A Direct Assessment of the Slit Pore Model - ACS Publications

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Simulating Local Adsorption Isotherms in Structurally Complex Porous Materials: A Direct Assessment of the Slit Pore Model Jeremy C. Palmer,*,† Joshua D. Moore,‡ John K. Brennan,‡ and Keith. E. Gubbins† †

Department of Chemical and Biomolecular Engineering, North Carolina State University, 911 Partners Way, Raleigh, North Carolina 27695-7905, United States, and ‡Weapons and Materials Research Directorate, U.S. Army Research Laboratory, Aberdeen Proving Ground, Maryland 21005-5069, United States

ABSTRACT A fundamental understanding of the behavior of fluids confined in structurally complex nanoporous materials is crucial to the development of improved technologies for environmental remediation and energy storage. We present a computational method for assessing the impact that confinement has on the properties of fluids in model porous materials. The proposed method is demonstrated by calculating pore-size-specific adsorption isotherms and adsorption selectivites in a structurally heterogeneous nanoporous carbon (NPC) model. The results from this method are used to test the predictions made by the ubiquitous slit pore (SP) model. In general, we find that the SP model does not qualitatively capture the behavior of the pore-size-specific adsorption isotherms and selectivites in the NPC structure. These qualitative differences provide significant insight into the origins of the well-known deficiencies of the SP model to predict the adsorption behavior of real NPCs. SECTION Statistical Mechanics, Thermodynamics, Medium Effects

C

omplex porous materials, such as nanoporous carbons, metal-organic frameworks, zeolites, mesoporous silicas, metal oxides, and controlled porous glasses, play a crucial role in a wide range of emerging technologies applied to CO2 capture, water purification, chromatography, heterogeneous catalysis, gas storage, and electrode materials, among others. Guest nanophases confined within the pores of these materials experience numerous confinement effects that may significantly impact their phase behavior,1 transport properties,2-4 and chemical reactivity.5,6 Understanding these behaviors and the extent to which they are influenced by the physical properties of the host materials is of critical importance in optimizing existing applications as well as developing new technologies altogether. However, the complexity of these phenomena and the nanoscopic length scales over which they occur pose formidable challenges to achieving this understanding with current empirical and theoretical methods. These challenges are compounded in many porous materials of interest such as nanoporous carbons7,8 and controlled porous glasses9,10 because they have amorphous microstructures. While the features of crystalline materials can be determined precisely using standard crystallographic methods, the structures of amorphous materials can only be described in a statistical manner.7,9 This intrinsic heterogeneity significantly complicates both the structural characterization of these materials as well as the interpretation of empirical observations related to the behavior of confined guest phases.

r 2011 American Chemical Society

A concept that is nearly ubiquitous among the many theoretical methods used to study porous materials is that the size and number of the pores in the material dictate the phenomenological properties of confined fluids. For the case of adsorption, the amount adsorbed, N, at a given temperature, T, and pressure, P, can be written as H max X ^ NðT, PÞ ¼ NðH; T, PÞ ð1Þ Hmin

^ (H) is the amount adsorbed in regions of the material where N where the characteristic pore size is H. To make evaluating eq 1 tractable, the independent pore approximation is usually used H max X NðH; T, PÞf ðHÞ dH ð2Þ NðT, PÞ ¼ Hmin

where N(H) is the amount adsorbed in an isolated pore of size H and f(H) is the pore size distribution (PSD) of the material.11,12 In a typical application of eq 2, an ideal slit-, cylindrical-, or spherical-shaped pore morphology is assumed, and a suitable theoretical method, such as classical density functional theory or molecular simulation, is used to construct a kernel of N(H) values by calculating isotherms in a number of

Received Date: November 18, 2010 Accepted Date: January 4, 2011 Published on Web Date: January 10, 2011

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DOI: 10.1021/jz1015668 |J. Phys. Chem. Lett. 2011, 2, 165–169

