- Email: [email protected]
Study of atomic displacement fields in shape memory alloys by high-resolution electron microscopy
Materials Science and Engineering A273 – 275 (1999) 266 – 270 www.elsevier.com/locate/msea
Study of atomic displacement fields in shape memory alloys by high-resolution electron microscopy M.J. Hy¨tch a, Ph. Vermaut a,b,*, J. Malarria b,1, R. Portier a,b a
Centre d’Etudes de Chimie Me´tallurgique-CNRS, 15 rue G. Urbain, 94407 Vitry-sur-Seine, France b Laboratoire de Me´tallurgie Structurale, ENSCP, 11, rue P. et M. Curie, 75231 Paris, France
Abstract The distortion of the atomic lattice is determined using a recently developed method of analysing high resolution electron microscope images. The analysis is carried out in terms of the individual lattice fringes which contribute to the image contrast. A perfect set of fringes has an amplitude and phase given by the corresponding Fourier component. A distorted lattice can be analysed by introducing the concept of a ‘local’ Fourier component, representing the local values of amplitude and phase as a function in position of the image. The local phase is used to determine the line of a perfectly coherent interface between two sets of lattice planes. The degree of self-accommodation of a series of martensitic plates in a Cu – Zn – Al shape-memory-alloy is studied by the phase analysis of an experimental high resolution image. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Shape memory alloys; High resolution electron microscopy; Cu – Zn – Al; Atomic displacement fields; Image processing
1. Introduction The martensitic transformation is a first order phase transformation with a homogeneous lattice deformation. Even in the case of thermoelastic transformations for which the formation of self accommodated groups associated with the inhomogeneous lattice deformation drastically reduce the shape-change, the kinetics and the morphology of this transformation are dominated by the deformation energy. Moreover, some specific mechanical properties are directly related to the effect of a stress field, external or internal. The two-way shape memory effect can be related to the existence of defects induced during the training of the specimen, dislocations for the thermomechanical or superelastic cycling [1 –4], or oriented precipitates for the all round shape memory effect [5 – 7]. The microstructure also has a strong influence on the behaviour of the specimen. For instance in nickel titanium alloys, the double transformation on cooling is attributed to the stress field around precipitates in the austenite: the transformation is then assisted by the strain field in the area around the * Corresponding author. Tel.: + 33-1-4354-3276; fax: + 33-1-44276-710. 1 On leave from the Universidad Nacional of Rosario, Argentina.
precipitates (first transformation) and not for areas further away (second transformation) [8]. All these aspects mean that experimental data on the local displacement fields are of considerable interest. High resolution electron microscopy (HREM) is considered to be a very efficient tool for local investigation. In particular, there is increasing interest in the measurement of atomic displacement fields following the work on the expansion of grain-boundaries [9] and more recently on strained semi-conductor multilayers [10]. Unlike the interpretation of image intensities in terms of chemical composition, which requires an accurate knowledge of a great number of experimental parameters (thickness and orientation of the specimen, defocus, beam alignment etc.), it is relatively easy to correlate image features with atomic column positions. Nevertheless, there are a number of points which need to be considered. For example, although very thin, the object is three dimensional and the image is only two dimensional: the information along the incident electron beam direction is therefore lost. More crucially, if the displacement in a direction perpendicular to the incident beam direction is not constant across the thickness of the sample, then the contrast is blurred and cannot be interpreted. Another important point, related to the thinness of the specimen, is the effect of the free sur-
0921-5093/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 9 9 ) 0 0 3 5 2 - 4
M.J. Hy¨tch et al. / Materials Science and Engineering A273–275 (1999) 266–270
faces which allows the relaxation of stresses. We must remember that the observed results coming from the thin part can be different from the bulk. Finally, in a thin crystal some special martensite can appear due to the local stresses which cannot be avoided during the preparation and manipulation of the specimen and due to the effect of the incident electron beam. In this article, we will present a recently developed method of analysing HREM images which allows the characterisation of the local strain of the lattice [11,12]. The method relies on the accurate measuring of the positions of lattice fringes in the image. As an illustration of its potential, we will present an analysis of a series of self-accommodated martensite plates in a Cu– Zn–Al shape memory alloy.
