Polymer membranes are widely used for separating gas mixtures due to their reduced energy cost compared to traditional separation methods, reliable mechanical stability under operational conditions, and ease of integration into existing industrial processes. 1,2 Many industrial applications of polymer membranes rely on the size-selective permeability enabled by the polymer nanostructure. 3 For instance, glassy polymers exhibit high selectivity in response to minor variations in penetrant size. 3 This characteristic is advantageous in applications such as the separation of high-purity nitrogen from the air, the extraction of hydrogen from mixtures containing larger components, and the purification of natural gas. 3,4 Often, membrane separation performance is characterized by the gas permeability and selectivity. 5 Within the framework of the solution-diffusion model, the gas permeability, directly related to gas flux, is the product of the gas solubility and diffusion in the polymer. The selectivity of a membrane is determined by comparing the permeability coefficients of two gases, expressed as the ratio α A/B = P A /P B , where P A and P B denote the permeabilities of the more and less permeable gases, respectively.
The desired goal of designing polymer membranes with high permeability and high selectivity has prompted extensive research efforts. However, a well-established trade-off exists whereby polymers with higher permeability generally display less selectivity, and vice versa. [5][6][7][8] Robeson notably characterized this relationship and its upper bound of the best available polymers for a given gas pair separation. 5,6,9,10 While numerous methods have been investigated to improve the performance of traditional polymer membranes, [11][12][13] polymer nanocomposites have emerged as promising membrane materials due to their novel polymer structures and superior separation properties.
In the context of gas separation, polymer-grafted nanoparticles (PGNs) are nanocomposite materials that have been shown to exhibit tunable transport properties. PGNs also possess enhanced mechanical properties, such as strength and durability, enabled by the superior dispersion effects resulting from grafting chains onto the nanoparticle surfaces. [14][15][16][17][18][19] Moreover, chemically grafting the polymers onto the particles overcomes significant immiscibility challenges that typically hinder the formation of polymer nanocomposite membranes through the physical mixing of the constituents. [20][21][22][23] Several studies in the literature have demonstrated that the permeability and separation performance of PGN membranes can be tuned by manipulating their structural parameters, such as NP size, grafted chain length, and grafting density, and free volume distribution. [23][24][25] Bilchak et al. demonstrated that adding free polymer chains to the interstitial pockets between PGNs restricts the transport of larger gas molecules, thereby increasing gas selectivity with only a minor reduction in small gas permeability. 23 This finding suggests the presence of spatial heterogeneity within PGN membranes, characterized by regions with distinct polymer conformations and free volume distributions. While experimental work has suggested a rationale for the enhanced performance of PGN membranes, there lacks a theoretical framework to understand how spatial heterogeneities in the polymer nanocomposite structure influence size-based gas selectivity.
This work presents a model that attributes transport performance enhancements of PGNs to their heterogeneous structure, characterized by regions with unique polymer conformations and free volume distributions. The work of Bilchak et al. [23][24][25] supplements this hypothesis by suggesting that the enhanced permeability arises from an increased diffusion coefficient, facilitated by distinct polymer conformations within regions where gas molecule transport occurs. This model integrates established theories of diffusion and permeability to explain the kinetics of the enhanced gas molecule transport through the heterogeneous nanostructure of PGNs. The model is demonstrated by fitting its parameters to data acquired from existing literature on gas transport through traditional linear polymer membranes. Subsequently, the fitted model is used to analyze permeability data acquired for PGN membranes with identical chemistry to those in the fitting data. The significance of this model lies in its ability to identify contributions to transport properties such as permeability and activation energy barriers that arise from the heterogeneous architecture of PGNs.
