- Email: [email protected]
Modeling the formation of twins and stacking faults in the ag-cu system
Acta mater. 49 (2001) 1537–1540 www.elsevier.com/locate/actamat
MODELING THE FORMATION OF TWINS AND STACKING FAULTS IN THE Ag-Cu SYSTEM K. HAN1†, J. P. HIRTH2 and J. D. EMBURY3 1
National High Magnetic Field Laboratory/FSU, 1800 E Paul Dirac Dr. Tallahassee, Florida, FL 32310, USA, 2Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA and 3Department of Materials Science and Engineering, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4L7, Canada ( Received 2 June 2000; received in revised form 31 January 2001; accepted 31 January 2001 )
Abstract—In forming new grains, both in solid state reactions and in solidification, the formation of twins or stacking-faults is of considerable interest. In the present work, a terrace/ledge nucleation model is used to explore the relationship between the degree of undercooling and the probability of forming twins or stacking-faults, and compared with experimental results for rapid solidification of several alloys in the twophase Ag-Cu system. The model is consistent with the experimental results and indicates that at an undercooling between 37 K and 55 K, twins or stacking-faults form first in the Ag-rich solid solution. Larger undercooling brings about the formation of twins or stacking-faults in the Cu-rich solid solution. The degree of undercooling required for defect formation can be related to the stacking fault or twin boundary energies of the materials. 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Casting; Rapid solidification; Copper; Phase transformations; Twin
1. INTRODUCTION
The formation of twins or stacking-fault defects is of interest in forming new grains, both in solid state reactions [1, 2] and in solidification from the liquid. The Taylor macro-wire technique entails melting materials in a glass tube and drawing the composite to a small diamter [3, 4] with solidification occurring during the drawing process. In an alloy of Cu-Ag, wires with diameters of 10–1000 µm were produced with corresponding cooling rates of 苲102–105 K/s [5] and large undercoolings. The decrease of the wire diameters resulted in a higher cooling rate and refinement of the microstructure, as shown in Fig. 1 (see reference [5] for detailed experimental technique descriptions). In the largest diameter wire with the lowest cooling rate, proeutectic Cu and eutectic Cu/Ag were observed, Fig. 1(a). For the finest wire with the highest cooling rate, fine grains of a supersaturated Cu-Ag solid solution were formed as shown in Fig. 1(c), while at intermediate cooling rates, a fine lamellar product with successive layers in twin orientation was formed, as shown in Fig. 1(b). This figure also shows the presence of extensive stacking faults
† To whom all correspondence should be addressed. Tel: +1-850-644-6746; fax: +1-850-644-0867. E-mail address: [email protected] (K. Han)
in addition to the twins. Twins have also been observed in pure Ag and Cu rapidly cooled from liquid [6] At large undercoolings, the growth occurs by a ledge mechanism, with monatomic ledges traveling over low index terrace planes [7]. In the absence of defects, the mechanism of ledge formation is the successive nucleation of disc-shaped embryos. The structure of the g⬘ phase in the Cu-Ag case is an example of this repeated nucleation mechanism. In the present paper, the process of repeated nucleation is analyzed. Our work states that the probability of the nucleation of embryos in the faulted orientation increases with increasing undercooling. There is also an example of this trend in the case of nucleation of silver particles from the vapor phase [8]. The current work predicts a threshold supercooling below which the formation of stacking faults by a mechanism of nucleation cannot occur.
2. MODELS
In the nucleation of crystals of pure Ag or pure Cu or of a Cu-Ag solid solution, i.e. the g⬘ phase, from a supersaturated Cu-Ag liquid, we consider growth to occur by the repeated nucleation of ledges on {111} terraces (Fig. 2). There are two possibilities for a new layer to nucleate on a (111) plane. (a) The atoms can
1359-6454/01/$20.00 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 1 ) 0 0 0 5 7 - X
1538
HAN et al.: FORMATION OF TWINS AND STACKING FAULTS
Fig. 2. Schematic diagram of the nucleation of a new layer of atoms on a terrace.
