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The microstructure and electrical conductivity of aluminum alloy foams
Materials Chemistry and Physics 78 (2002) 196–201
The microstructure and electrical conductivity of aluminum alloy foams Yi Feng a,b,∗ , Haiwu Zheng a , Zhengang Zhu b , Fangqiou Zu a,b b
a Department of Materials Science and Engineering, Hefei University of Technology, Hefei 230009, China Laboratory of Internal Friction and Defects in Solid, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, China
Received 16 January 2002; received in revised form 25 April 2002; accepted 30 May 2002
Abstract Closed-cell aluminum alloy foams were produced by applying the powder metallurgical technique, i.e. by mixing metal powders and foaming agent (titanium hydride powder) and compacted them into a dense foamable precursor and the foamable precursor material expanded by heating it up to above its melting point. The important geometric parameter that affects the properties of foams, such as cell diameter, edge thickness and edge length, has been examined. The effect of porosity and cell diameter on the electrical conductivity of foams was investigated and the results were compared with a number of models. It was found that the percolation theory can be successfully applied to describe the dependence of the electrical conductivity of aluminum alloy foams on the relative density. The cell diameter has a negligible effect on the electrical conductivity of foams. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Porous materials; Powder metallurgy; Microstructure; Electrical conductivity
1. Introduction Structural efficiency and cost requirements have created increasing interest in cellular metals, especially for aluminum alloy foams. Metallic foams combine the advantages of a metal (strong, tough, deformable and conductive to both electricity and heat) with the structural advantages of a foam (light, stiff and adjustable cell structure) that make them attractive in a number of engineering applications including energy absorption devices, acoustical damping panels, compact heat exchangers, electrical insulating material and electromagnetic wave shields [1–3]. These applications have been made possible by the recent advances in manufacturing processes that lead to improved performance at relatively low cost and by the better understanding about the fundamental behaviors of metal foams having either open- or closed-cell. The mechanical, sound absorption and damping properties of the foam have already been studied and the understandings of the behavior of aluminum alloy foams have been provided in the literatures. Some of these are listed below [4,5]: 3/2 σ ρ ρ (1) = C2 ϕ 3/2 + C2 (1 − ϕ) σs ρs ρs ∗ Corresponding author. Present address: Department of Materials Science and Engineering, Hefei University of Technology, Hefei 230009, China. E-mail address: [email protected] (Y. Feng).
E = C1 ϕ 2 Es
ρ ρs
2
ρ εd = 1 − 1.4 ρs
ρ + C1 (1 − ϕ) ρs
(2)
(3)
where σ , σ s are the yield strength of foam and solid, E, Es are the elastic modulus of foam and solid, ε d is the compaction strain of foam, C1 , C2 are constants, ϕ is the fraction of solid in the foam that is contained in the cell edges and (ρ/ρ s ) is the relative density. It is obvious that the properties of the foam are significantly affected by the properties of solid and cell structure. The transport properties (thermal, acoustic properties) of the foam have been an interesting subject for a long time. The problem relating the conductivity in a two-phase composite to the conductivity of the component phases—a continuous solid phase and a continuous or discontinuous gas phase— has been approached in a number of ways by different authors [9,10]. Some expressions are empirical generalization, while others emerge from consideration of the structure of foam. However, the data and the formulae that described the dependence of electrical properties on the cell structure and on the porosity are still limited. Based on the electrodeposition nickel foams (open-cell), Langlois and Coeuret [11] found the empirical formula connecting electrical conductivity with porosity for porous nickel with porosities ranging from 97 to 97.8%:
0254-0584/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 2 5 4 - 0 5 8 4 ( 0 2 ) 0 0 3 3 4 - 6
Y. Feng et al. / Materials Chemistry and Physics 78 (2002) 196–201
λ = 41 (1 − θ )λs
(4)
where λ and λs are the electrical conductivity of foam and solid, respectively, and θ is the mean porosity of the foam. A theoretical model based on octahedral arrays of wires has been proposed by Liu and Liang [12]: λ=
1−θ λs 3[1 − 0.121(1 − θ )1/2 ]
(6)
