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A theoretical study of Gasarite eutectic growth
Scripta Materialia 49 (2003) 379–386 www.actamat-journals.com
A theoretical study of Gasarite eutectic growth Liu Yuan *, Li Yanxiang Department of Mechanical Engineering, Tsinghua University, Room 204, Welding Hall, Beijing 100084, PR China Received 15 October 2002; received in revised form 22 May 2003; accepted 28 May 2003
Abstract A theoretical model has been developed to get the relation between the porosity, the inter-pore spacing and the processing parameters in the metal–gas eutectic (Gasarite eutectic) solidification. The theoretical relation between the inter-pore spacing and the solidification rate also can be described as a simple expression (v L2 ¼ A) as that presented in the classical Jackson–Hunt eutectic growth model. 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Materialia; Metal–gas eutectic; Directional solidification; Porous material; Medeling
1. Introduction Porous materials with a regular distribution of pores can be produced by using a novel method named Gasar developed by the Dnepropetrovsk Metallurgical Institute in Ukraine [1]. Most metal– hydrogen binary systems also have a eutectic decomposition as the classic eutectic systems. These metals can be melted, saturated with hydrogen, and then directionally solidified into a porous structure, as shown in Fig. 1 [2]. During solidification, solid metal and hydrogen simultaneously form by a gas-eutectic reaction, resulting in a porous structure with elongated gaseous pores filled by hydrogen. In Japan, castings with elongated pores were named Lotus-type porous metals, because they look like the lotus roots.
* Corresponding author. Tel.: +86-10-6277-3640; fax: +8610-6277-3637. E-mail address: [email protected] (Y. Liu).
Gasar structures offer a number of attractive properties [3], such as significantly improved thermal conductivity over conventionally processed porous materials. In addition, these materials combine good mechanical properties with reduced weight. The ultimate tensile strength and the yield strength of Gasar materials with the cylindrical pores orientated parallel to the tensile direction decrease linearly with increasing porosity. In other words, the gas pores do not cause stress concentration during the stretching process [4]. Due to these attractive properties, the Gasar materials have many potential applications. These initial studies about the Gasar process are focused on the nucleation mechanism of gaseous hydrogen bubbles in Gasar solidification [5,6], and the experimental studies on the processing conditions to produce Gasar structures [7–12]. However, no systematic investigations have been made to determine the relation between the porosity, the inter-pore spacing and the processing parameters such as the gas pressure during melting and
1359-6462/03/$ - see front matter 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-6462(03)00330-0
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Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386
2. No fluid convection occurs in the direction parallel to the solidifying front. 3. There is a uniform temperature in the gas pores that equals the melting point of the metal. Fig. 2 is a schematic diagram of the Gasarite growth and the corresponding cylindrical coordinate system used for solution of the solute field. rg is the radius of the gas pore and rs is one-half of the inter-pore spacing. The porosity can be expressed as: e ¼ rg2 =rs2 Fig. 1. Gasar process and the obtained Gasar structure [2].
solidification processes, the solidification rate and the melt temperature. In this paper, this problem is examined through a solution of the solute distribution in front of the solidifying front. The solution procedure is similar to that adopted in Jackson–Hunt model [13]. These theoretical analyses are helpful for understanding the Gasarite eutectic growth.
2. Model analysis 2.1. Solute field To simplify the model analysis, three hypotheses are supposed: 1. The pore size and inter-pore spacing are homogeneous.
ð1Þ
When the escaping rate of H2 from the melt is negligible, under the steady-state solidification conditions the porosity, e, is only dependent upon the gas pressure and the melt temperature. The solute distribution in the melt can be described with the following steady-state diffusion equation: o2 C 1 oC o2 C v oC þ 2 þ ¼0 þ 2 or r or oz DL oz
ð2Þ
in which v is the solidification rate, DL is the diffusion coefficient of the gas atom in the melt. The corresponding boundary conditions are expressed as: oC v P ¼ ðat the gas phase front; 06r 6rg Þ oz z¼0 DL RTm ð3Þ oC v ¼ ðC ls C s Þ oz z¼0 DL ðat the solid phase front; rg < r 6 rs Þ
Fig. 2. A schematic diagram of the Gasarite growth and the corresponding coordinate selection.
