# Gas Permeation Model of Mixed-Matrix Membranes with Embedded Impermeable Cuboid Nanoparticles

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Mixed-Matrix Membranes (MMMs) as Barrier Materials

#### 1.2. Main Prediction Models of the Relative Permeability of MMMs

_{r}):

_{eff}is the effective permeability of the MMM and P

_{c}is the permeability of the continuous phase, i.e., that of a neat polymeric membrane. In all these models, P

_{r}depends on the filler volume fraction $\varphi $, the permeability of the continuous phase P

_{c}, and the permeability of the dispersed phase P

_{d}. For a MMM with impermeable particles, P

_{d}= 0, the mathematical expressions of Table 1 are significantly reduced. The first model used to predict the permeation properties of MMM was the Maxwell’s model, a model initially proposed to estimate the dielectric properties of polymer composites. Among all models listed, the Maxwell’s model is the most commonly used. Besides the permeability coefficients of the continuous phase (the polymer) and the dispersed phase (the fillers), the Maxwell’s model uses only the volume fraction $\varphi $ as a model parameter, regardless of the particle shape, size distribution, and particle dispersion. The model is only applicable to a dilute two-phase MMM system containing spherical particles with $\varphi $ < 0.2. Bruggeman [11], also attempting to predict the dielectric properties of composite materials with randomly dispersed particles, modified the Maxwell’s model to predict the relative permeability over a larger range of the volume fraction. Lewis and Nielsen [12,13] studied the mechanical properties of composite materials by experimentally exploring the relationship between the relative elastic modulus of composite materials and the volume fraction of its spherical fillers. The equation introduced an additional function ψ taking into account the maximum packing fraction ${\varphi}_{m}$. The model was often applied to predicting the P

_{r}of MMMs. Pal’s model was originally developed to determine the effective thermal conductivity of composite materials using the differential effective medium approach. This model also introduced an additional parameter, the maximum packing fraction ${\varphi}_{m}$. The Bruggeman’s model is a special case of Pal’s model when ${\varphi}_{m}$ = 1.0. Most models in Table 1 idealized the filler geometry and investigated only the effect of the filler volume fraction $\varphi $. To understand the impact of the filler’s geometry, models such as the one proposed by Cussler’s group relate the relative permeability to an aspect ratio and the volume fraction $\varphi $ [15,16]. They extended the Maxwell’s model by studying regular and random arrays of high aspect ratio impermeable particles, such as flakes and lamellae. They finally verified their model by performing experiments with the measurement of electrical resistance of salt solutions. Another well-known model is the two-dimensional model proposed by Bharadwajl [17]. Bharadwaj modified Nielsen’s model and performed a theoretical study on the effects of filler sheet length, concentration, orientation, and degree of delamination on the relative permeability of MMMs. He concluded that dispersing a long sheet of inorganic filler in a polymer matrix was particularly beneficial for barrier properties.

_{r}of MMMs with dispersed layered nanoparticles over a wide range of volume fractions. Secondly, to develop a simple, yet general, mathematical model to predict P

_{r}of MMMs for different geometries and dimensions of the filler particles in a 3D setting. Considering that a general and easily identifiable analytical model, which covers all geometries of nanoparticles, may or may not exist, as an intermediate step between the numerical solution and an analytical model to predict P

_{r}, we also demonstrate the applicability of an artificial neural network model for the prediction of P

_{r}.

## 2. Gas Transport in an MMM

_{2}nanoparticles, and an example of the latter is montmorillonite (MMT) clay nanoparticles. The dimensions of each cubic elementary unit were L

_{x}, L

_{y}, and L

_{z}(Figure 1), whereas the spherical nanoparticle was defined with its diameter d

_{p}, and the cuboid nanoparticle dimensions were x

_{p}, y

_{p}, and z

_{p}.

