By using Equation (9.11) with the measured value of the fractal dimension determined from mobility-mass measurements one obtains AL = 1.87. The difference between the two values can be accounted for by experimental error and the assumption that the projected area diameter nearly equals the mobility diameter.

Wentzel et al. [28] also used the projected maximum length method in order to determine the morphology of various soots. The fractal properties of 37 diesel soot aggregates were determined from TEM images. The average fractal dimension DfL of diesel soot derived from TEM was 1.70 ± 0.13. A second independent approach to determine the fractal properties of soot, based on computer simulations of the aerosol dynamics, was also used [82]. A good reproduction of the time evolution of mass and number concentrations and of the mobility size distribution was achieved. The primary particle diameters obtained from the computer simulations (25 ± 3 nm for diesel soot) were in excellent agreement with the TEM results. The fractal dimension of diesel soot calculated by computer simulation was 1.9 ± 0.2, consistent with the value obtained from TEM image analysis.

Fractional Brownian motion (FBM) theory can, because of its self-similarity and long correlation properties, provide both a description and a mathematical model for many highly complex natural shapes and textures. FBM analysis can derive the fractal parameter, the Hurst coefficient H, to represent the characteristics of a nonstationary zero-mean Gaussian random function such as the fluctuation of soot aggregate textures. The value of H ranges from zero to one and is related to the fractal dimension by Df = De + 1 - H. The parameter De represents the Euclidean dimension, which equals one for linear data. Luo et al. [83] first extracted the multiscale H values to represent the fractal texture of airborne particles by applying FBM combined with the Fourier-domain maximum likelihood estimator. They then explored the application of FBM analysis to SEM micrographs of soot aggregates emitted by a dynamometer [84]. By directly quantifying the surface texture of fractal-like aggregates to extract their Hurst coefficients, the fractal dimension of such particles was found to be in the range 1.6 to 1.7. The impact of image properties on Df measurements due to digital image processing and data recording was also investigated. A twofold change in SEM magnification size gave rise to a 7 % deviation in the fractal dimension, and scaling up from the original image increased the discrepancy compared with miniaturisation. Brightness was not a serious interference factor, as its variance did not exceed the grey level value of 80. These results give more confidence to earlier measurements based on image analysis.

Gwaze et al. [85] have recently shown that the fractal dimension of aggregates from biomass combustion is, on average, 1.83 and so consistent with aggregate formation by DLCA. Importantly, the fractal dimension was determined by three different techniques: Df = 1.84 ± 0.05 from projected surfaces in the SEM images; Df = 1.80 ± 0.13 from the relationship between mobility and number of primary particles; and Df = 1.83 ± 0.05 from the mass and radius of gyration relationship. Although each technique has its limitations, this paper does give confidence in the different approaches to the determination of the fractal dimension of aggregates.

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