Mass fractal dimension determination

The exponent df is denoted mass fractal dimension or simply fractal dimension. It characterizes the mass distribution in three dimensional space and can vary between lfractal analysis of furnace blacks was performed, e.g., by Herd et al. [108] or Gerspacher et al. [109, 110]. The solid volume Vp of primary aggregates is normally determined (ASTM 3849) from the cross-section area A and the perimeter P of the single carbon black aggregates by referring to a simple Euclidean relation [108] ... 

Koylu et al. used extensive TEM studies to derive the mass fractal dimension, Df, of flame-made soot particles from Equation A1 (see the appendix) by extracting the number of primary particles per agglomerate, N, the primary particle size, (ip, and the radius of gyration of the agglomerate, R. The constant fractal prefactor, A, as well as the fractal dimension, Dj, were then determined by regression. 

Ultra small-angle x-ray scattering (USAXS) is capable of determining all characteristics of agglomerate and primary particles from a single scattering experiment mass fractal dimension, Dj, radii of gyration (primary particles and... 

The mass-fractal dimension of the ramified agglomerates is determined from the slope of the weak power law decay in between the power law regimes that follow Porod s law (Equation 10.9) ... 

Electrical sensing zone technique commonly used to determine equivalent volume diameter, required in Eq. (19), might be problematic. The error associated with this technique is contributed by the breakup of aggregates and inclusion of pores in volume measurement. With this technique, an aggregate will have to be suspended in a liquid. The challenge is to preserve the structure of aggregates. Hence the first method is preferred to obtain the mass fractal dimension of aggregates in situ. 

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) ... 

Determination of mass fractal dimension using mercury porosimetry is less accurate when the pores are interconnected by narrow channels because mercury cannot access these channels. ... 

Mass fractal dimension is able to determine the extent of gravitational coagulation and sedimentation that take place as the particles traverse the airway. 

The nonlinear form of the mass exponent in Fig. 9a, the convex form of the singularity spectmm/(/i) in Fig. 9b, and the fit to f(q) in Fig. 13, are all evidence that the interstride interval time series are multifractal. This analysis is further supported by the fact that the maxima of the singularity spectra coincide with the fractal dimensions determined using the scaling properties of the time series using the allometric aggregation technique. 

Techniques have been developed to determine the fractal dimension of experimental aggregate particles in solution using small-angle scattering techniques, from x-ray, light, or even neutron sources. In these techniques the scattering intensity, I q), is proportional to the scattering vector, q, raised to the mass fractal dimension by ... 

We recommend that future research in this area focus on establishing theoretical and/or empirical relations between the prefactor and exponent in the power law relating K to Pn, and quantitative pore characteristics, such as the mass fractal dimension and Betti numbers. In particular, there is an urgent need for additional experimental studies, in which K is determined over a wide range of Pn on undisturbed samples of natural porous media that are well-characterized in terms of their pore space geometry. Such data are required for model testing. 

Dinsmore and Weitz (2002) [60] examined a model system of polymethylmethacrylate particles dyed with rhodamine in a density- and refractive-index-matched solution of decalin and cyclohexylbromide. They used CLSM to follow the (very slow) aggregation in real time and determined the particle positions in full three-dimensional detail. From this they performed a comprehensive structural analysis of the gels, including measurement of coordination numbers and backbone fractal dimension of the structures, as well as the much more commonly measured mass fractal dimension. 

Surface fractal dimensions of a number of Cambisols and Luvisols were determined using the FHH equation from data obtained from N2 and water vapor adsorption isotherms. Values were compared with those obtained from the mercury intmsion method and with mass fractal dimensions that were evaluated from particle-size distributions using a modified number-based method [108] (Figure 6.3). This method was proposed by Kozak et al. [116] in order to correct some inconsistencies of previous approaches... 

Figure 7.3 Effect of ionic strength (as NaCl or CaCL concentration) and pH on the mass fractal dimension of a soil humic acid. Fractal dimensions are taken from Senesi et al. [35], and were determined hy turhidimetry (see text for details).

The only parameters required for these determinations are an accurate estimation of the sedimentation velocity and various balance voltages in the cell. As a verification of the technique and equations, polyvinyltoluene particles were used as calibration standards. Mass fractal dimensions vary from 2.96 to 2.99 and show excellent agreement to the theoretical value of 3, hence verifying the method. 

Besides the determination of the porosity, pore size distribution and surface area, the (surface or mass) fractal dimension of dry gels may be of interest. To this purpose, small-angle X-ray scattering can be used nitrogen sorption and mercury porosimetry also offer possibilities to extract this structural information (see, for example, Blacher et al. (2000)). 

---