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Structural change of rapidly solidified 2024 aluminium alloy powders in mechanical milling and subsequent consolidation process
Journal of
E LSE VIE R
Journal of Materials Processing Technology 58 (1996) 247-250
Materials Processing Technology
Structural change of rapidly solidified 2024 aluminium alloy powders in mechanical milling and subsequent consolidation process Liang Guoxian *, Li Zhichao, Wang Erde, Wang Zhongren College of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, People's Republic of China Received 18 December 1994
Industrial summary
Rapidly solidified 2024 aluminium powders were mechanically milled, and then consolidated by vacuum hot-pressing and hot hydrostatic-extrusion. The grain-growth and phase-separation behaviors during the consolidation process were studied by X-ray diffraction and TEM observation. The results show that mechanical milling (MM) produces a supersaturated solid solution with a fine microstructure of nanometer-sized grains. In the consolidation process, the grain size increased, and AlzCu and Alz(Cu, Mg, Si, Fe, Mn) intermetallic-phases precipitated. A deformation-enhanced grain growth behavior was observed. Vacuum hot-pressing results in more obvious grain growth than simple annealing does, and hot hydrostatic-extrusion results in more serious grain-size coarsening than hot pressing does. Keywords: Aluminium alloy powders; Mechanical milling; Consolidation process
1. Introduction
Rapid solidification (RS) techniques have been used widely for the development of new alloys having extended solubility and for the refinement of dispersion or inclusion particles [1,2]. However, owing to thermodynamic restrictions, the intermetallic particles may have a large size, and the solubility may not be extendable further [3]. Mechanical alloying (MA) has the unique advantages of extending the solubility and producing fine dispersions and grainsizes with rapid solidification in some alloy systems [4-6]. However, with mechanical alloying, starting from mixed elemental powders, it is difficult to obtain a real homogeneous microstructure [7]. Thus a composite RS + M M technique may overcome the drawbacks of both techniques and enable the obtaining of fully-homogeneous fine microstructures. Recently, nanocrystalline materials have been the subject of intense research, motivated by their unusual physical and mechanical properties [8 10]. Mechanical milling has been used to synthesize nanocrystalline metals and alloys, and the stability of the nanostructure on annealing in some alloy systems has been investigated [11,12]. However, the thermal stability and consolidation of mechanically-milled nanocrystalline alloys have not yet been adequately studied. * Corresponding author. 0924-0136/96/$15.00 © 1996 Elsevier Science S.A. All rights reserved SSD1 0924-0136(95)02121-2
In the present work, the application of the mechanicalmilling process to rapidly solidified 2024 aluminium powders is described, and the grain growth and precipitation behavior of the mechanically-milled metastable powders in the consolidation process is studied.
2. Experimental procedure
2024 aluminium powders with a normal composition of 4.48 Cu-1.45 Mg-0.61 Mn-0.31 Fe-0.15 Si-balance A1 (wt.%) were prepared using the argon gas atomization method. Mechanical milling of powders was performed in an attritor ball-mill under the protection of pure argon gas. The milling vial was made of stainless steel and the balls were made of hardened chrome steel. No process control agent was added. Chemical analysis shows that the Fe content increased by 0.5 wt.% after 10 h of milling. The processed powders were consolidated by two processing routes: (A) cold compacted to a cylindrical billet under a pressure of 1 GPa, canned in a lubricant medium, then pre-heated for 10 min at 450 °C before hot hydrostatic-extrusion as in Ref. [13]; (B); hot pressed under a pressure of 500 MPa at 200-400 °C in a vaccum of better than 4 x 10 2pa for 0.5h, and then hot hydrostatically extruded to rod with an extrusion ratio of 14:1.
248
G. Liang et al. /."Journal of Materials Processing Technology 58 (1996) 247-250
The processed powders and consolidated materials were characterized by X-ray diffraction using a Rigakud/max-rB diffractometer with CuK~ radiation. Microstructural observations were made by transmission electron microscopy with a EM420 microscope. Thin foils for T E M observation were prepared by twin-jet electropolishing with an electrolyte of 1:3 nitric acid/methanol at - 3 0 °C.
