Composite materials are a very common and important class of substances. Many theories have been developed to help explain how their properties depend upon their composition. One class of models are called effective medium theories since they provide an average estimate of a material's property based on its overall composition and a geometrical picture of its structure. Assuming a random, heterogeneous composite composed of freely overlapping spherical inclusions, a representation for the thermal conductivity of such a composite can be obtained using the Hashin-Strikman formalism [45].

\[k_{m}\left(1+\frac{3 \phi\left(k_{f}-k_{m}\right)}{3 k_{m}+(1-\phi)\left(k_{f}-k_{m}\right)}\right)

where \(k_{m}\) is the thermal conductivity of the matrix material and \(k_{f}\) is the thermal conductivity of the filler material. Assuming the matrix is an epoxy \((k=0.2 \mathrm{~W} / \mathrm{mK})\) and the filler is silver \((k=428 \mathrm{~W} / \mathrm{mK})\), plot both bounds for filler volume fractions between \(10 \%\) and \(90 \%\). If a real material is made that has a thermal conductivity of \(40 \mathrm{~W} / \mathrm{mK}\) at a filler volume fraction of \(48 \%\), does the material follow the upper or lower bound you plotted?