Analytical representation of micropores for predicting gas adsorption in porous materials

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Abstract

A straightforward method for the prediction of the gas storage capabilities of porous materials has been established. The Topologically Integrated Mathematical Thermodynamic Adsorption Model (TIMTAM) combines analytical surface potential energies with classical physisorption thermodynamics in a computationally inexpensive fashion. Experimental and simulated isotherms from leading sorbent candidates such as metal–organic frameworks (MOFs), zeolitic imidazolate frameworks (ZIFs), carbon nanotubes and activated carbons have been used to verify the model. Furthermore, the effect of pore size and shape upon gas storage characteristics is explored using the TIMTAM routine.

Highlights

► Micropores are represented as geometric constructs. ► Analytical potential energy formulations are used to predict gas adsorption. ► Parameter landscapes are explored and optimized to maximize gas storage capacity. ► Critical parameters include pore size, shape, atomic density and heat of adsorption. ► As a result, the optimal pore size is found as function of pressure and temperature.

Introduction

One of the challenges of the 21st century is the generation, storage and delivery of energy in an affordable, renewable and clean fashion [1], [2]. The ability to efficiently capture, store and release gases plays an important role in addressing such issues, for example: carbon capture and storage (CCS) for clean coal, petroleum and natural gas combustion [3], [4], [5]; methane capture and storage for climate control or safe energy transport [6], [7], and hydrogen capture and storage from photolytic water splitting or coal gasification for clean combustion or fuel cell technologies [1]. Porous materials are poised to meet such challenges.

The leading candidates for efficient high capacity gas storage porous materials are metal–organic frameworks (MOFs) [8], [9], [10], [11], zeolitic imidazolate frameworks (ZIFs) [12], [13], [14], nanotubes [15], [16], [17], and graphenic carbons [18], [19], [20], [21], [22], which can each undergo physisorption, with capacity closely related to surface area. The usage of such adsorbents depends strongly on key factors such as the capacity, operating temperature, cyclability and kinetics of operation. Within a porous gas adsorbent, the nature of the pores is integral to addressing these issues, with pore shape, size, concentration and surface chemistry all important to the overall performance.

Experimental materials discovery has led to substantial improvements in gas storage performance. However, the formulation of improved guidelines for further development of new materials could greatly speed this process. Whilst detailed, atomic level resolution modeling can elucidate the fundamental atomic interactions, a model that focuses on the overall impact of pore topology on overall performance is pertinent. Furthermore, a quick, accurate, yet accessible model would strengthen the feedback loop between experimentalists and modelers.

Modeling techniques for addressing physisorption range in complexity, accuracy and utility. Firstly there are fundamental equations such as the Langmuir single-layer model [23] and the Brunauer, Emmett and Teller (BET) multi-layer model [24] that have simplicity, such that anyone can calculate adsorption properties without the need for extensive computational or modeling expertise. Then, on a more complex level, there are the simulation (or computational) techniques such as Molecular Dynamics (MD) [25], [26], [27], [28], [29], [30], Monte Carlo (MC) [16], [30], [31], [32], [33], [34] and those based on ab initio principles [35], [36], [37], [38], [39], [40], [41] or mean-field density functional theories [42], [43]. Each of these techniques offer useful functions that have significant impact on the field of materials science and more specifically on the current demand for high performing membranes and adsorbents [44]. The simple equations offer quick estimates of surface area and heat of adsorption from experimental adsorption isotherms, while the computationally-expensive simulation techniques offer in-depth detail and prediction of all aspects of adsorption within specific atomic structures. A tool that is missing for the porous materials community is a simple predictive model that can speedily estimate gas uptakes based on obtainable details such as pore size, shape and composition. Therefore we present a technique that exists on the middle ground – it is as straightforward as the BET type models and as predictive as the simulation techniques. As such it can be readily applied by those without extensive modeling experience. We envisage that this technique will also be utilized for computational screening of large databases of potential adsorbents where detailed simulation techniques are not suitable.

By exploiting thermodynamic and kinetic principles we develop a model that can be used to investigate the gas adsorption phenomenon and which encompasses the main characteristics found by statistical simulation studies. Essential input factors for the model include temperature, pressure, and pore geometry (which is used to derive the potential energy landscape, free volume and surface area). For complex atomic structures, algorithms for calculating potential energy, free volume and surface area are readily available [45], [46], [47], [48]. For structures with simple pore geometries that can be approximated by spheres, cylinders or slits, we provide analytical formulations which are used throughout this study. These formulations are ideal as they do not rely on specific atom positions for which computationally expensive methods are needed to calculate the potential energy at each point within the structure. Therefore, by approximating the topology, integrating the potential energy within the cavity and incorporating thermodynamic theory we establish the Topologically Integrated Mathematical Thermodynamic Adsorption Model (TIMTAM). The major advantage of our TIMTAM approach is that it provides researchers with analytical formulae that are computationally instantaneous, and therefore many distinct scenarios can be rapidly investigated to accelerate material design [11]. The primary function of the model will be the simulation of adsorption isotherms from estimates of pore volume, size and shape.

The following section outlines the basic theory and formulations for the model. Experimental and simulation results are then used to validate the model, followed by an investigation into the effect that pore shapes have on adsorption. This approach has already been used to investigate the performance of a new class of adsorbents, MOFs impregnated with nanostructures [11]. The model successfully described the uptake within existing MOFs and predicted an enhanced hydrogen and methane uptake within MOFs infused with fullerenes and decorated fullerenes. Here we fully present the model in its complete form, demonstrating its ability to model gas uptake in a range of adsorbents such as MOFs, ZIFs, nanotubes and activated carbons, and direct the reader to a graphical user interface, Adsorb IT, that we have developed to demonstrate the speed, versatility, usability and accuracy of the TIMTAM approach.

Section snippets

Theory and mathematical formulation

The interactions between a gas molecule and a surface, arising from van der Waals forces, are well described by the Lennard-Jones (L-J) potential energy function. This potential is known in this context as the potential energy for adsorption, found by integrating the atom–atom L-J function over the surface. Here we assume a continuous surface and formulate the potential energy within cavities that can be represented as analytical geometric shapes. For conventional Monte Carlo algorithms, the

Results and discussion

To demonstrate the capability of TIMTAM to accurately predict adsorption performance within a range of porous adsorbents we reproduce the results found from MC simulations for hydrogen uptake within carbon slits and carbon nanotubes by Rzepka et al. [54], experimental results for methane uptake in carbon slits from Aukett et al. [61], MC simulations for hydrogen uptake within MOFs by Ryan et al. [62], experimental results for hydrogen uptake within MOFs from Kaye et al. [63], and experimental

Conclusions

The TIMTAM formulation has been established to provide a simple method for predicting gas uptake capacities within a variety of porous materials for which the Langmuir or BET adsorption isotherms are inappropriate. Good agreement between experiment and simulation for uptake of H2, CH4 and CO2 has been demonstrated, with variations typically less than 15%. The continuum-based model can be readily applied to cylindrical, spherical or slit-shaped pores. The model is able to account for the effects

Acknowledgements

This work is partially supported by the Australian Research Council Discovery Project scheme. J.M.H. is grateful to the Australian Research Council for provision of an Australian Professorial Fellowship. A.W.T. gratefully acknowledges the CSIRO Division of Materials Science and Engineering and the OCE Science team. A.J.H. and M.R.H. thank the OCE Science Leader Scheme for funding.

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