ABSTRACT
In the fluidized regime there is an attractive force F0 between the dry and uncharged particles mainly arising from the van der Waals interaction F0 = FvdW Ada/(24z20), where z0 4 Å is the distance of closest approach between two molecules, A is the Hamaker constant, and da is the typical size of the surface asperities (typically A ∼ 10−19J and da ∼ 0.2 µm) (Rietema 1991). The typical size of surface asperities at contact can be decreased down to the size of silica agglomerates covering the particle surface for large enough SAC, thus reducing FvdW . (SEM micrographs show that silica nanoparticles are aggregated
in agglomerates of estimated diameters dag ∼ 50 nm (for the 8 nm nanoparticles) and dag ∼ 200 nm (for the 40 nm nanoparticles). Because of the still strong interparticle attractive force as compared to particle weight, toner particles are clustered in the fluidlike regime (Castellanos et al. 2001). According to our previous experimental results the typical number of particles per cluster N and typical ratio of cluster size to particle size κ depend on the ratio of attractive force to particle weight F0/(mpg) ≡ Bog (granular Bond number). In particular we found N ∼ Boαg (α 0.7) and D = ln N/ ln κ 2.5 for the fractal dimension, in agreement with the Diffusion-Limited-Aggregation (DLA) model prediction. Cluster growth in the initial fluidized state is limited by the interplay of gravitational and attractive forces with flow shear effects. The drag acts mainly at the surface of the cluster whereas gravity is a body force acting uniformly through the cluster. This results in shear forces distributed across the cluster that increase with its size and therefore curtail its growth. A theoretical estimation (Valverde et al. 2005) of cluster size leads to Bog ∼ κD+2, thus the maximum number of particles per cluster should be N = κD ∼ BoD/(D+2)g . For DLA clusters (D = 2.5) we obtain N ∼ Bo0.6g , in close agreement with our experimental results (Castellanos et al. 2001). In our typical clusters (κ< 10) the intercluster Bond number is Bo∗g = Bog/N = κ2 < 100, i.e. intercluster cohesiveness is small.
