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
625
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
J. Mater. Res., Vol. 17, No. 3, Mar 2002
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.
J. Mater. Res., Vol. 17, No. 3, Mar 2002
627
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
J. Mater. Res., Vol. 17, No. 3, Mar 2002
629
K.P. Dharmasena et al.: Electrical conductivity of open-cell metal foams
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.
630
J. Mater. Res., Vol. 17, No. 3, Mar 2002
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.
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