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Analysis of the local excess density of dislocations in Y2O3-stabilized ZrO2 by means of X-ray Berg-Barrett topography
Volume 7. number
4
MATERIALS
ANALYSIS OF THE LOCAL EXCESS DENSITY OF DISLOCATIONS IN Y,O,-STABILIZED ZrO, BY MEANS OF X-RAY BERG-BARRETT F.L. CUMBRERA,
October
LETTERS
I988
TOPOGRAPHY
A. DOMINGUEZ-RODRIGUEZ
Departamento de Fislca de la Materia Condensada, Facultad de Fisrca, Apostado 1065, Seville, Sparn
F. GUIBERTEAU Departamento de Fisica, Faculiad de Ciencias. 06071 Badajoz, Spain
and E. FRIES Lahoratorre de Physique des MatPriaux, C.N.R.S., Bellewe. France Received
IS July 1988
This Letter deals with the application of X-ray Berg-Barrett topography to the Y,O,-stabilized cubic zirconia. The long-scale dislocation arrangement is observed. Characteristic cusps appear in some topographs at the intersection of dislocation walls and dislocation layers. An easy way to calculate the local excess density of dislocations of one sign. associated with the cusps, is proposed. The obtained values are consistent with the results of TEM experiments.
1. Introduction and experimental X-ray Berg-Barrett topography allows one to study the long-range correlations in the dislocation arrangements of deformed single crystals. This experimental method is complementary to other techniques as transmission electron microscopy (TEM ) and etching of individual dislocations, leading to an understanding of the plastic deformation of materials. Recently, we have examined by this technique single crystals of YzO,-stabilized cubic zirconia plastically deformed at 1400°C [ 11. The experimental arrangement used for the X-ray Berg-Barrett topography consists of an X-ray source (Cu tube) at a distance of = 350 mm from the specimen mounted on a goniometer. The beam divergence in the incidence plane ranges between 2’ and 24’ and is controlled by a calibrated slit. The topographs must be analyzed in terms of different mechanisms of contrast formation, which have already been studied by Newkirk [ 2 ] and Wilkens [ 3 1. The characteristic patterns we 0167-517x/88/$ ( North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
observed were also described when deforming Cu [ 3,4] and NaCl [ 51 single crystals. The main features are dislocation layers parallel to a slip plane and dislocation walls perpendicular to the slip direction. These dislocation arrangements are imaged by extinction contrast [ 6 1. At the intersection of walls and layers, cusps can be observed: they arise from lattice rotations with the rotational axis parallel to the plane of incidence (displacement contrast ). In our case, the Berg-Barrett patterns are dominated by the traces of ( 1f0) polygon walls resulting from glide polygonization of dislocations on a primary (001) [ liO] slip system (fig. la). Dislocation walls associated with a secondary ( 111) [ 10 1 ] slip system are observed in weak contrast (fig. 1b). Figs. 2a and 2b sketch these relevant features. The cusps observed in fig. la arise from an excess of edge dislocations of one sign on each side of the intersection of ( 1i0) walls and dislocation layers parallel to the (00 1) primary slip plane. The cusps formed by interactions between the secondary ( ii 1) [ 10 1 ] slip system and the ( 10 1) walls, were never observed with B.V.
143
Volume 7, number
4
MATERIALS
LETTERS
October
1988
2. Results and interpretation In fig. 3a we represent a sketch of a single contrast line with several cusps; the horizontal coordinate x is parallel to the trace of the surface with the ( 1TO) plane. We know experimentally the vertical shift accounting for the cusps, y(x), which is related to the angle of lattice rotation, a(x), as 500 pm
Ill 21
where the geometrical factors df, 8 and y are, respectivley, the distance crystal-film, the Bragg angle and the angle between the rotation axis and the intersection of the reflecting and incidence planes. Any change Ay(x) due to a local density of excess dislocations ANo( can be accounted for the corresponding angular variation Aa( in this way dy(x)/dx=2df
sin Bcos rda(x)/dx.