ð4Þ
Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386
where P is the gas pressure in the pore. R is the gas constant and Tm is the melting point of the metal. C ls is the average content of H2 in the melt at the solidifying front. C s is the average content of H2 in the solid metal, which depends upon the gas pressure, and the average temperature of the solidified metal. After a strict mathematical manipulation, the general solute distribution can be written as: 1 X C ¼ C1 þ Bn exn z J0 ðkn r=rs Þ ð5Þ
R rs
C ls ¼ ¼
rg
381
2prCjz¼0 dr
pðrs2 rg2 Þ R rs P1 2pr C1 þ B0 þ n¼1 Bn J0 ðkn r=rs Þ dr rg
pðrs2 rg2 Þ v P ðC ls C s Þ þ ¼ C 1 þ B0 þ DL RTm p ffiffi 1 X 4eJ12 ð ekn Þ
2 2 n¼1 ð1 eÞkn xn J0 ðkn Þ
ð8Þ
Substituting B0 into Eq. (8) above gives:
n¼0
C1 ð1 eÞC s RTPm e þ DvL C ls ¼ P1 e DvL n¼1
xn ¼ ðv=2DL Þ þ
P 1 Cs n¼1 pffi 4eJ 2 ð ek Þ
P RTm
1
pffi 4eJ12 ð ekn Þ ð1eÞk2n xn J02 ðkn Þ
n
ð9Þ
ð1eÞk2n xn J02 ðkn Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðv=2DL Þ2 þ ðkn =rs Þ2
ð6Þ
where C1 is the solute concentration at z ! 1 and equal to Cm ––the saturated solute concentration in the melt (see Section 3.1). J0 ðxÞ is the zero-order Bessel function and kn is the root of J1 ðxÞ ¼ 0 (J1 ðxÞ is the first-order Bessel function). Bn can be derived from the following boundary conditions: h i 8 < B0 ¼ ðC ls C s Þð1 eÞ P e ðn ¼ 0Þ m pffi pffi h RT i 2 e J ð e k Þ n v P 1 : Bn ¼ ðC ls C s Þ þ RTm ðn P 1Þ DL kn xn J 2 ðkn Þ 0
ð7Þ The classical Jackson–Hunt model simplified the solute field by assuming v=2DL 0 in Eq. (6), and C ls CE (the eutectic composition) in Eq. (4). In this paper, these assumptions are relaxed. 2.2. Average content of hydrogen at solid phase growth front The average content of H2 at the growing solid front can be obtained through integrating the solute field C at z ¼ 0.
It can be seen from Eq. (9) that the solidification rate, the gas pressure and the melting temperature all have effects on C ls . 3. Results and discussion The Cu–H2 system is used as an example to test and evaluate the model established above. Fig. 3 is the phase diagram of Cu–H2 eutectic system. Table 1 lists the used parameters related to Cu–H2 system. 3.1. Porosity If all of the formed bubbles are trapped as pores during the steady-state solidification process, a simple equation based on the ideal gas law can be used to calculate the porosity in the final casting [10]: ðCm Cs ÞRTm e¼ 1000000 P MH2 ðCm Cs ÞRTm þ qs
ð10Þ
where Cm and Cs are the hydrogen concentrations (in ppm) in the liquid and the solid at the
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Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386 1360 5
7
1.01x10 Pa
1.01x10 Pa
Temperature (K)
1355
L S+L
1350
1345
S
1340
0.000
0.001
0.002 0.003 XH
0.004
0.005
Fig. 3. Phase diagram of copper-hydrogen eutectic system [6].
solidification front. qs is the density of the solid metal and MH2 is the molecular weight of H2 . P is the gas pressure (in Pa) in the pore. For Cu–H2 system, Cm and Cs can be expressed as [8]: Cm ¼ nðT Þ
pffiffiffiffiffiffiffi P H2
It can be seen from Fig. 4 that, if the pure hydrogen is used singly, the porosity first increases up to a maximum, then starts to drop with increasing hydrogen pressure. In most cases, the Gasar process uses the mixture of hydrogen and argon because that the hydrogen bubbles are not easy to form and float away before the solidifying front under this kind of condition [10]. In other words, the use of the mixed gases favors the coupled growth between the solid and the gaseous phase, therefore favors the achievement of Gasar structures. In this situation, the porosity increases continually with rising hydrogen partial pressure at a constant argon partial pressure. This dependence of porosity on the partial pressures of argon and hydrogen arises because the amount of hydrogen dissolved in the melt is proportional to the square root of hydrogen pressure, whereas the density of hydrogen in the pore is proportional to the total applied pressure. As for the influence of the melt temperature on the porosity, it is apparent that the porosity increases continually with the rise of the melt temperature because that a higher melt temperature
pffiffiffiffiffiffiffi P H2
ð11Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pg þ Pc þ Ph
ð12Þ
0.6
where PH2 is the partial pressure of H2 . Pg is the total gas pressure in the furnace atmosphere. In general, the height of the casting is not higher than 0.2 m in the Gasar solidification. If the average height is selected as 0.125 m, then the average hydrostatic pressure Ph is about 104 Pa for Cu–H2 system (Ph ¼ ql gh ¼ 8000 10 0:125 104 Pa). The pore diameter ranges from tens of microns to several millimeters in the Gasar structures [17]. If the average radius is selected as 30 microns, the average capillary pressure Pc is 8.5 · 104 Pa for Cu–H2 system (Pc ¼ 2rl =rg ¼ 2 1:28=30 106 8:5 104 Pa).