However, Eq. (6) shows that λ equals to 0 when θ is 66%. Thus, it is not in agreement with the metal foam feature, which has high porosity. The electrical conductivity of a two-phase composite proposed by Huang is [13] λ = λ1
λ2 (1 + 2K) − 2Kρ1 (λ2 − λ1 ) λ1 (1 + 2K) + ρ1 (λ2 − λ1 )
Table 1 The composition and properties of the powders Powders
Composition (wt.%)
Purity (%)
Mean diameter (m)
Al Si TiH2
88.5 10 1.5
99.6 99.4 99
<43 <43 <43
(5)
where the symbols have the same meaning as above. But these formulae are based on the prerequisite that the cells of the porous body are the open-type, i.e. interconnected, and this is not in agreement with the cell feature of the aluminum foam produced by the powder metallurgical foaming method, i.e. isolated. For closed-cell foams, the expression connecting the electrical conductivity with the porosity of a closed-cell material is [13] λ = (1 − 1.5 θ )λs
197
(7)
where λ, λ1 and λ2 are the electrical conductivities of composite, first component and second component, respectively, ρ 1 is the volume fractions of first component and K is a constant. When second component is cell, λ2 = 0, ρ1 = 1 − θ , so 2K(1 − θ ) λ= (8) λs 2K + θ In Eq. (8), K is a constant determined by the cell structure, when the shape of the cell is spherical and K = 0.3. As summarized above, the dependence of electrical properties on the cell structure and the relative density that characterize the metal foam is lacking. The value of the effective electrical conductivity proves to be useful in structural foam enclosures, in insulating coatings, in mounting panel for electrical components, and in the protective dome of a radar guidance system that is frequently a sandwich structure with a foam core [6,7]. In these and other regards, the electrical property of foam is of critical importance. The objective of this present work is to reveal the geometrical characterization (such as porosity, cell diameter, etc.) and their effects on the electrical behavior of foams.
2. Experimental Metal foams can be produced by a variety of different methods, such as those reviewed by Davies and Zhen [8]. Among these, powder metallurgical methods are especially attractive because they allow for manufacturing inexpensive
closed-cell materials with attractive properties. The composition and properties of the powders used are presented in Table 1. Mixing was conducted in a Turbula powder mixer for 20–30 min. Specimens were uniaxially cold-pressed in a steel die into a cylinder of 95% theoretical density and compaction pressure was 350 MPa. Sintering was conducted in a high-purity nitrogen atmosphere, at temperatures between 600 and 640 ◦ C. This transfers the metal into a semiliquid viscous state and simultaneously causes the foaming agent (TiH2 ) to decompose, thus releasing gas and creating a highly porous structure. By varying the heating temperature and the holding time, the foaming agent could generate the desired amount of gas bubbles and the molten aluminum alloy possesses a certain viscosity (too low a viscosity leads to rapid floating of the bubbles while too high a viscosity results in suppression of the formation of bubbles). So we can obtain a series of samples with different porosity and different cell size. The electrical conductivity, λ, of the aluminum alloy foam was calculated from the geometry and the electrical resistance of the sample. The electrical resistance of foams was measured using a double bridge circuit and dc currents (intensity up to 50 A) with air ventilation at constant temperature (25 ◦ C). The dimension of samples is 200 mm × 30 mm × 30 mm and three or more tests are conducted for each condition.
3. Results and discussion 3.1. Microstructure Micrographs of several aluminum foams are shown in Fig. 1. Foams are described as open or closed-cell. Those used in this investigation are closed-cell, with faces separating the voids of each cell. A hierarchy of structures exists in cellular solids and these can be grouped into two categories: cell structure (geometrical) and material structure. The geometrical features include cell shape and size, distribution in cell size, and imperfections in the cell structure, etc. The material features include the nature and detailed microstructure of the cell wall material. The overall response of the foam is determined by the interaction between these factors for a given condition. So these important geometric parameters and their effects on the properties of foams were investigated in this study. In order to characterize the structure of foams, optical microscope has been used and the appar-
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Y. Feng et al. / Materials Chemistry and Physics 78 (2002) 196–201
Fig. 2. Schematic illustration of a cross-sectional structure in an aluminum alloy foam.
