The Mark-Houwink equation [η]=KW~α is used to determine the mass fractal dimensions of Ancelles. Here [η] being intrinsic viscosity, M beging molecular weight of micellar cluster. K, α, constants. The viscosity exponent α=(3/D-1), and the fractal dimension D may be obtained by ln[η]-InM plot, e.g. D=1.73 for Brij 35 in water and D=1.68, 1.62 for polyepichlorohydrin in tetrahydrofuran (THE) and cyclohexanone.
To determine with easy the fractal dimensions of micelles and macromolecules, a new approach, dubbed GPC-LALLS (gel permeation chromatography coupled with LALLS), has been work out by us, and the value of fractal dimension for polyepichloropydrin in THF obtained by GPC-LALLS is 1.667. The results indicate that the micellar clusters resemble the DLA. According to the Laplace fractal approach. The theoretical values of fractal dimensions of micellar clusters are obtained as D=1.54 (second order). D=1.67 (higer order) in agreement with computer simulation. In the same time. the multifractal spectrum f(α)-α of micelles are obtained and f_(max)(α)=1.75.
The multifractal thermodymics is employed to study the formation and transition of micelle. The partition function is defined as Z(β.L)=Σ_αC_αΣ_iP~β_(i,α) where C_α is the weight of configuration α. P_i, α is the growth probability of site i of configuration α, L is the system size, and β is the parameter measuring the heterogeneity of surfactant aggregation. Thereby two transition points, β_c=-4. β_c=1, for micelle aare found and set β_c=-4 corresponds the transition between surfactants and fractal micelles. and β_c=-1, the transition between fractal micelles and Euclidean structures (i.e. spherical, cylindrical, layer micelles etc.). These transitions correspond to the Brigg (5-5 nm). Porod (5-20 nm) and Guinier (20-100 nm).