2. Experimental details Tensile specimens (gauge length = 10 mm and diameter 6 mm) were spark machined from b phase single crystals. The composition of the alloy was Zn 15.34 at.%, Al 16.33 at.%, Cu-rest, corresponding to e/a = 1.48, with a Ms= −10°C. The samples were heated to 850°C for 20 min and air quenched. They were then left at room temperature for at least 4 days to anneal out excess vacancies and to allow the ordering of the alloy. Under a tensile stress, the samples were pseudoelastically cycled 3000 times in a closed loop at 0.8 Hz. They were deformed, keeping the temperature constant at 25.59 0.5°C, between a strain which still corresponds to pure b phase, and a strain that lead to a 100% martensitic transformation [13]. Several discs parallel to the family of planes {001}b were cut from the cycled specimens for transmission electron microscopy (TEM) analysis. Thin foils were obtained by double jet electro-polishing at room temperature (30% orthophosphoric acid in distilled water, V =2 V, I = 30 mA). HRTEM observations were carried out on a Topcon 002B electron microscope operating at 200 kV with a point to point resolution of 0.18 nm (Cs= 0.4 mm). Negatives were digitised with an Agfa Duoscan scanner and the resulting images have a resolution of 0.038 nm/pixel. The image processing was carried out using routines specially written within the software package Digital Micrograph [14] running on a G3 Macintosh Power PC.
2.1. Local phase images The contrast of a high resolution image is due to several sets of overlapping lattice fringes. The image intensity for a perfect crystal, I(r), can be expressed as a Fourier sum:
I(r) = % Ig exp{2pig · r}
267
(1)
g
where g are the lattice fringe periodicities and Ig, the corresponding Fourier components. It has been shown that variations in crystal structure can be described by allowing the Fourier components Ig to acquire a ‘local’ value Ig (r) depending on the position in the image [11]: I(r) = % Ig (r)exp{2pig · r}
(2)
g
As for the Fourier components in a perfect crystal, the local Fourier components have an amplitude and phase such that: Ig (r) =Ag (r)exp{iPg (r)}
(3)
The amplitude Ag (r) represents the local degree of contrast of a particular set of lattice fringes g, and Pg (r) determines their position with respect to the perfect lattice. The local phase images, Pg (r), have particularly interesting properties. It can be shown, for example, that the displacement field, u(r), associated with a particular set of lattice planes is directly related to the phase [11]: Pg (r) = −2pg · u(r)
(4)
and by combining the information from two lattice planes, g1 and g2, the two dimensional displacement field can be obtained: u(r) = −
1 [P (r)a1 + Pg2(r)a2] 2p g1
(5)
where a1 and a2 represent the lattice in real-space corresponding to the reciprocal lattice vectors g1 and g2 [12]. In this paper we will be using the phase images rather to help understand the geometry and the self-accommodation of different austenitic and martensitic variants. A simplified case is shown in Fig. 1a of a perfectly coherent interface between two sets of lattice fringes g1 and g2. To obtain the local phase image, Pg (r), a mask is placed around the two spots in Fourier space (see Fig. 1b) and the inverse Fourier transform performed. This operation is equivalent to selecting relevant terms in Eq. (2). In crystal 1 the phase of the resulting complex image is 2pg1 · r and in crystal 2, 2pg2 · r (see Fig. 1c). The local phase image Pg (r), shown in Fig. 1d, is the phase with respect to a given reference, in this case crystal 1, and is calculated by subtracting 2pg1 · r from the previous image. It is therefore zero in crystal 1 (the grey contrast) but in crystal 2 the phase is given by: Pg (r) =2p(g2 − g1) · r= 2p Dg · r
(6)
with Dg the difference between the local reciprocal lattice vector and the reference g1.
268
M.J. Hy¨tch et al. / Materials Science and Engineering A273–275 (1999) 266–270
We see that the relative phase is zero along the interface plane, which is why the lattice match is perfect. Turning this argument around, we can say that for a perfect match between lattice planes at an interface the phase must be continuous. In the general case, constant phase terms P1 and P2 for the two lattice planes must be included and the condition for a coherent interface, ri, becomes: 2pg1 · ri + P1 = 2pg2 · ri +P2 [ 2p Dg · ri =P2 −P1
(7)
which is the equation of a plane with normal Dg. This is in agreement with a reinterpretation of the O-lattice theory [15] in terms of differences in reciprocal lattice vectors [16]. We will be investigating whether this is the case experimentally.