Fig. 3. SEM micrograph of the joint of the cell walls.
unit area S of the foam was counted and was then used to calculate the √ average cell diameter D according to the equation: D = 4S/π N . Fig. 4 shows a histogram of the pore diameter distribution for foams A–C. The characterization data of foams A–C are listed in Table 2. 3.2. Electrical conductivity From Fig. 1 we know, the foam is created by an “infinite” aluminum cluster. This cluster has a complicated structure, defined by the randomly distributed gas pores with various sizes in the metallic matrix. The presence of such a cluster is essential for the existence of the foam. If the cluster does not exist the foam disintegrates and the effective properties Fig. 1. Micrographs of aluminum alloy foams with various cell sizes: (a) foam A; (b) foam B; (c) foam C.
ent edge length (L) and the thickness of cell walls (t) were measured for over 200 edges. The schematic illustration for the measurement is shown in Fig. 2. Fig. 3 shows the joint of three walls; it can be seen that this section is triangular and its thickness is nearly constant except at the immediate vicinity of a node. The number of the cells N in a
Table 2 Characterization data for foams A–C Microstructure parameters
Foam A
Foam B
Foam C
Average density ρ (g cm−3 ) Porosity (1 − ρ/ρs ) (%) Mean cell diameter D (mm) Mean edge thickness t (m) Mean edge length L (mm)
0.512 80.3 1.7 79 0.93
0.507 80.5 2.5 91 1.65
0.505 80.6 3.6 109 2.09
Y. Feng et al. / Materials Chemistry and Physics 78 (2002) 196–201
199
Fig. 5. The dependence of electrical conductivity of aluminum alloy foam on the relative density.
of the relevant component, the constant t is often called the critical exponent [15]. If the foam structure is considered, the threshold can be fixed at zero density (pc = 0). Eq. (9) can then be written as follows: t ρ λ =K (10) λs ρs where λ, λs are the electrical conductivity of foam and solid, ρ/ρs is the relative density (defined as the density of foam, ρ, divided by the density of solid of which it is made, ρ s ) and the constant K ought to be 1 because for ρ = ρs the effective property λ = λs . Introducing the experimental data and λs = 10.87 × 106 ( m)−1 into Eqs. (8) and (10), K and t values can be obtained for each and their mean values are K = 0.3403 and t = 1.504. On substituting the values of K = 0.3403 and t = 1.5 back into Eqs. (8) and (10), we get 0.6806(1 − θ) λs 0.6806 + θ 3/2 ρ λ= λs ρs
λ=
Fig. 4. Distribution of the cell diameter in aluminum alloy foams: (a) foam A (1 − ρ/ρs = 80.3%); (b) foam B (1 − ρ/ρs = 80.5%); (c) foam C (1 − ρ/ρs = 80.6%).
are zero. This situation agrees well with the behavior of effective properties according to percolation theory [14]. The effective property becomes zero at the percolation threshold pc while near pc it behaves as a power of p − pc : π ∝ (p − pc )t
(9)
where π is the effective property and p is the volume fraction
(11)
(12)
Fig. 5 shows that the electrical conductivity of foams (mean cell diameter: 2.5 mm) increases with the increase of relative density. The behavior can be qualitatively understood by noting that, as the relative density decreases, the average cross-section available for conduction decreases and the tortuosity of the current path increases, thus the resistivity increases. Fig. 5 also shows that the experimental results of electrical conductivity are fundamentally consistent with the calculation results of Eq. (12). That is to say, the relationship between electrical conductivity and relative density for the aluminum alloy foams is in agreement with the percolation theory. Table 3 gives the experimental and calculated values of electrical conductivity from the Eqs. (11) and (12). In
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Table 4 Electrical conductivity of aluminum alloy foams with different cell diameter Sample
Density (g cm−3 )
Porosity (%)
Cell diameter (mm)
Electrical conductivity λ × 106 (m)−1
Relative deviation |λ/λT | (%)
Foam A Foam B Foam C
0.512 0.507 0.505
80.3 80.5 80.6
1.7 2.5 3.6
1.003 0.988 0.981
5.19 5.26 5.30