0.5
¼ 0:72159 expð5234=T Þ pffiffiffi Cs ¼ n0 ðT Þ P ¼ 0:43399 expð5888=T Þ
Pg=PH , PAr=0 Pa, T=1523K
Porosity (e)
2
0.4 0.3 5
Pg=PH +PAr=8.08x10 Pa, T=1523K
0.2
2
0.1 0.0 0.0
2.0x10
5
5
4.0x10 6.0x10 PH /Pa
5
8.0x10
5
2
Fig. 4. Influence of the partial pressure of hydrogen on the porosity of Gasar Cu.
Table 1 The parameters of Cu–H2 system used for calculation ml (K/mol m3 )
DHðliquid
0.01026 [6]
1:46 102 expð18900=RT Þ [14]
CuÞ
(cm2 /s)
rls (1356 K) (J/m2 )
DHm (J/g)
qs (g/cm3 )
ql (1356 K) (g/cm3 )
0.178 [15]
211.85 [16]
8.9
8.0
Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386
will lead to a higher solubility of hydrogen in the molten metal (see Eq. (11)). 3.2. Solute distribution According to Eqs. (5)–(10), the solute distribution in front of the solidifying front and the average hydrogen content at the solidifying front can be calculated. The two-dimensional solute distributions at different solidification rates are shown in Fig. 5(a), (b). It can be found that there exists a slight hydrogen concentration gradient in the radial direction, but the strongest concentration gradient develops in the z axial direction when the solidification rate is as small as 50 lm/s. With increasing solidification rate, a sharp concentration gradient also occurs in the radial direction. It is obvious that the diffusion of the hydrogen atom in the radial direction will become a restrictive factor for the growth of the hydrogen pore. In this situation, rs and rg will be automatically adjusted to maintain the cooperative growth between the solid and the gaseous phase. This point is the major reason why the inter-pore spacing decreases with rising solidification rate discussed in Section 3.3. 3.3. Inter-pore spacing By using the minimum undercooling criterion as that in the Jackson–Hunt model, the required undercooling for the growth of the solid phase and the inter-pore spacing can be calculated. However, for the metal–gas eutectic solidification, the
383
growth of the gas pore does not need any undercooling, so that only the undercooling for the growth of the solid phase is calculated here. Fig. 6 shows the interfacial energies balance between the solid, the liquid and the gas phases. Generally speaking, the surface energy of the solid phase ðrs Þ is much higher than that of the liquid (rl ) and the interfacial energy of liquid/solid (rls ). With rs P rls þ rl , the average curvature radius (rsl ) of the solid/liquid interface should be written pffiffi as rsl ¼ ðrs rg Þ ¼ ð1 eÞrs . The required undercooling for the growth of the solid can be described as:
DT ¼ ml ðCE C ls Þ þ
rls TE pffiffi DHm ð1 eÞrs qs
ð13Þ
where ml is the slope of liquidus in metal–gas eutectic phase diagram. DHm and qs are the fusion enthalpy and the density of the solid metal, respectively. CE and TE are the eutectic composition and the eutectic temperature of the metal–gas eutectic system, respectively. It can be seen from Eq. (9) that C ls is not a simple function of rs , so that it is not possible to obtain a simple relationship––L ¼ A vn (L ¼ 2rs , L is the inter-pore spacing) as that in the Jackson– Hunt model through directly applying oDT =ors ¼ 0 (minimum undercooling criterion) to Eq. (13). But the calculated results in Fig. 7 show that, al though the form of Eq. (9) is complex, actually, C ls has a linear relation with rs at a constant solidification rate. In this case:
Fig. 5. Solute distribution in front of the solidification front (T ¼ 1400 K, rs ¼ 200 lm, PH2 ¼ PAr ¼ 5:05 105 Pa, e ¼ 0:25).
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Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386
only has a linear relation with v, but also the interceptions are zero, so Eq. (14) can be simplified as:
C ls ¼ E þ H v rs
ð15Þ
in which H is a constant dependent on the gas pressure (P ) and the melt temperature (T ). Substituting Eq. (15) into Eq. (13) and applying the minimum undercooling criterion gives: Fig. 6. A schematic diagram of the interfacial energies balance.
34.0 v=500µm/s
Cls* /ppm
33.5 33.0 32.5
v=200µm/s
32.0 31.5
v=100µm/s
31.0 0
20
40
60
rs /µ m
Fig. 7. Relationship between C ls and rs (z ¼ 0, PH2 ¼ PAr ¼ 5:05 105 Pa, T ¼ 1400 K).
0.075
T=1500K, 5 x10 Pa PH =PAr =5.05x10
0.060
2
T=1400K,
0.045 PH =PAr=5.05x105Pa 2
0.030
T=1400K, 5 PH =4.04x10 Pa
0.015
PAr=2.02x10 Pa
2
0.000 0
5
100
200
300
400
500
Fig. 8. Relationship between the parameter, F , in Eq. (14) and the solidification rate.