## 3. Methodology

_{x}by L

_{y}by L

_{z}) with a single nanoparticle at its center. To solve numerically for the temporal variation of the concentration at all points within the membrane, it was necessary to define the initial and boundary conditions. Concerning the initial condition, i.e., before the concentration step-change on one side of the membrane at time t = 0, a nil concentration was assumed throughout the membrane (Equation (17)). Concerning the boundary conditions, twelve relations were required to define the problem: Six boundary conditions at the periphery of the polymeric elementary unit and six boundary conditions at the polymer-solid interfaces. Equation (18) provided the boundary conditions on both sides of the membrane (y-axis was the permeation direction). It was assumed that at the onset of permeation, a step-change in the gas pressure was applied to the upstream side of the membrane (y = 0) whereas the gas pressure in the downstream side of the membrane (y = L

_{y}) was kept under perfect vacuum. The resulting concentrations in the membrane at both surfaces were simply the product of the neighboring pressure and the solubility S. Because all elementary units were identical, symmetry conditions prevailed at the other four faces of the polymer parallelepiped as expressed in Equation (19) for BC

_{3–6}. Since the nanoparticle in the center of the elementary unit was impermeable, the mass flux at each of the six faces of the nanoparticle, assuming a cuboid nanoparticle, was zero (see BC

_{7–12}in Equation (20)). The latter boundary conditions imply that the concentration of the migrating species inside the nanoparticle was zero.

_{eff}) was calculated from the steady flux, the thickness L

_{y}of the elementary unit and the feed pressure difference between the two sides of the elementary unit of the membrane (Δp), as expressed in Equation (23).

_{y}

_{=0}is the upstream permeation flux and J

_{y=Ly}is the downstream permeation flux.

## 4. Results and Discussion

#### 4.1. Numerical Simulation Results

_{2}and the MMT nanoparticles, respectively. For a spherical nanoparticle, the diameter was varied whereas, for a cuboid nanoparticle, the relative thickness, y

_{p}/L

_{y}, and the aspect ratio q were varied in order to assess the effect of the size and the shape of nanoparticles on the relative permeability (P

_{r}) of MMMs. Similar to the Cussler’s definition but in a 3D setting, the aspect ratio (q) is defined in a dimensionless form using Equation (24):

_{r}, the results for the cuboid nanoparticles were grouped over a series of narrow ranges of the aspect ratio: from 0.51–0.56 to 7.00–7.29. Many more numerical results were obtained than those presented in Figure 3, and they will all be used in developing a model to predict P

_{r}. When q is large, the cuboid nanoparticle is a thin flat sheet with a large projected area. The migrating species must circumvent to reach the permeate side of the membrane, thereby reducing the permeability of the membrane. On the other hand, when q is small, the cuboid has a small projected area with a large thickness (y

_{p}).

_{r}as a function of $\varphi $ for MMMs with an impermeable sphere and an impermeable cube (having an aspect ratio of unity) was nearly identical. For impermeable cuboid nanoparticles with q smaller than unity, the relative permeability is greater than the relative permeability of MMMs with spherical and cubic nanoparticles. In contrast, when q is larger than unity, the situation is exactly the opposite. For a membrane with cuboid nanoparticles with the same $\varphi $, the greater the value of q, the smaller the value of P

_{r}. For all results presented in this investigation, the permeability of the continuous phase was assumed to equal to 5 × 10

^{−12}m

^{2}/s (5.0 × 10

^{−11}m

^{2}/s diffusivity and 0.10 solubility). However, it is important to note that the relative permeability was independent of the permeability of the continuous polymeric phase since the filler was impermeable. A different polymer permeability would only affect the time required to reach steady-state permeation and the steady-state permeation flux.

