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Deformation mechanisms for high-temperature creep of high yttria content stabilized zirconia single crystals
Acfamater.Vol.44, No. 3, pp. 991-999,1996 ElsevierScienceLtd
Pergamon
Copyright 0 1996Acta Metallurgica Inc. Printed in Great Britain.All rightsreserved
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DEFORMATION MECHANISMS FOR HIGH-TEMPERATURE CREEP OF HIGH YTTRIA CONTENT STABILIZED ZIRCONIA SINGLE CRYSTALS D. GOMEZ-GARCfA’, J. MARTfNEZ-FERNANDEZ’, A. DOMfNGUEZ-RODRfGUEZ’, P. EVENO’ and J. CASTAING*t ‘Dpto de Fisica de la Materia Condensada, Instituto de Ciencia de Materiales de Sevilla, Universidad de Sevilla, C.S.I.C., Apdo. 1065, 41080 Sevilla, Spain and ZLaboratoire de Physique des Mattriaux, (C.N.R.S.) Bellevue, 92195 Meudon Cedex, France (Received 21 October 1994; in revised form 22 May 1995)
Abstract-Creep of 21 mol.% yttria-stabilized zirconia single crystals has been studied between 1400and 1800°C. The creep parameters have been determined indicating a change of the controlling mechanism around 1500°C.At higher temperatures recovery creep is found to be the rate controlling mechanism, with a stress exponent 13 and an activation energy g 6 eV. Transition to glide controlled creep occurs below 15OO”C,associated with larger stress exponents (~5) and activation energies (~8.5 eV). TEM observations of the dislocation microstructure confirm this transition. The influence of the hiah vttria content, which is at the origin of the high creep resistance of these crystals, is discussed fir dach range of temperatures.
orientation [5-71. Several orientations and temperatures have been used with the 9.4 mol.% composition [8,9]. Creep of this composition has also been studied between 1300 and 1550°C and two regimes have been found depending on the temperature. At lower temperature, cross slip occurs and at high temperature climb becomes the rate controlling mechanism [lo]. In this work, we report the creep resistance of 21 mol.% Y-FSZ single crystals, including study of the dislocation substructures. Previous creep work on 9.4mol.O~ is compared with the present data in order to obtain a consistent picture of creep in this system [lo].
1. INTRODUCTION Since the discovery of the transformation toughening in ZrO, [l] a considerable effort has been performed to understand the microstructure and mechanical properties of ZrO,-based ceramics. These efforts have mainly been concerned with the system MgO-ZrO,, which shows excellent properties of strength and toughness at low temperatures because of the stressinduced martensitic transformation of the tetragonal to the monoclinic phase. ZrO-Y,O, is probably the most promising system in zirconia-based ceramics for high temperature applications, not only as monolithic material but as ceramic fibre reinforcement for composites, owing to the good stability when submitted to a strongly oxidizing environment and to the good creep resistance. Significant research has been performed over the years on the high temperature deformation of ZrO,-Y,O, single crystals as a function of the composition. Y,O, partially-stabilized zirconia single crystals (Y-PSZ) exhibit a potent high temperature precipitation hardening with flow strength depending on several factors and that can lead to a flow stress over 700 MPa at 1400°C [2-4]. Yz03 fully stabilized ZrOz (Y-FSZ) deformed at constant strain rate shows a potent solid solution strengthening with an increase in flow stress (from 150 to 360 MPa at 1600°C) when the solute content increases from 9.4 to 21 mol.% for the easy glide
2. EXPERIMENTAL
PROCEDURE
2.1. Sample preparation Single crystals of 21-mol.%-Y,O,-stabilized cZrO,, grown by skull melting, (Ceres Corporation, Massachussets) were used. Crystals were oriented using the Laue back reflexion technique and cut into parallelepipeds of dimensions 2 x 2 x 4.5 mm with a diamond saw. The faces were mechanically polished with 9.6 and 3 pm diamond pastes. The longer axis, was taken parallel to the [I121 direction, and the lateral faces were cut parallel to (711) and (110) planes. This orientation activates the single slip system (001) [ITO] with a Schmid factor of 0.47 [l 11. 2.2. Creep tests
tAlso affiliated to: Universitt EVE, Blvd Coquibus, 91025 Evry Cedex, France.
Creep tests were carried out in compression under constant load, using a prototype machine described
991
G6MEZ-GARCfA et al.: CREEP DEFORMATION MECHANISMS in the literature [12]. Tests were performed between 1400 and 18OO”C,under nominal stresses between 30 and 240 MPa, and the corresponding strain rates were in the interval 2 x lo-‘-3 x 10-6s-L. Silicon carbide pads were interposed to prevent the Sic rams from indentation; indentation on the pads was not detected in any test. Tests were done in argon in order to protect the rams and pads from oxidation at high temperatures. Samples turned dark in these conditions because of the colour centres created by zirconium reduction [ 131. Some tests at 1400°C were performed in these conditions (PO, = IO-‘atm) and in air, using alumina rams and Y-PSZ pads, to check a dependence of the creep rate with partial oxygen pressure. The specimens were loaded incrementally to avoid fractures and let dislocations multiply to reach the steady state deformation. A quantitative estimation of creep transients is therefore difficult. The data recorded during creep were the instantaneous length 1 vs time at constant load and temperature, and they are plotted as l vs 6, with 6 = ln(Z/l,), where 1, is the initial length. One of the tests plotted in this way is shown in Fig. 1. Data were fitted to the general constitutive equation for creep, where we assumed that diffusion is the controlling mechanism [14]:
In this expression, A is a dimensionless constant, D, is the preexponential factor for a suitable diffusion coefficient, Q is the activation energy for this process (in practice, AD, is the constant measured from the tests), p is the shear modulus, b is the Burgers vector for cubic zirconia (b = 3.63 A), and n is the stress exponent.