C ls ¼ E þ F rs
ð14Þ
in which E and F are constants dependent upon P and T . As shown in Fig. 8, the parameter F not
v L2 ¼ A
ð16Þ
where A is a constant dependent on the selected system, the gas pressure (P ) and the melt temperature (T ). It is apparent that this equation is similar to the relation between the lamellar spacing and the solidification rate (L ¼ A vn ) presented in the classical Jackson–Hunt eutectic growth model. Fig. 9(a)–(c) show the effects of the solodification rate, the gas pressure and the melt temperature on the inter-pore and the pore size. It is obvious that the inter-pore spacing and the pore size decrease with increasing solidification rate. Besides the solidification rate, the gas pressure and the melt temperature also influence the inter-pore spacing and the pore size. However, their influences are small, compared with that of the solidification rate. The dropping trend of the inter-pore spacing with increasing hydrogen partial pressure is arrested at high partial pressures. However, the rising trend of the pore size continues with increasing hydrogen partial pressure at a constant total pressure. The inter-pore spacing and the pore size all increase with rising melt temperature. Comparatively, the solidification rate is the major factor influencing the inter-pore spacing and the pore size in the Gasarite eutectic solidification. Only limited quantitative measurements [18] at a solidification rate of 500 lm/s show the average pore size in Gasarite copper is about 30 lm. This value is close to the predicted values of 40 lm under the same condition. In addition, much more qualitative experimental results [19] also show that the inter-pore spacing decreases with increasing solidification rate. However, the lack of more quantitative experimental results prevents the evaluation of the accuracy of Eq. (16).
Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386
385
Fig. 9. Effects of the solidification rate, the gas pressure and the melt temperature on the inter-pore spacing (L) and the pore size (2rg ): (a) effect of the solidification rate, (b) effect of the gas pressure and (c) effect of the melt temperature.
Unlike the classical solid–solid eutectics, the metal–gas eutectics present a special characteristic in which the inter-pore spacing and the pore size are determined not only by the diffusion selfadjustment, but also are functions of the gas pressure and the melt temperature. In addition, the nucleation conditions, the temperature gradient GT and the different thermal conductivities of the solids and the gaseous phases also should have an important influence on the solute distribution and the inter-pore spacing [6,9]. The latter two factors may lead to the solute transferring along the solidification front, therefore enhance the inter-pore spacing. The nucleation will adjust the inter-pore spacing to a smaller level. Further detailed investigations on the influence of these factors remain for future work.
4. Conclusions 1. The gas pressure and the melt temperature in Gasar solidification determine the porosity of the produced Gasar structures. If the pure hydrogen is used singly, the porosity first increases up to a maximum, then starts to drop with increasing hydrogen pressure. If the mixture of hydrogen and inert gas are used, the porosity increases continually with rising hydrogen partial pressure. The porosity increases continually with the melt temperature. 2. A theoretical model has been developed to describe the Gasarite eutectic growth. The calculated results show that the relation between the inter-pore spacing and the solidification rate can be described as a simple expression
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Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386
(v L2 ¼ A) as that presented in the Jackson– Hunt model. 3. Unlike the classical solid–solid eutectics, the metal–gas eutectics present a special characteristic in which the inter-pore spacing and the pore size are determined not only by the diffusion self-adjustment, but also are functions of the gas pressure and the melt temperature. The inter-pore spacing decreases and the pore size increases with increasing hydrogen partial pressure. The inter-pore spacing and the pore size all increase with rising melt temperature. Acknowledgements The authors are very grateful to the Postdoctoral Foundation and the Doctoral Foundation of China for the financial support. References [1] Shapovalov VI. US Patent No. 5,181,549 (January 26, 1993). [2] http://www.msm.cam.ac.uk/mmc/people/dave/method2. html.
