Rapid Prediction of Adsorption Isotherms of a Diverse Range of

Article Cite This: J. Phys. Chem. C 2019, 123, 17884−17893

pubs.acs.org/JPCC

Rapid Prediction of Adsorption Isotherms of a Diverse Range of Molecules in Hyper-Cross-Linked Polymers Dai Tang,† Grit Kupgan,‡ Coray M. Colina,‡,§ and David S. Sholl*,† †

School of Chemical & Biomolecular Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0100, United States Department of Materials Science and Engineering and §Department of Chemistry, University of Florida, Gainesville, Florida 32611-7200, United States

‡

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S Supporting Information *

ABSTRACT: Hyper-cross-linked polymers (HCPs) are materials with permanent porosity that have potential applications in adsorption-based chemical separations. We use molecular simulations to predict the largest set of adsorption isotherms available to date for HCPs. Specifically, we predict room-temperature isotherms for 24 distinct molecules in 9 different HCPs that vary in both the monomer composition and cross-link density. This information allows us to develop efficient approximate models that predict adsorption isotherms in HCPs based only on the Henry’s constant for adsorption, textural information about the HCP, and bulk phase properties of the adsorbing molecule. This approximate approach will be useful in considering the adsorption of diverse groups of molecules in HCPs in the future. Our data also enables us to describe the selectivity of adsorption for the 276 binary mixtures that can be formed from the 24 molecules we simulated. This information greatly expands the information that is available about adsorption selectivity in HCPs.

1. INTRODUCTION Microporous materials have many potential applications in chemical separations. This broad class of materials can be further categorized into ordered and amorphous structures. Metal−organic frameworks (MOFs),1−4 covalent−organic frameworks (COFs),5 and zeolites6,7 are well-known examples of ordered microporous materials. Examples of amorphous microporous materials include activated carbons,8 hyper-crosslinked polymers (HCPs),9−13 polymers of intrinsic microporosity (PIMs),14−16 and conjugated microporous polymers (CMPs).17−19 All of these materials have their own advantages and disadvantages in terms of costs, stability, performance, processability, etc. Hyper-cross-linked polymers (HCPs) have a number of characteristics that make them interesting for chemical separations. A wide range of HCPs can be synthesized from widely used and easily accessible monomers, and their production is readily scalable. Due to their high density of covalent cross-links, HCPs have excellent physical, thermal, and chemical stability, a feature that is critical in many applications.20−23 Motivated by these observations, HCPs have been explored for potential applications in gas separation and storage, chromatography, catalysis, sensors, and drug delivery,11,13 and commercialized as sorbents for water treatment, food, and chemical applications.24−27 The majority of work that has been reported with HCPs has been experimental.11,13 Computational modeling of HCPs is challenging because of their complex structure. Recently, © 2019 American Chemical Society

molecular simulation techniques have made it possible to describe properties of HCPs that are not readily accessible by experiments.28,29 For instance, molecular simulation has been used to study several dichloroxylene (DCX) networks. Wood et al. showed that the constructed DCX models could accurately represent structural properties such as pore volume, pore width, and bulk density but overestimated the absolute uptake of H2.30 Trewin et al. proposed a Monte Carlo method for polymer construction and were able to study surface areas, pore structure, and reproduce experimentally observed H2 and N2 isotherms.31 Abbott and Colina used a simulated polymerization approach to construct polyDCX structures without imposing experimentally observed densities and were able to reproduce density, pore volume, H 2 and CH4 isotherms.32 For hyper-cross-linked polystyrene systems (e.g., styrene, vinylbenzyl chloride, or divinylbenzene), modeling has also been used to probe the atomic scale structure of HCPs to understand the formation of microporosity in various stages of polymerization as well as the degree of cross-linking and to predict H2/CO2 adsorption.33,34 Other studies used molecular simulations to complement experimental data for poly(tri(4ethynylphenyl)amine) networks,35 triptycene-based microReceived: May 9, 2019 Revised: June 26, 2019 Published: June 27, 2019 17884

