Geometrical determination of the lacunarity of agglomerates with integer fractal dimension.

Different agglomerates composed by a variable number of spherical primary particles corresponding to extreme and intermediate values of fractal dimension (D(f)=1, D(f)=2 and D(f)=3) are analysed in this work. In each case, the moment of inertia, diameter of gyration and prefactor of the power-law relationship are determined as a function of the number of composing primary particles. The obtained results constitute the geometrical data base for the development of a method for the determination of the fractal dimension of individual agglomerates from their planar projections, although it is not the aim of this paper to describe the method itself. As a result of these calculations, the prefactor of the power-law relationship was shown not to be a constant parameter, but to tend asymptotically to a limit value with increasing number of primary particles. This limit value is closely related with the compactness of the initial geometrical arrangement in the agglomerate, this justifying the historical association of this parameter with the lacunarity of the agglomerate. A correlation for the determination of the prefactor as a function of the fractal dimension and the number of elementary structures is proposed and compared with other methods proposed in the literature.