Packing in multimodal crystals and particle assemblies as models for dynamic packing during spontaneous and enforced particle movement: Settling, viscosity and elasticity

Abstract

Colloid particles (CP, 10⁻⁸–10⁻⁶ m = 10–1000 nm) are used as models for atom scale processes, such as crystallization since the process is experimentally observable. Packing of atoms in crystals resemble mono-, bi-, and trimodal packing of noncharged hard spheres (particles). When the size of one particle exceeds the two others an excluded volume consisting of small particles is created around large particles. This is also the case when colloid particles are dispersed in water. The formation of an excluded volume does not require attraction forces, but it is enforced by the presence of dissolved primary (cations) and secondary (protons of surface hydroxyls) potential determining ions. The outcome is an interfacial solid-liquid charge. This excluded volume, denoted Stern layer is characterized by the surface potential and charge density. Charge neutrality is identified by point of zero charge (pH_pzc and pc_pzc). Outside Stern layer another excluded volume is formed of loosely bound counterions which interact with Stern layer. The extent of this diffuse layer is given by inverse Debye length and effective ζ-potential. The overall balance between attractive and repulsive energies is provided by Derjaguin-Landau-Veerwey-Overbeek (DLVO) model. Charge neutrality is identified at isoelectric point (pH_iep and pc_iep). The dependence of viscosity and yield stress on shear rate may be modeled by von Smoluchowski's volumetric collision frequency multiplied by some total interaction energy given by DLVO model. Equilibrium and dynamic models for settling and enforced particle movement (viscosity) are presented. Both compressive yield stress (sedimentation) and cohesive energy (viscoelasticity) are characterized by power law exponents of volume fraction. The transition of disperse suspensions (sols) to spanning clusters (gels) is identified by oscillatory rheology. The slope of linear plots of logarithmic storage (G´) and loss (G") moduli against logarithm of frequency or logarithm of volume fraction provide power law exponents from the slopes. These exponents relate to percolation and fractal dimensions characterizing the particle network. Moreover, it identifies the structure formation process either as diffusion limited cluster-cluster (DLCCA) or as reaction limited cluster-cluster (RLCCA) aggregation.