250
200
" 150
o ~
200"C
e~
300"C
~
400"C
100
50 o
3. R e s u l t s and d i s c u s s i o n 0
3.1. Mechanical milling
X-ray diffraction patterns of 2024 aluminium powders milled for various times are shown in Fig. 1. In the rapidly-solidified powders, diffraction peaks of A12Cu intermetallic compound were found. With an increase in the milling time, the diffraction peaks of A12Cu phase disappeared gradually, suggesting that this phase was dissolved in the aluminium matrix, and a supersaturated solid-solution formed. Increasing the milling time also produced broadening of the aluminium peaks and a decrease in the peak height, indicating a continuous decrease in the grain size. Indeed, measurements made of the width at the half-maximum of the peaks [14] have confirmed that the grain-size values decrease with increasing milling time. After 10 h milling, the grain size was 25 nm and the microstrain was 0.39%. 3.2. Grain-growth behavior
In order to simulate the influence of the high temperature to which the powders would be exposed during consolidation, powders that had been mechanically milled for 10 h were annealed at various temperatures. The grain-size values, measured by the X-ray method, are shown in Fig. 2. It is observed, that the grain size increased slowly at 200 °C, but increased rapidly at 400 °C. Fig. 3 shows the average grain-size of the 10 h-milled
• - - AI2Cu
.)
I 6
I 8
10
Fig. 2. Grain sizes of powders mechanically-milledfor 10h, then isothermally annealed for 1 h to 10 h at various temperatures. powders after annealing or hot pressing and subsequent hydrostatic extrusion at various temperatures. During simple vacuum-annealing, the grain size increased slowly with increasing temperature, whilst hot pressing was found to result in serious grain size coarsening at the same temperature in a relatively short time. The grain size of the extruded materials is much larger than that of the hot-pressed materials, indicating that the shear strain in hot hydrostatic-extrusion accelerates the coarsening of the grain size of nanocrystalline alloy. Fig. 4 shows dark-field transmission electron micrographs of some consolidated materials, all of the samples showing a uniform structure with a fairly narrow grain-size distribution. 3.3. Precipitation o f intermetallic compound
Fig. 5 shows the transmission electron micrographs of some materials consolidated by process (A). In the 250 200
e,,,
150
100
50 0
------.
I 4
Annealing time, h
,. ,
I 2
0
j
0
I
I
I
I
100
200
300
400
500
Temperature, "C 20
40
60
80
100
20, deg
Fig. 1. XRD patterns of powders milled for various times.
Fig. 3. Grain sizes of powder mechanically-milled for 10 h, then consolidated at various temperatures: (1) annealed for 1 h; (2) hot pressed for 0.5 h; (3) hot hydrostatically-extruded from the hotpressed billet.
249
G. Liang et al. /Journal of Materials Processing Technology 58 (1996) 247-250
A - A;2(Oa,Mg,Si,Fe,Mn) • m A12Cu
~o
2.5h
j
6h / Fig. 4. TEM dark-field images of some consolidated materials: (a) cold consolidated; (b) hot pressed at 250 °C and extruded at 250 °C.
~_........_._...._~
2O
rapidly-solidified aluminium alloy, A12Cu and A12 (Cu, Mg, Si, Mn, Fe) intermetallic compounds in the laminated structure of about 43 nm thickness were noted from the T E M structure (Fig. 5(a)). In the mechanically-milled materials, fine dispersoids with sizes of 10-20 nm were observed, as shown in Fig. 5(b), the volume fraction of the dispersoids being smaller than that in the rapidly-solidified materials: this result can be seen also in the X-ray diffraction patterns of the consolidated materials, as shown in Fig. 6. In the rapidly-solidified powder materials, A12Cu and A12 (Cu, Mg, Si, Fe, Mn) phases are present, whilst in the materials consolidated from 10 h-milled powders, no A12 (Cu, Mg, Si, Fe, Mn) phase was detected. Compared with Fig. 1, it is seen that A12 (Cu, Mg, Si, Fe, Mn) phase is formed in the consolidation process. After 10 h milling, the Cu, Mg, Si, Fe, Mn elements were distributed homogeneously in the aluminium matrix, whilst the A12 (Cu, Mg, Si, Fe, Mn) phase could not be formed in the consolidation process because it needs the diffusion of Mg, Si, Fe, Mn elements. Since copper has a larger coefficient of diffusion in an aluminium matrix than Mg and other elements, the A12Cu intermetallic compound is more easily precipitated. Fig. 7(a) shows the microstructure of consolidated 10 h-milled powder alloy after having been heat treated at 400 °C x 3 h. Many A12 (Cu, Mg, Si, Fe, Mn) phases of size of about 3 0 - 5 0 n m have precipitated. EDs analyses of this phase indicates Mg, Si, Fe, Mn and Cu to be present in the following approximate atomic percentages: A166.3 Mg6. 2 Sis. 8 Mn7. 4 Fes.8 Cu7.5, this composition seems to indicate that this phase is proba-
Fig. 5. TEM bright-field images of materials consolidated by process (A): (a) non-mechanicallymilled; (b) milled for 10 h.
