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The relationship between effective entropy change and volume fraction of the eutectic phases in eutectic microstructures
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Materials Letters 62 (2008) 984 – 987 www.elsevier.com/locate/matlet
The relationship between effective entropy change and volume fraction of the eutectic phases in eutectic microstructures Guanghui Meng ⁎, Xin Lin, Weidong Huang State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi'an 710072, People's Republic of China Received 28 January 2007; accepted 15 July 2007 Available online 20 July 2007
Abstract The characterization of eutectic microstructures is related to the effective entropy change and the volume fraction of eutectic phases. For eutectic growth, both parameters can be inter-associated as long as solidification occurs at small Peclet numbers. There is no determinate relationship between the volume fraction and the effective entropy change of either eutectic phases, and the volume fraction is only related to the ratio of the effective entropy change of the two phases. By comparing with well-known eutectic systems, the results indicate that the classification depends on rather the volume fraction than the effective entropy change. In addition, there exists a certain relationship between the eutectic arrangement and the symmetric characteristic of phase diagram. © 2007 Elsevier B.V. All rights reserved. Keywords: Solidification; Eutectic; Microstructures
1. Introduction Eutectics can exhibit a wide variety of geometrical arrangements. Many attempts have been made to characterize eutectic systems in the past. Scheil [1] proposed a eutectic structure will be regular, rod like or lamellar if the eutectic phases are present in nearly equal proportions, and the structure will be anomalous when the volume fraction differ widely or if the minor phase exhibit a remarkable anisotropy. Following Scheil's work, a number of classification treatments have emerged, thereinto a few investigations are more noteworthy [2–5]. The Hunt and Jackson's classification of binary eutectics microstructures [2] is based on the Jackson's theory [6] on the classification between non-faceted and faceted solid-liquid interfaces. However, the Jackson factor is derived for an interface at equilibrium, and this theory is incomplete [7]. In addition, eutectic phases grow from melt at a much lower temperature than the melting point of the pure component, and their entropy value may different from that of the pure component due to temperature and composition effects. A
⁎ Corresponding author. Tel.: +86 29 88494001; fax: +86 29 88494001. E-mail address: [email protected] (G. Meng). 0167-577X/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2007.07.026
more appropriate expression for an alloy is proposed by Taylor et al. [3,4]. They classified the eutectic systems based on the interface roughness parameter. The final equation for the interface free energy is similar to Jackson's original solution, except that the solution entropy replaces the fusion entropy. In the analysis of Croker et al. [5], according to solution entropy of solid eutectic phases in the liquid, volume fraction of phases, and growth rate, eutectic microstructures are broadly divided into normal and anomalous groups, and a solution entropy above 23 J/mol·K is used to indicate typical faceting behavior and distinguish two groups. Recently, the eutectic microstructures were characterized by Ramachandrarao and Dubey again [8]. In their work the solution entropy and the volume fraction of the two phases are believed to be inter-related. To date, as a rule of thumb, one can suppose that when the volume fraction of one phase is between zero and 0.28, the eutectic will probably be fibrous, especially if both phases are non-faceted type. If it is between 0.28 and 0.5, the eutectic will tend to be lamellar [9]. In practice, the solution entropy is sometimes difficult to determine as either the heat of solution [4] or the partial molar entropies [8] should be first known, which are often not available. In this letter, the aim is directed to obtaining of the volume fraction of eutectic phases, which can be related to the
G. Meng et al. / Materials Letters 62 (2008) 984–987
985
ratio of their effective entropy change that can be obtained more easily than solution entropy.
