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Electrical conductivity of open-cell Fe–Ni alloy foams
Journal of Alloys and Compounds 479 (2009) 898–901
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jallcom
Electrical conductivity of open-cell Fe–Ni alloy foams X.L. Huang a,∗ , G.H. Wu a , Z. Lv b , Q. Zhang a , P.C. Kang a , J.F. Leng a a b
School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150080, China Center for Condensed Matter Science and Technology, Harbin Institute of Technology, Harbin 150001, China
a r t i c l e
i n f o
Article history: Received 25 September 2008 Received in revised form 20 January 2009 Accepted 22 January 2009 Available online 6 February 2009 Keywords: Metals and alloys Chemical synthesis Electrical transport
a b s t r a c t The electrical conductivity of open-cell Fe–Ni alloy foams over a range of relative density from 2.3% to 12.8% is investigated in this paper. Data from the present Fe–Ni alloy foam and from the literature are compared to predict the conductivity of porous Fe–Ni foams. All the results show that electrical conductivity of Fe–Ni alloy foams increases with the relative density in the range from 2.3% to 12.8%. Comparison of data with theory shows that the electrical conductivity of the present open-cell Fe–Ni alloy foams is best predicted by Kf /K0 = k(1 − Vp )/[3(1 − 0.121(1 − Vp )1/2 )], which is proposed by Liu based on octahedral array of open-cell nickel foam in the relative density range lower than 10.7%, for that a parameter k is used to eliminate the effect of different production methods and the porosity. In addition, data predictions from Kf /K0 = 0.94428Al /l3 proposed by Dharmasena and Kf /K0 = (1 − Vp )3/2 proposed by Ondracek provide an effective upper and lower bound for experimental data, respectively. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Metallic foams combine the advantages of metal and structural foams that make them attractive in a number of engineering applications such as energy absorption devices, acoustical damping panels, compact heat exchangers, electrical insulating material and electromagnetic wave shields [1–3]. Moreover, studies on mechanical behavior of the aluminum foams have resulted in a significantly profound understanding of the foams performance [4,5]. The most generally used is the model based on single beams in bending and plates in tension proposed by Gibson and Ashby [6]. In this model, pores are considered as cubes or hexagons and result in a quadratic dependence of the modulus of elasticity on the relative density. Recently, the application of alloy foam is also used in secondary batteries, such as Zn–Ni and MH–Ni batteries [7–10]. The electrical conductivity of open-cell foams, which is generally produced by electrodeposition, has been paid more attentions [11]. Kováˇcik and Simancik report the effective mechanics and transport properties of foam with respect to the percolation theory [12,13]. Langlois and Coeuret find the empirical formula between conductivity and relative density with porosities ranging from 97% to 97.8% [14]. Also, some expressions on electrical conductivity of foams produced by powder metallurgical foaming method are suitable to be use in high porosity foams [15,16]. In the present study, open-cell Fe–Ni alloy foams are prepared by electrodeposition method, and the electrical
∗ Corresponding author. Tel.: +86 451 86402375; fax: +86 451 86412164. E-mail address: [email protected] (X.L. Huang). 0925-8388/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2009.01.099
conductivity is also investigated with the relative density ranging from 2.3% to 12.8%. Based on the magnetic solid materials Fe and Ni and the three-dimension structure, this foam has the possible use in the field like the shielding materials in the wide frequency range, the heat exchangers or the batteries.
2. Theory Problems relating to the transport properties of closed- and open-cell foams have been approached in a number of ways by different authors [6,17–21]. Models that have been proposed for closed-cell foams are almost based on the analysis of two-phase composite—a continuous solid phase and a continuous or discontinuous gas phase [17–19]. As for open-cell foams, the cell structure is always simplified as a network of parallel series with respect to the current/heat flow paths [15,20–22]. Goodall points out the Lemlich model on electrical conductivity of microcellular metals that are suitable for the Duocel and Incofoam foams [21]: 1 − Vp Kf = K0 3
(1)
the parameter K refers to the electrical conductivity, with subscript ‘f’ for the foam and 0 for the solid, nonporous material. Parameter Vp is the volume fraction of porous in the foam. In Lemlich model (Eq. (1)), the conductivity is then calculated from the component of the average orientation of the struts parallel to the direction of the electrical field.
