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ENERGY COMPUTATIONS OF PLANAR DEFECTS IN SOME OXIDES
J. Rabier, P. Veyssière, J. Grilhé
To cite this version:
J. Rabier, P. Veyssière, J. Grilhé. ENERGY COMPUTATIONS OF PLANAR DEFECTS IN SOME OXIDES. Journal de Physique Colloques, 1973, 34 (C9), pp.C9-373-C9-377.
�10.1051/jphyscol:1973964�. �jpa-00215439�
JOURNAL DE PHYSIQUE Co//oque C9, supplknzent au no 11-12, T0117e 34, Novetnbre-DPcornbre 1973, page C9-373
ENERGY COMPUTATIONS OF PLANAR DEFECTS IN SOME OXIDES
J. R A B I E R , P. VEYSSIERE a n d J. GRILHE Laboratoire d e Metallurgie Physique (LA 131) 40, avenue d u Recteur-Pineau, 86022 Poitiers, France
ResumC. - L'energie des dcfauts d'empilement dans les plans (loo), ( 1 10) et ( I I I) est calculee dans I'approximation, satisfaisante pour les oxydes, de la liaison ionique. Les variations des interactions coulombiennes sont donnccs suivant dcs interactions entre plans selon la rnethode de Fontaine.
Les resultats nunieriques montrcnt qu'on doit s'attcndre i rencontrer des parois d'antiphase resultant du contact de donlaines nuclees differeniment plut6t qile des fautes d'empilernent forrnees par LIII processus de dissociation par glisscrnent dc dislocations parfaites, ce q i ~ i est en bon accord avec les observations par microscopic electronique de Lewis.
Le mecanisme de cis:iillemcnt sycichronc developpe par Hor~istra pour expliquer les proprietes du glissement des dislocations dans la structure spinelle cst discute en fonction des energies des defauts associes. On conclut que ce I)1-ocessus semble plus probable dans le corindon que dans la spinelle.
Abstract. - The energies of (loo), (1 10) and (1 I I) planar defects in the MgAlzOl spinel were c o m p ~ ~ t e d using the ionic bond approximation which is a nearly good a s s ~ ~ m p t i o n for oxides.
The coulombic interactions can be summed to give the interplanar interactions used in the compu- tation according to Fontaine's treatment.
The nun~erical results show that those defects should be considered as antiphase boundaries resulting from the impingement of adjacent domains rather than stacking faults formed by a process o f glide from the dissociation of perfect dislocations, in good agreement with the experimental conclusions o f Lewis based on electron microscope observations.
The synchro-shear mechanism developed by Hornstra to explain the glide properties of the dislocations in the spincl structure is disc~~ssed taking into account the associated defect energies in both 2-Al2O3 and MgAI2O4. I t is concluded that this process seems to be more realistic for rorunduni than for spinels.
1. Introduction. - T h e particular glide properties of dislocations in s o m e oxydes (1-A120,. MgA1,04, ...) showing structural vacancies in the cation sublattice, were explained o n the basis o f the dissociation abilities o f perfect dislocations a n d of the cooperative dis- placements o f t h e t w o kinds of ions (the synchro-shear process) [I], [2].
F o r example, in the spinel structure, the formation of a stacking fault by splitting of tlie perfect dislo- cation o f Burgers vector AB into two partial dislo- cations following the reaction :
results f r o m tlie existence of cation vacancies. T h e shortest u n i t translation for the conlplete structure is t h a t between t w o vacant cation sites and is twice the unit translation for the anion sublattice (see Fig. I).
Following the Kronberg's model for c o r u n d u m [ I ] . Hornstra [2] assumes in tlie MgAI,O, case that a new dissociation into Sour partials m a y occur from the a b o v e configuration ( F i g . 2 sho\vs this predicted configuration)
AB + Aa + ay + yb + b B .
FIG. 1. -- Laryc opcn circles rcpresent ~~nderlying oxide ions.
Sninll filled circles represent the upper layer of cations.
