To confirm and help understand our experimental results, we also performed COMSOL simulations of models of the composite material under several different sets of assumptions. The results of the simulations are shown in the figures below.

The above figure shows the experimental results from the pucks in circles, while the curves show COMSOL simulations under the following assumptions. In the air case (blue curve), the microspheres are treated as incompressible bubbles of air. In the factory case (red curve), the microspheres are treated as incompressible bubbles of effective resistivity equal to the factory-stated resistivity of the microspheres. In the glass shell case (orange curve), the microspheres are treated as air bubbles inside glass shells, with glass shell thickness set to the average stipulated by the factory. In all three conditions, what was modeled was a single microsphere centered inside a cubic cell. Since resistivity is a property of the overall material, the idea was that the material could be viewed as a lattice of such cells stacked in a Cartesian lattice. Then the resistivity of a single cell would be the same as the resistivity of the overall material, at the same volumetric density, calculated from the corresponding ratio of the microsphere diameter and the length of the side of the cell.

Overall, the results show reasonably good agreement for the datapoints of the higher volumetric content, with the factory case describing the overall situation best. This is not a surprise, because the factory-stated effective resistivity plus an incompressible bubble model ought to perform best, as they would be the closest representation of the actual situation among the three models considered.

Curiously, low-density pucks seem to outperform the model predictions experimentally. The likely explanation is that low-density pucks likely had far more air bubble content due to imperfect evacuation. The air content would be greater volumetrically when the polymer share of the total volume is greater. Conversely, high-density pucks would have far less polymer and thus much less percentile volume taken up by non-evacuated air bubbles. The presence of these extra bubbles would boost the performance of the low-volumetric-density samples. So, the deviations from the models at low percentages are understandable.

We also performed simulations of resistivity of composite materials built by the use of uniform-size microspheres arranged in various lattices. The figure below shows the results.

To minimize the combinatorics, the material assumptions were limited to two possible options: either “air” (microspheres treated as incompressible air bubbles) or “3M rating” (microspheres treated as incompressible bubbles of effective resistivity as listed by the manufacturer). The experimental results are shown in circles, while the curves correspond the COMSOL models. For each material assumption, three curves were produced: “single bead” (one microsphere centered in a cubic cell); “body centered” (cubic cell contains a single bead in the center and 8 one-eighths of a bead at each corner of the cube); “face centered” (face centered cubic arrangement, where 6 halves are arranged at the centers of the cube sides while 8 one-eighths are positioned at the cube corners). The results again show best agreement for the factory-rated bead resistivity models, as well as overperformance of low-percentage samples likely due to non-evacuated air in the polymer.

Overall, the COMSOL results confirmed that our experimental measurements made sense from theoretical perspective. The agreement also meant that we likely did not make mistakes in the experimental measurements and calculated values. This confirmed that our conclusions and predictions based on these measurements were valid and a good representation of the material behavior.