X.L. Huang et al. / Journal of Alloys and Compounds 479 (2009) 898–901
899
Table 1 Chemical compositions of Fe–Ni alloy foam. Content (wt%)
Before annealed After annealed
Ni
Fe
Cr
P
S
Other elements (Al, Si, Cu, etc.)
47.7 46.5
50.4 51.1
0.97 0.98
0.2 0.03
0.06 0.01
Balance Balance
A theoretical model based on octahedral arrays of open-cell nickel foam has been proposed by Liu [23,24]: 1 − Vp Kf =k K0 3[1 − 0.121(1 − Vp )1/2 ]
(2)
where k is a constant determined by the preparation technology and the material type. Regardless of the effect of microstructure and defects of ligaments on electrical conductivity, equivalent circuit of the unit octahedron is composed of four series-links connected in parallel (Fig. 1 in Ref. [15]). Dharmasena gives a tetrakaidecahedral model of aluminum foams on electrical conductivity, and hence the open-cell foam is treated as a network of series and parallel resistors (Fig. 5 in Ref. [20]). The relative electrical conductivity is then given by Kf 0.9428 = Al K0 l2
(3) Fig. 1. The SEM of open-cell Fe–Ni alloy foam with a relative density of 12.8%.
where Al (mm2 ) and l (mm) represent the cross-sectional area and the length of the ligament, respectively. Other model related to conductivity and foam structure refers to Ondracek: Kf = (1 − Vp )3/2 K0
(4)
Aluminum foam in Eq. (4) produced by the powder metallurgical foaming method is considered isotropic and can be used in high porous materials [16]. 3. Experimental The precursor polymer foam was coated with nickel and iron successively, the subsequent heat treatment removed the polymer, leaving foam with hollow struts [25]. The as-received Fe–Ni foams were sealed in quartz capsules in a vacuum and heated according to the following protocol: heating the materials at 1100 ◦ C for 5 h, and furnace cooling at 150 ◦ C/h to 350 ◦ C, followed by air cooling to ambient temperature. Then the chemical compositions measured by X-ray fluorescence spectrometry (XRF) before and after annealing were listed in Table 1. It is obvious that the content of impurities, like phosphorus and sulfur after annealing decreased rapidly. The microstructures and cell morphological characteristics were investigated by ZESIS optical microscope (OM) and scanning electron micrographs (SEMs). The length of ligaments l is statistics analyzed from SEM picture. The electrical conductivity of the Fe–Ni open-cell foam was calculated from the geometry and the electrical resistance of the samples. The electrical resistance of foams were measured using four-electrode measurement method with air ventilation at constant temperature (25 ◦ C). Dimension of samples size was 60 mm × 5 mm × 3 mm, and three or more tests were conducted for each condition. The electrical conductivity of solid phase Fe–Ni alloy with the same chemical composition was also tested with the same method, and the electrical conductivity value is K0 = 2.1 × 106 −1 m−1 .
In the present study, hollow ligaments are thought as cylindrical solid ones with equivalent radius r, since that the cylindrical specimens were confirmed to give more accurate results on electrical properties [13]. The equivalent diameter in a tetrakaidecahedral (K) unit cell is then expressed as [26]:
r (mm) = 0.0316
Structure of open-cell Fe–Ni alloy foam with a relative density of 12.8% is shown in Fig. 1. The cell geometry is uniform and the mean diameter of pores is about 0.6 mm. The schematic illustration and optic-micrographs of cross-section of ligament is presented in Fig. 2. As mentioned above, the heat treatment remove polymer, leaving foam with solid nodes and hollow struts.