⌬Gb = 2prhg/2⫺pr 2h⌬Gv/2 + pr 2gt/2
(2)
where subscripts a and b indicate case (a) and (b); gt is the twin fault energy and g is the interface energy between solid and liquid for the disc periphery; equivalently, 2prhg is the total interfacial energy of the periphery of the disc. This g value undoubtedly differs from the value for an extended planar interface and includes edge energies. In the absence of measured values, however, we assume the bulk value for g as a typical example. Here, ⌬Gv is the bulk free energy per unit volume driving the nucleation ⌬Gv = ⌬Hv
(Tm⫺T) ⌬Hm ⌬T = Tm ⍀ Tm
(3)
⌬Hm is the enthalpy change per unit ⍀ volume; ⌬Hm is the enthalpy of fusion; ⍀ is the molar volume; Tm is the melting point and ⌬T = Tm⫺T. In the most likely case, the nucleus is a disc of monatomic height, for which gt may differ from the bulk twin boundary energy and similarly for the successive layer formation creating adjacent stacking fault. Again, in the absence of atomic calculations, we assume bulk values for gt, a reasonably approximation in a hard ball model [10, p. 939]. For the monatomic height embryo case (a), h is equal to the lattice spacing of {111} planes; d(111). With these parameters, one can determine the critical size r* by setting the differential of ⌬Gv with respect to r, equal to zero. where ⌬Hv =
Fig. 1. Electron microscopic images of Taylor wires of different diameters formed with various cooling rates. The detailed experimental procedures were reported in reference [5]. (a) A Scanning Electron Microscopy image of a 100 µm diameter Taylor wire showing both the proeutectic Cu and eutectic (Cu and Ag) regions. (b) A Transmission Electron Microscopy (TEM) bright field image of an eutectic region in a 30 µm diameter Taylor wire showing that both the Cu and Ag phases exist in two variants interrelated by a twin relationship. The double white or dark arrows indicate the defects in Ag crystals. The g vector used for dark field images is indicated by a white arrow. (c) A TEM dark field image of a 10 µm diameter Taylor wire showing a nano-crystalline solid solution.
r∗a = g/(⌬Gv)
(4)
Similarly, for case b; deposit in the normal fcc ABCABCAB sequence for the close-packed planes. (b) The atoms can deposit in a twinned ABCABACB sequence, where the bold letter B indicates the fault position. Two successive twins form an intrinsic stacking fault. The nucleation rates for those two cases are different because of the creation of a twin or a stacking fault in case (b). The total free energies of a half disc-shaped nucleus with diameter r and height h in the two cases are, ⌬Ga = 2prhg/2⫺pr2h⌬Gv/2
(1)
r∗b = hg/(h⌬Gv⫺gsf)
(5)
Equation (5) can be rewritten as: r∗b = g/(⌬Gv⫺gsf/h). Because both gsf and h have positive values, the 1/(⌬Gv) is smaller than 1/(⌬Gv⫺gsf/h) for a given ⌬Gv. Evidently, for a given value of g and ⌬Gv, r∗a ⬍r∗b , and correspondingly, from equations (1) and (2), there is a larger free energy barrier, ⌬G∗b for nucleation in the twinning case. Combining equations (3) and (4) or (5), we find
HAN et al.: FORMATION OF TWINS AND STACKING FAULTS
Tm ⌬Hv⌬T
(6)
hg [h⌬Hv⌬T/Tm]⫺gsf
(7)
r∗a = g r∗b = ∗ a
∗ b