3. Results and discussion A first analysis of the microstructure of the specimen has shown that it is still a single crystal of the ordered b1 phase (L21, a= 5.86 A, ) with large dislocation bands introduced by the pseudo elastic tensile cycling. In addition to these bands, in thin parts of the specimen, small plates of martensite phase are observed (see the high resolution image taken in [001] orientation shown in Fig. 2). Their formation is eased by the Ms temperature close to room temperature (− 10°C). When the
Fig. 2. [001] high resolution image of austenitic phase. The two martensitic variants M1 and M2 of widths L1 and L2 are indicated.
Fig. 1. Phase analysis of an ideal interface; (a) simulated image of lattice fringes; (b) power spectrum of the image indicating the mask used to calculate the phase; (c) phase of lattice fringes; and (d) ‘local’ phase relative to crystal 1, Pg (r), showing the interface line is perpendicular to Dg= g2 −g1. Grey levels for phase images correspond to black = −p white=p.
electron beam is focused on such plates, they are observed to grow, with the austenite/martensite interface at the tip of the plates gliding towards the inner parts of the specimen (an enlargement of a tip region is shown in Fig. 3). Although this phenomenon seems to go opposite to the driving force of the phase transformation, caused by the heating of the electron beam, the growth of the martensite plates can be attributed to an accommodation of local stresses introduced by the bending of the thin foil under the electron beam. According to the electronic concentration of the alloy (e/a= 1.48) a 18R martensite is expected to be formed [17]. No confusion can be made with surface martensite which does not form on (100) surfaces [18]. Nevertheless, the exact nature of the martensitic phase has not yet been determined. Unfavourable geometrical conditions prevented a detailed analysis since the area under observation is quite small and, furthermore, the sample appears bend on displacing the observation towards the hole. The latter feature constitutes a very strong obstacle to obtaining a planar interface, well suited for the structure determination as, for instance, the (128( ) twin interface in the 18R martensite [19]. An exhaustive study with more appropriate samples is now in progress. In this paper we focus our attention on the potential of the method to determine the local strain of the lattice.
M.J. Hy¨tch et al. / Materials Science and Engineering A273–275 (1999) 266–270
269
Table 1 Differences between lattice vectors in the austenitic and martensitic phases measured from the gradient of the local phase images. Reciprocal lattice lengths, Dg, are expressed as a fraction of gA, and the orientation with respect to the horizontal
Fig. 3. Enlargement of the boxed region on Fig. 2 showing the tips of the martensitic plates.
To calculate the local phase image, a Fourier transform was taken of the region shown in Fig. 3. A mask was applied around the periodicity corresponding to the 110 planes in the austenite (see Fig. 4a) and the phase image calculated from the inverse Fourier transform (Fig. 4b). Where the phase image has a gradient of zero, the local lattice vector g corresponds exactly to that of the reference lattice, as in the region marked as the austenite. By measuring the gradient of the phase, the deviation from the austenitic lattice, Dg, can be determined via Eq. (6). It was found from this analysis that the bands in Fig. 2 were of two different martensitic variants M1 and M2, as marked on Figs. 3 and 4b. The results are summarised in Table 1 and the positions
110
Dg/0.134 gA
Angle (°)
gM 2−gM 1 gM 2−gA gA−gM 1
1 0.65 9 0.01 0.36 9 0.02
59.390.2 65.990.3 47.2 90.5
1( 10
Dg/0.104 gA
gM 2−gM 1 gM 2−gA gA−gM 1
1 0.66 9 0.01 0.34 9 0.02
58.090.3 53.8 90.4 66.090.7
of the periodicities in reciprocal space indicated on Fig. 4a. The angle of the interface line between the two martensitic variants was measured from Fig. 2 and found to be − 309 0.5°. The differences in lattice vectors Dg = gM 1 − gM 2 for both the {110} lattice planes are almost exactly perpendicular to this line at 59.390.2 and 58.0 9 0.3° (see Table 1). This means from Eq. (7) that the {110} lattice planes in the two martensitic phases are almost perfectly coherent and form a near invariant line [20]. Though the conditions for both lattice planes cannot be satisfied exactly, a deviation of 1° from the ideal interface line would only mean a dislocation every 110 nm. In the direction perpendicular to the plates of martensite, it is interesting to calculate if any extra planes, and hence dislocations, are necessary with respect to the austenitic phase. For no extra planes, the phase difference across two martensitic plates must equal that for the austenite. Therefore: 2pg1 · L1 + 2pg2 · L2 = 2pgA · (L1 + L2) [ (g1 −gA ) · L1 = (g2 − gA ) · L2
(8)
and since the vectors are almost perpendicular to the interface line: L1 gM 2 − gA = L2 gM 1 − gA
(9)
From Fig. 2, the ratio of the widths of the two plates, L1 to L2, was measured to be approximately 1:2. Comparing with the values given in Table 1, this means that the plates are in a ratio which auto-accommodates with the austenitic matrix. Fig. 4. Phase analysis of the region shown in Fig. 3: (a) Fourier transform of the image showing the mask used to calculate the phase image in (b) and positions of the different image periodicities; (b) local phase image Pg (r) relative to austenite for the 110 planes, the two martensitic plates are revealed by the presence of gradients in the phase. Grey levels for phase image correspond to black = −p white =p.
