Variations in GNP core size.

To investigate the effect of NP size on gas transport, we studied two additional cases: (i) PMA chains densely grafted to R_c = 25 nm silica NPs (σ ≈ 0.47 chains/nm²) and (ii) PMMA chains grafted to R_c = 2 nm ZrO₂ NPs (σ ≈ 0.25 chains/nm², Figure 3C). We first consider R_c = 25 nm silica NPs grafted with PMA. For M_n < 50 kDa the P_CO2 values are below that of the pure PMA (Figure 3C), and there is no significant change in CO₂/CH₄ selectivity (see supporting information, Figure S6). Beyond this molecular weight, the CO₂ permeability of the GNP increase monotonically with increasing M_n; however, we do not see a maximum for the molecular weights considered.

Experiments conducted with the R_c = 2 nm ZrO₂ NPs grafted with PMMA show similar behavior to the high graft density PMA brushes on the R_c = 8 nm silica NPs discussed above. However, while there is a local maximum in permeability, the graft M_n at which it occurs is significantly smaller than for the larger silica GNPs (M_n ≈ 7-10 kDa vs. M_n ≈ 60-90 kDa for the R_c = 8 nm NPs).

While we only have limited data on variations in NP core size, grafted chain chemistry, M_n and σ, below we will show that the data are supportive of theoretical ideas which appear to suggest that permeability enhancement values only depend on the NP volume fraction. Clearly, however, much more data need to be obtained to place these conclusions on a more solid footing.

Suppressed gas transport in sparsely grafted GNP.

In contrast to the trends discussed above for densely-grafted GNPs, very low graft density materials (PS grafted R_c = 8nm NPs with σ ≈ 0.03 chains/nm²) show very different behavior. In these materials, we observe a suppression of gas permeability regardless of chain M_n, though longer chains (i.e., smaller NP loadings) exhibit less of a suppression (Figure 3D). The trends for both CO₂ and CH₄ permeability are well-described by the Maxwell equation, suggesting GNPs with a grafting density below a threshold value behave as expected by macroscopic theories where the (impermeable) NPs hinder gas transport. Where the crossover to hindered transport occurs, and why this occurs, are important underpinning questions we shall consider below.

Empirical unifying trends in the high graft density data.

We have conjectured above that the peak permeability occurs at an apparently universal value of the NP volume fraction, namely ϕ_NP,max = 0.049. Inspired by this prediction, we have re-examined all of our CO₂ permeation data (except for the lowest graft density of σ ≈ 0.03 chains/nm², where the two-layer model does not apply) as functions of ϕ_NP in Figure 4A; Figure 4B shows the corresponding data for CH₄. Note that the GNP permeability data are normalized by the corresponding values in the pure polymer – if the theory is correct then this takes in to account the effects of chain chemistry on gas permeability. Within the relatively large scatter, it is apparent that the data for each gas follow an essentially universal non-monotonic dependence of permeability on ϕ_NP. The permeability of both gases initially increases with ϕ_NP for small ϕ_NP < 0.04, but then it decreases with increasing ϕ_NP for ϕ_NP> 0.04. There is a peak in relative permeability in the vicinity of ϕ_NP ≈ 0.04-0.05, which is close to the predicted value (ϕ_NP,max = 0.049) where the (unfavorable) chain extension free energy is maximum. While these are satisfying general trends, there are important exceptions to these ideas which we discuss below. Regardless, it is clear that the permeability enhancements in these GNP layers apparently follow a universal dependence on the ϕ_NP.

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Figure 4:

Permeability enhancements of (A) CO₂ and (B) CH₄ as a function ϕ_NP for all GNP membranes discussed in Table 1 except for the lowest grafting density.

We use the two-layer model to calculate the molecular weight at which maximum chain extension occurs and from there the overlap parameter at this chain length, x_max. Table 1 shows that the experimentally studied systems fall into three distinct classes. The last three rows in this Table, with x_max >1, are in the regime where the brushes are extended. These are the systems that are most likely to follow the two-layer model, and indeed they show enhanced CO₂ and CH₄ permeation behavior when examined as a function of ϕ_NP (Figure 4).