40
60
80
100
20,deg
Fig. 6. XRD patterns of some materials consolidated from powders milled for various times by process (A). bly A12 (Cu, Mg, Si, Mn, Fe), where the silicon, magnesium, iron and manganese atoms substitute for the copper atoms in the intermetallic compound. The growth of A12 (Cu, Mg, Si, Mn, Fe) needs the diffusion of Si, Mn, Mg, Fe, so that it slowly coarsens, as shown in Fig. 7(b), the A12Cu phase growing to about 3/~m and swallowing many A12 (Cu, Mg, Si, Mn, Fe) phases. The A12 (Cu, Mg, Si, Fe, Mn) phase is stable and only about 0.3/~m after being exposed to 480 °C for 20 h, and it also inhibits the growth of grain and of the A12Cu phase (see Fig. 7(b)).
4. C o n c l u s i o n s
1. Mechanical milling produces a fine microstructure with nanometer-sized grains and makes the A12Cu phase dissolve in the aluminium matrix, leading to a supersaturated solid-solution. 2. The nanometer-sized grains coarsen slowly at a temperature of lower than 200 °C, the grain-growth rate increasing with increase of temperature. 3. Vacuum hot-pressing and hot hydrostatic-extrusion of nanocrystalline powders result in more serious coarsening of the grain size than simple annealing and exhibit deformation-enhanced grain-growth behavior.
Fig. 7. TEM bright-field images of consolidated 10 h-milled powder materials after having been heat treated at: (a) 400 °C × 3 h; (b) 480 °C x 20 h. (S: AI2Cu; P: AI2 (Cu, Mg, Si, Fe, Mn)).
250
G. Liang et al./Journal of Materials Processing Technology 58 (1996) 247-250
4. D u r i n g the c o n s o l i d a t i o n process, p h a s e s e p a r a tion o f s u p e r s a t u r a t e d s o l i d - s o l u t i o n occurs, a n d A12Cu a n d A12 (Cu, M g , Si, Fe, M n ) phases o f n a n o m e t e r size are p r e c i p i t a t e d . T h e l a t t e r phase has a slower c o a r s e n ing rate t h a n does the f o r m e r phase, a n d can inhibit g r a i n growth.
References [1] M. Van Rooyen, P.F. Colijn, Th.H. de Keijser and E.J. Mittemeijer, J. Mater. Sci., 21 (1986) 2373. [2] N. Kuroishi, Y. Odani and Y. Takeda, MPR, 11 (1985) 642. [3] R.D. Schelleng, J. Met., 41 (1989) 132.
[4] J.S. Benjamin, Metall. Trans., 1 (1970) 2943. [5] R. Sundaresan and F.H. Froes, J. Met., 3 (1987) 22. [6] R. Sundaresan and F.H. Froes, MPR, 4 (1989) 185. [7] A. Layous, S, Nadir and I.J. Lin, PMI, 19(1) (1987) 11. [8] R. Birringer, Mater. Sci. Eng., A l l 7 (1989) 33. [9] F. H. Froes and C. Suryanarayana, JOM, 6 (1989) 12. [10] R. W. Siegel, Mater. Sci. Eng., A168 (1993) 189. [11] J. Eckert, J.C. Holzer and W.L. Johnson, J. Appl. Phys., 73 (1993) 131. [12] S. Kawanishi, K. Isonishi and K. Okazaki, Mater. Trans. JIM, 34 (1993) 49. [13] G.X. Liang, Z.M. Li and E.D. Wang, J. Mater. Sci. Technol., •0(4) (1994) 285. [14] M. Atzmon, K.M, Vnrub and W.L. Johnson, J. Appl. Phys., 58 (1985) 3865.