4. Conclusions A coherent interface between two sets of lattice planes is characterised by the difference in the reciprocal lattice vectors. The phase images allow the accurate
270
M.J. Hy¨tch et al. / Materials Science and Engineering A273–275 (1999) 266–270
measurement of lattice periodicities in different regions of an image. It was therefore possible to identify the different austenitic and martensitic phases. It was shown that the interface between the two martensitic phases was almost perfectly coherent for the {110} lattice fringes. The relative widths of the two martensitic phases were such that the plates were auto-accommodated with the austenitic phase. Future work will concern the measurement of the strain fields around the tips of the martensitic plates.
References [1] M. Sade, A. Uribarri, F. Lovey, Philos. Mag. A 58 (1987) 881. [2] J. Pons, F.C. Lovey, E. Ce´sari, Acta Metall. Mater. 38 (1990) 2733. [3] A. Amengual, E. Ce´sari, J. Pons, J. Phys. (France) IV 5 (1995) C2 – 87. [4] F.C. Lovey, P.L. Rodriguez, J. Malarria, M. Sade, V. Torra, J. Phys. (France) IV 5 (1995) C2–287. [5] N. Nishida, T. Honna, Scripta Metall. 18 (1984) 1293.
.
[6] T. Honna, Proceedings of ICOMAT 86, The Japan Institute of Metals, 1986, p. 709. [7] R. Kainuma, M. Matsumoti, T. Honna, Proceedings of ICOMAT 86, The Japan Institute of Metals, 1986, p. 717. [8] L. Bataillard, Transformation martensitique multiple dans un alliage a` me´moire de forme Ni – Ti, Ph.D. Thesis, Ecole Polytechnique Fe´de´rale de Lausanne, Switzerland, 1996. [9] W.M. Stobbs, G.J. Woods, D.J. Smith, Ultramicroscopy 14 (1984) 145. [10] R. Bierwolf, H. Hohenstein, F. Philipp, O. Brandt, G.E. Crook, K. Ploog, Ultramicroscopy 49 (1993) 273. [11] M.J. Hy´tch, Microsc. Microanal. Microstruct. 8 (1997) 41–57. [12] M.J. Hy´tch, E. Snoeck, R. Kilaas, Ultramicroscopy 74 (1998) 131. [13] J. Malarria, M. Sade, F. Lovey, Z. Metallkd. 87 (1996) 953. [14] Digital Micrograph 2.5, Gatan Inc., phase routines available at http://ncem.lbl.gov. [15] W. Bollman, Crystal Defects and Crystalline Interfaces, Springer, Berlin, 1970. [16] W.-Z. Zhang, G.R. Purdy, Phil. Mag. A 68 (1993) 279. [17] M. Ahlers, J. Pelegrina, Acta Metall. Mater. 40 (1992) 3213– 3220. [18] F.C. Lovey, M. Chandrasekaran, M. Ahlers, Z. Metalld. 71 (1980) 37. [19] F.C. Lovey, G. Van Tendeloo, J. Van Landuyt, S. Amelincks, Scripta Metall. 19 (1985) 1223 – 1228. [20] U. Dahmen, Acta Metall. 30 (1982) 63.
