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Modelling of swelling of Fe–Cu compacts sintered at temperatures above the copper melting point
Journal of Materials Processing Technology 152 (2004) 131–135
Modelling of swelling of Fe–Cu compacts sintered at temperatures above the copper melting point Zongyin Zhang∗ , Rolf Sandström, Lingna Wang1 Department of Materials Science and Engineering, Brinell Centre, Royal Institute of Technology, SE-100 44 Stockholm, Sweden Received 20 March 2003; accepted 15 March 2004
Abstract Swelling of Fe–Cu sintered alloy at temperatures above the copper melting point during sintering has been studied for many years. It has been shown that the penetration of liquid copper into iron interparticle boundaries is the dominant mechanism. Based on this mechanism, a model of swelling of Fe–Cu sintered alloys at the copper melting point is established. The particle coordination number is introduced to take into account the porosity. Different heating rates lead to different thicknesses of the diffusion layer between iron and copper particles, which also influences the swelling. In the model, the combined effect of copper content, porosity, particle size and heating rate on the swelling is analysed. The calculated volume and dimensional growths show a qualitative agreement with the published data. The calculated results can be used to predict swelling of Fe–Cu compacts with different copper contents and green densities. © 2004 Elsevier B.V. All rights reserved. Keywords: Powder metallurgy; Fe–Cu alloy; Swelling modelling; Green density; Heating rate; Particle size
1. Introduction Swelling of Fe–Cu compacts during sintering at temperatures above the melting point of copper has been extensively studied during the past four decades [1–5]. The swelling mechanism accepted today is that liquid copper penetrates into iron interparticle boundaries, and leads to the swelling. Pores are left at the copper original sites [6–10]. The swelling that increases with increasing copper content reaches a maximum value, and then decreases with further increasing content. Lawcock and Davies [11] reported that maximum growth occurs at 8 wt.% Cu compact sintered at 1150 ◦ C. Wanibe et al. [12] found that a compact sintered at 1150 ◦ C showed maximum swelling at 10 wt.% Cu. Several authors [1,11–13] suggested that this critical copper content corresponds to the solubility limit of copper in ␥-iron at the sintering temperature. Trudel and Angers [13] proposed that during isothermal sintering, solid phase sintering mechanism is dominant at copper contents lower than the solubility limit of copper in iron. For copper content larger than the solubility limit, a permanent liquid phase sintering occurs. ∗ Corresponding author. Tel.: +46-8-7906544; fax: +46-8-207681. E-mail address: [email protected] (Z. Zhang). 1 Present address: Department for Food Science, Electrolux AB, Stockholm, Sweden.
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.03.025
Other authors [14,15] also studied the mechanism of liquid phase sintering in the Fe–Cu system. However, Lawcock and Davies [11] reported that maximum growth at 8 wt.% Cu is due to boundary coverage rather than excess liquid phase sintering. Actually, results obtained using dilatometer technique [1,2,11,13,16–18] showed that swelling increases significantly when compacts contain low copper content, while swelling increases only slowly when copper content is larger than a certain value. The maximum swelling occurs at a copper content between 10 and 20%, which is determined by the processing parameters and powder characteristics. The swelling is affected by many factors, such as green density of the compact, heating rate, particle size, particle shape and particle size distribution of iron and copper powders, purity of the powders, degree of mixing of powders, and micropores on the surface of iron particles. Efforts have been made [19–25] to quantitatively describe swelling of Fe–Cu and Fe–Cu–C alloys. On the basis of a diffusion mechanism, Pelzel [19] and Elliott [20] calculated the dimensional change and volume change with copper content. Although it is well-known that the mechanism of swelling of Fe–Cu compacts is penetration of liquid copper into iron interparticle boundaries, some points are still useful for today’s consideration. Pelzel [19] calculated the dimensional change from the volume change. Elliott [20] assumed that all of the copper contributed to the swelling when the copper content
132
Z. Zhang et al. / Journal of Materials Processing Technology 152 (2004) 131–135
is lower than 8 wt.%, and gave a relationship between volume swelling (V) and copper content (VCu ): V = 1.08VCu
(1)
A similar equation was also used by Kaysser et al. [21] in calculating penetration of liquid copper along the grain boundaries of iron particles. VCu is in their calculation the measured volume of dissolved Cu in Fe. Griffo et al. [22–24] quantitatively described the dimensional change of Fe–Cu–C compacts after cooling as a function of the specific surface area of iron powder and particle size of copper powder, using regression analysis. In their studies, green density of the compacts, and sintering time were also considered. The computer simulation of sintering of Fe–Cu–C alloy was made based on the mechanism that carbon and copper diffuses into iron, and that molten copper penetrates into interparticle pore spaces and then into iron grain boundaries [25]. The aims of this paper are to use the particle coordination number in the compacts with different green densities to establish a model in which the effects of copper content and green density on swelling of Fe–Cu sintered alloy at temperatures above the copper melting point are considered.