1.OOEb
and by means of changes of load at constant temperature for the case of the stress exponent: In z n =-. 0
(3)
In 3 0 01 The activation energy was also determined from the plot In i vs l/T, where the strain rates were normalized to 100 MPa using the values of n calculated above. Values obtained for Q are in agreement with those from instantaneous changes of temperature (Fig. 2). 2.3. Microstructural observations TEM foils were prepared to characterize the dislocation microstructure. Slices were cut from deformed samples at high temperatures (T > 1SOOC) and from samples deformed at low temperatures (T G 15OOC). For each sample, several slices were obtained: one of them was parallel to the primary slip plane (OOl), and the others were cut parallel to the (110) plane which is perpendicular to the slip one, as well as the cross-slip (111) plane. The slices were thinned to electron transparency by mechanical polishing, and ion milling. Once thinned, the foils were carbon coated to avoid charging during observation. TEM observations were performed by conventional tech21 mole% YFSZ
1 Q= 1 n=
Q and n depend on the details of the deformation mechanism and conclusions about them are obtained from the whole analysis of the experimental data. These parameters were calculated by means of changes of the temperature keeping the load constant during the tests, in the case of the activation energy:
5.3
8.7
8.1
4.9 2.8
2.8
5i
z r
l.OOE-6
d
l.OOE-7
39.1 MPa 61.3MPa 165OoC 1600% 16OOoC
0.00
0.04
51.3 MPa
63.6 MPa 1660%
0.12
0.08
160~~
0.16
0.20
STRAIN
Fig. 1. Typical creep curve of 21 mol.% Y,O, stabilized ZrO, plotted as strain rate (6) vs strain (6).
GOMEZ-GARCIA
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MECHANISMS
993
l.OOE-4
l.OOE-5 z 5 e
1mE-6
s I l.OOE-7 3 Iu) l.OOE-8
21 MOLE% l.OOE-8
104K
I 6.40
I 6.00
I 5.80
I 5.20
4.80
9.4 tiOLE% (MARTINEZ ET AL.)
(K-l)
Fig. 2. Steadv-state creeo rate. normalized to 100 MPa. dotted versus the recinrocal of the temuerature. ?he solid kne represents tde results obtained by M&nez et al. in the cask of 9.4 mol.%-Y-FSZ.
H-800 (Electron Microscopy Service, University of Seville, Spain) electron microscope operating at 200 kV. The three-dimensional nature of the dislocation arrangement was studied using stereo pairs obtained with g = (220). All the samples studied had been cooled down at a rate of 80O’C/h under reduced load (2-5 MPa).
niques using a Hitachi
3. RESULTS 3.1. Mechanical behaviour
A typical creep test is shown in Fig. 1, where T and cr are changed incrementally to determine Q and n respectively. The values of n and Q obtained are summarized in Table 1. It can be seen that there are two different regions, for temperatures lower than approx. 15OO”C, n and Q reach values of 2 5 and g 8.5 eV respectively. The values of these parameters decrease to g 3 and s 6 eV for higher temperatures. For all the temperature range the crystal showed high creep resistance, with a strain rate up to 10 times lower than in the 9.4mol.% for the same test conditions (T and u). These results are confirmed when normalization of i to 100 MPa was done (Fig. 2), using the n values of Table 1. Q was calculated using the least square method. From this plot, a change of the deformation mechanism is encountered around 1500°C. For lower temperatures, high values of the
activation energy are obtained (Q = 8.9 + 0.6 eV), whereas these values are next to 6eV (Q = 5.6 f 0.4eV) at higher temperature. In fact if we apply the least square method to plot all the data to a single straight line, a poor correlation factor r is found (Q = 6.8 eV, r = 0.508), when r was higher than 0.98 for the previous analysis. Up to this point, it must be said that the least square method applied is in fact an oversimplification, as the slope of the plot does not change sharply, but smoothly around 1500°C. In any case, the least square method improvement found when two different slopes are considered indicates that the fit using two straight lines is a much better approximation than the fit to only one, and it is a proof of the existence of the transition commented above. No dependence with the oxygen partial pressure was found on the tests performed at 14OO”C,in good agreement with previous results from Martinez ef al. in 9.4 mol.% Y-FSZ [lo, 151. 3.2. Dislocation substructure Dislocation substructures in foils of different orientations were examined in all the conditions studied. However we will restrict ourselves to analyze the microstructure for samples deformed at 1400 and
Table 1. Creep-law parameters determined by incremental changes during deformation. In parenthesis, we have written the number of determinations considered to calculate the mean value of the parameters shown. We have observed no difference in n and Q for positive or negative steps of Q or T Interval
of temperature
140~1500°C 1500-1800”c
(“C)
Stress exponent 5.4 * 0.4 (5) 2.9 f 0.2 (13)
(n)
Activation
energy (Q) in eV
6 _+ 1 (6) 5.8 5 0.5 (13)
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G6MEZ-GARCfA et al.:
CREEP DEFORMATION MECHANISMS
Fig. 3. TEM micrograph of the primary slip plane (OOl),showing the microstructure at 1400°C.
16Oo”C, corresponding to the two regions observed (Fig. 2). At 14OO”C, a high dislocation density was found. Dislocations react with each other, forming nodes and they are frequently pinned (Fig. 3). Extinction studies indicate that most of the dislocations have l/2(110) type Burgers vector and lie on the primary (001) slip plane, as discussed in [ 161. Stereopair analysis on the (001) zone axis using g = (220) showed that some cross slip is present. Analysis of the (111) cross-slip plane is shown in
Fig. 4. TEM micrograph
showing
Fig. 4. It can be observed that many dislocations are lying on this plane, showing that the cross-slip of the primary dislocations is operating as a mechanism to decrease the internal stress. The same result has been reported in samples of 9.4 mol.% Y-FSZ single crystals deformed at 1400°C
[161. In order to confirm these results, TEM has been made on the (1 IO) plane perpendicular to the primary (001) and parallel to the Burgers vector. By using the same g-conditions, we have found that only very
the dislocation microstructure in the case of a slice parallel cross-slip plane (11 l), at 1400°C.
to the
G6MEZ-GARCIA et al.:
CREEP DEFORMATION MECHANISMS
995
Fig. 5. TEM micrograph of the dislocation structure in the case of a slice cut parallel to the plane (110) normal to the primary slip plane. Notice that the dislocation segments are very short compared with those seen in the previous Figs 3 and 4.
short dislocation segments are lying on this plane (Fig. 5). This fact indicates that the dislocation structure lies essentially either on the primary slip plane or the secondary cross-slip one. Two planes (001) and (110) have been observed by TEM in the case of samples deformed at 1600°C. The picture is rather different: (i) The dislocation density is lower, as is expected from the annealing effect during the cooling
and the low stresses used while deforming the samples. (ii) In both cases, very long dislocations with Burgers vector type l/2(110) were observed. Figures 6 and 7 correspond to the dislocation in (001) and (110) planes respectively. Stereopair analysis for Figs 6 and 7 shows that the dislocations are lying on the respective planes. The observation of Fig. 7 shows clearly that
Fig. 6. TEM micrograph of the dislocation structure of the primary slip plane (001) in the case of one sample deformed at 1600°C.
