Polymer genome–based prediction of gas permeabilities in polymers

2.1 Data sets

In this work, we used experimentally collected gas permeability data for six different gases (CH₄, CO₂, H₂, He, N₂, and O₂) from more than 300 publications with more than 1000 entries [10]. The testing conditions vary across the literature with most of the tests executed at 25°C to 35°C and 1–10 atm on dense membranes with a thickness of 10–200 μm. This extensive collection of experimental gas permeability data enables us to construct a universal model for polymers spanning across polyimides, polyamides, poly(amide-imides), polyarylates, polycarbonates, polyesters, polypyrrolones, polynorbornenes, vinyl and vinylidene polymers, poly(aryl ethers) and poly(aryl ether ketones), polyphosphazenes, perfluorinated polymers, polysulfones, parylenes, high-temperature polymers, polypropynes, substituted polyacetylenes, and polypentynes. A total of 315 polymers were considered in this work of which the number of permeability of measurements recorded for CH₄, CO₂, H₂, He, N₂, and O₂ was 258, 293, 174, 183, 293, and 300, respectively. This collection represents a total of 1501 permeability measurements.

One of the benefits of using experimental data is that hidden information on polymer processing and testing is integrated into the data, which will be used in model regression. In this work, the model input only contains a chemical structure. Other factors that will affect polymer performance, such as polymer process history and testing method, are not explicitly used as parameters in the model. However, because experimental data in the database were generated under realistic processing and testing methods, the effect of these hidden factors will be correlated in ML. This process also indicates that the model we generated from ML will predict an experimental value of the polymer permeability that is expected with current “averaged” processing and testing method.

2.2 Polymer fingerprinting

Gas permeation properties of a polymer are related to complex contributions that can be broadly categorized into physical and chemical interactions. Chemical interactions will affect both diffusion and sorption of guest molecules in the polymer, which will be captured by the building blocks of the polymer. Physical interactions originate from the contributions of free volume and polymer chain arrangement that will affect the diffusion of guest molecules in the polymer. We believe that the packing of the polymer affected by the guest molecules can be captured by using experimental data and higher level descriptors such as morphological descriptors.

A hierarchical polymer fingerprinting scheme was used to comprehensively capture the descriptors that may control the gas permeability of polymers [23]. Briefly, the fingerprint building process consists of four hierarchical levels of descriptors. The first level is at the atomic scale wherein the occurrence of atomic triples, which is a set of three contiguous atoms (e.g. C2–C3–C4, made up of a twofold coordinated oxygen, a threefold coordinated carbon, and a fourfold coordinated carbon), was calculated [33]. For the 315 polymers considered in this study, there are 93 such components. The second set of fingerprint components captures a population of predefined chemical building blocks (e.g. –C₆H₄–, –CH₂–, –C(=O)–). We include a fixed set of 148 block level components. The third hierarchical level deals with quantitative structure-property relationship descriptors, such as van der Waals surface area [34], topological surface area [35], and the fraction of rotatable bonds, implemented in the RDKit cheminformatics library [36]. Such descriptors, 39 in total, form the next set of components of the overall fingerprint. To these third level descriptors, morphological features such as the topological distance between rings, fraction of atoms that are part of side chains and length of the largest side chain were added [17]. We include a fixed set of 19 such morphological descriptors.

As a result of the fingerprinting method, some limitations on polymer structures that can be included and predicted by the model were introduced. As the fingerprinting method digests a single repeating unit of the polymer, polymers with structures that cannot be represented by a single repeating unit with single end points are not included (i.e. polymer blends, random copolymers, random substituted polymers, ladder polymers, among others). Chemical bonds across repeating units are also limited (i.e. crosslinked polymer, hyperbranched polymer, thermal rearranged polymer, among others).

2.3 Machine learning model development

In this work, we utilize Gaussian process regression, which generates a probabilistic surrogate model of a specific property. We used a radial basis function kernel defined as

k(xi, xj)=σ2exp[−(xi−xj)22l2]+σn2δ(xi, xj)

where σ, l, and σ_n are hyperparameters to be determined during the training process (in the ML parlance, these hyperparameters are referred to as signal variance, length scale parameter, and noise level parameter, respectively). x_i and x_j are the fingerprint vectors for two polymers i and j. Performance of the model was evaluated based on the coefficient of determination (R2).