From Eqs. (1), (2), (4) and (5), the effective entropy change per unit volume for a binary eutectic alloy is given by
2. Theory
DS ¼
The solid-liquid interfacial tension originates from the excess energy associated with the solid-liquid interface area. It causes the change in equilibrium melting point, leads capillary (also curvature or Gibbs–Thomson) undercooling during solidification. For pure substance, the capillary undercooling is given by: DTr ¼ Cj
ð1Þ
and C¼
r DSf
ð2Þ
where Γ is the Gibbs–Thomson coefficient, κ the curvature, σ the solid–liquid interface energy, and ΔSf is the fusion entropy per unit volume. Eqs. (1) and (2) can also be used to estimate the curvature undercooling for an alloy, this is done merely by replacing fusion entropy by the effective value, which is given by h i DS ¼ ð1 CS Þ SaL SaS þ CS SbL SbS =VS ð3Þ where SαL , SαS, SβL, SβS are partial molar entropies, the superscripts L and S represent the liquid and solid, and the subscripts α and β denote the α and β phases respectively, CS is the solid composition, and VS is the molar volume. Actually, the entropy terms are generally not available. Following Hunt and coworkers' work [10,11], the undercooling at a constant composition may be related to the change in composition at a constant temperature. For a sphere [12] DCr ¼
2rVS ð1 CL ÞCL rRTm CS CL
DTr ¼ mDCr
ð5Þ
where m is the liquidus slope.
ð6Þ
We restrict our analysis only to alloys of eutectic composition. As can be seen from Eq. (6), if the eutectic reaction occurs under thermodynamic equilibrium conditions, ΔS would be determined more easily, otherwise the problem will become complicated, because CS and CL are related to the growth processes. However, in the case of very small Peclet numbers, according to Magnin and Trivedi's analysis [13], CL = CE + δC, where CE is the eutectic composition, δC, whose value is very small, is an average corrective factor. Therefore, the boundary layer can maintain an average composition of the liquid at solid–liquid interface close to the eutectic concentration. Consequently, CL is approximately equal to CE, under such conditions. It is expected that the interfacial temperature is close to the eutectic temperature, TE, thus DSa ¼
RTE Ca CE CE ð1 CE Þ ma VSa
ð7aÞ
DSb ¼
RTE Cb CE CE ð1 CE Þ mb VSb
ð7bÞ
in which Cα and Cβ are concentration of α phase and β phase, respectively, at TE. Dividing Eq. (7a) by Eq. (7b), it can be found that there is a relationship between volume fractions and effective entropy change of the eutectic phases, as is given by fb ma DSa ¼ : fa mb DSb
ð4Þ
where r is the radius, R the gas constant, Tm the melting point, and CS and CL are the compositions of the equilibrated solid and liquid phases, respectively. For small undercooling, the capillary undercooling can be obtained by
RTm CS CL : mVS CL ð1 CL Þ
ð8Þ
Then the volume fraction of α phase can be derived as fa ¼
1 : DSa 1 mmba DS b
ð9Þ
From Eqs. (8) and (9), it can be seen that the volume and the entropy are inter-related for both eutectic phases. In form, Eq. (9) is identical with Eq. (12) in Ref. [7]. The difference is merely that the entropy terms of Eq. (12) in Ref. [7] denote the solution entropy. Note that there is a typographical
Table 1 Physical constants and calculated results for different eutectic systems System
TE
CL
Cα
Cβ
mα
mβ
VSα × 10− 6 VSβ × 10− 6 ΔSmαa 3
(K) (mol%) (mol%) (mol%) (K/mol%) (K/mol%) (m /mol) Al–CuAl2[10,11] Al–Si[10] Pb–Sn[10] Al–NiAl3[11]
821 17.3 850 12.1 456 26.1 913 3.06
2.5 1.59 71.0 0.023
32.0 100.0 1.45 25.0
Note: ΔSmi (i = α, β) is the molar entropy change.
