(75d) A general approach for the characterization of fragmentation problems

In many technical processes as grinding, dispersing, adhesion or fluidization the breakage of particles or/and particulate structures is of main importance. If particles or agglomerates with well defined characteristics regarding their shape, size, material and morphology are mechanically stressed, then they differ in their behaviour with respect to their breakage strength. This different behaviour of macroscopically homogeneous particles is based on the existence of microscopic failures in the volume of the particles, i.e. on the distribution of the particle strength. Thereby the failures at the particle surface are of special importance.

Based on the Weibull statistics, an approach for the quantitative and systematic characterization of fragmentation problems is presented. A model, initially developed for materials stressed under impact with respect to their breakage probability, has been successfully applied to compressive comminution, fragmentation of nanoparticle agglomerates and destroying of adhesive bonds. The main emphasis of the model development has been placed on the separation of material and machine function. This approach opens up two ways for the characterization of the grinding process. On one side the material properties relevant to comminution can be determined from grinding experiments at defined conditions in a mill with an already known machine function. On the other side the properties of the mill material, i.e. the material function, can be applied to the prediction of the comminution result in other mill types.

Based on two different and independent approaches, dimensional analyses and fracture mechanical considerations, the following analytical expression for the breakage probability S of impacted particles has been derived [1]:

Herein x denotes the particle diameter and W_m,kin is the mass specific impact energy. Two material parameters, f_Mat and W_m,min, arise in the model. f_Mat is a material property which describes the fracture behaviour of the particles under investigation. The higher f_Mat, the weaker the material. W_m,min denotes the material specific threshold energy which has to be provided for fracture generation. The comparison between the fracture mechanical considerations and the dimensional analyses has shown that x*W_m,min = const. Both parameters f_Mat and x*W_m,min describe quantitatively the comminution behaviour of the material under investigation in respect to its breakage probability. The breakage probability of over 12 investigated materials of different sizes and morphology has been described by a unique master curve using the experimentally determined material parameters (Fig. 1). A similar relationship indicates also the breakage function. The plot of the normalised mass related median particle size of the broken particles as a function of the dimensionless group f_Mat x (W_m,kin-W_m,min) results for different materials also approximately in a single curve. Hence both material parameters f_Mat and x*W_m,min, initially deduced for the characterization of the breakage probability, crucially influence the mean product particle size.

Furthermore the application of the model on single particle compression comminution is presented. The measuring points from the compression experiments match exactly to the master curve for impact comminution. However the values of the material parameters differ for the two stress mechanisms because of the different stress distributions formed within the particle.

The failure of adhesive bonds can be also characterized similarly by a Weibull distribution [2]. The "breakage" is here also released by stresses, which must be larger than the local strength. It therefore physically makes sense that breakage and adhesion can be represented by similar mathematical functions.

The concept, initially developed for the impact comminution of particles, has been transferred to the characterisation of the impact fragmentation of nanoparticle agglomerates [3]. This is illustrated on the example of Ag, Ni and Pt agglomerate structures. As expected, the values obtained for f_Mat are some orders of magnitude higher compared to the values of f_Mat in the case of the single particle impact. This result confirms the fact that much higher energy is needed for the fragmentation of a single particle than for the fragmentation of weak nanoparticle agglomerates. The values determined for the threshold energy W_m,min lie in the case of the fragmentation of the nanoparticle agglomerates in the range of the energy necessary for destroying of the adhesive bond between the primary nanoparticles.

[1] L. Vogel, W. Peukert, Powder Technol., 129 (2003), 101-110

[2] M. Goetzinger, W. Peukert , Langmuir, 20 (2004), 5298-5303

[3] M. Seipenbusch, P. Toneva, W. Peukert, A. Weber, J. Colloid Interface Sci., 2005, submitted