Quenched solid density functional theory and pore size analysis of micro-mesoporous carbons

Abstract

We present a new model of adsorption on micro-mesoporous carbons based on the quenched solid density functional theory (QSDFT). QSDFT quantitatively accounts for the surface geometrical inhomogeneity in terms of the roughness parameter. We developed the QSDFT models for pore size distribution calculations in the range of pore widths from 0.4 to 35 nm from nitrogen at 77.4 K and argon at 87.3 K adsorption isotherms. The QSDFT model improves significantly the method of adsorption porosimetry: the pore size distribution (PSD) functions do not possess gaps in the regions of ∼1 nm and ∼2 nm, which are typical artifacts of the standard non-local density functional theory (NLDFT) model that treats the pore walls as homogeneous graphite-like plane surfaces. The advantages of the QSDFT method are demonstrated on various carbons, including activated carbons fibers, coal based granular carbon, water purification adsorbents, and mirco-mesoporous carbon CMK-1 templated on MCM-48 silica. The results of PSD calculations from nitrogen and argon are consistent, however, argon adsorption provides a better resolution of micropore sizes at low vapor pressures than nitrogen adsorption.

Introduction

Over the last decade, a significant progress has been achieved in understanding the underlying mechanisms of adsorption in micro- and mesoporous solids and, consequently, in elaborating the theoretical foundations of adsorption characterization. This progress has been related, to a large extent, to the application of microscopic methods such as the density functional theory (DFT) of inhomogeneous fluids, which allows one to describe adsorption and phase behavior of fluids in pores on a molecular level [1], [2], [3]. DFT has helped qualitatively classify the specifics of adsorption and capillary condensation in pores of different geometries [2], [4], [5], [6]. It has been shown that the non-local density functional theory (NLDFT) with suitably chosen parameters of fluid–fluid and fluid–solid interactions quantitatively predicts the positions of capillary condensation and evaporation transitions of argon and nitrogen in cylindrical and spherical pores of ordered mesoporous molecular sieves such as MCM-41, SBA-15, SBA-16, and hierarchically structured silica materials [7], [8], [9]. The NLDFT method has been commercialized by the producers of adsorption equipment for the interpretation of experimental data and the pore size distribution (PSD) calculation from adsorption isotherms. The NLDFT method is widely applied, and it is featured in a recent standard by ISO [10].

While NLDFT has been demonstrated to be a reliable method for characterization of ordered silica materials, pore size analysis of carbons remains difficult. Although the first DFT methods were suggested for activated carbons [11], [12], [13], the inherent complexity and heterogeneity of pore structures in carbonaceous materials make the development of improved adsorption isotherm models and new characterization methods a topical problem. Current implementations of NLDFT for carbon materials are based on a model of independent slit-shaped pores with ideal graphitic walls. Such a model has a significant drawback; starting from pore widths of more than a few molecular diameters, theoretical adsorption isotherms exhibit multiple steps associated with layering transitions related to the formation of a monolayer, second adsorbed layer, and so on [14], [15], [16]. Experimentally, stepwise adsorption isotherms are observed only at low temperatures for fluids adsorbed onto molecularly smooth surfaces, such as mica or graphite. However, in disordered carbon materials (e.g. active carbons, activated carbon fibers, etc.), layering transitions are hindered due to inherent energetic and geometrical heterogeneities of real surfaces. The layering steps on the theoretical isotherms cause artificial gaps on the calculated pore size distributions, because the computational scheme, which fits the experimental isotherm as a linear combination of the theoretical isotherms in individual pores, attributes a layering step to a pore filling step in a pore of a certain size. For example, in the case of nitrogen at 77.4 K on graphite, the monolayer formation step in NLDFT occurs at the same relative pressure of ∼0.3 × 10⁻⁴ p/p₀ as the pore filling in ∼1 nm wide slit. This coincidence results in a prominent false gap on the pore size distribution histograms [15], [16]. The second false gap around ∼2 nm may appear due to the artificial first-to-second layer transition predicted by NLDFT at ∼0.2 p/p₀. This mismatch between the theoretical assumption of a smooth and homogeneous surface and the experimental situation is especially pronounced for materials with broad PSDs that is typical for many microporous carbons (see detailed discussion with examples below). It is characteristic for novel nanoporous carbons, which were specifically designed for pore size sensitive applications, such as carbide derived carbons [17], [18] and exfoliated graphite nanofibers [19]. In addition, the fit of the low-pressure part of experimental isotherms is rarely satisfactory – calculated isotherms exhibit unavoidable swings reflecting layering transitions, see examples given in Section 4).

