Texture analysis, equation

If the quantitative texture analysis is not of interest the sample is not rotated on a goniometer and only one or a small number of patterns are recorded. Because the number of points in the space (T, y) is not sufficient, one expects that the refined harmonic coefficients give only a rough description of the texture, even if the texture correction is very good. An extreme case is the Bragg-Brentano geometry. In this case in Equations (41-43) we must take (0 = 9, x =

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There is of course a relation between the ODF of a given material and the pole densities or pole figures. This relation, called the fundamental equation of the texture analysis, has an integral form. 

A mathematical method for texture analysis has therefore to define the way of solving this integral equation in view of determining the unknown, i.e. f g). In 1965 Bunge [8] and Roe [9] independently proposed to use series expansions to solve this. Since then many improvements and variants have been proposed for this method which is referred to as the harmonic method in the literature. Other... 

If the expansion of f g), (16), is inserted into the fundamental equation of texture analysis [15], the integration of the functions (g) leads to y) functions. By identifying the terms of both sides, this fundamental equation thus becomes a system of linear equations which link the and the i (hi) coefficients. 

At the end of the 1970s it was recognized that the fundamental equation of texture analysis has no unique solution even when a large number of pole figures is available [14]. This has been understood by considering very carefully the role of the elements of the crystal symmetry group, especially the role of the center of symmetry which is present in most of the materials studied [15]. An ODF, f g), is in fact, a sum of two parts,... 

Textural analysis was performed by physical adsorption of N2 (BET method) with a Micromeritics ASAP 2000 equipment. Prior to analysis, the samples were outgassed for 24 h at 300°C. The average particle size (Dbet) was calculated assuming the presence of spherical particles, by means of the equation ... 

The value of the angle of tilting of the texture can be determined from the analysis of x-ray fiber diagrams. As Urbanczyk noted [18], the position of layer reflexes oil and ill or the position of equatorial reflexes 010 and 100 can be analyzed. In the first case, the tilting angle of the texture (v ) can be determined from the equation ... 

The classical Kelvin equation assumes that the surface tension can be defined and that the gas phase is ideal. This is accurate for mesopores, but fails if appUed to pores of narrow width. Stronger sohd-fluid attractive forces enhance adsorption in narrow pores. Simulation studies [86] suggest that the lower limit of pore sizes determined from classical thermodynamic analysis methods hes at about 15 nm. Correction of the Kelvin equation does lower this border to about 2 run, but finally also the texture of the fluid becomes so pronounced, that the concept of a smooth hquid-vapor interface cannot accurately be applied. Therefore, analysis based on the Kelvin equation is not applicable for micropores and different theories have to be applied for the different ranges of pore sizes. 

Chemical composition of fresh HTs was determined in a Perkin Elmer Mod. OPTIMA 3200 Dual Vision by inductively coupled plasma atomic emission spectrometry (ICP-AES). The crystalline structure of the solids was studied by X-ray diffraction (XRD) using a Siemens D-500 diffractometer equipped with a CuKa radiation source. The average crystal sizes were calculated from the (003) and (110) reflections employing the Debye-Scherrer equation. Textural properties of calcined HTs (at 500°C/4h) were analyzed by N2 adsorption-desorption isotherms on an AUTOSORB-I, prior to analysis the samples were outgassed in vacuum (10 Torr) at 300°C for 5 h. The specific surface areas were calculated by using the Brunauer-... 

Textural characterisation of the samples was carried out by measuring apparent density (mercury at 0.1 MPa), mercury porosimetry and N2 and CO2 adsorption isotherms, at -196 and 0 °C, respectively. The apparent surface areas of the samples were obtained by using the BET equation [5]. The micropore size analysis was performed by means of the t-plot and the Dubinin-Astakhov methods [6]. 

A complete analysis of the reaction would require measurements of the variations with time of all the phases participating. The product giving unusual textures identified by Brown et al. [18] may, perhaps, be K3(Mn04)2. The fit of kinetic data to the Avrami-Erofeev equation (n = 2) [18], together with the appearance of nuclei, illustrated in Figure 14.1., can be regarded now [17] as only an incomplete representation of this more complicated reaction. 