2. Modelling of swelling of Fe–Cu compacts A model was established based on the following conditions: 1. Iron and copper powders have about the same particle size, spherical shape, and uniform deformation during compaction. 2. The swelling mechanism of Fe–Cu alloy is mainly due to penetration of Cu liquid into the boundaries between iron particles. 3. Most of the copper liquid contributes to the swelling except for that consumed by filling the pores close to copper particles and the pores on the surface of iron particles as well as that diffusing into the iron. 4. The amount of volume growth of compacts is equal to the volume of liquid copper, which penetrates into iron interparticle boundaries. When Fe–Cu compacts are heated to the copper melting point, copper particles melt, and a small amount of copper is consumed by diffusion during heating. Liquid copper will fill the pores, which are close to the original copper particles and micropores on the surface of iron particles before penetration occurs. The volume (V) of liquid copper, which penetrates into iron interparticle boundaries, can be expressed as: V = 1.04∗ VCu (1 − fVP)(1 − fVPFe)(1 − fVdiff)(1 − fPCu) (2) where V is thus at the same time the amount of volume growth of compacts, VCu the volume fraction of copper,
fVP the pore factor, the fraction of liquid copper filling the pores close to copper particles, fVPFe the surface factor, the fraction of liquid copper filling the micropores on the surface of iron particles, fVdiff the fraction of copper consumed by diffusion, and fPCu the probability that copper particles are next to each other. The amount of liquid copper, which penetrates into iron interparticle boundaries, is determined by the four factors, fVP , fVPFe , fVdiff , and fPCu . The value of fVPFe is small for iron powders without micropores on the surface as for atomized powder, and fVPFe is neglected. During melting of solid copper at the copper melting point, the volume changes by around 4% (two sources give 4.2 [26] and 3.96% [27], respectively). This is the reason for the factor 1.04. To consider interaction between or among copper particles at higher copper content, a particle coordination number N is used to express the effect of porosity in the compacts. During the compaction of loose powder to certain green density, particles get in contact after elastic and plastic deformation. One particle is surrounded by other particles. With increasing compaction pressure, porosity in the compacts decreases, and the particle coordination number increases. The coordination number is generally around 6–7 in the loose powder. Coordination number N has a relationship with fractional porosity P, which can be expressed as [28]: N = 14 − 10.4P 0.38
(3)
Liquid copper will fill partial pores before it penetrate into the interparticle boundaries of iron. The fraction of liquid copper filling the pores close to copper particles does not contribute to the swelling. This fraction of Cu is given by: fPCu =
NVCu (N + 1)
(4)
The effect of the porosity of copper consumption can be expressed using the porosity of the compacts (P) and copper content: fVP = NPVCu
(5)
The thickness of Fe–Cu diffusion layer is dependent on the particle size of copper as well as the heating rate. The amount of copper consumed by diffusion (fVdiff ) per unit volume can be expressed as: fVdiff =
3d r
(6)
where d is the thickness of Fe–Cu diffusion layer, and r the particle radius. The following equation was used to calculate the thickness of the Fe–Cu diffusion layer: AD0 2 d = A D dt = eQw /RT dT (7) Hr where A ≈ 0.45 is a constant, t the time, D the diffusion coefficient, Hr the heating rate, Qw the activation energy, and T the temperature. The following data was used for Eq. (7):
Z. Zhang et al. / Journal of Materials Processing Technology 152 (2004) 131–135 Table 1 Diffusion distances for different concentrations at different heating rate Diffusion distance Fe into Cu (m)
Diffusion distance Cu into Fe (m)
5 10 20 50
6.9 4.9 3.4 2.2
7.2 5.1 3.6 2.3
1. Diffusion data of copper into iron: D0Cu→Fe = 3.0 × 10−4 m2 /s, QCu→Fe = 255 kJ/mol [29]. 2. Diffusion data of iron into copper: D0Fe→Cu = 1.4 × 10−4 m2 /s, QFe→Cu = 217 kJ/mol [29]. 3. The temperature range used was from 950 ◦ C (above ␣-iron to ␥-iron temperature) to 1083 ◦ C (Cu melting point). The calculated thicknesses of the diffusion layers are shown in Table 1. Majima and Mitani [17] studied Fe–8.0% Cu–0.6% C and Fe–8.0% Cu–0.6% C alloys by heating the compacts to 1080 ◦ C at a heating rate of 10 ◦ C/min, and then quenching them into ice water. Diffusion layers (from 98.5% Fe to 98.5% Cu) with thickness of 6–15 m were observed in the compacts, which is in agreement with the calculated result, 10 m (4.9 + 5.1) in the present study. By combining Eqs. (2)–(6), the volume growth of the compacts can be obtained: 3d NVCu V = 1.04∗ × VCu 1 − [1 − NPVCu ] 1 − r (N + 1) (8)
0.04 0.035 0.03
Linear Swelling
Heating rate (◦ C/min)
