Electrical conductivity of open-cell metal foams K.P. Dharmasena and H.N.G. Wadley Department of Materials Science and Engineering, School of Engineering and Applied Science, University of Virginia, Charlottesville, Virginia 22903 (Received 16 January 2001; accepted 2 January 2002)

Cellular metal foams are of interest because of the ability to tailor their mechanical, thermal, acoustic, and electrical properties by varying the relative density and cell morphology. Here, a tetrakaidecahedral unit-cell approach is used to represent an open-cell aluminum foam and a simplified electrical resistor network derived to model low frequency current flow through the foam. The analysis indicates that for the range of relative densities studied (4–12%), the conductivity of tetrakaidecahedral foams has a linear dependence upon relative density. The distribution of metal in the cell ligaments was found to significantly affect the conductivity. Increasing the fraction of metal at the ends of the ligaments resulted in a decrease in electrical conductivity at a fixed relative density. Low frequency electrical conductivity measurements of an open-cell aluminum foam (ERG Duocel) confirmed the linear dependence upon density, but the slope was smaller than that predicted by the unit-cell model. The difference between the model and experiment was found to be the result of the presence of a distribution of cell sizes and types in real samples. This effect is due to the varying number of ligaments, ligament lengths, and the cross-sectional areas available for current conduction across the cellular structure.

I. INTRODUCTION

The ability to create open cellular metal foams with properties that are dependent on the relative density and cell morphology has led to interest in their use as multifunctional, light-weight, impact and energy/ absorbing structures with high heat transfer coefficients.1 Other uses as load-supporting electrochemical storage structures also appear feasible. Recent studies of the mechanical behavior of metal foams has resulted in a significantly improved understanding of the performance of closed- and open-cell foams.2 However, the electrical properties of metal foams and their dependence upon the foam’s relative density and cell morphology are less well understood. Present-day metal foams are predominantly produced by one of several liquid-phase (melt foaming) or solidphase (powder metallurgical) methods.3,4 Both open- and closed-cell metal foams with a wide range of relative densities and cell morphologies can be made. Cellular metal structures may be characterized by the porosity (relative density), the average pore size, pore shape, the pore orientation, and the degree of pore interconnectivity (open-cell versus closed-cell foams). Here we used a four-point probe method to measure the low-frequency electrical conductivity of ERG Duocel (ERG Materials and Aerospace Corporation, Oakland, CA) open-cell J. Mater. Res., Vol. 17, No. 3, Mar 2002

foams of varying relative density and cell size. A unit cell model was then used to investigate the dependence of the conductivity upon these parameters. II. MATERIALS

A set of open cellular aluminum (6101) samples was obtained from ERG Inc. The electrical properties of an aluminum cellular metal sample of fixed size are influenced by the metal fraction within the sample volume (relative density) and the cell morphology. The samples ranged in relative density from 4–12% and cell size from 5–40 pores per inch. Figure 1 shows a micrograph of one of the ERG Duocel aluminum foams with a relative density of 7.5% and a linear pore density of 20 ppi. It appears that each cell is composed of a collection of cusp-shaped cross-sectional edges or ligaments, which form hexagonal, pentagonal, or quadrilateral faces. The two common polyhedral structures used to represent cellular foams are the dodecahedron and the tetrakaidecahedron (Kelvin cell). The dodecahedron-based structure does not fill space entirely. In this study, a completely space-filling tetrakaidecahedral unit-cell representation of the aluminum foam was used to create a simplified model, which enables the calculation of the “apparent” electrical conductivity of the foam from its ligament properties and relative density. © 2002 Materials Research Society

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K.P. Dharmasena et al.: Electrical conductivity of open-cell metal foams

III. CONDUCTIVITY MEASUREMENTS

Very little experimental data for the electrical resistivity of metal foams have been reported. One study has reported data for closed-cell aluminum foams;5 a second reports data for an open-cell nickel foam.6 A more recent study reported the characterization of open-cell aluminum foam (relative density and pore size) based on multifrequency electrical impedance measurements.7 Here a four-probe method was used to measure the resistivity of ERG Duocel open-cell aluminum foams. In the four-probe method, an inline four-point probe is placed on the surface of a sample sufficiently thick so it can be approximated to be semi-infinite (Fig. 2). A direct current is passed through the specimen between the outer probes (P1 and P4), and the resulting potential difference is measured between the inner probes (P2 and P3). If the sample is sufficiently thick, the electrical resistivity r is given by 2␲ r=

冋

冉冊 V I

1 1 1 1 + − − s1 s3 s1 + s2 s2 + s3

册

,

(1)

where s1, s2, and s3 are the probe spacings as shown in Fig. 2. The electrical conductivity ␴ (units, ⍀−1 ⭈ m−1) is the reciprocal of the measured resistivity. The resistance Rf (in ⍀) of a piece of foam of length l and cross-sectional area A normal to the direction of current flow is given by Rf = r

l l = A ␴A

.