996
G~MEZ-GAIK~A
et ~1.: cm3 DEFORMATION MECHANISMS
Fig. 7. TEM micrograph of the microstructure found in the (110) plane (the climb plane). Notice that the dislocation segments are quite long. This result is clearly in contrast with the result seen in the same material in the case of deformation at 1400°C (Fig. 5). The dark spots are due to radiation damage.
climb is an active mechanism temperatures.
in this range
of
the slowest species, which control the deformation (i.e. cationic diffusion for zirconia). a is the ratio of the mobilities for glide and climb, respectively
4. DISCUSSION (5)
4.1. Creep model
From the results previously shown, it is clear that a deformation mechanism change happens around 1500°C in c-ZrOl with high yttria concentration. In order to explain this, we shall do a detailed analysis of the mechanical parameters, as well as of the microstructural ones determined by TEM observations. We shall use a model designed for the deformation of alloys by Burton [17-201. According to this model, the steady state strain rate is the result of a dynamic equilibrium between the rates of dislocation creation and annihilation. The rate of dislocation creation is controlled by the mean dislocation glide velocity from the sources. The rate of annihilation is controlled by a diffusion controlled process, in which dislocation climb is taking place. By writing these two rates as a function of the glide and climb dislocation mobilities respectively, and making use of the Orowan equation, which is the conventional expression joining up microscopic and macroscopic descriptions of plastic deformation conducted by dislocation dynamics, the creep equation rises:
Values of a < 1 are found if dislocation glide is controlling the deformation, and values higher than one mean that dislocation glide becomes easier than climb, which is the limiting step. Consequently, this parameter helps us to analyse the experimental data in relation to the dominant deformation mechanism. bi is a characteristic parameter known as the internal stress. This is the average stress exerted on one dislocation by the others. Following closely Burton’s conclusions, the internal stress (ai) is related to the applied stress by means of the equation: a
( 1
ui = \l-cr
(6)
Joining (4) and (6), the steady state equation for creep is given by: i =$(&)‘(;Lr.
(7)
where the diffusion coefficient for creep in zirconia can be written as [21]: Ddia=
where all the previously used symbols have their usual meaning and D is the diffusion coefficient for
I O.
Da. DY xzr4 + +&r
(8)
In order to build up a mathematical expression for the diffusion coefficient previously shown, we have
G6MEZ-GARCIA
et al.:
CREEP DEFORMATION
997
MECHANISMS
made use of the experimental results given by Solmon et al. for the diffusion of yttrium and zirconium in FSZ for different concentrations [22]. Using our experimental results and D,, we can calculate the dimensionless parameter A of the creep law equation (1): A=-
ikT
p3 Dcmpb 0 0
(9)
The empirical values of A can be related directly to the ratio of glide and climb mobilities (a) by means of the Burton relationship previously shown [equations (7) and (l)]:
(10) In Fig. 8 the values of tl vs Tare plotted, using the diffusion coefficient for 21 mol.%-Y-FSZ from Solmon et al. [22]. The values of c( spread from well above 1 to less than 1 from high to low temperatures. We have to admit that no final interpretation of the absolute values of c1 can be done, one of the difficulties being the scatter in these values. This happens because of the restrictive assumptions made in such general models and the scattering of the mechanical data measured. So, we shall make use of these data in relative terms, to compare different physical situations, in conjunction with the conclusions from the experimental results. 4.2. Mechanical behaviour for T < 1500°C The dislocation microstructure in this temperature range (Figs 3 and 4) indicates that the dislocation glide is slower than climb, so glide is the mechanism controlling creep. In this range of temperatures, the activation energy (8.5 eV) is higher than the one measured in the case of cationic diffusion (5-6 eV), and high stress exponents are reached (6). Unfortunately, no theoretical model can explain accurately such higher values for the activation energy and stress
.
0.10
I
4.00
5.20
5.60
6.00
lOOOO/T(l/K)
Fig. 8. LX,ratio of mobilities vs l/Tfor the 21 mol.%-YFSZ.
4.00
5.20
5.60
6.00
104/T(1/K)
Fig. 9. Effective activation energy, vs l/T (x 104) for 21 mol.%-YFSZ. In order to plot this function, couples of values of a associated to temperatures which differed in 50°C were considered, as well as the equation (11).
exponent although they are common in oxide ceramics at intermediate temperatures [23]. Up to this point, we can add a possible explanation for this fact which is in good agreement with recent Burton’s considerations [20] and it is physically reasonable. Admitting that dislocation glide is the mechanism controlling the deformation at intermediate temperatures (T < 15OO”C),we will have to make use of the general Burton’s equation, in which the variation of the internal stress versus the temperature can be considerable. After some simple algebra, and remembering how the activation energy is calculated, equation (2) the relation between the diffusional activation energy and the measured energy would be:
where the indices 1 and 2 refer to the lower and the higher temperatures considered in order to calculate the activation energy, respectively. From the fitted plot of the dependence of c1vs T -’ (Fig. 8), it can be seen that CIis equal or lower than 1 along the low temperature region. An accurate mathematical analysis of equation (11) shows that the second term in the second member cannot be neglected at all when compared to the first one at this stage. Therefore, the effective activation energy measured in a creep test is higher than the diffusional one when a < 1. For values higher than one, the asymptotical value for Q is the diffusional activation energy because the second term cited above vanishes, A plot of the “effective activation energy” as a function of the temperature has been carried out and is shown in Fig. 9. The agreement with the experimental data in Table 1 is good enough and acts as a proof of the self-consistency of the model proposed. (Values as high as 9.5 eV may be reached when Q is
998
G6MEZ-GARCfA
et al.:
CREEP DEFORMATION
measured.) In general terms, the dependence of the effective activation energy with temperature can be expressed by-means of the following law:
au
3
+cl(lfa)d(llkT):(
(14
=
3+ 3
q:ii:
1:il)
In l+$ (
(13)
)
where (Tand a + Aa are the two stresses considered when a stress jump is made, and the indexes of a’ are correlated with these two stresses, respectively. By physical considerations, a’ is an increasing function of c ; SO %Tative is higher than 3, in agreement with the experimental values. Here again, the dependence of the effective stress exponent with the stress and temperature can be written in a general way, as:
n effective =
8 lni
0
alna
T
3 =3+----x(1+x)
4.3. Mechanical behaviour for T 3 1500°C An accurate analysis of the plot a vs l/T (Fig. 8) shows that ratio of mobilities increases with the temperature, reaching values higher than 1 for T > 1500°C. According to the equation (5), this fact means that the glide mobility is higher than the climb one in this range of temperatures. In this situation, we can approach
)
Now it can be seen that the slope of the plot In < - l/T must change smoothly from very high values (9 eV) to the energy for cationic diffusion value, when the additional term in the second member of equation (12) vanishes. By reasoning in the same way, the apparent exponent stress calculated when equation (3) is used, results to be:
%ffective
MECHANISMS
aff
(14) ( alna 1 T’
We should notice that the apparent stress exponent depends on both the stress and temperature, and it tends to 3 at high temperature, when the dislocation glide is easy and a has a weak dependence with stress. At low temperature, the glide mobility is usually a non-linear function of stress, and no term in equation (14) can be neglected. Unfortunately, no explicit dependence of a as a function of the stress can be plotted from our data. This happens because an explicit dependence of n as a function of both the stress and the temperature would be required. Consequently, dislocation glide controlled by solute drag can explain this mechanical behaviour. Dislocations have to drag the mobile defects made of segregation of yttrium associated with point defects. The stress exponent, around 5, and the activation energy, higher than the diffusional one, confirm this assumption. As for the a values, it can be seen that this parameter is included in the range of values permitted for the Burton’s theory for drag controlled creep (Fig. 5).