In the case of pure tilt walls we can write Fig. I. Y,O,-stabilized cubic 210~ deformed to 0.03 along [ 1121 at 1400°C. Berg-Barrett topograph of a (li I ) plane, g=224, beam divergence 24’. (a) Coarse traces parallel to [ 1121 correspond to ( 1i0) walls. (b) Coarse traces parallel to [ i21] correspond to ( 10 1) walls.
contrast good enough to perform any analysis abom them. The aim of this Letter is to perform a quantitative analysis of the situation presented above.
me(x)
=B dy(x)ldx,
where B= (2bdf sin 0 cos y)-‘, b being the module of the Burgers vector (0.36 nm in our case). A sketch of m,(x) is given in fig. 3b. From an enlarged view of fig. la we have digitalized 12 cusps by means of a Kontron MOP-30 semiautomatic image analyser. The specimen surface is ( 11 i ) and the reflecting plane (224). In a previous work [ 1] the walls ( 1 i0) were shown to be pure tilt walls, formed of [ liO] edge dislocations, with a
[iio]
t
b) I
Fig. 2. Sketch of slip systems and polygonization from (iil). (a) (001) [liO],(b) (iii) [loll.
144
walls viewed
Fig. 3. (a) Representation of a single contrast excess density, Ah’,, of dislocations of one sign.
x
line. (b) Local
Volume 7. number 4
MATERIALS LETTERS
1
x(aloPm) -1s 0
2
I
0
0
10
Fig. 4. Calculated variation of M’,,(x) along a wall
[ 1 lo] rotation axis. As the reflecting plane is (224) and [ 1TO] being set vertical, the displacement axis is [ 1111. We have performed the numeric derivation of y( x) by means of an interpolation and smoothing routine based on cubic spline functions and part of the results is shown in fig. 4. We have determined a maximum value of the local density of excess edge dislocations of 1.5x 1OL3m-’ and an average value of 6 x 1012 m-’ for these maxima. On another way, we could calculate from the height of the cusps an average value for the maximum lattice rotation of 13’ ?I 3’ which corresponds to an excess of dislocations of one sign of about (1 kO.3) x lo-’ m-’ in the thickness of the wall. The arrangement of polygon walls observed in fig. la is much more regular than in the other materials cited above, which must result from dislocations in the walls being able to climb into their lowest energy positions, resulting in the regular arrangement just cited. However, the cusp structure is very irregular along a wall (see figs. la and 4). Comparing the thickness of the bend region with the distances between cusps (about 3 urn), the region with excess dislocations is found to be slightly larger than one half of the lattice volume. Besides the short distance
October 1988
between cusps, overlapping cusps give a wide region with high dislocation density. For the same material deformed under the same conditions, Dominguez et al. [ 7 ] found by TEM a density of dislocations of 2.5 x lOI rne2, while our TEM measurements [ 81 gave a density between 1 x lOI and 2x 10” m-‘, close to the present results. Wilkens reported values of about 40’ for the kink angle of polygonization walls in Cu single crystals [ 3,4]. This corresponds to a greater excess of dislocations of one sign than here, but a smaller local density of 1 x lOI m-’ was found as the bend lattice region extended over about 15 urn and the distance between cusps ranged between 30 and 100 nm. The polygon walls and cusps are generally [ 3-51 much more irregular than in our case. These differences can be due to the fact that these materials were deformed to low temperature whereas our crystals were deformed at ~0.67’,,,, where climb of dislocations occurred as was observed by TEM [ 7 1.
Acknowledgement One of us (FG) thanks the Hispano-Frances curio Scientific Program for support.
Mer-
References [ 11E. Fries, F. Guiberteau, .A. Dommguez-Rodriguez.
D.S. Cheong and A.H. Heuer. Phil. Mag. A, submitted for publication. [ 21 J.B. Newkirk, Trans. Metall. Sot. AIME 2 I5 ( 1959) 483. [3] M. Wilkens, Can. J. Phys. 45 (1967) 567. [4] B. Obst, H. Auer and M. Wilkens. Mater. Sci. Eng. 3 ( 1968/ 69) 41. [5] H. Strunk, Mater. Sci. Eng. 26 (1976) 231. [ 61 E. Fries, M. Spendel and J. Philibert. J. Appl. Cryst. 14 ( 1981 ) 285. [7] A. Dominguez-Rodriguez. K.P.D. Lagerlof and A.H. Heuer. J. Am. Ceram. Sot. 69 ( 1986) 28 1. [ 8] F.L. Cumbrera, A. Dominguez-Rodriguez. F. Guiberteau and E. Fries. unpublished results.