(2)

For comparison of calculated conductivities with experimental values, four samples in the 4–12% relative density range with an inverse pore size of 5 ppi were

selected for measurements. Figure 2 shows the relative placement of the two current probes and two voltage probes used in the four-point measuring technique. It was observed that the smaller cell size samples (10, 20, and 40 ppi) had much thinner ligaments, which did not have sufficient strength for the direct physical attachment of probes. A current of 1 amp was passed through the outer probes using the constant current source of a HAMEG Instruments (Frankfurt, Germany) power supply unit (HM 8142). The voltage drop across the two inner probes was measured with a Hewlett Packard HP34401A multimeter. Table I shows the measured electrical conductivities of four Duocel aluminum samples of different relative densities. IV. CONDUCTIVITY MODEL

Figure 3 shows a tetrakaidecahedral unit-cell representation for an open-cell foam. Each cell has fourteen faces (eight hexagonal and six square), thirty-six edges and twenty-four vertices. A closer look at the micrograph in Fig. 1 shows that the cross sections of the ligaments are actually cusp-shaped. However, since the degree of concavity is small, the curvatures of the edge faces can be ignored, and each ligament can be assumed to be of triangular cross-section. Each cell edge ligament is shared by three cells, and four of these edges meet at a node. There is a tendency for metal to concentrate at the (twenty-four) nodes resulting in a thinning at the center of the ligament. To approximate this condition, I

P1

I

V

s1

P2

s2

P3

s3

P4

Semi-infinite sample medium

Equipotential surface

Current flow line

FIG. 2. Four-point probe method for measuring the electrical conductivity of metal foams.

TABLE I. Measured electrical conductivities of 5 ppi Duocel aluminum foam.

FIG. 1. Scanning electron micrograph of an ERG Duocel aluminum (6101) foam, with a relative density of 7.5% and a linear cell density of 20 ppi. 626

Relative density

Measured electrical conductivity (×106 S/m)

0.049 0.072 0.092 0.116

0.655 0.974 1.368 1.787

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K.P. Dharmasena et al.: Electrical conductivity of open-cell metal foams

where ␳s is the theoretical density of the solid metal or alloy. The volume of a tetrakaidecahedral unit cell is obtained from a truncated octahedron. The unit cell volume Vc is given by Vc = 8公2l3 = 11.314l3 .

(8)

From Eqs. (7) and (8) the density of the foam ␳ can be calculated as ␳=

Mc ␳s关12Vl + 6Vn兴 = Vc Vc = ␳s

FIG. 3. Tetrakaidecahedral unit-cell representation of the ERG opencell foam.

the cross-sectional area of the ligament can be allowed to increase linearly from the middle of the ligament to its two ends (Fig. 4). It can be assumed that the intersection of four of the cell ligaments at a node results in a tetrahedral node volume element. To compare the electrical conductivity to the relative density first requires a relation between the geometry of the cell and its density. Let t1 be the width of a triangular cross-sectional ligament at the two ends (where the vertices form), t2 the width at the ligament’s midpoint and l is its length (Fig. 4). The cross-sectional areas of the ligament at the nodes (A1) and midpoint (A2) are then given by

公3t21

A1 =

4

公3t22

A2 =

4

,

(3)

.

(4)

2.598共t21 + t22兲l + 0.707t31

(9)

.

11.314l3

The relative density of the foam is given by

冉 冊

冉冊

t21 + t22 t1 ␳ = 0.2296 + 0.0625 2 ␳s l l

3

.

(10)

For the foams studied here, t1 Ⰶl. As a result, the last term in Eq. (10) is much less than the first and can be ignored. To calculate the electrical resistance, the open-cell foam can be treated as a network of series and parallel resistors (Fig. 5). For a ligament of constant cross section (A) and resistivity (rs), the ligament resistance Rl is given by Rl =

rs ⭈ l . A

(11)

The volume of a ligament, Vl, is given by, A1 + A2 l A1 + A2 l ⭈ + ⭈ 2 2 2 2 公3l = 0.2165l 共t21 + t22兲 . = 共t21 + t22兲 ⭈ 8

Vl =

l

(5)

The volume, Vn, at each node or vertex is calculated assuming a tetrahedral volume element for which Vn =

公2 12

t31 = 0.1178t31 .