a
−10.50 −8.20 4.08 −8.50
4.88 23.50 − 1.74 34.00
9.9068 9.9647 18.2587 9.9068
3
ΔSmβa
ΔSα × 106 ΔSβ × 106 fα 3
3
fαr
(m /mol)
(J/mol·K) (J/mol·K) (J/m ·K)
(J/m ·K) (–)
(–)
9.0005 11.6000 16.5024 8.8214
6.72 8.52 21.63 9.14
1.60 2.14 1.69 1.87
0.54 [13] 0.873 [13] 0.37 [13] 0.89 [17]
14.37 24.85 27.85 17.45
0.68 0.85 1.19 0.92
0.52 0.878 0.38 0.89
986
G. Meng et al. / Materials Letters 62 (2008) 984–987
eutectic research owing to its lower eutectic temperature and knowledge of its thermo-physical parameters. Table 1 presents their thermo-physical parameters, and these values can be considered as accurate [10,11], because of the phase diagram of these alloys is wellknown. It is well recognized that except for the Al–Si alloy system, others fall into the category of the regular eutectics. The calculated values of ΔSα and ΔSβ (according to Eqs. (7a) and (7b), respectively) are given in Table 1. For comparison, the molar entropic changes of eutectic phases as well as the calculated volume fraction of the α phase are represented. The results show that in the anomalous Al–Si system, the calculated values agree well with that predicted by Croker et al. [5], i.e. it belongs to irregular categories since the molar entropy change of the Si phase (ΔSβm in Table 1) is high and there is small volume fraction of the faceted phase (Fig. 1a). In contract, in the case of the Pb–Sn alloy, the molar entropy change of Sn-rich phase is larger than the critical value, 23 J/mol K, however, such alloy still belongs to regular eutectics (Fig. 1b). This means the entropy criterion is incomplete to predict the eutectic microstructure even for well-known regular eutectic systems. Moreover, for regular eutectics, the volume fraction appears to merely determine whether a lamellar or a rod structure ensues. The results also represent that the rule of thumb [9] is consistent with experimental observation. For example, Al–CuAl2 (its microstructure is similar to Pb–Sn eutectic) and Pb–Sn show lamellar structures owing to a larger volume fraction, and Al–NiAl3 (eutectic composition Al–3.06 at.%Ni) shows a rod arrangement or broken lamellar eutectic due to its lower volume fraction (Fig. 1c). In order to verify the reliability of the calculated results, the calculated volume fractions based on the calculated molar entropy change of eutectic phases were further compared with the accurately referenced values (fαr in Table 1). It is found that, there is a very good agreement between them as denoted in Table 1. It also can be indicated that from Eq. (8), the volume fraction ratio of the eutectic phases is inversely proportional to the ratio of their liquidus slope. It is well-known that the ratio of liquidus slope, in general, represents the symmetrical characteristic of the phase diagram. So it is expected that alloys, whose phase diagram is approximately symmetrical, will tend towards a regular eutectic structure. Finally, it is interesting to note that the volume fraction of either α phase or β phase depends on the ratio of entropic changes, and not single entropy of the eutectic phases. Therefore, a simple and unique relationship between fα or fβ and ΔSα or ΔSβ is not to be expected.
4. Conclusion
Fig. 1. Eutectic microstructures of Al–Si (a) [14] Pb–Sn (b) [15] and Al–NiAl3 (c) [16] eutectics.
error in Eq. (12) for negative sign in Ramachandrarao and Dubey's paper [8]. 3. Results and Discussion We consider several kinds of industrially important eutectic aluminum alloy systems: Al–CuAl2, Al–Si and Al–NiAl3. Besides, Pb–Sn alloy system is also investigated since it is often encountered in
The effective entropy change and the volume fraction of the eutectic phases are inter-related during eutectic solidification at low velocity, and the eutectic microstructures can be characterized based on the relationship between them. It is found that the volume fraction of either two phases only depends on the ratio of effective entropy change of the one phase to that of the other. The results indicate the entropy criterion for classification of eutectic microstructures is incomplete on comparison with the calculated values of the well-known systems, in which the thermo-physical parameters are accurately determined. The rule of thumb of volume fraction is more appropriate to characterize eutectic system. Furthermore, the symmetrical characteristic of the phase diagram also affects the eutectic arrangement.