Two other sets of data, the one corresponding to the R_c = 8 nm and σ ≈ 0.11 chains/nm² and the other to R_c = 2 nm with σ ≈ 0.25 chains/nm², have x_max ≈0.5-1 but these GNPs have extension free energies less than thermal energy. These systems, which are not at high enough graft density, thus are in a regime where the two-layer model is not an effective descriptor of chain conformations in these grafted layers. While the permeability data for these weakly stretched layers qualitatively follow the trends obtained from the brushes with more overlap, there evidently are differences relative to the higher graft density samples especially at ϕ_NP > 0.1.

The data at the lowest grafting density σ ≈ 0.03 chains/nm² demonstrate permeability reductions for all volume fractions up to ϕ_NP = 0.8 (Figure 3D). The x values for this case are always << 1, regardless of graft chain length. Since the two-layer model is clearly inapplicable for these cases, we expect chains from neighboring NPs to readily interpenetrate. These sparsely grafted NPs behave more akin to star polymers in this context with little chain extension, i.e., the chains assume their unperturbed Gaussian conformations. In these cases, the role of NPs on gas transport follows the Maxwell model. While the two-layer model is thus not applicable to explain the trends observed for this low grafting density, the discussion to this point lends credence to our underpinning picture that chain extension is likely the cause of the permeability increases in this class of GNPs.

We now proceed to understand the non-monotonic behavior seen in Figure 4. We have attempted to use extended Maxwell models as suggested in the literature, separately above and below the maximum permeability: we assume that the dry region has significantly higher permeability than the interpenetrated zone, while the NP core is not permeable. This model does not capture the trends observed in Figure 4, in agreement with previous conclusions of Bocharova and coworkers⁴².

We thus resorted to a more empirical approach. For ϕ_NP > ϕ_NP,max we used a variety of forms, including an exponential, a hyperbolic tangent and a power law to see which one adequately describes the data. The exponential fit (i.e., Ce^{−ϕ_NP/−_NP,max}), for example, only fits data in the mid ϕ_NP range (i.e., 0.049 < ϕ_NP < 0.1) but fails at larger ϕ_NP. Instead, we found that a power law form best described the data for both CO₂ and CH₄. In particular, the line in Figure 5 for the CO₂ is a fit that follows:

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Figure 5:

Comparison of experimental permeability increases with scaling predictions for (A) CO₂ and (B) CH₄. We postulate that chain extension dominates gas transport behavior for ϕ_NP > 0.05; The interpenetration zone, where chains segments behave akin to a melt, dominates for sufficiently long chains (i.e., ϕ_NP < 0.049).

While the fit power law dependence of P on ϕ_NP is slightly different from −1, we do not see any reason to use a more complicated dependence than that illustrated in Eq. (11). The CH₄ data in Figure 5 follow the same dependence but with a slightly different prefactor, i.e., 0.41 vs. 0.31 for CO₂. Here we note that these numbers have significant uncertainties, of order a factor of 2. However, experimental CH₄ permeability enhancements are always significantly higher than the CO₂ permeability enhancements and hence these uncertainties must be seen as conservative estimates. This result suggests that the transport of CO₂ is increased by as large as a factor of ≈ 10 relative to the neat polymer (without NPs), with this increase occurring in the vicinity of the NP volume fraction where the chain extension free energy is the largest. The increase of NP loading leads to a decrease in permeability, so that for large NP loadings, ϕ_NP > 0.31, GNP permeabilities are predicted to fall below that of the neat polymer. Verification of this conjecture is found experimentally for large NPs with R_c = 25 nm, σ ≈ 0.47 chains/nm² and M_n = 7.5 kDa, where for example ϕ_NP = 0.45 we experimentally find $\frac{\mathrm{P}}{{\mathrm{P}}_{\mathrm{n}}}=0.3$. Eq. (11) predicts $\frac{\mathrm{P}}{{\mathrm{P}}_{\mathrm{n}}}=0.65$.