145
4
MATERIALS
ANALYSIS OF THE LOCAL EXCESS DENSITY OF DISLOCATIONS IN Y,O,-STABILIZED ZrO, BY MEANS OF X-RAY BERG-BARRETT F.L. CUMBRERA,
October
LETTERS
I988
TOPOGRAPHY
A. DOMINGUEZ-RODRIGUEZ
Departamento de Fislca de la Materia Condensada, Facultad de Fisrca, Apostado 1065, Seville, Sparn
F. GUIBERTEAU Departamento de Fisica, Faculiad de Ciencias. 06071 Badajoz, Spain
and E. FRIES Lahoratorre de Physique des MatPriaux, C.N.R.S., Bellewe. France Received
IS July 1988
This Letter deals with the application of X-ray Berg-Barrett topography to the Y,O,-stabilized cubic zirconia. The long-scale dislocation arrangement is observed. Characteristic cusps appear in some topographs at the intersection of dislocation walls and dislocation layers. An easy way to calculate the local excess density of dislocations of one sign. associated with the cusps, is proposed. The obtained values are consistent with the results of TEM experiments.
1. Introduction and experimental X-ray Berg-Barrett topography allows one to study the long-range correlations in the dislocation arrangements of deformed single crystals. This experimental method is complementary to other techniques as transmission electron microscopy (TEM ) and etching of individual dislocations, leading to an understanding of the plastic deformation of materials. Recently, we have examined by this technique single crystals of YzO,-stabilized cubic zirconia plastically deformed at 1400°C [ 11. The experimental arrangement used for the X-ray Berg-Barrett topography consists of an X-ray source (Cu tube) at a distance of = 350 mm from the specimen mounted on a goniometer. The beam divergence in the incidence plane ranges between 2’ and 24’ and is controlled by a calibrated slit. The topographs must be analyzed in terms of different mechanisms of contrast formation, which have already been studied by Newkirk [ 2 ] and Wilkens [ 3 1. The characteristic patterns we 0167-517x/88/$ ( North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
observed were also described when deforming Cu [ 3,4] and NaCl [ 51 single crystals. The main features are dislocation layers parallel to a slip plane and dislocation walls perpendicular to the slip direction. These dislocation arrangements are imaged by extinction contrast [ 6 1. At the intersection of walls and layers, cusps can be observed: they arise from lattice rotations with the rotational axis parallel to the plane of incidence (displacement contrast ). In our case, the Berg-Barrett patterns are dominated by the traces of ( 1f0) polygon walls resulting from glide polygonization of dislocations on a primary (001) [ liO] slip system (fig. la). Dislocation walls associated with a secondary ( 111) [ 10 1 ] slip system are observed in weak contrast (fig. 1b). Figs. 2a and 2b sketch these relevant features. The cusps observed in fig. la arise from an excess of edge dislocations of one sign on each side of the intersection of ( 1i0) walls and dislocation layers parallel to the (00 1) primary slip plane. The cusps formed by interactions between the secondary ( ii 1) [ 10 1 ] slip system and the ( 10 1) walls, were never observed with B.V.
143
Volume 7, number
4
MATERIALS
LETTERS
October
1988
2. Results and interpretation In fig. 3a we represent a sketch of a single contrast line with several cusps; the horizontal coordinate x is parallel to the trace of the surface with the ( 1TO) plane. We know experimentally the vertical shift accounting for the cusps, y(x), which is related to the angle of lattice rotation, a(x), as 500 pm
Ill 21
where the geometrical factors df, 8 and y are, respectivley, the distance crystal-film, the Bragg angle and the angle between the rotation axis and the intersection of the reflecting and incidence planes. Any change Ay(x) due to a local density of excess dislocations ANo( can be accounted for the corresponding angular variation Aa( in this way dy(x)/dx=2df
sin Bcos rda(x)/dx.
