Polyetherimide-Montmorillonite Nano-Hybrid Composite Membranes: CO 2 Permeance Study via Theoretical Models

: The incorporation of aminolauric acid modiﬁed montmorillonite ( f -MMT) in polyetherimide (PEI) has been implemented to develop hollow ﬁbre nano-hybrid composite membranes (NHCMs) with improved gas separation characteristics. The aforementioned characteristics are caused by enhanced f -MMT spatial dispersion and interfacial interactions with PEI matrix. In this study, existing gas permeation models such as, Nielsen, Cussler, Yang–Cussler, Lape–Cussler and Bharadwaj were adopted to estimate the dispersion state of f -MMT and to predict the CO 2 permeance in developed NHCMs. It was found out that the average aspect ratio estimated was 53, with 3 numbers of stacks per unit tactoid, which showed that the intercalation f -MMT morphology is the dominating dispersion state of ﬁller in PEI matrix. Moreover, it was observed that Bharadwaj model showed the least average absolute relative error (%AARE) values till 3 wt. % f -MMT loading in the range of ± 10 for a pressure range of 2 to 10 bar. Hence, Bharadwaj was the best ﬁt model for the experimental data compared to other models, as it considers the platelets orientation.


Introduction
Today, among world community, the concerns over global warming have increased due to its impact on climate and weather change. Nevertheless, the global warming has changed the human life and the risk can be even more in future [1]. Carbon dioxide, a major greenhouse gas, contributes a lot to global warming. As a result, increasing efforts have been observed worldwide in order to counter the dangers of CO 2 emission [2]. The research community around the globe put serious efforts in order to separate CO 2 from its emission source. The conventional separation methods like absorption, adsorption, and cryogenic distillation have been used industrially; however, the newer technologies like membrane-based processes are getting overwhelming response owing to cleaner, simpler and cost-effective technique [3,4]. In membrane-based separation technology, gas-gas separations are rapidly growing industrial processes, and increasingly used for nitrogen separation from air, hydrogen recovery in ammonia synthesis, petroleum cracking plants and natural gas purification [5,6].

Background of Gas Permeation Models
The gas molecules passage through permeation membranes is governed by viscous flow, Knudsen diffusion, surface diffusion, molecular sieving or solution diffusion. The permeation of gas molecules across membranes is dependent on the membrane morphology which is either porous or non-porous in nature. For porous morphology, the size and shape of the gas molecules and pore size are responsible for gas molecules transport.
To the contrary, the gas molecules permeation through non-porous clay particles is governed by solution-diffusion mechanism. The gas molecules permeation through polymer phase takes place in four steps; the sorption of penetrant gas molecules at the membrane surface, gas dissolution inside membrane, diffusion through membrane, and desorption from the other surface [18]. Hence, the gas molecules mobility across polymer phase depends on both thermodynamic and kinetic properties of Processes 2020, 8,118 3 of 11 penetrant gas [19]. Subsequently, the gas permeation in polymer matrix depends on both diffusion (D) and solubility (S) coefficients as explained by Fickian diffusion as given in Equation (1) [20].
However, the presence of impermeable inorganic filler like f -MMT in polymer phase increases the diffusion path for penetrant molecules. This obstruction in the diffusion pathways depend on filler's loading, aspect ratio and platelets orientation.
The increase in impermeable filler phase decreases the diffusion coefficient due to tortuosity, and consequently the permeability of membrane decreases [21]. The f -MMT orientation in polymer matrix is another key aspect that defines the membrane performance. For instance, the parallel orientation of layered fillers to the diffusion path results in increment of diffusion coefficient of penetrant molecules, whereas, opposite behavior is observed with perpendicular orientation. In addition, aspect ratio plays an important role in layered structure filler's intercalation and exfoliation in polymer matrix. The higher aspect ratio of filler favors exfoliation due to weak inter-layer bonding and eventually improved dispersion can be achieved [22]. Delamination state of filler represents the intercalation of polymer chain in filler inter-layer spacing, whereas, exfoliation shows the complete separation of filler layers. Exfoliation is the most desired state as the enhanced membrane performance can be ascertained at lower loadings. Generally, complete exfoliation state of the clay is not achieved in polymer matrix, however, mixed morphology containing intercalated and exfoliated state are often observed [23].