_{r}obtained numerically in this investigation, it was desired to verify if any correlation available in the literature could adequately predict the relative permeability of cuboids with large values of the q. Figure 4 compares the relative permeability obtained numerically with the one obtained using the six correlations of Table 1 for spherical nanoparticles for a value of $\varphi $ in the range [0.00, 0.52], and cuboid nanoparticles for four different narrow interval ranges of q: [2.0, 2.3], [3.0, 3.2], [5.0, 5.2], and [7.0, 7.3]. Results confirm that the relative permeability of spherical nanoparticles predicted by the model proposed by Maxwell is nearly identical to the one obtained numerically. The predicted P

_{r}by the Maxwell model starts to deviate for a volume fraction in the vicinity of 0.25. On the other hand, three models (Bruggeman (BGM), Lewis–Nielsen (LN), and Pal (PAL)) accurately predict P

_{r}over the whole range of $\varphi $. Only the models proposed by Cussler [15,16] and Bharadwajl [17], which were not developed for spheres, show more significant deviations. Based on the results of Figure 3, it is not surprising that the same four models were equally able to predict the relative permeability of the cuboid nanoparticle with an aspect ratio of unity (results not shown). For the prediction of P

_{r}for cuboid nanoparticles with a q value other than unity, Cussler’s model predicts better than the four models for the two larger q values [5.0, 5.2] and [7.0, 7.3]; however, the deviations are still significant. The model proposed by Bharadwajl [17], which includes some geometrical parameters, can reasonably represent the data for small $\varphi $. However, this model becomes less accurate in predicting P

_{r}for higher $\varphi $. All the other models over-predicted P

_{r}of MMMs with cuboid nanoparticles when q was greater than unity and under-predicted P

_{r}when q was smaller than unity. It is evident that the available analytical models fail to predict accurately the relative permeability of MMMs with dispersed impermeable nano-cuboids with q > 1, in particular when $\varphi $ becomes larger. The greater the value of q, the more significant the deviation between the simulated and predicted relative permeability is even at very small $\varphi $. For example, for q = 5, the deviation between the simulated P

_{r}and the one predicted by the best existing model (BWD) becomes evident at $\varphi $ ~ 0.02.

#### 4.2. Artificial Neural Network Model for the Prediction of the Relative Permeability

_{r}), an artificial neural network can be used. Artificial neural networks are now commonly used for a myriad of engineering applications. The high degree of plasticity of its structure is the main reason for its ability to efficiently represent the underlying causal relationship between input and output data. In this investigation, a three-layer feedforward neural network (FFNN) was used to predict P

_{r}as a function of some input variables. Cybenko [22] showed that a three-layer FFNN was sufficient to encapsulate any input-output relationship if a sufficient number of neurons are used.

_{r}. The two simplest structures of the FFNN were the ones that used only the relative dimensions of the nanoparticles within an elementary unit. In one case, the three relative lengths, x

_{p}/L

_{x}, y

_{p}/L

_{y}and z

_{p}/L

_{z}, were used. In the other case, the relative projected area (x

_{p}z

_{p}/L

_{x}L

_{z}) and the relative thickness (y

_{p}/L

_{y}) as shown in Figure 5, were used. To determine the neural model for the prediction of P

_{r}, 359 data points obtained by solving numerically the governing partial differential equation were divided equally into a training and a validation data set. The quasi-Newton nonlinear regression algorithm was used to adjust the weights of the FFNN that minimize the sum of squares of the training data set. At each iteration, the sum of squares of the validation data set was also evaluated, and the set of weights that minimize the sum of squares of the validation data set was retained. The coefficient of regression for the FFNN with six hidden neurons (including the bias) was 0.9998 for both neural network structures mentioned above. The parity plot based on the FFNN of Figure 5 is presented in Figure 6.

_{r}of MMMs containing impermeable cuboid nanoparticles. The accuracy of the neural model was excellent over the entire range of x

_{p}z

_{p}/L

_{x}L

_{z}and y

_{p}/L

_{y}as the parity plot of Figure 6 shows. The most significant deviation was observed for thin cuboids covering nearly the entire x-z area of the elementary unit, thereby associated with low relative permeability. The latter condition was, however, extreme and will not be encountered in reality. The thinner the cuboid, the larger the deviation. It is believed that the FFNN can be used with confidence to predict P

_{r}. The mathematical model of the FFNN is given in Equations (26)–(28). The results obtained with the neural network suggest that a strong relationship exists between the relative projected area and the relative thickness. It is, therefore, hopeful of developing an analytical model with the two geometrical parameters. It differs from the traditional modelling approach mostly based on the volume fraction $\varphi $ [23].

