## 3 Selected Methods and Experimental Approaches for Analysis of HS Structure
### Liquid State Investigations
#### 3.3.1 Scattering Techniques
Mass fractals are defined as those molecular agglomerates having a reticular structure, likewise trees where the branches are dendritic networks made up of smaller networks. In contrast, surface fractals have dense internal structure in which the fractal aspect is evident only at the particle surface. Depending on the type of investigation technique employed, the network would assume a collapsed spherical shape while on closer inspection it could result that it is irregular and not space filling. Between mass and surface fractals, partially penetrable hybrid structures may be found, likewise a porous sponge (see Figure 6). For an aggregate or cluster of \(N\) particles with radius (\(R_{p}\)), the fractal dimension d\({}_{f}\) is defined as the exponent of the power law [166]\(N\propto\left(\frac{R_{g}}{R_{p}}\right)^{d_{f}}\), which roughly reflects the shape of the aggregate itself (Figure 6). For large \(Q\), a power-law dependence of the scattering intensity \(I(Q)\propto Q^{-n}\), which in double logarithmic scale looks like a linear dependence within some range in \(Q\) called the fractal region, is a main indicator of the fractal structure. The power law exponent describes how the mass or surface in the system scales with the linear dimensions. In particular, if \(n<3\) the measured sample is a mass fractal, which often represents open aggregates of smaller subunits with a dendritic morphology, and \(n\) coincides with its respective fractal dimension d\({}_{f}\). However, if \(3<n<4\) then the particles are surface fractals, in which only the surface of the material exhibits fractal properties, and d\({}_{f}\)satisfies \(n=2D-\)d\({}_{f}\), where D = 3 for three-dimensional space. Both R\({}_{g}\) and d\({}_{f}\) values are used to calculate surface-to-volume ratios to identify a general shape for the particles under investigation. More details on the fractal properties obtained from small angle scattering techniques can be found, e.g., in refs. [162,167]. The fractal nature of HS has been extensively reported from various studies based on the application of small angle scattering techniques and turbidimetry (see, e.g., [168]). The major weakness of the fractal approach is in providing results and interpretations that can be valid only in an average or statistical sense for finite natural systems such as HS. HS particles in solution that have been characterized to date have demonstrated mass fractal behaviour in solution [84,85] and surface fractal in the solid state [169]. The fractal properties of aquatic HA and FA were recognized by analyzing the power law exponents of SAXS intensities acquired on the humus samples, which were previously extracted, converted to their acid form and lyophilized [169]. The finding of the exponents falling within the range 3-4 identified the humic materials in solid state as surface fractals with d\({}_{f}\) = 2.2-2.8. However, when both the aquatic HS were measured in aqueous solution they behaved as mass fractals with dimensions in the range d\({}_{f}\) = 1.6-2.5. Using scattering methods, the fractal dimensions of both fractal types have been generally found within the range \(2\leq\) d\({}_{f}\)\(\leq\) 3. Particles with mass fractal dimensions that fall within this range can be visualized as open 3D aggregates, which get denser as d\({}_{f}\)\(\to\) 3. Former SANS studies reported by [PERSON] et al. [170] indicated that SRHA could form fractal aggregates in D\({}_{2}\)O at pH 5 and solute concentration of 3.6 mg/mL, characterized by d\({}_{f}\) = 2.3 \(\pm\) 0.1. In absence of a clear Guinier region, the primary particles were  
Figure 6: Pictorial representation of mass and surface fractal objects. A mass fractal is depicted by a dendritic structure while a surface fractal is an object similar to a filled tennis ball. Partially penetrable structures (hybrid fractals) can be modelled as a porous sponge.