Article pubs.acs.org/Langmuir
NLDFT Pore Size Distribution in Amorphous Microporous Materials Grit Kupgan,† Thilanga P. Liyana-Arachchi,‡ and Coray M. Colina*,†,‡ †
Department of Materials Science and Engineering and ‡Department of Chemistry, University of Florida, Gainesville, Florida 32611, United States S Supporting Information *
ABSTRACT: The pore size distribution (PSD) is one of the most important properties when characterizing and designing materials for gas storage and separation applications. Experimentally, one of the current standards for determining microscopic PSD is using indirect molecular adsorption methods such as nonlocal density functional theory (NLDFT) and N2 isotherms at 77 K. Because determining the PSD from NLDFT is an indirect method, the validation can be a nontrivial task for amorphous microporous materials. This is especially crucial since this method is known to produce artifacts. In this work, the accuracy of NLDFT PSD was compared against the exact geometric PSD for 11 different simulated amorphous microporous materials. The geometric surface area and micropore volumes of these materials were between 5 and 1698 m2/g and 0.039 and 0.55 cm3/g, respectively. N2 isotherms at 77 K were constructed using Gibbs ensemble Monte Carlo (GEMC) simulations. Our results show that the discrepancies between NLDFT and geometric PSD are significant. NLDFT PSD produced several artificial gaps and peaks that were further confirmed by the coordinates of inserted particles of a specific size. We found that dominant peaks from NLDFT typically reported in the literature do not necessarily represent the truly dominant pore size within the system. The confirmation provides concrete evidence for artifacts that arise from the NLDFT method. Furthermore, a sensitivity analysis was performed to show the high dependency of PSD as a function of the regularization parameter, λ. A higher value of λ produced a broader and smoother PSD that closely resembles geometric PSD. As an alternative, a new criterion for choosing λ, called here the smooth-shift method (SSNLDFT), is proposed that tuned the NLDFT PSD to better match the true geometric PSD. Using the geometric pore size distribution as our reference, the smooth-shift method reduced the root-mean-square deviation by ∼70% when the geometric surface area of the material is greater than 100 m2/g.
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INTRODUCTION The pore size distribution (PSD) is an important characteristic of microporous materials with pore sizes of less than 2 nm because it is related to the transport of molecules, which is correlated to the efficiency of separation processes1 (e.g., membranes or adsorbents). PSD is a crucial design criterion for gas storage and separation applications because having an optimal pore size is one of the key aspects to obtaining both the maximum deliverable storage capacity and selectivity.2−5 In other applications, optimizing the pore size can lead to a maximum stored energy density in a nanoporous supercapacitor.6 In short, obtaining accurate PSDs for specific microporous materials is critical. Unfortunately, obtaining and verifying the pore size distribution in amorphous microporous materials is still a challenging task because of the highly disordered and complex molecular arrangements and the indirect techniques used to obtain such properties.7 © XXXX American Chemical Society
Although several methods can be used to acquire the pore size distribution (e.g., small-angle X-ray scattering, mercury porosimetry, nuclear magnetic resonance, thermoporosimetry, and positron annihilation lifetime spectroscopy),8,9 molecular adsorption10−12 with nonlocal density functional theory (NLDFT) is one of the most widely used methods today for nanoscale-resolution PSD. The NLDFT method uses a classical fluid density functional theory to construct the adsorption isotherms in ideal pore geometries (e.g., N2 adsorption in the slit-pore model at 77 K). The PSD result can be obtained by Special Issue: Tribute to Keith Gubbins, Pioneer in the Theory of Liquids Received: June 9, 2017 Revised: August 1, 2017
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DOI: 10.1021/acs.langmuir.7b01961 Langmuir XXXX, XXX, XXX−XXX
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Figure 1. Microporous materials investigated in this work. See the Supporting Information for structural details.