Models Used for Developed Membranes
Various models are reported in the literature to predict the gas permeation behavior of membranes incorporating layer structured fillers in polymer phase, a few are provided in Table 1. These models are based on specific assumptions; therefore, each model has certain limitations. Nielsen model is the simplest model as it considers the rectangular shaped layers arranged in the gas diffusion pathways. Since the model is based on diffusion path obstruction due to platelets. Therefore, this model is also known as "tortuous path" model [24]. This model assumes the layered filler rectangular shaped nanoparticles are dispersed in polymer phase. Table 1. Phenomenal models used to calculate gas permeance through nano-hybrid composite membranes (NHCM) using layered silicates as fillers.

Models Equation Ref Description
a.ϕ 2 [24] Where, P c , Permeability of composite, P 0 , Permeability of neat polymeric membrane ϕ, Volume fraction of filler a, Aspect ratio of dispersed filler S, is the order factor S = 0, Random orientation S = −1/2, Parallel orientation to the gas diffusion path S = 1, Horizontal orientation to the gas diffusion path [28] Cussler considered the nano-filler arrangement in layered form and each layer is separated by narrow slits [25]. Since, this model encompasses the effect of filler aspect ratio along with volume fraction; it is more reliable in permeability prediction than the Nielsen model. Furthermore, Yang-Cussler model was developed by considering the mono-dispersion of nanoparticles with varied aspect ratio and parallel alignment of platelets to the gas diffusion path [26]. On the other hand, Lape-Cussler assumed the rectangular shaped filler platelets that have uniform aspect ratio, random dispersion in polymer phase and are in parallel alignment to each other along the gas flow direction [27]. The common considerations in all of the above-stated models are the complete platelets exfoliation and their horizontal arrangement to the penetrant diffusion path. Later, Bharadwaj introduced the platelets orientation factor in his permeation model [28]. The orientation factor includes various platelets arrangement such as horizontal, parallel or random distribution to the diffusion path. Unlike others, Bharadwaj model describes both intercalation and exfoliation states of the dispersed layered nanoparticles.

Estimation of f-MMT Aspect Ratio
In this section, the f -MMT aspect ratio and its dispersion state are determined by opting phenomenological models. In addition, the permeance at 2 bar pressure is considered for uncoated PEI-f -MMT NHCM. The rationale behind choosing low pressure was to eliminate the effect of surface defects and plasticization. Furthermore, uncoated membranes carried the effect of f -MMT only and independent of PDMS coating characteristics.
The aspect ratio, dispersion state and the length of the f -MMT flakes together with its phase interactions with PEI matrix defines the permeation properties of the developed NHCMs. The average aspect ratio of the clay platelets can either determined by using TEM analysis or estimated through theoretical permeation models [17,29,30]. In the latter case, the experimentally determined permeation values are inserted in permeation models. The theoretical aspect ratio value, at which the experimental permeation values are in close agreement to the theoretically determined values, is considered as clay's average aspect ratio. Zulhairun and Ismail developed NHCM and estimated the average aspect ratio of cloisite flakes in polysulfone by adopting theoretical permeation models [31].