Study of atomic displacement fields in shape memory alloys by high-resolution electron microscopy M.J. Hy¨tch a, Ph. Vermaut a,b,*, J. Malarria b,1, R. Portier a,b a
Centre d’Etudes de Chimie Me´tallurgique-CNRS, 15 rue G. Urbain, 94407 Vitry-sur-Seine, France b Laboratoire de Me´tallurgie Structurale, ENSCP, 11, rue P. et M. Curie, 75231 Paris, France
Abstract The distortion of the atomic lattice is determined using a recently developed method of analysing high resolution electron microscope images. The analysis is carried out in terms of the individual lattice fringes which contribute to the image contrast. A perfect set of fringes has an amplitude and phase given by the corresponding Fourier component. A distorted lattice can be analysed by introducing the concept of a ‘local’ Fourier component, representing the local values of amplitude and phase as a function in position of the image. The local phase is used to determine the line of a perfectly coherent interface between two sets of lattice planes. The degree of self-accommodation of a series of martensitic plates in a Cu – Zn – Al shape-memory-alloy is studied by the phase analysis of an experimental high resolution image. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Shape memory alloys; High resolution electron microscopy; Cu – Zn – Al; Atomic displacement fields; Image processing
1. Introduction The martensitic transformation is a first order phase transformation with a homogeneous lattice deformation. Even in the case of thermoelastic transformations for which the formation of self accommodated groups associated with the inhomogeneous lattice deformation drastically reduce the shape-change, the kinetics and the morphology of this transformation are dominated by the deformation energy. Moreover, some specific mechanical properties are directly related to the effect of a stress field, external or internal. The two-way shape memory effect can be related to the existence of defects induced during the training of the specimen, dislocations for the thermomechanical or superelastic cycling [1 –4], or oriented precipitates for the all round shape memory effect [5 – 7]. The microstructure also has a strong influence on the behaviour of the specimen. For instance in nickel titanium alloys, the double transformation on cooling is attributed to the stress field around precipitates in the austenite: the transformation is then assisted by the strain field in the area around the * Corresponding author. Tel.: + 33-1-4354-3276; fax: + 33-1-44276-710. 1 On leave from the Universidad Nacional of Rosario, Argentina.
precipitates (first transformation) and not for areas further away (second transformation) [8]. All these aspects mean that experimental data on the local displacement fields are of considerable interest. High resolution electron microscopy (HREM) is considered to be a very efficient tool for local investigation. In particular, there is increasing interest in the measurement of atomic displacement fields following the work on the expansion of grain-boundaries [9] and more recently on strained semi-conductor multilayers [10]. Unlike the interpretation of image intensities in terms of chemical composition, which requires an accurate knowledge of a great number of experimental parameters (thickness and orientation of the specimen, defocus, beam alignment etc.), it is relatively easy to correlate image features with atomic column positions. Nevertheless, there are a number of points which need to be considered. For example, although very thin, the object is three dimensional and the image is only two dimensional: the information along the incident electron beam direction is therefore lost. More crucially, if the displacement in a direction perpendicular to the incident beam direction is not constant across the thickness of the sample, then the contrast is blurred and cannot be interpreted. Another important point, related to the thinness of the specimen, is the effect of the free sur-
0921-5093/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 9 9 ) 0 0 3 5 2 - 4
M.J. Hy¨tch et al. / Materials Science and Engineering A273–275 (1999) 266–270
faces which allows the relaxation of stresses. We must remember that the observed results coming from the thin part can be different from the bulk. Finally, in a thin crystal some special martensite can appear due to the local stresses which cannot be avoided during the preparation and manipulation of the specimen and due to the effect of the incident electron beam. In this article, we will present a recently developed method of analysing HREM images which allows the characterisation of the local strain of the lattice [11,12]. The method relies on the accurate measuring of the positions of lattice fringes in the image. As an illustration of its potential, we will present an analysis of a series of self-accommodated martensite plates in a Cu– Zn–Al shape memory alloy.