#### 4.3. New Analytical Model for Pr of MMMs with Impermeable Cuboid Nanoparticles

_{r}as the analytical models listed in Table 1 do. On the other hand, these models, except for the Bharadwajl’s correlation [17], fail to accurately predict P

_{r}for cuboid nanoparticles with q different from unity. Therefore, it is desired to propose a new analytical model that will be valid for the widest possible ranges of $\varphi $ and the geometrical parameters of the cuboid nanoparticles.

_{r}, multivariate covariance analysis was performed to assess the underlying correlation between all possible geometrical factors and P

_{r}. For this analysis, the Pearson correlation coefficient (PCC), defined in Equations (29) to (31), was used [24].

_{A}, E

_{B}) is the covariance of E

_{A}and E

_{B}, E

_{A}and E

_{B}are the average values of E

_{A}and E

_{B}, and PCC(E

_{A}, E

_{B}) is the Pearson correlation coefficient between E

_{A}and E

_{B}. In this analysis, E

_{A}corresponded to one geometrical variable to be tested, and E

_{B}was P

_{r}determined numerically. The same 359 data points used for developing the neural network were used for this analysis. The results of this statistical analysis for five potential geometrical factors are presented in Table 2. The results show that x

_{p}z

_{p}/L

_{x}L

_{z}had, in agreement with the results of the previous section, the highest negative PCC with P

_{r}. The x

_{p}/L

_{x}ranked second because it was equivalent to the square root of the x

_{p}z

_{p}/L

_{x}L

_{z}. The volume fraction $\varphi $ also had a significant correlation factor with P

_{r}. The aspect ratio and the relative thickness also had some impact on P

_{r}but to a lesser degree. It is important to note that the five selected geometrical variables are not all mutually independent.

_{r}, one needs to examine the permeation process. Gas molecules entering a membrane will diffuse freely through the polymer matrix in the absence of impermeable nanoparticles. In that case, based on an elementary unit, the entire surface area of the polymer L

_{x}L

_{z}is available for diffusion. When a nanoparticle is introduced into an elementary unit, the gas molecules have to adopt a more tortuous path to diffuse around the impermeable nanoparticle. Figure 7 presents a plot of the isoconcentration lines within the polymer where the concentration within the cuboid particle is zero. Since the diffusion streamlines of gas molecules in the presence of an impermeable cuboid nanoparticle run perpendicular to the isoconcentration lines, one can easily imagine the diffusion path of these gas molecules. Indeed, the diffusion streamlines are perpendicular to the x-z plane of the elementary unit at the gas-membrane interface and deviate progressively as they approach the impermeable nanoparticle. The density of the isoconcentration lines is indicative of the intensity of the local flux. The flux increases when the diffusion channel narrows down and decreases when it widens up.

_{x}L

_{z}− x

_{p}z

_{p}) and the maximum surface area for diffusion (L

_{x}L

_{z}). It is logical to postulate that an increase in A* will lead to an increase in P

_{r}of the MMM, and vice versa. Figure 8 clearly shows the strong relationship between P

_{r}of a MMM with cuboid nanoparticles and the dimensionless parameter A*.

_{p}z

_{p}is the projected area of the nanoparticle, while L

_{x}L

_{z}is the total permeation area of an elementary membrane unit.

_{r}, Figure 8 illustrates the impact of the relative thickness y

_{p}/L

_{y}over the range spanning from 0.0500 to 0.9833. For a nanoparticle with a fixed y

_{p}/L

_{y}, a decrease in x

_{p}z

_{p}leads to an increase of A* and an increase in P

_{r}. For a fixed A*, P

_{r}decreases when y

_{p}/L

_{y}increases. The plot of P

_{r}versus A* at the largest y

_{p}/L

_{y}is approaching the 45° line. In other words, P

_{r}approaches A* when y

_{p}/L

_{y}tends to 1.0. It is clear that the model to be developed must include the strong linear relationship of P

_{r}with A* and a nonlinear component to account for the effect of y

_{p}/L

_{y}.