DOI: 10.1021/acs.jpcc.9b04413 J. Phys. Chem. C 2019, 123, 17884−17893

Article

The Journal of Physical Chemistry C

Figure 1. (a) Four types of monomers used for the construction of HCP models and the united-atom types of monomers. The united atoms Cp0, Ccp, Ccp2, Lc1, Lc2, and Cl are depicted as the brown, black, white, orange, green, and purple spheres, respectively. Cl atoms are not included in our model and are being depicted in this figure for clarity. The spheres show the relative positions of united atoms composing monomers but not the sizes of united atoms. (b) The atomic structure of a single sample of HCP with 100% cross-linking and 10% DVB. The simulation volume shown in (b) has a volume of 51.65 nm3 and contains 1820 united atoms.

porous polymers,36 benzene-based knitted networks,37 and aromatic heterocyclic polymers.38 When taking a broad view of chemical separations, the extreme diversity that exists among molecular species creates a significant challenge. Crude oil, for example, contains > 104 distinct molecules.39 Atmospheric aerosols contain enormous numbers of distinct dissolved volatile organic compounds (VOCs).40,41 Pharmaceutical development has long used molecular libraries containing vast numbers of different molecules.42 The challenge of understanding the vast number of possible separations that could potentially be tackled with porous adsorbents has been described as “exploring adsorption space”.43 In this paper, we describe some initial steps toward exploring adsorption space for molecular separations in HCPs. Although molecular simulations can be used to make detailed predictions about equilibrium adsorption in HCPs, the computational demands associated with describing a single species adsorbing in a HCP are high. It is, therefore, not feasible to tackle the task of fully exploring adsorption space in HCPs with molecular simulations. Our work is motivated by a study of molecular adsorption in MOFs by Tang et al.43 In that work, molecular simulations were used to generate adsorption isotherms for a diverse set of 24 molecules adsorbing in 471 different experimentally accessible MOFs. This collection of data was then used to develop a simplified model based on Langmuir adsorption isotherms that can predict the adsorption of new molecules in MOFs extremely rapidly. The accuracy of this simplified model is, of course, lower than performing a detailed molecular simulation of a full isotherm, but the accuracy was shown to be high enough to allow meaningful conclusions to be drawn. In this paper, we examine whether a similar approach can be used to rapidly predict equilibrium isotherms of a diverse range of molecules in HCPs. Below, we report single-component adsorption isotherms from molecular simulations for 24 chemical species in a set of 9 HCPs with varying degrees of cross-linking and monomer compositions. The HCPs we considered have accessible surface area varying from 44 to 1577 m2/g. The information from these simulations greatly expands the variety of examples for which adsorption isotherms in HCPs are available. We use this large collection of molecular simulation data in three ways. First, we look at whether the prediction methods previously introduced by Tang et al.43 for MOFs can be extended to molecular

adsorption in HCPs. Second, we explore the chemical separations that can be achieved in HCPs by considering the 276 distinct adsorbate binary mixtures that can be defined using the 24 species included in our molecular simulations. We also ask whether the model developed by Freeman for molecular solubility in rubbery and glassy polymers can be also used for molecular adsorption in HCPs.44,45

2. SIMULATION DETAILS Molecular simulations of adsorption isotherms need reasonable representative structures of adsorbents. We simulated virtual synthesis of hyper-cross-linked polymers using the polymatic code,46 which has previously been validated for a variety of polymers for providing structural and adsorptive properties that are consistent with experiments. The HCPs we investigated consisted of four monomers, styrene (ST), divinylbenzene (DVB), and vinylbenzyl chloride (p-VBC and m-VBC), each shown in Figure 1 as united-atom models. Details regarding the simulation methodology have been provided elsewhere.33,34 Briefly, the construction of HCP models with ST, DVB, m-VBC, and p-VBC involved two steps of simulated polymerization, first linear polymerization to mimic a free-radical process (i.e., linking Lc1 and Lc2 atoms), then cross-linking to mimic Friedel−Craft alkylation (i.e., linking Cc2 to Ccp atoms). After a bond is formed between linker atoms, they can no longer participate in further bond formation. This modeling approach does not truly simulate free-radical polymerization or Friedel−Craft alkylation, instead, it makes bonds between specified linker atoms based on a cutoff distance. After the bond formation process is completed, the HCPs were equilibrated using a series of NVT and NPT molecular dynamics. The scheme includes multiple stages of MD simulation, as described elsewhere.47 Briefly, the scheme includes compression and slow decompression between 50 000 to 1 atm as well as heating and cooling between 800 and 300 K to allow the simulated HCP structures to relax. We investigated a family of ST-VBC-DVB HCPs, where 4 materials have 0% DVB but various degrees of cross-linking (25, 50, 75, and 100%), and another 5 materials have 100% cross-linking but different percentages of DVB (10, 20, 30, 40, 50%). Since both VBC and DVB can act as cross-linkers, the degree of cross-linking is defined as the percentage of VBC and DVB (i.e., % cross-linking = mol % VBC + mol % DVB). For amorphous materials, an ensemble of configurations needs to 17885