Table 3 |λ| is the deviation of electrical conductivity between experimental results and calculation results (|λ| = |experimental value − calculated value|) and |λ/λT | is the relative deviation of electrical conductivity experimental value − calculated value × 100 experimental value From Table 3 we can see that the relative deviation range of Eq. (11) is small (from 8.98 to 0.95%) when the porosity is low (from 68.7 to 82.9%) and the relative deviation increases rapidly when the porosity is more than 82.9%. Eq. (11) originally developed for powder metallurgy materials and if we regard the isolated second component as cell, Eq. (11) is obtained. As the isolated second component of the traditional powder materials is not too high, so Eq. (11) is suitable for aluminum alloy foams with low porosity and does not cause a large relative deviation. For aluminum alloy foams with high porosity, the experimental values do not conform well to Eq. (11). The conductivity critical exponent of Eq. (9), t, is a dynamic exponent. The value of 1.3 is relatively well established for two dimensions. In the three-dimensional case, the reported results are more controversial: Monte Carlo gave the value for t = 1.6 ± 0.1 [16], Isichenko [15] shifted the value of t up to 1.7–2.0. Gibson and Ashby [6] reported t = 1.5 which is in agreement with the exponent of 1.504, experimentally obtained from this study for the electrical conductivity of the aluminum alloy foam. Table 3 shows that the variation range of the relative deviation of Eq. (12) (11.77–0.15%) is smaller than that of Eq. (11) (28.32–0.95%) and the mean relative deviation of Eq. (12) (3.51%) is also smaller than that of Eq. (11) (9.47%). It is important to note that the actual state is not the same as that of theoretical model, for example the porosity of the foam is far above the proposed percolation threshold (zero density) and the aluminum cluster in the foam is not really infinite. Typical morphological defects, including broken cell walls, missing cells, inclusion, cell wall curvature and corrugations, also affects the electrical conductivity of foams. These factors are found to be responsible for the deviation between experimental results and calculation results of the electrical conductivity of aluminum alloy foams. Miyoshi et al. [17] reported that the mechanical properties are also affected by the geometry of cells. However, there are few experimental data to correlate the cell size with the electrical conductivity. The electrical conductivity of foams
with different cell diameter is shown in Table 4. Table 4 shows that when the sample possesses a roughly equal density, the electrical conductivity is almost identical for foams A–C and there is little difference in the relative deviation in these foams. Therefore, it is suggested that the cell diameter had a minor influence on the electrical conductivity of foams. 4. Conclusion Aluminum alloy foams with different densities and different cell diameters have been fabricated by using powder metallurgy technique. The morphology of foams was examined using optical microscope and the results revealed relative uniform, closed-cell structures. The electrical conductivity of aluminum alloy foams with different relative density and cell diameter was investigated. It was shown that the relative density is the most important variable and the power law function can be successfully applied to describe the dependence of electrical conductivity of aluminum alloy foams on the relative density, confirming the suitability of the percolation theory approach. Whereas, the cell diameter has a minor influence on the electrical conductivity of foams and appears to have a negligible effect. References [1] E. Andrews, W. Sanders, L.J. Gibson, Mater. Sci. Eng. A 270 (1999) 113. [2] Y. Sugimura, J. Meyer, A.G. Evans, Acta Mater. 45 (1997) 5245. [3] H. Kanahashi, T. Mukai, Y. Yamada, K. Higashi, Mater. Sci. Eng. A 308 (2001) 287. [4] J. Banhart, J. Baumeister, J. Mater. Sci. 33 (1998) 1431. [5] L.J. Gibson, Annu. Rev. Mater. Sci. 30 (2000) 191. [6] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, Cambridge University Press, Cambridge, UK, 1997. [7] A.G. Evans, J.W. Hutchinson, M.F. Ashby, Prog. Mater. Sci. 43 (1999) 171. [8] G.J. Davies, S. Zhen, J. Mater. Sci. 18 (1983) 1899. [9] T.J. Lu, C. Chen, Acta Mater. 47 (1999) 1469. [10] A.G. Leacht, J. Phys. D: Appl. Phys. 26 (1993) 733. [11] S. Langlois, F. Coeuret, J. Appl. Electrochem. 19 (1989) 43. [12] P.S. Liu, K.M. Liang, Mater. Sci. Technol. 16 (2000) 341. [13] P.Y. Huang, Principles of Powder Metallurgy, Metallurgical Industry Press, Beijing, 1997. [14] J. Kovacik, F. Simancik, Scripta Mater. 39 (1998) 1055. [15] M.B. Isichenko, Rev. Mod. Phys. 64 (1992) 961. [16] S. Kirkpatrick, Rev. Mod. Phys. 45 (1973) 574. [17] T. Miyoshi, M. Itoh, T. Mukai, Scripta Mater. 41 (1999) 1055.