to 1, and the general creep equation (7), will be:
(15) Consequently, the stress exponent expected is 3, and that is really what happens, as was shown in Table 1. On the other hand, the expected activation energy should be close to the activation energy for cationic diffusion, as the main temperature dependence in the creep equation comes from the diffusion coefficient. The experimental values are in good agreement with this prediction, when compared with those given by Solmon et al. [22]. Their values are close to 5 eV for cationic diffusion, whatever the yttrium or zirconium one. As a consequence, we can conclude that a conventional recovery creep is the deformation mechanism at high temperature. TEM observations are in good agreement with these points. Stereo pairs made at 1400 and 1600°C show that dislocations are lying on (001) and (ill), and on (001) and (110) respectively, indicating that climb is activated at the higher temperature region (T > lSOO’C), whereas dislocation glide is the mechanism for the lower one (T < 1500°C). This hypothesis is in full agreement with the considerations made by Martinez et al. [lo] for the case of the high temperature plastic deformation of 9.4 mol.%-YFSZ. In Fig. 2, the solid squares plotted show their results compared with ours. It is clear that the same behaviour is found at high temperature, and also a similar transition in mechanical properties was reported. Finally, we must emphasize that the mechanical strength enhances with the yttrium content along the whole range of temperatures. For the same load and temperature, the strain rate decreases up to one order of magnitude when the yttria content increases from 9.4 to 21 mol.%. This higher strength can be explained in terms of a yttria dependence of the diffusion coefficient. The only results in the literature concerning the dependence of the cation diffusion coefficient as a function of the yttria content [22] agree with this explanation. 5. CONCLUSIONS
Creep of 21 mol.%-Y,O, stabilized c-ZrO, single crystals has been studied between 1400 and 1800°C.
GGMEZ-GARCIA
et al.:
CREEP DEFORMATION
A deformation mechanism change has been found, and a transition temperature around 1500°C has been established. For the lower temperatures, solute dragcontrolled creep occurs, whereas at the higher ones dislocation climb is controlling. Theoretical considerations by Burton have been applied, explaining the phenomena described. These conclusions are compared with those given for 9.4mol.% Y-FSZ which shows a similar behaviour, although the material is weaker than the 21 mol.% Y-FSZ solid solution. Acknowledgements-We
would like to thank very gratefully the technical support given by Bernard Pellissier (L.P.M.C.N.R.S.) as well as the financial one given by the Spanish authorities by means of both the research project C.I.C.Y.T. (MAT94-0120-C03) and la Action Integrada HispanoFrancesa no. H.F.35B.
REFERENCES D. J. Green, R. H. J. Hannick and M. V. Swain, Transformation Toughening of Ceramics. CRC Press, FL (1989). A. H. Heuer, V. Lanteri and A. Dominguez-Rodriguez, Acta metall. 37, 559 (1989).
J. Martinez-Femandez, M. Jimenez-Melendo, A. Dominguez-Rodriguez, K. P. D. Lagerliif and A. H. Heuer, Actn metafl. muter. 41, 3171 (1993). J. Martinez-Femandez, M. Jimenez-Melendo, A. Dominguez-Rodriguez, P. Cordier, K. P. D. Lagerliif and A. H. Heuer, Acta metall. mater. 43, 2469 (1995).
MECHANISMS
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A. Dominguez-Rodriguez, K. P. D. Lagerlof and A. H. Heuer, J. Am. Cerum. Sot. 69, 281 (1986). A. Dominguez-Rodriguez, J. Castaing and A. H. Heuer, Rad. E’cts Defects in Solids 119-121, 759 (1991). K. J. McClellan, H. Sayir, A. H. Heuer, A. Sayir, J. S. Haggerty and J. Sigalovsky, Cerum. Engng Sci. Proc. 14, 651 (1993). 8. A. Dominguez-Rodriguez and A. H. Heuer, Cryst. Latt. Amorphous Mater. 16, 117 (1987).
9. D. S. Cheong, Ph.D. dissertation. Case Western Reserve Univ., Cleveland, OH (1989). 10. J. Martinez Femandez, M. Jimenez Melendo, A. Dominguez Rodriguez and A. H. Heuer, J. Am. Ceram. Sot. 73, 2452 (1990). 11. A. Dominguez Rodriguez, D. S. Cheong and A. H. Heuer, Phil. Mag. A 64, 923 (1991). 12. H. Gervais, B. Pellicier and J. Castaing, Rev. Znt. Huutes Temp. Refract. 15, 43 (1978). 13. G. M. Ingo, J. Am. Cerum. Sot. 74, 381 (1991). 14. J. P. Poirier. Creep in Crystals. Cambridge Univ. Press (1985). 1.5. A. Bravo-Leon, M. Jimenez-Melendo, A. DominguezRodriguez and A. H. Chokshi, J. Muter. Sci. Left. 13, 1169 (1994).
16. D. S. Cheong, A. Dominguez-Rodriguez and A. H. Heuer, Phil. Mag. A 60, 123 (1989). 17. B. Burton, Phil. Mag. A 45, 657 (1982). 18. B. Burton, Phil. Mag. A 46, 607 (1982). 19. B. Burton, Phil. Mag. Lett. 62, 383 (1990). 20. B. Burton, Mater. Sci. Technol. 5, 1005 (1989). 21. J. Philibert, La Diffusion dans les Solides. Les editions de physique, Paris (1990). 22. H. Solmon, Ph.D. dissertation. Universitt de Paris VI (1992). 23. T. Bretheau, J. Castaing, J. Rabier and P. Veyssiere, Adv. Phys. 28, 835 (1979).
