Dear Quahiba,

Attached please find two ppt and one pdf files. in addition, I have copied a text from a paper entitled " Pore Size Analysis by Gas Adsorption and the Density Functional Theory – Quantachrome Instruments" however, there was no possibility to copy the figures since the paper is available only in HTML format.

These publications cover the whole project and answer your question.  

Source: Pore Size Analysis by Gas Adsorption

For more information on this source, please visit Quantachrome Instruments.

Date Added: Apr 26, 2010 | Updated: Jun 11, 2013

Topics Covered

Introduction

Pore Size Analysis by DFT and Monte-Carlo Simulation Methods in Comparison with Classical, Macroscopic Methods like BJH, SF etc.

DFT and Monte Carlo Simulation Methods: A Single Method for a Combined Micro/Mesopore Analysis as HTML format.

DFT Methods in Quantachrome’s Data Reduction Software

Summary and Conclusions

Introduction

Gas adsorption is a prominent method to obtain a comprehensive characterization of porous materials with respect to the specific surface area, pore size distribution and porosity. This requires, however, a detailed understanding of the fundamental processes associated with the sorption and phase behavior of fluids in porous materials and their influence on the shape of sorption isotherms, which serves as a basis for surface and pore size analysis. Pore width, pore shape and the effective adsorption potential are the factors that determine the pore filling. In case of so-called micropores (pore width < 2 nm, according to IUPAC classification) the pore filling occurs in a continuous way, whereas in case of mesopores (pore widths in the range from 2 nm – 50 nm) pore filling occurs by pore condensation, which reflects a first order gas-liquid phase transition.

So-called classical macroscopic, thermo-dynamic concepts are based on the assumption of a certain pore filling mechanism. Methods based on the Kelvin equation (e.g. BJH-method) are linked to the pore condensation phenomena, i.e, they are applicable for mesopore size analysis, but they fail to describe the pore filling of micropores and even narrow mesopores in a correct way. Other classical theories, like for instance the Dubinin-Radushkevich approach, and semiempirical treatments such those of Horvath and Kawazoe (HK), and Saito and Foley are dedicated to describe micropore filling but cannot be applied for mesopore size analysis. Hence, in case a material contains both, micro- and mesopores, at least two different methods have to be used to obtain the pore size distribution(s) from such an adsorption/ desorption isotherm. In addition the accuracy of such thermodynamic, macroscopic methods is limited, because of the assumption that the pore fluid has similar thermophysical properties as the bulk fluid. Recent theoretical and experimental work has shown, that the thermodynamic properties of a confined fluid can be considerably different as compared to the bulk fluid (shifts in critical point, freezing point, and triple point etc.)

In contrast to these macroscopic approaches, methods like the Density Functional Theory (DFT) or methods of molecular simulation (Monte Carlo simulation methods (MC), Molecular Dynamics methods (MD)) provide not only a microscopic model of adsorption but also a more realistic description of the thermodynamic properties of the pore fluid. These theories, which are based on statistical mechanics, connect macroscopic properties to the molecular behavior. Therefore, in order to achieve a more realistic description of adsorption phenomena and an accurate and comprehensive pore size analysis, methods such as the DFT of inhomogeneous fluids and Monte Carlo simulations, which bridge the gap between the molecular level and macroscopic approaches are needed. These methods allow to calculate equilibrium density profiles of a fluid adsorbed on surfaces and in pores from which properties such as the adsorption/desorption isotherm, heats of adsorption, neutron scattering patterns and transport properties for model systems can be derived. The density profiles obtained by applying MC simulations and DFT-theory are of course based on the intermolecular fluid-fluid and fluid-solid interactions used in the calculations. The parameters of the fluid-fluid interactions are determined in a way that they allow to reproduce the bulk properties (e.g., of nitrogen and argon at low temperatures). The parameters of the solid-fluid interactions can then be obtained by fitting the calculated adsorption isotherms on a planar surface to the standard nitrogen and argon isotherms.

The most prominent computer simulation method for the study of adsorption phenomena etc. is the Grand Canonical Monte Carlo simulation method (GCMC), which simulates an open system at fixed temperature T, volume V and chemical potential µ. This technique simulates the situation of an adsorbed fluid (or mixture) in equilibrium with a bulk fluid reservoir, which reflects usually the situation encountered in experimental studies of confined systems. A random number generator is used to move and rotate the molecules in a random fashion, which leads to particular configurations. Such movements and the resulting configurations are then accepted or rejected according to thermodynamic criteria (i.e, based on the temperature and chemical potential). After generating a long sequence of such moves (so-called Markov chain, typically in the order of several millions), they can be averaged (based on equations of statistical mechanics) to obtain the equilibration density profiles, and hence the adsorption isotherm.