G. Meng et al. / Materials Letters 62 (2008) 984–987
Acknowledgements This work was supported by National Natural Science Foundation of China under grant Nos. 50201012 and 50471065. References [1] [2] [3] [4] [5]
E.Z. Scheil, Z. Met.kd. 45 (1954) 298. J.D. Hunt, K.A. Jackson, Trans. Met. Soc. AIME 236 (1966) 843. M.R. Taylor, R.S. Fidler, R.W. Smith, J. Cryst. Growth 3/4 (1968) 666. M.R. Taylor, R.S. Fidler, R.W. Smith, Metall. Trans. 2 (1971) 1793. M.N. Crocker, R.S. Fidler, R.W. Smith, Proc. R. Soc. Lond., A 335 (1973) 15. [6] K.A. Jackson, Liquid Metals and Solidification, ASM, Cleveland, 1958. [7] R. Elliott, Eutectic Solidification Processing Crystalline and Glassy Alloys, Butterworths, London, 1983.
987
[8] P. Ramachandrarao, K.S. Dubey, Metall. Trans., A 23 (1992) 2675. [9] W. Kurz, J.D. Fisher, Fundamentals of Solidification, 4th edition, Trans Tech Pub., Aedermannsdorf, Switzerland, 1998. [10] M. Gündüz, J.D. Hunt, Acta Metall. 33 (1985) 1651. [11] N. Maraşli, J.D. Hunt, Acta Mater. 44 (1996) 1085. [12] J.W. Christian, The Theory of Transformation in Metals and Alloys, Part I, 2nd edition, Pergamon, Oxford, 1975. [13] P. Magnin, R. Trivedi, Acta Metall. Mater. 39 (1991) 453. [14] K.A. Jackson, Kinetic Processes: Crystal Growth, Diffusion, and Phase Transformation in Materials, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2004. [15] B. Ma, Mater's thesis, Northwestern Polytechnical University, 2005. [16] Y.X. Zhuang, X.M. Zhang, L.H. Zhu, Z.Q. Hu, Sci. Technol. Adv. Mater. 2 (2001) 37. [17] H.B. Smartt, T.H. Courtney, Metall. Trans. 2 (1972) 2000.
Materials Letters 62 (2008) 984 – 987 www.elsevier.com/locate/matlet
The relationship between effective entropy change and volume fraction of the eutectic phases in eutectic microstructures Guanghui Meng ⁎, Xin Lin, Weidong Huang State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi'an 710072, People's Republic of China Received 28 January 2007; accepted 15 July 2007 Available online 20 July 2007
Abstract The characterization of eutectic microstructures is related to the effective entropy change and the volume fraction of eutectic phases. For eutectic growth, both parameters can be inter-associated as long as solidification occurs at small Peclet numbers. There is no determinate relationship between the volume fraction and the effective entropy change of either eutectic phases, and the volume fraction is only related to the ratio of the effective entropy change of the two phases. By comparing with well-known eutectic systems, the results indicate that the classification depends on rather the volume fraction than the effective entropy change. In addition, there exists a certain relationship between the eutectic arrangement and the symmetric characteristic of phase diagram. © 2007 Elsevier B.V. All rights reserved. Keywords: Solidification; Eutectic; Microstructures
1. Introduction Eutectics can exhibit a wide variety of geometrical arrangements. Many attempts have been made to characterize eutectic systems in the past. Scheil [1] proposed a eutectic structure will be regular, rod like or lamellar if the eutectic phases are present in nearly equal proportions, and the structure will be anomalous when the volume fraction differ widely or if the minor phase exhibit a remarkable anisotropy. Following Scheil's work, a number of classification treatments have emerged, thereinto a few investigations are more noteworthy [2–5]. The Hunt and Jackson's classification of binary eutectics microstructures [2] is based on the Jackson's theory [6] on the classification between non-faceted and faceted solid-liquid interfaces. However, the Jackson factor is derived for an interface at equilibrium, and this theory is incomplete [7]. In addition, eutectic phases grow from melt at a much lower temperature than the melting point of the pure component, and their entropy value may different from that of the pure component due to temperature and composition effects. A