The I\\'() outer pairs of partials a r c bounding a st:~cking f ; ~ u l t in [lie oxygen subl:~ltice in which concurrently
~ l l c ' cations would li;~vc movcd t o their favoured crystal coordir?alion. This process is called (( . y ~ . / ~ r l l i o - .cII(,NI. )) by Kronbcr-g. T h e t w o inner partials bound
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973964
C9-374 J. RABIER, P. VEYSSIERE A N D J. G R I L H E
in wliicli tlie partial filling of interstices by tlie cations is nucleated diferently [6], [7].
Under the above experimental facts a calculation of the stacking fault energy is interesting.
g ~ " ] 2. Stacking fault energy calculation. - 2.1 ASSUMP-
TIONS. - We use the ionic approximation wliich is nearly good as well as for Al,03 than for MgAI,O,.
Indeed following Pauling [8], the bond is about 73 o/:, FIG. 2. - Dissociation into four partials in the spincl str~~cttlrc. ionic for ,410 and 63 % for MgO.
We suppose tliat the close range repulsion of ions a fault in the cation stacking only. His argunients
are based on tlie requirement of local electroneu- trality and on the observation of twinning on ( 1 11) planes. This last dissociation may only occur in ( 1 I I ) planes, the first reaction being possible in different planes. The syncliro-shear process obviously acts during the glide of the splitted dislocation (see Fig. 3).
FIG. 3. - The synchro-shear process.
The upper layer of oxygen ions are displaced of Aa.
The cations are displaced to a new octahedral interstice.
The same treatment leads to analogous I-esult in the a-AI,O, case. In this structure, the splitting of the perfect dislocation in two partials is limited to the (0001) plane only (see Fig. 4).
between second neiglibouring planes may be neglected.
This is justified by the exponential decreasing of this interaction energy with the distance between two ions.
Futliermore in the calculated stacking fault types.
tlie cations still have their favoured crystal coordi- nation so tliat the change of the Born repulsive energy is zero for tlie first neiglibouring planes.
We take into account tlie change of electronic polarization of ions close to tlic defect by introducing the H F dielectric constant c, of the material [9].
So tlie coulombic interaction energies are divided by c , . We will discuss later tlic validity of tliis approximn- tion.
The structures are idealized so tliat the oxygen ions occupy a perfect close packed lattice : FCC for spinel, HCP for corundu~ii.
2 . 2 CALCULATION OF THE C O U L O M U I C E N E R G Y CHANGE TAKING PART I N A STACKING FAULT. -
2 . 2 . 1 The variation of coulombic interaction energy between an ion with a charge Z , and a plane of periodic charges Z , after a translation parallel to this plane of a vector b, may be written as (Fontaine [9]) :
z is the distance between Z , and tlie plane.
r is the projection in the plane of the vector Z, Z,.
1, is a reciprocal lattice vector associated to the plane with periodic charges Z , .
Q is the surface of the unit cell of tliis plane (see Fig. 5).
FIG. 4. - The splitting in four partials in both corunduln and spincl.
The plastic deformation of stoichionietric MgAI,O, and T-AI,O, can be understood using this assumption.
Unfortunately, no experimental evidence has been reported to confirm the glide dissociation yet [3], [4], [5]. Futliermore electron microscope observations in spinels have revealed large extended defects which present a faulting in the stacking of cation sublattice, these extended defects having fault vectors not lying in the defect plane [6]. They have been interpreted in terms of antiphase boundaries probably formed
during crystal growth by the impingement of domains FIG. 5 .
ENERGY COMPUTATIONS O F PLANAR DEFECTS IN SOME OXIDES C9-3 75 2 . 2 . 2 Each plane of the structures studied here
may be colisidered as the superposition of elementary planes caracterized by an identical unit cell. For a given stacking, the elementary plane corresponds to a common cell over all the stacking planes. The variation of coulombic interaction energy per unit surface between two planes A and B, respectively translated of b, and b, is :
N M
2 7T
AVAB = C C C Za Zll X
R~ 1 + 0 a = l , , = I
features, each plane containing one kind of ions (cf. Fig. 6)
- Oxygen type with a FCC stacking (b, b2 b, b, type).
- Kagomt layer type made of cations with six fold coordination in octahedral interstices (Z = 3 ) (c, c2 c4 type).
- (( M i x e d layer )) type. The mixed layer is a stacking constituted by three simple planes (a, type).
