Diversity-driven, efficient exploration of a MOF design space to optimize MOF properties

Abstract

Metal-organic frameworks (MOFs) promise to engender technology-enabling properties for numerous applications. However, one significant challenge in MOF development is their overwhelmingly large design space, which is intractable to fully explore even computationally. To find diverse optimal MOF designs without exploring the full design space, we develop Vendi Bayesian optimization (VBO), a new algorithm that combines traditional Bayesian optimization with the Vendi score, a recently introduced interpretable diversity measure. Both Bayesian optimization and the Vendi score require a kernel similarity function, we therefore also introduce a novel similarity function in the space of MOFs that accounts for both chemical and structural features. This new similarity metric enables VBO to find optimal MOFs with properties that may depend on both chemistry and structure. We statistically assessed VBO by its ability to optimize three NH₃-adsorption dependent performance metrics that depend, to different degrees, on MOF chemistry and structure. With ten simulated campaigns done for each metric, VBO consistently outperformed random search to find high-performing designs within a 1000-MOF subset for (i) NH₃ storage, (ii) NH₃ removal from membrane plasma reactors, and (iii) NH₃ capture from air. Then, with one campaign dedicated to finding optimal MOFs for NH₃ storage in a "hybrid" ∼10 000-MOF database, we identify twelve extant and eight hypothesized MOF designs with potentially record-breaking working capacity ΔN _NH3 between 300 K and 400 K at 1 bar. Specifically, the best MOF designs are predicted to (i) achieve ΔN _NH3 values between 23.6 and 29.3 mmol g^-1, potentially surpassing those that MOFs previously experimentally tested for NH₃ adsorption would have at the proposed operation conditions, (ii) be thermally stable at the operation conditions and (iii) require only ca. 10% of the energy content in NH₃ to release the stored molecule from the MOF. Finally, the analysis of the generated simulation data during the search indicates that a pore size of around 10 Å, a heat of adsorption around 33 kJ mol^-1, and the presence of Ca could be part of MOF design rules that could help optimize NH₃ working capacity at the proposed operation conditions.

Fig. 1. Applications for which NH₃ adsorption-based MOF performance metrics were optimized to test the efficacy of our Vendi Bayesian optimization (VBO) framework. (a) Adsorptive NH₃ storage at ambient conditions with release at 400 K. (b) Membrane-based NH₃ removal from plasma reactors during NH₃ synthesis at 400 K and 1 bar. (c) Dilute NH₃ capture from air in adsorbent traps at ambient conditions. Gas-phase composition relevant to each application indicated at the top. The three chosen metrics present different levels of dependence on MOF chemistry and structure.

Fig. 2. Workflow for our VBO framework. an initial GP, trained with data for two randomly chosen MOFs, is used to predict the performance metric in the starting database. k + 1 MOFs are selected for molecular simulation evaluation based on the upper confidence bound (UCB) acquisition function. One MOF is chosen as the MOF scoring the highest UCB just as in standard Bayes optimization. The remaining k MOFs are selected based on UCB but only after 10% of the database is pruned. The MOFs pruned from the database are the MOFs that would increase the least the Vendi score of the cumulative set of MOFs evaluated by molecular simulation. The top k + 1 MOFs selected are then evaluated using molecular simulations. To perform a new iteration, the molecular simulation data for the newly evaluated k + 1 MOFs are added to the data for training the GP, and the MOF selection process is repeated.

Fig. 3. Schematic representation of methods to calculate kernel similarity between MOFs. (a) Chemical similarity (K_node and K_linker kernels) obtained by decomposing two MOFs into their building blocks, and calculating the Tanimoto index between the Morgan fingerprints of their building blocks. (b) Global textural properties similarity (K_global kernel) obtained by calculating the radial basis function kernel of the Euclidean distance between the property vectors of two MOFs. (c) Detailed pore structure similarity (K_PSD kernel) obtained by calculating the difference between one and the Jensen–Shannon divergence between the pore size distributions (PSDs) of two MOFs. The different kernels cover different aspects of MOFs, and by tuning the weights of each kernel, the representation is adaptable to prediction of properties with different level of dependence on MOF chemistry and structure.

Fig. 4. MOF mapping onto two-dimensional plots by using multidimensional scaling (MDS) representations. (a) All MOFs in the hybrid database colored by their origin (either the ToBaCCo database or the CoRE database). (b–d) 1000 random MOF subset, colored by range of ΔN_NH3 (b), M_ATS (c), and M_ATSTH (d) performance metrics. The extent of segregation observed is a harbinger of the efficacy of our MOF kernel similarity as input to train the GP.

