Metal Foams: A Design Guide

192 Metal Foams: A Design Guide
with that of the edges, and since only one third of these conduct in a body
containing a fraction / s of conducting material, the relative conductivity is
simply
�
�
D
s
1
⊲14.5⊳
3 s
where s is the conductivity of the solid from which the foam was made.
As the relative density increases, the nodes make an increasingly large
contribution to the total volume of solid. If the node volume scales as t3 and
that of the edges as t2ℓ, then the relative contribution of the nodes scales as
t/ℓ, oras⊲ / s⊳1/2 . We therefor expect that the relative conductivity should
scale such that
s
D 1
3
�
D 1
�
3
s
s
� � �
1 C 2
�
C 2
�
3
s
s
� 3/2
� 1/2 �
⊲14.6⊳
where the constant of proportionality 2 multiplying ⊲ / s⊳1/2 has been chosen
to make / s D 1when / s D 1, as it obviously must.
Real foams differ from the ideal of Figure 14.3 in many ways, of which
the most important for conductivity is the distribution of solid between cell
edges and nodes. Surface tension pulls material into the nodes during foaming,
forming thicker ‘Plateau borders’, and thinning the cell edges. The dimensionality
of the problem remains the same, meaning that the fraction of material in
the edges still scales as / s and that in the nodes as ⊲ / s⊳3/2 , but the multiplying
constants depend on precisely how the material is distributed between
edges and nodes. We therefore generalize equation (14.6) to read
� � � �3/2 D ˛ C ⊲1 ˛⊳
⊲14.7⊳
s
s
s
retaining the necessary feature that / s D 1when / s D 1. This means that
the conductivity of a foam can be modeled if one data point is known, since
this is enough to determine ˛.
Equation (14.7) is plotted on Figure 14.2, for two values of ˛. The upper
line corresponds to ˛ D 0.33 (the ‘ideal’ behavior of equation (14.6)), and
describes the open-cell Duocell results well. The lower line corresponds to
˛ D 0.05, meaning that the edges make a less than ideal contribution to
conductivity. It fits the data for Alulight well.
Alulight is an closed-cell foam, yet its behavior is describe by a model
developed for open cells. This is a common finding: the moduli and strengths
of closed-cell foams also lie close to the predictions of models for those with