Pore size distribution function

The pore size distribution function (a) appears parametrically in the flux relations of Feng and Stewart, so their models certainly cannot be completely predictive in nature unless this distribution is known. It is... 

For a percolating medium the generalized exponential pore-size distribution function of the scale for porous medium can be written as... 

In this paper we present a new characterisation method for porous carbonaceous materials. It is based on a theoretical treatment of adsorption isotherms measured in wide temperature (303 to 383 K) and pressure ranges (0 to 10000 kPa) and for different adsorbates (N2, CH4, Ar, C3H8 and n-C4Hio). The theoretical treatment relies on the Integral Adsorption Equation concept. We developed a local adsorption isotherm model based on the extension of the Redlich-Kwong equation of state to surface phenomena and we improved it to take into account the multilayer formation. The pore size distribution fimction is assumed to be a bi-modal gaussian. By a minimisation procedure, it is possible to determine the bi-modal pore size distribution function witch can be used for purely characterisation purposes or to predict adsorption isotherms. 

The methods depend on the theoretical treatment which is used. A majority of them are based on the Generalised Adsorption Isotherm (GAI) also called the Integral Adsorption Equation (LAE). The more recent approaches use the Monte Carlo simulations or the density functional theory to calculate the local adsorption isotherm. The analytical form of the pore size distribution function (PSD) is not a priori assumed. It is determined using the regularization method [1,2,3]. Older methods use the Dubinin-Radushkevich or the Dubinin-Astakhov models as kernel with a gaussian or a gamma-type function for the pore size distribution. In some cases, the generalised adsorption equation can be solved analytically and the parameters of the PSD appear directly in the isotherm equation [4,5,6]. Other methods which do not rely on the GAI concept are sometimes used the MP and the Horvath-Kawazoe (H-K) methods are the most well known [7,8]. 

Figures 1 to 4 show the pore size distribution functions (obtained by the H-K and lAE) methods and the comparison between the experimental results and the recalculated isotherms for three of the five adsorbates. The highest mean deviation is 5.66% for nitrogen. Consequently, our characterization method appears to be an efficient modelling tool as it allows to simulate adsorption isotherms of five different adsorbates in wide temperature and pressure conditions (subcritical and supercritical isotherm temperatures) using a unique pore size distribution flmction of the adsorbent.

Figure 1 Pore size distribution functions obtained by the lAE (-)and H-K (+)methods...

Fig. 5 Pore size distribution function of microporous membranes, parameterization with logarithmic normal distribution, for the parameter sets specified in Table 4. (a) Differential psd,...

Figure 3. Pore size distribution functions (mercury porosimetry data) for A) initial CPG (D = 30.5nm) (solid line) and the same material heated for 20 hrs at 650°C (dashed line) 20 hrs at 720°C (dotted line) 20 hrs at 805°C (dash-dotted line), B) initial silica gel (D = 32.8 nm) (solid line) and the same material heated for 20 hrs at 580°C.

Figure 5. Pore size distribution functions (mercury porosimetry data) for CPG (D = (solid line) and sihca gel (D 58.0 nm) (dashed line).

Figure 12. A) SAXS scattering curves, B) size heterogeneity distribution functions Fv = f(D) obtained from SAXS data and C) pore size distribution functions obtained by means of mercury porosimetry for CPG (mean D = 31.8 nm) (white rings or solid line) CPG heated at 650 C over 100 hrs, (mean D = 28.1 nm) (white triangles or dashed line) and CPG heated at 650°C over 100 hrs and then leached to remove borate clusters (mean D = 28.0 nm) (black rings or dotted line).

Figure 16. Differential pore size distribution function of pore radius (R) for the following adsorbents (1) Y, (2)Yh r, (3) Yh-a-...

The probability P(x) can be easily transformed Into the pore size distribution function N(r) of the membrane with the aid of the following approximation In which x Is assumed to be continuous... 

We can conclude the following from an Inspection of Figures 20, 21 and 22. Equation 32 gives an accurate pore size distribution function for the porous polymeric membrane prepared by the microphase separation method. The mean radius Increases and the pore size distribution broadens with S. and Pr. The reduced pore distribution N(r)S vs. r/S curve is Independent of S. but dependent on Pr. The effect of Pr on N(r) Is more remarkable than that of S. The reduced pore size distribution curves widen with an Increase In Pr. 

The pore size distribution function is an important characteristic of a porous solid. Given a pore size distribution J H) and a set of local isotherms pip, H) determined by any methods presented in Sections 11.3—11.5, the overall amount adsorbed is given by... 

Measurement of the porosity of plasma-sprayed ceramic coatings can be accomplished by a wide variety of methods that can be divided into those yielding as a result a simple number, the porosity or pore volume related to the total volume of the coating in cm3 g-1, and those that yield a pore size distribution function. In many cases, the former methods are sufficient to characterise the porosity of a coating. 

Pore Size Distribution. The pore structure is sometimes interpreted as a characteristic pore size, which is sometimes ambiguously called porosity. More generally, pore structure is characterized by a pore size distribution, characteristic of the sample of the porous medium. The pore size distribution/ ) is usually defined as the probability density function of the pore volume distribution with a corresponding characteristic pore size 6. More specifically, the pore size distribution function at 5 is the fraction of the total pore volume that has a characteristic pore size in the range of 5 and 5 + dd. Mathematically, the pore size distribution function can be expressed as... 

It was shown [150] that the applicability of the BEurett, Joyner and Halenda (BJH) computational method based on the Kelvin equation could be extended significantly towards small mesopores and large micropores when a proper t-curve was used to represent the fihn thickness of nitrogen adsorbed on the carbon surface. The t-curve proposed in the work [150] gave the pore-size distribution functions for the carbons studied that reproduce the total pore volume and show realistic behavior in the range at the borderline between micropores and mesopores. 

In equation (21-6) for the void fraction Sp, the pore-size distribution function is given by /(r), and fir) dr represents the fraction of the total volume of an isolated catalytic peUet with pore radii between r and r +dr. This is not a normalized distribution function because... 

However, the pore-size distribution function /(r) is not normalized, so the zeroth moment must be included in the expression for the average pore radins ... 

Kowalczyk, P. et al., Estimation of the pore size distribution function from the nitrogen adsorption isotherm. Comparison of density functional theory and the method of Do and co-workers. Carbon. 2003,41(6), 1113-1125. 

Many processes and structures that are difficult to describe by means of traditional Euclidean geometry can thus be precisely characterized using fiactal geometry, for example the complex and disordered microstructures of advanced materials, adsorbents, polymers and minerals. Recent studies have shown that using fiactal dimensions enables the real sizes of pore radii to be determined and pore-size distribution functions to be calculated from the data of programmed thermodesorption of liquids [35],... 

Fig. 9. Pore-size distribution functions of Na- (left) and La-montmorillonite (right) samples from the porosimetry technique.

From such measurements, surface areas (normalized cumulative and relative), pore radii (choice of three measuring units), pore volumes (raw, normalized, cumulative and relative) and pore-size distribution functions of samples can calculated. Figure 8 presents the graphs of mercury-penetrated volume versus pressure in pores of Na- and La-montmorillonite samples. Figure 9 shows pore-size distribution functions from porosimetry data. 

Figure 14. Isotherm of nitrogen adsorption and desorption (left side) and pore-size distribution functions of sample A5 (right side).

Figure 21 presents the graphs of mercury-penetrated volume versus pressure in the pores of Na- and La-montmorimllonitamples. Figure 22 shows the pore-size distribution functions calculated from the porosimetry data. 

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