⁎ Corresponding author. Tel.: +86 29 88494001; fax: +86 29 88494001. E-mail address: [email protected] (G. Meng). 0167-577X/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2007.07.026
more appropriate expression for an alloy is proposed by Taylor et al. [3,4]. They classified the eutectic systems based on the interface roughness parameter. The final equation for the interface free energy is similar to Jackson's original solution, except that the solution entropy replaces the fusion entropy. In the analysis of Croker et al. [5], according to solution entropy of solid eutectic phases in the liquid, volume fraction of phases, and growth rate, eutectic microstructures are broadly divided into normal and anomalous groups, and a solution entropy above 23 J/mol·K is used to indicate typical faceting behavior and distinguish two groups. Recently, the eutectic microstructures were characterized by Ramachandrarao and Dubey again [8]. In their work the solution entropy and the volume fraction of the two phases are believed to be inter-related. To date, as a rule of thumb, one can suppose that when the volume fraction of one phase is between zero and 0.28, the eutectic will probably be fibrous, especially if both phases are non-faceted type. If it is between 0.28 and 0.5, the eutectic will tend to be lamellar [9]. In practice, the solution entropy is sometimes difficult to determine as either the heat of solution [4] or the partial molar entropies [8] should be first known, which are often not available. In this letter, the aim is directed to obtaining of the volume fraction of eutectic phases, which can be related to the
G. Meng et al. / Materials Letters 62 (2008) 984–987
985
ratio of their effective entropy change that can be obtained more easily than solution entropy.
From Eqs. (1), (2), (4) and (5), the effective entropy change per unit volume for a binary eutectic alloy is given by
2. Theory
DS ¼
The solid-liquid interfacial tension originates from the excess energy associated with the solid-liquid interface area. It causes the change in equilibrium melting point, leads capillary (also curvature or Gibbs–Thomson) undercooling during solidification. For pure substance, the capillary undercooling is given by: DTr ¼ Cj
ð1Þ
and C¼
r DSf
ð2Þ
where Γ is the Gibbs–Thomson coefficient, κ the curvature, σ the solid–liquid interface energy, and ΔSf is the fusion entropy per unit volume. Eqs. (1) and (2) can also be used to estimate the curvature undercooling for an alloy, this is done merely by replacing fusion entropy by the effective value, which is given by h i DS ¼ ð1 CS Þ SaL SaS þ CS SbL SbS =VS ð3Þ where SαL , SαS, SβL, SβS are partial molar entropies, the superscripts L and S represent the liquid and solid, and the subscripts α and β denote the α and β phases respectively, CS is the solid composition, and VS is the molar volume. Actually, the entropy terms are generally not available. Following Hunt and coworkers' work [10,11], the undercooling at a constant composition may be related to the change in composition at a constant temperature. For a sphere [12] DCr ¼
2rVS ð1 CL ÞCL rRTm CS CL
DTr ¼ mDCr
ð5Þ
where m is the liquidus slope.