(5)
where Zf represents the factor of connection [6]. The parameter Sa (g/m2 ) and l (mm) is the area density of foams and average length of ligaments, respectively. Parameter d (mm) is the depth of foams and n is the number of ligaments contained in a Kelvin cell (here n = 36 [6]). The s (g/cm3 ) refers to the density for the solid. The characteristics of Fe–Ni open-cell foams used in this paper are listed in Table 2. 4.2. Electrical conductivity Fig. 3 shows the comparison of electrical conductivity between the data predicted by the models above (Eqs. (1)–(4)) and experiment. As seen, predictions from Dharmasena and Ondracek provide an effective upper and lower bound for data, respectively. Also, there is little difference between the predictions from Eqs. (1), (2) Table 2 Characteristic data of Fe–Ni alloy open-cell foams. Samples
Equivalent radius, r (mm)
Length of ligaments, l (mm)
Aspect ratio, r/l
2.3/80 2.5/40 0.3/13 3.2/30 3.9/90 4.3/40 4.7/30 5.0/40 10.7/90 12.8/110
0.03 0.061 0.126 0.06 0.035 0.072 0.071 0.071 0.048 0.063
0.38 0.58 1.2 0.6 0.33 0.58 0.6 0.58 0.28 0.33
0.079 0.086 0.105 0.1 0.106 0.124 0.118 0.122 0.171 0.191
4. Result and discussion 4.1. Cell structure
11.3Zf Sa l2 dns
900
X.L. Huang et al. / Journal of Alloys and Compounds 479 (2009) 898–901
Fig. 3. Relative electrical conductivity of calculated and experimental value of Fe–Ni foams.
stood by noting that, as the relative density decreases, the average cross-section available for conduction decreases, and lead to the increase of resistivity. Table 3 gives the experimental and calculated relative deviation of electrical conductivity from Eqs. (1)–(4). |K| is the deviation of electrical conductivity between measured and calculated results, and the relative deviation of electrical conductivity is expressed as [27]:
K measured value − calculated value K = measured value f
From Table 3 we can see that the relative deviation of Liu based on octahedral arrays of open-cell nickel foam agrees well with the experimental data than the others in the relative density lower than 10.7%. In fact, model of Liu (Eq. (2)) is from the Eq. (4) of Ondracek that the aluminum foam is produced by the powder metallurgical foaming. However, Liu introduced a parameter k to eliminate the effect of different production methods and the porosity on the electrical conductivity [15,28]. The relationships established between effective properties and relative density is more applicable for the sake of the variable cell feature of different porous materials. The relative deviation range of Lemlich is also small in the relative density range of 3.9–5.0%. Goodall has pointed out the Lemlich model on electrical conductivity of microcellular metals that are suitable for the Duocel and Incofoam foams where the struts are slender, straight, and randomly oriented [21]. In this model, the quantity of material contained in the nodes is negligible compared to the volume contained in the struts. But as shown in Fig. 2, the quantity of material contained in the nodes could not be neglected otherwise it would lead to the deviation of data especially in lower Table 3 Relative deviation of electrical conductivity using different models. Samples Fig. 2. (a) Schematic illustration of a cross-sectional structure of ligament in a Fe–Ni alloy foam. Optic-micrographs of cross-section at point A—the node of ligament (b) and at point B—the struts of ligament (c).