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force field (FF).64,65 The choice of this FF has been validated against experimental data in previous work. 33,34 The parameters of the nonbonded potentials are reported in Table S2. No charges were assigned on the united atoms making up the HCPs. We verified that this approximation was reasonable by performing DFT calculations for isolated monomers, as described in the Supporting Information. The 24 molecules used as adsorbates included linear and cyclic organic molecules and noble gases. The organic molecules were described by flexible models with FF parameters for bonds, angles, and torsions from TraPPE-UA, except for the two cyclic molecules, thiophene and toluene, which were described by rigid models.65−71 The two noble gases, Kr and Xe, were modeled with the universal force field (UFF).72,73 Nonbonded terms in adsorbate−adsorbate interactions were modeled by Lennard-Jones interactions. For propan-1-ol (alcohols), acetone (ketones), dimethylether (ethers), acetonitrile (nitriles), methanethiol (thiols), etc. considered in this study, Coulombic terms are also used to model the nonbonded terms because partial charges are assigned on oxygen/ nitrogen/sulfur atoms and their associated neighbor pseudoatoms, and accordingly the Ewald summation was used for computing electrostatic interactions. The adsorbate−HCP nonbonded interactions were modeled with Lennard-Jones interactions defined using the Lorentz−Berthelot combining rule and a tail correction with a cutoff of 14 Å. All simulated adsorption isotherms are available in the Supporting Information. Henry’s constants, K, were calculated by the Widom insertion method using the RASPA code,74 and the heats of adsorption at zero coverage, ΔH, were also calculated for each molecule−HCP pair.75 We calculated the void fraction of each HCP sample with the RASPA code using a probe size of 1.4 Å to approximate a He probe.

be examined in simulations to give a reliable description of a bulk material. For this reason, we used 5 independent samples of each HCP. In all of our calculations, the HCPs were approximated as being rigid in GCMC calculations due to a large number of covalent cross-links.9 A study by Ahn et al. showed that the swelling ratios for these type of HPCs immersed in a good solvent are about 1.8 for 2% DVB and 1.2 for 20% DVB.48 For gas adsorption well below the gas-phase vapor pressure, however, the swelling ratio should be significantly lower. Additionally, the small hysteresis loops observed in N2 and H2 adsorption isotherms at 77 K for most HCPs also suggest that swelling should not occur to a significant extent in the systems studied in this work especially at the conditions studied in this work.49−51 However, swelling could be relevant in materials with a low percentage of crosslinking under high pressure. This effect can be accounted for by using hybrid molecular dynamics (MD) and Monte Carlo (MC) calculations, for instance, but we have not used this approach in this work.52−56 Structural properties of each HCP structure were calculated using Poreblazer.57 Pore volumes were calculated based on the integral of interaction potential as proposed by Myers and Monson.58 The surface area was obtained based on the geometric method similar to a probe being “rolled” on the polymer surface.59 Pore size distributions were determined by the approach of Gelb and Gubbins, which is based on the construction of a pore size histogram.60 In our previous work, we have validated simulated HCP structures with the available experimental data.34 For structural properties, BET surface area and micropore volume as a function of cross-linking percentage from 25 to 100% were found in good agreement with the experimental data. For adsorption properties, the simulated isotherm of excess H2 was compared with the experimental isotherm for up to 5000 kPa and showed good agreement. Finally, CO2 loadings at 1 bar were only a few weight percent below the value observed experimentally. Single-component adsorption isotherms at 300 K were calculated by grand canonical Monte Carlo (GCMC) calculations using the RASPA code.61,62 To get well-converged results, 105 Monte Carlo cycles were used in each simulation for the equilibration and production period, with each cycle including translation, rotation, and reinsertion with equal probability. In total, 216 adsorption isotherms were obtained, representing 24 different molecules in 9 HCPs, where each isotherm was averaged over 5 independent HCP samples. We used the same set of molecules as Tang et al. in studying adsorption in MOFs.43 The identities of these molecules and some of their physical characteristics are listed in Table S1. Each isotherm was characterized in terms of relative pressure, P/P0, where P0 is the vapor pressure of the adsorbing molecule at 300 K. Several of the molecules are supercritical at 300 K, including methane, ethane, krypton, and xenon. For supercritical species, P0 was estimated using the vapor pressure of the species at their critical temperature using data from the NIST webbook.63 The critical temperatures associated with all 24 molecules and the resulting estimates for P0 are listed in Table S1. For each molecule, the lowest pressure simulated was chosen such that Henry’s law for adsorption could be defined (i.e., loading linearly increases with pressure). In every case, the highest relative pressure in each GCMC simulation was P/P0 = 10. During GCMC simulations, the HCPs were described by united-atom models using nonbonded terms from TraPPE-UA