The microstructure and electrical conductivity of aluminum alloy foams Yi Feng a,b,∗ , Haiwu Zheng a , Zhengang Zhu b , Fangqiou Zu a,b b
a Department of Materials Science and Engineering, Hefei University of Technology, Hefei 230009, China Laboratory of Internal Friction and Defects in Solid, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, China
Received 16 January 2002; received in revised form 25 April 2002; accepted 30 May 2002
Abstract Closed-cell aluminum alloy foams were produced by applying the powder metallurgical technique, i.e. by mixing metal powders and foaming agent (titanium hydride powder) and compacted them into a dense foamable precursor and the foamable precursor material expanded by heating it up to above its melting point. The important geometric parameter that affects the properties of foams, such as cell diameter, edge thickness and edge length, has been examined. The effect of porosity and cell diameter on the electrical conductivity of foams was investigated and the results were compared with a number of models. It was found that the percolation theory can be successfully applied to describe the dependence of the electrical conductivity of aluminum alloy foams on the relative density. The cell diameter has a negligible effect on the electrical conductivity of foams. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Porous materials; Powder metallurgy; Microstructure; Electrical conductivity
1. Introduction Structural efficiency and cost requirements have created increasing interest in cellular metals, especially for aluminum alloy foams. Metallic foams combine the advantages of a metal (strong, tough, deformable and conductive to both electricity and heat) with the structural advantages of a foam (light, stiff and adjustable cell structure) that make them attractive in a number of engineering applications including energy absorption devices, acoustical damping panels, compact heat exchangers, electrical insulating material and electromagnetic wave shields [1–3]. These applications have been made possible by the recent advances in manufacturing processes that lead to improved performance at relatively low cost and by the better understanding about the fundamental behaviors of metal foams having either open- or closed-cell. The mechanical, sound absorption and damping properties of the foam have already been studied and the understandings of the behavior of aluminum alloy foams have been provided in the literatures. Some of these are listed below [4,5]: 3/2 σ ρ ρ (1) = C2 ϕ 3/2 + C2 (1 − ϕ) σs ρs ρs ∗ Corresponding author. Present address: Department of Materials Science and Engineering, Hefei University of Technology, Hefei 230009, China. E-mail address: [email protected] (Y. Feng).
E = C1 ϕ 2 Es
ρ ρs
2
ρ εd = 1 − 1.4 ρs
ρ + C1 (1 − ϕ) ρs
(2)
(3)
where σ , σ s are the yield strength of foam and solid, E, Es are the elastic modulus of foam and solid, ε d is the compaction strain of foam, C1 , C2 are constants, ϕ is the fraction of solid in the foam that is contained in the cell edges and (ρ/ρ s ) is the relative density. It is obvious that the properties of the foam are significantly affected by the properties of solid and cell structure. The transport properties (thermal, acoustic properties) of the foam have been an interesting subject for a long time. The problem relating the conductivity in a two-phase composite to the conductivity of the component phases—a continuous solid phase and a continuous or discontinuous gas phase— has been approached in a number of ways by different authors [9,10]. Some expressions are empirical generalization, while others emerge from consideration of the structure of foam. However, the data and the formulae that described the dependence of electrical properties on the cell structure and on the porosity are still limited. Based on the electrodeposition nickel foams (open-cell), Langlois and Coeuret [11] found the empirical formula connecting electrical conductivity with porosity for porous nickel with porosities ranging from 97 to 97.8%:
0254-0584/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 2 5 4 - 0 5 8 4 ( 0 2 ) 0 0 3 3 4 - 6
Y. Feng et al. / Materials Chemistry and Physics 78 (2002) 196–201
λ = 41 (1 − θ )λs
(4)
where λ and λs are the electrical conductivity of foam and solid, respectively, and θ is the mean porosity of the foam. A theoretical model based on octahedral arrays of wires has been proposed by Liu and Liang [12]: λ=
1−θ λs 3[1 − 0.121(1 − θ )1/2 ]
(6)
However, Eq. (6) shows that λ equals to 0 when θ is 66%. Thus, it is not in agreement with the metal foam feature, which has high porosity. The electrical conductivity of a two-phase composite proposed by Huang is [13] λ = λ1