Pergamon
Copyright 0 1996Acta Metallurgica Inc. Printed in Great Britain.All rightsreserved
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1359~6454/96 $15.00+ 0.00
DEFORMATION MECHANISMS FOR HIGH-TEMPERATURE CREEP OF HIGH YTTRIA CONTENT STABILIZED ZIRCONIA SINGLE CRYSTALS D. GOMEZ-GARCfA’, J. MARTfNEZ-FERNANDEZ’, A. DOMfNGUEZ-RODRfGUEZ’, P. EVENO’ and J. CASTAING*t ‘Dpto de Fisica de la Materia Condensada, Instituto de Ciencia de Materiales de Sevilla, Universidad de Sevilla, C.S.I.C., Apdo. 1065, 41080 Sevilla, Spain and ZLaboratoire de Physique des Mattriaux, (C.N.R.S.) Bellevue, 92195 Meudon Cedex, France (Received 21 October 1994; in revised form 22 May 1995)
Abstract-Creep of 21 mol.% yttria-stabilized zirconia single crystals has been studied between 1400and 1800°C. The creep parameters have been determined indicating a change of the controlling mechanism around 1500°C.At higher temperatures recovery creep is found to be the rate controlling mechanism, with a stress exponent 13 and an activation energy g 6 eV. Transition to glide controlled creep occurs below 15OO”C,associated with larger stress exponents (~5) and activation energies (~8.5 eV). TEM observations of the dislocation microstructure confirm this transition. The influence of the hiah vttria content, which is at the origin of the high creep resistance of these crystals, is discussed fir dach range of temperatures.
orientation [5-71. Several orientations and temperatures have been used with the 9.4 mol.% composition [8,9]. Creep of this composition has also been studied between 1300 and 1550°C and two regimes have been found depending on the temperature. At lower temperature, cross slip occurs and at high temperature climb becomes the rate controlling mechanism [lo]. In this work, we report the creep resistance of 21 mol.% Y-FSZ single crystals, including study of the dislocation substructures. Previous creep work on 9.4mol.O~ is compared with the present data in order to obtain a consistent picture of creep in this system [lo].
1. INTRODUCTION Since the discovery of the transformation toughening in ZrO, [l] a considerable effort has been performed to understand the microstructure and mechanical properties of ZrO,-based ceramics. These efforts have mainly been concerned with the system MgO-ZrO,, which shows excellent properties of strength and toughness at low temperatures because of the stressinduced martensitic transformation of the tetragonal to the monoclinic phase. ZrO-Y,O, is probably the most promising system in zirconia-based ceramics for high temperature applications, not only as monolithic material but as ceramic fibre reinforcement for composites, owing to the good stability when submitted to a strongly oxidizing environment and to the good creep resistance. Significant research has been performed over the years on the high temperature deformation of ZrO,-Y,O, single crystals as a function of the composition. Y,O, partially-stabilized zirconia single crystals (Y-PSZ) exhibit a potent high temperature precipitation hardening with flow strength depending on several factors and that can lead to a flow stress over 700 MPa at 1400°C [2-4]. Yz03 fully stabilized ZrOz (Y-FSZ) deformed at constant strain rate shows a potent solid solution strengthening with an increase in flow stress (from 150 to 360 MPa at 1600°C) when the solute content increases from 9.4 to 21 mol.% for the easy glide
2. EXPERIMENTAL
PROCEDURE
2.1. Sample preparation Single crystals of 21-mol.%-Y,O,-stabilized cZrO,, grown by skull melting, (Ceres Corporation, Massachussets) were used. Crystals were oriented using the Laue back reflexion technique and cut into parallelepipeds of dimensions 2 x 2 x 4.5 mm with a diamond saw. The faces were mechanically polished with 9.6 and 3 pm diamond pastes. The longer axis, was taken parallel to the [I121 direction, and the lateral faces were cut parallel to (711) and (110) planes. This orientation activates the single slip system (001) [ITO] with a Schmid factor of 0.47 [l 11. 2.2. Creep tests
tAlso affiliated to: Universitt EVE, Blvd Coquibus, 91025 Evry Cedex, France.
Creep tests were carried out in compression under constant load, using a prototype machine described
991
G6MEZ-GARCfA et al.: CREEP DEFORMATION MECHANISMS in the literature [12]. Tests were performed between 1400 and 18OO”C,under nominal stresses between 30 and 240 MPa, and the corresponding strain rates were in the interval 2 x lo-‘-3 x 10-6s-L. Silicon carbide pads were interposed to prevent the Sic rams from indentation; indentation on the pads was not detected in any test. Tests were done in argon in order to protect the rams and pads from oxidation at high temperatures. Samples turned dark in these conditions because of the colour centres created by zirconium reduction [ 131. Some tests at 1400°C were performed in these conditions (PO, = IO-‘atm) and in air, using alumina rams and Y-PSZ pads, to check a dependence of the creep rate with partial oxygen pressure. The specimens were loaded incrementally to avoid fractures and let dislocations multiply to reach the steady state deformation. A quantitative estimation of creep transients is therefore difficult. The data recorded during creep were the instantaneous length 1 vs time at constant load and temperature, and they are plotted as l vs 6, with 6 = ln(Z/l,), where 1, is the initial length. One of the tests plotted in this way is shown in Fig. 1. Data were fitted to the general constitutive equation for creep, where we assumed that diffusion is the controlling mechanism [14]:
In this expression, A is a dimensionless constant, D, is the preexponential factor for a suitable diffusion coefficient, Q is the activation energy for this process (in practice, AD, is the constant measured from the tests), p is the shear modulus, b is the Burgers vector for cubic zirconia (b = 3.63 A), and n is the stress exponent.