Results and Discussion
Figure 1 depicts various theoretical permeation curves generated via Nielsen, Cussler, Yang-Cussler, Lape-Cussler and Bharadwaj models. Where, P c , represent the permeance of NHCM; P o , permeance of neat membrane; and, a, is the aspect ratio of f -MMT inside PEI matrix. For all the models, it is observed that, the relative permeation curves fall-off as the volume fraction and aspect ratio of the nano-filler increases. In addition, the experimental data points and theoretical curves are developed against several aspect ratios of the nano-clay. Furthermore, it is observed that the experimental data points are not fully fit the theoretical curves; nevertheless, they exhibit the same trend at volume fraction above 0.005 for CO 2 gas.
Moreover, the calculated relative permeance using theoretical models displays a trend same as the experimentally determined relative permeation data. Nevertheless, an abrupt increase in relative permeance is noted at 0.020 filler volume fraction, which might show the clay agglomeration at this loading. Table 2 shows the relative permeance data of CO 2 gas determined via experiments and calculated by using phenomenological models at aspect ratios 40, 40, 60, 15 and 110 respectively. where P cal i and P exp i are the relative permeance for the ith experimental value and NDP represents the number of data points.   The average aspect ratio calculated by all these models is 53 for PEI-f -MMT. Equation (2) describes the average absolute relative error (%AARE), which is the tuning parameter and relates the difference between experimental and calculated values. The lower the %AARE, the closer agreement exists between experimental and calculated data.
where P cal i and P exp i are the relative permeance for the ith experimental value and NDP represents the number of data points.
Furthermore, Table 2 (1). Moreover, for volume fraction 0.002, the agglomeration took place and abrupt increase in permeation is observed. This agglomeration behavior of filler was not considered in theoretical permeation models.
Moreover, in order to approximate the number of stacks of flakes per unit tactoid, Bharadwaj proposed a method for layered nanoparticles [28]. In order to approximate the average aspect ratio, the lateral dimension of f -MMT is required. Yano et al. studied the polyimide-Na-MMT for water permeability and determined the lateral dimension as 218 nm and width of 1 nm [29]. In addition, the polymer chains expanded by f -MMT interlayer space to 1 nm as suggested by Bharadwaj. The number of stacks per unit tactoid calculated using average aspect ratio as explained in Figure 2, and is approximately 4 for volume fraction of 0.005 to 0.020. This number shows superior dispersion properties of f -MMT for 1 to 3 wt. % loading in PEI matrix. Though, it does not show complete f -MMT exfoliation, however, the intercalation consists of only 3 stacks per unit tactoid, and this in agreement to reported literature. Zulhairun and Ismail determined the intercalation morphology of cloisite 15A via theoretical models and reported 2 to 4 number of stacks in a tactoid ranged from 2 to 4 [15]. Similarly, in another study, 2-3 stacks in a tactoid for 5 wt. % cloisite 15A loadings [21]. Furthermore, Kim et al. used high shear mixer to develop cellulose-AMH-3 NHCM owing to achieve enhanced filler dispersion and reported 2-3 stacks per unit tactoid [16]. Picard et al. experimentally estimated the average aspect ratio of f -MMT in Nylon 6, and the experimental value of 20 was then validated via Lape-Cussler model, which provides closer value of 17.2 [17]. Therefore, it can be concluded that promising dispersion and improved interactions exists between f -MMT and PEI. Similarly, in another study, 2-3 stacks in a tactoid for 5 wt. % cloisite 15A loadings [21]. Furthermore, Kim et al. used high shear mixer to develop cellulose-AMH-3 NHCM owing to achieve enhanced filler dispersion and reported 2-3 stacks per unit tactoid [16]. Picard et al. experimentally estimated the average aspect ratio of f-MMT in Nylon 6, and the experimental value of 20 was then validated via Lape-Cussler model, which provides closer value of 17.2 [17]. Therefore, it can be concluded that promising dispersion and improved interactions exists between f-MMT and PEI.

Validation of Developed Membranes CO2 Permeance via Theoretical Models
In order to validate the experimental CO2 permeation values with existing theoretical models, Nielsen, Cussler, Yang-Cussler, Lape-Cussler and Bhardwaj models were considered at pressure range from 2 to 10 bar. The above stated models are available models that considered the layered form of dispersed filler. Since f-MMT is a type of clay and consist of alumino silicates in layered form, the aforementioned models were adopted for this study. Table 3 shows the %AARE calculated for PDMS coated PEI-f-MMT composite membranes from theoretically generated curves.