[3] Walukas DM. DMK TEK INC., 1992. [4] Yamamura S, Shiota H, Murakami K. Mater Sci Eng 2001;A318:137–43. [5] Russell KC, Sridhar S. J Mater Synth Process 1995;3(4):215–22. [6] Sridhar S, Zheng Y. In: Nash P, Sunman B, editors. Applications of Thermodynamics in the Synthesis and Processing of Materials. Warrendale, PA: TMS Publications; 1995. p. 259–69. [7] Hyun KS, Murakami K, Nakajima H. Mater Sci Eng 2001;A299:241–8. [8] Nakajima H, Hyun KS, Ohashi K. Colloid Surface A 2001;179:209–14. [9] Shapovalov VI. Mat Res Soc Symp Proc 1998;521:281–90. [10] Apprill JM, Poirier DR. Mat Res Soc Symp Proc 1998;521:291–6. [11] Paradies CJ, Tobin A. Mat Res Soc Symp Proc 1998;521:297–302. [12] Bonenberger RJ, Kee AJ. Mat Res Soc Symp Proc 1998;521:303–14. [13] Jackson KA, Hunt JD. TMS of ASME 1966;236:1129–42. [14] Chernega DF, Vashchenko KI, Ivanchuk DF. Izvest VUZ Tsvetnaya Met 1973;4:120–2. [15] Jones H. Mater Lett 2002;53(4–5):364–6. [16] Knacke O. Thermochemical Properties of Inorganic Substances. 2nd ed. Berlin: Springer-Verlag; 1991. [17] Banhart J. Prog Mater Sci 2001;46:559–632. [18] Pattnaik A, Sanday SC. Mat Res Soc Symp Proc 1995;371:371–6. [19] Chanman P, Steven RN. Mat Res Soc Symp Proc 1998;521:315–20.
A theoretical study of Gasarite eutectic growth Liu Yuan *, Li Yanxiang Department of Mechanical Engineering, Tsinghua University, Room 204, Welding Hall, Beijing 100084, PR China Received 15 October 2002; received in revised form 22 May 2003; accepted 28 May 2003
Abstract A theoretical model has been developed to get the relation between the porosity, the inter-pore spacing and the processing parameters in the metal–gas eutectic (Gasarite eutectic) solidification. The theoretical relation between the inter-pore spacing and the solidification rate also can be described as a simple expression (v L2 ¼ A) as that presented in the classical Jackson–Hunt eutectic growth model. 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Materialia; Metal–gas eutectic; Directional solidification; Porous material; Medeling
1. Introduction Porous materials with a regular distribution of pores can be produced by using a novel method named Gasar developed by the Dnepropetrovsk Metallurgical Institute in Ukraine [1]. Most metal– hydrogen binary systems also have a eutectic decomposition as the classic eutectic systems. These metals can be melted, saturated with hydrogen, and then directionally solidified into a porous structure, as shown in Fig. 1 [2]. During solidification, solid metal and hydrogen simultaneously form by a gas-eutectic reaction, resulting in a porous structure with elongated gaseous pores filled by hydrogen. In Japan, castings with elongated pores were named Lotus-type porous metals, because they look like the lotus roots.
* Corresponding author. Tel.: +86-10-6277-3640; fax: +8610-6277-3637. E-mail address: [email protected] (Y. Liu).
Gasar structures offer a number of attractive properties [3], such as significantly improved thermal conductivity over conventionally processed porous materials. In addition, these materials combine good mechanical properties with reduced weight. The ultimate tensile strength and the yield strength of Gasar materials with the cylindrical pores orientated parallel to the tensile direction decrease linearly with increasing porosity. In other words, the gas pores do not cause stress concentration during the stretching process [4]. Due to these attractive properties, the Gasar materials have many potential applications. These initial studies about the Gasar process are focused on the nucleation mechanism of gaseous hydrogen bubbles in Gasar solidification [5,6], and the experimental studies on the processing conditions to produce Gasar structures [7–12]. However, no systematic investigations have been made to determine the relation between the porosity, the inter-pore spacing and the processing parameters such as the gas pressure during melting and
1359-6462/03/$ - see front matter 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-6462(03)00330-0
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Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386
2. No fluid convection occurs in the direction parallel to the solidifying front. 3. There is a uniform temperature in the gas pores that equals the melting point of the metal. Fig. 2 is a schematic diagram of the Gasarite growth and the corresponding cylindrical coordinate system used for solution of the solute field. rg is the radius of the gas pore and rs is one-half of the inter-pore spacing. The porosity can be expressed as: e ¼ rg2 =rs2 Fig. 1. Gasar process and the obtained Gasar structure [2].
solidification processes, the solidification rate and the melt temperature. In this paper, this problem is examined through a solution of the solute distribution in front of the solidifying front. The solution procedure is similar to that adopted in Jackson–Hunt model [13]. These theoretical analyses are helpful for understanding the Gasarite eutectic growth.
2. Model analysis 2.1. Solute field To simplify the model analysis, three hypotheses are supposed: 1. The pore size and inter-pore spacing are homogeneous.
ð1Þ
When the escaping rate of H2 from the melt is negligible, under the steady-state solidification conditions the porosity, e, is only dependent upon the gas pressure and the melt temperature. The solute distribution in the melt can be described with the following steady-state diffusion equation: o2 C 1 oC o2 C v oC þ 2 þ ¼0 þ 2 or r or oz DL oz
ð2Þ
in which v is the solidification rate, DL is the diffusion coefficient of the gas atom in the melt. The corresponding boundary conditions are expressed as: oC v P ¼ ðat the gas phase front; 06r 6rg Þ oz z¼0 DL RTm ð3Þ oC v ¼ ðC ls C s Þ oz z¼0 DL ðat the solid phase front; rg < r 6 rs Þ
Fig. 2. A schematic diagram of the Gasarite growth and the corresponding coordinate selection.