ð6Þ
We restrict our analysis only to alloys of eutectic composition. As can be seen from Eq. (6), if the eutectic reaction occurs under thermodynamic equilibrium conditions, ΔS would be determined more easily, otherwise the problem will become complicated, because CS and CL are related to the growth processes. However, in the case of very small Peclet numbers, according to Magnin and Trivedi's analysis [13], CL = CE + δC, where CE is the eutectic composition, δC, whose value is very small, is an average corrective factor. Therefore, the boundary layer can maintain an average composition of the liquid at solid–liquid interface close to the eutectic concentration. Consequently, CL is approximately equal to CE, under such conditions. It is expected that the interfacial temperature is close to the eutectic temperature, TE, thus DSa ¼
RTE Ca CE CE ð1 CE Þ ma VSa
ð7aÞ
DSb ¼
RTE Cb CE CE ð1 CE Þ mb VSb
ð7bÞ
in which Cα and Cβ are concentration of α phase and β phase, respectively, at TE. Dividing Eq. (7a) by Eq. (7b), it can be found that there is a relationship between volume fractions and effective entropy change of the eutectic phases, as is given by fb ma DSa ¼ : fa mb DSb
ð4Þ
where r is the radius, R the gas constant, Tm the melting point, and CS and CL are the compositions of the equilibrated solid and liquid phases, respectively. For small undercooling, the capillary undercooling can be obtained by
RTm CS CL : mVS CL ð1 CL Þ
ð8Þ
Then the volume fraction of α phase can be derived as fa ¼
1 : DSa 1 mmba DS b
ð9Þ
From Eqs. (8) and (9), it can be seen that the volume and the entropy are inter-related for both eutectic phases. In form, Eq. (9) is identical with Eq. (12) in Ref. [7]. The difference is merely that the entropy terms of Eq. (12) in Ref. [7] denote the solution entropy. Note that there is a typographical
Table 1 Physical constants and calculated results for different eutectic systems System
TE
CL
Cα
Cβ
mα
mβ
VSα × 10− 6 VSβ × 10− 6 ΔSmαa 3
(K) (mol%) (mol%) (mol%) (K/mol%) (K/mol%) (m /mol) Al–CuAl2[10,11] Al–Si[10] Pb–Sn[10] Al–NiAl3[11]
821 17.3 850 12.1 456 26.1 913 3.06
2.5 1.59 71.0 0.023
32.0 100.0 1.45 25.0
Note: ΔSmi (i = α, β) is the molar entropy change.
a
−10.50 −8.20 4.08 −8.50
4.88 23.50 − 1.74 34.00
9.9068 9.9647 18.2587 9.9068
3
ΔSmβa
ΔSα × 106 ΔSβ × 106 fα 3
3
fαr
(m /mol)
(J/mol·K) (J/mol·K) (J/m ·K)
(J/m ·K) (–)
(–)
9.0005 11.6000 16.5024 8.8214
6.72 8.52 21.63 9.14
1.60 2.14 1.69 1.87
0.54 [13] 0.873 [13] 0.37 [13] 0.89 [17]
14.37 24.85 27.85 17.45
0.68 0.85 1.19 0.92
0.52 0.878 0.38 0.89
986
G. Meng et al. / Materials Letters 62 (2008) 984–987
eutectic research owing to its lower eutectic temperature and knowledge of its thermo-physical parameters. Table 1 presents their thermo-physical parameters, and these values can be considered as accurate [10,11], because of the phase diagram of these alloys is wellknown. It is well recognized that except for the Al–Si alloy system, others fall into the category of the regular eutectics. The calculated values of ΔSα and ΔSβ (according to Eqs. (7a) and (7b), respectively) are given in Table 1. For comparison, the molar entropic changes of eutectic phases as well as the calculated volume fraction of the α phase are represented. The results show that in the anomalous Al–Si system, the calculated values agree well with that predicted by Croker et al. [5], i.e. it belongs to irregular categories since the molar entropy change of the Si phase (ΔSβm in Table 1) is high and there is small volume fraction of the faceted phase (Fig. 1a). In contract, in the case of the Pb–Sn alloy, the molar entropy change of Sn-rich phase is larger than the critical value, 23 J/mol K, however, such alloy still belongs to regular eutectics (Fig. 1b). This means the entropy criterion is incomplete to predict the eutectic microstructure even for well-known regular eutectic systems. Moreover, for regular eutectics, the volume fraction appears to merely determine whether a lamellar or a rod structure ensues. The results also represent that the rule of thumb [9] is consistent with experimental observation. For example, Al–CuAl2 (its microstructure is similar to Pb–Sn eutectic) and Pb–Sn show lamellar structures owing to a larger volume fraction, and Al–NiAl3 (eutectic composition Al–3.06 at.%Ni) shows a rod arrangement or broken lamellar eutectic due to its lower volume fraction (Fig. 1c). In order to verify the reliability of the calculated results, the calculated volume fractions based on the calculated molar entropy change of eutectic phases were further compared with the accurately referenced values (fαr in Table 1). It is found that, there is a very good agreement between them as denoted in Table 1. It also can be indicated that from Eq. (8), the volume fraction ratio of the eutectic phases is inversely proportional to the ratio of their liquidus slope. It is well-known that the ratio of liquidus slope, in general, represents the symmetrical characteristic of the phase diagram. So it is expected that alloys, whose phase diagram is approximately symmetrical, will tend towards a regular eutectic structure. Finally, it is interesting to note that the volume fraction of either α phase or β phase depends on the ratio of entropic changes, and not single entropy of the eutectic phases. Therefore, a simple and unique relationship between fα or fβ and ΔSα or ΔSβ is not to be expected.
