This direct numerical study investigated the effect of orientational randomness on the barrier properties of flake-filled composites. Over 2500 simulations have been conducted in two-dimensional, doubly periodic unit cells, each containing 500 individual flake cross-sections which, besides being spatially random, assume random orientations within an interval [−ɛ, +ɛ] ([ ]). We consider long flake systems (aspect ratio α = 50, 100, and 1000) from the dilute (αϕ = 0.01) to the concentrated (αϕ = 15) regime, where (ϕ) is the flake volume fraction. At each (ɛ) and (αϕ), several realizations are generated. At each of those, the steady-state diffusion equation is solved, the mass flux across a boundary normal to the diffusion direction is computed and an effective diffusivity D_eff calculated from Fick’s Law. The computational results for D_eff are analyzed and the effects of (ɛ) and (αϕ) are quantified. These differ in the dilute (αϕ < 1) and in the concentrated regimes (1 < αϕ < 15). In the dilute regime, the barrier improvement factor is a linear function of (ɛ) and a power function of (αϕ), with the exponent (∼1.07) independent of orientation. In concentrated systems, we find that for aligned flakes or flakes showing small deviations from perfect alignment, the barrier improvement factor approaches the quadratic dependence on (αϕ) predicted by theory. However, the power exponent is found to decrease as (ɛ) increases, from 1.71 in the aligned system (ɛ = 0) to ∼0.9 in the fully random system (ɛ = π/2). We propose a scaling which incorporates the effects of both (αϕ) and (ɛ) on the barrier improvement factor, resulting in a master curve for all (αϕ) and (ɛ). Our results suggest that the anticipated barrier property improvement may not be realized if the flake orientations exhibit a significant scatter around the desired direction.

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.

Fibrous biomaterials are essential in biomedical engineering, tissue engineering, and filtration due to their specific transport and mechanical properties. Fluid flow through these materials is critical for their function. However, many biological fluids exhibit non-Newtonian behavior, characterized by micro-rotational effects, which traditional models often overlook. The current study presents a machine learning (ML) framework for the prediction and understanding of hydraulic permeability in fibrous biomaterials with a micropolar fluid. A dataset of 1000 numerical simulations was generated by varying the micropolar fluid properties and the fiber volume fraction in a periodic porous structure with nine parallel cylindrical fibers in a square lattice. Six powerful ML algorithms were deployed: Decision Trees (DT), Random Forests (RF), XGBoost, LightGBM, Support Vector Regression (SVR), and k-Nearest Neighbors (kNN). The balance of predictive capacity to unseen data values (tracking $R^2$ values and error metrics) with computational efficiency for all algorithms was assessed. The best-performing ML algorithm was subsequently used to interpret the decisions made by the model using Shapley Additive exPlanations (SHAP) analysis and understand the role of feature importances. The SHAP findings highlight the potential of ML in capturing complex fluid interactions and guiding the design of advanced fibrous biomaterials with optimized hydraulic permeability.