Multi-Scale Modeling of Polymer Nanocomposites
L. Adam, K. Delaere, J.-S. Gérard, R. Assaker, I. Doghri, M. Kaszacs
eXstream engineering S.A., BE
Keywords: nanocomposites, multi-scale, mean-field homogenization, finite element analysis, micromechanics, microstructure
Abstract:Due to the small size of nano-inclusions, their surface becomes an important factor determining the composite properties. For “microcomposites” (with inclusions larger than one micron) it is often sufficient to consider the phases’ bulk properties, while for nanocomposites enhanced modeling is needed to capture this surface effect, or other “nano-effects”. Due to these nano-effects, it is possible to obtain a given improvement of matrix properties (mechanical, electrical, thermal, ...) with a much smaller filler weight fraction (lower cost), raising the interest of manufacturers. Mineral nano-scale inclusions are provided as a powder of particles with average size 5–150nm. During production, nano-filler is dispersed in the matrix. Full dispersion is rarely reached and clusters of nano-filler appear. Clustering can be advantageous, e.g. when clusters of densely packed nano-particles have the same stiffness as a solid micron-sized inclusion, while using less filler material. This beneficial use of voids in between particles induces great property enhancements. Mean-field (MF) homogenization techniques describe the composite rheological behavior based on the average stress and strain tensors on phase level and on composite level. These analytical formulae are applied in a numerical simulation. This MF approach is successful for a wide range of composites and can also handle coated inclusions, which is important for nanocomposites because the nano-particles interact with other nano-particles and with the matrix. For matrix-particle interaction, the coating represents a real, new phase (“interphase”), e.g. matrix material with density and stiffness variations. For particle-particle interaction, the coating properties emulate the force between particles. Both coating applications are presented: particle-particle interaction, and reverse-engineering the properties of the matrix-particle interphase. While MF analysis is size-independent, it is possible by “design of experiment” to obtain curves representing the size-effect, i.e. the effect of coating thickness for a given particle size. If the inclusion is coated, then inclusions are first homogenized with their coating, and this result is then homogenized with the matrix. A similar, two-level approach is used for densely packed clusters: nano-filler is first homogenized with voids in the cluster, and the resulting “effective cluster material” is homogenized with the matrix. To verify the MF predictions, FE models are constructed that represent the real microstructure geometry. Models and results are presented for polymer matrix with mono- and polydisperse nano-filler, including coating, clustering and using a cluster size distribution as determined from image analysis (Figure 1). The MF results, obtained in CPU-seconds, are less than 5% off the FE results, obtained in CPU-days. FE analysis reveals that clustering of particles may increase the maximum stress in the matrix with 25%. In FEA, the coating can be used to represent electron tunneling and emulate the percolation effect that occurs above a critical volume fraction, increasing electrical conductivity with many orders of magnitude. Results have been compared successfully with experimental data from industrial partner. Figure 1. Geometric model of nanocomposite microstructure representing cluster size distribution for (a) 5% mass fraction, and (b) 15% mass fraction. Smallest spheres represent single nanoparticles, larger spheres represent nanoparticle clusters.