is unlikely to be suitable for amorphous microporous materials that have highly heterogeneous surfaces and no well-defined pore geometry. In computer simulations, the atomic coordinates of materials are known, and thus the exact pore size distribution can be obtained using a geometric insertion method.26 The geometric method represents a true pore size distribution because it is based on actual insertions and quantifications of various probe sizes into the microporous material structures. Therefore, geometric PSD can be used as a reference that can help evaluate the suitability of the NLDFT PSD method and its regularization parameter in experiments. The insight from geometric PSD is important because, in experiments, NLDFT PSD is an indirect method, and the validation of PSD for amorphous samples for a given λ is a challenging task. Several methods can be used to obtain the smoothing parameter such as the generalized cross-validation method (GCV),27 effective biased analysis,15 or L-curve;28 however, it is difficult to verify that the choice of λ truly represents the system of interest. Moreover, some of these methods do not always guarantee the optimal choice of λ.29−31 Therefore, it is crucial to investigate the suitability of λ obtained from these methods and compare it with known PSD references (i.e., geometric PSD). On the basis of the current capability of generating amorphous structures in silico (e.g., free volume polymers, crosslinked polymers, conjugated polymers, and porous discrete molecules), we can now take advantage of molecular simulations to study the accuracy of NLDFT PSD by comparing it with the true geometric PSD directly obtained from molecular simulations. The use of NLDFT to determine the pore size distribution in microporous carbons,11 the development of constant-pressure Gibbs ensemble Monte Carlo for simulated adsorption,32 and the utilization of the geometric pore size
solving an adsorption integral equation (eq 1, adsorption integral contribution, first term), which is an ill-posed problem, using regularization techniques (eq 1, regularization contribution, second term) such as the discrete Tikhonov regularization with non-negative least squares13,14 or the B-spline numerical technique.15 Thus, the PSD solution will be dependent upon the chosen regularization parameter, λ (also known as the smoothing parameter). In this equation, Nexp is the experimental N2 adsorption at 77 K, NNLDFT represents the theoretical N2 isotherms assuming an ideal pore geometry such as slit pore and cylindrical, PSD is the pore size distribution, P/Po is the pressure ratio with respect to the N2 saturation pressure, and D is the pore diameter. ⎛P⎞ Nexp⎜ ⎟ = ⎝ Po ⎠ +λ
∫D
∫D
Dmax
min
Dmax
min
⎛P ⎞ NNLDFT⎜ , D⎟PSD(D) dD ⎝ Po ⎠
[PSD″(D)]2 dD
(1)
The NLDFT PSD result may also be different depending on the chosen adsorption kernel using various gases and assumed pore geometries.16−19 Although PSD obtained from NLDFT has given valuable insights into material characteristics, it can exhibit artifacts such as artificial gaps, which has been welldocumented in the past for carbonaceous materials.20−22 To minimize these artifacts, several groups have devised elegant methods that account for energetic and surface heterogeneity, such as 2D-NLDFT23,24 and quenched solid density functional theory (QSDFT).25 These methods have been shown to be promising in eliminating artifacts for PSD within the specific cases studied. Even though these improved models are available, the recent literature is still reporting PSD using the simple slit-pore model. This simple slit-pore model B
DOI: 10.1021/acs.langmuir.7b01961 Langmuir XXXX, XXX, XXX−XXX
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geometric pore size distribution are presented and artifacts were clearly identified. As an alternative, we propose a new criterion for choosing λ in slit-pore NLDFT for amorphous materials that can be applied easily by both experimental and simulation groups in order to obtain a pore size distribution similar to that obtained from the geometric insertion method.
