The connectedness of packed circles and spheres with application to conductive cellular materials

In this paper, we examine the static connectivity of 2D and 3D arrays of spherical cells with conductive paths, and the associated power dissipation in the individual cells. Herein, we use the term "cellular material" to describe the ensemble of many cells, in contrast to the more traditional use of the term for foams and honeycomb materials. Using a numerical analytical approach from highly parallel resistor arrays, we examine the cells and ensemble structures in terms of their connectivity, defined as the number of cells that are dissipating power, as well as the redundancy and robustness to localized cell failure. We examine how the connectivity changes with the geometry of the conductive cell surface area, and in particular, the percentage of the cell half that is conductive and makes contact with neighboring cells. We find that the best connectivity exists when the conductive surface of the cell is approximately 80% of the hemisphere surface, addressing the tradeoff of maximizing contact with neighboring cells while minimizing shorts in the structure. In terms of robustness, the results show that, for the proposed circular and spherical cell design, the connectivity is a nearly linear function of the number of disconnects, indicating that there is not a catastrophic effect of isolated cell failures. In terms of structure size, the connectivity appears to plateau at around 60% for the planar structures and around 50% for the cubic structures of around 500 cells or greater with random cell orientation.