Abstract
We derive closed-form solutions for the barrier factor of flake-filled composites, in which the flakes are randomly placed and oriented, with the orientation angles uniformly distributed in an interval [−ɛ, +ɛ], 0 < ɛ < π/2. Our solutions are based on the arithmetic and harmonic averaging of the diffusivity of unidirectional misaligned flake systems. Using large-scale 2D simulations, some involving up to 50,000 individual flakes in one unit cell and spanning the regime from dilute to highly concentrated, the proposed solutions are tested against and confirmed by computational results. We use both, traditional 2D and also 1D representations of the flake cross sections. The 1D representation is suitable for very high aspect ratio flakes, such as exfoliated nano-platelets and allows us to model flake nano-composites (α > 1000) in which (αϕ) can reach levels in excess of 100. Comparison of the derived closed-form solutions to computational results reveals that both the harmonic and the arithmetic averages are adequate in dilute systems; however, at large values of (αϕ) and (ɛ), only the harmonic average is in good agreement with the computational results. The predictions of the proposed solution are also compared to those of existing literature models for the effective diffusivity of flake composites. We find that discrepancies become very significant at large (ɛ) and also at large values of (αϕ), pointing further to the conclusion that the proposed solution is currently the only accurate one to predict the effective diffusivity of randomly oriented and highly concentrated nano-flake composites.