and (4) except that from Dharmasena in relative density lower than 12.8%. Furthermore, this difference between measured value and calculated value from Dharmasena grows closer when the relative density is more than 10.7%. It is also evident that the electrical conductivity of open-cell Fe–Ni alloy foams increases with the relative density from 2.3% to 12.8%. This behavior can be qualitatively under-
2.3/80 2.5/40 0.3/13 3.2/30 3.9/90 4.3/40 4.7/30 5.0/40 10.7/90 12.8/110
Measured electrical conductivity, Kf (×106 −1 m−1 )
Relative deviation |K/Kf | (%) Lemlich
Liu
Dharmasena
Ondracek
0.099 0.012 0.025 0.043 0.028 0.029 0.028 0.032 0.107 0.184
59.39 46.97 16.87 48.49 4.62 2.97 14.77 9.07 31.54 52.45
9.71 17.60 1.44 46.84 8.44 10.24 7.25 17.56 18.34 55.29
283.99 286.07 171.24 42.93 144.02 226.94 201.94 188.359 66.123 20.32
27.48 30.28 56.81 72.36 43.49 35.94 25.52 26.83 32.82 48.97
X.L. Huang et al. / Journal of Alloys and Compounds 479 (2009) 898–901
relative density, in which the electrical conductivity changes greatly with small change in porosity [28]. It is known that calculated data from Eq. (4) agree well with the experimental data of aluminum foam in relative density 4–12% [20], but it is also evident in Table 3 that prediction from Dharmasena deviates much to the measured electrical conductivity of Fe–Ni alloy foam. In Eq. (3), the ratio of cross-sectional area to length of ligament, furthermore the aspect ratio of ligament is thought to play an important role in calculation of electrical conductivity. This seems more reasonable according to Ohm’s Law. In open-cell foams, the change of structure parameter, like the description of the shape of cross-sectional area, the thickness and the length of ligament would influence the aspect ratio of ligament and thus change the current path of open-cell foams. In this paper, the change of thickness of ligament and other parameters are attributed to the change of equivalent radius in a tetrakaidecahedral (K) unit cell to simplify the calculation. Also, in the present study, it is difficult to represent the cross-section of Fe–Ni ligament with solid nodes and hollow struts. Consequently, this would result in a deviation in the representation of equivalent circuit of the unit cell on electrical conductivity in Eq. (3) and thus the deviation from predictions to the value of measured data. Moreover, recent study of Gong indicates that representation of the ligaments as simple uniform cross-section Bernoulli–Euler beams is an oversimplification appropriate for foams of lower relative density [29]. 5. Conclusion The electrical conductivity of open-cell Fe–Ni alloy foams over a range of relative density from 2.3% to 12.8% was investigated in this paper. Comparison of predictions for electrical conductivity from classical open-cell aluminum and nickel foams applied to Fe–Ni highly porous materials. The conclusions can be drawn as follows: (1) all the results indicated that electrical conductivity increased with the relative density in the range of 2.3–12.8% for open-cell Fe–Ni alloy foams; (2) predictions from Dharmasena and Ondracek
901
provided an effective upper and lower bound for experimental data, respectively. Data for open-cell Fe–Ni alloy foam were shown closer to the prediction from Liu than the others in the relative density lower than 10.7%, in which a parameter k was introduced to eliminate the effect of different production methods and the porosity on the electrical conductivity. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
E. Andrews, W. Sanders, L.J. Gibson, Mater. Sci. Eng. A 270 (1999) 113. Y. Sugimura, J. Meyer, A.G. Evans, Acta Mater. 45 (1997) 5245. H. Kanahashi, T. Mukai, Y. Yamada, K. Higashi, Mater. Sci. Eng. A 308 (2001) 287. Y. Yamada, K. Shimojima, et al., Mater. Sci. Eng. A 272 (1999) 455–458. B. Vamsi Krishna, et al., Mater. Sci. Eng. A 452–453 (2007) 178–188. L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, second ed., Cambridge University Press, 1997, pp. 25–37. M. Ma, J.P. Tu, Y.F. Yuan, X.L. Wang, K.F. Li, et al., J. Power Sources 179 (2008) 395–400. J.Y. Qi, P. Xu, Z.S. Lv, X.R. Liu, A.H. Wen, J. Alloys Compd. 462 (2008) 164–169. X.L. Wang, J.P. Tu, S.F. Wang, Y.F. Yuan, et al., J. Alloys Compd. 462 (2008) 220–224. D.H. Yang, B.Y. Hu, S.R. Yang, J. Alloys Compd. 461 (2008) 221–227. Y. Maeda, T. Kawakoe, Japanese Patent 109 597A (1995). J. Kováˇcik, F. Simancik, Scripta Mater. 39 (1998) 239–246. J. Kováˇcik, Scripta Mater. 39 (1998) 153–157. S. Langlois, F. Coeuret, J. Appl. Electrochem. 19 (1989) 43. P.S. Liu, T.F. Li, C. Fu, M. Lü, Rare Mater. Eng. 28 (1999) 260–263. P.Y. Huang, Principles of Powder Metallurgy, Metallurgical Industry Press, Beijing, 1997. G.H. Wu, Z.Y. Dou, D.L. Sun, L.T. Jiang, B.S. Ding, et al., Scripta Mater. 