## 143 Electrical conductivity and relative density

Figure 14.3 shows an idealization of a low-density open cell foam. The cell edges have length i and cross-section t x t, meeting at nodes of volume t3. The relative density of an open-cell foam is related to the dimensions of the cells (omitting terms of order (t/i)4) by

Ps i2 Figure 14.3 An idealized open-cell foam consisting of cell edges of length I and cross-section t2, meeting at nodes of volume t3. In real foams the nodes are larger ('Plateau borders') and the edges thinner at their mid-points, because of the effects of surface tension

The dependence of electrical conductivity on relative density can be understood in the following way. The cell edges form a three-dimensional network. If a potential gradient is applied parallel to one set of cell edges, the edges which lie parallel to the gradient contribute to conduction but those which lie normal to it do not, because there is no potential difference between their ends. The network is linked at nodes, and the nodes belong to the conducting path. At low relative densities the volume of the nodes is negligible compared with that of the edges, and since only one third of these conduct in a body containing a fraction p/ps of conducting material, the relative conductivity is simply where os 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 f3 and that of the edges as f2l, then the relative contribution of the nodes scales as f/l, or as (p/ps)1/2. We therefor expect that the relative conductivity should scale such that where the constant of proportionality 2 multiplying (p/ps)1/2 has been chosen to make a/as = 1 when p/ps = 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 p/ps and that in the nodes as (p/ps)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 retaining the necessary feature that a/as = 1 when p/ps = 1. This means that the conductivity of a foam can be modeled if one data point is known, since this is enough to determine a.

Equation (14.7) is plotted on Figure 14.2, for two values of a. The upper line corresponds to a = 0.33 (the 'ideal' behavior of equation (14.6)), and describes the open-cell Duocell results well. The lower line corresponds to a = 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

open cells, perhaps because the cell faces are so thin and fragile that they contribute little to the properties, leaving the edges and nodes to determine the response.

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