Fractal pore volume

Characteristic for a fractal structure is self-similarity. Similar to the mentioned pores that cover all magnitudes , the general fractal is characterized by the property that typical structuring elements are re-discovered on each scale of magnification. Thus neither the surface of a surface fractal nor volume or surface of a mass fractal can be specified absolutely. We thus leave the application-oriented fundament of materials science. A so-called fractal dimension D becomes the only absolute global parameter of the material. 

Specimen BET surface area - I l .rr Micro- Meso- Macro- porosity" porosity porosity" Average pore diameter, Dave Standard Pore radius of the deviation of maximum differential PSD, a pore volume, Surface fractal dimension, tfsurf,SP Low-end thickness, High-end thickness, ad. max... 

Since these centers are distributed in the pore volume, the excitation transmits through the volume and is not related to the hydration centers located on the pore surface of the connective pores. Due to the large number of hydration centers, and the short distance between the neighboring centers, the path can be approximated by a line with a fractal dimension close to unity (see Fig. 29a). 

The objective of this research work was to use the surface fractal theory to analyze the adsorbent from industrial sludge at various conditions. Effects of activated temperature, activated time and activated agent on the specific surface area and pore volume surface of adsorbents were investigated. 

In addition to surface fractal dimension ( Mercury Intrusion Porosimetry, under Boundary and Surface Fractal Dimensions ), this method can also be employed to determine mass fractal dimension of porous particles. Once the relative density of the particle at different pore volume, p, is obtained, then Dm can be deduced according to Eq. (20) ... 

N2 adsorption is also used to estimate the micro pore volume and the pore size distribution (see e. g. Glasauer et al. 1999) whieh ean be derived from a plot of adsorbed N2 vs. the thiekness of a statistieal monolayer, t, whieh is a function of the relative gas pressure (t-plot method). Mereury porosimetiy serves the same purpose (Celis et al. 1998). N2 adsorption isotherms have also been used to determine the fractal dimensions of Fe oxide particles (c. f.. Celis et al. 1998 Weidler et al. 1998). 

The surface property of a solid is characterized by the nature of the surface boundary. The surface boimdary is expected to be related to the underlying geometric nature of the surface, hence its fractal dimension. Many properties of the solid depend on the scaling behavior of the entire solid and of the pore space. The distribution of mass in the porous solid and the distribution of pore space may also reflect the fractal nature of the surface. If the mass and the surface scale are alike, that is, have the same power-law relationship between the radius of a particle and its mass, then the system is referred to as a mass fractal. In a similar manner if the pore volume of porous material has the same power-law relationship between the pore volume and radius as that of the surface, then it is described as a pore fractal. 

The surface area, pore volume, and pore size of the deposited film depend on such factors as the size and structure (fractal dimension) of the entrained inorganic species, the relative rates of condensation and evaporation during deposition, and the magnitude of the capillary pressure (122). The fractal dimension influences porosity through steric control. Mandelbrot (47) showed that if two objects of radius R are placed independently in the same region of space, the number of intersections (Mi,2) is expressed as... 

Fractal dimensions Fd were used for the calculation of the pore volume distribution functions in relation to their radii from equation ... 

ABSTRACT Based on low-temperature nitrogen adsorption principle, the pore structure of coal particles is tested and adsorption isotherms of coal particles with different size are obtained by Quantachrome Autosorb-iQ automatic specific surface area and pore size distribution analyzer. Then, microstructure characteristic parameters such as specific surface area, pore volume and average pore size of coal particles are calculated. Besides, fractal dimension of the internal surface of coal particles is calculated with FHH fractal theory. The relationship between fractal dimension and pore structure parameters together with the adsorption capacity of coal particles is analyzed. Studies show that fractal dimension can characterize the variation of characteristic parameters such as specific surface area and total pore volume of coal particles. In addition, with the increase of fractal dimension, the surface heterogeneity of pore structure is strengthened and so is adsorption capacity. The findings can provide a certain theoretical foundation for mechanism study on coal gas adsorption, desorption and seepage. 

It can be known from Fig. 4 and Fig. 5 that the fractal dimension of coal particles increase with the increase of coal surface area and total pore volume. The increase of surface area inside coal... 

Figure 5. Correlation between fractal dimension and total pore volume.

Environmental particles, microbial colonies, and even patterns of movement of organisms can be characterized in terms of several different fractal dimensions [14] (Table 1.1). The fractal dimension of the surface Ds (boundary/interface) of a solid structure is obviously an important characteristic, but many of the physical properties of solids also depend on the scaling behavior of the entire solid and/or of its pores. Systems where surface and mass scale similarly are termed mass fractal systems, those where surface and the pore volumes scale similarly are described as pore fractals and systems where only the surface is fractal are designated as surface fractals (Table 1.1 Figure 1.2). 

Notes The specific surface area (in mVg) and the pore volumes (in cmVg) of nanopores (Sna o and at radius for the model of cylindrical pores for LiChrolut EN (or half-width for carbon adsorbents) R<1 nm, mesopores (S eso and V eso) at 1 < / < 25 nm and macropores (5 acro and V acro) at / > 25 nm were determined by integration of the fs(R) and/v(/ ) functions, respectively. The fractal dimension Djj2 and D 2o values were determined from the nitrogen and water adsorption isotherms. The Aw value determines the average error of the model of pores due to roughness of the pore walls. 

In all the cases examined so far, it is the matter distribution of the object that has exhibited the property of self-similarity. These objects are called mass fractals. Other situations are encountered, where it is not the matter distribution which has self-similarity, but rather the pore distribution in these cases, we speak of pore volume fractals. Some structures are found in which only the contour or surface manifests scale invariance these are called boundary or surface fractals, and the exponent we need to know is the boundary fractal dimension. To obtain the corresponding exponents, we calculate the autocorrelation function, the mass distribution or the number of boxes, restricting ourselves to the relevant subsets (the points occupied by matter, the points in the pore volume, or the points lying in the interface). 

From the data in Table 12.2, the bulk density values p( ) and pore volumes of the fractal aggregates can be calculated for the different samples studied. From the bulk density, one obtains the pore volume defined as... 

Figure 0.15. (a) Aggregation model for PEGeoo-mediated silica at PBO/ silica ratios 0.5 and 2.0. (b) Schematic presentation of the relation of PEO/silica ratio and hydrophobicity (which increases with size of the PEO monomer) and the physicochemical characteristics of the silica formed specific surface area (ssa in m g ) determined by N2 sorption-desorption, fractal dimension (Ds) determined by SAXS, and pores diameter (d) determined from SAXS and pore volume contributions (derived from sorption-desorption isotherm)... 

Fig. 3 LHS Euclidean macropore array in p-type silicon (Kim et al. 2009) RHS Fractal-like oxide replica of tire pore volume in n-type silicon (Tondare et al. 2008)...

The pore specific surface area an the average radius values could be used to determined the fractal properties of TVEX resins following the mathematical models describing the disorder and polydispersity of materials. The fractal dimension Df, defines self-similarity of porous materials [8]. It was calculated based on differential function of pore volume distribution of radius [9] ... 

Inspection of die fluid structure obtained from the GCMC simulations ows that the difference between the lower end of the PSDs in Figure 4(c) is in fact caused by SFe not being able to access all the nooks and cranni of individual porra that are accessible to the CF4 rather than entire pores - such variation in accessible pore volume as a fimction of the molecule size is in fact die province of fractals. 

---