0.025
Heating rate
0.02
12 10 20 60 mod 10 mod 20 mod 60
0.015 0.01 0.005 0
0
0.15
0.2
0.25
with decreasing iron particle size and green density. The calculated results for the compacts with different iron powder sizes and porosities using a particle size of 100 m and the comparison with Bockstiegel’s results are shown in Fig. 2 for a heating rate of 5 ◦ C/min and in Fig. 3 for a heating rate of 60 ◦ C/min. The calculated results for compacts with very fine iron particle show shrinkage in Fig. 2, in agreement with Bockstiegel’s results. The calculated results can only partially describe the observations at higher copper contents, since shrinkage due to the liquid phase sintering is neglected. 0.21 45 0.21 150 0.21 300 0.35 45 0.35 20 0.35 7 mod 0. 21 45 mod 0. 21 150 mod 0. 21 300 mod 0. 35 7 mod 0. 35 20 mod 0. 35 45
0.07
where L is the relative dimensional change.
0.06
Linear Swelling
0.05
According to Eqs. (8) and (9), the volume and dimensional changes of Fe–Cu compacts with particle size, green density, heating rate and copper content can be calculated. Different heating rates result in different thickness of the Fe–Cu layer at the iron–copper particle boundaries. The effect of heating rate on the dimensional swelling at varying copper content for heating rates, 10, 20 and 60 ◦ C/min using a particle size of 100 m and 15% porosity compact is calculated and presented in Fig. 1. It can be seen that the predicted influence of the heating rate is about the same as in the experiments. Bockstiegel [2] studied the effect of iron particle size and porosity on the swelling of Fe–Cu compacts using a copper particle size of less than 150 m. When the copper content was less than 8%, the same swelling was observed for the compacts with particle size between 45 and 400 m, while at higher copper contents, the swelling significantly decreases
0.1
Fig. 1. Comparison of the calculated dimensional swelling with observed data for compacts with different copper contents and heating rates, and with a particle size of 100 m and 15% porosity. The legend gives the heating rate in ◦ C/min, and “mod” is short for model.
(9)
3. Results and discussion
0.05
Volume fraction of Cu
Dimensional growth is then given by: L = (1 + V)1/3 − 1
133
0.04 0.03 0.02 0.01 0 -0.01
0
0.05
0.1
0.15
0.2
0.25
Volume fraction of Cu Fig. 2. Comparison of the calculated dimensional swelling with observed data from Bockstiegel [2] for compacts with different copper contents, porosities and particle sizes, at a heating rate of 5 ◦ C/min. In the legend the left hand number represents the porosity, the right hand number the particle size in m. “mod” is short for model. The three highest and lowest represent a porosity of 0.21 and 0.35, respectively.
134
Z. Zhang et al. / Journal of Materials Processing Technology 152 (2004) 131–135 0.07 0.21 45 0.21 150 0.21 300 0.35 45 0.35 20 0.35 7 mod 0. 21 45 mod 0. 21 150 mod 0. 21 300 mod 0. 35 7 mod 0. 35 20 mod 0. 35 45
0.06
Linear Swelling
0.05
0.04
0.03
0.02
0.01
0
0
0.05
0.1
0.15
0.2
0.25
Volume fraction of Cu Fig. 3. Same as Fig. 2 for a heating rate of 60 ◦ C/min.
Krantz [30] measured the dimensional change of Fe–Cu compacts with two porosity ranges (16–23 and 28–30%), and found that when the copper content was less than 5% the porosity did not affect the dimensional change, while when copper content was larger than 6% the dimensional change was significantly influenced by porosity. At 12% Cu content, the difference of dimensional change between low and high porosity was 0.8%. The calculated results for the compacts with 5–30% porosities using a particle size of 100 m and the comparison with Krantz’ results are shown in Fig. 4 for a heating rate of 60 ◦ C/min. When the copper content exceeds 6%, the calculated results show the significant swelling dif-
ference for different porosity compacts, and the same order of magnitude as observed in Krantz’ study. Iron particles completely surround copper particles at lower copper content, and a small of amount of liquid copper fills to the pores close to the copper particles. Hence, the porosity in the compacts has a small effect on the swelling. For the compacts with higher copper content, copper particles contact each other, form a copper network, and the interaction between copper particles increases, which leads to that the swelling at higher copper content varies largely with green density of the compacts. At high copper content, swelling and liquid phase sintering occur at the same time, and the shrinkage resulting from liquid phase sintering offsets a part of the swelling, which explains why the experimental values are lower than the calculated ones. In the model, the shrinkage due to liquid phase sintering has not been taken in account. That the compacts with the finer iron powders and low green density showed small swelling was considered to be the result of the greater ratio of surface to volume involved, and more copper is consumed by diffusion.