Applying the DFT approach allows calculating the equilibration density profile for all locations in the pore. The equilibrium density profiles are obtained by minimizing a free-energy functional, which is the grand potential or grand free energy for a pore system in equilibrium with a bulk phase (i.e. the situation when an adsorption experiment is performed). This free-energy potential consists also of terms that describe the attractive and repulsive parts of the fluid- fluid and fluid-wall interactions. Difficulties are associated with a proper description of fluid-fluid interactions and also because of this, different DFT-approaches were suggested during the last decade, like for instance the so-called Local Density Functional Theory (LDFT) and the Non-Local Density Functional Theory (NLDFT). The LDFT approach is often used, but is not able to produce the strong oscillation characteristics of a fluid density profile at a solid fluid interface, which leads in particular for narrow micropores pores, to an inaccurate description of the sorption isotherms, and correspondingly to an inaccurate pore size analysis. In contrast, the Non-Local Density Functional Theory and the Monte Carlo Computers simulation techniques provide a more accurate structure of a fluid confined to narrow pores. Such characteristic, oscillating, density profiles are shown in figure 1. These density profiles refer to coexisting gas- and liquid states of a fluid in a slit-like mesopore. The density of coexisting gas (circles) and liquid (squares) is shown as a function of distance to the pore walls. The layering close to the pore walls reflects multilayer adsorption. The density decreases with increasing distance to the pore walls. The density profiles shown in figure 1 also clearly indicate, that pore condensation essentially occurs in the core of the pore, which leads in the case of larger mesopores (as here, pore width is 20 molecular diameters) to a bulk-like core liquid as indicated by essentially no oscillations of the density profile in the core region of the pore.

Figure 1. Density Profiles of coexisting gas (circles)- and liquid densities (squares) in a slit pore of pore width 20σ, where σ denotes the molecular diameter.

Pore Size Analysis by DFT and Monte-Carlo Simulation Methods in Comparison with Classical, Macroscopic Methods like BJH, SF etc.

As indicated above, the Non-Local Density Functional Theory (NLDFT) and the Grand Canonical Monte Carlo simulation (GCMC) methods correctly describe the local fluid structure near curved solid walls; adsorption isotherms in model pores are determined based on the intermolecular potentials of the fluid-fluid and solid-fluid interactions. The relation between isotherms determined by these microscopic approaches and the experimental isotherm on a porous solid can be interpreted in terms of a Generalized Adsorption Isotherm (GAI) equation:

Where:

N(P/P0) = experimental adsorption isotherm data

W = pore width

N(P/P0,W) = isotherm on a single pore of width W

f(W) = pore size distribution function

The GAI equation reflects the assumption that the total isotherm consists of a number of individual “single pore” isotherms multiplied by their relative distribution, f(W), over a range of pore sizes. The set of N(P/P0W) isotherms (kernel) for a given system (adsorbate/adsorbent) can be obtained, as indicated above, by either Density Functional Theory or by Monte Carlo computer simulation. The pore size distribution is then derived by solving the GAI equation numerically via a fast non-negative least square algorithm.

The validity of the DFT and Monte Carlo methods could be assessed by comparing the DFT/Monte Carlo pore size distribution obtained form sorption isotherms obtained in so-called mesoporous molecular sieves (e.g., MCM 41, which consist of an array of independent pores) with the results of other, independent methods like XRD, TEM etc, which are applicable in order to obtain the pore size from these materials. On the other hand, the macroscopic, thermodynamic methods (e.g., Saito Foley, BJH etc..) cannot provide a correct correlation between the pore size and filling pressure for micro- and narrow mesopores. This is illustrated in figures 2 and 3.

Figure 2. Pore filling pressures for nitrogen in cylindrical oxide pores at 77 K, as predicted by DFT, Saito-Foley equation (SF), Kelvin equation (K) and Gibbs Ensemble Monte Carlo simulations (points).

Figure 3. The pore size dependence of the relative pressure of equilibrium condensation/desorption transition for N2 in cylindrical pores at 77 K.

According to the results shown in figures 2 and 3, one expects, that classical methods underestimate the pore diameter up to ca. 20 %. This is illustrated in figures 4a and 4b.

Figure 4a. Nitrogen sorption isotherm at 77 K in a MCM 41 material, comparison between experimental and NLDFT isotherm.