E LSE VIE R
Journal of Materials Processing Technology 58 (1996) 247-250
Materials Processing Technology
Structural change of rapidly solidified 2024 aluminium alloy powders in mechanical milling and subsequent consolidation process Liang Guoxian *, Li Zhichao, Wang Erde, Wang Zhongren College of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, People's Republic of China Received 18 December 1994
Industrial summary
Rapidly solidified 2024 aluminium powders were mechanically milled, and then consolidated by vacuum hot-pressing and hot hydrostatic-extrusion. The grain-growth and phase-separation behaviors during the consolidation process were studied by X-ray diffraction and TEM observation. The results show that mechanical milling (MM) produces a supersaturated solid solution with a fine microstructure of nanometer-sized grains. In the consolidation process, the grain size increased, and AlzCu and Alz(Cu, Mg, Si, Fe, Mn) intermetallic-phases precipitated. A deformation-enhanced grain growth behavior was observed. Vacuum hot-pressing results in more obvious grain growth than simple annealing does, and hot hydrostatic-extrusion results in more serious grain-size coarsening than hot pressing does. Keywords: Aluminium alloy powders; Mechanical milling; Consolidation process
1. Introduction
Rapid solidification (RS) techniques have been used widely for the development of new alloys having extended solubility and for the refinement of dispersion or inclusion particles [1,2]. However, owing to thermodynamic restrictions, the intermetallic particles may have a large size, and the solubility may not be extendable further [3]. Mechanical alloying (MA) has the unique advantages of extending the solubility and producing fine dispersions and grainsizes with rapid solidification in some alloy systems [4-6]. However, with mechanical alloying, starting from mixed elemental powders, it is difficult to obtain a real homogeneous microstructure [7]. Thus a composite RS + M M technique may overcome the drawbacks of both techniques and enable the obtaining of fully-homogeneous fine microstructures. Recently, nanocrystalline materials have been the subject of intense research, motivated by their unusual physical and mechanical properties [8 10]. Mechanical milling has been used to synthesize nanocrystalline metals and alloys, and the stability of the nanostructure on annealing in some alloy systems has been investigated [11,12]. However, the thermal stability and consolidation of mechanically-milled nanocrystalline alloys have not yet been adequately studied. * Corresponding author. 0924-0136/96/$15.00 © 1996 Elsevier Science S.A. All rights reserved SSD1 0924-0136(95)02121-2
In the present work, the application of the mechanicalmilling process to rapidly solidified 2024 aluminium powders is described, and the grain growth and precipitation behavior of the mechanically-milled metastable powders in the consolidation process is studied.
2. Experimental procedure
2024 aluminium powders with a normal composition of 4.48 Cu-1.45 Mg-0.61 Mn-0.31 Fe-0.15 Si-balance A1 (wt.%) were prepared using the argon gas atomization method. Mechanical milling of powders was performed in an attritor ball-mill under the protection of pure argon gas. The milling vial was made of stainless steel and the balls were made of hardened chrome steel. No process control agent was added. Chemical analysis shows that the Fe content increased by 0.5 wt.% after 10 h of milling. The processed powders were consolidated by two processing routes: (A) cold compacted to a cylindrical billet under a pressure of 1 GPa, canned in a lubricant medium, then pre-heated for 10 min at 450 °C before hot hydrostatic-extrusion as in Ref. [13]; (B); hot pressed under a pressure of 500 MPa at 200-400 °C in a vaccum of better than 4 x 10 2pa for 0.5h, and then hot hydrostatically extruded to rod with an extrusion ratio of 14:1.
248
G. Liang et al. /."Journal of Materials Processing Technology 58 (1996) 247-250
The processed powders and consolidated materials were characterized by X-ray diffraction using a Rigakud/max-rB diffractometer with CuK~ radiation. Microstructural observations were made by transmission electron microscopy with a EM420 microscope. Thin foils for T E M observation were prepared by twin-jet electropolishing with an electrolyte of 1:3 nitric acid/methanol at - 3 0 °C.