In the case of pure tilt walls we can write Fig. I. Y,O,-stabilized cubic 210~ deformed to 0.03 along [ 1121 at 1400°C. Berg-Barrett topograph of a (li I ) plane, g=224, beam divergence 24’. (a) Coarse traces parallel to [ 1121 correspond to ( 1i0) walls. (b) Coarse traces parallel to [ i21] correspond to ( 10 1) walls.
contrast good enough to perform any analysis abom them. The aim of this Letter is to perform a quantitative analysis of the situation presented above.
me(x)
=B dy(x)ldx,
where B= (2bdf sin 0 cos y)-‘, b being the module of the Burgers vector (0.36 nm in our case). A sketch of m,(x) is given in fig. 3b. From an enlarged view of fig. la we have digitalized 12 cusps by means of a Kontron MOP-30 semiautomatic image analyser. The specimen surface is ( 11 i ) and the reflecting plane (224). In a previous work [ 1] the walls ( 1 i0) were shown to be pure tilt walls, formed of [ liO] edge dislocations, with a
[iio]
t
b) I
Fig. 2. Sketch of slip systems and polygonization from (iil). (a) (001) [liO],(b) (iii) [loll.
144
walls viewed
Fig. 3. (a) Representation of a single contrast excess density, Ah’,, of dislocations of one sign.
x
line. (b) Local
Volume 7. number 4
MATERIALS LETTERS
1
x(aloPm) -1s 0
2
I
0
0
10
Fig. 4. Calculated variation of M’,,(x) along a wall
[ 1 lo] rotation axis. As the reflecting plane is (224) and [ 1TO] being set vertical, the displacement axis is [ 1111. We have performed the numeric derivation of y( x) by means of an interpolation and smoothing routine based on cubic spline functions and part of the results is shown in fig. 4. We have determined a maximum value of the local density of excess edge dislocations of 1.5x 1OL3m-’ and an average value of 6 x 1012 m-’ for these maxima. On another way, we could calculate from the height of the cusps an average value for the maximum lattice rotation of 13’ ?I 3’ which corresponds to an excess of dislocations of one sign of about (1 kO.3) x lo-’ m-’ in the thickness of the wall. The arrangement of polygon walls observed in fig. la is much more regular than in the other materials cited above, which must result from dislocations in the walls being able to climb into their lowest energy positions, resulting in the regular arrangement just cited. However, the cusp structure is very irregular along a wall (see figs. la and 4). Comparing the thickness of the bend region with the distances between cusps (about 3 urn), the region with excess dislocations is found to be slightly larger than one half of the lattice volume. Besides the short distance
October 1988
between cusps, overlapping cusps give a wide region with high dislocation density. For the same material deformed under the same conditions, Dominguez et al. [ 7 ] found by TEM a density of dislocations of 2.5 x lOI rne2, while our TEM measurements [ 81 gave a density between 1 x lOI and 2x 10” m-‘, close to the present results. Wilkens reported values of about 40’ for the kink angle of polygonization walls in Cu single crystals [ 3,4]. This corresponds to a greater excess of dislocations of one sign than here, but a smaller local density of 1 x lOI m-’ was found as the bend lattice region extended over about 15 urn and the distance between cusps ranged between 30 and 100 nm. The polygon walls and cusps are generally [ 3-51 much more irregular than in our case. These differences can be due to the fact that these materials were deformed to low temperature whereas our crystals were deformed at ~0.67’,,,, where climb of dislocations occurred as was observed by TEM [ 7 1.
Acknowledgement One of us (FG) thanks the Hispano-Frances curio Scientific Program for support.
Mer-
References [ 11E. Fries, F. Guiberteau, .A. Dommguez-Rodriguez.
D.S. Cheong and A.H. Heuer. Phil. Mag. A, submitted for publication. [ 21 J.B. Newkirk, Trans. Metall. Sot. AIME 2 I5 ( 1959) 483. [3] M. Wilkens, Can. J. Phys. 45 (1967) 567. [4] B. Obst, H. Auer and M. Wilkens. Mater. Sci. Eng. 3 ( 1968/ 69) 41. [5] H. Strunk, Mater. Sci. Eng. 26 (1976) 231. [ 61 E. Fries, M. Spendel and J. Philibert. J. Appl. Cryst. 14 ( 1981 ) 285. [7] A. Dominguez-Rodriguez. K.P.D. Lagerlof and A.H. Heuer. J. Am. Ceram. Sot. 69 ( 1986) 28 1. [ 8] F.L. Cumbrera, A. Dominguez-Rodriguez. F. Guiberteau and E. Fries. unpublished results.
145