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DISLOCATION STUDIES.Atomistic calculations of interaction energies between point defect complexes and
dislocations in ionic crystals
M. Puls
To cite this version:
M. Puls. DISLOCATION STUDIES.Atomistic calculations of interaction energies between point defect complexes and dislocations in ionic crystals. Journal de Physique Colloques, 1980, 41 (C6), pp.C6-135-C6-138. �10.1051/jphyscol:1980635�. �jpa-00220073�
JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 7, Tome 41, Juillet 1980, page C6-135
DISLOCATION STUDIES.
Atomistic calculations of interaction energies between point defect complexes and dislocations in ionic crystals
M. P. Puis
Materials Science Branch, Atomic Energy of Canada Limited, Whiteshell Nuclear Research Establishment, Pinawa, Manitoba, ROE 1L0 Canada
Résumé. — Les défauts ponctuels, en particulier les impuretés aliovalentes, ont une action réciproque avec les dislocations et peuvent changer les propriétés mécaniques. Spécifiquement, la force maximale, fm, entre le défaut et la dislocation et la portée de cette action réciproque, w, sont des paramètres importants dans les théories de durcissement en solution solide.
Nous avons fait des calculs atomistiques à l'aide du programme d'ordinateur, PDINT, pour déterminer les énergies d'interaction entre les complexes F e3 + et les dislocations coin, a/2 < 110 > { lTO }. Deux potentiels interatomiques, F e3 +- 02~ , étaient utilisés. Le centre du complexe était situé dans un des deux plans de glissement, si bien que le défaut était soumis au cisaillement en étant traversé par la dislocation. fm varie avec le potentiel interatomique choisi et l'orientation du défaut de 1,2 x 1 0 "1 0 N à 5,4 x 1 0 "1 0 N pour w m 2 b (le vecteur de Burgers). En comparaison, la valeur trouvée expérimentalement est donnée par fm = 2,02 x 10~9 N.
Abstract. — Point defects, particularly aliovalent impurities, by interacting with dislocations, can affect the mechanical properties of ionic crystals. Specifically, the maximum defect-dislocation interaction force, fm, and the range of this interaction, w, are important parameters in theories of solution hardening.
We have carried out atomistic calculations using the program PDINT to determine interaction energies of ( F e3 +- cation vacancy-Fe3+) point-defect complexes with a/2 < 110 > { lTO } edge dislocations in MgO. Two Fe3 + - 02"
impurity potentials were used. The center of the complex was positioned along one of the two slip planes. This ensured that the defect was sheared as the dislocation passed through it. Depending on defect orientation and potential,/m varied from 1.2 x 1 0 ~1 0N to 5.4 x 1 0 "1 0N , with w ~2b (b, Burgers vector). This compares with an experimental value of fm = 2.02 x 10 ~9 N.
1. Introduction. — In the theory of solution har- dening of a crystal by solute impurities, the maximum force of interaction fm between the solute and the dislocation, and the range of this force w are impor- tant parameters [1]. In MgO, iron is a common impurity. The solute can occur either in the di- or trivalent state. However, studies [2] have shown that hardening is associated with the ions only in their trivalent state (Fe3 +). Solute impurities such as Fe3 +
having valences higher than those of the solvent atoms are frequently associated with charge-compensating vacancies. The neutral defect responsible for the low temperature solution hardening of MgO is thus thought to be the Fe3+-cation vacancy F e3 + trimer.
Two different linear configurations of this defect are possible depending on whether it is oriented along the [100] or [110] directions. A detailed theoretical study [3] and our own present calculations, however, suggest that the [100] configuration has a slightly lower energy and is probably the most commonly occurring defect complex. This differs from previous suggestions [4] which presumed the [110] configuration to have the lower energy.
Previous theoretical evaluations of solution harden- ing in ionic crystals [4], have determined fm and w
on the basis of linear continuum models for the interaction. Contributions to the dislocation-defect interaction from both elastic and electrostatic dis- tortions were evaluated. The elastic interaction arises from the presumed tetragonal distortion of the defect whereas the electrostatic interaction is a consequence of the shearing of the defect as the dislocation passes through it. In general, since we require the value of the interaction within the very non-linear core region of the dislocation, we expect the accuracy of these continuum calculations to be fairly low. The following results, based on an atomistic model, should be an improvement on the above, since these types of cal- culations take explicit account of the non-linear effects. The principal limitations of the present method are the accuracy with which the atomic force laws are known and the cost of the calculations.
