Gradient of mechanical properties in polymer nanocomposites: From atomistic scale to the strain gradient effective continuum

Abstract

It is well known that glassy polymer nanocomposites (PNC) exhibit heterogeneous mechanical properties, with a distinct polymer/nanoparticle (NP) interfacial region. The interphase rigidity in particular can significantly influence the overall mechanical response of glassy PNCs. Continuum theories typically describe the mechanical behavior of the interphase such to depend only on the deformation and, thus, they discard the strong internal deformation gradients within the interphase, that are necessary for enriched continuum models of the PNC’s. Here we formulate a higher-gradient elasticity constitutive law for the polymer/NP interphase region through a systematic micro–meso–macro coupling, using data from atomistic molecular dynamics (MD) simulations. From a physical standpoint, we show that in order to accurately consider the polymer/NP interphase is necessary to describe it as a strain gradient material, whereas the bulk material may still behave as a Cauchy material, experiencing much lower strain gradient effects, compared to the polymer/NP interphase. Using atomistic MD simulations, the polymer/NP interphase thickness in PNCs is examined by probing the density profile distribution at equilibrium. By upscaling the molecular response within the interphase towards a continuum model, a strain gradient effective model for the interphase emerges, whereby the hyperstress tensor and the conjugate strain gradient, together with the stress and strain, are computed for each atom using the MD simulations. In a further upscaling step, a continuum-based homogenization scheme based on the unfolding method is elaborated for the identification of the strain-gradient elastic moduli of polymer composite materials, according to which the interphase obeys a second-gradient linear elastic constitutive law, with an inner reinforcement and an outer surrounding matrix obeying a first-gradient elasticity model.