56 (2007) 221–224. T.J. Lu, C. Chen, Acta Mater. 47 (1999) 1469. A.G. Leacht, J. Phys. D: Appl. Phys. 26 (1993) 733. K.P. Dharmasena, H.N.G. Wadley, J. Mater. Res. 17 (2002) 625–631. R. Goodall, et al., J. Appl. Phys. 100 (2006), 044912-1-7. ˜ N. Dukhan, P.D. Quinones-Ramos, E. Cruz-Ruis, M. Vélez-Reyes, E.P. Scott, Int. J. Heat Mass Transfer 48 (2005) 5112–5120. P.S. Liu, K.M. Liang, Mater. Sci. Technol. 16 (2000) 341. P.S. Liu, T.F. Li, C. Fu, Mater. Sci. Eng. A 268 (1999) 208–215. J. Banhart, Prog. Mater. Sci. 46 (2001) 599. X.L. Huang, G.H. Wu, Q. Zhang, Z.Y. Dou, S. Chen, Mater. Sci. Eng. A 497 (2008) 231–234. Y. Feng, H.W. Zheng, Z.G. Zhu, F.Q. Zu, Mater. Chem. Phys. 78 (2002) 196–201. P.S. Liu, H. Chen, K.M. Liang, et al., J. Appl. Electrochem. 30 (2000) 1183–1186. L. Gong, S. Kyriakides, W.-Y. Jiang, Int. J. Solids Struct. 42 (2005) 1355–1379.
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jallcom
Electrical conductivity of open-cell Fe–Ni alloy foams X.L. Huang a,∗ , G.H. Wu a , Z. Lv b , Q. Zhang a , P.C. Kang a , J.F. Leng a a b
School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150080, China Center for Condensed Matter Science and Technology, Harbin Institute of Technology, Harbin 150001, China
a r t i c l e
i n f o
Article history: Received 25 September 2008 Received in revised form 20 January 2009 Accepted 22 January 2009 Available online 6 February 2009 Keywords: Metals and alloys Chemical synthesis Electrical transport
a b s t r a c t The electrical conductivity of open-cell Fe–Ni alloy foams over a range of relative density from 2.3% to 12.8% is investigated in this paper. Data from the present Fe–Ni alloy foam and from the literature are compared to predict the conductivity of porous Fe–Ni foams. All the results show that electrical conductivity of Fe–Ni alloy foams increases with the relative density in the range from 2.3% to 12.8%. Comparison of data with theory shows that the electrical conductivity of the present open-cell Fe–Ni alloy foams is best predicted by Kf /K0 = k(1 − Vp )/[3(1 − 0.121(1 − Vp )1/2 )], which is proposed by Liu based on octahedral array of open-cell nickel foam in the relative density range lower than 10.7%, for that a parameter k is used to eliminate the effect of different production methods and the porosity. In addition, data predictions from Kf /K0 = 0.94428Al /l3 proposed by Dharmasena and Kf /K0 = (1 − Vp )3/2 proposed by Ondracek provide an effective upper and lower bound for experimental data, respectively. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Metallic foams combine the advantages of metal and structural foams that make them attractive in a number of engineering applications such as energy absorption devices, acoustical damping panels, compact heat exchangers, electrical insulating material and electromagnetic wave shields [1–3]. Moreover, studies on mechanical behavior of the aluminum foams have resulted in a significantly profound understanding of the foams performance [4,5]. The most generally used is the model based on single beams in bending and plates in tension proposed by Gibson and Ashby [6]. In this model, pores are considered as cubes or hexagons and result in a quadratic dependence of the modulus of elasticity on the relative density. Recently, the application of alloy foam is also used in secondary batteries, such as Zn–Ni and MH–Ni batteries [7–10]. The electrical conductivity of open-cell foams, which is generally produced by electrodeposition, has been paid more attentions [11]. Kováˇcik and Simancik report the effective mechanics and transport properties of foam with respect to the percolation theory [12,13]. Langlois and Coeuret find the empirical formula between conductivity and relative density with porosities ranging from 97% to 97.8% [14]. Also, some expressions on electrical conductivity of foams produced by powder metallurgical foaming method are suitable to be use in high porosity foams [15,16]. In the present study, open-cell Fe–Ni alloy foams are prepared by electrodeposition method, and the electrical
∗ Corresponding author. Tel.: +86 451 86402375; fax: +86 451 86412164. E-mail address: [email protected] (X.L. Huang). 0925-8388/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2009.01.099
conductivity is also investigated with the relative density ranging from 2.3% to 12.8%. Based on the magnetic solid materials Fe and Ni and the three-dimension structure, this foam has the possible use in the field like the shielding materials in the wide frequency range, the heat exchangers or the batteries.