_{r}in Figure 8 into its linear (P

_{r}= A*) and NLP as a function of A*. This data exploration is essential in searching for the right form of the analytical equation in the development of an accurate predictive model. The contribution of the NLP that is related to y

_{p}/L

_{y}is clearly illustrated in Figure 9. The nonlinear term first increases rapidly with A* up to a maximum value before decreasing more gently to zero as A* tends to unity. The location and the magnitude of the maximum are a function of the relative thickness y

_{p}/L

_{y}. The maximum value is located in the interval of A* between 0.16 to 0.38. The contribution of the nonlinear part can be as high as 0.37 of the value of P

_{r}at y

_{p}/L

_{y}= 0.05 and A* = 0.18.

_{r}is now presented in Equations (33) to (35). The model depends strictly on two simple geometrical parameters: A* and y

_{p}/L

_{y}. Equation (33) is the sum of the linear and nonlinear parts of the estimated P

_{r}. The parameters a and b in Equation (33), given by Equations (34) and (35), respectively, are a function of only the relative thickness. It is interesting to note that $\varphi $ is not used explicitly in Equation (33). On the other hand, both A* and y

_{p}/L

_{y}indirectly determine the value of $\varphi $.

_{A}is the average prediction error, n is the number of data for the average error calculation, and P

_{r}and ${\widehat{P}}_{r}$ are the relative permeability obtained numerically and the one calculated by the proposed model, respectively. Table 3 summarizes the average prediction error for the combination of the geometrical parameters presented in Figure 10. In this investigation, 359 numerically predicted values of the relative permeability were used to fit the model by minimizing Equation (36) using the Levenberg–Marquardt algorithm.

_{r}accurately over a wide range of A* and y

_{p}/L

_{y}. Table 3 shows that for each group, the relative particle thickness, ε

_{A}is always below 0.03. The proposed model resorting to two simple geometrical parameters can be used with confidence to predict the permeability of MMMs with impermeable cuboid nanoparticles.

#### 4.4. Prediction of Experimental Data with the Proposed Model

^{3}[26] for the specific thermoplastic polyurethane used, the filler volume fraction was calculated. A void fraction of the nanofiller powder flakes of 0.38 [27] was used in the conversion from mass fraction to volume fraction. The comparison of the experimental relative permeability P

_{r}obtained by Zahid et al. [25] and the P

_{r}predicted by the proposed model is presented in Figure 11. The geometrical parameters A

^{*}and y

_{p}/L

_{y}depend on the spacing between the particles for fixed particle dimensions; however, this information was not provided by Zahid et al. [25]. Therefore, for each set of experimental results, three values of A

^{*}were used in the model to obtain reasonable estimations. Correspondingly, y

_{p}/L

_{y}was decreased for an increase of A

^{*}to maintain constant the filler volume fraction ϕ. Results show that the experimental data fall among the predicted P

_{r}curves for the three values of A

^{*}. The proposed model assumed ideal MMMs whereas, in an actual membrane, the dimensions, orientation, and spatial distribution may show significant variability. Despite these potential discrepancies, the comparison of Figure 11 shows that the model can adequately represent the relative permeability of the four types of MMMs presented in the paper of Zahid et al. [25].

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic diagram of a MMM and one elementary unit with a nanoparticle located at its center; y is the direction of the gas permeation. The dimensions of the cuboid nanoparticle are x

_{p}, y

_{p}, and z

_{p}, and the spherical particle is d

_{p}. The dimensions of the polymer elementary unit are L

_{x}, L

_{y}, and L

_{z}.

**Figure 2.**Nomenclature of an interior mesh point with its six neighboring mesh points. x, y, and z directions are represented by i, j, and k, respectively.