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METHODS
Simulated Structures. The atomic structures for amorphous microporous materials were obtained from previous work. As shown in Figure 1, the investigation included several amorphous materials such as a polymer of intrinsic microporosity (PIM-1),34 hyper-cross-linked polymers (HCP1, HCP2, and HCP3),35,36 conjugated microporous polymers (PY1 and PY2),37 polyolefin (OS-DVB1 and OS-DVB2),38 microporous polymer networks (DCX),39 and trigonal and octahedral organic molecules of intrinsic microporosity (OMIM1 and OMIM2).40 All frameworks except for OMIM1 and OMIM2 were constructed using the Polymatic41,42 simulated polymerization algorithm and the compression/decompression scheme.43 The simulated polymerization algorithm consisted of two steps: (1) random packing of respective repeat monomers into a simulation box under periodic boundary conditions at low densities followed by polymerization via Polymatic and (2) compression/decompression of the polymer framework samples to generate final polymer structures. In the case of OMIM1 and OMIM2, the synthesis process included the random packing of respective monomers followed by compression/ decompression. All molecular dynamics (MD) simulations conducted during both the Polymatic simulated polymerization algorithm and compression/decompression were done using the LAMMPS software package.44 More detailed descriptions of the virtual synthesis methodology and validation protocols used for these models were provided in the original articles as listed above. Five independent samples were used for the data analysis of each amorphous material. Structural details are provided in the Supporting Information. The ranges of the geometric surface area and micropore volume of these materials were between 5 and 1698 m2/g and 0.039 and 0.55 cm3/g, respectively (Figure S1). The geometric surface area was calculated by the summation of nonoverlapping areas accessible to a nitrogen probe. The micropore volume was calculated from the adsorption second virial coefficient obtained from simulated helium insertion. All of these values were obtained using the PoreBlazer45 code. Simulated Adsorption. Nitrogen adsorption isotherms at 77 K in amorphous materials were obtained with isobaric−isothermal Gibbs ensemble Monte Carlo (NPT-GEMC)32,46,47 as implemented in the Cassandra Monte Carlo (MC) simulation software.48,49 Several force fields were employed for amorphous microporous materials; the details are reported in the Supporting Information (Table S1). The transferable potential for the phase equilibria force field (TraPPE) was used for N2 molecules.50 Lorentz−Berthelot combining rules were employed to calculate the cross-interaction Lennard-Jones parameters. The equilibration and production periods were performed for at least 5 × 106 MC steps each. The probabilities for translation, rotation, volume change, and particle swap between simulation boxes were included during MC simulations. The frameworks were fixed throughout the simulation. The nitrogen isotherms at 77 K for these materials are reported in Figure S2. Pore Size Distribution. Pore size distributions were obtained using both NLDFT and geometric methods. The NLDFT pore size distributions (nldftPSD) were obtained using a demo version of SAIEUS software (www.nldft.com/download/) (Micromeritics, GA). The standard slit-pore model for N2 adsorption at 77 K on carbon was selected because it is widely used in experiments to characterize amorphous microporous materials. The L-curve method, which balances the roughness of the solution and goodness of the fit, is implemented in SAIEUS when choosing λ as a user-adjustable parameter. The geometric PSD (geoPSD) was obtained by differentiating the cumulative volume, V(d), with respect to the probe size, d, which is the general definition of the pore size distribution (PSD = −dV(d)/dd). The cumulative pore volume, V(d), was obtained by multiplying the micropore volume, Vpore, by the volume fraction, v(d), from
Figure 2. Comparison between NLDFT PSDs (nldftPSD, red) using the slit-pore model versus geometric PSDs (geoPSD, black).
distribution to characterize simulated porous glasses26 were pioneered by the Gubbins group in the 1990s. These methods have provided major advancement in the field of simulation and materials characterization ever since. In this work, nitrogen adsorption isotherms at 77 K were obtained from Monte Carlo simulations, which are widely used to predict molecular adsorption in porous materials.33 The N2 adsorption data were subsequently analyzed by SAIEUS software (Micromeritics, GA), which outputs NLDFT PSD with adjustable λ. The discrepancies between NLDFT and the C
DOI: 10.1021/acs.langmuir.7b01961 Langmuir XXXX, XXX, XXX−XXX
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Figure 3. Artifacts from NLDFT PSD in PIM-1 and HCP1. The evidence of an artificial peak is shown for PIM-1, and artificial gaps are shown for HCP1. Note that the probes were not drawn to scale.
