During the compounding operation, the dispersion is generally achieved in four steps: incorporation, wetting, agglomerate breakup, and aggregate spatial distribution in a polymer matrix [109]. It must be ensured that the particulate fillers can be wetted and distributed before they can agglomerate. In the field of solid-liquid dispersion, a series of works have been devoted to the breakup of agglomerates. Agglomerate breakup occurs when hydrodynamic forces induced by the flow overcome the cohesion forces mentioned in the previous chapter. Consequently, the mixing intensity needed is strongly dependent on the affinity between the polymer matrix and the additives.

The mechanism of agglomerate dispersion in various media has been studied by several researchers. Two major types of cluster breakup have been identified; in particular, Parker et al. proposed the occurrence of (1) large-scale splitting, also called the rupture process, and (2) fine particle erosion [110]. Rwei et al. suggested the existence of critical conditions for the different dispersion mechanisms [111, 112]. Erosion, which is initiated at lower shear stresses, is characterized by a continuous detachment of small fragments from the outer cluster surface. The rupture mechanism is characterized by an abrupt splitting of the agglomerate into a small number of large aggregates; it tends to take place at relatively high shear stresses. Analysis of the evolution of carbon black agglomerates in the melts of PS and high-density polyethylene (HDPE) indicated that a coarse rupturing of the agglomerates occurred during the early stages of the dispersion. This was followed by a more gradual erosion of small aggregates from large fragments [113]. Pandya and Spielman investigated the rupture process of cohesive agglomerates and found that fragment size and configuration were affected by the shape and internal structure of the parent agglomerate [114]. Erosion of cohesionless clusters of spherical particles was studied by Mason et al.; they also modeled the effect of shear stress and particle size on the erosion rate [115, 116]. Shiga and Furuta suggested that the dominant mechanism of carbon black agglomerate dispersion in elastomers is the scraping of individual constituent particles from the surface of the agglomerates [117]. Most of the models based on experimental investigations show that the variation of the agglomerate size as a function of time follows a first-order kinetic law regardless of the matrix viscosities [112]:

where R(t) is the radius of the cluster at time t, Ro is the initial cluster radius, K is the erosion rate constant, y is the applied shear rate, and ty is a dimensionless erosion time (Fig. 19). It is believed that K is dependent on the flow geometry, the applied shear stress, and the cohesive strength of the agglomerate.

For a short erosion time, Eq. (3) leads to

In this case, the erosion kinetics was found to be independent of both the agglomerate size and the applied shear rate. When interfacial interactions between particulate agglomerates and the polymer matrix are taken into consideration, variation of the agglomerate size can be described by

where K' is the erosion rate constant corrected for the work of adhesion, Tc is the cohesive strength of the agglomerate, and / is the fluid viscosity [118]. This model has been confirmed by the radius reduction of ultrafine TiO2 agglomerates in linear low-density polyethylene (LLDPE) melts with shearing time.

Since particulate agglomerates have a permeable structure, they can be subjected to a fluid infiltration. The infiltration of a liquid polymer to a porous agglomerate

0 2000 4000 6000 8000 10000

Dimensionless time (yt)

Figure 19. Carbon black agglomerate size reduction in PS melt as a function of dimensionless shearing time under a shear stress of 19,500 Pa. Reprinted with permission from [113], S. P. Rwei et al., Polym. Eng. Sci. 32, 130 (1992). © 1992, Society of Plastics Engineering.

□ From light absorption test Ù. From image analysis corresponds to a transition from a dry agglomerate (pores filled with air) to a wetted one (pores filled with polymer melt). This process is driven by the capillary pressure (pressure difference across the interface between the two filling media) and the hydrostatic pressure, although the latter can be shown to be negligible compared with the former. Liquids characterized by a contact angle on particulate fillers of less than a critical value can infiltrate the agglomerates, provided the air displaced from the pores is allowed to escape. Otherwise, the air pressure inside the agglomerate rises to a point where it counterbalances the capillary pressure, and consequently the infiltration stops.

The presence of a liquid inside an agglomerate affects its cohesiveness. The creation of liquid bridges leads to a substantial increase in the tensile strength of moist agglomerates [119]. Calcium carbonate agglomerates, for example, were found to be strengthened by matrix infiltration during dispersion in polymer melts under simple shear flow conditions. Hence assessing the kinetics of matrix infiltration constitutes a valuable tool in the analysis of the dispersion behavior of fine particle agglomerates in molten polymers.

The role of matrix infiltration in the hydrodynamic dispersion of particulate clusters was highlighted during the study of the dispersion of carbon black agglomerates in liquid polymers. Yamada and co-workers investigated the dispersion mechanisms of carbon black agglomerates in simple shear flows. They found that the agglomerate erosion kinetics could be interpreted in terms of the degree of matrix infiltration and agglomerate permeability [120, 121]. The agglomerate dispersion behavior could be correlated with the ratio of the depth of the infiltrated layer to the square root of the agglomerate permeability, a characteristic length related to the thickness of the agglomerate region that can be drained by viscous flow.

According to the model proposed by Yamada et al. for the kinetics of agglomerate infiltration [122], it is assumed that matrix infiltration in spherical agglomerates occurs symmetrically, leading to an infiltrated outer shell of constant thickness and a dry core. The principle of a spherically symmetric infiltration has been confirmed experimentally by Bohin et al. by phase-contrast microscopy [123]. The kinetics of infiltration of CaCO3 agglomerates by liquid polymers can be well described by the theoretical relationship based on Darcy's law [124]. Reducing the agglomerate radius, polymer viscosity, and/or agglomerate density leads to higher infiltration kinetics. A stearic acid surface treatment, for example, was observed to have a dramatic influence on the latter. Infiltration was much slower for the surface-coated CaCO3 (80 nm) than for the untreated powder when other parameters were kept identical. It was suspected that the surface treatment could have affected the morphology and the packing characteristics of the nanoparticles.

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