ð4Þ
Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386
where P is the gas pressure in the pore. R is the gas constant and Tm is the melting point of the metal. C ls is the average content of H2 in the melt at the solidifying front. C s is the average content of H2 in the solid metal, which depends upon the gas pressure, and the average temperature of the solidified metal. After a strict mathematical manipulation, the general solute distribution can be written as: 1 X C ¼ C1 þ Bn exn z J0 ðkn r=rs Þ ð5Þ
R rs
C ls ¼ ¼
rg
381
2prCjz¼0 dr
pðrs2 rg2 Þ R rs P1 2pr C1 þ B0 þ n¼1 Bn J0 ðkn r=rs Þ dr rg
pðrs2 rg2 Þ v P ðC ls C s Þ þ ¼ C 1 þ B0 þ DL RTm p ffiffi 1 X 4eJ12 ð ekn Þ
2 2 n¼1 ð1 eÞkn xn J0 ðkn Þ
ð8Þ
Substituting B0 into Eq. (8) above gives:
n¼0
C1 ð1 eÞC s RTPm e þ DvL C ls ¼ P1 e DvL n¼1
xn ¼ ðv=2DL Þ þ
P 1 Cs n¼1 pffi 4eJ 2 ð ek Þ
P RTm
1
pffi 4eJ12 ð ekn Þ ð1eÞk2n xn J02 ðkn Þ
n
ð9Þ
ð1eÞk2n xn J02 ðkn Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðv=2DL Þ2 þ ðkn =rs Þ2
ð6Þ
where C1 is the solute concentration at z ! 1 and equal to Cm ––the saturated solute concentration in the melt (see Section 3.1). J0 ðxÞ is the zero-order Bessel function and kn is the root of J1 ðxÞ ¼ 0 (J1 ðxÞ is the first-order Bessel function). Bn can be derived from the following boundary conditions: h i 8 < B0 ¼ ðC ls C s Þð1 eÞ P e ðn ¼ 0Þ m pffi pffi h RT i 2 e J ð e k Þ n v P 1 : Bn ¼ ðC ls C s Þ þ RTm ðn P 1Þ DL kn xn J 2 ðkn Þ 0
ð7Þ The classical Jackson–Hunt model simplified the solute field by assuming v=2DL 0 in Eq. (6), and C ls CE (the eutectic composition) in Eq. (4). In this paper, these assumptions are relaxed. 2.2. Average content of hydrogen at solid phase growth front The average content of H2 at the growing solid front can be obtained through integrating the solute field C at z ¼ 0.
It can be seen from Eq. (9) that the solidification rate, the gas pressure and the melting temperature all have effects on C ls . 3. Results and discussion The Cu–H2 system is used as an example to test and evaluate the model established above. Fig. 3 is the phase diagram of Cu–H2 eutectic system. Table 1 lists the used parameters related to Cu–H2 system. 3.1. Porosity If all of the formed bubbles are trapped as pores during the steady-state solidification process, a simple equation based on the ideal gas law can be used to calculate the porosity in the final casting [10]: ðCm Cs ÞRTm e¼ 1000000 P MH2 ðCm Cs ÞRTm þ qs
ð10Þ
where Cm and Cs are the hydrogen concentrations (in ppm) in the liquid and the solid at the
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Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386 1360 5
7
1.01x10 Pa
1.01x10 Pa
Temperature (K)
1355
L S+L
1350
1345
S
1340
0.000
0.001
0.002 0.003 XH
0.004
0.005
Fig. 3. Phase diagram of copper-hydrogen eutectic system [6].
solidification front. qs is the density of the solid metal and MH2 is the molecular weight of H2 . P is the gas pressure (in Pa) in the pore. For Cu–H2 system, Cm and Cs can be expressed as [8]: Cm ¼ nðT Þ
pffiffiffiffiffiffiffi P H2
It can be seen from Fig. 4 that, if the pure hydrogen is used singly, the porosity first increases up to a maximum, then starts to drop with increasing hydrogen pressure. In most cases, the Gasar process uses the mixture of hydrogen and argon because that the hydrogen bubbles are not easy to form and float away before the solidifying front under this kind of condition [10]. In other words, the use of the mixed gases favors the coupled growth between the solid and the gaseous phase, therefore favors the achievement of Gasar structures. In this situation, the porosity increases continually with rising hydrogen partial pressure at a constant argon partial pressure. This dependence of porosity on the partial pressures of argon and hydrogen arises because the amount of hydrogen dissolved in the melt is proportional to the square root of hydrogen pressure, whereas the density of hydrogen in the pore is proportional to the total applied pressure. As for the influence of the melt temperature on the porosity, it is apparent that the porosity increases continually with the rise of the melt temperature because that a higher melt temperature
pffiffiffiffiffiffiffi P H2
ð11Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pg þ Pc þ Ph
ð12Þ
0.6
where PH2 is the partial pressure of H2 . Pg is the total gas pressure in the furnace atmosphere. In general, the height of the casting is not higher than 0.2 m in the Gasar solidification. If the average height is selected as 0.125 m, then the average hydrostatic pressure Ph is about 104 Pa for Cu–H2 system (Ph ¼ ql gh ¼ 8000 10 0:125 104 Pa). The pore diameter ranges from tens of microns to several millimeters in the Gasar structures [17]. If the average radius is selected as 30 microns, the average capillary pressure Pc is 8.5 · 104 Pa for Cu–H2 system (Pc ¼ 2rl =rg ¼ 2 1:28=30 106 8:5 104 Pa).