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RESULTS AND DISCUSSION Comparison of NLDFT and Geometric Pore Size Distributions. The comparison between NLDFT using the slit-pore shape model and geometric pore size distributions is shown in Figure 2. The regularization parameters, λ, are selected using the L-curve method available in SAIEUS. Pores less than 3.6 Å will not be observed experimentally using the NLDFT method because it is the lower limit of the nitrogen probe size. However, even beyond 3.6 Å our result indicates that there are significant differences between NLDFT and the geometric pore size distribution obtained from our simulated samples. Artificial peaks and gaps from the NLDFT method are found throughout various amorphous materials as shown by differences in the NLDFT and geometric distributions. The presence of the artifacts is confirmed, and as has been discussed before,20−22 they are ill-defined. Most importantly, the dominant peaks from NLDFT typically reported in the literature do not necessarily represent the truly dominant pore size within a system. Moreover, the organic materials presented in this work also have surface heterogeneity that can be drastically different from the basic carbon slit-pore model usually adopted with NLDFT. Other DFT models23−25 that take into account the chemical surface heterogeneity and different pore geometries18,19 might prove more appropriate for amorphous porous organic polymers. Although choosing other pore geometries can affect the pore size distribution, the elimination of artifacts is still not guaranteed. NLDFT PSD features obtained from simulated N2 adsorption using Monte Carlo simulations can be used to partially validate simulated structures. For instance, an NLDFT PSD for PIM-1 obtained experimentally has been reported with two
Figure 4. NLDFT PSDs for PIM-1 (left) and PY1 (right) as a function of the regularization parameter, λ. An overlapped figure is available in the Supporting Information (Figure S4). PoreBlazer45 (i.e., V(d) = Vpore × v(d)). In this case, PSD values are normalized by the pore volume of the sample; consequently, the PSD intensity is proportional to the pore volume. For the PoreBlazer PSD, a total of 500 test points were used. For each test point, 106 attempts were performed in order to find the largest probe diameter between 0.1 and 20 Å using a 0.02 Å bin size. D
DOI: 10.1021/acs.langmuir.7b01961 Langmuir XXXX, XXX, XXX−XXX
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Figure 5. (A) Comparison of the L-curve method (left) and smooth-shift method (right) for choosing λ. The star is the solution from the L-curve method, and the circle is the solution for the smooth-shift method. (B) Transformation of PSD from the L-curve to the smooth-shift method. When using the smooth-shift method (SSNLDFT), PSD becomes broader. Then, a better representation of PSD can be obtained by shifting the distribution to the origin.
roughness of the solution. In contrast, the low value of λ gives the lowest fitting error and highest roughness. If the expected distribution is known a priori, users can adjust λ to obtain the desirable distribution. If the distribution is unknown, the appropriate choice of λ can be obtained from methods described previously (e.g., GCV,27 effective biased,15 or L-curve28). However, the validation process remains a difficult task. From this investigation, we found that the higher value of λ gives a more comparable representation of the PSD obtained using the geometric method because, using the geometric method, disordered materials typically have a broader PSD and have no gaps or dominant peaks (Figure 2). Thus, choosing the appropriate λ is extremely crucial because it can significantly alter the shape of the PSD. In future work, we recommend that the method used for determining λ should be reported to increase the consistency and reproducibility. Improved Method When Choosing a Regularization Parameter. As an alternative, a new criterion is devised to assist in the determination of λ for amorphous materials by applying the basic slit-pore shape NLDFT of 77 K N2 isotherms. As mentioned previously, a regularization parameter, λ, is typically obtained from techniques such as GCV,27 effective biased analysis,15 or the L-curve.28 Jagiello and Tolles found that effective biased and L-curve methods usually yield similar values of λ.53 In the L-curve method, the roughness of the solution and the goodness of the fit are balanced, which implies that the left corner of the roughness versus the RMS error plot is the optimal solution (star in Figure 5A, left). The location of this optimal solution in RMS error versus the λ plot is also shown on the right of Figure 5A (dotted star). By using this method, however, the shape of the NLDFT PSD distribution remains significantly different from that of the geometric PSD as shown in Figure 2. The new criterion is based on the premise, as shown by geometric PSD, that highly disordered materials typically have broad PSDs, i.e., no gaps or dominant spikes. From this argument, we propose that the regularization parameter should be maximized (i.e., maximize the smoothness and
peaks at around 6 and 13 Å.51 Experimental HCPs samples typically have few to no pores