The variations of r and r of Ag with respect to temperature are shown in Fig. 3. The following data were used: hAg = 0.236 nm (the lattice spacing of Ag111) for Ag, hCu = 0.209 nm (the lattice spacing of Cu111) for Cu, and hg⬘ = 0.222 nm (the lattice spacing of g⬘111) for the g⬘ solid solution. ⌬Hv = 1.95×109 Jm⫺3 [9] and Tm = 1052 K. The stacking fault energies used were [10] 0.016 J/m2 for Ag, 0.045 J/m2 for Cu, and the assumed mean value of 0.031 J/m2 for the supersaturated solid solution. Similarly, the twin fault energies, gt were 0.008 J/m2 for Ag, 0.024 J/m2 for Cu, and 0.016 J/m2 for the solid solution. Figure 3 shows that above a critical temperature Tc, r∗b ⬍0 and, consequently, the nucleation of a twin is impossible. At a temperature close to Tc but smaller than Tc, r∗b r∗a . In order to have a positive radius of nuclei for nucleation of a stacking fault or a twin, the system
1539
has to meet a requirement that can be expressed quantitatively as: ⌬Gv>gsf /h
(8)
Thus, there is a critical undercooling, below which the nucleation of a twin or a stacking fault is impossible. The critical undercooling corresponds to a critical temperature for the nucleation of twins. From a combination of equations (3) and (8), one can calculate the value of Tc for various cases. The results indicate that successively larger critical undercoolings are required for Cu, the g⬘ solid solution and Ag, respectively. The nucleation rates of the unfaulted and faulted structures are Ia = I0exp(⫺⌬G∗a /RT)
(9)
Ib = I0exp(⫺⌬G∗b /RT)
(10)
If the pre-exponential factors I0 are assumed to be similar, Ib /Ia = exp[⫺(⌬G∗b ⫺⌬G∗a )/RT]
(11)
The logarithm of equation (11) is: ln(Ib /Ia) = ⫺(⌬G∗b ⫺⌬G∗a )/RT
(12)
and the free energy difference is ⌬G∗b ⫺⌬G∗a =
冋
pg 2gsf 1 2⌬Gv ⌬Gv⫺gsf /h
册
(13)
For the case of Ag, ⫺(⌬G∗b ⫺⌬G∗a )/T or R ln(Ib/Ia) can be obtained for nucleation of stacking fault and twins and is shown in Fig. 4 as a function of temperature. Evidently, the ratio Ib/Ia, which represents the relative probability of nucleation of faults increases with increased undercooling.
Fig. 3. Calculated critical size r* of nucleation of faulted and unfaulted embryos versus temperature (K). (a) Formation of stacking fault; the dashed and solid lines are for a nucleation of a monatomic layer with unfaulted or faulted atoms. (b) Formation of twin; the dashed and solid lines are for a nucleation of a layer with twinned or untwinned atoms.
Fig. 4. Calculated ⫺(⌬G∗b ⫺⌬G∗a )/T or Rln(Ib/Ia) versus temperature (K). The dashed and solid lines are from the values of ⫺(⌬G∗b ⫺⌬Ga)∗/T pertinent to the nucleation of stacking fault and twins, respectively.
1540
HAN et al.: FORMATION OF TWINS AND STACKING FAULTS
Table 1. Calculated critical and undercooling temperatures (K)a Tm (K) Stacking fault Ag Cu g⬘ (solid solution) Twin Ag Cu g⬘ (solid solution) a
Tc (K)
⌬Tc (K)
1052 1052 1052
1015 935 981
37 117 71
1052 1052 1052
1034 997 1011
18 55 42
Tc = Tm⫺Tmgsf/(h⌬Hv) was used to calculate Tc.
Cooperative growth of Cu and Ag phases with certain twins or (stacking) faults may occur in addition to nucleation of such interfaces. Equation 14 describes the minimum possible spacing l of the CuAg composite [11]: 2TmgCu/Ag l= ⌬Hv⌬T
(14)
where gCu/Ag is the interface energy, which is assumed to be 0.5 J/m2. At the undercooling temperatures calculated for formation of faults of twins in Ag, as shown in Table 1, the minimum spacings are estimated to be 17 and 9 nm, respectively. Therefore, in order to have a cooperative growth, a large undercooling is required. The observed spacing is about 50 nm, larger than the minimum spacing as must be the case for consistency.