4. Conclusions A model for the swelling of Fe–Cu alloys sintered at a temperature above the copper melting point based on the mechanism of penetration of liquid copper into iron interparticle boundaries has been proposed. In the model, the effect of copper content, particle size and heating rate on the swelling is described by using particle coordination number to represent the role of the porosity. The model can qualitatively describe the observed influence of copper content, porosity, particle size and heating rate on the swelling.
References
0.06
Porosity 0.05 0.1 0.15 0.2 0.25 0.3 0.16-0.23 0.28-0.30
Linear Swelling
0.05
0.04
0.03
0.02
0.01
0
0
0.05
0.1
0.15
0.2
0.25
Volume fraction of Cu
Fig. 4. Comparison of the calculated dimensional swelling with observed data from Krantz [30] for the compacts with different copper contents and porosities, and with a particle size of 100 m at a heating rate of 60 ◦ C/min. The experimental values are shown with markers, the model values with curves.
[1] G. Bockstiegel, Dimensional changes during sintering of iron–copper powder mixes and means to reduce them, Metallurgie III (4) (1962) 67–78. [2] G. Bockstiegel, Erscheinungsbild und ursachen von volumenänderungen beim sintern von presslingen aus eisen-kupfer- und einsen-kupfer-graphit-pulvermischungen, Stahl und Eisen 79 (8) (1959) 1187–1201 (in German). [3] K. Tabeshfar, G.A. Chadwick, Dimensional changes during liquid phase sintering of Fe–Cu compacts, Powder Metall. 27 (1) (1984) 19–24. [4] H. Fredriksson, K. Hansson, A. Olsson, On the mechanism of liquid copper penetration into iron grain boundaries, Scand. J. Metall. 30 (2001) 41–50. [5] H. Danninger, Pore formation during sintering of Fe–Cu and its effects on mechanical properties, Powder Metall. Int. 19 (1) (1987) 19–23. [6] F.V. Lenel, T. Pecanha, Observations on the sintering of compacts from a mixture of iron and copper powders, Powder Metall. 16 (32) (1973) 351–365. [7] D. Berner, H.E. Exner, G. Petzow, Swelling of iron–copper mixtures during sintering and infiltration, in: H.H. Hausner, W.E. Smith (Eds.),
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Modern Developments in Powder Metallurgy, Princeton, NJ, vol. 6, 1974, pp. 237–250. N. Dautzenberg, Verfolgung von sintervorgängen im system eisen-kupfer durch dilatometrie und heiztischmikroskopie, Arehiv für das Eisenhüttenwesen 10 (1970) 1005–1010 (in German). B.E. Magee, J. Lund, Mechanisms of liquid-phase sintering in iron–copper powder compacts, Zeitschrift für metallkunde 67 (1) (1976) 596–602. S.J. Jamil, G.A. Chadwick, Investigation and analysis of liquid phase sintering of Fe–Cu and Fe–Cu–C compacts, Powder Metall. 28 (2) (1985) 65–71. R.L. Lawcock, T.J. Davies, Effect of carbon on dimensional and microstructural characteristics of Fe–Cu compacts during sintering, Powder Metall. 33 (2) (1990) 147–150. Y. Wanibe, H. Yokoyama, T. Itoh, Expansion during liquid phase sintering of iron–copper compacts, Powder Metall. 33 (1) (1990) 65–69. Y. Trudel, R. Angers, Properties of iron copper alloys made from elemental or prealloyed powders, Int. J. Powder Metal. Powder Technol. 11 (1) (1975) 5–16. P. Ramakrishnan, R. Lakshminarasimhan, Mechanism of liquid phase sintering in the iron–copper system, Int. J. Powder Metall. 3 (2) (1967) 63–68. W.A. Kaysser, S. Takajo, G. Petzow, Skeleton dissolution and skeleton formation during liquid phase sintering of Fe–Cu, in: H.H. Hausner, H.W. Antes, G.D. Smith (Eds.), Modern Developments in Powder Metallurgy, Metal Powder Industries Federation, Princeton, NJ, vol. 12, 1980, pp. 473–482. H. Mitani, K. Majima, Y. Hanatate, Effect of graphite addition on the abnormal expansion of mixed powder compacts of the Fe–Cu binary system during sintering, J. Jpn. Inst. Met. 40 (4) (1976) 320–327 (in Japanese). K. Majima, H. Mitani, Sintering mechanism in mixed powder compacts of the Fe–Cu–C ternary system, Trans. Jpn. Inst. Met. Appl. 18 (1977) 663–672. N. Dautzenberg, H.J. Dorweiler, Dimensional behaviour of copper–carbon sintered steels, Powder Metal. Int. 17 (6) (1985) 279– 282.