The diffusion of gas molecules through polymeric media is postulated to be an activated process with an Arrhenius temperature dependence:
where D is the diffusion coefficient, D 0 is a pre-exponential factor, E D is the activation energy required for diffusion through the medium, R is the gas constant, and T is the temperature. Bilchak et al. identified that the primary source of permeability enhancements in PGN membranes likely stems from an increase in the diffusion coefficient. 24 Consequently, the permeability of gas molecules through PGN membranes is expected to be captured by a similar Arrhenius temperature dependence as that described in Equation 1. Equation 2provides a rewritten formulation of the Arrhenius temperature dependence of permeability for the context of this study:
where P is the permeability and P 0 is a pre-exponential factor. Brandt demonstrated that the activation energy required for permeation (E a ) is related to the kinetic diameter (d) of a penetrating gas molecule: 5,26
where c and f are polymer-dependent constants, which can be determined by fitting Equation 3 with neat polymer melt data. These quantities are extensively reported throughout the published literature. Typically, large values of c and f correspond to high diffusivity selectivity and stiffer polymers, respectively. 5,27 In this study, the polymer constants c and f , derived from the fitting procedure, are directly applied to PGN membranes possessing identical grafted polymer chain chemistry as the neat polymer melt used in the fitting of Equation 3. A relationship describing P 0 of Equation 2 has been previously identified by Barrer and Amerongen. 28,29 This "linear free energy" relationship is rationalized by an enthalpy-entropy compensation effect 30 and is herein modified for permeability in Equation 4:
where a and b are constants. For diffusion through pure polymers, the first term of the linear free energy relationship typically appears as
, where a D has been demonstrated to be independent of polymer type. 30 Additionally, throughout the literature, larger values of b D correspond to stiffer polymers. 31 In this study, Equation 4 is fitted using permeability data of neat polymer melts to determine the constants a and b, where a holds the unit of Kelvin -1 to ensure unit consistency.
The structural attributes of PGNs such as grafted polymer molecular weight, grafting density, and nanoparticle surface curvature have been extensively shown to correlate with the formation of polymer brushes. 14,[32][33][34][35][36] At moderate-to-high grafting densities polymer chains assume concentrated polymer brush (CPB) conformations due to confinement effects near the nanoparticle surface. Surface curvature and neighboring grafted chain restrictions facilitate highly stretched and well ordered polymer chain segments in this region. Further away from the nanoparticle surface, polymer chains assume a semidilute polymer brush (SDPB) conformation. In this region, the segments of the grafted chains assume their Gaussian conformations reminiscent of the melt state. 32,[34][35][36][37][38][39] The extent of each region is characterized by the "concentrated polymer brush to semidilute polymer brush" (CPB-SDPB) transition that has been well studied in PGN systems. 34,36,40 Previous experimental studies on the permeability of neat PGN membranes have demonstrated that adding free polymer chains to the membrane can enhance gas selectivity. 23 Long free polymer chains added to the neat PGN membrane reside in interstitial pockets between the NPs, while shorter free chains disperse throughout the grafted chains. The addition of long chains is found to inhibit the transport of large gases, resulting in large increases in selectivity. These results suggest that large gas molecules are likely solubilized within the interstitial regions of PGN membranes. The transport of smaller gases, for example, CO 2 , is found to be independent of the addition of free polymer chains, suggesting that small gases likely solubilize throughout the entire membrane.
The fundamental premise of the model devised to capture these attributes is that the heterogeneous structure of a composite membrane can be idealized as two channels characterized by distinctly different permeability. This work hypothesizes that these channels correspond to the CPB and SDPB regions, each holding a distinct energy barrier. Recent experimental work demonstrated that the activation energy (E a ) of PGN melts derived from the temperature dependence of relaxation times displays a pronounced minimum at intermediate grafted chain molecular weights (50 -75 [kDa] for poly(methyl acrylate) PMA at a grafting density of ≈ 0.47 [chains/nm 2 ]. 25 This finding is rationalized by the theoretically established notion that these chains exhibit the highest degree of extension, which also appears to correlate with a minimum in the normalized monomer friction coefficients. Similar observations throughout the literature have been made on the decrease in the effective monomeric friction coefficient of partially stretched and aligned chains. 41 As a consequence of the activation energy minimum accompanied by a decreased monomeric friction coefficient for extended chain conformations, the CPB region of PGN membranes is expected to yield locally accelerated dynamics, leading to increased permeabilities. 42 Based on this rationale, a two channel model for gas transport in PGNs is proposed. The first channel, representing the SDPB region, is postulated to behave akin to an ungrafted linear polymer membrane. This channel is designated as the high-barrier channel (HBC).