**Figure 3.**Plots of the relative permeability P

_{r}of MMMs as a function of the solid volume fraction $\varphi $ for spherical and cuboid impermeable nanoparticles, grouped by narrow ranges of the aspect ratio q. Data points were obtained numerically, and the solid lines are trend lines. The permeability of the continuous phase was kept constant at 5 × 10

^{−12}m

^{2}/s (5.0 × 10

^{−11}m

^{2}/s diffusivity and 0.10 solubility).

**Figure 4.**A comparison of the relative permeability of MMMs obtained numerically with the ones predicted by the six models presented in Table 1 as a function of the filler volume fraction ($\varphi $). The comparison is made for a polymeric elementary unit containing an impermeable nanoparticle: A sphere and cuboids with four different ranges of the aspect ratio (q) ([2.0, 2.3], [3.0, 3.2], [5.0, 5.2], [7.0, 7.3]). Acronyms of each model are defined in Table 1.

**Figure 5.**Feedforward neural network for the prediction of the relative permeability (P

_{r}) for impermeable cuboid nanoparticles as a function of the normalized projected area (x

_{p}z

_{p}/L

_{x}L

_{z}) and the relative thickness (y

_{p}/L

_{y}).

**Figure 6.**Parity plot of the predicted P

_{r}and the numerically-determined P

_{r}for the feedforward neural network (FFNN) of Figure 5 with six hidden neurons, including the bias.

**Figure 7.**The steady-state isoconcentration lines of the migrating species on the half-cut plane of an elementary unit of a MMM with a cuboid nanoparticle with an aspect ratio q = 2.07 and a relative thickness y

_{p}/L

_{y}= 0.25.

**Figure 8.**P

_{r}of MMMs as a function of A* for five values of y

_{p}/L

_{y}ranging from 0.0500 to 0.9833.

**Figure 9.**Plots of the linear and nonlinear portions of the P

_{r}of a MMM as a function of A* for five values of y

_{p}/L

_{y}ranging from 0.05 to 0.98.

**Figure 10.**Plots of the P

_{r}of MMMs embedding cuboid nanoparticles as a function of A* for five different values of y

_{p}/L

_{y}.

**Figure 11.**Comparison of the experimental data of Zahid et al. [25] and the prediction of the proposed model for the relative permeability of the oxygen transport across four types of MMMs consisting of thermoplastic polyurethane (TPU), embedding few-layer (FLG), and multi-layer (MLG) graphene flakes.

**Table 1.**Predictive models for the relative permeability (P

_{r}) of a migrating species in a mixed-matrix membrane (MMM) with nanoparticles [19].

Model | Equation | Equation # |
---|---|---|

Maxwell (MXW) [10] | ${P}_{r}=\frac{{P}_{d}+2{P}_{c}-2\varphi ({P}_{c}-{P}_{d})}{{P}_{d}+2{P}_{c}+\varphi ({P}_{c}-{P}_{d})}$ | (2) |

$\begin{array}{cc}{P}_{d}=0:& {P}_{r}=\frac{1-\varphi}{1+\varphi /2}\end{array}$ | (3) | |

Bruggeman (BGM) [11] | ${P}_{r}{}^{1/3}\left(\frac{{P}_{d}-{P}_{c}}{{P}_{d}-{P}_{r}{P}_{c}}\right)={\left(1-\varphi \right)}^{-1}$ | (4) |

$\begin{array}{cc}{P}_{d}=0:& {P}_{r}={\left(1-\varphi \right)}^{3/2}\end{array}$ | (5) | |

Lewis–Nielsen (LN) [12,13] | ${P}_{r}=\frac{1+2\varphi \left({P}_{d}-{P}_{c}\right)/\left({P}_{d}+2{P}_{c}\right)}{1-\psi \varphi \left({P}_{d}-{P}_{c}\right)/\left({P}_{d}+2{P}_{c}\right)},\psi =1+\left(\frac{1-{\varphi}_{m}}{{\varphi}_{m}{}^{2}}\right)\varphi $ | (6) |