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[19] E. Pelzel, Massänderungen bei eisen-kupfer-sinterlegierungen, Metallwissenschaft und Technik. 9 (7) (1955) 565–569 (in German). [20] J.E. Elliott, Growth of sintered metal compacts, Metallurgia 59 (1) (1959) 17–27. [21] W.A. Kaysser, W.J. Huppmann, G. Petzow, Analysis of dimensional changes during sintering of Fe–Cu, Powder Metall. 2 (1980) 86– 91. [22] A. Griff, J. Ko, R.M. German, Critical assessment of variables affecting the dimensional behaviour in sintered iron–copper–carbon alloys, in: C. Lall, A.J. Neupaver (Eds.), Advances in Powder Metallurgy and Particulate Materials, Metal Powder Industries Federation, Princeton, NJ, vol. 3, 1994, pp. 221–236. [23] A. Griff, R.M. German, Dimensional control in the sintering of iron–copper–carbon via particle surface area, Int. J. Powder Metall. 30 (4) (1994) 399–407. [24] A. Griff, Y. Liu, R.M. German, The effect of green density and particle surface area on the dimensional behaviour of Fe–2Cu–0.8C, in: C. Lall, A.J. Neupaver (Eds.), Advances in Powder Metallurgy and Particulate Materials, Metal Powder Industries Federation, Princeton, NJ, vol. 3, 1994, pp. 207–220. [25] R. Raman, A. Griffo, T.F. Zahrah, R.M. German, Computer simulation of Fe–2Cu–0.8C sintering, in: J.J. Oakes, J.H. Reinshagen (Eds.), Advances in Powder Metallurgy and Particulate Materials, Metal Powder Industries Federation, Princeton, NJ, vol. 3, 1998, pp. 1089–1096. [26] J.C. Bailar, H.J. Emeleus, S.R. Nyholm, A.F. Trotman-Dickenson (Eds.), Comprehensive Inorganic Chemistry, Pergamon Press, 1973, p. 12. [27] T. Iida, R.I.L. Guthrie, The Physical Properties of Liquid Metals, Oxford University Press, New York, 1993, p. 14. [28] R.M. German, Powder Metallurgy Science, 2nd ed., Metal Powder Industries Federation, Princeton, NJ, 1994. [29] C.J. Smithells (Ed.), Metals Reference Book, 4th ed., vol. II, Butterworths, London, 1967. [30] T. Krantz, Effect of density and composition on the dimensional stability and strength of iron–copper alloys, Int. J. Powder Metall. 5 (3) (1969) 35–43.
Modelling of swelling of Fe–Cu compacts sintered at temperatures above the copper melting point Zongyin Zhang∗ , Rolf Sandström, Lingna Wang1 Department of Materials Science and Engineering, Brinell Centre, Royal Institute of Technology, SE-100 44 Stockholm, Sweden Received 20 March 2003; accepted 15 March 2004
Abstract Swelling of Fe–Cu sintered alloy at temperatures above the copper melting point during sintering has been studied for many years. It has been shown that the penetration of liquid copper into iron interparticle boundaries is the dominant mechanism. Based on this mechanism, a model of swelling of Fe–Cu sintered alloys at the copper melting point is established. The particle coordination number is introduced to take into account the porosity. Different heating rates lead to different thicknesses of the diffusion layer between iron and copper particles, which also influences the swelling. In the model, the combined effect of copper content, porosity, particle size and heating rate on the swelling is analysed. The calculated volume and dimensional growths show a qualitative agreement with the published data. The calculated results can be used to predict swelling of Fe–Cu compacts with different copper contents and green densities. © 2004 Elsevier B.V. All rights reserved. Keywords: Powder metallurgy; Fe–Cu alloy; Swelling modelling; Green density; Heating rate; Particle size
1. Introduction Swelling of Fe–Cu compacts during sintering at temperatures above the melting point of copper has been extensively studied during the past four decades [1–5]. The swelling mechanism accepted today is that liquid copper penetrates into iron interparticle boundaries, and leads to the swelling. Pores are left at the copper original sites [6–10]. The swelling that increases with increasing copper content reaches a maximum value, and then decreases with further increasing content. Lawcock and Davies [11] reported that maximum growth occurs at 8 wt.% Cu compact sintered at 1150 ◦ C. Wanibe et al. [12] found that a compact sintered at 1150 ◦ C showed maximum swelling at 10 wt.% Cu. Several authors [1,11–13] suggested that this critical copper content corresponds to the solubility limit of copper in ␥-iron at the sintering temperature. Trudel and Angers [13] proposed that during isothermal sintering, solid phase sintering mechanism is dominant at copper contents lower than the solubility limit of copper in iron. For copper content larger than the solubility limit, a permanent liquid phase sintering occurs. ∗ Corresponding author. Tel.: +46-8-7906544; fax: +46-8-207681. E-mail address: [email protected] (Z. Zhang). 1 Present address: Department for Food Science, Electrolux AB, Stockholm, Sweden.