3. RESULTS AND DISCUSSION 3.1. Structure Properties of HCPs. A representative HCP structure with 100% cross-linking and 10% DVB is shown in Figure 1b, which illustrates the irregular porous structure of these materials. Figure 2 shows the computed average pore size

Figure 2. Computed pore size distribution as a function of degrees of cross-linking and the percentage of DVB. Each curve shows results averaged over 5 samples.

distributions of all of the HCPs we considered. The structures have broad-range pore sizes with pore width up to 17 Å. The variation of the pore size distribution based on the monomer composition is the major motivation for exploring the adsorption space defined by these materials. The pore volumes, surface areas, density, and pore size distributions of each simulated HCP are summarized in Figures S1−S3. As 17886


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previous work of Tang et al. for molecular adsorption in MOFs and explored approaches in which we assume that each isotherm has a type I Langmuir form.43 This is clearly a strong assumption, since the heterogeneous nature of the adsorption sites in amorphous materials such as HCPs contradicts the central assumption of the Langmuir approach that every binding site for an adsorbed molecule is identical. Nevertheless, earlier simulation studies of adsorption of H2,9 CH4,76 and CO2,77 in several HCPs suggested that the resulting isotherms can be fitted with Langmuir isotherms with reasonable accuracy. A key advantage of the Langmuir form is that it includes only two physical parameters, the Henry’s constant, K, and the saturation loading, qsat, giving

expected, there is a strong correlation between the pore volume and surface area (see Figure S1), i.e., the pore volume and the surface area increase with the degree of cross-linking and decrease with the percentage of DVB. We validated our results against experimental data available in our previous work.33,34 The largest standard deviations of pore volumes, surface areas, and density among the five samples of each HCP are 20, 37, and 28%, respectively. 3.2. Adsorption Isotherms from GCMC Simulations. Several representative adsorption isotherms from GCMC calculations are shown in Figure 3, where qsim is the averaged

qpre(P) = KP /(1 + KP /qsat)

(1)

where qpre is the predicted adsorption loading at pressure P. The Henry’s constant for an adsorbing molecule can be accurately computed for molecular models such as those we have introduced above by the Widom insertion method much more rapidly than the computations needed for a full GCMC simulation of an adsorption isotherm. We therefore seek to predict adsorption isotherms based on Henry’s constants obtained from the Widom insertion method and saturation loadings predicted using physical properties of the adsorbing molecules and the HCPs. We summarize the Henry’s constants at 300 K for 216 adsorbate−HCP pairs in Figure 4. There is an Figure 3. Adsorption isotherms from GCMC simulations at 300 K for (a) methane and (b) octan-1-ol in 9 HCPs. For each molecule, the upper panel shows 4 HCPs with 0% DVB and various degrees of cross-linking, whereas the lower panels show 5 HCPs with 100% cross-linking and different percentages of DVB. Error bars show the standard deviation of loadings among 5 HCP samples. For some cases, error bars are smaller than the symbol size. The error bars for the octan-1-ol adsorption loadings in the HCPs with 25% crosslinking and 0% DVB overlap with the horizontal-axis, so only onesided error bars are shown.