2. Experimental details Tensile specimens (gauge length = 10 mm and diameter 6 mm) were spark machined from b phase single crystals. The composition of the alloy was Zn 15.34 at.%, Al 16.33 at.%, Cu-rest, corresponding to e/a = 1.48, with a Ms= −10°C. The samples were heated to 850°C for 20 min and air quenched. They were then left at room temperature for at least 4 days to anneal out excess vacancies and to allow the ordering of the alloy. Under a tensile stress, the samples were pseudoelastically cycled 3000 times in a closed loop at 0.8 Hz. They were deformed, keeping the temperature constant at 25.59 0.5°C, between a strain which still corresponds to pure b phase, and a strain that lead to a 100% martensitic transformation [13]. Several discs parallel to the family of planes {001}b were cut from the cycled specimens for transmission electron microscopy (TEM) analysis. Thin foils were obtained by double jet electro-polishing at room temperature (30% orthophosphoric acid in distilled water, V =2 V, I = 30 mA). HRTEM observations were carried out on a Topcon 002B electron microscope operating at 200 kV with a point to point resolution of 0.18 nm (Cs= 0.4 mm). Negatives were digitised with an Agfa Duoscan scanner and the resulting images have a resolution of 0.038 nm/pixel. The image processing was carried out using routines specially written within the software package Digital Micrograph [14] running on a G3 Macintosh Power PC.
2.1. Local phase images The contrast of a high resolution image is due to several sets of overlapping lattice fringes. The image intensity for a perfect crystal, I(r), can be expressed as a Fourier sum:
I(r) = % Ig exp{2pig · r}
267
(1)
g
where g are the lattice fringe periodicities and Ig, the corresponding Fourier components. It has been shown that variations in crystal structure can be described by allowing the Fourier components Ig to acquire a ‘local’ value Ig (r) depending on the position in the image [11]: I(r) = % Ig (r)exp{2pig · r}
(2)
g
As for the Fourier components in a perfect crystal, the local Fourier components have an amplitude and phase such that: Ig (r) =Ag (r)exp{iPg (r)}
(3)
The amplitude Ag (r) represents the local degree of contrast of a particular set of lattice fringes g, and Pg (r) determines their position with respect to the perfect lattice. The local phase images, Pg (r), have particularly interesting properties. It can be shown, for example, that the displacement field, u(r), associated with a particular set of lattice planes is directly related to the phase [11]: Pg (r) = −2pg · u(r)
(4)
and by combining the information from two lattice planes, g1 and g2, the two dimensional displacement field can be obtained: u(r) = −
1 [P (r)a1 + Pg2(r)a2] 2p g1
(5)
where a1 and a2 represent the lattice in real-space corresponding to the reciprocal lattice vectors g1 and g2 [12]. In this paper we will be using the phase images rather to help understand the geometry and the self-accommodation of different austenitic and martensitic variants. A simplified case is shown in Fig. 1a of a perfectly coherent interface between two sets of lattice fringes g1 and g2. To obtain the local phase image, Pg (r), a mask is placed around the two spots in Fourier space (see Fig. 1b) and the inverse Fourier transform performed. This operation is equivalent to selecting relevant terms in Eq. (2). In crystal 1 the phase of the resulting complex image is 2pg1 · r and in crystal 2, 2pg2 · r (see Fig. 1c). The local phase image Pg (r), shown in Fig. 1d, is the phase with respect to a given reference, in this case crystal 1, and is calculated by subtracting 2pg1 · r from the previous image. It is therefore zero in crystal 1 (the grey contrast) but in crystal 2 the phase is given by: Pg (r) =2p(g2 − g1) · r= 2p Dg · r
(6)
with Dg the difference between the local reciprocal lattice vector and the reference g1.
268
M.J. Hy¨tch et al. / Materials Science and Engineering A273–275 (1999) 266–270
We see that the relative phase is zero along the interface plane, which is why the lattice match is perfect. Turning this argument around, we can say that for a perfect match between lattice planes at an interface the phase must be continuous. In the general case, constant phase terms P1 and P2 for the two lattice planes must be included and the condition for a coherent interface, ri, becomes: 2pg1 · ri + P1 = 2pg2 · ri +P2 [ 2p Dg · ri =P2 −P1
(7)
which is the equation of a plane with normal Dg. This is in agreement with a reinterpretation of the O-lattice theory [15] in terms of differences in reciprocal lattice vectors [16]. We will be investigating whether this is the case experimentally.