- A tetrahedral cation plane (2 = 2 ) ; - An octahedral cation plane ( Z = 3) ;
e-l 1 1 2 - A tetrahedral cation 'lane ( Z = 2).
e ; U r ~ ~ - r-1 ei2.(ho - h ~ )
1 1 Between the oxygen layers in a [111] direction are found alternately kagome layers and mixed layers.
A and B containing respectively N and M elementary elementary plane used in the calculation is a planes, 1 and 52 are respectively a reciprocal lattice simple plane of (( ,17ixec/
,
aJ,c?,. ,,.vector and the unit cell surface associated to the The i n energy between two planes resulting choosen elementary plane. from the creation of a stacking fault may be written
The term : as follows :
depends on tlie geometrical and electrostatic features
of the A a n d B planes. x [exp(ikb)- l] .exp J6(l2 + nz2- lm)li2 z
I
2 . 3 CALCULATION OF T H L S T A C K I N G FAULT ENER- With
GIES. - A stacking fault is introduced in a crystal -
when one half of the crystal is translated of a vector b = - ( I + ,?,) + J ~ ( ~ , ~ - - 1) 1, n1 Z * with respect to the other half. T o calculate tlie stacking
defect energy, we have to know the change in tlze
:: [P I
and G,,,, is a coefficient depending on the nature and interaction energy between the different planes of
on the charges of tlie A and B planes. For example, the two half-crystals. The change in coulon>bic energy GAB = Z , 2, when A and B are two mixed layers and between two planes has been given in 2 . 2 . 2 .
We have computed tlie stacking fault energies in
the ( 1 1 I), ( I 10) and (100) planes of the spinel struc- GAB = Z,,,. Z,
(
1 + 4 cos(5
(1 - m ) ) xture and in the basal plane of corundum.
2 . 3 . 1 Spi17el s t r ~ l c t ~ n . e . - + ( l l 1 ) plane. - In x [cOS
(5
( I + 111)) + cos(;
(1 - t n ) ) ]1
tlie [I 1 I] direction the s t a c k c ~ ~ is conlpletely des-
cribed by 18 consecutive planes of three different for two kagome layers (2, = Z, = 3).
Stacking i n t h e (1 1 1 ) p l a n e - oxygen b l b r b : bi Iyp? k a ~ o r n k layer c i cz c4 type mixed layer a , type.
C9-376 J. RABIER, P. VEYSSIERE AND J. GRILHE We have considered the synchroshear process where
both in a kagomC layer and in a mixed layer. 1 = --- 271 J2
+ (110) plane. - The stacking has a period of (lx + m y ) .
a0
4 atomic planes; each of them being constituted by
different ions 2 . 3 . 2 Corundum. - Along the C axis, the stacking
period is 12 planes, each plane containing one kind
x [exp(ihb) - 11 exp
[-
2 (2 I' + m2)li2 z]a0
where
of ion
x [exp(ihb) - I] exp
[
- -- 4 n (1' + in ' - lni) l"J3 a0
2 n where
h = -(JZlx + my) .
a0 1 ) y .
+ (100) plane. - The period is 8 atomic planes, each plane containing different ions
I
2 . 3 . 3 One may notice that all the values of stacking
2 J2 1 fault energy may be expressed in e2/ai E , units, so
AVAB = T 1 GA,(h) x that the corresponding stacking fault energy can be
U1.m)
l,m#O (12 + m2)'I2 evaluated for similar structures only by substituting the appropriate numerical values.
x [exp(ip(ilb) - I] exp
[-
- Results of the computations of stacking faultao energies are summarized in the table. The width rl MgA1204
a, = 8.08 i% : ,u = 1.170 x 1012 d y n e ~ . c m - ~ ; v = 0.250 ; E , = n2 = 3.0.
-
b 7 y caract. d (4 dl6
(ergs. ~ m - ~ )
(?
e2 units) screw edge screw edgeEm a0
%
[lll] 2 417 16.579 - - - -mixed layers
ENERGY COMPUTATIONS OF PLANAR DEFECTS IN SOME OXIDES C9-377 of dissociation between the two partial dislocations
has been evaluated taking into account only the repulsive elastic force acting on them. The two extreme values of d given in the table correspond to the widths of dissociation found starting from a perfect screw dislocation or from a perfect edge dislocation.