Similar to the ODF for texture, SODF can be subjected to a Fourier analysis by using generalized spherical harmonics. However, there are three important differences. The first is that in place of one distribution (ODF), six SODFs are analyzed simultaneously. The components of the strain, or the stress tensor can be used for analysis in the sample or in the crystal reference system. The second difference concerns the invariance to the crystal and the sample symmetry operations. The ODF is invariant to both crystal and sample symmetry operations. By contrast, the six SODFs in the sample reference system are invariant to the crystal symmetry operations but they transform similarly to Equation (65) if the sample reference system is replaced by an equivalent one. Inversely, the SODFs in the crystal reference system transform like Equation (65) if an equivalent one replaces this system and remain invariant to any rotation of the sample reference system. Consequently, for the spherical harmonics coefficients of the SODF one expects selection rules different from those of the ODF. As the third difference, the average over the crystallites in reflection (83) is structurally different from Equations (5)+ (11). In Equation (83) the products of the SODFs with the ODF are integrated, which, in comparison with Equation (5), entails a supplementary difficulty. 

The difficulty in the case of microporous materials stems from the porefilling mechanism. For this reason, the surface area of such materials is often determined by other methods than BET, which is based on layer formation. From the Dubinin equation the micropore volume Wo can be converted to the surface area. The as isotherm comparison method is an independent method for estimating the micropore volume and the surface area (20). The reference isotherm is a plot of the measured isotherm normalized by the amount of gas adsorbed at a fixed relative pressure, typically at p/po = 0.4. High resolution as analysis (21) yields more information about the characteristic texture of the adsorbent. Further methods (MP (22), -plot (23), Dubinin-Astakhov (11), Dubinin-Stockli (12), and so on) are also available for more reliable estimates of the micropore volume and surface area. 

Monitoring the amount of material removed by the laser and transported to the ICP is conqjlicated making normalization of data difficult Conditions such as the texture of the sanq>le, location of the sample in the laser cell, surface topography, laser energy, and other hictors affect e amount of material diat is introduced to the ICP torch and thus the intensity of die signal monitored for the various atomic masses of interest In addition, instrumental drift affects count rates. With liquid sanqiles internal standards typically are used to counteract instrument drift, but this approach is not feasible when material for the analysis is ablated from an intact solid sanqile. If one or more elements can be determined by another analytic technique, dien these can serve as internal standards. In the case of rhyolitic obsidian, which has relatively consistent silicon concentrations (ca. 36%), we have determined that silicon count rates can be normalized to a common value. Likewise, standards are normalized to their known silicon concentrations. This value, divided by the actual number of counts produces a normalization factor ftom i ch all the odier elements in that san le can be multiplied. A regression of blank-subtracted normalized counts to known elemental concentrations in the standards yields a calibration equation that can be used to calculate elemental concentrations in the samples analyzed. 

Two standard methods (mercury porosimetry and helium pycnometry) together with liquid expulsion permporometry (that takes into account only flow-through pores) were used for determination of textural properties. Pore structure characteristics relevant to transport processes were evaluated fiom multicomponent gas counter-current difhision and gas permeation. For data analysis the Mean Transport-Pore Model (MTPM) based on Maxwell-Stefan diffusion equation and a simplified form of the Weber permeation equation was used. 

Mercury porosimetry is a method currently used to characterize the texture of porous materials. It enables determining pore volume, specific surface area and also distributions of pore volume and surface area versus pore size. It can be applied to powders, as weU as to monolithic porous materials. The basic hypothesis usually accepted is that mercury penetrates into narrower and narrower cavities or pores as pressure increases. Data analysis is performed using the intrusion equation proposed by Washburn (1921) ... 

Aerogels composed of 60% silica and 40% zirconia were calcined in air, at 400°C, 800°C and 1000°C. After thermal treatment, the porous texture of the samples have been analyzed by mercury porosimetry (Fig. 11-15) (Pirard, 1997c). The three samples are irreversibly densified by isostatic pressure in the whole pressure domain, from 0.01 to 200 MPa. The data analysis (Fig. 11-16) has been done using equation (11-7), with a constant k estimated at 48 nm MPa°-, by nitrogen adsorption-desorption isotherm analysis. The volume distributions versus pore size obtained show that the pore volume decreases for all pore sizes during aerogel calcination at increasing temperatures. This is... 

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