Fractal dimension instrument

The properties characteristic to fractal objects were mentioned first by Leonardo da Vinci, but the term fractal dimension appeared in 1919 in a publication by Felix Hausdorff [197], a more poetic description of fractals was given by Lewis Richardson in 1922 [198] (cited by [199]), but the systematic study was performed by Benoit B. Mandelbrot [196], Mandelbrot transformed pathological monsters by Hausdorff into the scientific instrument, which is widely used in materials science and engineering [200-202]. Geometrical self-similarity means, for example, that it is not possible to discriminate between two photographs of the same object taken with two very different scales. 

Particle size and aggregate fractal dimension were measured with a Malvern Instruments Mastersizer/E. A 100 mm lens was used to measure particle sizes between 200 nm and 110 im (the 45 mm lens allowed particle analysis from 50 nm to 80 pm). Measurements were taken immediately after filling the cell to avoid settling effects, and no stirring was applied. 

All these methods require quite complex and laborious measurements [3-5]. The simplest of these methods, which requires no sophisticated instrumentation, is measurement of [q], which can be performed in virtually any laboratory. Therefore, in this work, we propose a simple rapid method of estimating the fractal dimension (D) of macromolecular coils in solutions, which is based on the same principles as applied in deriving Equations (16.4)-(16.6). The coefficient of swelling of a macromolecular coil is known to be defined as [6] ... 

Here, C is a constant and dy the volume equivalent diameter. Since 4n can be measured with instruments such as a differential mobility analyser (DMA) or scanning mobility particle sizer (SMPS) and dy can be estimated as a function of N (for a given d ) as described above, then Equation (9.3) can be used to estimate a fractal dimension based on the mass-mobility relationship with known values of Jp and N. The fractal dimension may be obtained from Equation (9.2) as the slope of log(Mp) versus log(fi ni)-An alternative method for estimating Df (for values of 2 or larger) can be derived using the results of Rogak et al. [57] and Schmidt-Ott [50]. For Df > 2.0 ... 

Nitrogen adsorption and desorption isotherms for the sludge adsorbent were measured using the standard N2-BET test (Micromeritics Instrument Corporation ASAP2000). The properties of the sludge adsorbents were also characterized by the BET surface area method. The surface fractal dimension D was calculated from their nitrogen isotherms using both the fiactal isotherm equations derived from the FHH theory. 

Note DFT/PaSD with void model, self-consistent model of a mixture of voids, cylindrical and slit-shaped pores, self-consistent regularization with respect to both PoSD (fy(i p)) and PaSD ((l)( z)) with the model of voids, (5 ), Aw=Sgg j-/(5 ) - 1, f)pHH e fractal dimension with Frenkel-Halsey-Hill equation accounting for adsorbate surface tension effects (Quantachrome Instruments software), Ag j is the gelatin adsorption in mg per gram of silica. 

Thus the fractal dimension of aggregates can be easily measured at small scattering angles using a laser diffraction instrument [86]. 

In addition, the fractal dimension seems to be an useful parameter for dispersed materials (ref. 1). To facilitate the exchange of data and to compare industrial products two approaches have been taken Standardisation of measuring methods and instruments and certification of reference materials. 

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