1.OOEb
and by means of changes of load at constant temperature for the case of the stress exponent: In z n =-. 0
(3)
In 3 0 01 The activation energy was also determined from the plot In i vs l/T, where the strain rates were normalized to 100 MPa using the values of n calculated above. Values obtained for Q are in agreement with those from instantaneous changes of temperature (Fig. 2). 2.3. Microstructural observations TEM foils were prepared to characterize the dislocation microstructure. Slices were cut from deformed samples at high temperatures (T > 1SOOC) and from samples deformed at low temperatures (T G 15OOC). For each sample, several slices were obtained: one of them was parallel to the primary slip plane (OOl), and the others were cut parallel to the (110) plane which is perpendicular to the slip one, as well as the cross-slip (111) plane. The slices were thinned to electron transparency by mechanical polishing, and ion milling. Once thinned, the foils were carbon coated to avoid charging during observation. TEM observations were performed by conventional tech21 mole% YFSZ
1 Q= 1 n=
Q and n depend on the details of the deformation mechanism and conclusions about them are obtained from the whole analysis of the experimental data. These parameters were calculated by means of changes of the temperature keeping the load constant during the tests, in the case of the activation energy:
5.3
8.7
8.1
4.9 2.8
2.8
5i
z r
l.OOE-6
d
l.OOE-7
39.1 MPa 61.3MPa 165OoC 1600% 16OOoC
0.00
0.04
51.3 MPa
63.6 MPa 1660%
0.12
0.08
160~~
0.16
0.20
STRAIN
Fig. 1. Typical creep curve of 21 mol.% Y,O, stabilized ZrO, plotted as strain rate (6) vs strain (6).
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l.OOE-4
l.OOE-5 z 5 e
1mE-6
s I l.OOE-7 3 Iu) l.OOE-8
21 MOLE% l.OOE-8
104K
I 6.40
I 6.00
I 5.80
I 5.20
4.80
9.4 tiOLE% (MARTINEZ ET AL.)
(K-l)
Fig. 2. Steadv-state creeo rate. normalized to 100 MPa. dotted versus the recinrocal of the temuerature. ?he solid kne represents tde results obtained by M&nez et al. in the cask of 9.4 mol.%-Y-FSZ.
H-800 (Electron Microscopy Service, University of Seville, Spain) electron microscope operating at 200 kV. The three-dimensional nature of the dislocation arrangement was studied using stereo pairs obtained with g = (220). All the samples studied had been cooled down at a rate of 80O’C/h under reduced load (2-5 MPa).
niques using a Hitachi
3. RESULTS 3.1. Mechanical behaviour
A typical creep test is shown in Fig. 1, where T and cr are changed incrementally to determine Q and n respectively. The values of n and Q obtained are summarized in Table 1. It can be seen that there are two different regions, for temperatures lower than approx. 15OO”C, n and Q reach values of 2 5 and g 8.5 eV respectively. The values of these parameters decrease to g 3 and s 6 eV for higher temperatures. For all the temperature range the crystal showed high creep resistance, with a strain rate up to 10 times lower than in the 9.4mol.% for the same test conditions (T and u). These results are confirmed when normalization of i to 100 MPa was done (Fig. 2), using the n values of Table 1. Q was calculated using the least square method. From this plot, a change of the deformation mechanism is encountered around 1500°C. For lower temperatures, high values of the
activation energy are obtained (Q = 8.9 + 0.6 eV), whereas these values are next to 6eV (Q = 5.6 f 0.4eV) at higher temperature. In fact if we apply the least square method to plot all the data to a single straight line, a poor correlation factor r is found (Q = 6.8 eV, r = 0.508), when r was higher than 0.98 for the previous analysis. Up to this point, it must be said that the least square method applied is in fact an oversimplification, as the slope of the plot does not change sharply, but smoothly around 1500°C. In any case, the least square method improvement found when two different slopes are considered indicates that the fit using two straight lines is a much better approximation than the fit to only one, and it is a proof of the existence of the transition commented above. No dependence with the oxygen partial pressure was found on the tests performed at 14OO”C,in good agreement with previous results from Martinez ef al. in 9.4 mol.% Y-FSZ [lo, 151. 3.2. Dislocation substructure Dislocation substructures in foils of different orientations were examined in all the conditions studied. However we will restrict ourselves to analyze the microstructure for samples deformed at 1400 and
Table 1. Creep-law parameters determined by incremental changes during deformation. In parenthesis, we have written the number of determinations considered to calculate the mean value of the parameters shown. We have observed no difference in n and Q for positive or negative steps of Q or T Interval
of temperature
140~1500°C 1500-1800”c
(“C)
Stress exponent 5.4 * 0.4 (5) 2.9 f 0.2 (13)
(n)
Activation
energy (Q) in eV
6 _+ 1 (6) 5.8 5 0.5 (13)
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Fig. 3. TEM micrograph of the primary slip plane (OOl),showing the microstructure at 1400°C.
16Oo”C, corresponding to the two regions observed (Fig. 2). At 14OO”C, a high dislocation density was found. Dislocations react with each other, forming nodes and they are frequently pinned (Fig. 3). Extinction studies indicate that most of the dislocations have l/2(110) type Burgers vector and lie on the primary (001) slip plane, as discussed in [ 161. Stereopair analysis on the (001) zone axis using g = (220) showed that some cross slip is present. Analysis of the (111) cross-slip plane is shown in
Fig. 4. TEM micrograph
showing
Fig. 4. It can be observed that many dislocations are lying on this plane, showing that the cross-slip of the primary dislocations is operating as a mechanism to decrease the internal stress. The same result has been reported in samples of 9.4 mol.% Y-FSZ single crystals deformed at 1400°C
[161. In order to confirm these results, TEM has been made on the (1 IO) plane perpendicular to the primary (001) and parallel to the Burgers vector. By using the same g-conditions, we have found that only very
the dislocation microstructure in the case of a slice parallel cross-slip plane (11 l), at 1400°C.
to the
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995
Fig. 5. TEM micrograph of the dislocation structure in the case of a slice cut parallel to the plane (110) normal to the primary slip plane. Notice that the dislocation segments are very short compared with those seen in the previous Figs 3 and 4.
short dislocation segments are lying on this plane (Fig. 5). This fact indicates that the dislocation structure lies essentially either on the primary slip plane or the secondary cross-slip one. Two planes (001) and (110) have been observed by TEM in the case of samples deformed at 1600°C. The picture is rather different: (i) The dislocation density is lower, as is expected from the annealing effect during the cooling
and the low stresses used while deforming the samples. (ii) In both cases, very long dislocations with Burgers vector type l/2(110) were observed. Figures 6 and 7 correspond to the dislocation in (001) and (110) planes respectively. Stereopair analysis for Figs 6 and 7 shows that the dislocations are lying on the respective planes. The observation of Fig. 7 shows clearly that
Fig. 6. TEM micrograph of the dislocation structure of the primary slip plane (001) in the case of one sample deformed at 1600°C.
