentration, indicating that the assumed localized interaction
81
4
Ca)
23~
4N-8500]3640 4,N-4300/1520 4N-2300/6-20
l
I (b) 3N
4N
15 I
20 25 --Ill(K)'~2 I
I
30 I
Fig. 7. The temperature dependence of the CRSS of Fe-doped MgO (a) and of undoped MgO (b), represented in a x/To -- r G -- ~/T diagram after Frank, eqn. (4). The data points symbolized by • are shifted by 0.5 kp mm -2. Open symbols: ~ = 6 X 10-4 s-l; full symbols: ~ = 1.2 x 10 -2 s-1. is justified. T h e slope o f the linear Fleischer F r a n k p l o t in Fig. 8 includes n o w t h e electrostatic c o n t r i b u t i o n . If o n e separates this as described above, the t e t r a g o n a l i t y o f t h e defects is calculated b y use o f the slope, determ i n e d using eqn. (4), t o be 5X* = 0.43, setting d = 0.5. With this value t h e free a c t i v a t i o n e n t h a l p y is given b y eqn. (6) as E0 = 1.37 eV.
4.2. The plateau (T > 700 K) In t h e t e m p e r a t u r e - i n d e p e n d e n t r e g i o n elastic and e l e c t r o s t a t i c dipole i n t e r a c t i o n s result in a linear d e p e n d e n c e o f the CRSS on the dipole c o n c e n t r a t i o n (eqns. (9) and (10)). T h e e s t i m a t i o n o f F r a n k [16] shows t h a t the e l e c t r o s t a t i c c o n t r i b u t i o n t o the p l a t e a u stress a m o u n t s t o 12 - 18%. Using t h e l o w e r value, the expressions (9) a n d (10) can be c o m b i n e d t o re = (to - - r a ) = 1.48 (1.0 + 0 . 1 2 ) G AX*c D. T h e e x p e r i m e n t s d e m o n s t r a t e t h a t this relat i o n s h i p is fulfilled (Fig. 6b). As d e s c r i b e d in o w h e r e roo is t h e s u b s e c t i o n 3.4, re = ro -- To,
CRSS o f t h e u n d o p e d 4N MgO or 3N MgO, which already includes rG. Insertion o f the slope o f t h e p l o t o f Fig. 6b into t h e above form u l a gives AX* = 0.40, in g o o d a g r e e m e n t with AX* q u o t e d in the l o w - t e m p e r a t u r e region. If we p r e f e r the AX* values d e t e r m i n e d f r o m t h e dipole d e p e n d e n c e o f ro in the p l a t e a u and in t h e l o w - t e m p e r a t u r e region r a t h e r t h a n t h o s e AX* values o b t a i n e d f r o m the T and d e p e n d e n c e b y use o f the adjustable p a r a m e t e r d, t h e best value f o r the dipole s t r e n g t h is AX* (MgO : F e 3÷) = 0.42 in c o m p a r i s o n with previous estimates f o r alkali halides, AX* (NaC1 : Ca 2÷) = 0.57 [ 1 ] , and AX* (LiF : Mg 2÷) = 0.53 [ 2 ] .
4.3. The high-temperature region (T > 1300 K) The t h e o r e t i c a l l y p r e d i c t e d decrease o f the CRSS at high t e m p e r a t u r e s was f o u n d experim e n t a l l y (Fig. 3) and marks the end o f t h e plateau. If we c o m b i n e this result w i t h the
82 100
i
1
i
I
9O I 8O ~00
70 6O ~00
(kprnrr}2y/2
)2 -2.2
± 0 Davidge(1967) ~7 Moon&Pratt(1970) [ ] Srinivasan& Stoebe (1970) • present study
•
I
0
,
1000
~
I 2OOO
~
r00
I 300O
Co (rnol.ppm )
Fig. 8. T h e d e p e n d e n c e o f t h e C R S S o f F e d o p e d 4 N M g O at R T o n t h e d i p o l e c o n c e n t r a t i o n : b o t t o m : t h e F l e i s c h e r - F r a n k p l o t (I"o - - TG) 2 v s . CD, e q n . ( 2 ) o r e q n s . (4) a n d (7); t o p : t h e L a b u s c h p l o t o f t h e s a m e d a t a (T o - - TG) 312 vs. CD, e q n . (1). T h e c D values f o r d a t a p o i n t s c i t e d w e r e e s t i m a t e d f r o m t h e given t o t a l i r o n c o n t e n t as d e s c r i b e d in s u b s e c t i o n 3.1.