λ2 (1 + 2K) − 2Kρ1 (λ2 − λ1 ) λ1 (1 + 2K) + ρ1 (λ2 − λ1 )
Table 1 The composition and properties of the powders Powders
Composition (wt.%)
Purity (%)
Mean diameter (m)
Al Si TiH2
88.5 10 1.5
99.6 99.4 99
<43 <43 <43
(5)
where the symbols have the same meaning as above. But these formulae are based on the prerequisite that the cells of the porous body are the open-type, i.e. interconnected, and this is not in agreement with the cell feature of the aluminum foam produced by the powder metallurgical foaming method, i.e. isolated. For closed-cell foams, the expression connecting the electrical conductivity with the porosity of a closed-cell material is [13] λ = (1 − 1.5 θ )λs
197
(7)
where λ, λ1 and λ2 are the electrical conductivities of composite, first component and second component, respectively, ρ 1 is the volume fractions of first component and K is a constant. When second component is cell, λ2 = 0, ρ1 = 1 − θ , so 2K(1 − θ ) λ= (8) λs 2K + θ In Eq. (8), K is a constant determined by the cell structure, when the shape of the cell is spherical and K = 0.3. As summarized above, the dependence of electrical properties on the cell structure and the relative density that characterize the metal foam is lacking. The value of the effective electrical conductivity proves to be useful in structural foam enclosures, in insulating coatings, in mounting panel for electrical components, and in the protective dome of a radar guidance system that is frequently a sandwich structure with a foam core [6,7]. In these and other regards, the electrical property of foam is of critical importance. The objective of this present work is to reveal the geometrical characterization (such as porosity, cell diameter, etc.) and their effects on the electrical behavior of foams.
2. Experimental Metal foams can be produced by a variety of different methods, such as those reviewed by Davies and Zhen [8]. Among these, powder metallurgical methods are especially attractive because they allow for manufacturing inexpensive
closed-cell materials with attractive properties. The composition and properties of the powders used are presented in Table 1. Mixing was conducted in a Turbula powder mixer for 20–30 min. Specimens were uniaxially cold-pressed in a steel die into a cylinder of 95% theoretical density and compaction pressure was 350 MPa. Sintering was conducted in a high-purity nitrogen atmosphere, at temperatures between 600 and 640 ◦ C. This transfers the metal into a semiliquid viscous state and simultaneously causes the foaming agent (TiH2 ) to decompose, thus releasing gas and creating a highly porous structure. By varying the heating temperature and the holding time, the foaming agent could generate the desired amount of gas bubbles and the molten aluminum alloy possesses a certain viscosity (too low a viscosity leads to rapid floating of the bubbles while too high a viscosity results in suppression of the formation of bubbles). So we can obtain a series of samples with different porosity and different cell size. The electrical conductivity, λ, of the aluminum alloy foam was calculated from the geometry and the electrical resistance of the sample. The electrical resistance of foams was measured using a double bridge circuit and dc currents (intensity up to 50 A) with air ventilation at constant temperature (25 ◦ C). The dimension of samples is 200 mm × 30 mm × 30 mm and three or more tests are conducted for each condition.
3. Results and discussion 3.1. Microstructure Micrographs of several aluminum foams are shown in Fig. 1. Foams are described as open or closed-cell. Those used in this investigation are closed-cell, with faces separating the voids of each cell. A hierarchy of structures exists in cellular solids and these can be grouped into two categories: cell structure (geometrical) and material structure. The geometrical features include cell shape and size, distribution in cell size, and imperfections in the cell structure, etc. The material features include the nature and detailed microstructure of the cell wall material. The overall response of the foam is determined by the interaction between these factors for a given condition. So these important geometric parameters and their effects on the properties of foams were investigated in this study. In order to characterize the structure of foams, optical microscope has been used and the appar-
198
Y. Feng et al. / Materials Chemistry and Physics 78 (2002) 196–201
Fig. 2. Schematic illustration of a cross-sectional structure in an aluminum alloy foam.
