Network particle yielding

The fractal dimension D is used to quantify the micro structure of the fat crystal networks, where d is the Euclidean dimension, x is the backbone fractal dimension that is estimated between 1 and 1.3. The backbone fractal dimension describes the tortuosity of the effective chain of stress transduction within a cluster of particles yielding under an externally applied stress (Shih et al. 1990 Kantor and Webman 1984). 

Colloidal dispersions often display non-Newtonian behaviour, where the proportionality in equation (02.6.2) does not hold. This is particularly important for concentrated dispersions, which tend to be used in practice. Equation (02.6.2) can be used to define an apparent viscosity, happ, at a given shear rate. If q pp decreases witli increasing shear rate, tire dispersion is called shear tliinning (pseudoplastic) if it increases, tliis is known as shear tliickening (dilatant). The latter behaviour is typical of concentrated suspensions. If a finite shear stress has to be applied before tire suspension begins to flow, tliis is known as tire yield stress. The apparent viscosity may also change as a function of time, upon application of a fixed shear rate, related to tire fonnation or breakup of particle networks. Thixotropic dispersions show a decrease in q, pp with time, whereas an increase witli time is called rheopexy. 

If the dispersion particles are attracted to each other, they tend to flocculate and form a stmcture. At low concentrations the particles form open aggregates, which give a fractal stmcture (93,94). At higher concentrations a network stmcture results, which can be so pronounced that the mixture has a yield point and behaves like a soHd when at rest. Shearing breaks up this stmcture, and viscosity decreases. 

When the void space in an agglomerate is completely filled with a Hquid (Fig. Ic), the capillary state of wetting is reached, and the tensile strength of the wet particle matrix arises from the pressure deficiency in the Hquid network owing to the concave Hquid interfaces at the agglomerate surface. This pressure deficiency can be calculated from the Laplace equation for chcular capillaries to yield, for Hquids which completely wet the particles ... 

The interpretation of the relationships obtained here is based on the same principles of polyfunctional interaction between CP and organic ions which are considered in sections 3.1-3.3. The dispersion of CP grains to a certain size (1-10 pm) yields particles retaining the ability of polyfunctional interaction with organic ions. Simultaneously with increasing dispersion, the mobility of elements of the crosslinked structure also increases, which favors additional interaction. Further dispersion of CP (d 0.1 pm) gives so weak networks that the spatial effect of polyfunctional interaction with organic ions drastically decreases similar to linear polyelectrolytes [64]. 

The gel point is defined as the point at which the entire solid mass becomes interconnected. The physical characteristics of the gel network depends upon the size of particles and extent of cross-linking prior to gelation. Acid-catalysis leads to a more polymeric form of gel with linear chains as intermediates. Base-catalysis yields colloidal gels where gelation occurs by cross-linking of the colloidal particles. 

While the surface modification is not effective to suppress cavitation, Yee and coworkers performed an experiment to suppress the cavitation mechanically in a rubber-modified epoxy network. They applied hydrostatic pressure during mechanical testing of rubber toughened epoxies [160]. At pressures above BOSS MPa the rubber particles are unable to cavitate and consequently no massive shear yielding is observed, resulting in poor mechanical properties just like with the unmodified matrix. These experiments proved that cavitation is a necessary condition for effective toughening. 

Dimethacrylate monomers were polymerized by free radical chain reactions to yield crosslinked networks which have dental applications. These networks may resemble ones formed by stepwise polymerization reactions, in having a microstructure in which crosslinked particles are embedded in a much more lightly crosslinked matrix. Consistently, polydimethacrylates were found to have very low values of Tg by reference to changes in modulus of elasticity determined by dynamic mechanical analysis. 

Concentrated particle suspensions may also show a yield point which must be exceeded before flow will occur. This may result from interaction between irregularly shaped particles, or the presence of water bridges at the interface between particles which effectively bind them together. Physical and chemical attractive forces between suspended particles can also promote flocculation and development of particle network structures, which can be broken down by an applied shear stress [2]. 

An overview of the origins of yield stress and parameters which can lead to variations in behaviour with highly filled polymer dispersions is given by Malkin [1]. Much of the following literature, describing experimental work undertaken, demonstrates that yield phenomena can be correlated with the extent of interaction between the filler particles and the formation of a network structure. However, the actual behaviour observed during experimentation may also depend on the deformation history of the material, or the time and temperature of imposed deformation, especially if the material exhibits thixotropic properties. 

The durability of the particle network structure imder the action of a stress may also be time-dependent. In addition, even at stresses below the apparent yield stress, flow may also take place, although the viscosity is several orders of magnitude higher than the viscosity of the disperse medium. This so-called creeping flow is depicted in Fig. 11 where r (. is the creep viscosity. In practice this phenomenon is insignificant in the treatment of filled polymer melts, but may be relevant, for example, in consideration of cold flow of filled elastomers. 

Several studies have considered the influence of filler type, size, concentration and geometry on shear yielding in highly loaded polymer melts. For example, the dynamic viscosity of polyethylene containing glass spheres, barium sulfate and calcium carbonate of various particle sizes was reported by Kambe and Takano [46]. Viscosity at very low frequencies was found to be sensitive to the network structure formed by the particles, and increased with filler concentration and decreasing particle size. However, the effects observed were dependent on the nature of the filler and its interaction with the polymer melt. 

It has been suggested that the three-dimensional network structures discussed above, which are believed to occur from particle interactions at high filler loadings, may, in the case of plate-like particles, lead to anisotropic shear yield values [35]. Although this effect has not been substantiated experimentally, further theoretical interpretation of shear yield phenomena in talc- and mica-filled thermoplastics has been attempted [31,35]. 

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