observation of Wicks and Lewis [9], namely a reduction of the number of Fe z÷ vacancy centres coupled by a simultaneous increase of the concentration of isolated F e 3÷ ions, the high-temperature decrease of ro could be interpreted informally by thermal dipole dissociation. But the fact, d o c u m e n t e d b y the To values above 1300 K in Fig. 3, that the dipole dissociation begins earlier for higher dipole concentrations, is not compatible with this explanation. Possibly the decrease of the plateau stress due to dipole dissociation is superposed b y other opposite effects [ 2 ] . Factors in the temperature region considered which can contribute to ro are: (a) Parelastic as well as dielastic interaction of dislocations with non-associated Fe z* ions by Cottrell locking [ 1 4 ] , which requires an extra stress for a dislocation to break loose from the Cottrell clouds formed in the stress field around the dislocation. At high temperatures and not t o o high strain rates the Fe 3÷ ions are sufficiently mobile so
that the Cottrell clouds can follow steadily the moving dislocations resulting in friction drag in analogy with the microcreep [14, 35]. Note the analogy with the friction stress in the plateau due to Snoek clouds consisting of oriented (Fe a÷ V Fe 3÷) dipoles. (b) Electrostatic interaction of charged dislocations or of dislocations having charged jogs with Fe a÷ ions [ 8 ] . On the other hand, all these mechanisms predict an increase of the CRSS with inc r e a s i n g F e 3+ concentration in contrast with the experimental results showing the inverse dependence. Eventually, a "doping-induced solid solution softening" plays an important role, as discussed for FeO [ 37, 3 8 ] .
CONCLUSION
The model of the combined elastic and electrostatic dipole hardening proposed by Frank for cubic crystals is applied on low-
83 i r o n - d o p e d MgO. T h e p r e s e n t c o n c e p t rests o n t h e a s s u m p t i o n t h a t t h e MgO o f s u f f i c i e n t l y high p u r i t y c o n t a i n s t h e d o p e d Fe 3÷ ions in t h e f o r m o f disperse d i s t r i b u t e d , n o n - a g g l o m e r a t e d s y m m e t r i c ( F e 3÷ V Fe a÷) dipoles, w h i c h h a v e a clear t e n d e n c y to act as s t r o n g "indiv i d u a l " o b s t a c l e s b u t n o t as o b s t a c l e groups. A c c o r d i n g l y , t h e critical resolved shear stress ro is c o n t r o l l e d at low t e m p e r a t u r e s b y therm a l l y a c t i v a t e d Fleischer i n t e r a c t i o n , while at elevated t e m p e r a t u r e s in t h e " p l a t e a u " ro is determined by the induced Snoek effect of o r i e n t e d dipoles in t h e stress field o f m o v i n g dislocations. In b o t h regions the elastic/ t e t r a g o n a l c o n t r i b u t i o n t o ro d o m i n a t e s (Fig. 1). T h e m e a s u r e d values o f ro w h i c h s h o w p r o n o u n c e d solid s o l u t i o n h a r d e n i n g e x h i b i t the theoretically expected dependencies upon t e m p e r a t u r e (Figs. 2, 3 a n d 7), d e f o r m a t i o n rate (Fig. 4) a n d dipole c o n c e n t r a t i o n (Figs. 5, 6 and 8). In b o t h regions q u a n t i t a t i v e agreem e n t w i t h t h e m o d e l is f o u n d . T h e d i p o l e s t r e n g t h was e s t i m a t e d at Ah* = 0.42 similar t o values o f d o p e d alkali halides. D i v a l e n t iron ions h a v e n o m e a s u r a b l e i n f l u e n c e o n to. T h e r a p i d h i g h - t e m p e r a t u r e fall o f ro is a t t r i b u t e d t o t h e t h e r m a l b r e a k i n g - u p o f the dipoles i n t o Fe 3÷ ions a n d c a t i o n vacancies. T h e solid s o l u t i o n s o f t e n i n g o b s e r v e d in this region c a n n o t be e x p l a i n e d .
p e r i m e n t s , a n d Dr. I. V a r n e r f o r r e a d i n g a n d c o r r e c t i n g t h e English m a n u s c r i p t . F i n a n c i a l support by the Deutsche Forschungsgemeins c h a f t is g r a t e f u l l y a c k n o w l e d g e d .
Note added in proof (December 22, 1975)
19
R e c e n t l y G i l m a n [ 3 9 ] has p r o p o s e d a n e w i n t e r e s t i n g m o d e l o f s h o r t r a n g e dipole i n t e r a c t i o n w h i c h is b a s e d o n c u t t i n g o f t h e dipoles b y gliding dislocations. A h l q u i s t [40] has applied this m o d e l t o d o p e d MgO a n d c o n c l u d e s t h a t also t h e C R S S o f MgO at R T is d e t e r m i n e d b y d i p o l e shearing a n d n o t b y t h e e l a s t i c / t e t r a g o n a l d i p o l e passing m o d e l o f Fleischer a n d F r a n k discussed here. But, it c a n be d e m o n s t r a t e d t h a t t h e G i l m a n - A h l q u i s t m e c h a n i s m explains 10% o f t h e m e a s u r e d values o n l y [41] a n d t h e r e f o r e c a n b e e x c l u d e d as t h e c o n t r o l l i n g p r o c e s s f o r t h e solid s o l u t i o n h a r d e n i n g in d o p e d MgO.
