Constructing conductive filler networks with high efficiency is essential to fabricating functional polymer composites. Although two-dimensional (2D) sheets have prevailed in nanocomposites, their efficiency in enhancing conductive functions seems to reach a limit, as if merely addressing the dispersion homogeneity. Here, we exploit the unrecognized geometric curvature of 2D sheets to break the efficiency limit of filler systems. We introduce the hyperbolic curvature concept to mediate the incompatibility between 2D planar topology and 3D filler space and hold the efficient conductive path through face-to-face contact. The hyperbolic graphene framework exhibits a record efficiency in enhancing electrically and thermally conductive functions of nanocomposites. At a volume loading of only 1.6%, the thermal and electrical conductivities reach 31.6 W/(mK) and 13 911 S/m, respectively. We demonstrate that the conductive nanocomposites with a hyperbolic graphene aerogel framework are useful for thermal management, sensing, and electromagnetic shielding. Our work provides a solution to reconcile the incompatibility between the 2D planar structure of sheets and the highly expected 3D conductive path, presenting a geometrically optimal filler system to break the efficiency limit of multifunctional nanocomposites and broaden the structural design space of 2D sheets by curvature modulation to meet more applications.
Keywords: composites; graphene framework; hyperbolic structure; optimally enhancing efficiency; thermal and electrical conductivity.