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.03.025
Other authors [14,15] also studied the mechanism of liquid phase sintering in the Fe–Cu system. However, Lawcock and Davies [11] reported that maximum growth at 8 wt.% Cu is due to boundary coverage rather than excess liquid phase sintering. Actually, results obtained using dilatometer technique [1,2,11,13,16–18] showed that swelling increases significantly when compacts contain low copper content, while swelling increases only slowly when copper content is larger than a certain value. The maximum swelling occurs at a copper content between 10 and 20%, which is determined by the processing parameters and powder characteristics. The swelling is affected by many factors, such as green density of the compact, heating rate, particle size, particle shape and particle size distribution of iron and copper powders, purity of the powders, degree of mixing of powders, and micropores on the surface of iron particles. Efforts have been made [19–25] to quantitatively describe swelling of Fe–Cu and Fe–Cu–C alloys. On the basis of a diffusion mechanism, Pelzel [19] and Elliott [20] calculated the dimensional change and volume change with copper content. Although it is well-known that the mechanism of swelling of Fe–Cu compacts is penetration of liquid copper into iron interparticle boundaries, some points are still useful for today’s consideration. Pelzel [19] calculated the dimensional change from the volume change. Elliott [20] assumed that all of the copper contributed to the swelling when the copper content
132
Z. Zhang et al. / Journal of Materials Processing Technology 152 (2004) 131–135
is lower than 8 wt.%, and gave a relationship between volume swelling (V) and copper content (VCu ): V = 1.08VCu
(1)
A similar equation was also used by Kaysser et al. [21] in calculating penetration of liquid copper along the grain boundaries of iron particles. VCu is in their calculation the measured volume of dissolved Cu in Fe. Griffo et al. [22–24] quantitatively described the dimensional change of Fe–Cu–C compacts after cooling as a function of the specific surface area of iron powder and particle size of copper powder, using regression analysis. In their studies, green density of the compacts, and sintering time were also considered. The computer simulation of sintering of Fe–Cu–C alloy was made based on the mechanism that carbon and copper diffuses into iron, and that molten copper penetrates into interparticle pore spaces and then into iron grain boundaries [25]. The aims of this paper are to use the particle coordination number in the compacts with different green densities to establish a model in which the effects of copper content and green density on swelling of Fe–Cu sintered alloy at temperatures above the copper melting point are considered.
