Peclet number suspensions

The electroviscous effect present with solid particles suspended in ionic liquids, to increase the viscosity over that of the bulk liquid. The primary effect caused by the shear field distorting the electrical double layer surrounding the solid particles in suspension. The secondary effect results from the overlap of the electrical double layers of neighboring particles. The tertiary effect arises from changes in size and shape of the particles caused by the shear field. The primary electroviscous effect has been the subject of much study and has been shown to depend on (a) the size of the Debye length of the electrical double layer compared to the size of the suspended particle (b) the potential at the slipping plane between the particle and the bulk fluid (c) the Peclet number, i.e., diffusive to hydrodynamic forces (d) the Hartmarm number, i.e. electrical to hydrodynamic forces and (e) variations in the Stern layer around the particle (Garcia-Salinas et al. 2000). 

The shear rate range over which the shear thinning of hard sphere suspensions occurs can be determined from the equations due to Krieger26 or Cross27 in conjunction with the reduced stress or Peclet number respectively ... 

Note that err = y (crr)a3/k Tand recall that in a concentrated dispersion the Peclet number is Pe = 67ry (crr)a3/k T. The use of the suspension viscosity implies that the particle diffusion can be estimated from an effective medium approach. Both Krieger and Cross gave the power law indices (n and m) as 1 for monodisperse spherical particles. In this formulation, the subscript c indicates the characteristic value of the reduced stress or Peclet number at the mid-point of the viscosity curve. The expected value of Pec is 1, as this is the point at which diffusional and convective timescales are equal. This will give a value of ac 5 x 10 2. Figure 3.15 shows a plot of Equation (3.57a) with this value and n = 1... 

Other dimensionless groups similar to the Deborah number are sometimes used for special cases. For example, in a steady shearing flow of a polymeric fluid at a shear rate y, the Weissenberg number is defined as Wi = yr. This group takes its name from the discoverer of some unusual effects produced by normal stress differences that exist in polymeric fluids when Wi 1, as discussed in Section 1.4.3. Use of the term Weissenberg number is usually restricted to steady flows, especially shear flows. For suspensions, the Peclet number is defined as the shear rate times a characteristic diffusion time to [see Eq. (6-12) and Section 6.2.2]. 

Dimensional analysis implies that for a given value of , all monodisperse hard-sphere suspensions ought to show an onset of shear thickening at a universal value of the Peclet number Pe, or reduced stress Or. Thus, the critical shear rate Yc foi shear thickening ought... 

The crossover from Brownian to non-Brownian behavior in a flowing suspension is controlled by a rotational Peclet number. 

The term Peclet number is common in the suspension literature, while the corresponding quantity is usually called the Deborah number or Weissenberg number in tbe polymer literature. From Eqs. (6-30) through (6-33) we find, in general, for a solvent of viscosity 1 cP, that Drd /b, where b — d or L) is the particle s longest dimension in units of pm, and is in sec. Since typical shear rates are in the range 10 > 10 sec , ... 

Brenner (1974) has presented numerical results for the suspension stresses in various flows. Figure 6-14 plots the intrinsic viscosity [defined in Eq. (6-6)] for oblate and prolate spheroids of various aspect ratios as functions of the Peclet number. Note that as the aspect ratio of the spheroid increases, the zero-shear viscosity increases, and the suspension shows more shear thinning. The suspension also becomes more elastic when the aspect ratio p for prolate or 1/ for oblate spheroids) is large see Fig. 6-15, which plots Ni N2 versus Pe for prolate spheroids of various aspect ratios p. Typically, N2 is roughly an order of magnitude less than Ni, so this plot of Nj, mainly reflects the behavior of V,. 

Changes in microstructure of the suspension become important when the diffusion time fj becomes long compared to the characteristic time of the process, fp. This number hcis been discussed earlier as the De number. The importance of convection relative to diffusion is compared in the Peclet number Pe (in which u is the fluid velocity). The importance of convection forces relative to the dispersion force is compared in Nf just as the dispersion force compared to the Brownian force. The electrical force compared to the dispersion or Brownian force is given by N. The particle size compared to the range of the electrical force is compared in UK. 

Figure 16 shows the steady shear relative viscosity variation with the effective Peclet number, Peeg, based on the effective particle diameter at each temperature level, and the temperature for a PMMA suspension. The particles of 0.8 pm are sterically stabilized by a thick layer of terminally anchored poly(dimethylsiloxane) and suspended in n-hexadecane at the volume fraction of 0 = 0.282. The data points are... 

Suspensions, even in Newtonian liquids, may show elasticity. Hinch and Leal [1972] derived relations expressing the particle stresses in dilute suspensions with small Peclet number, Pe = y/D 1 (D is the rotary diffusion coef-hcient) and small aspect ratio. The origin of elastic effect lies in the anisometry of particles or their aggregates. Rotation of asymmetric entities provides a mechanism for energy storage. Brownian motion for its recovery. Eor suspensions of spheres, this mechanism does not exist. 