Several approaches were suggested to account for the heterogeneity of carbon materials. New molecular structural models of porous carbons have been developed by reverse Monte Carlo techniques [20], [21]. Although very promising, these models are still too complex to be implemented for routine pore size analyses. Within the framework of the standard slit-pore model of carbons, a variability of pore wall thickness has been introduced [22], [23], [24](a), [24](b), but it led to just a marginal improvement over the standard NLDFT approach [16]. Molecular simulations have demonstrated that the surface roughness and defects affect significantly the shape of adsorption isotherms on heterogeneous surfaces [25], [26], [27]. In particular, Do and Do [27] simulated argon adsorption on the surfaces of carbon blacks and achieved quantitative agreement with experimental data by introducing various levels and sizes of surface defects. Several modifications of the solid–fluid potential within the Tarazona’s version of NLDFT were proposed [14], [28] to account effectively for the surface heterogeneity and generate smoothened adsorption isotherms. Ustinov et al. [28] developed a model for the pore size analysis of carbons, which is based on a fit to the reference isotherm on nongraphitized carbon black.

Recently, two of us have suggested the quenched solid density functional theory (QSDFT) [29]. QSDFT was devised for modeling adsorption in heterogeneous materials with corrugated amorphous walls. It has been successfully applied to siliceous materials of MCM-41 and SBA-15 type [29]. QSDFT is a multicomponent DFT, in which the solid is treated as one of the components of adsorbate–adsorbent system. In contrast to the conventional NLDFT models that assumed structureless graphitic pore walls, the solid is modeled using the distribution of solid atoms rather than the source of the external potential field. OSDFT allows one to account explicitly for the effects of surface heterogeneity. The surface heterogeneity in the QSDFT model is characterized by a single roughness parameter that represents the characteristic scale of surface corrugation.

In this work, the QSDFT approach is extended to adsorption of nitrogen at 77.4 K and argon at 87.3 K on carbon adsorbents. Although nitrogen adsorption is traditionally considered as a standard technique for pore size characterization, argon adsorption at 87.3 K has advantages for ultra-microporous materials, since argon fills micropores of dimensions <0.7 nm at higher pressures than nitrogen at 77.4 K that leads to faster adsorption kinetics and allows one to obtain high resolution low pressure adsorption isotherms [30]. The paper is structured as follows. In Section 2, we describe the basic equations of the QSDFT model and its parameters. It is worth noting that the QSDFT model is capable of describing adsorption isotherms on carbon materials with various degree of pore wall roughness/corrugation and/or surface defects. The roughness parameter and the fluid–solid interaction parameters can be customized for a given class of carbons by fitting the reference experimental adsorption isotherm on a well-characterized reference surface, e.g., on graphitized and non-graphitized carbon blacks [31], [32], [33], [34]. In this work, we have chosen Cabot BP-280 carbon black with a partial degree of graphitization [33], [34] as the reference surface. The characteristic features of adsorption on the reference surface and capillary condensation in pores with molecularly rough walls are presented in Section 3. We show that the QSDFT model quantitatively describes adsorption of nitrogen and argon on Cabot BP-280 carbon black, and that QSDFT adsorption isotherms do not exhibit layering transitions. We developed the QSDFT models for pore size distribution calculations in the range of pore widths from 0.4 to 35 nm. The QSDFT model improves significantly the method of adsorption porosimetry: the pore size distribution (PSD) functions do not possess artificial gaps in the regions of ∼1 nm and ∼2nm, which are typical artifacts of the standard NLDFT model that treats the pore walls as homogeneous graphite-like plane surfaces. The validity of the QSDFT method is demonstrated in Section 4 on various carbons, including activated carbons fibers, coal based granular carbon, water filtration adsorbents, and mirco-mesoporous CMK-1 carbon templated on MCM-48 silica. The results of PSD calculations from nitrogen and argon are consistent, however, argon adsorption provides a better resolution of micropore sizes at low vapor pressures than nitrogen adsorption. Conclusions are summarized in Section 5. Additional examples of calculations are given in Supplementary Information.