Additive manufacturing, especially in the form of 3D printing, offers the exciting possibility of generating articles with precisely controlled internal microstructure. One area in which this feature can be of significant advantage is in diffusion control, specifically in the design and fabrication of microstructures which allow for minimization of the transport of a solute to/from a contained fluid. Flake-filled polymeric composites, incorporating mica, glass or metallic flakes have found many uses in this direction, as they offer significant processing and property advantages, namely high dimensional stability and low warpage in molding, uniform in-plane mechanical properties, corrosion protection, sound insulation as well as appearance and color control [3] [4] [5] [6] [7] . Micron-sized flakes of inorganic materials such as mica, nano-scale platelets of clay minerals such as hectrite, saponite and montmorillonite and more recently graphene-oxide platelets of aspect ratios well over 1000, have been used for this purpose [8] . It has been demonstrated that incorporation of such fillers aligned perpendicular to the direction of macroscopic diffusion can be very effective in increasing the tortuosity of the diffusion path of the diffusing species. When the flakes are randomly placed, as would be the case in a flake composite manufactured from the melt, the predicted improvement in barrier efficiency ranges from being ~() in dilute systems, where () is the aspect ratio and () the volume fraction of the flakes, to being ~ in more concentrated dispersions [1] [9] - [16] .

One notable disadvantage of traditional processing methods vis-à-vis flake- filled composites is the fact that flake orientation cannot be precisely controlled. In such operations (extrusion, compression or injection molding, thermoforming and others) flake orientation is achieved due to the propensity of the flakes to orient in accordance to the prevailing flow field―either in the main direction of flow when the flow is shear or transverse to it when the flow is extensional [17] . An additional shortcoming of traditional flow-processing routes is the inability to utilize high flake loadings since, in that case, the viscosity of the resulting melt becomes prohibitively high. Given the capability afforded by 3D printing to fully control flake orientation as well as generate articles with flake loadings approaching those at maximum packing, it is desirable to predict the effective diffusion coefficient (or its inverse, the barrier improvement factor, BIF) as an explicit function of the flake orientation angle and for very high, previously untenable, concentrations.

The two main approaches which have been used in the literature to-date for this purpose are (i) an ad-hoc incorporation of orientation metrics in existing models for the BIF [8] [18] and (ii) derivation of BIF models from diffusion path calculations [19] [20] [21] . In both cases, the proposed models have been derived for low or very-low flake concentrations and have not been extensively tested in the moderate to high-concentration regime, which will be of importance in any 3D printing application. In addition, by not respecting the rotational properties of the diffusivity tensor, these models are not grounded on a sound theoretical footing. This paper addresses the above issues both computationally and theoretically, by proposing a model based on the two principal diffusivities of a flake composite. We also show that the implications of our theoretical model are fully supported by extensive computational results.

We carry out steady-state diffusion computations in doubly-periodic unit cells containing up to 3000 individual flake cross-sections. These are added in the domain sequentially, using a Random Sequential Addition (RSA) procedure. Specifically, at each flake placement attempt, two random numbers are used to assign the coordinates of the flake center while its orientation angle () is fixed and the same for all flakes. If, after placement, no overlap with other flakes is detected, the process continues with the next flake, until the desired number of flakes has been placed, or, until no flake can be placed after 50,000 attempts; in this case no geometry is generated. In order to enable subsequent meshing of the computational domain, a minimum allowable distance between flakes is imposed; this is () where () is the thickness of the flake. Since in this work we have dealt with flakes of high aspect ratio (and), this minimum distance requirement is deemed reasonable so as to not result in excessive local mesh refinement. In a rectangular unit cell of dimensions (H) and (L) containing (N) flakes of dimensions (), the flake area fraction () is and the length (l) of each flake. We have looked at systems in which. At higher values of () it becomes impossible to fill the space with non-overlapping flake crossections. This not-withstanding, the present study is to our knowledge the first to investigate systems of such large concentration.

