Introduction of an Adhesion Factor to Cube in Cube Models and its Effect on Calculated Moduli of Particulate Composites

<p>The cube&nbsp;in&nbsp;cube approach was used by Paul and Ishai-Cohen to model and derive formulas for filler content dependent Young&acute;s moduli of particle filled composites assuming perfect filler matrix adhesion. Their formulas were chosen because of their simplicity, recalculated using an elementary volume approach which transforms spherical inclusions to cubic inclusions. The EV approach led to expression for the composites moduli that allow for introducing an adhesion factor <em>k</em>_adh ranging from 0 and 1 to take into account none perfect reduced filler matrix adhesion. This adhesion factor scales the edge length of the cubic inclusions, thus, reducing the stress transfer area between matrix and filler. Fitting the experimental data with the modified Paul model provides reasonable <em>k</em>_adh for PA66, PBT, PP, PE-LD and BR which are in line with their surface energies. Further analysis showed that stiffening only occurs if <em>k</em>_adh exceeds&nbsp;<span class="math-tex">\( { \ \sqrt{E^M/E^F} \ }\)</span> and depends on the ratio of matrix modulus and filler modulus. The modified model allows for a quick calculation of any particle filled composites for known matrix modulus <em>E</em>_M, filler modulus <em>E</em>_F, filler volume content <em>v</em>_F and adhesion factor <em>k</em>_adh. Thus, finite element analysis (FEA) simulations of any particle filled polymer parts as well as materials selection are significantly eased. FEA of cubic and hexagonal EV arrangements show that stress distributions within the EV exhibit more shear stresses if one deviates from the cubic arrangement. At high filler contents the assumption that the property of the EV is representative for the whole composite, holds only for filler volume contents up to 15 or 20 % (corresponding to 30 to 40 weight %). Thus, for vast majority of commercially available particulate composites, the modified model can be applied. Furthermore, this indicates that the <em>cube&nbsp;in&nbsp;cube</em> approach reaches two limits: i) the occurrence of increasing shear stresses at filler contents above 20 % due to deviations of EV arrangements or spatial filler distribution from cubic arrangements (singular), and ii) increasing interaction between particles with the formation of particle network within the matrix violating the EV assumption of their homogeneous dispersion.</p>