3.1 Model performance

In the data collected from the literature, not all six gases (He, H₂, CO₂, O₂, N₂, and CH₄) were measured for every polymer. To increase the number of data available to the model, we assigned a distinct vector to each gas permeability data depending on the type of gas (i.e. one-hot encoding). As a result, a total number of 1501 data points can be used in the model fitting, which greatly improves the goodness of fitting. In addition, by doing this, we can easily predict the missing permeability data in the literature for gases that were not measured. The learning curve (Figure 1) shows that the size of the dataset (1501 permeability measurements) is sufficient to train the model as the R2 of training set and test set converges.

Figure 1:

Learning curves constructed from the R2 of the machine learning models for gas permeability. Total gas permeability datasets of 5%, 10%, 20%, …, 90% and 95% were used for training the model. For each model, the data were obtained from 50 independent runs with a different selection of train and test sets.

To test the ability for predicting unknown polymers, we randomly selected polymers from each polymer class as the test set and used the remainder of the database as the training set (Supplementary Figures S1 and S2). The polymers selected are listed in Supplementary Table S1. The correlation between the predicted and experimental permeability values for polymers in the test set are shown in Figure 2. Good agreement was achieved in general with several outlier points, which include one case in each of polynorbornenes, vinyl and vinylidene polymers, and parylenes. The outliers can be attributed to either data scarcity for that polymer class during training, or uncertainties within the experimental data.

Figure 2:

Comparison of predicted and experimental permeability values for 31 polymers that were not used for model training.

3.2 Comparison of experimental and calculated permeability coefficients

Figure 3 shows the correlations between experimental and predicted permeability values. The correlation coefficients for each individual gases range from 0.98 to 0.99. This is by far the most accurate prediction model for gas permeabilities based on fitted experimental data. Compared with earlier group contribution and bond contribution methods based on a much larger database (R2=0.9), our fingerprinting method showed better prediction performance [32].

Figure 3:

Performance of the permeability prediction model. Parity plots showing comparison between experimental and predicted gas permeability for (A) CH₄, (B) CO₂, (C) H₂, (D) He, (E) N₂, and (F) O₂ are generated using one unified predictive model for six gases trained on 100% dataset (total 1501 permeability data points associated with 315 polymers, illustrated in all figures as gray points).

3.3 Analysis of individual polymer classes

The inclusion of different polymer classes in the same model sacrifices accuracy to a certain extent because of the distinct physical properties of different polymer classes. Some polymer classes in the database are presented in larger numbers and thus are more heavily weighted in the model. As a result, the ability to predict permeability from minority polymer classes (e.g. polynorbornenes and polyesters vide infra) can be questionable.

Among the various polymer classes in the database, we picked two over-represented classes and two under-represented classes. As the over-represented classes, one is the combined group of polyimides and polypyrrolones; the other is the combined group of polypropynes, substituted polyacetylenes, and polypentynes. The two groups constitute approximately 50% of the data in the entire database. The two under-represented classes are polynorbornenes and polyesters, which only have six and three distinct structures, respectively. Figure 4A–D shows the parity plot of experimental and predicted permeability values from the four polymer classes. As expected, for the over-represented groups, there is good agreement between the predicted and experimental permeability values. Surprisingly, a similar agreement was observed for polynorbornenes and polyesters. This ability to unify drastically different polymers in one model shows the versatility and transferability characteristics of the Polymer Genome approach, which has proven successful for predicting other polymer properties from sparse data sets.

Figure 4:

Performance of model to predict permeability for different over- and under-represented polymer classes including (A) polyimides and polypyrrolones, (B) polypropynes, substituted polyacetylenes, polypentynes, (C) polynorbornenes, and (D) polyesters.

3.4 Comparison of experimental and predicted selectivities

Although accurately predicting permeabilities is useful, the permselectivity of a potential polymer membrane is often the final arbiter in whether that polymer will go forward in the design process or not. To this end, we calculated the predicted selectivities (α_ij=P_i/P_j) and compared with experimental selectivities in Figure 5A–F.

Figure 5:

Comparison of experimental and predicted selectivities for (A) O₂/N₂, (B) CO₂/CH₄, (C) CO₂/N₂, (D) H₂/CH₄, (E) H₂/CO₂, and (F) N₂/CH₄.

We also compared the model results with the 2008 Robeson upper bound (Supplementary Figure S3). It can be seen that the predicted gas pair permeability and selectivity closely resemble the experimental data and the 2009 upperbound [10].