Rheology dilute suspension viscosity

FIGURE 12S Schematic of the low shear viscosity of TiOj as a function of pH. Near the zero point of charge (ZPC) the rheology is non-Newtonian for dilute suspensions, conforming to the Cross equation, which suggests that aggregation is responsible for this increase in viscosity. Away from the ZPC, the rheology is Newtonian for dilute suspensions. 

Dilute Suspension Rheology - The Einstein Viscosity Formula... 

The rheological behavior of multiphase systems within the linear, dilute region ((() < 0.05) is relatively well described. For example, for dilute suspensions of spherical particles in Newtonian liquids, Eq 7.2 reduces to Einstein s formula for the relative viscosity, T ... 

According to Eq. (2.8) or (2.10) the concentration dependence of has two parameters, [rj and 0m- Both are measures of a specific physical quantity (respectively, shape and packing) and may be independently determined, for example, [rj] from viscosity of diluted suspensions and 0m from dry packing of solid particles. For anisometric particles, the magnitude of these parameters may also be theoretically predicted see the rheological summary in a quite recent monograph [64]. Once [rj] and 0m are known, Eq. (2.8) will correctly describe the rjr versus 0 dependence for complex industrial systems, for example, PVC (poly(vinyl chloride)) emulsions and plastisols, mica-reinforced polyolefins, and sealant formulations [44,65]. However, in some suspensions and blends, r] and 0m may vary with composition [66]. 

A range of methods are available for making rheological measurements (qv) (39-42). A frequently encountered problem involves knowing the parti-cle/droplet/bubble size and concentration in a dispersion and the need to predict the suspension, emulsion, or foam viscosity. Many equations have been advanced for this purpose. In the simplest case, a colloidal system can be considered Einsteinian. Here, the viscosity of the colloidal system depends on that of the continuous phase, r]o, and the volume fraction of colloid, 0, according to the Einstein equation, which was derived for a dilute suspension of noninteracting spheres ... 

These types of phenomena can t be described in terms of simple rheological models with constant parameters. Systems that reveal the dependence of the viscosity on the flow rate are referred to as anomalous or non-Newtonian. In dilute suspensions, changes in the viscosity associated with the orientation and deformafion of the particles in the absence of particle-partide interactions are typically not too large. 

We have shown in the preceding section that the rheological properties of particulate-filled molten thermoplastics and elastomers depend on many factors (1) particle size (t/p), (2) particle shape (a), (3) volume fraction of filler (f)), and (4) applied shear rate (y) or shear stress a). The situation becomes more complicated when interactions exist between the particulates and polymer matrix. There is a long history for the development of a theory to predict the rheological properties of dilute suspensions, concentrated suspensions, and particulate-filled viscoelastic polymeric fluids. As early as 1906, before viscoelastic polymeric fluids were known to the scientific community, Einstein (1906,1911) developed a theory predicting the viscosity of a dilute suspension of rigid spheres and obtained the following expression for the bulk (effective) viscosity of a suspension ... 

The rheological behavior of storage XGs was characterized by steady and dynamic shear rheometry [104,266]. Tamarind seed XG [266] showed a marked dependence of zero-shear viscosity on concentration in the semi-dilute region, which was similar to that of other stiff neutral polysaccharides, and ascribed to hyper-entanglements. In a later paper [292], the flow properties of XGs from different plant species, namely, suspension-cultured tobacco cells, apple pomace, and tamarind seed, were compared. The three XGs differed in composition and structural features (as mentioned in the former section) and... 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

Consequently, the rheological measurements of MPSs should be carried out such that the dimension of the flow channel is significantly larger than the size of the flow element. For example, the relative viscosity, jjr, of diluted spherical suspensions measured in a capillary instrument depends on the (d/D) factor, where 7) is the sphere diameter and d that of the capillary—for d 107), the error is around 1% [Happel and Brenner, 1983]. Thus, if 1% error is acceptable, the size of the dispersion should be at least 10 times smaller than the characteristic dimension of the measuring device (e.g., diameter of a capillary in capillary viscometers, distance between stationary and rotating cylinders or plates). Following this recommendation is not always possible, which lead to the decline and fall of continuum mechanics [Tanner, 2009]. 

The rheological behavior of PMS suspensions was investigated to elucidate the influence of PMS concentration on interaction between particles. Diluted PMS hydrogel (i.e., suspension or colloidal solution) is characterized by the viscosity increasing with concentration (Figure 1.260a). 

The intrinsic viscosity discussed in Section 11.3.1 is, strictly speaking, an infinite dilution value. To account for increasing solution viscosity with increasing concentration, an expansion in powers of concentration is usually used, as is also done to account for concentration effects in suspension rheology ... 

The rheological properties are described by relations between stress and strain (elasticity) or between stress and rate of strain (flow). In the simplest cases these relations are a simple proportionality as in the viscosity of NEWTONfian liquids or the elasticity of HooiCEan solids (sec chapter I, 4c, p. 22, 4d, p. 28). In the following sections we shall, however, mainly be interested-in non-linear behaviour, the only exception being the NkwTONian viscosity of dilute stable suspensions. 

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