Fig. 5. Prediction performance of GP models (trained on a subset of 1000 random MOFs extracted from the hybrid database) to predict (a) ΔN_NH3, (b) M_ATS, and (c) M_ATSTH. GP predictions appear on the vertical axis, while the ground truth (from molecular simulation) appears on the horizontal axis. The parity line is presented in red. Each point represents the prediction for a MOF, with the corresponding error bar representing the uncertainty of the predictions based on the prediction standard deviation. The observed prediction performance was found on subsequent statistical testing to be sufficient to make VBO effective.

Fig. 6. Efficacy of VBO (blue) applied on a 1000 subset of random MOFs compared to Bayesian optimization (green) and random search (orange). Top row presents the evolution of the highest value of the performance metric as the number of MOF evaluations increases for (a) ΔN_NH3 for ammonia storage, (b) M_ATS for ammonia removal from plasma reactor, and (c) M_ATSTH for ammonia capture from air. Bottom row presents the evolution of the Vendi score for the set of evaluated MOFs as the number of MOF evaluations increases for (d) ΔN_NH3 for ammonia storage, (e) M_ATS for ammonia removal from plasma reactor, and (f) M_ATSTH for ammonia capture from air. Results in (a)–(f) are averaged across 10 repeat runs, the average value is indicated by the solid line, whereas the standard deviation is indicated by the shaded area. Both VBO and Bayesian optimization outperformed random search, but VBO provided higher diversity of MOF “solutions.”

Fig. 7. Evolution of VBO campaign (blue) in the ∼10 000 MOF database, when searching for MOFs for NH₃ storage, compared to the evolution of the random search (orange). (a) Evolution of the highest ΔN_NH3 found among evaluated MOF at a given point in the campaign. (b) Evolution of the average ΔN_NH3 among the top-20 evaluated MOFs at a given point in the campaign. (c) Evolution of the Vendi score of evaluated MOFs at a given point in the campaign. Note that the VBO campaign was ended early due to negligible changes in the highest ΔN_NH3 since the 80th evaluation. Once again VBO greatly outperformed random search.

Fig. 8. Plots of structure–performance relationships for NH₃ storage. Each square bin corresponds to a combination of the ΔN_NH3 performance metric and MOF property, where the transparency of each square bin is indicative of the number of MOFs in the bin, and the color of each bin reflects the average value of the property in the side color scale across all MOFs in the bin. (a) ΔN_NH3versus MOF average pore diameter (APD), with each bin colored by MOF void fraction. (b) ΔN_NH3versus heat of adsorption Q_st, with each bin colored by gravimetric surface area. Optimal APD and Q_st appears to be 10 Å and 33 kJ mol⁻¹, respectively.

Fig. 9. Statistical significance for comparison of elemental compositions between the top-14 MOFs and the entire database based on the p-values derived from the t-test. The dashed line represents our chosen critical value for the one-sided t-test. Bars that fall below this threshold indicate elements that are statistically significantly more abundant in the top-performing MOFs. Ca is a metal that appears significantly more frequently in the top-14 MOFs than in the full database.

Fig. 10. Top-6 MOFs ranked by ΔN_NH3 value. Hypothesized (a–c) and extant MOFs (d–f) are in the top and bottom rows, respectively. V_F, GSA, and APD represent, respectively, the void fraction, gravimetric surface areas, and average pore diameter from pore size distribution. The CSD refcode and corresponding publication can be found in Table S5 with (a) n = 1, (b) n = 2, (c) n = 4, (d) n = 3, (e) n = 5, (f) n = 6, where n is the MOF ranking. The three hypothesized MOFs are potentially synthesizable per the free energy criterion by Anderson and Gómez-Gualdrón (Table S6†). Ca MOFs appear in the top-14 but not in top-6 presumably due to suboptimal textural properties.

Fig. 11. (a) Thermal stability in top-20 MOFs from VBO campaign. Blue diamonds indicate ΔN_NH3 (left-axis) and red circles indicate predicted thermal decomposition temperature (right-axis). Top-20 MOFs appear likely to withstand operation conditions. (b) Estimated energy penalty to release stored NH₃ as percentage of the hydrogen-based energy content of NH₃ (22.5 MJ kg_NH3⁻¹) in the top-20 MOFs. Penalty hovers around 8 to 12 percent on NH₃ energy content.