Fig. 4a shows a nitrogen sorption isotherm obtained at 77 K in a MCM-41 silica material in comparison with the theoretical NLDFT-isotherm. Pore size distributions obtained by applying the BJH-method and the NLDFT theory are shown in figure 4b. The widths of the pore size distribution curves is similar in both cases, but the mode diameter obtained by applying the BJH theory is ca. 10 Angstrom smaller than compared to the NLDFT result.

Figure 4b. Pore size distribution calculated from the nitrogen/MCM-41 isotherm shown in Fig.4a by applying the NLDFT and the BJH method.

DFT and Monte Carlo Simulation Methods: A Single Method for a Combined Micro/Mesopore Analysis

As indicated before, a great advantage of DFT- and Monte Carlo simulation methods is that they can be used for a combined micro-mesopore analysis. This is illustrated by the following data: Figure 5 shows argon sorption isotherms obtained at 87 K on ZSM 5, a zeolite with a cylindrical pore geometry, on MCM 41, and on a mixture of ZSM5 and MCM 41, which resembles a combined micro/mesoporous material. In order to obtain the pore size distribution, Quantachrome’s NLDFT hybrid kernel for Ar-zeolite/silica at 87 K is applied, which is dedicated to describe the adsorption of argon at liquid argon temperature on siliceous materials of cylindrical pore geometry. This kernel can be used to determine the pore diameter of pores in the range between 0.35 nm and 100 nm, i.e., it allows a combined micro/mesopore analysis using a single method.

Figure 6 compares the experimental isotherm with the calculated DFT isotherm; the fit to the isotherm is excellent. The NLDFT pore size distribution of the “combined” material (see figure 7) shows two distinct groups of pores: micropores of the same size as in ZSM-5 and mesopores of the size as in MCM-41. Please note, that the reported average pore diameter of ZSM-5 zeolite obtained from structural considerations is 0.51-0.55 nm, which agrees very well with the pore size distribution obtained from argon adsorption by the NLDFT method The pore size obtained by independent methods (XRD) for the mesoporous MCM 41 is 3.2 nm, which is again in excellent agreement with the result obtained with the NLDFT method.

Figure 5. Argon adsorption isotherms at 87 K on MCM-41, ZSM-5 and their mixture.

Figure 6. NLDFT fit of the isotherm on a mixture of ZSM-5 and MCM-41 materials.

Figure 7. Pore size distribution of a mixture of ZSM-5 and MCM-41 materials.

DFT Methods in Quantachrome’s Data Reduction Software

Quantachrome’s data analysis software contains a comprehensive library of DFT-and M.C. methods, which allow a micro/ mesopore analysis of carbon, zeolites and siliceous materials. These methods were developed for Quantachrome by the authors Prof. A.V Neimark and Dr. P. Ravikovitch. The basis of these methods are kernels consisting of individual N(P/P0W) isotherms derived for the system: nitrogen-carbon, argon-carbon, carbon dioxide-carbon, nitrogen-silica, and argon-zeolite/silica by the Non-Local Density Functional Theory and Grand Canonical Monte Carlo computer simulation. Please find below a summary of the kernels implemented in the Autosorb and NOVAWin software, and the pore diameter range, where these methods can be applied.

Please note, that in the case of a GCMC-CO2-carbon kernel a three-center potential function has been developed with interactions between the sites of different molecules modeled as a sum of Lennard-Jones and electrostatic contributions. Hence, the GCMC model may serve as a benchmark for quantitative estimates. The CO2-NLDFT-kernel is based on a common, one-center Lennard-Jones model; this kernel is in particular important in case a comparison with appropriate DFT-results in the literature is needed.

As indicated in table 1, the nitrogen-silica NLDFT and argon-zeolite/silica models implemented in the Autosorb software allow for an accurate pore size analysis from both, adsorption and desorption branches of the sorption hysteresis loop. The phenomenon of sorption hysteresis and the consequences for pore size analysis will be discussed in a separate Technical Note.

Table 1: NLDFT / GCMC Kernels available in Autosorb and NOVAWin software

NLDFT / GCMC Kernel File

Applicable Pore Diameter Range

NLDFT – N2 - silica at 77 K, based on a cylindrical pore model. In case of sorption hysteresis pore size analysis is possible from both adsorption and desorption branches of the hysteresis loop.

1.8 nm - 100 nm

NLDFT – N2 - carbon at 77 K based on a slit-pore model

0.35 nm - 8 nm

NLDFT - Ar-zeolite/silica at 87 K, based on a cylindrical pore model. In case of sorption hysteresis pore size analysis is possible from both adsorption and desorptionbranches of the hysteresis loop.