250
200
" 150
o ~
200"C
e~
300"C
~
400"C
100
50 o
3. R e s u l t s and d i s c u s s i o n 0
3.1. Mechanical milling
X-ray diffraction patterns of 2024 aluminium powders milled for various times are shown in Fig. 1. In the rapidly-solidified powders, diffraction peaks of A12Cu intermetallic compound were found. With an increase in the milling time, the diffraction peaks of A12Cu phase disappeared gradually, suggesting that this phase was dissolved in the aluminium matrix, and a supersaturated solid-solution formed. Increasing the milling time also produced broadening of the aluminium peaks and a decrease in the peak height, indicating a continuous decrease in the grain size. Indeed, measurements made of the width at the half-maximum of the peaks [14] have confirmed that the grain-size values decrease with increasing milling time. After 10 h milling, the grain size was 25 nm and the microstrain was 0.39%. 3.2. Grain-growth behavior
In order to simulate the influence of the high temperature to which the powders would be exposed during consolidation, powders that had been mechanically milled for 10 h were annealed at various temperatures. The grain-size values, measured by the X-ray method, are shown in Fig. 2. It is observed, that the grain size increased slowly at 200 °C, but increased rapidly at 400 °C. Fig. 3 shows the average grain-size of the 10 h-milled
• - - AI2Cu
.)
I 6
I 8
10
Fig. 2. Grain sizes of powders mechanically-milledfor 10h, then isothermally annealed for 1 h to 10 h at various temperatures. powders after annealing or hot pressing and subsequent hydrostatic extrusion at various temperatures. During simple vacuum-annealing, the grain size increased slowly with increasing temperature, whilst hot pressing was found to result in serious grain size coarsening at the same temperature in a relatively short time. The grain size of the extruded materials is much larger than that of the hot-pressed materials, indicating that the shear strain in hot hydrostatic-extrusion accelerates the coarsening of the grain size of nanocrystalline alloy. Fig. 4 shows dark-field transmission electron micrographs of some consolidated materials, all of the samples showing a uniform structure with a fairly narrow grain-size distribution. 3.3. Precipitation o f intermetallic compound
Fig. 5 shows the transmission electron micrographs of some materials consolidated by process (A). In the 250 200
e,,,
150
100
50 0
------.
I 4
Annealing time, h
,. ,
I 2
0
j
0
I
I
I
I
100
200
300
400
500
Temperature, "C 20
40
60
80
100
20, deg
Fig. 1. XRD patterns of powders milled for various times.
Fig. 3. Grain sizes of powder mechanically-milled for 10 h, then consolidated at various temperatures: (1) annealed for 1 h; (2) hot pressed for 0.5 h; (3) hot hydrostatically-extruded from the hotpressed billet.
249
G. Liang et al. /Journal of Materials Processing Technology 58 (1996) 247-250
A - A;2(Oa,Mg,Si,Fe,Mn) • m A12Cu
~o
2.5h
j
6h / Fig. 4. TEM dark-field images of some consolidated materials: (a) cold consolidated; (b) hot pressed at 250 °C and extruded at 250 °C.
~_........_._...._~
2O
rapidly-solidified aluminium alloy, A12Cu and A12 (Cu, Mg, Si, Mn, Fe) intermetallic compounds in the laminated structure of about 43 nm thickness were noted from the T E M structure (Fig. 5(a)). In the mechanically-milled materials, fine dispersoids with sizes of 10-20 nm were observed, as shown in Fig. 5(b), the volume fraction of the dispersoids being smaller than that in the rapidly-solidified materials: this result can be seen also in the X-ray diffraction patterns of the consolidated materials, as shown in Fig. 6. In the rapidly-solidified powder materials, A12Cu and A12 (Cu, Mg, Si, Fe, Mn) phases are present, whilst in the materials consolidated from 10 h-milled powders, no A12 (Cu, Mg, Si, Fe, Mn) phase was detected. Compared with Fig. 1, it is seen that A12 (Cu, Mg, Si, Fe, Mn) phase is formed in the consolidation process. After 10 h milling, the Cu, Mg, Si, Fe, Mn elements were distributed homogeneously in the aluminium matrix, whilst the A12 (Cu, Mg, Si, Fe, Mn) phase could not be formed in the consolidation process because it needs the diffusion of Mg, Si, Fe, Mn elements. Since copper has a larger coefficient of diffusion in an aluminium matrix than Mg and other elements, the A12Cu intermetallic compound is more easily precipitated. Fig. 7(a) shows the microstructure of consolidated 10 h-milled powder alloy after having been heat treated at 400 °C x 3 h. Many A12 (Cu, Mg, Si, Fe, Mn) phases of size of about 3 0 - 5 0 n m have precipitated. EDs analyses of this phase indicates Mg, Si, Fe, Mn and Cu to be present in the following approximate atomic percentages: A166.3 Mg6. 2 Sis. 8 Mn7. 4 Fes.8 Cu7.5, this composition seems to indicate that this phase is proba-
Fig. 5. TEM bright-field images of materials consolidated by process (A): (a) non-mechanicallymilled; (b) milled for 10 h.