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et ~1.: cm3 DEFORMATION MECHANISMS
Fig. 7. TEM micrograph of the microstructure found in the (110) plane (the climb plane). Notice that the dislocation segments are quite long. This result is clearly in contrast with the result seen in the same material in the case of deformation at 1400°C (Fig. 5). The dark spots are due to radiation damage.
climb is an active mechanism temperatures.
in this range
of
the slowest species, which control the deformation (i.e. cationic diffusion for zirconia). a is the ratio of the mobilities for glide and climb, respectively
4. DISCUSSION (5)
4.1. Creep model
From the results previously shown, it is clear that a deformation mechanism change happens around 1500°C in c-ZrOl with high yttria concentration. In order to explain this, we shall do a detailed analysis of the mechanical parameters, as well as of the microstructural ones determined by TEM observations. We shall use a model designed for the deformation of alloys by Burton [17-201. According to this model, the steady state strain rate is the result of a dynamic equilibrium between the rates of dislocation creation and annihilation. The rate of dislocation creation is controlled by the mean dislocation glide velocity from the sources. The rate of annihilation is controlled by a diffusion controlled process, in which dislocation climb is taking place. By writing these two rates as a function of the glide and climb dislocation mobilities respectively, and making use of the Orowan equation, which is the conventional expression joining up microscopic and macroscopic descriptions of plastic deformation conducted by dislocation dynamics, the creep equation rises:
Values of a < 1 are found if dislocation glide is controlling the deformation, and values higher than one mean that dislocation glide becomes easier than climb, which is the limiting step. Consequently, this parameter helps us to analyse the experimental data in relation to the dominant deformation mechanism. bi is a characteristic parameter known as the internal stress. This is the average stress exerted on one dislocation by the others. Following closely Burton’s conclusions, the internal stress (ai) is related to the applied stress by means of the equation: a
( 1
ui = \l-cr
(6)
Joining (4) and (6), the steady state equation for creep is given by: i =$(&)‘(;Lr.
(7)
where the diffusion coefficient for creep in zirconia can be written as [21]: Ddia=
where all the previously used symbols have their usual meaning and D is the diffusion coefficient for
I O.
Da. DY xzr4 + +&r
(8)
In order to build up a mathematical expression for the diffusion coefficient previously shown, we have
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MECHANISMS
made use of the experimental results given by Solmon et al. for the diffusion of yttrium and zirconium in FSZ for different concentrations [22]. Using our experimental results and D,, we can calculate the dimensionless parameter A of the creep law equation (1): A=-
ikT
p3 Dcmpb 0 0
(9)
The empirical values of A can be related directly to the ratio of glide and climb mobilities (a) by means of the Burton relationship previously shown [equations (7) and (l)]:
(10) In Fig. 8 the values of tl vs Tare plotted, using the diffusion coefficient for 21 mol.%-Y-FSZ from Solmon et al. [22]. The values of c( spread from well above 1 to less than 1 from high to low temperatures. We have to admit that no final interpretation of the absolute values of c1 can be done, one of the difficulties being the scatter in these values. This happens because of the restrictive assumptions made in such general models and the scattering of the mechanical data measured. So, we shall make use of these data in relative terms, to compare different physical situations, in conjunction with the conclusions from the experimental results. 4.2. Mechanical behaviour for T < 1500°C The dislocation microstructure in this temperature range (Figs 3 and 4) indicates that the dislocation glide is slower than climb, so glide is the mechanism controlling creep. In this range of temperatures, the activation energy (8.5 eV) is higher than the one measured in the case of cationic diffusion (5-6 eV), and high stress exponents are reached (6). Unfortunately, no theoretical model can explain accurately such higher values for the activation energy and stress
.
0.10
I
4.00
5.20
5.60
6.00
lOOOO/T(l/K)
Fig. 8. LX,ratio of mobilities vs l/Tfor the 21 mol.%-YFSZ.
4.00
5.20
5.60
6.00
104/T(1/K)
Fig. 9. Effective activation energy, vs l/T (x 104) for 21 mol.%-YFSZ. In order to plot this function, couples of values of a associated to temperatures which differed in 50°C were considered, as well as the equation (11).
exponent although they are common in oxide ceramics at intermediate temperatures [23]. Up to this point, we can add a possible explanation for this fact which is in good agreement with recent Burton’s considerations [20] and it is physically reasonable. Admitting that dislocation glide is the mechanism controlling the deformation at intermediate temperatures (T < 15OO”C),we will have to make use of the general Burton’s equation, in which the variation of the internal stress versus the temperature can be considerable. After some simple algebra, and remembering how the activation energy is calculated, equation (2) the relation between the diffusional activation energy and the measured energy would be:
where the indices 1 and 2 refer to the lower and the higher temperatures considered in order to calculate the activation energy, respectively. From the fitted plot of the dependence of c1vs T -’ (Fig. 8), it can be seen that CIis equal or lower than 1 along the low temperature region. An accurate mathematical analysis of equation (11) shows that the second term in the second member cannot be neglected at all when compared to the first one at this stage. Therefore, the effective activation energy measured in a creep test is higher than the diffusional one when a < 1. For values higher than one, the asymptotical value for Q is the diffusional activation energy because the second term cited above vanishes, A plot of the “effective activation energy” as a function of the temperature has been carried out and is shown in Fig. 9. The agreement with the experimental data in Table 1 is good enough and acts as a proof of the self-consistency of the model proposed. (Values as high as 9.5 eV may be reached when Q is
998
G6MEZ-GARCfA
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measured.) In general terms, the dependence of the effective activation energy with temperature can be expressed by-means of the following law:
au
3
+cl(lfa)d(llkT):(
(14
=
3+ 3
q:ii:
1:il)
In l+$ (
(13)
)
where (Tand a + Aa are the two stresses considered when a stress jump is made, and the indexes of a’ are correlated with these two stresses, respectively. By physical considerations, a’ is an increasing function of c ; SO %Tative is higher than 3, in agreement with the experimental values. Here again, the dependence of the effective stress exponent with the stress and temperature can be written in a general way, as:
n effective =
8 lni
0
alna
T
3 =3+----x(1+x)
4.3. Mechanical behaviour for T 3 1500°C An accurate analysis of the plot a vs l/T (Fig. 8) shows that ratio of mobilities increases with the temperature, reaching values higher than 1 for T > 1500°C. According to the equation (5), this fact means that the glide mobility is higher than the climb one in this range of temperatures. In this situation, we can approach
)
Now it can be seen that the slope of the plot In < - l/T must change smoothly from very high values (9 eV) to the energy for cationic diffusion value, when the additional term in the second member of equation (12) vanishes. By reasoning in the same way, the apparent exponent stress calculated when equation (3) is used, results to be:
%ffective
MECHANISMS
aff
(14) ( alna 1 T’
We should notice that the apparent stress exponent depends on both the stress and temperature, and it tends to 3 at high temperature, when the dislocation glide is easy and a has a weak dependence with stress. At low temperature, the glide mobility is usually a non-linear function of stress, and no term in equation (14) can be neglected. Unfortunately, no explicit dependence of a as a function of the stress can be plotted from our data. This happens because an explicit dependence of n as a function of both the stress and the temperature would be required. Consequently, dislocation glide controlled by solute drag can explain this mechanical behaviour. Dislocations have to drag the mobile defects made of segregation of yttrium associated with point defects. The stress exponent, around 5, and the activation energy, higher than the diffusional one, confirm this assumption. As for the a values, it can be seen that this parameter is included in the range of values permitted for the Burton’s theory for drag controlled creep (Fig. 5).
