Systematic upscaling for multiscale modeling of multifunctional polymer nanocomposites

Multifunctional polymer nanocomposites derive their superior mechanical and physical properties from embedded nanoinclusions, e.g., carbon nanotubes or piezoelectric nanoparticles. Processing conditions are critical for achieving desired dispersion/aggregation of inclusions and, therefore, for developing materials with tailored properties. Control of in-field processing conditions may include controlling polymer flow viscosity, applying pressure gradient or shear strain (as in extrusion flows and sputtering), as well as electric field orientation of inclusions and poling of inclusions/polymer composites. A number of complexities are identified such as disparate length scales of inclusions, strong relations between small-scale phenomena and macroscale materials properties, coupling between electrical and mechanical properties, etc. Here, we discuss the emerging systematic upscaling (SU) methodology with applications to multiscale multiphysics modeling of in-field processing of polymer composites. The SU methods provide the means for methodical derivation, scale after scale, of increasingly-larger-scale numerical "laws", starting at the known numerical laws at a fine scale and leading to the processing rules of collective variables at much larger scales. The SU methods combine two processes: (1) finescale processing (relaxation) and (2) repeated coarsening that creates a description of the same physical system that contains fewer (by a factor between 2 and 10) unknowns and is derived from the current fine-scale formulation. The numerical computations show that the SU models are very efficient and accurately account for small-scale features that cannot be directly represented on the coarse scales.