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Additive manufacturing, especially in the form of 3D printing, offers the exciting possibility of generating heterogeneous 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 optimize the rate of transport of a solute to and from a contained fluid. In this work we focus on the use of flakes as diffusion-control agents and study computationally and theoretically the effect of orientation on the barrier properties of flake-filled composites. We conducted over 1500 simulations in two-dimensional, doubly-periodic unit cells each containing up to 3000 individual flake cross-sections which are randomly placed and with their axes forming an angle (
) with the direction of macroscopic diffusion. We consider long-flake systems of aspect ratio (
) 100 and 1000, from the dilute (
) and into the concentrated (
) regime. Based on the rotation properties of the diffusivity tensor, we derive a model which is capable of accurately reproducing all computational results (
and
). The model requires as inputs the two principal diffusivities of the composite, normal and parallel to the flake axis. In this respect, we find the models of Lape
*et al.* [1] and Nielsen [2] form an excellent combination. Both our model and our computational data predict that at
the quadratic dependence of the Barrier Improvement Factor (BIF) on (
) is lost, with the BIF approaching a plateau at higher values of (
). This plateau is lower as (
) 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 (
).

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 [

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 [

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 [

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 (

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 [

The boundary conditions are cyclic on the right and left boundaries, namely

placed flakes. The computational meshes are created by the mesh generating program Salome through an in-house automated procedure and contain ~10^{6} triangular elements.

These meshes are then used by Open Foam to solve the steady-state diffusion equation

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

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 (_{eff} were practically indistinguishable; this conclusion extended for (

Representative results of the distribution of (C), also showing the corresponding flake distributions, are shown in

We define as D_{11} the diffusivity of such a system when q = 0˚ (all flakes oriented perpendicular to the direction of macroscopic diffusion) and D_{22} the diffusivity when q = 90˚ (all flakes oriented parallel to the direction of diffusion). D_{11} and D_{22} are the principal values of the two-dimensional diffusivity tensor, D. The diffusivity tensor

Hence

Therefore, the effective diffusivity of this system in the direction (y) forming

an angle (

We will investigate the use of Equation (2) to determine_{11} and D_{22} are known. By comparing its predictions to our computational results we will identify which models for D_{11} and D_{22} give the best agreement with computation.

In the first instance we have compared the computational results for dilute cases (

Nielsen’s [_{11} and D_{22}, namely

Extensive comparisons have shown that predictions of Equation (2) based on Nielsen’s model for D_{11} and D_{22} are close to the computational results only for the very dilute regime (

It is of course possible to use diffusivity models for D_{11} and D_{22} more suitable for concentrated suspensions. A review and evaluation of available models has been carried out by Chen and Papathanasiou [

In deriving this model, the tortuosity factor was taken to be 1 + αφ/3 and it was further assumed that the ratio of the areas available for diffusion is

Implicit in the above derivation is the assumption that the diffusion paths around a flake form straight lines; therefore it is not unreasonable to treat the factor “3” in the expression above as a geometrical parameter that may be adjusted if so suggested by the data. Since that was found to be the case in analyzing our data, we use the model of Lape et al. [

in which (

where (

comparison between the computational results, for flakes with _{11} is taken from [_{22} from [_{11} more suitable for concentrated systems results in significantly improved predictions of D_{eff} for all (θ). The model of Lape et al. [_{22} dominates, this shows that Nielsen’s model for diffusion parallel to the flakes is a reliable one, even for (αφ) as high as 10.

Additional comparisons for

In summary, our computational results and the comparisons presented above have shown that the effective diffusivity D_{eff} of a system of randomly placed flakes oriented at an angle (

where

in which _{11} is the diffusivity of the fully-aligned system, the BIF implied by the above statement will be

From Equation (2) it can be seen that the BIF implied by our model, (setting, without loss of generality or relevance, D_{22}~D_{0}~1) is

As shown in _{11} (that is, for more dilute systems) as (_{11}) suggests a qualitatively different behavior for the BIF. Our computational results support this prediction, as will be elaborated upon in the following section. With reference to _{11}, a value of

In aligned systems, it is known [

plateau value; this plateau is lower the larger the misalignment angle (θ) is. The implication of this result is that for the full potential of large-

In light of the excellent agreement between computational results and Equation (5) it is possible to use the latter to obtain analytical estimates of the leveling-off values of the BIF at each (θ), by observing that the first term of Equation (5) becomes negligible at high (

The above comments and results are particularly pertinent to high aspect ratio

flakes, such as found in exfoliated nanoclay or graphene composites, for which even at low (

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. [

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