to 1, and the general creep equation (7), will be:
(15) Consequently, the stress exponent expected is 3, and that is really what happens, as was shown in Table 1. On the other hand, the expected activation energy should be close to the activation energy for cationic diffusion, as the main temperature dependence in the creep equation comes from the diffusion coefficient. The experimental values are in good agreement with this prediction, when compared with those given by Solmon et al. [22]. Their values are close to 5 eV for cationic diffusion, whatever the yttrium or zirconium one. As a consequence, we can conclude that a conventional recovery creep is the deformation mechanism at high temperature. TEM observations are in good agreement with these points. Stereo pairs made at 1400 and 1600°C show that dislocations are lying on (001) and (ill), and on (001) and (110) respectively, indicating that climb is activated at the higher temperature region (T > lSOO’C), whereas dislocation glide is the mechanism for the lower one (T < 1500°C). This hypothesis is in full agreement with the considerations made by Martinez et al. [lo] for the case of the high temperature plastic deformation of 9.4 mol.%-YFSZ. In Fig. 2, the solid squares plotted show their results compared with ours. It is clear that the same behaviour is found at high temperature, and also a similar transition in mechanical properties was reported. Finally, we must emphasize that the mechanical strength enhances with the yttrium content along the whole range of temperatures. For the same load and temperature, the strain rate decreases up to one order of magnitude when the yttria content increases from 9.4 to 21 mol.%. This higher strength can be explained in terms of a yttria dependence of the diffusion coefficient. The only results in the literature concerning the dependence of the cation diffusion coefficient as a function of the yttria content [22] agree with this explanation. 5. CONCLUSIONS
Creep of 21 mol.%-Y,O, stabilized c-ZrO, single crystals has been studied between 1400 and 1800°C.
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A deformation mechanism change has been found, and a transition temperature around 1500°C has been established. For the lower temperatures, solute dragcontrolled creep occurs, whereas at the higher ones dislocation climb is controlling. Theoretical considerations by Burton have been applied, explaining the phenomena described. These conclusions are compared with those given for 9.4mol.% Y-FSZ which shows a similar behaviour, although the material is weaker than the 21 mol.% Y-FSZ solid solution. Acknowledgements-We
would like to thank very gratefully the technical support given by Bernard Pellissier (L.P.M.C.N.R.S.) as well as the financial one given by the Spanish authorities by means of both the research project C.I.C.Y.T. (MAT94-0120-C03) and la Action Integrada HispanoFrancesa no. H.F.35B.
REFERENCES D. J. Green, R. H. J. Hannick and M. V. Swain, Transformation Toughening of Ceramics. CRC Press, FL (1989). A. H. Heuer, V. Lanteri and A. Dominguez-Rodriguez, Acta metall. 37, 559 (1989).
J. Martinez-Femandez, M. Jimenez-Melendo, A. Dominguez-Rodriguez, K. P. D. Lagerliif and A. H. Heuer, Actn metafl. muter. 41, 3171 (1993). J. Martinez-Femandez, M. Jimenez-Melendo, A. Dominguez-Rodriguez, P. Cordier, K. P. D. Lagerliif and A. H. Heuer, Acta metall. mater. 43, 2469 (1995).
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A. Dominguez-Rodriguez, K. P. D. Lagerlof and A. H. Heuer, J. Am. Cerum. Sot. 69, 281 (1986). A. Dominguez-Rodriguez, J. Castaing and A. H. Heuer, Rad. E’cts Defects in Solids 119-121, 759 (1991). K. J. McClellan, H. Sayir, A. H. Heuer, A. Sayir, J. S. Haggerty and J. Sigalovsky, Cerum. Engng Sci. Proc. 14, 651 (1993). 8. A. Dominguez-Rodriguez and A. H. Heuer, Cryst. Latt. Amorphous Mater. 16, 117 (1987).
9. D. S. Cheong, Ph.D. dissertation. Case Western Reserve Univ., Cleveland, OH (1989). 10. J. Martinez Femandez, M. Jimenez Melendo, A. Dominguez Rodriguez and A. H. Heuer, J. Am. Ceram. Sot. 73, 2452 (1990). 11. A. Dominguez Rodriguez, D. S. Cheong and A. H. Heuer, Phil. Mag. A 64, 923 (1991). 12. H. Gervais, B. Pellicier and J. Castaing, Rev. Znt. Huutes Temp. Refract. 15, 43 (1978). 13. G. M. Ingo, J. Am. Cerum. Sot. 74, 381 (1991). 14. J. P. Poirier. Creep in Crystals. Cambridge Univ. Press (1985). 1.5. A. Bravo-Leon, M. Jimenez-Melendo, A. DominguezRodriguez and A. H. Chokshi, J. Muter. Sci. Left. 13, 1169 (1994).
16. D. S. Cheong, A. Dominguez-Rodriguez and A. H. Heuer, Phil. Mag. A 60, 123 (1989). 17. B. Burton, Phil. Mag. A 45, 657 (1982). 18. B. Burton, Phil. Mag. A 46, 607 (1982). 19. B. Burton, Phil. Mag. Lett. 62, 383 (1990). 20. B. Burton, Mater. Sci. Technol. 5, 1005 (1989). 21. J. Philibert, La Diffusion dans les Solides. Les editions de physique, Paris (1990). 22. H. Solmon, Ph.D. dissertation. Universitt de Paris VI (1992). 23. T. Bretheau, J. Castaing, J. Rabier and P. Veyssiere, Adv. Phys. 28, 835 (1979).