4. Conclusion
Fig. 1. Eutectic microstructures of Al–Si (a) [14] Pb–Sn (b) [15] and Al–NiAl3 (c) [16] eutectics.
error in Eq. (12) for negative sign in Ramachandrarao and Dubey's paper [8]. 3. Results and Discussion We consider several kinds of industrially important eutectic aluminum alloy systems: Al–CuAl2, Al–Si and Al–NiAl3. Besides, Pb–Sn alloy system is also investigated since it is often encountered in
The effective entropy change and the volume fraction of the eutectic phases are inter-related during eutectic solidification at low velocity, and the eutectic microstructures can be characterized based on the relationship between them. It is found that the volume fraction of either two phases only depends on the ratio of effective entropy change of the one phase to that of the other. The results indicate the entropy criterion for classification of eutectic microstructures is incomplete on comparison with the calculated values of the well-known systems, in which the thermo-physical parameters are accurately determined. The rule of thumb of volume fraction is more appropriate to characterize eutectic system. Furthermore, the symmetrical characteristic of the phase diagram also affects the eutectic arrangement.
G. Meng et al. / Materials Letters 62 (2008) 984–987
Acknowledgements This work was supported by National Natural Science Foundation of China under grant Nos. 50201012 and 50471065. References [1] [2] [3] [4] [5]
E.Z. Scheil, Z. Met.kd. 45 (1954) 298. J.D. Hunt, K.A. Jackson, Trans. Met. Soc. AIME 236 (1966) 843. M.R. Taylor, R.S. Fidler, R.W. Smith, J. Cryst. Growth 3/4 (1968) 666. M.R. Taylor, R.S. Fidler, R.W. Smith, Metall. Trans. 2 (1971) 1793. M.N. Crocker, R.S. Fidler, R.W. Smith, Proc. R. Soc. Lond., A 335 (1973) 15. [6] K.A. Jackson, Liquid Metals and Solidification, ASM, Cleveland, 1958. [7] R. Elliott, Eutectic Solidification Processing Crystalline and Glassy Alloys, Butterworths, London, 1983.
987
[8] P. Ramachandrarao, K.S. Dubey, Metall. Trans., A 23 (1992) 2675. [9] W. Kurz, J.D. Fisher, Fundamentals of Solidification, 4th edition, Trans Tech Pub., Aedermannsdorf, Switzerland, 1998. [10] M. Gündüz, J.D. Hunt, Acta Metall. 33 (1985) 1651. [11] N. Maraşli, J.D. Hunt, Acta Mater. 44 (1996) 1085. [12] J.W. Christian, The Theory of Transformation in Metals and Alloys, Part I, 2nd edition, Pergamon, Oxford, 1975. [13] P. Magnin, R. Trivedi, Acta Metall. Mater. 39 (1991) 453. [14] K.A. Jackson, Kinetic Processes: Crystal Growth, Diffusion, and Phase Transformation in Materials, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2004. [15] B. Ma, Mater's thesis, Northwestern Polytechnical University, 2005. [16] Y.X. Zhuang, X.M. Zhang, L.H. Zhu, Z.Q. Hu, Sci. Technol. Adv. Mater. 2 (2001) 37. [17] H.B. Smartt, T.H. Courtney, Metall. Trans. 2 (1972) 2000.