By analogy with transport in a linear polymer, it is proposed that the permeability of a gas molecule through the HBC follows:
where P 0H is the pre-exponential factor of the HBC and E H is the energy barrier required for permeation through the HBC. ln[P 0H ] is described by the linear free energy relation observed by Barrer and Amerongen (Equation 6) :
where a relationship with the high energy barrier is employed. Combining these variables in the construction of Equation 5 allows for modeling permeability through the HBC.
The second transport pathway, the low barrier channel (LBC), represents the CPB region and is associated with a reduced energy barrier. 25 It is proposed that
where k is a small constant assumed here to be 0 to signify a barrierless channel for transport. This assumption is motivated by the desire to explore the theoretical extreme case of no energy barrier to transport through the channel, providing an idealized baseline for the two channel model. The permeability of a gas molecule through the LBC thus follows P L = P 0L where P 0L is the pre-exponential factor of the LBC which follows the linear free energy relation observed by Barrer and Amerongen ln[P 0L ] = -b.
The total permeability through PGN membranes with this model combines the permeability of the HBC and LBC into an effective permeability. This is done by using a fractional contribution written by Equation 7:
where Φ L and Φ H are introduced terms that denote the fractional occupancy of the LBC and the HBC, respectively. Φ L + Φ H =1. The fractional occupancy terms are regarded as the solubility of gas molecules in each region of the PGN membrane, and are fitted to empirical data to complete the two channel permeability model. c, f , a, and b are assumed to be identical for both the LBC and HBC as they are solely polymer-dependent constants.
To First, high temperature data (above the glass transition temperature of PMA) is fitted to 4The HBC and LBC fractional occupancy terms (Φ H and Φ L ) of the effective permeability are unknown through the derivation of the two channel model. Temperature dependent permeability data for a gas molecule (of a specified kinetic diameter) through a PGN membrane are used to determine Φ L as a function of gas size, PGN chain molecular weight, and temperature. To demonstrate this procedure, the focus is placed on data for different gases at a single temperature of 308.15 [K] 23,24 (Figure 3). Figure 4a provides the determined Φ H as a function of the molecular weight of the grafted PMA chains for the available data.
Table 1 has been added to display the individual values of each point. (a) (b) Figure 4: (a) The fractional occupancy of the HBC, and (b) the contribution of the LBC's permeability to the effective PGN membrane permeability, as a function of grafted PMA chain molecular weight Figure 4a illustrates several important points. First, a clear change in behavior is observed when transitioning from small gases (H Examining the individual permeability contributions of each channel offers greater insight into the relative role of the two channels in gas transport. To accomplish this, Ψ = Φ L •P 0L P ef f is defined as the fractional contribution of the LBC to gas permeability. Ψ is plotted as a function of grafted polymer molecular weight in Figure 4b and tabulated in Table 2. The small gases (He, H 2 , and CO 2 ) are primarily transported through the LBC. This is an expected result based on the concept illustrated in Figure 4a. Analysis of Ψ for large gases reveals the underlying significance of this model. Large gases (O 2 , N 2 , and CH 4 ) generally display large values of Ψ. Although this result appears to contrast the observations presented in Figure 4a, it indeed follows the theoretical expectation of this model. That is, large gases primarily utilize the LBC for transport. Despite large gases having access to only a small fraction of the LBC, they demonstrate a significantly higher inclination to utilize the lowbarrier pathway when the opportunity arises. This phenomenon is depicted by Figure 4b, where large gases tend to traverse the LBC even when Φ L is small. Another pertinent observation regarding these results is the marginal increase in Ψ at 92 [kDa] correlating to a significantly amplified effect on the contribution of the LBC to the total permeability. The findings suggest that there must be a change in transport behavior with increasing gas size in which larger gas molecules transporting through the membrane depict a lower Φ L . To further understand these results, a range of Φ H values is selected, and for each value, P ef f is calculated as a function of gas size using the two channel model. Writing P ef f =
, the effective activation energy barrier (E ef f ) is solved as a function of the kinetic gas diameter squared (d 2 ) (Figure 5). This figure portrays the expected transition in Figure 5: E ef f solution of a range of values for Φ H activation energy as a function of the gas diameter squared. For small gases, E ef f follows the behavior of transport through a linear polymer membrane. More importantly, once a critical gas size is exceeded and extends to larger gas molecules, E ef f plateaus. This result reflects the assumptions and expectations of this two channel model. That is, small gases transport through both channels of PGN membranes, whereas large gases transport primarily through the LBC phase in which they experience a significantly lowered E ef f than what is expected of a linear ungrafted polymer membrane. More generally, in a heterogeneous polymer system containing a small fraction of a lower barrier phase, the presence of this phase leads to a critical gas size, beyond which the effective activation energy barrier remains constant.