To demonstrate the effect of Peclet number, Krieger (1972), in a series of classic experiments, measured the relative viscosity of suspensions of monomod-al spheres with sizes from about 0.1 to 0.5 (im. By adjusting the solution... 

The reduced shear stress is recognized to be essentially our Peclet number defined by Eq. (9.2.10). As can be seen from Fig. 9.2.3, as the Peclet number is increased, the viscosity reaches a stationary value and in this limit the suspension behaves as a Newtonian fluid. In the opposite limit, as the Peclet number tends to zero, the relative viscosity approaches a higher stationary value. The transition is seen to take place in the neighborhood of a Peclet number close to unity, consistent with our earlier discussion. 

In the last section we introduced the concept of two asymptotic viscosity limits for shear thinning colloidal suspensions as a function of shear rate. One is the high shear limit which corresponds to high values of the Peclet number where viscous forces dominate over Brownian and interparticle surface forces. Generally this limit is attained with non-colloidal size particles since to achieve large Peclet numbers by increase in shear rate alone requires very large values for colloidal size particles. In this limit, non-Newtonian effects are negligible for colloidal as well as non-colloidal particles. 

The rheological properties of the suspension are strongly influenced by the spatial distributiOTi of the particles. The relationship between microstructure and rheology of suspensions has been smdied extensively (Brader 2010 Morris 2009 Vermant and Solomon 2005). Most of earlier smdies dealt with the simplest form of suspensions, in which dilute hard-sphere suspensions are subjected only to hydro-dynamic and thermal forces near the equilibrium state (i.e., Peclet number << 1) (Bergenholtz et al. 2002 Brady 1993 Brady and Vicic 1995). In shear flows of such suspensions, the structure is governed only by the particle volume fraction and the ratio of hydrodynamic to thermal forces, as given by the Peclet number. 

The main problem in extending the microstructural theories to high Peclet number and volume fraction is related to the formulation of the many-body interactions. Recently, based on the Smoluchowski equation, Nazockdast and Morris (2012) developed a theory for concentrated hard-sphere suspensions under shear. The theory resulted in an integro-differential equation for the pair distribution function. It was used to capture the main features of the hard sphere structure and to predict the rheology of the suspension, over a wide range of volume fraction (<0.55) for 0 < Pe < 100 (Nazockdast and Morris 2012). 

We can take fg to represent a characteristic time required to restore the structure of the suspension from a disturbance caused by Brownian motion. The time scale ts for viscous flow due to a shear stress can be taken as the reciprocal of the strain rate y. The dimensionless parameter equal to the ratio tslts, which is referred to as the Peclet number (or the reduced strain rate)... 

Figure 4.37 Schematic representation of the relative viscosity versus the Peclet number for suspensions with different particle concentration f. (From Ref. 53.)...

Frank et al. [13] investigated particle migration in concentrated Brownian suspensions both by experiment and by modeling of flow in a mixer. The flow rate was quantified by the dimensionless Peclet number, which, conceptually, is the ratio of the time required for Brownian diffusion to move a particle by its own size, a /D = cP j kTl itrjci), to the time required for shear flow to move it by the same distance, y (where y is the shear rate of the surrounding flow field), yielding... 

Figure 7 A schematic diagram of die viscosity as a dinction of shear rate or Peclet number for large and small particle size suspensions.

The hydrodynamic forces are always proportional to the viscosity of the medium. Therefore, suspension viscosities are scaled with the viscosity of the suspending medium, meaning that relative viscosities are used. As for dilute systems, the balance between Brownian motion and flow can be expressed by a Peclet number. Here the translational diffusivity D, has to be used, but that does not change the functionality (for spheres. Dr is proportional to D,). A dimensionless number is obtained by taking the ratio of the time scales for diffusion (D,) and convective motion (y). This is again a Peclet number ... 

The Peclet number of eq. 10.4.6 is based on the difiusivity of an isolated sphere (i.e., Stokes law for the hydrodynamic effect). This is obviously incorrect in a concentrated suspension, where the presence of other particles can have an enormous effect on the mobility. The viscous resistance a particle encounters will be the sum of all hydrodynamic interactions with all neighboring particles. This resistance is much higher than that given by Stokes law and should be comparable with the global viscosity of the suspension. 

Lee, et al. measured viscosity as affected by shear rate for silica sphere suspensions, finding shear thinning at lower shear rates(57). In some but not all systems and volume fractions above 0.5, a reproducible abrupt transition to shear tbickening was found at elevated shear rates. The transition shear rate depended on concentration and temperature. In contrast, Jones, et al plot only a shear thinning region. A possible explanation for this difference is provided by the Peclet number Pe,... 

Section 10.6 examined viscoelastic properties of hard-sphere suspensions. Hard-sphere suspensions show shear thinning at sufficiently elevated Peclet numbers. Based on limited measurements, the loss and storage moduli of sphere suspensions have a stretched-exponential frequency dependence at smaller frequency and a power-law frequency dependence at larger frequency, as predicted by the temporal... 

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