0.35 nm - 100 nm

NLDFT – Ar - carbon at 77 K based on a slit-pore model

0.35 nm - 8 nm

NLDFT – co2 - carbon at 273 K based on a slit-pore model

0.35 nm - 1.5 nm

GCMC – co2 - carbon at 273 K based on a slit-pore model

0.35 nm - 1.5 nm

Summary and Conclusions

DFT and Monte Carlo simulation (MC) methods provide a microscopic and accurate description of fluids in confined geometries in contrast to macroscopic, thermodynamic approaches like the methods of Barrett-Joyner-Halenda, Dubinin-Radushkevich etc.

⇒In contrast to the classical, macroscopic approaches, the DFT and MC methods allow to obtain a more accurate pore size analysis for narrow micro- and mesopores

No assumptions are necessary concerning the nature of pore filling (e.g., first order phase transition or continuous pore filling). The equilibration density profiles of the confined fluid at given temperature and pressure can be directly obtained by DFT and MC calculations.

⇒DFT and MC methods can be applied to obtain a pore size analysis over the complete micro-and mesopore diameter range by using a single method.

The Quantachrome Autosorb and NOVA software contains in addition to the commonly used classical methods a unique library of NLDFT and Monte Carlo kernels, which can be applied for pore size analysis of micro- and mesoporous silica, active carbons, and zeolites etc. based on sorption isotherms obtained with nitrogen, Argon and carbon dioxide as adsorptive.

⇒Quantachrome’s data reduction software allows a state of the art pore size and surface area analysis based on gas adsorption/desorption data.

A complete list of references is available by referring to the source document.

Source: Pore Size Analysis by Gas Adsorption

For more information on this source, please visit Quantachrome Instruments.

Date Added: Apr 26, 2010 | Updated: Jun 11, 2013

Hoping this will be helpful,

Rafik

 thank you for this best information

Salam,

Basically BJH method is used only for the determination of mesopores distribution, but DFT method can used for micro and mesopores distribution. Others methods can be also used for exemple  t-plot method.

Best regards

dear salim Bousba thank you 

I have attached a pdf of the paper reproduced by Rafik.

For the casual reader who might come across this thread, I'd just like to add a very short and simple differentiation:

BJH is based on the Kelvin equation (aka Young-Laplace) which relates the pressure to the radius of curvature (pore size) of a liquid meniscus... albeit modified by taking into account the thickness of the adsorbed film of adsorbate. In large mesopores this is approximately reasonable (notwithstanding hysteresis which is not modeled in BJH), but in small mesopores (as in the material known as MCM-41 for example) the error of this simplistic model can be very significant. And for micropores (in which no liquid meniscus exists) it is totally inappropriate.  

DFT (Density Functional Theory) on the other hand is based on molecular modelling and takes into account direct interaction of adsorbate with the adsorbent surface, micropore filling process (not a condensation process) the development of the adsorbed film thickness, and capillary condensation (adsorption) and capillary evaporation (desorption). Hence, it is able to model hysteresis in the ads/des mesopore region of the isotherm.

Agree with Prof. Rafik, Prof. Salim and Dr. Martin. You should follow them.

thank you  rof. Rafik, Prof. Salim and Dr. Martin

@Dr. Martin

Does DFT model consider desorption branch also while doing pore size distribution analysis like BJH model where we can do the PSD analysis using either of the branch?

regards

raeesh

@Raeesh,

Yes, DFT can calculate same pore size distribution from adsorption and desorption. The only caveat is that the mesopore hyteresis should be of type H1. If there is pore blocking effect then adsorption is used. 

@Dr. Martin

Thank you. How can we measure the PSD analysis separately from adsorption and desorption branch of isotherm using ASiQwin software supplied by Quantachrome with autosorb instrumnet.

@Raeesh,

In "data reduction parameters" for the sample file already open (get there with right mouse click) select the DFT tab and choose a DFT model based on gas, temperature, sample type, expected pore shape, and adsorption or "equilibrium". Equilibrium  = desorption. Then you need to select the graph and table from the dropdown list. If you are unsure of pore shape, just try one then look at the fitting error. Try a different shape model. It should be obvious which model to choose from the lowest fitting error. 

Obtan PSD for adsorption, then go back and change the same model to equilibrium and the software will automatically recalculate. 

I can assist directly if you contact me privately and attach data file of interest. 

Thank you very much Dr. Martin.

regards

raeesh