# An evaluation of models and computational approaches for the barrier properties of coatings containing flakes of high aspect ratio

- 67 Downloads

## Abstract

We report on the results of a comprehensive two-dimensional computational study of diffusion across disordered flake composites. Our objective is (1) the evaluation of existing literature models for the effect of flake orientation and (2) the evaluation of the influence of boundary conditions and unit-cell types on the predicted barrier properties. Flake orientation is an important parameter affecting barrier properties in flake-filled composites, as the barrier efficacy of such systems depends significantly on the extent to which the flakes have been oriented as close as possible to being perpendicular to the direction of macroscopic diffusion. Our comparisons rely on an extensive set of computational results in two-dimensional, doubly periodic unit cells, each containing up to 3000 individual unidirectional flake cross-sections which are randomly placed and with their axes forming an angle (*π*/2 − θ) with the direction of macroscopic diffusion. A unique feature of our study is the consideration of high aspect ratio (*α*) systems with *α* = 100 and *α* = 1000, from the dilute (*αφ* = 0.01) and into the very concentrated (*αφ* = 40) regime. The effective diffusivity of the corresponding unit cells is computed from the imposed concentration difference and the computed mass flux, using Fick’s Law. We show that use of cyclic boundary conditions and doubly periodic unit cells results in effective diffusivities which are in agreement with theory and invariant of the shape of the unit cell. In addition, we show that the use of adiabatic boundary conditions produces erroneous results at high flake concentrations. Comparison of our results with existing theoretical models revealed several shortcomings of the latter concerning both the effect of flake concentration (*αφ*) and the effect of the orientation angle (*θ*). The principal reason for the latter shortcoming is the fact that said models do not respect the rotational invariance of the diffusivity tensor.

## Keywords

Coatings Flakes Misalignment Barrier properties## References

- 1.Lagaron, JM, Nunez, E, “Nanocomposites of Moisture-Sensitive Polymers and Biopolymers with Enhanced Performance for Flexible Packaging Applications.”
*J. Plastic Film Sheet.*,**28**(1) 79–89 (2012)CrossRefGoogle Scholar - 2.Lee, K-H, Hong, J, Kwak, SJ, Park, M, Son, JG, “Spin Self- assembly of Highly Ordered Multilayers of Graphene-Oxide Sheets for Improving Oxygen Barrier Performance in Polyolefins.”
*Carbon*,**83**40–47 (2015)CrossRefGoogle Scholar - 3.Cussler, EL, Hughes, SE, Ward, WJ, Aris, R, “Barrier Membranes.”
*J. Membr. Sci.*,**38**161–174 (1988)CrossRefGoogle Scholar - 4.Lape, NK, Nuxoll, EE, Cussler, EL, “Polydisperse Flakes in Barrier Films.”
*J. Membr. Sci.*,**236**29–37 (2004)CrossRefGoogle Scholar - 5.Nyflott, A, Mericer, C, Minelli, M, Moons, E, Jarnstrom, L, Lestelius, M, Baschetti, MG, “The Influence of Moisture Content on the Polymer Structure of PVOH in Dispersion Barrier Coatings and Its Effect on The Mass Transport of Oxygen.”
*J. Coatings Technol. Res.*,**14**(6) 1345–1355 (2017)CrossRefGoogle Scholar - 6.Chen, X, Papathanasiou, TD, “Barrier Properties of Flake-Filled Membranes: Review and Numerical Evaluation.”
*J. Plastic Film Sheet.*,**23**319–346 (2007)CrossRefGoogle Scholar - 7.Papathanasiou, TD, Tsiantis, A, “Orientational Randomness and Its Influence on the Barrier Properties of Flake-Filled Composite Films.”
*J. Plastic Film Sheet.*,**33**(4) 438–456 (2017)CrossRefGoogle Scholar - 8.Dondero, M, Tomba, JP, Cisilino, AP, “The Effect of Flake Orientational Order on the Permeability of Barrier Membranes: Numerical Simulations and Predictive Models.”
*J. Membr. Sci.*,**514**95–104 (2016)CrossRefGoogle Scholar - 9.Bharadwaj, RK, “Modeling the Barrier Properties of Polymer-Layered Silicate Nanocomposites.”
*Macromolecules*,**34**9189–9192 (2001)CrossRefGoogle Scholar - 10.Greco, A, “Simulation and Modeling of Diffusion in Oriented Lamellar Nanocomposites.”
*Comput. Mater. Sci.*,**83**164–170 (2014)CrossRefGoogle Scholar - 11.Greco, A, Maffezzoli, A, “Two-dimensional and Three-dimensional Simulation of Diffusion in Nanocomposite with Arbitrarily Oriented Lamellae.”
*J. Membr. Sci.*,**442**238–244 (2013)CrossRefGoogle Scholar - 12.Sorrentino, A, Tortora, M, Vittoria, V, “Diffusion Behavior in Polymer-Clay Nanocomposites.”
*J. Polym. Sci. Part B Polym. Phys.*,**44**265–274 (2006)CrossRefGoogle Scholar - 13.Nielsen, LE, “Models for the Permeability of Filled Polymer Systems.”
*J. Macromol. Sci. Part A Chem.*,**5**(1) 929–942 (1967)CrossRefGoogle Scholar - 14.Papathanasiou, TD, Guell, D, (Eds.)
*Flow-Induced Alignment in Composite Materials.*Woodhead Publishing (1997)Google Scholar - 15.Tsiantis, A, Papathanasiou, TD, “The Barrier Properties of Flake-Filled Composites with Precise Control of Flake Orientation.”
*Materials Sciences and Applications*,**8**234–246 (2017)CrossRefGoogle Scholar - 16.Roache, PJ,
*Verification and Validation in Computational Science and Engineering*. Hermosa Publishers, Albuquerque, NM (1998)Google Scholar - 17.Geuzaine, C, Remacle, JF, “Gmsh: a Three-Dimensional Finite Element Mesh Generator with Built-in Pre- and Post-Processing Facilities.”
*Int. J. Numer. Methods Eng.*,**79**(11) 1309–1331 (2009)CrossRefGoogle Scholar