adsorption loading over 5 samples of each HCP. Adsorption loadings increase with increased cross-linking due to the wider range of pore sizes and larger pore volume that occur with more cross-linking. For example, the averaged loading of methane at P/P0 = 10 (that is, P = 452 bar) increases from 1.01 ± 0.42 up to 11.04 ± 0.96 mol/kg as the degree of crosslinking of HCPs is increased from 25 to 100%. Increasing the amount of DVB causes the pore size and pore volume to decrease and, as a result, the saturation loadings of adsorbates also decrease. As an example, the saturation loadings of octan1-ol decrease from 2.74 ± 0.34 to 0.04 ± 0.06 mol/kg as the amount of DVB is varied from 10 to 50%. The entire set of adsorption isotherms from our GCMC simulations is available in the Supporting Information. In general, larger molecules reach saturation at lower relative pressures than smaller molecules. This is particularly notable for the two cyclic molecules we simulated, thiophene and toluene. Saturation of the adsorption isotherms in HCPs is observed above relative pressures of approximately P/P0 = 10−6 for thiophene and 10−4 for toluene. 3.3. Rapid Prediction of Adsorption Isotherms in HCPs. Computing the adsorption isotherms shown above using GCMC requires significant computational resources. It is therefore interesting to explore whether approximate forms for these isotherms can be developed to predict adsorption properties more efficiently. To this end, we followed the

Figure 4. Henry’s constants as a function of molecular weight in HCPs with different degrees of cross-linking and percentages of DVB. The error bars show the standard deviation in Henry’s constant among 5 samples for each type of HCP. The error bars are large for methanethiol, thiophene, toluene, dipropylether, octanal, and octan-1ol in the densest HCPs where there is considerable sample to sample variation in the accessible pore volume.

approximately exponential relationship between Henry’s constants and the molecular weight of the adsorbates, with cyclic organic molecules and noble gases being notable exceptions. Consistent with this observation, there is an approximately linear correlation between the heat of adsorption and molecular weight (see Figure S5). For octanal and octan-1-ol, the two largest molecules we considered, the standard deviations in heat of adsorption among samples are significant in the HCP composed of 25% cross-linking and 0% DVB. As might be expected, Henry’s constants are much more strongly dependent on the identity of the adsorbing species than on differences between the HCP structures. The error bars in Figure 4 indicate the standard deviation in Henry’s constants among 5 independent samples of each HCP. For a small number of materials, especially for adsorption of large molecules in HCPs with denser networks (25% cross-linking 17887


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Using these predicted isotherms, we can average the relative error over all of the pressures used in our GCMC simulations; this gives an average error of about 81%. This level of agreement is extremely poor, so it is important to understand the sources of the inaccuracy in these predictions. One possible reason is that our prediction of saturation loadings based on helium void fraction of each sample is not accurate enough. This idea is supported by Figure 5a, which plots the predicted and simulated saturation loading for each molecule−HCP pair. If eq 1 uses both Henry’s constants and saturation loadings directly from simulations, our data indicate that most of simulated adsorption loadings are approximately equal to the predicted results, as shown in Figure S6. In other words, using a Langmuir model in these HCPs is a reasonable choice. Additionally, Figure 5a also shows that the predicted saturation loadings can be categorized into two groups. The differing slopes for these two groups suggest that systematic improvement in predicting saturation loadings may be possible. In Figure 5a, group 1 contains the 5 molecules we considered with critical temperatures less than or slightly larger than 300 K (methane, ethene, ethane, krypton, and xenon), whereas the other 19 molecules are designated group 2. In group 1, the predicted saturation loadings are systematically smaller than the simulated results, in part, because the adsorbed phase is compressible to densities higher than the bulk phase critical density. In group 2, in contrast, the predicted saturation loadings are systematically larger than the simulated results. A similar effect was observed by Tang et al. for adsorption in MOFs,43 because the accessible pore volume defined using a He probe overestimates the pore volume available to larger molecules. We calculated the slope for group 1 and group 2 in Figure 5a by assuming a simple linear relationship between the simulated and predicted values. The resulting slopes show a clear trend that is dependent on the degree of cross-linking and the percentage of DVB (see Figure S7). We, therefore, fitted the slopes with the simple expression z = C1x + C2y + C3, where x is the degree of cross-linking, y is the percentage of DVB, and z is the slope (see Figure 5c,d). The dimensionless coefficients C1 = 0.0072 and C2 = −0.0069 of group 1 are roughly the same of those of group 2, C1 = 0.0087 and C2 = −0.0066, but for group 1, C3 = 0.084, whereas for group 2, C3 = 0.85. The resulting scaling factor s, which is calculated using the fitted expressions, for each group in each of the HCPs is listed in Table S4. We can now predict the saturation loadings for every molecule−HCP sample pair by estimating the void fraction, the helium void volume VHe multiplied by s:

and 0% DVB, 100% cross-linking and 50% DVB), the variation in Henry’s constants among samples is significant, especially for large molecules. The low porosity in these materials leads to higher uncertainties in adsorption properties (see Figures S1−S3). One natural way to estimate the saturation loading of adsorbing molecules in specific HCPs is to combine the helium void volume, VHe, of the HCP with the bulk phase liquid density, ρ, of the adsorbate, using the idea that the adsorbed phase is liquid-like at saturation (qsat ≈ ρVHe). To this end, we calculated the helium void fraction f He for each sample, as described in the Simulation Details section. The resulting data is listed in Table S3. Saturation loadings were estimated using this void volume and bulk liquid phase density at 300 K. Bulk densities (Table S1) were obtained from the experimental data or the NIST webbook.63 For molecules that are in the supercritical state for the bulk at 300 K, we used the density at the critical point. We compare the adsorption loadings, qpre, predicted by eq 1, with the simulated adsorption loadings, qsim, at our highest simulated pressure, P/P0 = 10, in Figure 5a. To quantify the performance of the model, we define the relative error by σ(P) = |qpre(P) − qsim(P)| /qsim(P)

(2)

The average relative error, σavg, for the entire set of molecule−HCP pairs at P/P0 = 10 is 108%. Combining these predicted saturation loadings and the simulated Henry’s constants gives a Langmuir isotherm defined at all pressures.

qsat = sρVHe

(3)

We show a comparison between predictions with this approach and our GCMC data in Figure 5b. The average relative error, σavg, for these predictions at the same pressures shown in Figure 5a is reduced to ∼25%. We now turn to the feasibility of predicting full adsorption isotherms (qpre) in HCPs. As noted above, we assumed that Henry’s constant for each molecule−HCP pair was available from molecular simulations. We can then consider three different ways to obtain the saturation loadings: (1) directly from GCMC simulations at high relative pressure, (2) using the HCP helium void fraction and the molecule’s liquid phase density, and (3) using the helium void fraction modified with the scaling factors defined above and the molecule’s liquid phase density. We predicted the full adsorption isotherms for

Figure 5. Comparison of simulated and predicted adsorption loading at P/P0 = 10 for all molecule−HCP sample pairs with two approaches to the loading prediction: (a) without correction to the predicted saturation loading for group 1 (orange) and group 2 molecules (green) and (b) after applying a scaling factor, s, for each group to the predicted saturation loading. (c) and (d) show the slopes, z, for the data in (a) and (b) as a function of the percentage of cross-linking and DVB for group 1 (orange triangles) and group 2 (green circles). The linear fits in (c) and (d) are used to calculate the scaling factor, s. 17888


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ments. We are not aware of similar literature meta-analysis for adsorption in polymers, so we are unable to directly compare with data for HCPs or similar polymers in the same way. The method we have introduced here is useful in that it allows predictions for a new molecule−HCP pair to be made using molecular simulations only to compute Henry’s constant of the adsorbing species. 3.4. Selectivity of HCPs for Adsorption of Binary Mixtures. Now that we have information on singlecomponent adsorption of a variety of molecules in HCPs, it is useful to consider the adsorption selectivity of these materials. Previous work by Kupgan et al. showed that although varying the monomer composition of HCPs could significantly affect the CO2 working capacity and sorbent selection parameter of H2/CO2 separation, it has a minor effect on adsorption selectivity.34 From that previous work, it was uncertain if the composition-independent selectivity was specifically for H2/CO2 pair or if it was a general behavior of the HCPs that would be observed for other gas pairs. Here, we consider 276 unique molecular pairs based on 24 adsorbates. We take advantage of the observation that under dilute conditions, the adsorption selectivity, Spair, of binary mixture is exactly equal to the ratio of Henry’s constants of the two adsorbing molecules.79,80 This observation is convenient given the modeling approach we have introduced above where Henry’s constants are computed directly by Widom particle insertions. To consider the trends in adsorption selectivity among HCPs, we calculated the selectivity for each of the 276 molecular pairs in each HCP and then ranked the HCPs for each pair in descending order by selectivity. By this approach, we could find the HCPs that give the largest and the smallest selectivity, Spair,max and Spair,min, for each molecular pair. These results are summarized in Figure 7a. A feature of Figure 7a is that the outcomes for Spair,max and Spair,min are very similar. That is, the variation in selectivity among different molecular pairs is larger than the variation among HCPs in selectivity for a particular pair. The color scale in Figure 7a is logarithmic, emphasizing the very large selectivities that can be achieved by adsorbing dissimilar molecules in HCPs. This general observation is similar to what has been reported by Tang et al. for MOFs.43 The large range of values in Figure 7a masks the variation that occurs between different HCPs. This variation is

