Pore size density function

To avoid excessive shrinkage a freeze drying process was used to prepare the partially saturated samples (Delage et al. 1982). Figure 2 shows the measured pore size density functions for the as-compacted state, which correspond to a packing having /C(( = 1.35 Mg/m for pellets having /C(i= 1.95 Mg/m ... 

Figure 2. Pore size density functions for the as-compacted state (pj = 1.35 Mg/m j and for the high-density pellets (Pd = 1.95 Mg/m j.

One approach to define a pore size is in the following way the pore diameter S at a given point within the pore space is the diameter of the largest sphere that contains this point, while still remaining entirely within the pore space. To each point of the pore space such a diameter can be attached rigorously, and the pore-size distribution can be derived by introducing the pore-size density function d S) defined as the fraction of the total void space that has a pore diameter comprised between S and S+dS. This distribution is normalized by the Eq. (16) of Appendix ... 

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]. 

Lastoskie, M., Gubbins, K.E., and Quirke, N. (1993). Pore size heterogeneity and the carbon slit pore A density functional theory model. Langmuir, 9, 2693—702. 

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... 

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. 

Nitrogen adsorption isotherms were measured with a sorbtometer Micromeretics Asap 2010 after water desorption at 130°C. The distribution of pore radius was obtained from the adsorption isotherms by the density functional theory. Electron microscopy study was carried out with a scanning electron microscope (SEM) HitachiS800, to image the texture of the fibers and with a transmission electron microscope (TEM) JEOL 2010 to detect and measure metal particle size. The distribution of particles inside the carbon fibers was determined from TEM views taken through ultramicrotome sections across the carbon fiber. 

Thus, either type I or type IV isotherms are obtained in sorption experiments on microporous or mesoporous materials. Of course, a material may contain both types of pores. In this case, a convolution of a type I and type IV isotherm is observed. From the amount of gas that is adsorbed in the micropores of a material, the micropore volume is directly accessible (e.g., from t plot of as plot [1]). The low-pressure part of the isotherm also contains information on the pore size distribution of a given material. Several methods have been proposed for this purpose (e.g., Horvath-Kawazoe method) but most of them give only rough estimates of the real pore sizes. Recently, nonlocal density functional theory (NLDFT) was employed to calculate model isotherms for specific materials with defined pore geometries. From such model isotherms, the calculation of more realistic pore size distributions seems to be feasible provided that appropriate model isotherms are available. The mesopore volume of a mesoporous material is also rather easy accessible. Barrett, Joyner, and Halenda (BJH) developed a method based on the Kelvin equation which allows the calculation of the mesopore size distribution and respective pore volume. Unfortunately, the BJH algorithm underestimates pore diameters, especially at... 

In the main body, this sechon presents recently employed mesoscale computational methods that can be uhlized to evaluate structural factors during fabrication of PEMs. These simulations provide density distributions or maps and structural correlahon functions that can be employed to analyze the sizes, shapes, and connectivihes of phase domains of water and polymer the internal porosity and pore size distributions and the abundance and wetting properties of polymer-water interfaces. 

Comparison of the pore size distribution determined by the present method with that from the classical methods such as the BJH, the Broekhoff-de Boer and the Saito-Foley methods is shown in Figure 4. Figure 5 shows a close resemblance of the results of our method with those from the recent NLDFT of Niemark et al. [16], and XRD pore diameter for their sample AMI. The results clearly indicate the utility of our method and accuracy comparable to the much more computationally demanding density functional theory. There are several other methods published recently (e. g. [21]), however space limitations do not permit comparison with these results here. It is hoped to discuss these in a future publication. 

The formalism of nonlocal functional density theory provides an attractive way to describe the physical adsorption process at the fluid - solid interface.65 In particular, the ability to model adsorption in a pore of slit - like or cylindrical geometry has led to useful methods for extracting pore size distribution information from experimental adsorption isotherms. At the moment the model has only been tested for microporous carbons and slit - shaped materials.66,67 It is expected that the model will soon be implemented for silica surfaces. 

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