2. Theory Problems relating to the transport properties of closed- and open-cell foams have been approached in a number of ways by different authors [6,17–21]. Models that have been proposed for closed-cell foams are almost based on the analysis of two-phase composite—a continuous solid phase and a continuous or discontinuous gas phase [17–19]. As for open-cell foams, the cell structure is always simplified as a network of parallel series with respect to the current/heat flow paths [15,20–22]. Goodall points out the Lemlich model on electrical conductivity of microcellular metals that are suitable for the Duocel and Incofoam foams [21]: 1 − Vp Kf = K0 3
(1)
the parameter K refers to the electrical conductivity, with subscript ‘f’ for the foam and 0 for the solid, nonporous material. Parameter Vp is the volume fraction of porous in the foam. In Lemlich model (Eq. (1)), the conductivity is then calculated from the component of the average orientation of the struts parallel to the direction of the electrical field.
X.L. Huang et al. / Journal of Alloys and Compounds 479 (2009) 898–901
899
Table 1 Chemical compositions of Fe–Ni alloy foam. Content (wt%)
Before annealed After annealed
Ni
Fe
Cr
P
S
Other elements (Al, Si, Cu, etc.)
47.7 46.5
50.4 51.1
0.97 0.98
0.2 0.03
0.06 0.01
Balance Balance
A theoretical model based on octahedral arrays of open-cell nickel foam has been proposed by Liu [23,24]: 1 − Vp Kf =k K0 3[1 − 0.121(1 − Vp )1/2 ]
(2)
where k is a constant determined by the preparation technology and the material type. Regardless of the effect of microstructure and defects of ligaments on electrical conductivity, equivalent circuit of the unit octahedron is composed of four series-links connected in parallel (Fig. 1 in Ref. [15]). Dharmasena gives a tetrakaidecahedral model of aluminum foams on electrical conductivity, and hence the open-cell foam is treated as a network of series and parallel resistors (Fig. 5 in Ref. [20]). The relative electrical conductivity is then given by Kf 0.9428 = Al K0 l2
(3) Fig. 1. The SEM of open-cell Fe–Ni alloy foam with a relative density of 12.8%.
where Al (mm2 ) and l (mm) represent the cross-sectional area and the length of the ligament, respectively. Other model related to conductivity and foam structure refers to Ondracek: Kf = (1 − Vp )3/2 K0
(4)
Aluminum foam in Eq. (4) produced by the powder metallurgical foaming method is considered isotropic and can be used in high porous materials [16]. 3. Experimental The precursor polymer foam was coated with nickel and iron successively, the subsequent heat treatment removed the polymer, leaving foam with hollow struts [25]. The as-received Fe–Ni foams were sealed in quartz capsules in a vacuum and heated according to the following protocol: heating the materials at 1100 ◦ C for 5 h, and furnace cooling at 150 ◦ C/h to 350 ◦ C, followed by air cooling to ambient temperature. Then the chemical compositions measured by X-ray fluorescence spectrometry (XRF) before and after annealing were listed in Table 1. It is obvious that the content of impurities, like phosphorus and sulfur after annealing decreased rapidly. The microstructures and cell morphological characteristics were investigated by ZESIS optical microscope (OM) and scanning electron micrographs (SEMs). The length of ligaments l is statistics analyzed from SEM picture. The electrical conductivity of the Fe–Ni open-cell foam was calculated from the geometry and the electrical resistance of the samples. The electrical resistance of foams were measured using four-electrode measurement method with air ventilation at constant temperature (25 ◦ C). Dimension of samples size was 60 mm × 5 mm × 3 mm, and three or more tests were conducted for each condition. The electrical conductivity of solid phase Fe–Ni alloy with the same chemical composition was also tested with the same method, and the electrical conductivity value is K0 = 2.1 × 106 −1 m−1 .
