Analytical Model of Fractal Aggregate Stability and Restructuring in Shear Flows

Abstract

We combine the analytical theory of nonaffine deformations of noncrystalline solids with numerical Stokesian dynamical simulations to obtain analytical closed-form expressions for the shear modulus of fractal aggregates in shear flows. The proposed framework also provides analytical predictions for the evolution of the fractal dimension d f of the aggregate during the aggregation process. This leads to a lower bound on d f below which aggregates are mechanically unstable (they possess floppy modes) and cannot survive without restructuring into more compact, higher-d f configurations. In the limit of large aggregates, the predicted lower bound is d f = 2.407. This result provides the long-sought explanation as to why all experimental and simulation studies in the past consistently reported d f ≳ 2.4 for shear-induced colloidal aggregation. The analytical expressions derived here can be used within population balance calculations of colloidal aggregation in shear flows whereby until now the fractal dimension evolution was treated as a free parameter. These results may open up the possibility of developing new microscopic mechanical manipulation techniques to control nanoparticle and colloidal aggregates at the nanoscale.

Fractal-like structures in colloid science

“…Notably, the d f values reached in this frame are larger than those observed in stagnant conditions reaching values in the range of 2.4-2.7 [68]. The lower limit of such range, i.e., d f = 2.4, has been recently supported through a mathematical model providing analytical predictions of the aggregate d f during the aggregation process [69].…”

“…Accordingly, the interplay of aggregation and breakage in shear depends strongly on these parameters that clearly affect the resulting cluster d f . An aggregate-stability map, discriminating between clusters restructuring and breakup as a function of the parameters d f , R/a, and _ γ (fractal dimension, cluster size over primary particle radius and shear rate), has been recently presented [69]. Further efforts in this direction, e.g., focusing on an experimental validation of these results would be of great value.…”

Population-balance description of shear-induced clustering, gelation and suspension viscosity in sheared DLVO colloids

“…With the simulation approach used for these calculations, this value has been set by the limiting of the cluster structure, which is somehow elusive and usually semi-empirical 45 . Such simulations would also be extremely time-consuming.…”

Stability of restructured non-fractal aggregates in simple shear flow

“…They found that in shear flow, the aggregates tended to have high fractal dimension by means of experiment [1], numerical simulation [6] and analytical model [7]. Further studies showed that the breakup behavior of the high fractal dimension aggregate was significantly diverse [4,[7][8][9].…”

“…Further studies showed that the breakup behavior of the high fractal dimension aggregate was significantly diverse [4,[7][8][9]. One of the reason is that the aggregate with high space-filling properties may have the connectivity of the particle vary, leading to different ways in force distribution and propagation.…”