20
REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
21 22 23 24 25 26 27 28
ACKNOWLEDGEMENTS T h e a u t h o r wishes t o t h a n k Mr. B. K u m m e r f o r the p e r f o r m a n c e o f t h e d e f o r m a t i o n ex-
29
W. Frank, Phys. Status Solidi, 29 (1968) 391. B. Reppich, Acta Met., 20 (1972) 557. W. Frank, Z. Naturforsch., 22a (1966) 365. W. Frank, Z. Naturforsch., 22a (1966) 377. R. L. Fleischer, Acta Met., 10 (1962) 835. R. L. Fleischer, J. Appl. Phys., 33 (1962) 3504. G. Schoeck and A. Seeger, Acta Met., 7 (1959) 469. R. L. Moon and P. L. Pratt, Proc. Brit. Ceram. Soc., 15 (1970) 203. B. J. Wicks and M. E. Lewis, Phys. Status Solidi, (a), 6 (1970) 281. G. D. Miles, F. J. P. Clarke, B. Henderson and R. D. King, Proc. Brit. Ceram. Soc., 6 (1966) 325. R.W. Davidge, J. Mater. Sci., 2 (1967)339. G. W. Groves and M. E. Fine, J. Appl. Phys., 35 (1964) 3587. M. Srinivasan and T. G. Stoebe, J. Appl. Phys., 41 (1970) 3726. P. Haasen, in R. W. Cahn (ed.), Physical Metallurgy, North-Holland Publishing Co., Amsterdam, 1965, p. 821. R. L. Fleischer, in D. Peckner (ed.), The Strengthening of Metals, Reinhold, 1964, p. 93. W. Frank, personal communication. R. Labusch, Acta Met., 20 (1972) 917. B. Reppich, Mater. Sci. Eng., 19 (1975) 51; see also B. Reppich, Habil. Thesis, Univ. ErlangenNfirnberg, 1973. H. Knoch and B. Reppich, Acta Met., 23 (1975) ],055 and 1061. A. B. Lidiard, Handbuch der Physik, Band 20, Springer, 1957, p. 246. J. Brynestadt and H. Flood, Z. Elektrochem., (1958) 954. J. F. Wertz, J. W. Orton and P. Auzins, J. Appl. Phys., Suppl., 33 (1962) 322. A. L. Schawlow, J. Appl. Phys., Suppl., 33 (1962) 395. H. Wiedersich, Proc. Second Symp. on Low Energy, X- and Gamma-Sources and Applications, ORNL II C-10, Tennessee, U.S.A., 1967, p. 101. B. Ilschner, in O. Madelung (ed.), FestkSrperprobleme X, Advances in Solid State Physics, Pergamon, Vieweg, Braunschweig, 1970, p. 415. E. KrSner, Kontinuumstheorie der Versetzungen und Eigenspannungen, Springer, 1958. F. R. N. Nabarro, Rept. of Conf. on the Strength of Solids, Phys. Soc., London, 1958. A. Cochardt, G. Schoeck and H. Wiedersich, Acta Met., 3 (1955) 533. R. F. S. Hearmon, in K. H. Hellwege, LandoldBSrnstein, Zahlenwerte und Funktionen aus Naturwissenschaften und Technik, Bd. 1, Springer, 1966, p. 29.
84 30 R. Strumane and R. De Batist, Phys. Status Solidi, 6 (1958) 569. 31 R . W . Whithworth, Phil. Mag., 15 (1967) 305. 32 P. D. Southgate, Bull. Am. Phys. Soc., 7 (1962) 501. 33 J. Snoek, Physica, 8 (1941) 711. 34 A. Portevin and F. Le-Chatelier, Compt. Rend., Paris, 176 (1923) 507. 35 J. Weertman and J. R. Weertman, in R. W. Cahn (ed.), Physical Metallurgy, North-Holland Publishing Co., Amsterdam, 1965, p. 793. 36 B. Reppich, W. Blum and B. Ilschner, Ber. Deut.
Keram. Ges., 44 (1967) 41. 37 B. Ilschner, B. Reppich and E. Riecke, Discussions Faraday Soc., 38 (1964) 243. 38 B. Reppich, Phys. Status Solidi, 20 (1967) 69. 39 J. J. Gilman, J. Appl. Phys., 45 (1974) 508. 40 C. N. Ahlquist, in R.C. Bradt and R. E. Tressler (ed.), Proc. Symposium on Plastic Deformation of Ceramic Materials, held at Pennsylvania State Univ., July 1974, Plenum Press, New York, 1975, p. 233. 41 B. Reppich and A. Straub, in preparation.