(6)

There are thirty-six ligaments in each tetrakaidecahedral unit cell, each of which is shared between three unit cells. Similarly, the twenty-four vertices of each cell are shared between four unit cells. The mass of metal in each unit cell, Mc, is therefore given by

冋冱

1 Mc = ␳s 3

1 Vl + 4

冱V 册 = ␳ 关12V + 6 V 兴 n

s

l

n

,

(7)

t1

t2

t2 t1

t2

t1 FIG. 4. Triangular ligament of varying cross section.

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K.P. Dharmasena et al.: Electrical conductivity of open-cell metal foams

4

1

I 2

2

I

R

R

R

R

I 2

I 2

R

16

R

6

I 2

I 2

R 9

R

10

R

7

11

I 2

I 2

R

R

R

12

R

R 18

I

R

R

14

R 19

15

16

R R

R 20

I

R

21

I 2

R

I

R

8

R 13

R

R

24

1

I

5

17

4

I

R 15

3

I

I

R

22

R

23

21

24

Equivalent resistance between planes 1-2-3-4 and 21-22-23-24 =

3 R 4

FIG. 5. Resistor network equivalent to the tetrakaidecahedral unit cell.

For a ligament of varying cross section, the resistance is given by integration of the cross-sectional area along the ligament: Rl = rs

兰

l

0

dx = rs A共x兲

兰

l Ⲑ2

0

dx + rs A 共x兲

兰

l

lⲐ2

dx . A共x兲

(12)

If we assume a linear variation in the cross-sectional area of each ligament from the center (A2) to its two ends (A1), then l For 0 艋 x 艋 , 2

R2–22 = R2–6 +

A共x兲 = For

2共A2 − A1兲 ⭈ x + A1 , l

(13)

(14)

Substituting Eqs. (13) and (14) into (12) and integrating leads to the resistance Rl of a ligament: Rl =

| |

A1 rsl ln A2 共A1 − A2兲

.

(15)

The current path through a cellular structure is determined by the orientation of the unit cells with respect to the overall current-flow direction. In general, the orientation will be random, but to illustrate the method, 628

For ligaments of equal length and cross-sectional areas, R2–22 = Rl +

2共A1 − A2兲 ⭈ x + 共2A2 − A1兲 . l

共R6 –10 + R10 –18兲 ⭈ 共R6 –11 + R11–18兲

共R6 –10 + R10 –18兲 + 共R6 –11 + R11–18兲 + R18–22 . (16)

l 艋 x 艋 l, 2 A共x兲 =

consider the case where two opposing square faces have been chosen to apply a voltage difference. For the unit cell shown in Fig. 3 there are eight current paths. For the path starting at node 2, it can be seen that the series resistors from nodes 6 to 10 and 10 to 18 are in parallel with the series resistors, 6 to 11 and 11 to 18 (Fig. 5). The resistance from nodes 2 to 22 can then be computed from simple expressions for series and parallel resistor networks:

共Rl + Rl兲 ⭈ 共Rl + Rl兲 + Rl = 3Rl 共Rl + Rl兲 + 共Rl + Rl兲

.

(17)

From symmetry, R1–21 = R3–23 = R4–24 = R2–22 .

(18)

The equivalent unit cell resistance Ruc can then be calculated as 1 1 1 1 4 1 = + + + = Ruc R1–21 R2–22 R3–23 R4 –24 3Rl Ruc =

3Rl . 4

J. Mater. Res., Vol. 17, No. 3, Mar 2002

,

(19)

(20)

K.P. Dharmasena et al.: Electrical conductivity of open-cell metal foams

If the volume occupied by the tetrakaidecahedral unit cell is replaced by a homogenous conductor of the same equivalent resistance, rapp ⭈ a , Auc

Ruc =

(21)

where rapp is the apparent resistivity of the equivalent homogenous solid, a is the distance between two parallel square faces of the tetrakaidecahedron and Auc is the average unit-cell cross-sectional area. From geometrical relationships for the truncated octahedron, a = 2公2l Auc

(22)

,

Vuc 8公2l3 = 4l 2 = = a 2公2l

(23)

.

From Eqs. (20), (21), (22), and (23), the apparent electrical conductivity ␴app is given by ␴app =

1 rapp

=

4 2公2l 0.9428 ⭈ = 3Rl Rl ⭈ l 4l2

.