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pores that span the appropriate range of H for the material being considered. Using experimental adsorption data for N and a kernel for the same adsorbate, the PSD of the material is obtained by numerical inversion of eq 2.11,12 Adsorption isotherms of other adsorbates as well as mixtures may then be predicted for the material by numerically evaluating eq 2 using the PSD and an appropriate N(H) kernel.13 Expressions similar to eq 2 have also been used to successfully predict heats of adsorption14 and diffusion coefficients15 from the PSDs of porous materials. The independent pore approximation has played an essential role in the development of theoretical approaches for characterizing porous materials and predicting phenomenological properties of confined fluids. In addition, it provides a unified context for achieving a fundamental understanding of various confinement effects. However, despite the widespread use of this approach, its accuracy depends on several assumptions that may not be valid for many real materials. First, the independent pore approximation neglects correlations that arise between molecules occupying different pores. The absence of these correlations can result in the failure of the independent pore model to correctly predict phase transitions of the confined guest phase.16 Second, the calculation of N(H) requires simplifying assumptions of the pore morphology of the material and the atomic structure of the adsorbent.11 However, the pores in amorphous materials, in particular, are often irregularly shaped, highly tortuous, and interconnected. Finally, the local atomic structure of the pore walls is highly heterogeneous and contains numerous chemical defects. As a result, the physical properties of these materials may not be well-described by pores with simple morphologies that have crystalline or atomistically smooth microstructures. While these assumptions have been documented in the literature, their implications on the predicted behavior of confined fluids have received less attention. This is largely due to the difficulty in finding suitable methods to test eq 2 in model systems that do not adhere to these assumptions. In this ^ (H) in eq 1 for Letter, a method of directly evaluating N arbitrarily complex porous material models using molecular simulation is presented, and the validity of eq 2 is examined by comparing the predictions made by the independent pore model with those obtained with the new method. ^ (H) requires defining The proposed method for calculating N the regions in the model material by a characteristic pore size H, which is a geometrical measure of the extent of confinement imposed on a guest phase by the host framework of the material. In general, the pore size is a local quantity that varies as a function of the position, r, where in the proposed method, the local pore size, h(r), is evaluated at the center-of-mass position vector, rci , for each of the i = 1, ..., N adsorbed molecules in a molecular simulation configuration. After determining h(r), the amount adsorbed in regions where the local pore size is H is then calculated as the number of molecules P whose center-of-mass lies in such a region, that is, ^ (H) = i δ(H - h(rci )). However, for finite systems, the points N where h(r) is exactly equal to any given value of H are vanishingly small. It is therefore necessary in practice to calculate ^ (H) by considering the number of molecules for which h(rci ) N lies within some small but finite interval, (dH, around H. The


problem of evaluating h(r) in complex materials has been addressed previously by Gelb and Gubbins.10 In their method, h(r) is equal to the diameter of the largest sphere that covers r without overlapping with the host framework. For a material consisting of Lennard-Jones particles with a collision diameter σs, overlap occurs if the boundary of the sphere comes within a distance of σs/2 of the nucleus of an atom in the framework. This method provides an unambiguous definition of h(r) in complex materials, and for simple pores, such as slits, cylinders, and spheres, it leads to a commonly used definition of the pore size H=dn - σs, where dn is the distance between the nuclei of framework atoms on opposing walls of the pore. To estimate h(r), the simulation cell is divided into small cubic voxels, each with an edge length dL. The largest sphere that can be placed at the center of each voxel without overlapping with the framework is determined and stored. Then, h(r) is calculated by searching the voxels to find the diameter of the largest sphere that encompasses r. This procedure is computationally intensive, especially for large systems that contain many adsorbed molecules. An alternative approach, which we have employed, is to construct a pore size map by storing the value of h(r) at the center of each voxel. During the simulation, h(rci ) is then estimated using the tabulated value of h(r) at the voxel closest to rci . This pore size mapping is the key aspect of the method presented in this Letter, which makes ^ (H) computationally tractable for full-length sicalculating N mulations. By conducting a series of test simulations, we found that the adsorption isotherms were insensitive to the voxel width if a resolution of at least dL = 0.2 Å was used, which indicated that this resolution was sufficient to obtain accurate estimates of h(rci ). Calculation of the pore size map was carried out in a model of an amorphous nanoprous carbon (NPC) that was developed using the quench molecular dynamics (QMD) method of Shi17 in a previous study.8 The model structure, shown in the upper left panel of Figure 1, was produced by simulated thermal quenching of a system of 3821 C atoms in a cubic simulation box 4.33 nm in length from a gaseous phase into an amorphous solid at a rate of 3.125 1011 K/s. The reactive summation state potential for carbon17 was used to capture bond formation and destruction during the QMD simulation. The NPC model has a highly defective microstructure and consists of curved graphene fragments that are comprised of aromatic carbon rings. As previously demonstrated, these structural features are consistent with neutron diffraction and transmission electron microscopy measurements on real NPCs.8 In a previous study,8 the adsorption isotherms for Ar at 77 K for the NPC model were also shown to be in excellent qualitative agreement with experiment, exhibiting type-I reversible behavior according to the IUPAC classification system, which is typical for NPCs.18 The upper right panel of Figure 1 also shows the pore size map, which highlights the voxels in the simulation cell where the local pore size is 3.5, 7.5, and 11.1 Å. The map was generated using a collision diameter of σs = 3.4 Å for the C atoms in the structure, a voxel width of dL = 0.2 Å, and an interval width of dH = 0.1 Å. For example, for a pore size of H = 7.5 Å, all of the voxels in the simulation cell where h(r) is in the interval [7.4 Å, 7.6 Å] are shaded with a single color. The total PSD of the NPC calculated

