# Dimensional Transformation of Percolation Structure in Mixed-Matrix Membranes (MMMs)

^{*}

## Abstract

**:**

## 1. Introduction

_{2}and a selectivity of CO

_{2}/N

_{2}and CO

_{2}/CH

_{4}are observed. The permeability to other gases first increased and then decreased, i.e., it varied non-linearly in a narrow range.

_{2}first increases and then decreases with an increase in the content of CNT-PEG in the dry state [15]. Water vapor in the gas stream further improves the permeability to CO

_{2}. At a relative humidity of 100% for a hybrid membrane containing 3 wt.% CNT-PEG, a permeability to CO

_{2}of 369.1 barrels was obtained with a CO

_{2}/N

_{2}selectivity of 110.8, which exceeds the upper limit of Robson. At the same time, the permeability of CNT-PEG to CO

_{2}varies non-linearly depending on the concentration of CNTs. An increase in the mass concentration of CNTs from 0 to 3% leads to an increase in the permeability to CO

_{2}from 90 to 260 barrels, and an increase in the concentration of CNTs from 3 to 20% leads to a decrease in the permeability to CO

_{2}from 260 to 110 barrels.

_{2}O feed mixture at 50 °C, and an increase in the concentration from 3% to 5% led to a doubling of the flow. In the second, an increase in the CNT concentration from 0 to 3% led to a 3.5-fold increase in the flux, and a further increase in the flux led to a decrease in the flux from 0.14 to 0.08 kg/(m

^{2}∙h).

_{2}) were grafted on the surface of multi-walled carbon nanotubes (MWCNTs) which were then incorporated as fillers in the poly(ether-block-amide) (PEBA) polymeric matrix in the range of 0.1–1 wt% loading. The permeability of MMMs to all the studied gases first increased with an increase in the concentration of CNTs, and then decreased.

_{2}, H

_{2}, CO

_{2}) varied non-linearly with the CNT concentration. An increase in the concentration of CNTs from 0 to 1% led to an increase in the permeability to all gases under study, and a further increase in the concentration from 1 to 2% led to a decrease in the permeability to all gases except hydrogen; for hydrogen, the permeability did not change within the error. In recent studies on the introduction of CNTs into polymers, it is reported that the upper limit of Robson has been overcome [14,15,16,17,18].

## 2. Calculation

- (1)
- Through the polymer;
- (2)
- Through the polymer/CNT interfacial layer, which results from poor adhesion between polymer chains and CNTs;
- (3)
- Through the internal cylindrical channels of the CNTs.

_{x}is the fraction of the polymer volume occupied by the percolation cluster, K

_{x}is the permeability coefficient of the percolation cluster, K

_{p}is the polymer permeability coefficient, and dR = R – R

_{0}is the thickness of the interfacial layer. The thickness of polymeric membranes with embedded CNTs differs slightly from the length of the CNTs; such a system cannot be considered infinitely large when calculating the parameters of a CNT percolation cluster, and surface effects must be taken into account. Expression 1 has a general form for a composite material parameter and illustrates the linear correlation of properties in a two-component material based on the rule of mixtures. It shows that the contribution of each component is proportional to its concentration and the observed nonlinear changes in properties can be associated with a nonlinear change in the number of particles affecting the properties of the matrix.

_{0}, which forms regions of the polymer, the permeability of which increases due to the interaction with the surface of the nanotubes, and is determined by the interaction between the CNTs and the polymer. Spherocylinders form a cluster when shells overlap, and if a cluster connects opposite faces of a 3D matrix (upper and lower), then it is considered percolative. To achieve this, a three-dimensional matrix is randomly filled with spherical cylinders, the distance between them is determined, a list of formed clusters is formed, and it is checked whether there is a percolative one among them.

_{∞}is the number of spherical cylinders that make up the percolation cluster, and N is the number of spherocylinders added to the system.

_{0}is the radius and L is the length of the spherocylinders, N is the number of spherocylinders, and V is the volume of the matrix.

## 3. Results and Discussion

#### 3.1. Influence of CNT Length

_{0}was equal to 25 nm, and the outer radius R was equal to 125 nm. Figure 2 shows the results of the numerical simulation.

_{0}ranges from 0.1294 to 0.2742, A from 0.6809 to 0.6814, and t from 1.9458 to 1.9980.

#### 3.2. Effect of CNT Interfacial Layer Thickness

_{0}was equal to 25 nm, and the length L was equal to 4 µm. An increase in the thickness of the interfacial layer from 25 to 100 nm leads to a nonlinear decrease in the CNT concentration at which the percolation cluster is formed with a probability of 100%. This change in the thickness of the interfacial layer leads to a decrease in the concentration of the CNTs which must be introduced into the polymer to form a percolation cluster with a probability of 100% and a strength of 60% by a factor of 4.3 (Figure 4).

_{0}, A

_{1}ranges from 3.0383 to 1.7113, A

_{2}from 1.0812 to 6.8893, t

_{1}from 0.0088 to 0.0329, t

_{2}from 0.0371 to 0.0071, and y

_{0}from 0.1451 to 0.1973.