In multi-particle simulations, use of doubly-periodic cells is essential when dealing with elongated particles so as to eliminate artifacts of oriented (or, depleted) layers which would otherwise appear adjacent to cell boundaries [11] [16] . The effect of the RVE dimensions on the computed effective diffusivity is also eliminated when using periodic unit cells. A sample unit cell, showing flakes oriented at an angle θ = 0.8 rad with respect to the horizontal (x) axis (extended slightly outside the limits of the unit cell to show the doubly-periodic geometry) is shown in Figure 1.

The boundary conditions are cyclic on the right and left boundaries, namely. On the top and bottom boundaries, fixed values of concentration are prescribed. On the surface of each flake we impose indicating that the flakes are impermeable. At each level of () and () we generate ~10 different realizations, each containing up to 3000 randomly

placed flakes. The computational meshes are created by the mesh generating program Salome through an in-house automated procedure and contain ~10⁶ triangular elements.

These meshes are then used by Open Foam to solve the steady-state diffusion equation, (C) being the solute concentration, and provide its distribution in the domain of interest. The assumption of an isotropic matrix material is also made. The solution also supplies the value of at each point on the upper (or lower) boundary. As a result, the mass flux along this boundary can be calculated as

where the subscript (n) indicates numerically computed value, n is the outward unit vector and (L) is the width of the unit cell. Because of impermeable flakes crossing boundaries, which results in sudden local changes of the flux, care must be taken in performing this integration. In this work, we used adaptive intervals and only accepted values of the integral when these were convergent with refinement. Equating this flux with the one obtained from Fick’s law in a macroscopic equivalent cell (whose diffusivity is D_eff) we obtain

where is the macroscopically imposed concentration difference and (H) the height of the unit cell. These effective diffusivities will be presented and discussed for various values of (), () and () in the following sections.

In the following we present the results of a comprehensive computational study of diffusion across doubly-periodic unit cells, each containing up to N = 3000 randomly placed impermeable flakes of rectangular cross-section. In such a system, the orientation of each flake is defined by the orientation angle (θ) formed between the vertical axis (y), which is taken to be the direction of macroscopic diffusion, and the outward normal vector on the flake surface. The horizontal axis is indicated as (). Parametric studies have shown that for N > 200 the obtained D_eff were practically indistinguishable; this conclusion extended for () as large as 40; therefore most of our computations have been carried out in RVEs containing 500 flake cross sections. We look at systems ranging from dilute to concentrated and in which the fiber orientation (θ) changes between zero (flake orientation perpendicular to the direction of diffusion) to (fibers oriented along the direction of macroscopic diffusion). We have carried out computations in unit cells similar to those of Figure 1 for and and, as well as for and.

In this study we proposed a model for the Barrier Improvement Factor (BIF) of misaligned flake composites which is valid up to very high flake concentrations, as could be found in composites fabricated by 3D printing. The model requires as inputs the two principal diffusivities of the composite, normal and parallel to the flake axis. In this respect, we find that the models of Lape et al. [1] and Nielsen [2] form an excellent combination. The simple algebraic form of the proposed model makes it usable without recourse to special computing facilities. This model was tested exhaustively by comparing to predictions of 2D computer simulations which included up to 3000 randomly placed but uniformly oriented flake cross-sections in each RVE. Each cross-section forms an angle () with the direction of macroscopic diffusion. Over 1500 simulations were carried out and upon comparison the model was found in agreement with computational results for all misalignment angles and for values of () up to 40. Both our model and our computational data predict that at the quadratic dependence of the BIF on () is lost, with the BIF approaching a plateau at higher values of (). This plateau is lower as (θ) increases. We derive analytical estimates of this maximum achievable BIF at each level of misalignment; these are also shown to be in excellent agreement with the computational results. Finally we show that our computational results and model are in agreement with experimental evidence at small values of (θ). Future work involves extension to even higher values of () as well as comparison with 3D computations.

Tsiantis, A. and Papathanasiou, T.D. (2017) The Barrier Properties of Flake-Filled Composites with Precise Control of Flake Orientation. Materials Sciences and Applications, 8, 234-246. https://doi.org/10.4236/msa.2017.83016