While the form that is represented in Eq. (11) is empirically derived, a few points are noted. First, it is quite different from the Maxwell form, Eq. (2). A deeper analysis of the stretching free energy shows that, for ϕ_NP 0.05, it follows the form which is obtained by Taylor expanding the free energy as a function of ln (ϕ_NP) about ϕ_NP=1.

where the constant a is dependent on grafting density, while b=0. While the slope of this curve for 8 nm radii NPs assumes a value of a ≈1 for a grafting density of 0.47 chains/nm², a ≈−1.5 for a grafting density of 0.66 chains/nm². If we use this form for the free energy of stretching chains, with the assumption that gas motion is facilitated by the free energy of the extended chains with an Arrhenius dependence, we obtain:

Since theory predicts that the exponent a is strongly dependent on the grafting density and other chemical details of the grafted chains, more comprehensive data, especially for different NP sizes, are critically necessary to further explore the apparent “universality” that appears evident in Figure 5 at large ϕ_NP.

Of course, while the fact that Eq (11) predicts $\frac{\mathrm{P}}{{\mathrm{P}}_{\mathrm{n}}}=0.31$ in the limit of ϕ_NP → 1 evidently accounts for the reduction of permeability due to the impenetrability of the NPs, this is not captured by a simple model that includes only the chain extension free energy. We therefore conjecture that the blocking and increased tortuosity effects modeled in the traditional approaches, especially how they vary with NP loading, play a secondary role to the variation of chain extension free energy with increasing chain length. Apparently, these effects can be modeled by a constant value of 0.31 for CO₂ (0.41 for CH₄).

Gas permeabilities for NP loadings smaller than ϕ_NP,max are slowed relative to the maximum. The CO₂ data are fit by the empirical equation: $\frac{\mathrm{P}}{{\mathrm{P}}_{\mathrm{n}}}=6.4{\mathrm{e}}^{-\frac{{\varphi }_{\text{NP},\max }-{\varphi }_{\text{NP}}}{{\varphi }_{\text{poly,c}}}}$ where we use the requirement that the $\frac{\mathrm{P}}{{\mathrm{P}}_{\mathrm{n}}}$ must go to unity in the limit where the NP loading goes to zero to define ${\varphi }_{\text{poly,c}}=\frac{{\varphi }_{\text{NP},\max }}{\text{ln}(6.4)}\approx 0.026$ (for CO₂). The numerator in the exponential accounts for any chain segments outside the dry zone, i.e., those that are unperturbed. The exponential term thus identifies the polymer fragments that are outside the “dry” zone, i.e., the chain segments in the interpenetration zone, as being responsible for permeability decreases, while the prefactor accounts for the transport enhancement that is afforded by the chain segments in the dry region, Ih we assume to be at its maximum extension. The CH₄ data behave similarly, except that the peak permeability enhancements (≈ 10) are higher.

Physical picture:

Referring back to our schematic in Figure 2, we conclude that the gas transport in these media is driven by fact that the polymer layer in the GNP has two regions for long enough chains and high enough grafting densities where x > 1. There is an inner regime of extended chains where gas transport is speeded-up relative to the neat polymer, while the outer region of interpenetrated chains is akin to the neat polymer with slow gas transport. Thus, while the gas motion in the inner regions is facilitated by the presence of extended chain segments, penetrant molecules must pass through the slower melt regions to find other fast regions. The fit data suggest that CO₂ transport in the region with the most extended chain segments is ~ 6.4 times faster than in the regions with interpenetrated chain segments. The picture proposed is similar in spirit to the hop-like dynamics executed by gas molecules moving through dense polymer matrices with the additional condition that the heterogeneities in the case of the GNPs are created by differential chain extension in different spatial regions of the polymer brush.

Partial financial support for this research was from the Department of Energy under grant DOE-SC0021272. CRB thanks the National Science Foundation Graduate Research Fellowship Program (Grant # DGE-16-44869).