large undercoolings, the fine grains containing supersaturated solid solution with stacking fault and twin boundaries are generated. In conventional casting, which entails growth at small undercoolings, stacking faults or twins are rarely, if ever, observed except during subsequent deformation. The model also shows that nucleation of twins or stacking faults requires nuclei of larger critical sizes and higher free energy barrier than the normal structure does. Therefore, the volume fraction of normal nuclei is larger than that of twins or stacking faults. Hence, we have demonstrated, consistent with experimental observation, that in the case of crystal growth by repeated nucleation of layers by the terrace/ledge model there is a critical undercooling below which nucleation of a faulted structure is impossible. Acknowledgements—Financial support from the MST, Los Alamos National Laboratory, US Department of Energy and the US National Science Foundation is gratefully acknowledged. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
3. DISCUSSION AND SUMMARY
The results obtained for the solidification of Taylor wires are consistent with the proposed model. At intermediate undercoolings, twin related lamellae of Cu-rich and Ag-rich solid solution are formed. At
9. 10. 11.
Dash, S. and Brown, N., Acta metall., 1963, 13, 1067. Gleiter, N., Acta metall., 1969, 17, 1421. Taylor, G. F., Phys. Rev., 1924, 23, 655. Taylor, G. F., US Patent 1, 1931, 793, 529. Han, K., Embury, J. D., Petrovic, J. J. and Weatherly, G. C., Acta mater., 1998, 46(13), 4691. Stroering, R. and Conrad, H., Acta metall., 1969, 17, 933. van der Eerden J. P., Hurle D. J., Handbook of Crystal Growth, vol. 1, North Holland, Amsterdam, 1993, p. 307. Pilkington, G. and Hirth, J. P., Surface Science, 1972, 29, 363. Beck, D. G., Copley, S. M. and Bass, M., Metallurgical Transaction, 1879, 1982, 13A. Hirth, J. P. and Lothe, L., Theory of Dislocations, 2nd ed. Krieger, Melbourne, FL, 1992. Porter, D. A. and Eastering, K. E., Phase Transformations in Metals and Alloys, Chapman and Hall, 1981, p. 224.
MODELING THE FORMATION OF TWINS AND STACKING FAULTS IN THE Ag-Cu SYSTEM K. HAN1†, J. P. HIRTH2 and J. D. EMBURY3 1
National High Magnetic Field Laboratory/FSU, 1800 E Paul Dirac Dr. Tallahassee, Florida, FL 32310, USA, 2Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA and 3Department of Materials Science and Engineering, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4L7, Canada ( Received 2 June 2000; received in revised form 31 January 2001; accepted 31 January 2001 )
Abstract—In forming new grains, both in solid state reactions and in solidification, the formation of twins or stacking-faults is of considerable interest. In the present work, a terrace/ledge nucleation model is used to explore the relationship between the degree of undercooling and the probability of forming twins or stacking-faults, and compared with experimental results for rapid solidification of several alloys in the twophase Ag-Cu system. The model is consistent with the experimental results and indicates that at an undercooling between 37 K and 55 K, twins or stacking-faults form first in the Ag-rich solid solution. Larger undercooling brings about the formation of twins or stacking-faults in the Cu-rich solid solution. The degree of undercooling required for defect formation can be related to the stacking fault or twin boundary energies of the materials. 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Casting; Rapid solidification; Copper; Phase transformations; Twin
1. INTRODUCTION
The formation of twins or stacking-fault defects is of interest in forming new grains, both in solid state reactions [1, 2] and in solidification from the liquid. The Taylor macro-wire technique entails melting materials in a glass tube and drawing the composite to a small diamter [3, 4] with solidification occurring during the drawing process. In an alloy of Cu-Ag, wires with diameters of 10–1000 µm were produced with corresponding cooling rates of 苲102–105 K/s [5] and large undercoolings. The decrease of the wire diameters resulted in a higher cooling rate and refinement of the microstructure, as shown in Fig. 1 (see reference [5] for detailed experimental technique descriptions). In the largest diameter wire with the lowest cooling rate, proeutectic Cu and eutectic Cu/Ag were observed, Fig. 1(a). For the finest wire with the highest cooling rate, fine grains of a supersaturated Cu-Ag solid solution were formed as shown in Fig. 1(c), while at intermediate cooling rates, a fine lamellar product with successive layers in twin orientation was formed, as shown in Fig. 1(b). This figure also shows the presence of extensive stacking faults