#### 3.3. Effect of CNT Radius

_{0}. An increase in the radius from 25 to 40 nm leads to an increase in the volume concentration at which a percolation CNT cluster is formed in the polymer from 0.35 to 0.85% at a strength of 20%, from 0.4 to 0.84% at a strength of 40%, from 0.42 to 0.95% at a strength of 60%, and from 0.51 to 1.25% at a strength of 80%.

_{0}from −0.934 to −0.9693.

#### 3.4. Effect of Geometric Parameters of the System

## 4. Conclusions

_{0}= 25 nm, L = 4 µm) in a geometrically isotropic matrix, we obtained a percolation structure of a certain strength. As can be seen from Figure 4, with an assumed layer thickness of dR = 100 nm, we can expect the formation of a percolation cluster with a capacity of about 80% at a particle concentration of more than 0.35%. However, if the actual thickness of the modified layer is 65 nm, then the cluster strength at the same concentration will be about 20%. That is, the proportion of MMMs in which specific transport associated with embedded particles is implemented will decrease by four times, which will significantly worsen the transport properties of such MMMs.

_{0}= 25 nm, L = 5 µm, and dR = 100 nm) in a geometrically isotropic matrix, a CNT concentration of more than 0.2% was chosen, which guaranteed a 100% probability of forming a percolation cluster with a strength of about 50% (Figure 8). However, with a decrease in the actual thickness of the film, the cluster strength is reduced by more than two times already with the ratio of lateral and normal matrix sizes at 5 µm (Figure 8b). A further increase, despite the fulfillment of the concentration conditions of percolation (Figure 8b), may lead to zero strength of the percolation cluster, and in such systems there will be a strong heterogeneity of the transport properties of the MMMs.

_{0}= 25 nm, L = 2 µm, and dR = 100 nm) has been added to the isotropic matrix. With such particle sizes, a percolation cluster is formed at concentrations of about 0.4% (Figure 6); however, an error in determining the actual radius of the particles (for example, R

_{0}= 30 nm) leads to a situation where the concentration is insufficient to guarantee the formation of a percolation cluster, and the strength of such a percolation cluster is less than 20%. Thus, most of the formed MMMs will not contain a percolation structure. Errors in the actual CNT length lead to similar consequences. For example, for particles of a given size (R

_{0}= 25 nm, dR = 100 nm, and length L = 5 µm), we determine the required concentration of 0.2%, with some excess. However, if the actual length of the CNTs is equal to 4 µm, the percolation cluster is practically not formed, since the probability is less than 10% (Figure 2) and its capacity is about 10%. In such a system, the change in transport properties will be significantly lower than expected and in a small number of experiments. Thus, an error in the geometric dimensions of the particles by 20–25% leads to an erroneous choice of the required concentration and, as a consequence, the absence of the necessary percolation structure in the MMMs.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The probability of the formation and strength of a percolation cluster from the volume concentration of CNTs with a length from 2 to 5 µm.

**Figure 3.**The volume fraction of CNTs that must be embedded in order to obtain the strength of the percolation cluster equal to 20, 40, 60, and 80% of the CNT length.

**Figure 4.**The probability of the formation and the strength of a percolation cluster from the volume concentration of CNTs with the thickness of the interfacial layer dR from 100 to 25 nm.

**Figure 5.**The volume fraction of CNTs that must be embedded in order to obtain the strength of the percolation cluster equal to 20, 40, and 60% of the thickness of the interfacial layer.

**Figure 6.**The probability of the formation and the strength of the percolation cluster from the volume concentration of CNTs with an inner radius of 25 to 40 nm.

**Figure 7.**The volume fraction of CNTs that must be embedded in order to obtain the strength of the percolation cluster equal to 20, 40, 60, and 80% of the CNT radius.

**Figure 8.**(

**a**) The probability of the formation of a percolation cluster for anisotropic systems of different thicknesses (from 25 to 5). (

**b**) The probability of the formation and the strength of the percolation cluster on the volume concentration of CNTs for anisotropic systems of different thicknesses (from 25 and 5).

**Figure 9.**Correlation radius of percolation structures in the vicinity of percolation concentration for anisotropic systems.

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**MDPI and ACS Style**

Grekhov, A.; Eremin, Y.
Dimensional Transformation of Percolation Structure in Mixed-Matrix Membranes (MMMs). *Membranes* **2023**, *13*, 798.
https://doi.org/10.3390/membranes13090798

**AMA Style**

Grekhov A, Eremin Y.
Dimensional Transformation of Percolation Structure in Mixed-Matrix Membranes (MMMs). *Membranes*. 2023; 13(9):798.
https://doi.org/10.3390/membranes13090798

**Chicago/Turabian Style**

Grekhov, Alexey, and Yury Eremin.
2023. "Dimensional Transformation of Percolation Structure in Mixed-Matrix Membranes (MMMs)" *Membranes* 13, no. 9: 798.
https://doi.org/10.3390/membranes13090798