2. Modelling of swelling of Fe–Cu compacts A model was established based on the following conditions: 1. Iron and copper powders have about the same particle size, spherical shape, and uniform deformation during compaction. 2. The swelling mechanism of Fe–Cu alloy is mainly due to penetration of Cu liquid into the boundaries between iron particles. 3. Most of the copper liquid contributes to the swelling except for that consumed by filling the pores close to copper particles and the pores on the surface of iron particles as well as that diffusing into the iron. 4. The amount of volume growth of compacts is equal to the volume of liquid copper, which penetrates into iron interparticle boundaries. When Fe–Cu compacts are heated to the copper melting point, copper particles melt, and a small amount of copper is consumed by diffusion during heating. Liquid copper will fill the pores, which are close to the original copper particles and micropores on the surface of iron particles before penetration occurs. The volume (V) of liquid copper, which penetrates into iron interparticle boundaries, can be expressed as: V = 1.04∗ VCu (1 − fVP)(1 − fVPFe)(1 − fVdiff)(1 − fPCu) (2) where V is thus at the same time the amount of volume growth of compacts, VCu the volume fraction of copper,
fVP the pore factor, the fraction of liquid copper filling the pores close to copper particles, fVPFe the surface factor, the fraction of liquid copper filling the micropores on the surface of iron particles, fVdiff the fraction of copper consumed by diffusion, and fPCu the probability that copper particles are next to each other. The amount of liquid copper, which penetrates into iron interparticle boundaries, is determined by the four factors, fVP , fVPFe , fVdiff , and fPCu . The value of fVPFe is small for iron powders without micropores on the surface as for atomized powder, and fVPFe is neglected. During melting of solid copper at the copper melting point, the volume changes by around 4% (two sources give 4.2 [26] and 3.96% [27], respectively). This is the reason for the factor 1.04. To consider interaction between or among copper particles at higher copper content, a particle coordination number N is used to express the effect of porosity in the compacts. During the compaction of loose powder to certain green density, particles get in contact after elastic and plastic deformation. One particle is surrounded by other particles. With increasing compaction pressure, porosity in the compacts decreases, and the particle coordination number increases. The coordination number is generally around 6–7 in the loose powder. Coordination number N has a relationship with fractional porosity P, which can be expressed as [28]: N = 14 − 10.4P 0.38
(3)
Liquid copper will fill partial pores before it penetrate into the interparticle boundaries of iron. The fraction of liquid copper filling the pores close to copper particles does not contribute to the swelling. This fraction of Cu is given by: fPCu =
NVCu (N + 1)
(4)
The effect of the porosity of copper consumption can be expressed using the porosity of the compacts (P) and copper content: fVP = NPVCu
(5)
The thickness of Fe–Cu diffusion layer is dependent on the particle size of copper as well as the heating rate. The amount of copper consumed by diffusion (fVdiff ) per unit volume can be expressed as: fVdiff =
3d r
(6)
where d is the thickness of Fe–Cu diffusion layer, and r the particle radius. The following equation was used to calculate the thickness of the Fe–Cu diffusion layer: AD0 2 d = A D dt = eQw /RT dT (7) Hr where A ≈ 0.45 is a constant, t the time, D the diffusion coefficient, Hr the heating rate, Qw the activation energy, and T the temperature. The following data was used for Eq. (7):
Z. Zhang et al. / Journal of Materials Processing Technology 152 (2004) 131–135 Table 1 Diffusion distances for different concentrations at different heating rate Diffusion distance Fe into Cu (m)
Diffusion distance Cu into Fe (m)
5 10 20 50
6.9 4.9 3.4 2.2
7.2 5.1 3.6 2.3
1. Diffusion data of copper into iron: D0Cu→Fe = 3.0 × 10−4 m2 /s, QCu→Fe = 255 kJ/mol [29]. 2. Diffusion data of iron into copper: D0Fe→Cu = 1.4 × 10−4 m2 /s, QFe→Cu = 217 kJ/mol [29]. 3. The temperature range used was from 950 ◦ C (above ␣-iron to ␥-iron temperature) to 1083 ◦ C (Cu melting point). The calculated thicknesses of the diffusion layers are shown in Table 1. Majima and Mitani [17] studied Fe–8.0% Cu–0.6% C and Fe–8.0% Cu–0.6% C alloys by heating the compacts to 1080 ◦ C at a heating rate of 10 ◦ C/min, and then quenching them into ice water. Diffusion layers (from 98.5% Fe to 98.5% Cu) with thickness of 6–15 m were observed in the compacts, which is in agreement with the calculated result, 10 m (4.9 + 5.1) in the present study. By combining Eqs. (2)–(6), the volume growth of the compacts can be obtained: 3d NVCu V = 1.04∗ × VCu 1 − [1 − NPVCu ] 1 − r (N + 1) (8)
0.04 0.035 0.03
Linear Swelling
Heating rate (◦ C/min)
0.025
Heating rate
0.02
12 10 20 60 mod 10 mod 20 mod 60
0.015 0.01 0.005 0
0
0.15
0.2
0.25
with decreasing iron particle size and green density. The calculated results for the compacts with different iron powder sizes and porosities using a particle size of 100 m and the comparison with Bockstiegel’s results are shown in Fig. 2 for a heating rate of 5 ◦ C/min and in Fig. 3 for a heating rate of 60 ◦ C/min. The calculated results for compacts with very fine iron particle show shrinkage in Fig. 2, in agreement with Bockstiegel’s results. The calculated results can only partially describe the observations at higher copper contents, since shrinkage due to the liquid phase sintering is neglected. 0.21 45 0.21 150 0.21 300 0.35 45 0.35 20 0.35 7 mod 0. 21 45 mod 0. 21 150 mod 0. 21 300 mod 0. 35 7 mod 0. 35 20 mod 0. 35 45
0.07
where L is the relative dimensional change.