The two channel model is schematically illustrated in Figure 6. The large black circles represent nanoparticles. The polymer chains grafted onto the surface of the nanoparticles are split into two regimes, the CPB and SDPB which correspond to the LBC and HBC, respectively. Gases are depicted in green and are distinguished by large and small molecules. As demonstrated in Figure 4a, the large gases solubilize mostly within the interstitial region of the membrane, which is part of the HBC. This is illustrated by the accumulation of large gases in the interstitial space of the PGNs (HBC phase). Alternatively, small gases are dispersed throughout both the CPB and SDPB regions of the membrane, reflecting the inherent ability of a small gas to solubilize throughout the entirety of the PGN membrane.
The high energy barrier of the HBC regions results in an effective large resistance to gas permeation. Therefore, gas transport preferentially occurs through the LBC. Although large gases are not likely to solubilize within the CPB, unlike small gases, they will traverse through the LBCs to reach other SDPB regions of the membrane. This theory aligns with the intuitive notion that a gas will preferentially transport along the path offering the least resistance whenever possible. Thus gas molecules that solubilize within the SDPB will still preferentially transport through the LBC if provided the opportunity. Although large gases are unlikely to solubilize within the LBC phase of the membrane, the lower energy barrier effectively funnels these gases, making it the primary pathway for their permeation.
This work provides a simple analytical theoretical framework aimed at understanding the underlying mechanisms of gas molecule transport through a heterogeneous material, with a specific focus on polymer-grafted nanoparticle (PGN) membranes. The model defines two transport channels within the heterogeneous structure of a PGN melt. The first channel corresponds to the semidilute polymer brush (SDPB), which resembles the properties of a neat polymer melt and is referred to as the high-barrier channel (HBC). The second channel corresponds to the concentrated polymer brush (CPB) region, which significantly accelerates transport dynamics. This acceleration is due to a lowered effective activation energy barrier resulting from a reduction in local monomeric friction. Consequently, this is referred to as the low-barrier channel (LBC). By defining these distinct phases within the structure of a PGN membrane, the model characterizes the gas molecule size-dependent transport.
Leveraging simplistic parameterization from conventional linear polymer membranes, the permeability data of PGN membranes extracted from the literature serves as a basis for predicting size-dependent transport mechanisms. Large gas molecules are predicted to predominantly solubilize within the phase holding characteristics akin to neat polymer melts.
Small gases exhibit inherent ability to solubilize throughout the melt-like and accelerated dynamics phases. According to these predictions, the LBC contribution to the effective permeability suggests that both small and large gases prefer transport through the LBC. This preference is a consequence of the significantly reduced energy barrier necessary to overcome for permeation. The effective activation energy barrier derived from the model reinforces the results of transport preference by revealing the underlying bilinear nature of the two channel model. Gas molecules transport indiscriminately throughout the entire membrane up to a critical size, however, once this critical size is exceeded, gases preferentially utilize the channel with the reduced barrier. Finally, an illustration is provided to interpret the mechanisms of transport proposed by this model. By bridging the gap between experimental insights and a novel theoretical basis, this model contributes to refining the current understanding of gas transport phenomena through heterogeneous materials, particularly PGN membranes.
Furthermore, future research should seek to validate the underlying assumptions proposed in this work and expand on understanding the gas transport dependencies on other membrane factors, including chemistry and unconventional polymer conformation structures. Through comprehensive characterization of gas transport within emerging material structures, it will be possible to discern the mechanisms employed by tunable membranes to optimize future gas separation technologies.