3. Results and discussion A first analysis of the microstructure of the specimen has shown that it is still a single crystal of the ordered b1 phase (L21, a= 5.86 A, ) with large dislocation bands introduced by the pseudo elastic tensile cycling. In addition to these bands, in thin parts of the specimen, small plates of martensite phase are observed (see the high resolution image taken in [001] orientation shown in Fig. 2). Their formation is eased by the Ms temperature close to room temperature (− 10°C). When the
Fig. 2. [001] high resolution image of austenitic phase. The two martensitic variants M1 and M2 of widths L1 and L2 are indicated.
Fig. 1. Phase analysis of an ideal interface; (a) simulated image of lattice fringes; (b) power spectrum of the image indicating the mask used to calculate the phase; (c) phase of lattice fringes; and (d) ‘local’ phase relative to crystal 1, Pg (r), showing the interface line is perpendicular to Dg= g2 −g1. Grey levels for phase images correspond to black = −p white=p.
electron beam is focused on such plates, they are observed to grow, with the austenite/martensite interface at the tip of the plates gliding towards the inner parts of the specimen (an enlargement of a tip region is shown in Fig. 3). Although this phenomenon seems to go opposite to the driving force of the phase transformation, caused by the heating of the electron beam, the growth of the martensite plates can be attributed to an accommodation of local stresses introduced by the bending of the thin foil under the electron beam. According to the electronic concentration of the alloy (e/a= 1.48) a 18R martensite is expected to be formed [17]. No confusion can be made with surface martensite which does not form on (100) surfaces [18]. Nevertheless, the exact nature of the martensitic phase has not yet been determined. Unfavourable geometrical conditions prevented a detailed analysis since the area under observation is quite small and, furthermore, the sample appears bend on displacing the observation towards the hole. The latter feature constitutes a very strong obstacle to obtaining a planar interface, well suited for the structure determination as, for instance, the (128( ) twin interface in the 18R martensite [19]. An exhaustive study with more appropriate samples is now in progress. In this paper we focus our attention on the potential of the method to determine the local strain of the lattice.
M.J. Hy¨tch et al. / Materials Science and Engineering A273–275 (1999) 266–270
269
Table 1 Differences between lattice vectors in the austenitic and martensitic phases measured from the gradient of the local phase images. Reciprocal lattice lengths, Dg, are expressed as a fraction of gA, and the orientation with respect to the horizontal
Fig. 3. Enlargement of the boxed region on Fig. 2 showing the tips of the martensitic plates.
To calculate the local phase image, a Fourier transform was taken of the region shown in Fig. 3. A mask was applied around the periodicity corresponding to the 110 planes in the austenite (see Fig. 4a) and the phase image calculated from the inverse Fourier transform (Fig. 4b). Where the phase image has a gradient of zero, the local lattice vector g corresponds exactly to that of the reference lattice, as in the region marked as the austenite. By measuring the gradient of the phase, the deviation from the austenitic lattice, Dg, can be determined via Eq. (6). It was found from this analysis that the bands in Fig. 2 were of two different martensitic variants M1 and M2, as marked on Figs. 3 and 4b. The results are summarised in Table 1 and the positions
110
Dg/0.134 gA
Angle (°)
gM 2−gM 1 gM 2−gA gA−gM 1
1 0.65 9 0.01 0.36 9 0.02
59.390.2 65.990.3 47.2 90.5
1( 10
Dg/0.104 gA
gM 2−gM 1 gM 2−gA gA−gM 1
1 0.66 9 0.01 0.34 9 0.02
58.090.3 53.8 90.4 66.090.7
of the periodicities in reciprocal space indicated on Fig. 4a. The angle of the interface line between the two martensitic variants was measured from Fig. 2 and found to be − 309 0.5°. The differences in lattice vectors Dg = gM 1 − gM 2 for both the {110} lattice planes are almost exactly perpendicular to this line at 59.390.2 and 58.0 9 0.3° (see Table 1). This means from Eq. (7) that the {110} lattice planes in the two martensitic phases are almost perfectly coherent and form a near invariant line [20]. Though the conditions for both lattice planes cannot be satisfied exactly, a deviation of 1° from the ideal interface line would only mean a dislocation every 110 nm. In the direction perpendicular to the plates of martensite, it is interesting to calculate if any extra planes, and hence dislocations, are necessary with respect to the austenitic phase. For no extra planes, the phase difference across two martensitic plates must equal that for the austenite. Therefore: 2pg1 · L1 + 2pg2 · L2 = 2pgA · (L1 + L2) [ (g1 −gA ) · L1 = (g2 − gA ) · L2
(8)
and since the vectors are almost perpendicular to the interface line: L1 gM 2 − gA = L2 gM 1 − gA
(9)
From Fig. 2, the ratio of the widths of the two plates, L1 to L2, was measured to be approximately 1:2. Comparing with the values given in Table 1, this means that the plates are in a ratio which auto-accommodates with the austenitic matrix. Fig. 4. Phase analysis of the region shown in Fig. 3: (a) Fourier transform of the image showing the mask used to calculate the phase image in (b) and positions of the different image periodicities; (b) local phase image Pg (r) relative to austenite for the 110 planes, the two martensitic plates are revealed by the presence of gradients in the phase. Grey levels for phase image correspond to black = −p white =p.