0.5
¼ 0:72159 expð5234=T Þ pffiffiffi Cs ¼ n0 ðT Þ P ¼ 0:43399 expð5888=T Þ
Pg=PH , PAr=0 Pa, T=1523K
Porosity (e)
2
0.4 0.3 5
Pg=PH +PAr=8.08x10 Pa, T=1523K
0.2
2
0.1 0.0 0.0
2.0x10
5
5
4.0x10 6.0x10 PH /Pa
5
8.0x10
5
2
Fig. 4. Influence of the partial pressure of hydrogen on the porosity of Gasar Cu.
Table 1 The parameters of Cu–H2 system used for calculation ml (K/mol m3 )
DHðliquid
0.01026 [6]
1:46 102 expð18900=RT Þ [14]
CuÞ
(cm2 /s)
rls (1356 K) (J/m2 )
DHm (J/g)
qs (g/cm3 )
ql (1356 K) (g/cm3 )
0.178 [15]
211.85 [16]
8.9
8.0
Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386
will lead to a higher solubility of hydrogen in the molten metal (see Eq. (11)). 3.2. Solute distribution According to Eqs. (5)–(10), the solute distribution in front of the solidifying front and the average hydrogen content at the solidifying front can be calculated. The two-dimensional solute distributions at different solidification rates are shown in Fig. 5(a), (b). It can be found that there exists a slight hydrogen concentration gradient in the radial direction, but the strongest concentration gradient develops in the z axial direction when the solidification rate is as small as 50 lm/s. With increasing solidification rate, a sharp concentration gradient also occurs in the radial direction. It is obvious that the diffusion of the hydrogen atom in the radial direction will become a restrictive factor for the growth of the hydrogen pore. In this situation, rs and rg will be automatically adjusted to maintain the cooperative growth between the solid and the gaseous phase. This point is the major reason why the inter-pore spacing decreases with rising solidification rate discussed in Section 3.3. 3.3. Inter-pore spacing By using the minimum undercooling criterion as that in the Jackson–Hunt model, the required undercooling for the growth of the solid phase and the inter-pore spacing can be calculated. However, for the metal–gas eutectic solidification, the
383
growth of the gas pore does not need any undercooling, so that only the undercooling for the growth of the solid phase is calculated here. Fig. 6 shows the interfacial energies balance between the solid, the liquid and the gas phases. Generally speaking, the surface energy of the solid phase ðrs Þ is much higher than that of the liquid (rl ) and the interfacial energy of liquid/solid (rls ). With rs P rls þ rl , the average curvature radius (rsl ) of the solid/liquid interface should be written pffiffi as rsl ¼ ðrs rg Þ ¼ ð1 eÞrs . The required undercooling for the growth of the solid can be described as:
DT ¼ ml ðCE C ls Þ þ
rls TE pffiffi DHm ð1 eÞrs qs
ð13Þ
where ml is the slope of liquidus in metal–gas eutectic phase diagram. DHm and qs are the fusion enthalpy and the density of the solid metal, respectively. CE and TE are the eutectic composition and the eutectic temperature of the metal–gas eutectic system, respectively. It can be seen from Eq. (9) that C ls is not a simple function of rs , so that it is not possible to obtain a simple relationship––L ¼ A vn (L ¼ 2rs , L is the inter-pore spacing) as that in the Jackson– Hunt model through directly applying oDT =ors ¼ 0 (minimum undercooling criterion) to Eq. (13). But the calculated results in Fig. 7 show that, al though the form of Eq. (9) is complex, actually, C ls has a linear relation with rs at a constant solidification rate. In this case:
Fig. 5. Solute distribution in front of the solidification front (T ¼ 1400 K, rs ¼ 200 lm, PH2 ¼ PAr ¼ 5:05 105 Pa, e ¼ 0:25).