(24)

For a ligament of constant cross section, substituting Eqs. (11) and (3) into (24), ␴app =

0.9428 rs ⭈ l2

Al =

冉冊

0.4082 t rs l

2

.

(25)

The relative conductivity for a cell with ligaments of uniform cross section is given by

冉冊

t ␴app = 0.4082 ␴s l

2

(26)

.

For a ligament of varying cross section, substituting Eqs. (15), (3), and (4) into (24), ␴app =

0.9428 ⭈ 共A1 − A2兲

||

A1 rs ⭈ l2 ⭈ ln A2

=

0.4082 rs ⭈ ln

|| t 21

冋 册 t 21 − t 22 l2

t 22

,

(27)

and the relative conductivity is given by

冋 册

␴app 0.4082 t 21 − t 22 = ␴s l2 t 21 ln 2 t2

||

.

(28)

for the case of a uniform triangular ligament (t1 ⳱ t2) at relative densities of 0.04, 0.06, 0.08, 0.10, and 0.12. Since the side length of each ligament (t1 or t2) is less than the ligament length (l), the contribution from the first term on the right hand side of Eq. (10), with a squared dependence on the aspect ratio, was found to far exceed that of the second term. The contribution of the second term (although relatively insignificant) was found to increase as the relative density increased from 4% to 12%. The calculated aspect ratios were then used with Eq. (25) to compute the apparent electrical conductivities of the foam. An electrical resistivity of 3 × 10−8 ⍀m was used for the 6101 aluminum alloy contained in the Duocel aluminum foam.8 The effect of a varying cross section of the ligament was investigated for two other ratios of t1/t2 (representative of the ligament cross-sectional area change observed from Duocel aluminum samples) to first compute the equivalent aspect ratio for a known relative density [from Eq. (10)] and then the electrical conductivity [from Eq. (27)]. Table II gives the apparent conductivity values calculated for relative densities of 0.04, 0.06, 0.08, 0.10, and 0.12 for all three t1/t2 ratios. Figure 6 shows the comparison between the conductivity prediction for a uniform cross-section ligament model and the experimental data. Both the model and experiment indicate that the electrical conductivity increases linearly with relative density for this range of ERG Duocel aluminum foam densities. However, it is evident that the uniform cross-section ligament model significantly overestimates the electrical conductivity. The effect of a varying cross-section ligament on the electrical conductivity–relative density relationship is plotted in Fig. 7 for three ligament cross sections of end-to-middle area ratios of 1, 2.25, and 4, corresponding to t1/t2 ratios of 1, 1.5, and 2. [The high ratio of 4 is more representative of a smaller cell size sample (30–40 ppi) where metal mass is clustered more toward the nodes along with thin ligaments.] In all three cases, the conductivity shows a linear relationship with relative density. It is apparent that a change in ligament cross section has a smaller effect on apparent electrical conductivity TABLE II. Apparent electrical conductivities of Duocel aluminum foam. Apparent electrical conductivity (× 106 S/m)

V. DISCUSSION

For the open-cell foam, the relative density is determined by the metal contained in the ligament structure. Equation (10) shows that the relative density is dependent on the aspect ratio of the ligaments. Equation (10) was first solved iteratively for this unknown aspect ratio

Relative density

t1 ⳱ t2

t1 ⳱ 1.5t2

t1 ⳱ 2t2

0.04 0.06 0.08 0.10 0.12

1.140 1.695 2.248 2.792 3.331

1.062 1.563 2.068 2.569 3.044

0.954 1.406 1.849 2.308 2.739

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change at lower relative densities than at higher densities. For a relative density of 4%, a change from a uniform cross-section ligament to one that had a node to midlength area ratio of 4:1 resulted in a decrease of 16.3%, and for a relative density of 12%, the decrease was found to be 17.8%. For a constant relative density, the mass of the unit cell was preserved, and redistribution of mass

FIG. 6. Comparison of the electrical conductivity–relative density relationship for a uniform triangular ligament model with experimental data for a five-ppi ERG Duocel foam made from 6101 aluminum alloy.