Validation of Developed Membranes CO 2 Permeance via Theoretical Models
In order to validate the experimental CO 2 permeation values with existing theoretical models, Nielsen, Cussler, Yang-Cussler, Lape-Cussler and Bhardwaj models were considered at pressure range from 2 to 10 bar. The above stated models are available models that considered the layered form of dispersed filler. Since f -MMT is a type of clay and consist of alumino silicates in layered form, the aforementioned models were adopted for this study. Table 3 shows the %AARE calculated for PDMS coated PEI-f -MMT composite membranes from theoretically generated curves. Nielsen model considered the horizontal dispersion of f -MMT in the PEI matrix. The experimentally determined value showed deviation from the theoretical values as depicted in Table 3 Moreover, Cussler model showed a similar deviating trend from experimentally determined permeance till 8 bar for all developed PEI-f -MMT NHCMs. Furthermore, the experimentally determined CO 2 permeance values for PEI-f -MMT (1) showed close agreement to the theoretically generated values. In contrast, NHCM containing 4 wt. % of f -MMT showed the maximum deviation from the theoretical curve. This is mainly due to agglomeration of f -MMT in PEI matrix, which was not considered in the models.
In contrast to aforementioned models, Yang-Cussler model considered the mono dispersion of nano filler containing variations in aspect ratio. Therefore, it can be seen that the CO 2 permeance values follow a similar trend with their respective theoretical curves. For 1 and 2 wt. % f -MMT loadings in PEI polymer, the calculated %AARE values are less than ±10 as tabulated in Table 3. However, for PEI-f -MMT (3), the %AARE is ±20. This model is better suited to experimentally determined data compared to Nielsen and Cussler models. This is due to Yang-Cussler model incorporated the concept of flakes for filler, which is a relevant assumption to f -MMT structure. Thus, the theoretical and experimental values are in close agreement for lower filler loadings.
Since, Lape-Cussler model considered the rectangular shape of incorporated fillers with random dispersibility in host polymer matrix. It is noted that the experimental points are poorly suited with theoretical curves. The minimum %AARE for any membrane at any pressure is between ±20 to ±60, which indicates the poor performance of developed membrane based on Lape-Cussler's assumptions.
The assumption of parallel distribution of clay platelets in the diffusion path was unable to relate the actual distribution of clay platelets in PEI matrix. Table 3 shows that the Bharadwaj model is the most suitable model for developed membranes. It is observed that for all feed pressure range, the PEI membranes incorporated with 1 to 3 wt. % of f -MMT has shown close agreement with %AARE less than ±10. This showed excellent performance of Bharadwaj model in order to fit the experimentally determined CO 2 permeance values at feed pressure of 2 to 10 bar for PEI-f -MMT NHCMs. However, PEI-f -MMT (4) also showed close agreement to theoretically generated data points, the deviations for all pressure ranges are below 24 value of %AARE. This %AARE value shows lower deviation for 4 wt. % f -MMT for all the aforementioned models despite higher filler agglomeration at this loading. Figure 3 presents the experimental point for spun hollow fibre NHCMs in comparison to Bharadwaj model's generated curves. It is observed that CO 2 permeance trend against pressure by theoretical curve is the same as experimentally observed data points. Moreover, for 1 to 3 wt. % of f -MMT incorporation in PEI matrix shows closer CO 2 permeance behavior compared to their respective theoretical curves. However, the model does not consider the effect of filler agglomeration. As a result, due to high f -MMT agglomeration at 4 wt. %, the model showed significant deviation compared to theoretically generated curve.
Bharadwaj model is the most advanced and versatile which incorporates the orientation geometry of the dispersed filler. Due to this assumption, Bharadwaj model is the best suited to PEI-f -MMT hollow fibre NHCMs at various feed pressure and f -MMT loadings, compared to other considered theoretical models.
Bharadwaj model's generated curves. It is observed that CO2 permeance trend against pressure by theoretical curve is the same as experimentally observed data points. Moreover, for 1 to 3 wt. % of f-MMT incorporation in PEI matrix shows closer CO2 permeance behavior compared to their respective theoretical curves. However, the model does not consider the effect of filler agglomeration. As a result, due to high f-MMT agglomeration at 4 wt. %, the model showed significant deviation compared to theoretically generated curve. Bharadwaj model is the most advanced and versatile which incorporates the orientation geometry of the dispersed filler. Due to this assumption, Bharadwaj model is the best suited to PEIf-MMT hollow fibre NHCMs at various feed pressure and f-MMT loadings, compared to other considered theoretical models.

Conclusions
Existing phenomenological models, Nielsen, Cussler, Yang-Cussler, Lape-Cussler, and Bharadwaj models were opted in order to validate the experimental data. In the first place, the aspect ratio of the f-MMT filler was estimated by incorporating the experimentally determined permeation values in theoretical models. The average aspect ratio was found to be 53, with 3 numbers of stacks per tactoid, which showed that the intercalation morphology is the dominating dispersion state of f-MMT filler in PEI matrix. Afterwards, to validate the experimental CO2 permeation values with existing theoretical models were considered at pressure range from 2 to 10 bars. Bharadwaj model proved to be the most suitable against at all feed pressures. The Bharadwaj model is versatile in nature, which assumed both parallel and horizontal distribution of incorporated filler. For f-MMT loading of 1 to 3 at full pressure range, the %AARE range from −10.11 to 8.55, which showed closer

Conclusions
Existing phenomenological models, Nielsen, Cussler, Yang-Cussler, Lape-Cussler, and Bharadwaj models were opted in order to validate the experimental data. In the first place, the aspect ratio of the f -MMT filler was estimated by incorporating the experimentally determined permeation values in theoretical models. The average aspect ratio was found to be 53, with 3 numbers of stacks per tactoid, which showed that the intercalation morphology is the dominating dispersion state of f -MMT filler in PEI matrix. Afterwards, to validate the experimental CO 2 permeation values with existing theoretical models were considered at pressure range from 2 to 10 bars. Bharadwaj model proved to be the most suitable against at all feed pressures. The Bharadwaj model is versatile in nature, which assumed both parallel and horizontal distribution of incorporated filler. For f -MMT loading of 1 to 3 at full pressure range, the %AARE range from −10.11 to 8.55, which showed closer agreement to the experimentally determined permeation values. However, at further loading, f -MMT agglomeration started and deviation from the experimental data was observed.