each molecule−HCP sample pair using each of these methods. For each approach, we calculated the average error σavg from the relative error σ averaged over all loadings on the isotherm for each molecule−HCP pair. Figure 6 shows the fraction of

Figure 6. Cumulative plot of relative error of full adsorption isotherms for all comparisons between simulations (qsim) and predictions (qpre) using qsat directly from simulation (blue), with VHe (black), and with VHe and scaling factor s (red).

cases observed as a function of the average error for all of the molecule−HCP pairs we considered. As expected, the predictions using simulated saturation loadings directly from GCMC data have the highest accuracy. In this case, about 90% of the isotherms has an average relative error smaller than 32.5%. The deviations between this model and the true isotherm reflect the fact that the actual isotherms are not Langmuir in form, but the small average errors reiterate the point made above that the Langmuir form is an effective approximation for these systems. Using the predictions made with the scaled void fractions give isotherms where 75% of the cases has relative errors less than 34.5%. This result is considerably better than the outcome if the scaling factors for the void fractions are not used. Work examining the experimental reproducibility of adsorption isotherms in MOFs has indicated that experimental uncertainties of 10− 30% for CO2 adsorption are not unusual.78 This suggests that the approximate method we have demonstrated here can make predictions with accuracy similar to that of routine experi-

Figure 7. (a) Maximum (upper right triangle) and minimum (lower left triangle) HCP selectivities Spair,max and Spair,min, shown as an absolute value of logarithmic, for 276 unique molecular pairs. The molecules are ordered by molecular weight. (b) Molecular pairs (10) where the species that is favored upon mixture adsorption are dependent on HCP composition. 17889


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mixture separation at dilute conditions for each of the 276 molecular pairs that could adsorb in each HCP. One use of our data is to develop computationally efficient models that can rapidly predict molecular adsorption in HCPs without performing extensive molecular simulations. We showed that based on Henry’s constants determined from molecular simulations, the helium void fraction of the HCP, and bulk liquid phase density of the adsorbing molecule, we could predict isotherms with reasonable accuracy relative to the full GCMC simulations. This approach will allow predictions of adsorption in additional HCPs from different combinations of monomers and also for far larger sets of adsorbates than the 24 species, we have considered. We also examined the application of HCPs to molecular separations; the thousands of binary mixture−HCP pairs we considered greatly extend the number of examples for which information of this kind is available. Describing some caveats associated with our work highlights several potential directions for future improvements. Our approximate predictions of adsorption isotherms rely on assuming that each isotherm has the Langmuir form. It clearly may be possible to achieve more accurate results using more complex isotherms, although this would also likely increase the quantity of molecular simulation data needed before an isotherm can be predicted. Our approximate predictions also relied on empirical scaling factors that had to be determined for each HCP composition. These scaling factors would need to be established before our approximate methods could be fully applied to other HCP structures, although the GCMC calculations we reported here provide a natural avenue to achieve this task. Finally, we note that the aim of our approximate methods was to predict the adsorption isotherms that came from GCMC simulations. These simulations have assumptions of their own, perhaps most importantly that any effects of polymer flexibility were neglected in our simulations. Although this assumption seems plausible for the hyper-crosslinked polymers of interest in this paper, the impact of polymer flexibility or swelling would need to be examined if a high fidelity treatment of adsorption at high loadings or in polymers with low cross-link densities was desired.