4. Conclusions A coherent interface between two sets of lattice planes is characterised by the difference in the reciprocal lattice vectors. The phase images allow the accurate
270
M.J. Hy¨tch et al. / Materials Science and Engineering A273–275 (1999) 266–270
measurement of lattice periodicities in different regions of an image. It was therefore possible to identify the different austenitic and martensitic phases. It was shown that the interface between the two martensitic phases was almost perfectly coherent for the {110} lattice fringes. The relative widths of the two martensitic phases were such that the plates were auto-accommodated with the austenitic phase. Future work will concern the measurement of the strain fields around the tips of the martensitic plates.
References [1] M. Sade, A. Uribarri, F. Lovey, Philos. Mag. A 58 (1987) 881. [2] J. Pons, F.C. Lovey, E. Ce´sari, Acta Metall. Mater. 38 (1990) 2733. [3] A. Amengual, E. Ce´sari, J. Pons, J. Phys. (France) IV 5 (1995) C2 – 87. [4] F.C. Lovey, P.L. Rodriguez, J. Malarria, M. Sade, V. Torra, J. Phys. (France) IV 5 (1995) C2–287. [5] N. Nishida, T. Honna, Scripta Metall. 18 (1984) 1293.
.
[6] T. Honna, Proceedings of ICOMAT 86, The Japan Institute of Metals, 1986, p. 709. [7] R. Kainuma, M. Matsumoti, T. Honna, Proceedings of ICOMAT 86, The Japan Institute of Metals, 1986, p. 717. [8] L. Bataillard, Transformation martensitique multiple dans un alliage a` me´moire de forme Ni – Ti, Ph.D. Thesis, Ecole Polytechnique Fe´de´rale de Lausanne, Switzerland, 1996. [9] W.M. Stobbs, G.J. Woods, D.J. Smith, Ultramicroscopy 14 (1984) 145. [10] R. Bierwolf, H. Hohenstein, F. Philipp, O. Brandt, G.E. Crook, K. Ploog, Ultramicroscopy 49 (1993) 273. [11] M.J. Hy´tch, Microsc. Microanal. Microstruct. 8 (1997) 41–57. [12] M.J. Hy´tch, E. Snoeck, R. Kilaas, Ultramicroscopy 74 (1998) 131. [13] J. Malarria, M. Sade, F. Lovey, Z. Metallkd. 87 (1996) 953. [14] Digital Micrograph 2.5, Gatan Inc., phase routines available at http://ncem.lbl.gov. [15] W. Bollman, Crystal Defects and Crystalline Interfaces, Springer, Berlin, 1970. [16] W.-Z. Zhang, G.R. Purdy, Phil. Mag. A 68 (1993) 279. [17] M. Ahlers, J. Pelegrina, Acta Metall. Mater. 40 (1992) 3213– 3220. [18] F.C. Lovey, M. Chandrasekaran, M. Ahlers, Z. Metalld. 71 (1980) 37. [19] F.C. Lovey, G. Van Tendeloo, J. Van Landuyt, S. Amelincks, Scripta Metall. 19 (1985) 1223 – 1228. [20] U. Dahmen, Acta Metall. 30 (1982) 63.