The Finite Pore Volume GAB Adsorption Isotherm Model as a Simple Tool to Estimate a Diameter of Cylindrical Nanopores

Abstract

The finite pore volume Guggenheim-Anderson-de Boer (fpv-GAB) adsorption isotherm model has been considered as a simple tool which not only enables us to analyze the shape of isotherms theoretically, but also provides information about pore diameter. The proposed methodology is based on the geometrical considerations and the division of the adsorption space into two parts: the monolayer and the multilayer space. The ratio of the volumes of these two spaces is unambiguously related to the pore diameter. This ratio can be simply determined from the N₂ adsorption isotherm by its fitting with the use of fpv-GAB model. The volume ratio is equal to the ratio of the adsorption capacities in the monolayer and the multilayer-two of the best-fit parameters. The suggested approach has been verified using a series of isotherms simulated inside ideal carbon nanotubes. The adsorption data for some real adsorbents has also been used during tests. The studies performed have proven that diameters estimated with the use of the proposed method are comparable with the geometrical sizes or diameters published by others and based on the application of more sophisticated methods. For pores wider than 3 nm, the relative error does not exceed a few percent. The approach based on the fpv-GAB model reflects well the differences in pore sizes for the series of materials. Therefore, it can be treated as a convenient tool to compare various samples.

Keywords: N2 adsorption; cylindrical pores; isotherm models; type IV isotherm.

Figure 1

(a) The scheme of the division of cylindrical adsorption space into the monolayer and the multilayer. D_eff is the effective diameter and λ is the monolayer thickness; (b) the ratio of monolayer and multilayer volumes plotted as the function of the effective diameter. The diameters are reduced by the monolayer thickness.

Figure 2

Simulated isotherms of N₂ adsorption inside selected carbon nanotubes (CNTs) (points) and their fits by the models: fpv-GAB (Equations (4)–(8))—blue lines and fpv-GAB-li (Equations (5), (6) and (8)–(10))—red lines. Only selected simulated points are shown for clarity. The inserts in lower panels show the low-pressure part of isotherms, related to the formation of N₂ monolayer on the pore wall.

Figure 3

(a) Effective pore diameters D_eff,model calculated using Equation (2) and fpv-GAB (blue circles) or fpv-GAB-li (red triangles) fits of simulated isotherms of N₂ adsorption in CNTs. The diameters are plotted as the function of the geometrical diameter of CNTs (D_eff,geo). The dashed line represents D_eff,model = D_eff,geo; (b,c) the absolute (b) and relative (c) differences between D_eff,model and D_eff,geo. The horizontal dashed lines correspond to D_eff,model = D_eff,geo.

Figure 4

The results of the fitting of experimental N₂ adsorption isotherms by fpv-GAB (Equations (4)–(8)) and fpv-GAB-li (Equations (4)–(6) and (8)–(10)) models for the samples CNHs, Al-MCM-41(30) and MCM-41-16B. The insets show the same data in logarithmic scale of the relative pressure. The points represent experimental data and lines reflect the predictions of the models. Since fpv-GAB-li equation is simplified to fpv-GAB (A ≈ 0) only one theoretical line is plotted for each system. In the case of CNHs and MCM-41-16B only selected experimental points are shown for clarity.