† To whom all correspondence should be addressed. Tel: +1-850-644-6746; fax: +1-850-644-0867. E-mail address: [email protected] (K. Han)
in addition to the twins. Twins have also been observed in pure Ag and Cu rapidly cooled from liquid [6] At large undercoolings, the growth occurs by a ledge mechanism, with monatomic ledges traveling over low index terrace planes [7]. In the absence of defects, the mechanism of ledge formation is the successive nucleation of disc-shaped embryos. The structure of the g⬘ phase in the Cu-Ag case is an example of this repeated nucleation mechanism. In the present paper, the process of repeated nucleation is analyzed. Our work states that the probability of the nucleation of embryos in the faulted orientation increases with increasing undercooling. There is also an example of this trend in the case of nucleation of silver particles from the vapor phase [8]. The current work predicts a threshold supercooling below which the formation of stacking faults by a mechanism of nucleation cannot occur.
2. MODELS
In the nucleation of crystals of pure Ag or pure Cu or of a Cu-Ag solid solution, i.e. the g⬘ phase, from a supersaturated Cu-Ag liquid, we consider growth to occur by the repeated nucleation of ledges on {111} terraces (Fig. 2). There are two possibilities for a new layer to nucleate on a (111) plane. (a) The atoms can
1359-6454/01/$20.00 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 1 ) 0 0 0 5 7 - X
1538
HAN et al.: FORMATION OF TWINS AND STACKING FAULTS
Fig. 2. Schematic diagram of the nucleation of a new layer of atoms on a terrace.
⌬Gb = 2prhg/2⫺pr 2h⌬Gv/2 + pr 2gt/2
(2)
where subscripts a and b indicate case (a) and (b); gt is the twin fault energy and g is the interface energy between solid and liquid for the disc periphery; equivalently, 2prhg is the total interfacial energy of the periphery of the disc. This g value undoubtedly differs from the value for an extended planar interface and includes edge energies. In the absence of measured values, however, we assume the bulk value for g as a typical example. Here, ⌬Gv is the bulk free energy per unit volume driving the nucleation ⌬Gv = ⌬Hv
(Tm⫺T) ⌬Hm ⌬T = Tm ⍀ Tm
(3)
⌬Hm is the enthalpy change per unit ⍀ volume; ⌬Hm is the enthalpy of fusion; ⍀ is the molar volume; Tm is the melting point and ⌬T = Tm⫺T. In the most likely case, the nucleus is a disc of monatomic height, for which gt may differ from the bulk twin boundary energy and similarly for the successive layer formation creating adjacent stacking fault. Again, in the absence of atomic calculations, we assume bulk values for gt, a reasonably approximation in a hard ball model [10, p. 939]. For the monatomic height embryo case (a), h is equal to the lattice spacing of {111} planes; d(111). With these parameters, one can determine the critical size r* by setting the differential of ⌬Gv with respect to r, equal to zero. where ⌬Hv =
Fig. 1. Electron microscopic images of Taylor wires of different diameters formed with various cooling rates. The detailed experimental procedures were reported in reference [5]. (a) A Scanning Electron Microscopy image of a 100 µm diameter Taylor wire showing both the proeutectic Cu and eutectic (Cu and Ag) regions. (b) A Transmission Electron Microscopy (TEM) bright field image of an eutectic region in a 30 µm diameter Taylor wire showing that both the Cu and Ag phases exist in two variants interrelated by a twin relationship. The double white or dark arrows indicate the defects in Ag crystals. The g vector used for dark field images is indicated by a white arrow. (c) A TEM dark field image of a 10 µm diameter Taylor wire showing a nano-crystalline solid solution.
