Characterization of nanoporous materials from adsorption and desorption isotherms

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Abstract

We present a consistent method for calculation of pore size distributions in nanoporous materials from adsorption and desorption isotherms, which form the hysteresis loop H1 by the IUPAC classification. The method is based on the nonlocal density functional theory (NLDFT) of capillary condensation hysteresis in cylindrical pores. It is implemented for the nitrogen and argon sorption at their boiling temperatures. Using examples of MCM-41 type and SBA-15 siliceous materials, it is shown that the method gives the consonant pore size distributions calculated independently from the adsorption and desorption branches of the sorption isotherm. The pore size distributions, pore volumes and specific surface areas calculated from nitrogen and argon data are consistent. In the case of SBA-15 materials, the method evaluates also the amount of microporosity. The results of the NLDFT method are in agreement with independent estimates of pore sizes in regular nanoporous materials.

Introduction

Capillary condensation isotherms of vapors are widely used for calculations of pore size distributions (PSD) in nanoporous materials [1]. However, the standard methods for PSD calculations, such as the Barrett–Joiner–Halenda (BJH) method [2], are based on a simplified macroscopic description of the capillary condensation [3], which limitations at the nanoscale were repeatedly discussed in the literature [4], [5], [6], [7], [8], [9], [10], [11]. Albeit extensive theoretical studies of the confined fluids (see [12], [13], [14] for reviews), the lack of porous solids with well-defined structures hampered the direct assessment of the accuracy of the adsorption models and, consequently, the methods for PSD analysis.

Recent progress in synthesis of nanoporous materials [15], [16], [17], [18] lead to the development of adsorbents with highly uniform morphologies. The unique structure of MCM-41 and related materials with cylindrical pores of controlled size provides a long-desired opportunity for testing the theoretical models of capillary condensation against reliable experiments [10], [19], [20], [21], [22]. Experimental studies with mesoporous molecular sieves highlighted severe limitations of the standard methods for nanopore structure analysis, and prompted the development of new models and characterization methods [10], [11], [20], [23], [24], [25], [26], [27], [28], [29], [30], [31].

In this work, we focus on the theoretical description of the capillary condensation in nanoporous materials using nonlocal density functional theory (NLDFT). Adsorption–desorption hysteresis presents another well-known problem for the PSD analysis from experimental isotherms. Which branch of the experimental adsorption–desorption isotherm should be taken for PSD calculations? Here, we present a consistent method for PSD calculations in nanoporous materials from adsorption and desorption isotherms, which form the hysteresis loop H1 by the IUPAC classification [1]. The method is implemented for N2 adsorption at 77.4 K and for Ar adsorption at 87.3 K. The method is validated using experimental data on MCM-41 type and SBA-15 siliceous materials. The paper is organized as follows. In Section 2, we give a short description of the NLDFT model and discuss the correlations between the theoretical and experimental isotherms. In Section 3, we describe the method for PSD calculations with a special emphasis on the construction of the kernel of theoretical isotherms. In Section 4, we present several examples of PSD calculations and compare the NLDFT model with other methods. In Section 5, we offer recommendations on the application of the developed method to the PSD analysis from experimental adsorption and desorption isotherms.

Section snippets

Model

In the density functional theory [32], the local density, ρ(r), of the adsorbate confined in a pore at a chemical potential, μ, and temperature, T, is calculated by minimization of the grand thermodynamic potential, Ω:Ω[ρ(r)]=FHS[ρ(r)]+12∫∫rdr′ρ(r)ρ(r′)Φattr(|r−r′|)−∫drρ(r)[μ−Uext(r)]

To model the N2 and Ar adsorption–desorption isotherms in cylindrical pores, we employed Tarazona's version of the nonlocal density functional [33], FHS[ρ(r)], and Weeks–Chandler–Andersen [34] treatment of

The NLDFT method for pore size distribution calculations from adsorption and desorption isotherms

To calculate the pore size distributions, the experimental isotherm was represented as a combination of theoretical isotherms in individual pores, which is described by the Integral Adsorption Equation (IAE) [1]:NexpP/P0=DminDmaxNVexDin,P/P0ϕV(Din)dDinwhere NVex(Din, P/P0) is a kernel of the theoretical isotherms in pores of different diameters, ϕV(Din) is the pore size distribution function.

NLDFT pore size distribution calculations from adsorption and desorption branches

Fig. 4 presents a prominent example of PSD calculations for the MCM-41-like material [43], for which the isotherms exhibit H1 hysteresis loop. Calculations were performed from N2 and Ar adsorption and desorption isotherms. Good agreement between the results obtained from two gases and two branches of the isotherms is evident from the differential (Fig. 4 top) and cumulative PSDs (Fig. 4 bottom). The fit of the experimental isotherms is presented in Fig. 5.

Comparison with the macroscopic methods of PSD analysis

In our earlier publications it has been

Conclusions and recommendations

We developed a method for consistent pore size characterization of siliceous nanoporous materials with cylindrical pores from nitrogen and argon adsorption and desorption isotherms. The kernels of equilibrium capillary condensation isotherms and the kernels of metastable adsorption isotherms of N2 at 77 K and Ar at 87 K in individual pores (2–100 nm) were calculated by means of the nonlocal density functional theory (NLDFT). Comparison of the theoretical adsorption–desorption isotherms in

Acknowledgements

This work is supported in parts by the TRI/Princeton exploratory research program, EPA grant R825959-010, and Quantachrome Corp. We thank A. Gédéon for the table data of isotherms on SBA-15 materials.

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