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Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386
only has a linear relation with v, but also the interceptions are zero, so Eq. (14) can be simplified as:
C ls ¼ E þ H v rs
ð15Þ
in which H is a constant dependent on the gas pressure (P ) and the melt temperature (T ). Substituting Eq. (15) into Eq. (13) and applying the minimum undercooling criterion gives: Fig. 6. A schematic diagram of the interfacial energies balance.
34.0 v=500µm/s
Cls* /ppm
33.5 33.0 32.5
v=200µm/s
32.0 31.5
v=100µm/s
31.0 0
20
40
60
rs /µ m
Fig. 7. Relationship between C ls and rs (z ¼ 0, PH2 ¼ PAr ¼ 5:05 105 Pa, T ¼ 1400 K).
0.075
T=1500K, 5 x10 Pa PH =PAr =5.05x10
0.060
2
T=1400K,
0.045 PH =PAr=5.05x105Pa 2
0.030
T=1400K, 5 PH =4.04x10 Pa
0.015
PAr=2.02x10 Pa
2
0.000 0
5
100
200
300
400
500
Fig. 8. Relationship between the parameter, F , in Eq. (14) and the solidification rate.
C ls ¼ E þ F rs
ð14Þ
in which E and F are constants dependent upon P and T . As shown in Fig. 8, the parameter F not
v L2 ¼ A
ð16Þ
where A is a constant dependent on the selected system, the gas pressure (P ) and the melt temperature (T ). It is apparent that this equation is similar to the relation between the lamellar spacing and the solidification rate (L ¼ A vn ) presented in the classical Jackson–Hunt eutectic growth model. Fig. 9(a)–(c) show the effects of the solodification rate, the gas pressure and the melt temperature on the inter-pore and the pore size. It is obvious that the inter-pore spacing and the pore size decrease with increasing solidification rate. Besides the solidification rate, the gas pressure and the melt temperature also influence the inter-pore spacing and the pore size. However, their influences are small, compared with that of the solidification rate. The dropping trend of the inter-pore spacing with increasing hydrogen partial pressure is arrested at high partial pressures. However, the rising trend of the pore size continues with increasing hydrogen partial pressure at a constant total pressure. The inter-pore spacing and the pore size all increase with rising melt temperature. Comparatively, the solidification rate is the major factor influencing the inter-pore spacing and the pore size in the Gasarite eutectic solidification. Only limited quantitative measurements [18] at a solidification rate of 500 lm/s show the average pore size in Gasarite copper is about 30 lm. This value is close to the predicted values of 40 lm under the same condition. In addition, much more qualitative experimental results [19] also show that the inter-pore spacing decreases with increasing solidification rate. However, the lack of more quantitative experimental results prevents the evaluation of the accuracy of Eq. (16).
Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386
385
Fig. 9. Effects of the solidification rate, the gas pressure and the melt temperature on the inter-pore spacing (L) and the pore size (2rg ): (a) effect of the solidification rate, (b) effect of the gas pressure and (c) effect of the melt temperature.
Unlike the classical solid–solid eutectics, the metal–gas eutectics present a special characteristic in which the inter-pore spacing and the pore size are determined not only by the diffusion selfadjustment, but also are functions of the gas pressure and the melt temperature. In addition, the nucleation conditions, the temperature gradient GT and the different thermal conductivities of the solids and the gaseous phases also should have an important influence on the solute distribution and the inter-pore spacing [6,9]. The latter two factors may lead to the solute transferring along the solidification front, therefore enhance the inter-pore spacing. The nucleation will adjust the inter-pore spacing to a smaller level. Further detailed investigations on the influence of these factors remain for future work.
4. Conclusions 1. The gas pressure and the melt temperature in Gasar solidification determine the porosity of the produced Gasar structures. If the pure hydrogen is used singly, the porosity first increases up to a maximum, then starts to drop with increasing hydrogen pressure. If the mixture of hydrogen and inert gas are used, the porosity increases continually with rising hydrogen partial pressure. The porosity increases continually with the melt temperature. 2. A theoretical model has been developed to describe the Gasarite eutectic growth. The calculated results show that the relation between the inter-pore spacing and the solidification rate can be described as a simple expression
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Y. Liu, Y. Li / Scripta Materialia 49 (2003) 379–386
(v L2 ¼ A) as that presented in the Jackson– Hunt model. 3. Unlike the classical solid–solid eutectics, the metal–gas eutectics present a special characteristic in which the inter-pore spacing and the pore size are determined not only by the diffusion self-adjustment, but also are functions of the gas pressure and the melt temperature. The inter-pore spacing decreases and the pore size increases with increasing hydrogen partial pressure. The inter-pore spacing and the pore size all increase with rising melt temperature. Acknowledgements The authors are very grateful to the Postdoctoral Foundation and the Doctoral Foundation of China for the financial support. References [1] Shapovalov VI. US Patent No. 5,181,549 (January 26, 1993). [2] http://www.msm.cam.ac.uk/mmc/people/dave/method2. html.
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