40
60
80
100
20,deg
Fig. 6. XRD patterns of some materials consolidated from powders milled for various times by process (A). bly A12 (Cu, Mg, Si, Mn, Fe), where the silicon, magnesium, iron and manganese atoms substitute for the copper atoms in the intermetallic compound. The growth of A12 (Cu, Mg, Si, Mn, Fe) needs the diffusion of Si, Mn, Mg, Fe, so that it slowly coarsens, as shown in Fig. 7(b), the A12Cu phase growing to about 3/~m and swallowing many A12 (Cu, Mg, Si, Mn, Fe) phases. The A12 (Cu, Mg, Si, Fe, Mn) phase is stable and only about 0.3/~m after being exposed to 480 °C for 20 h, and it also inhibits the growth of grain and of the A12Cu phase (see Fig. 7(b)).
4. C o n c l u s i o n s
1. Mechanical milling produces a fine microstructure with nanometer-sized grains and makes the A12Cu phase dissolve in the aluminium matrix, leading to a supersaturated solid-solution. 2. The nanometer-sized grains coarsen slowly at a temperature of lower than 200 °C, the grain-growth rate increasing with increase of temperature. 3. Vacuum hot-pressing and hot hydrostatic-extrusion of nanocrystalline powders result in more serious coarsening of the grain size than simple annealing and exhibit deformation-enhanced grain-growth behavior.
Fig. 7. TEM bright-field images of consolidated 10 h-milled powder materials after having been heat treated at: (a) 400 °C × 3 h; (b) 480 °C x 20 h. (S: AI2Cu; P: AI2 (Cu, Mg, Si, Fe, Mn)).
250
G. Liang et al./Journal of Materials Processing Technology 58 (1996) 247-250
4. D u r i n g the c o n s o l i d a t i o n process, p h a s e s e p a r a tion o f s u p e r s a t u r a t e d s o l i d - s o l u t i o n occurs, a n d A12Cu a n d A12 (Cu, M g , Si, Fe, M n ) phases o f n a n o m e t e r size are p r e c i p i t a t e d . T h e l a t t e r phase has a slower c o a r s e n ing rate t h a n does the f o r m e r phase, a n d can inhibit g r a i n growth.
References [1] M. Van Rooyen, P.F. Colijn, Th.H. de Keijser and E.J. Mittemeijer, J. Mater. Sci., 21 (1986) 2373. [2] N. Kuroishi, Y. Odani and Y. Takeda, MPR, 11 (1985) 642. [3] R.D. Schelleng, J. Met., 41 (1989) 132.
[4] J.S. Benjamin, Metall. Trans., 1 (1970) 2943. [5] R. Sundaresan and F.H. Froes, J. Met., 3 (1987) 22. [6] R. Sundaresan and F.H. Froes, MPR, 4 (1989) 185. [7] A. Layous, S, Nadir and I.J. Lin, PMI, 19(1) (1987) 11. [8] R. Birringer, Mater. Sci. Eng., A l l 7 (1989) 33. [9] F. H. Froes and C. Suryanarayana, JOM, 6 (1989) 12. [10] R. W. Siegel, Mater. Sci. Eng., A168 (1993) 189. [11] J. Eckert, J.C. Holzer and W.L. Johnson, J. Appl. Phys., 73 (1993) 131. [12] S. Kawanishi, K. Isonishi and K. Okazaki, Mater. Trans. JIM, 34 (1993) 49. [13] G.X. Liang, Z.M. Li and E.D. Wang, J. Mater. Sci. Technol., •0(4) (1994) 285. [14] M. Atzmon, K.M, Vnrub and W.L. Johnson, J. Appl. Phys., 58 (1985) 3865.