2. Potentials. — We use the Model 1 shell model potential of Catlow et al. [5] for the MgO crystal.
For the F e3 + impurity, we assume that the cation-:
anion interaction dominates. Thus, a straightforward approach is to use a potential which describes this interaction in the pure solute oxide, i.e. F e302. The first potential, II, derived by Catlow and Fender [6] was
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980635
C6-136 M . P. PULS
first used by Gourdin and Kingery [3]. It is based on a Born-Mayer type of potential for ~ e ' + - 0 ' - with the same p
-
+ - but the pre-exponential multiplied by-.
YFe3 + - rFe2f
expl
-- I
where the r's are the ionic radii.P + -
~ h e t a l u e used bdCatlow and Fender and Gourdin and Kingery was-0.80
A
which is only slightly less than the radius of 0.82A
for Fez +.
The ionic radius for iron in Fe203 was originally derived by Dickens et al. [7]. They adjusted the radius to account for Fe3 + in an fcc structure obtaining values of 0.7-0.8A.
Thus scaling the FeO potential using rFe3+ = 0.7A,
weobtain the second iron impurity potential, 12. Since, as discussed in detail in [3], I1 is probably too weak, the calculations were repeated using this second potential. Table I summarizes the impurity potential data.
Table I. - Born-Mayer parameters for the Fe3+-02- interaction.
The potential cp+ - has the form (P+- = A + - exp(r+-lp+-).
A + - P + - Ionic radius
Type
- (eV)
(4 (4
- - -
I1 702.29 0.336 2 0.80
I2 1 003.53 0.336 2 0.70
3. Calculations. - The method of calculation is based on the computer program PDINT described in detail elsewhere [8]. The program calculates the energy of formation of a defect in a perfect or a dislo- cated lattice : E,? and E;, respectively. Thus the inter- action energy, El, = Eg - E:. This is attractive for negative interactions. Our objective is to calculate the variation in E,,, with separation R between defect and dislocation centers. Unfortunately, beca~se of the size of the defect and the two possible orientations, the total number of positions for which El, should be calculated to give a comprehensive picture is prohi- bitively large. We have chosen to calculate E,,, for the [loo] trimer (because it has the lower energy) and the three non-equivalent orientations with respect to the a12 [I101 edge dislocation shown schematically in figure 1. The center of the defect has been arbitrarily taken at its mid-point, i.e. the location of the cation vacancy V&,. The defect-dislocation separation R is thus the distance between VL, and the dislocation's center. As shown in figu're 1, the defect center is located on the lower of the two slip planes. Locating the defect center on this plane ensures that the complex is sheared as the dislocation passes it. Locating the defect center in the planes immediately above and below the upper and lower slip planes would also ensure that this objective is realized. Some of these calculations will be done at a future date.
Fig. 1. - Location of point-defect complex with respect to edge dislocation showing the shearing action of the dislocation on the defect for different defect orientations : a) [OlO] ; 6 ) [loo] ; c) [OOl].
The shaded circle and the square represent, respectively, the Fe3+
impurity and the cation vacancy. The host lattice is represented by the open circles with Mg2+ given by the small and 0'- by the large circles.
4. Results and discussion. - Figures 2-4 summarize the interaction energy results as a function of R for the three orientations given in figure 1. In figures 5-7 we have plotted the interaction force between the defects. These results were obtained by a graphical differentiation of the curves given in figures 2-4. The
SHEARED PERFECT .-LATTICE VALUE
FOR I 0 7
SHEARED PERFECT
,,- , "--,..- ..- 0.6
t
V-L*, ,ILL V,,LYC
FOR 2 0.5
t
Fig. 2. - Values of Ei,, versus defect-dislocation separation, R, for the [loo] orientation shown in figure lb. Curve 1) potential I1 ; curve 2) potential 12.
ATOMISTIC CALCULATIONS OF INTERACTION ENERGIES
SHEARED PERFECT
L A T T I C ~ ~ - - ~ VALUE
#.
SHEARED
Fig. 3. - Same as figure 2 for the [010] orientation shown in figure l a .
Fig. 4. - Same as figure 2 for the [001] orientation shown in figure l c .