from the ligament center to the nodes resulted in a thinning of the ligament at its mid-length (a reduction of the area of current conduction), causing an increase in ligament resistance and the apparent reduction of its conductance. In our model, a Kelvin (or tetrakaidecahedral) unit-cell representation of the foam structure is used. The Kelvin cell has thirty-six edges and twenty-four nodes, which form eight hexagonal and six square faces. The model assumes that the foam consists of a collection of monodispersed cells. A more accurate representation of the foam would require measurements of the individual cells and a statistical distribution of their sizes since the foam may have a polydispersed structure. In addition, cells may have missing or damaged ligaments, which would alter the current conduction paths. To illustrate this case, two (of the thirty-six) ligaments (2–6 and 4–8) were removed (Fig. 5). Since then there was no current flow from nodes 2 to 6 and 4 to 8, effectively ligaments 6–10, 6–11, 8–14, and 8–15 became inactive elements in the circuit. Thus current flow from nodes 1 and 3 had to take additional circuitous paths 1-5-9-10-18-22, 1-5-16-15-20-24, 3-712-11-18-22, and 3-7-13-14-20-24. The effective resistance between planes 1-2-3-4 and 21-22-23-24 was then 1.125 times the individual ligament resistance, which is a 50% increase from the Kelvin unit cell resistance. The effect of this variation on the relative electrical conductivity is plotted for the ligament varying in cross-section with an edge-to-center cross-sectional area ratio of 4 (Fig. 8). We observed that the model with two removed ligaments had a closer agreement with experimental

FIG. 7. Effect of varying ligament cross section on the electrical conductivity response.

FIG. 8. Effect of two missing ligaments on the electrical conductivity response.

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K.P. Dharmasena et al.: Electrical conductivity of open-cell metal foams

observations. The number of missing or damaged ligaments and their locations selected within the modelled unit cell contributed to the degree of reduction in the electrical conductivity from the idealized cell toward the measured foam conductivity. The analysis indicated that the available conduction paths (their resistance per unit length and tortuosity) had a significant effect on the electrical conductivity of stochastic metal foams. The relatively simple model of the metal foam utilized here reproduced basic trends with relative density but quantitative predictions would require the use of pore morphologies that more precisely model the polydispersity of real stochastic cellular structures. For example, a Weaire–Phelan-type structure, which has both dodecahedral and tetrakaidecahedral cells,9 combined with a realistic description of the stochastic nature of the distribution of cell sizes could lead to improved predictions.

VI. CONCLUSIONS

The electrical conductivity of open-cell aluminum foams was investigated using a unit-cell representation of the cellular structure. For low relative density open cell foam (4–12%), the electrical conductivity was found to have a linear dependence on the foam’s relative density. The calculated values were found to be higher than measurements obtained with a four-point probe. The simple model enabled the investigation and prediction of trends of the electrical conductivity response with relative density, cell size, and ligament material properties. A

more detailed morphological representation of the cellular structure is needed to capture effects of cell-size variations. ACKNOWLEDGMENTS

This work was performed as part of the research of the Multidisciplinary University Research Initiative (MURI) program on Ultralight Metals Structures conducted by a consortium that includes Harvard University, Massachusetts Institute of Technology, University of Virginia, and Cambridge University, United Kingdom. The consortium work was supported by an Office of Naval Research grant monitored by Dr. Steven Fishman. REFERENCES 1. M.F. Ashby, A.G. Evans, N.A. Fleck, L.J. Gibson, J.W. Hutchinson, and H.N.G. Wadley, Metal Foams: A Design Guide (ButterworthHeinemann, Boston, MA, 2000). 2. A.G. Evans, J.W. Hutchinson, and M.F. Ashby, Harvard University, Report MECH-323 (1998). 3. V. Shapovalov, MRS Bull. 19(4), 24 (1994). 4. G.J. Davies and S. Zhen, J. Mater. Sci. 18, 1899 (1983). 5. Mepura Data Sheet Metallpulvergesellschaft m.b.H., Randshofern, Austria (1995). 6. S. Langlois and F. Coeuret, J. Appl. Electrochem. 19, 43 (1989). 7. K.P. Dharmasena and H.N.G. Wadley, in Porous and Cellular Materials for Structural Applications, edited by D.S. Schwartz, D.S. Shih, A.G. Evans, and H.N.G. Wadley (Mater. Res. Soc. Symp. Proc. 521, Warrendale, PA, 1998), pp. 171–176. 8. Metals Handbook, Desk edition, edited by H.E. Boyer and T.L. Gall. (American Society for Metals, Metals Park, OH, 1985), pp. 6–32. 9. A.M. Kraynik, M.K. Neilsen, D.A. Reinelt, and W.E. Warren, in Foams and Emulsions, edited by J.F. Sadoc and N. Rivier (Kluwer, Boston, 1999) pp. 259–286.

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