Fig. 3. SEM micrograph of the joint of the cell walls.
unit area S of the foam was counted and was then used to calculate the √ average cell diameter D according to the equation: D = 4S/π N . Fig. 4 shows a histogram of the pore diameter distribution for foams A–C. The characterization data of foams A–C are listed in Table 2. 3.2. Electrical conductivity From Fig. 1 we know, the foam is created by an “infinite” aluminum cluster. This cluster has a complicated structure, defined by the randomly distributed gas pores with various sizes in the metallic matrix. The presence of such a cluster is essential for the existence of the foam. If the cluster does not exist the foam disintegrates and the effective properties Fig. 1. Micrographs of aluminum alloy foams with various cell sizes: (a) foam A; (b) foam B; (c) foam C.
ent edge length (L) and the thickness of cell walls (t) were measured for over 200 edges. The schematic illustration for the measurement is shown in Fig. 2. Fig. 3 shows the joint of three walls; it can be seen that this section is triangular and its thickness is nearly constant except at the immediate vicinity of a node. The number of the cells N in a
Table 2 Characterization data for foams A–C Microstructure parameters
Foam A
Foam B
Foam C
Average density ρ (g cm−3 ) Porosity (1 − ρ/ρs ) (%) Mean cell diameter D (mm) Mean edge thickness t (m) Mean edge length L (mm)
0.512 80.3 1.7 79 0.93
0.507 80.5 2.5 91 1.65
0.505 80.6 3.6 109 2.09
Y. Feng et al. / Materials Chemistry and Physics 78 (2002) 196–201
199
Fig. 5. The dependence of electrical conductivity of aluminum alloy foam on the relative density.
of the relevant component, the constant t is often called the critical exponent [15]. If the foam structure is considered, the threshold can be fixed at zero density (pc = 0). Eq. (9) can then be written as follows: t ρ λ =K (10) λs ρs where λ, λs are the electrical conductivity of foam and solid, ρ/ρs is the relative density (defined as the density of foam, ρ, divided by the density of solid of which it is made, ρ s ) and the constant K ought to be 1 because for ρ = ρs the effective property λ = λs . Introducing the experimental data and λs = 10.87 × 106 ( m)−1 into Eqs. (8) and (10), K and t values can be obtained for each and their mean values are K = 0.3403 and t = 1.504. On substituting the values of K = 0.3403 and t = 1.5 back into Eqs. (8) and (10), we get 0.6806(1 − θ) λs 0.6806 + θ 3/2 ρ λ= λs ρs
λ=
Fig. 4. Distribution of the cell diameter in aluminum alloy foams: (a) foam A (1 − ρ/ρs = 80.3%); (b) foam B (1 − ρ/ρs = 80.5%); (c) foam C (1 − ρ/ρs = 80.6%).
are zero. This situation agrees well with the behavior of effective properties according to percolation theory [14]. The effective property becomes zero at the percolation threshold pc while near pc it behaves as a power of p − pc : π ∝ (p − pc )t
(9)
where π is the effective property and p is the volume fraction
(11)
(12)
Fig. 5 shows that the electrical conductivity of foams (mean cell diameter: 2.5 mm) increases with the increase of relative density. The behavior can be qualitatively understood by noting that, as the relative density decreases, the average cross-section available for conduction decreases and the tortuosity of the current path increases, thus the resistivity increases. Fig. 5 also shows that the experimental results of electrical conductivity are fundamentally consistent with the calculation results of Eq. (12). That is to say, the relationship between electrical conductivity and relative density for the aluminum alloy foams is in agreement with the percolation theory. Table 3 gives the experimental and calculated values of electrical conductivity from the Eqs. (11) and (12). In
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Table 4 Electrical conductivity of aluminum alloy foams with different cell diameter Sample
Density (g cm−3 )
Porosity (%)
Cell diameter (mm)
Electrical conductivity λ × 106 (m)−1
Relative deviation |λ/λT | (%)
Foam A Foam B Foam C
0.512 0.507 0.505
80.3 80.5 80.6
1.7 2.5 3.6
1.003 0.988 0.981
5.19 5.26 5.30