r∗a = g/(⌬Gv)
(4)
Similarly, for case b; deposit in the normal fcc ABCABCAB sequence for the close-packed planes. (b) The atoms can deposit in a twinned ABCABACB sequence, where the bold letter B indicates the fault position. Two successive twins form an intrinsic stacking fault. The nucleation rates for those two cases are different because of the creation of a twin or a stacking fault in case (b). The total free energies of a half disc-shaped nucleus with diameter r and height h in the two cases are, ⌬Ga = 2prhg/2⫺pr2h⌬Gv/2
(1)
r∗b = hg/(h⌬Gv⫺gsf)
(5)
Equation (5) can be rewritten as: r∗b = g/(⌬Gv⫺gsf/h). Because both gsf and h have positive values, the 1/(⌬Gv) is smaller than 1/(⌬Gv⫺gsf/h) for a given ⌬Gv. Evidently, for a given value of g and ⌬Gv, r∗a ⬍r∗b , and correspondingly, from equations (1) and (2), there is a larger free energy barrier, ⌬G∗b for nucleation in the twinning case. Combining equations (3) and (4) or (5), we find
HAN et al.: FORMATION OF TWINS AND STACKING FAULTS
Tm ⌬Hv⌬T
(6)
hg [h⌬Hv⌬T/Tm]⫺gsf
(7)
r∗a = g r∗b = ∗ a
∗ b
The variations of r and r of Ag with respect to temperature are shown in Fig. 3. The following data were used: hAg = 0.236 nm (the lattice spacing of Ag111) for Ag, hCu = 0.209 nm (the lattice spacing of Cu111) for Cu, and hg⬘ = 0.222 nm (the lattice spacing of g⬘111) for the g⬘ solid solution. ⌬Hv = 1.95×109 Jm⫺3 [9] and Tm = 1052 K. The stacking fault energies used were [10] 0.016 J/m2 for Ag, 0.045 J/m2 for Cu, and the assumed mean value of 0.031 J/m2 for the supersaturated solid solution. Similarly, the twin fault energies, gt were 0.008 J/m2 for Ag, 0.024 J/m2 for Cu, and 0.016 J/m2 for the solid solution. Figure 3 shows that above a critical temperature Tc, r∗b ⬍0 and, consequently, the nucleation of a twin is impossible. At a temperature close to Tc but smaller than Tc, r∗b r∗a . In order to have a positive radius of nuclei for nucleation of a stacking fault or a twin, the system
1539
has to meet a requirement that can be expressed quantitatively as: ⌬Gv>gsf /h
(8)
Thus, there is a critical undercooling, below which the nucleation of a twin or a stacking fault is impossible. The critical undercooling corresponds to a critical temperature for the nucleation of twins. From a combination of equations (3) and (8), one can calculate the value of Tc for various cases. The results indicate that successively larger critical undercoolings are required for Cu, the g⬘ solid solution and Ag, respectively. The nucleation rates of the unfaulted and faulted structures are Ia = I0exp(⫺⌬G∗a /RT)
(9)
Ib = I0exp(⫺⌬G∗b /RT)
(10)
If the pre-exponential factors I0 are assumed to be similar, Ib /Ia = exp[⫺(⌬G∗b ⫺⌬G∗a )/RT]
(11)
The logarithm of equation (11) is: ln(Ib /Ia) = ⫺(⌬G∗b ⫺⌬G∗a )/RT
(12)
and the free energy difference is ⌬G∗b ⫺⌬G∗a =
冋
pg 2gsf 1 2⌬Gv ⌬Gv⫺gsf /h
册
(13)
For the case of Ag, ⫺(⌬G∗b ⫺⌬G∗a )/T or R ln(Ib/Ia) can be obtained for nucleation of stacking fault and twins and is shown in Fig. 4 as a function of temperature. Evidently, the ratio Ib/Ia, which represents the relative probability of nucleation of faults increases with increased undercooling.