In the present study, hollow ligaments are thought as cylindrical solid ones with equivalent radius r, since that the cylindrical specimens were confirmed to give more accurate results on electrical properties [13]. The equivalent diameter in a tetrakaidecahedral (K) unit cell is then expressed as [26]:
r (mm) = 0.0316
Structure of open-cell Fe–Ni alloy foam with a relative density of 12.8% is shown in Fig. 1. The cell geometry is uniform and the mean diameter of pores is about 0.6 mm. The schematic illustration and optic-micrographs of cross-section of ligament is presented in Fig. 2. As mentioned above, the heat treatment remove polymer, leaving foam with solid nodes and hollow struts.
(5)
where Zf represents the factor of connection [6]. The parameter Sa (g/m2 ) and l (mm) is the area density of foams and average length of ligaments, respectively. Parameter d (mm) is the depth of foams and n is the number of ligaments contained in a Kelvin cell (here n = 36 [6]). The s (g/cm3 ) refers to the density for the solid. The characteristics of Fe–Ni open-cell foams used in this paper are listed in Table 2. 4.2. Electrical conductivity Fig. 3 shows the comparison of electrical conductivity between the data predicted by the models above (Eqs. (1)–(4)) and experiment. As seen, predictions from Dharmasena and Ondracek provide an effective upper and lower bound for data, respectively. Also, there is little difference between the predictions from Eqs. (1), (2) Table 2 Characteristic data of Fe–Ni alloy open-cell foams. Samples
Equivalent radius, r (mm)
Length of ligaments, l (mm)
Aspect ratio, r/l
2.3/80 2.5/40 0.3/13 3.2/30 3.9/90 4.3/40 4.7/30 5.0/40 10.7/90 12.8/110
0.03 0.061 0.126 0.06 0.035 0.072 0.071 0.071 0.048 0.063
0.38 0.58 1.2 0.6 0.33 0.58 0.6 0.58 0.28 0.33
0.079 0.086 0.105 0.1 0.106 0.124 0.118 0.122 0.171 0.191
4. Result and discussion 4.1. Cell structure
11.3Zf Sa l2 dns
900
X.L. Huang et al. / Journal of Alloys and Compounds 479 (2009) 898–901
Fig. 3. Relative electrical conductivity of calculated and experimental value of Fe–Ni foams.
stood by noting that, as the relative density decreases, the average cross-section available for conduction decreases, and lead to the increase of resistivity. Table 3 gives the experimental and calculated relative deviation of electrical conductivity from Eqs. (1)–(4). |K| is the deviation of electrical conductivity between measured and calculated results, and the relative deviation of electrical conductivity is expressed as [27]:
K measured value − calculated value K = measured value f
From Table 3 we can see that the relative deviation of Liu based on octahedral arrays of open-cell nickel foam agrees well with the experimental data than the others in the relative density lower than 10.7%. In fact, model of Liu (Eq. (2)) is from the Eq. (4) of Ondracek that the aluminum foam is produced by the powder metallurgical foaming. However, Liu introduced a parameter k to eliminate the effect of different production methods and the porosity on the electrical conductivity [15,28]. The relationships established between effective properties and relative density is more applicable for the sake of the variable cell feature of different porous materials. The relative deviation range of Lemlich is also small in the relative density range of 3.9–5.0%. Goodall has pointed out the Lemlich model on electrical conductivity of microcellular metals that are suitable for the Duocel and Incofoam foams where the struts are slender, straight, and randomly oriented [21]. In this model, the quantity of material contained in the nodes is negligible compared to the volume contained in the struts. But as shown in Fig. 2, the quantity of material contained in the nodes could not be neglected otherwise it would lead to the deviation of data especially in lower Table 3 Relative deviation of electrical conductivity using different models. Samples Fig. 2. (a) Schematic illustration of a cross-sectional structure of ligament in a Fe–Ni alloy foam. Optic-micrographs of cross-section at point A—the node of ligament (b) and at point B—the struts of ligament (c).