0.06
Linear Swelling
0.05
According to Eqs. (8) and (9), the volume and dimensional changes of Fe–Cu compacts with particle size, green density, heating rate and copper content can be calculated. Different heating rates result in different thickness of the Fe–Cu layer at the iron–copper particle boundaries. The effect of heating rate on the dimensional swelling at varying copper content for heating rates, 10, 20 and 60 ◦ C/min using a particle size of 100 m and 15% porosity compact is calculated and presented in Fig. 1. It can be seen that the predicted influence of the heating rate is about the same as in the experiments. Bockstiegel [2] studied the effect of iron particle size and porosity on the swelling of Fe–Cu compacts using a copper particle size of less than 150 m. When the copper content was less than 8%, the same swelling was observed for the compacts with particle size between 45 and 400 m, while at higher copper contents, the swelling significantly decreases
0.1
Fig. 1. Comparison of the calculated dimensional swelling with observed data for compacts with different copper contents and heating rates, and with a particle size of 100 m and 15% porosity. The legend gives the heating rate in ◦ C/min, and “mod” is short for model.
(9)
3. Results and discussion
0.05
Volume fraction of Cu
Dimensional growth is then given by: L = (1 + V)1/3 − 1
133
0.04 0.03 0.02 0.01 0 -0.01
0
0.05
0.1
0.15
0.2
0.25
Volume fraction of Cu Fig. 2. Comparison of the calculated dimensional swelling with observed data from Bockstiegel [2] for compacts with different copper contents, porosities and particle sizes, at a heating rate of 5 ◦ C/min. In the legend the left hand number represents the porosity, the right hand number the particle size in m. “mod” is short for model. The three highest and lowest represent a porosity of 0.21 and 0.35, respectively.
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Z. Zhang et al. / Journal of Materials Processing Technology 152 (2004) 131–135 0.07 0.21 45 0.21 150 0.21 300 0.35 45 0.35 20 0.35 7 mod 0. 21 45 mod 0. 21 150 mod 0. 21 300 mod 0. 35 7 mod 0. 35 20 mod 0. 35 45
0.06
Linear Swelling
0.05
0.04
0.03
0.02
0.01
0
0
0.05
0.1
0.15
0.2
0.25
Volume fraction of Cu Fig. 3. Same as Fig. 2 for a heating rate of 60 ◦ C/min.
Krantz [30] measured the dimensional change of Fe–Cu compacts with two porosity ranges (16–23 and 28–30%), and found that when the copper content was less than 5% the porosity did not affect the dimensional change, while when copper content was larger than 6% the dimensional change was significantly influenced by porosity. At 12% Cu content, the difference of dimensional change between low and high porosity was 0.8%. The calculated results for the compacts with 5–30% porosities using a particle size of 100 m and the comparison with Krantz’ results are shown in Fig. 4 for a heating rate of 60 ◦ C/min. When the copper content exceeds 6%, the calculated results show the significant swelling dif-
ference for different porosity compacts, and the same order of magnitude as observed in Krantz’ study. Iron particles completely surround copper particles at lower copper content, and a small of amount of liquid copper fills to the pores close to the copper particles. Hence, the porosity in the compacts has a small effect on the swelling. For the compacts with higher copper content, copper particles contact each other, form a copper network, and the interaction between copper particles increases, which leads to that the swelling at higher copper content varies largely with green density of the compacts. At high copper content, swelling and liquid phase sintering occur at the same time, and the shrinkage resulting from liquid phase sintering offsets a part of the swelling, which explains why the experimental values are lower than the calculated ones. In the model, the shrinkage due to liquid phase sintering has not been taken in account. That the compacts with the finer iron powders and low green density showed small swelling was considered to be the result of the greater ratio of surface to volume involved, and more copper is consumed by diffusion.
4. Conclusions A model for the swelling of Fe–Cu alloys sintered at a temperature above the copper melting point based on the mechanism of penetration of liquid copper into iron interparticle boundaries has been proposed. In the model, the effect of copper content, particle size and heating rate on the swelling is described by using particle coordination number to represent the role of the porosity. The model can qualitatively describe the observed influence of copper content, porosity, particle size and heating rate on the swelling.
References
0.06
Porosity 0.05 0.1 0.15 0.2 0.25 0.3 0.16-0.23 0.28-0.30
Linear Swelling
0.05
0.04
0.03
0.02
0.01
0
0
0.05
0.1
0.15
0.2
0.25
Volume fraction of Cu
Fig. 4. Comparison of the calculated dimensional swelling with observed data from Krantz [30] for the compacts with different copper contents and porosities, and with a particle size of 100 m at a heating rate of 60 ◦ C/min. The experimental values are shown with markers, the model values with curves.
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