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Figure 2. Adsorption isotherms for Ar at 87.3 K. The top, middle, and bottom panels show results for pore sizes of 3.5, 7.5, and 11.1 Å, respectively. Filled circles and squares are adsorption isotherms calculated for the independent carbon SP and SP-H models, respectively, while open triangles indicate local, pore-size-specific isotherms in the NPC that were calculated from the number of Ar molecules occupying the voxels where the local pore size was within (0.1 Å of the specified value. The fractional loadings were calculated by normalizing each isotherm by its value at the saturation pressure of Ar. Isotherms for the NPC model were obtained by averaging the results of five statistically independent simulations. Symbol sizes are larger than the estimated uncertainties.

Figure 1. Upper left panel: simulation snapshot of the NPC with N2 and CO2 adsorbed from an equimolar bulk mixture at 300 K and 20 bar. Upper right panel: pore size map highlighting the voxels in the NPC framework where the local pore size is within (0.1 Å of the specified value (denoted by color). Lower panel: the total pore size distribution of the NPC framework, with contributions of the 3.5, 7.5, and 11.1 Å sized pores shaded in blue, green and pink, respectively.

using the method of Gelb and Gubbins10 is also shown in the lower panel of Figure 1. Note that the pore size map does not depict the morphology of the pores because h(r) may vary throughout the void space of a given pore. Thus, the pore size map merely provides a means to estimate h(rci ). As is evident from the molecular configuration in Figure 1, the irregular shape of the NPC framework gives rise to a complex network of pores with different sizes and morphologies that do not adhere to any simple geometry. Grand canonical Monte Carlo (GCMC) simulations19,20 were used to calculate adsorption isotherms of Ar at 87.3 K (the normal boiling point) in the model NPC. The isotherms ^ (H) taken from were calculated from ensemble averages of N five independent simulations at each state point, where each simulation was performed using a different random number sequence to sample configuration space. A discussion of the GCMC method and the details of its implementation are provided in the Supporting Information (SI). For comparison, N(H) was also calculated for Ar in atomistically detailed, graphitic slit pores (SP). A modified slit pore model (SP-H) with a heterogeneous pore surface was also considered. In the SP-H model, the interaction strength (Lennard-Jones well depth) of each carbon atom in the surface layers was scaled by a uniform random number between 0.5 and 1.0. This


slightly reduced the overall attractiveness of the surface layers and created an energetically heterogeneous pore surface that is analogous to the highly defective surfaces found in real NPCs. Figure 2 shows the adsorption isotherms for three different pore sizes. The pressures have been normalized by the saturation pressure of Ar at 87.3 K, Po = 1 bar, and the fractional loadings were calculated by normalizing each isotherm by its value at Po. For each of the pore sizes examined, the SP filled with Ar at lower relative pressures than the SP-H or NPC models. This is due to the ideal graphitic pore walls in the SP, which produce much stronger adsorbate-adsorbent interactions than the energetically heterogeneous pore surface of the SP-H model or the small, unstacked graphene fragments found in the NPC structure. For pore sizes of 3.5 and 7.5 Å, the SP-H model filled with Ar at a higher relative pressure than the NPC structure. However, for a pore size of 11.1 Å, the two models had approximately the same filling pressure. This suggests that the nature of the surface strongly influences the filling pressure in highly confined regions, where the

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Figure 3. Carbon dioxide adsorption selectivity from an equimolar bulk mixture of N2 and CO2 at 300 K as a function of bulk pressure and pore size in (a) independent carbon SPs and (b) the NPC model. The results in (b) were calculated from 15 statistically independent simulations. (c) Estimated uncertainties for (b).