highlighted by focusing on the binary mixtures for which the species that is favored upon adsorption varies as the HCP composition is changed. Out of 276 molecular pairs, 10 pairs from Figure 7a have this property, which is shown in more detail in Figure 7b. For example, the HCP with 25% crosslinking and 0% DVB selects toluene over octanal, but the other 8 HCPs show the reverse selectivity, selecting octanal over toluene (filled orange triangles in Figure 7b). Similar results are observed for 2-pentanone/dipropylether (filled red circles), diethylsulfide/dipropylether (filled black triangles), toluene/ dipropylether (filled green squares), and toluene/octan-1-ol (filled blue diamonds); in each case, the HCP with 25% crosslinking and 0% DVB shows a qualitatively different selectivity than the other HCPs. This observation is likely due to the relatively small and uniform pores in the former HCP. Considering toluene and octan-1-ol as an example, the Henry’s constant of octan-1-ol is larger than that of toluene in the other 8 HCPs. When the pore sizes of HCPs allow both of these two large molecules to enter, octan-1-ol has stronger affinity to those of HCPs, because its Henry’s constant is slightly larger (i.e., due to its larger molecular weight) than the one for toluene. However, in the HCP with 25% cross-linking and 0% DVB, toluene has an advantage over octan-1-ol for affinity, because its smaller size allows a better access to this HCP’s small pores. Setting aside the material with 25% cross-linking and 0% DVB, the adsorption selectivity among the remaining HCPs still varies by an order of magnitude or more in many cases. This kind of variation in selectivity is likely to have important consequences for practical separations processes, so efforts to choose appropriate HCP compositions for targeted separations of interest are clearly worthwhile. It is interesting to consider whether the selectivities discussed above can be predicted without detailed molecular simulations. Previous work by Freeman45,81,82 has indicated that ratios of gas solubilities, which are equivalent to adsorption selectivities in the dilute limit, for rubbery and glassy polymers can be approximated with the simple expression ln(KA /KB) = N (Tc,A − Tc,B)

■

(4)

where Tc,A and Tc,B are the bulk phase critical temperatures of the two adsorbing species. We found, perhaps surprisingly, that on average, this expression also applies to molecular adsorption in MOFs, although there are large variations around the average behavior fitted to eq 3.43 We considered whether this simple expression could also describe the selective adsorption of the diverse range of molecules we simulated in HCPs. As shown in Figure S8, however, this approach is seen to have little predictive power. The task of making accurate predictions of the adsorption selectivities for molecular adsorption in HCPs without using detailed molecular simulations remains an open problem.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.9b04413. Examples of checking the convergence of GCMC simulations are included in Simulation models of HCPs of the CIF format; structures of 4 monomers of XYZ format; computed structural properties of HCPs in the form of Excel files; simulated adsorption isotherms and predicted adsorption isotherms for each molecule− HCP pair in the form of Excel files; computed Henry’s law constant for each molecule−HCP pair in the form of Excel files; comparison of saturation loadings from simulation and prediction in the form of Excel files; data of relative error shown in Figure 6; data of selectivity shown in Figure 7 (ZIP) Structural properties of hyper-cross-linked polymers (HCPs) as functions of the percentage of DVB and degree of cross-linking computed by Poreblazer, computed pore volume (cm3/g), computed density (g/cm3), computed geometric surface area (m2/g),

4. SUMMARY This paper has presented the most diverse set of adsorption data for molecules adsorbing in hyper-cross-linked polymers available to date. We performed molecular simulations of 24 different molecules adsorbing in 9 HCPs at room temperature. For each of the 216 molecule−HCP pairs, we obtained the full adsorption isotherms using GCMC calculations. Our calculations also allowed us to describe the selectivity of binary 17890


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computed pore size distribution (cm3/(g·Å)); molecular weight (g/mol), bulk liquid density (g/cm3), critical temperature (K) and vapor pressure (Pa) are included; Lennard-Jones parameters for united atoms of HCPs are included; specific details of checking charge assignment are included; computed helium void fraction and heat of adsorption (kJ/mol) for each molecule−HCP pair; fitted expressions for scaling factors; comparison of simulated adsorption loadings and predicted adsorption loadings (with scaling factors) for all molecule−HCP pairs; estimated selectivities of molecular pairs using Freeman’s model (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Coray M. Colina: 0000-0003-2367-1352 David S. Sholl: 0000-0002-2771-9168 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Nanoporous Materials Genome Center, funded by the US Department of Energy, Office of Science, Basic Energy Sciences, under Award #DEFG02-17ER16362, as part of the Computational Chemical Sciences Program.

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