Gaussian pore size distribution

By assuming a Gaussian pore size distribution, Stoeckli was able to simplify... 

In order to estimate the pore size distributions in microporous materials several methods have been developed, which are all controversial. Brunauer has developed the MP method [52] using the de Boer t-curve. This pore shape modelless method gives a pore hydraulic radius r, which represents the ratio porous volume/surface (it should be realised that the BET specific surface area used in this method has no meaning for the case of micropores ). Other methods like the Dubinin-Radushkevich or Dubinin-Astakov equations (involving slitshaped pores) continue to attract extensive attention and discussion concerning their validity. This method is essentially empirical in nature and supposes a Gaussian pore size distribution. 

In the Dubinin-Stoeckli (DS) method, a Gaussian pore size distribution is assumed for 7(B) in Eq. (39), based on the premise that for heterogeneous carbons, the original DR equation holds only for those carbons that have a narrow distribution of micropore sizes. This assumption enables Eq. (39) to be integrated into an analytical form involving the error function [119] that relates the structure parameter B to the relative pressure A = -RT ln(P/Po)-The structure parameter B is proportional to the square of the pore halfwidth, for carbon adsorbents that have slit-shaped micropores. 

Fig. 5 Relations between with (solid line) and without (broken line) considering the wall effect, for logarithmic Gaussian pore size distributions. The shapes of the two curves get closer to each other when the standard deviation (o ) increases. The limiting displacement factor is ca. exp(1.40)=4.1.

Suppose a membrane has a Gaussian pore size distribution of 7 x and... 

On the assumption that the pore size distribution is Gaussian, Dubinin and Radushkevich arrived at the expression... 

Gel filtration chromatography has been extensively used to determine pore-size distributions of polymeric gels (in particle form). These models generally do not consider details of the shape of the pores, but rather they may consider a distribution of effective average pore sizes. Rodbard [326,327] reviews the various models for pore-size distributions. These include the uniform-pore models of Porath, Squire, and Ostrowski discussed earlier, the Gaussian pore distribution and its approximation developed by Ackers and Henn [3,155,156], the log-normal distribution, and the logistic distribution. 

The effects of various pore-size distributions, including Gaussian, rectangular distributions, and continuous power-law, coupled with an assumption of cylindrical pores and mass transfer resistance on chromatographic behavior, have been developed by Goto and McCoy [139]. This study utilized the method of moments to determine the effects of the various distributions on mean retention and band spreading in size exclusion chromatography. 

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]. 

Bimodal Gaussian distribution For solids having bimodal pore size distribution, we can use a combined Gaussian distribution as shown below. 

Krajeswska, B. and Olech, A. (1996). Pore structure of gel chitosan membranes II. Modelling of the pore size distribution Irom solute diffusion measurements. Gaussian distribution-Mathemat-ieal limitations. Polymer Gels Networks 4,45. 

The Gaussian normal distribution is used for the pore size distribution. Then, Equation 6.59 can be approximated by... 

Three commercial activated carbons were used BPL, CAL and GAe, manufactured by Chemviron, Calgon and CECA respectively. In addition, sample GAe-oxl was prepared by oxidation of GAe in aqueous solution of (NH4)2S20g and further pyrolysis in N2 flow at 773 K [5]. The specific surface areas were obtained applying the BET and Dubinin-Asthakov equations to the adsorption of N2 at 77 K and CO2 at 273 K respectively. Moreover, the C02 adsorption data permitted the evaluation of the micropore size distributions and the mean value of pore width using the Dubinin-Stoeckli equation [6] which supposes a gaussian distribution of pore sizes. 

Despite the difficulties of determining the actual nature of the asphaltenes, Sughrue et al. [39] have used size exclusion chromatography to determine molecular sizes of molecules associated with vanadium. As would be expected, the size varies from feed to feed, but - in general - a small amount of vanadium is contained in molecules of ca 3 nm diameter, while most is contained in molecules with a Gaussian size distribution centred on 8-10 nm diameter species. Given the relative diameters of the molecules (ca 8-10 nm) and the pores (ca 20 nm) it is not surprising that vanadium does not penetrate far into the pellets. 

The predictions of the model are normally expressed in terms of a curve of p versus (7OTN /Ti)(r - to), referred to as the reduced time. The curve is obtained as follows For a chosen value of p the parameter y is found from Eqs. (8.53) and (8.57). The function F/y) is then found from Eq. (8.59) and the reduced time is obtained from Eq. (8.58). The procedure is repeated for other values of p. The predictions of the model are shown by the full curve in Eig. 8.16. The curve has the characteristic sigmoidal shape observed for the density versus time data in many sintering experiments. The model has been extended by considering a Gaussian or a bimodal distribution of pore sizes (21). 

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