$\begin{array}{cc}{P}_{d}=0:& {P}_{r}=\frac{1-4\varphi}{1+2\psi \varphi}\end{array}$ | (7) | |

Pal (PAL) [14] | ${P}_{r}{}^{1/3}\left(\frac{{P}_{d}-{P}_{c}}{{P}_{d}-{P}_{r}{P}_{c}}\right)={\left(1-\frac{\varphi}{{\varphi}_{m}}\right)}^{-{\varphi}_{m}}$ | (8) |

$\begin{array}{cc}{P}_{d}=0:& {P}_{r}={\left(1-\frac{\varphi}{{\varphi}_{m}}\right)}^{3/2{\varphi}_{m}}\end{array}$ | (9) | |

Cussler (CSL) [15,16] | $\begin{array}{cc}{P}_{r}=\frac{1}{1+\alpha \varphi},& \alpha \varphi <1\end{array}$ | (10) |

$\begin{array}{cc}{P}_{r}=\frac{1-\varphi}{1-\varphi +\mu {\alpha}^{2}{\varphi}^{2}},& \alpha \varphi >1\end{array}$ µ = 1 for flakes as periodic ribbons, µ = 4/9 for flakes as periodic hexagons. | (11) | |

Bharadwaj (BDW) [17] | ${P}_{r}=\frac{1-\varphi}{1+\frac{{x}_{p}}{3{y}_{p}}\left(\mu +\frac{1}{2}\right)\varphi}$, µ = 0 for randomly dispersed fillers, µ = 1 for fillers perfectly aligned perpendicular to the gas flux | (12) |

Pearson Correlation Coefficient (PCC) | |||||
---|---|---|---|---|---|

Rank | Variable | Average | σ_{EA} | Cov(E_{A}, P_{r}) | PCC(E_{A}, P_{r}) |

1 | x_{p}z_{p}/L_{x}L_{z} | 0.3381 | 0.3154 | −0.0865 | −0.9629 |

2 | x_{p}/L_{x} | 0.4961 | 0.2923 | −0.0763 | −0.9156 |

3 | $\varphi $ | 0.1331 | 0.1821 | −0.0436 | −0.8401 |

4 | q | 2.5620 | 3.4326 | −0.1937 | −0.1981 |

5 | y_{p}/L_{y} | 0.4022 | 0.2753 | −0.0144 | −0.1831 |

- | P_{r} | 0.7395 | 0.2850 | - | - |

**Table 3.**Average prediction errors of the P

_{r}of Figure 10 using the proposed model evaluated over five values of the y

_{p}/L

_{y}.

y_{p}/L_{y} | A* | ε_{A} |
---|---|---|

0.0500 | 0.03–0.99 | 0.0098 |

0.2500 | 0.03–0.99 | 0.0070 |

0.5167 | 0.03–0.99 | 0.0110 |

0.7500 | 0.16–0.99 | 0.0281 |

0.9833 | 0.16–0.99 | 0.0048 |

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**MDPI and ACS Style**

Wu, H.; Zamanian, M.; Kruczek, B.; Thibault, J.
Gas Permeation Model of Mixed-Matrix Membranes with Embedded Impermeable Cuboid Nanoparticles. *Membranes* **2020**, *10*, 422.
https://doi.org/10.3390/membranes10120422

**AMA Style**

Wu H, Zamanian M, Kruczek B, Thibault J.
Gas Permeation Model of Mixed-Matrix Membranes with Embedded Impermeable Cuboid Nanoparticles. *Membranes*. 2020; 10(12):422.
https://doi.org/10.3390/membranes10120422

**Chicago/Turabian Style**

Wu, Haoyu, Maryam Zamanian, Boguslaw Kruczek, and Jules Thibault.
2020. "Gas Permeation Model of Mixed-Matrix Membranes with Embedded Impermeable Cuboid Nanoparticles" *Membranes* 10, no. 12: 422.
https://doi.org/10.3390/membranes10120422