Fig. 3. Calculated critical size r* of nucleation of faulted and unfaulted embryos versus temperature (K). (a) Formation of stacking fault; the dashed and solid lines are for a nucleation of a monatomic layer with unfaulted or faulted atoms. (b) Formation of twin; the dashed and solid lines are for a nucleation of a layer with twinned or untwinned atoms.
Fig. 4. Calculated ⫺(⌬G∗b ⫺⌬G∗a )/T or Rln(Ib/Ia) versus temperature (K). The dashed and solid lines are from the values of ⫺(⌬G∗b ⫺⌬Ga)∗/T pertinent to the nucleation of stacking fault and twins, respectively.
1540
HAN et al.: FORMATION OF TWINS AND STACKING FAULTS
Table 1. Calculated critical and undercooling temperatures (K)a Tm (K) Stacking fault Ag Cu g⬘ (solid solution) Twin Ag Cu g⬘ (solid solution) a
Tc (K)
⌬Tc (K)
1052 1052 1052
1015 935 981
37 117 71
1052 1052 1052
1034 997 1011
18 55 42
Tc = Tm⫺Tmgsf/(h⌬Hv) was used to calculate Tc.
Cooperative growth of Cu and Ag phases with certain twins or (stacking) faults may occur in addition to nucleation of such interfaces. Equation 14 describes the minimum possible spacing l of the CuAg composite [11]: 2TmgCu/Ag l= ⌬Hv⌬T
(14)
where gCu/Ag is the interface energy, which is assumed to be 0.5 J/m2. At the undercooling temperatures calculated for formation of faults of twins in Ag, as shown in Table 1, the minimum spacings are estimated to be 17 and 9 nm, respectively. Therefore, in order to have a cooperative growth, a large undercooling is required. The observed spacing is about 50 nm, larger than the minimum spacing as must be the case for consistency.
large undercoolings, the fine grains containing supersaturated solid solution with stacking fault and twin boundaries are generated. In conventional casting, which entails growth at small undercoolings, stacking faults or twins are rarely, if ever, observed except during subsequent deformation. The model also shows that nucleation of twins or stacking faults requires nuclei of larger critical sizes and higher free energy barrier than the normal structure does. Therefore, the volume fraction of normal nuclei is larger than that of twins or stacking faults. Hence, we have demonstrated, consistent with experimental observation, that in the case of crystal growth by repeated nucleation of layers by the terrace/ledge model there is a critical undercooling below which nucleation of a faulted structure is impossible. Acknowledgements—Financial support from the MST, Los Alamos National Laboratory, US Department of Energy and the US National Science Foundation is gratefully acknowledged. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
3. DISCUSSION AND SUMMARY
The results obtained for the solidification of Taylor wires are consistent with the proposed model. At intermediate undercoolings, twin related lamellae of Cu-rich and Ag-rich solid solution are formed. At
9. 10. 11.
Dash, S. and Brown, N., Acta metall., 1963, 13, 1067. Gleiter, N., Acta metall., 1969, 17, 1421. Taylor, G. F., Phys. Rev., 1924, 23, 655. Taylor, G. F., US Patent 1, 1931, 793, 529. Han, K., Embury, J. D., Petrovic, J. J. and Weatherly, G. C., Acta mater., 1998, 46(13), 4691. Stroering, R. and Conrad, H., Acta metall., 1969, 17, 933. van der Eerden J. P., Hurle D. J., Handbook of Crystal Growth, vol. 1, North Holland, Amsterdam, 1993, p. 307. Pilkington, G. and Hirth, J. P., Surface Science, 1972, 29, 363. Beck, D. G., Copley, S. M. and Bass, M., Metallurgical Transaction, 1879, 1982, 13A. Hirth, J. P. and Lothe, L., Theory of Dislocations, 2nd ed. Krieger, Melbourne, FL, 1992. Porter, D. A. and Eastering, K. E., Phase Transformations in Metals and Alloys, Chapman and Hall, 1981, p. 224.