Table 3 |λ| is the deviation of electrical conductivity between experimental results and calculation results (|λ| = |experimental value − calculated value|) and |λ/λT | is the relative deviation of electrical conductivity experimental value − calculated value × 100 experimental value From Table 3 we can see that the relative deviation range of Eq. (11) is small (from 8.98 to 0.95%) when the porosity is low (from 68.7 to 82.9%) and the relative deviation increases rapidly when the porosity is more than 82.9%. Eq. (11) originally developed for powder metallurgy materials and if we regard the isolated second component as cell, Eq. (11) is obtained. As the isolated second component of the traditional powder materials is not too high, so Eq. (11) is suitable for aluminum alloy foams with low porosity and does not cause a large relative deviation. For aluminum alloy foams with high porosity, the experimental values do not conform well to Eq. (11). The conductivity critical exponent of Eq. (9), t, is a dynamic exponent. The value of 1.3 is relatively well established for two dimensions. In the three-dimensional case, the reported results are more controversial: Monte Carlo gave the value for t = 1.6 ± 0.1 [16], Isichenko [15] shifted the value of t up to 1.7–2.0. Gibson and Ashby [6] reported t = 1.5 which is in agreement with the exponent of 1.504, experimentally obtained from this study for the electrical conductivity of the aluminum alloy foam. Table 3 shows that the variation range of the relative deviation of Eq. (12) (11.77–0.15%) is smaller than that of Eq. (11) (28.32–0.95%) and the mean relative deviation of Eq. (12) (3.51%) is also smaller than that of Eq. (11) (9.47%). It is important to note that the actual state is not the same as that of theoretical model, for example the porosity of the foam is far above the proposed percolation threshold (zero density) and the aluminum cluster in the foam is not really infinite. Typical morphological defects, including broken cell walls, missing cells, inclusion, cell wall curvature and corrugations, also affects the electrical conductivity of foams. These factors are found to be responsible for the deviation between experimental results and calculation results of the electrical conductivity of aluminum alloy foams. Miyoshi et al. [17] reported that the mechanical properties are also affected by the geometry of cells. However, there are few experimental data to correlate the cell size with the electrical conductivity. The electrical conductivity of foams
with different cell diameter is shown in Table 4. Table 4 shows that when the sample possesses a roughly equal density, the electrical conductivity is almost identical for foams A–C and there is little difference in the relative deviation in these foams. Therefore, it is suggested that the cell diameter had a minor influence on the electrical conductivity of foams. 4. Conclusion Aluminum alloy foams with different densities and different cell diameters have been fabricated by using powder metallurgy technique. The morphology of foams was examined using optical microscope and the results revealed relative uniform, closed-cell structures. The electrical conductivity of aluminum alloy foams with different relative density and cell diameter was investigated. It was shown that the relative density is the most important variable and the power law function can be successfully applied to describe the dependence of electrical conductivity of aluminum alloy foams on the relative density, confirming the suitability of the percolation theory approach. Whereas, the cell diameter has a minor influence on the electrical conductivity of foams and appears to have a negligible effect. References [1] E. Andrews, W. Sanders, L.J. Gibson, Mater. Sci. Eng. A 270 (1999) 113. [2] Y. Sugimura, J. Meyer, A.G. Evans, Acta Mater. 45 (1997) 5245. [3] H. Kanahashi, T. Mukai, Y. Yamada, K. Higashi, Mater. Sci. Eng. A 308 (2001) 287. [4] J. Banhart, J. Baumeister, J. Mater. Sci. 33 (1998) 1431. [5] L.J. Gibson, Annu. Rev. Mater. Sci. 30 (2000) 191. [6] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, Cambridge University Press, Cambridge, UK, 1997. [7] A.G. Evans, J.W. Hutchinson, M.F. Ashby, Prog. Mater. Sci. 43 (1999) 171. [8] G.J. Davies, S. Zhen, J. Mater. Sci. 18 (1983) 1899. [9] T.J. Lu, C. Chen, Acta Mater. 47 (1999) 1469. [10] A.G. Leacht, J. Phys. D: Appl. Phys. 26 (1993) 733. [11] S. Langlois, F. Coeuret, J. Appl. Electrochem. 19 (1989) 43. [12] P.S. Liu, K.M. Liang, Mater. Sci. Technol. 16 (2000) 341. [13] P.Y. Huang, Principles of Powder Metallurgy, Metallurgical Industry Press, Beijing, 1997. [14] J. Kovacik, F. Simancik, Scripta Mater. 39 (1998) 1055. [15] M.B. Isichenko, Rev. Mod. Phys. 64 (1992) 961. [16] S. Kirkpatrick, Rev. Mod. Phys. 45 (1973) 574. [17] T. Miyoshi, M. Itoh, T. Mukai, Scripta Mater. 41 (1999) 1055.