3. Discussion. - In order to take into account the change in electronic polarization energy of ions near the defect, we have used 8 , the H F dielectric constant. In fact, the variation in the neighbouring of the more polarizable ions (oxygen) with respect to the perfect crystal should be calculated better.
This change in electronic polarization will vary according to tlie introduced defect. In our calculations the use of c, leads to overestimate the polarization energy i. e. to lower tlie stacking fault energy and consequently to overestimate also the distance between the partial dislocations.
3 . 1 SPINEL. - For the spinel structure we may point out from the above calculations that the width of splitting in two partials is weak in MgAI,O, (about 1.5 to 2 b i n the ( 1 11) planes, 6 to 8 b in ( 1 10) and (100) planes). This fact is in rather good agreement with electron microscope observations.
However, under our approximations, the splitting in two partials appears to be more favourable in (100) and (110) planes. The stacking fault energy both of the anion and cation sublattices in the (1 11) planes is very high. In these planes, for so weak distances between partial dislocations, even with regard to the cation stacking fault, it seems difficult to speak in terms of perfect dislocation splitting. To be more significant a calculation of the atomic structure of the dislocation core should be performed.
3 . 2 CORUNDUM. - In this case, the estimate of the width of splitting gives more credibility to
the synchro-shear process which has still a high energy.
Electron microscope observations [3] has not brought forward the splitting into two partials when the order of magnitude found here enables us to think that the dissociation may be seen. However, it is important to notice in this particular case of the unsymmetric fault (which is more favourable as far as energy is concerned) that the neighbouring of an oxide ion is not modified in the defect [I].
(Only the direction of the electric polarizing field on the anion changes.) This would lead to a small difference in polarization energy, which would contri- bute to increase the stacking fault energy calculated here, so that the dissociation distance should be smaller.
4. Conclusion. - In order to obtain a better evaluation of the stacking fault energy, it would be valuable to perform first a precise calculation of the electronic polarization, secondly relaxations in the part of the crystal surrounding the defect. The para- meters used in the close-range repulsive energy between ions relevant to a crystal with three different ions shoud then be satisfactorily known. Another progress would be calculating the atomic core confi- guration of the whole dislocation using relaxation methods.
However, the calculation we have presented here enables us to say that the stacking fault energy in both cations and anions sublattices is very high, so that on the basis of the splitting in four partials, the synchro-shear process may still be expected but, chiefly for the spinel structure, in a more sophisticated form. At last, a general conclusion can be drawn from the above results i. e. the energies of cation sublattice stacking faults are high and lead to splitting in two partials with weak dissociation distances.
References
[I] KRONBERG, M . L., Acta M d . 5 (1957) 507. [6] LEWIS, M. H., Phil. Mag. 14 (1966) 1003 and Pl~il. Mag.
17 (1968) 481.
PI HORNSTRA, J., J . PIIJJS. Cllern. Solids 15 (J960) 311. See [71 TABATA, H., O K U D A , H,, ISHI[, E,, Japarr. J .
also, Proc. 4th I t ~ r . Synrposirrn~ 012 Reactivit~~ of Solids 12 (1973) 7.
(Elsevier), 1961, 563. [8] PAULING, L., The nature of Chemical Bond (Cornell Univ [3] BARBER, D. J. and TIGHE, N. J., Phil. Mag. 11 (1965) Press, Ithaca, New-York) 1960.
495. See also Phil. Mag. 14 (1966) 531. [9] FONTAINE, G., Thesis Orsay (1968), J. Phys. Chem. Sol.
28 (1967) 2199; J. Phys. Chem. Sol. 28 (1967) 2553.
[4] HOCKEY, B. J., J. Atn. Ceram. Soc. 54 (1971) 223. [lo] SIMMONS, G., WANG, H., Single Crystal, Elastic constants [5] DE JONGUE, L. C. and BELL, W. L., 7e Congrks Int. de and Calculated Aggregate Properties, A. Handbook
Microscope Electronique (Grenoble) (1970) 337. (the MIT Press) 1971, 278, 329.