Fig. 5. - Force on the dislocation due to the presence of the defect (X10) for the [loo] orientation using the I1 potential.
results summarized in the figures show that there are significant quantitative differences between the two impurity potentials with the weaker I1 potential
I+ 1
PEAK = -3.36
Fig. 6. -Same as figure 5 for the [OlO] orientation.
-10 (cev/A~
A 12.- 10.-
Fig. 7. - Same as figure 5 for the I2 potential. Curve 1) [loo]
orientation ; curve 2) [OlO] orientation.
exhibiting the stronger variation in energy. For both potentials, the variation of the force with R is less than that given by the continuum calculations [4].
Taking the largest forces, we obtain w
-
2 b (b, Bur- gers vector). With the trimer axis oriented parallel to the dislocation line, the variation in energy is less than that in the other two cases. This suggests that electro- static effects due to the shearing of the defect contri- bute significantly to the total interaction energy. This contrasts with the calculations of Mitchell and Heuer [4] who find this term to be unimportant.Using the expression for the increase in yield stress appropriate to the Friedel statistics approximation and the data of Ahlquist [9] extrapolated to low temperature [4], we obtain a value of the maximum force, f, = 2.02 x lo-' N. For this estimate we used a line-tension value calculated by Mitchell and Heuer [4] which agrees closely with our own atomistic determination [8]. Comparing the above experimental, Friedel-model, f,-value with the calculated ones given in figures 5-7, we see that the atomistic values are lower by an order of magnitude
( f , = 1.2-1.8 x 10-lo N for the I2 potential)
C6-138 M. P. PULS
to a factor of three between the two potentials also points out the impor- (fm = 4.8-5.4 x 10-lo N for the I1 potential)
.
The closest agreement to the experimental value is obtained with the I1 potential, but it generates results which, according to Gourdin and Kingery 131, appear to be questionable. More defect locations near the dislocation need to be evaluated before we can say whether the above disagreement is significant or not.
The difference of about a factor of three in fm-values
tance of obtaining reliable impurity potentials. Work is in progress to study the interaction of the Ca2+- cation vacancy dimer with the a12 [I101 edge dislo- cation in NaCl. For this impurity we have available a more rigorous potential of the Ca2+ - ~ 1 - interaction based on a calculation of the Hartree-Fock type [lo].
This work will be reported in a forthcoming publi- cation.
DISCUSSION Question. - T. E. MITCHELL.
Your calculations present an obviously vast irnpro- vement over traditional elasticity calculations (such as my own). Even so, how sensitive are the results to the difficult Fe3+ interionic potential ? And why are you an order of magnitude different from experiment ? Can you split up the interaction energy into an elastic component and an electrostatic component ?
order of magnitude estimate of the force, but to do better we would probably need a more rigorous potential based on a quantum mechanical model.
2) We do not have enough results as yet on the interaction energy (for instance locating the trimer's centres on other planes) to make it useful to speculate on the presently found discrepancy between experi- ment and calculation.
Reply. - M. P. PULS. 3) I think that would be difficult, but the limiting 1) There is, for the best case, about a factor of values for the perfect lattice for a sheared trimer four difference in the maximum force values using suggest that the electrostatic component might be the I1 Fe3+ potential. This suggests a better than important.
References
[I] K o c ~ s , U. D . , ARGON, A. S. and ASHBY, M. F., Prog. Mat.
Sci 19 (1975).
[2] REPPICH, B., Mat. Sci. Eng. 22 (1976) 71.
[3] GOURDIN, W. H . and KINGERY, W. D., J. Mat. Sci. 14 (1979) 2053.
[4] MITCHELL, T. E. and HEWER, A. H., Mat. Sci. Eng. 28 (1977) 81.
[5] CATLOW, C. R. A., FAUX, I. D. and NORGETT, M. J., J. Phys.
C 9 (1976) 419.
[6] CATLOW, C. R. A. and FENDER, B. E. F., J . Phys. C 8 (1975) 3267.
[7] DICKENS, P. G., HECKINGBOTTOM, R. and LINNETT, J. W., Trans. Farad. Soc. 64 (1968) 1489.
[8] PULS, M. P., WOO, C. H. and NORGETT, M. J., Phil. Mag.
36 (1977) 1457.
[9] AHLQUIST, C. N., J. Appl. Phys. 46 (1975) 14.
[lo] KIM, Y. S. and GORDON, R. G., J. Chem. Phys. 60 (1974) 4332.