and (4) except that from Dharmasena in relative density lower than 12.8%. Furthermore, this difference between measured value and calculated value from Dharmasena grows closer when the relative density is more than 10.7%. It is also evident that the electrical conductivity of open-cell Fe–Ni alloy foams increases with the relative density from 2.3% to 12.8%. This behavior can be qualitatively under-
2.3/80 2.5/40 0.3/13 3.2/30 3.9/90 4.3/40 4.7/30 5.0/40 10.7/90 12.8/110
Measured electrical conductivity, Kf (×106 −1 m−1 )
Relative deviation |K/Kf | (%) Lemlich
Liu
Dharmasena
Ondracek
0.099 0.012 0.025 0.043 0.028 0.029 0.028 0.032 0.107 0.184
59.39 46.97 16.87 48.49 4.62 2.97 14.77 9.07 31.54 52.45
9.71 17.60 1.44 46.84 8.44 10.24 7.25 17.56 18.34 55.29
283.99 286.07 171.24 42.93 144.02 226.94 201.94 188.359 66.123 20.32
27.48 30.28 56.81 72.36 43.49 35.94 25.52 26.83 32.82 48.97
X.L. Huang et al. / Journal of Alloys and Compounds 479 (2009) 898–901
relative density, in which the electrical conductivity changes greatly with small change in porosity [28]. It is known that calculated data from Eq. (4) agree well with the experimental data of aluminum foam in relative density 4–12% [20], but it is also evident in Table 3 that prediction from Dharmasena deviates much to the measured electrical conductivity of Fe–Ni alloy foam. In Eq. (3), the ratio of cross-sectional area to length of ligament, furthermore the aspect ratio of ligament is thought to play an important role in calculation of electrical conductivity. This seems more reasonable according to Ohm’s Law. In open-cell foams, the change of structure parameter, like the description of the shape of cross-sectional area, the thickness and the length of ligament would influence the aspect ratio of ligament and thus change the current path of open-cell foams. In this paper, the change of thickness of ligament and other parameters are attributed to the change of equivalent radius in a tetrakaidecahedral (K) unit cell to simplify the calculation. Also, in the present study, it is difficult to represent the cross-section of Fe–Ni ligament with solid nodes and hollow struts. Consequently, this would result in a deviation in the representation of equivalent circuit of the unit cell on electrical conductivity in Eq. (3) and thus the deviation from predictions to the value of measured data. Moreover, recent study of Gong indicates that representation of the ligaments as simple uniform cross-section Bernoulli–Euler beams is an oversimplification appropriate for foams of lower relative density [29]. 5. Conclusion The electrical conductivity of open-cell Fe–Ni alloy foams over a range of relative density from 2.3% to 12.8% was investigated in this paper. Comparison of predictions for electrical conductivity from classical open-cell aluminum and nickel foams applied to Fe–Ni highly porous materials. The conclusions can be drawn as follows: (1) all the results indicated that electrical conductivity increased with the relative density in the range of 2.3–12.8% for open-cell Fe–Ni alloy foams; (2) predictions from Dharmasena and Ondracek
901
provided an effective upper and lower bound for experimental data, respectively. Data for open-cell Fe–Ni alloy foam were shown closer to the prediction from Liu than the others in the relative density lower than 10.7%, in which a parameter k was introduced to eliminate the effect of different production methods and the porosity on the electrical conductivity. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
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