show SCO2/N2 as a function of pore size and pressure in the SP and NPC models, respectively. Figure 3c shows the uncertainties for SCO2/N2 in the NPC model; the calculation of the uncertainties is discussed in the SI. The selectivity in the SP model is generally much higher than that in the NPC because of strong interactions of the adsorbate with the graphitic pore walls. For small pores, both models show a large increase in SCO2/N2 that is due to a molecular sieving effect, where N2 is nearly excluded from adsorbing in the pores because of its larger size. The selectivity also has a local maximum in the SP model at H ≈ 6 Å, which occurs because of a minimum in the CO2-adsorbent potential energy that arises at that separation distance of the pore walls. This maximum in SCO2/N2 is not observed in the NPC model because pores of a given size are energetically heterogeneous and do not give rise to a distinct minimum in the CO2-adsorbent potential. A local maximum in SCO2/N2 with respect to pressure is also observed in the SP. The selectivity of CO2 increases at higher pressures due to the formation of CO2 clusters in the void space of the SP that selectively enhance the adsorption of CO2 because of its larger quadrupole moment. This maximum does not occur in the NPC model because clustering is prevented to a large extent by the complex connectivity and irregular shape of the void spaces. These qualitative differences provide significant insight into the reasons for the less-than-satisfactory agreement that has been found between experimental measurements of selectivities in real NPCs and the predictions made by eq 2.13 Using the pore size map approach presented here, the local extent of confinement imposed on individual molecules during a molecular simulation can be determined with minimal computational overhead. This method provides a means of directly examining the impact that the extent of confinement has on the behavior of fluids confined in complex nanoporous material models. Although the method was demonstrated for the case of adsorption, we anticipate its use in many other applications where the calculation of pore-size-specific physical properties may improve our understanding of confinement effects.

majority of adsorbate molecules are in close proximity to the pore surface. The shapes of the isotherms in the three models are also clearly different. The isotherms for the SP models have a sigmoidal shape (using a log scale for the abscissa), large and abrupt filling transitions, and smaller layering transitions in the 7.5 and 11.1 Å pores, which can accommodate two and three layers respectively. Similarly, the isotherms for the SP-H model are also sigmoidal and have abrupt filling transitions. In contrast, the isotherm for the pore size of 3.5 Å in the NPC model is nearly linear instead of sigmoidal. As is evident from Figure 1, the regions where the pore size is 3.5 Å are homogenously dispersed throughout the structure, which causes Ar molecules to adsorb in these regions in an uncorrelated fashion. For pore sizes of 7.5 and 11.1 Å in the NPC model, the regions are highly connected, and the isotherms exhibit a sigmoidal shape and filling transition, similar to that of the SP and SP-H models. In both the SP-H and NPC models, there is also an absence of layering transitions in the isotherms for pore sizes of 7.5 and 11.1 Å. This is due to the heterogeneous nature of these models, which causes adsorption to occur more gradually. These layering transitions are not observed in experimental isotherms measured on most real NPCs.21 Use of isotherm kernels based on the standard SP model, which exhibits these layering transitions, has been reported to give rise to false gaps in the PSDs estimated using eq 2.21 Recent developments in classical density functional theory have allowed kernels to be constructed from SPs with rough pore surfaces, which remove these false gaps21 and provide a reliable estimate of the geometric PSDs of NPCs.7 The isotherms from these improved models are qualitatively similar to those of the SP-H model presented in this Letter. The impact of pore size and the total bulk pressure on the adsorption selectivity, SCO2/N2, of CO2 from a bulk mixture containing equimolar amounts of N2 and CO2 at 300 K was also investigated in the SP and NPC models using the proposed method. Details of the calculations are provided in the SI. The adsorption selectivity, defined as SCO2/N2=(xCO2/xN2)/(yCO2/yN2), where x and y are the molar compositions in the adsorbed and bulk phase, respectively, is a key indicator of the ability of a material to separate two components in a mixture. The top left panel of Figure 1 shows a molecular configuration of the NPC with CO2 and N2 adsorbed at P=20 bar. Figures 3a and b


SUPPORTING INFORMATION AVAILABLE Details concerning the implementation of the grand canonical Monte Carlo method, including the potential models and sampling methods used, and the estimation of uncertainties. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION Corresponding Author:

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*To whom correspondence should be addressed. E-mail: jcpalmer@ ncsu.edu.

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ACKNOWLEDGMENT We gratefully acknowledge support from the

U.S. Defense Threat Reduction Agency (Project ID: AA07CBT011) and from the National Science Foundation (Grants CBET-0932656 and TG-CHE080046N). J.D.M. gratefully acknowledges support, in part, by an appointment to the Internship/Research Participation Program for the U.S. Army Research Laboratory administered by the Oak Ridge Institute for Science and Education through an agreement between the U.S. Department of Energy and USARL (Project ID: 201003211).

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A Direct Assessment of the Slit Pore Model - ACS Publications

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