Shear gradient

Some studies have been made of W/O emulsions the droplets are now aqueous and positively charged [40,41 ]. Albers and Overbeek [40] carried out calculations of the interaction potential not just between two particles or droplets but between one and all nearest neighbors, thus obtaining the variation with particle density or . In their third paper, these authors also estimated the magnitude of the van der Waals long-range attraction from the shear gradient sufficient to detach flocculated droplets (see also Ref. 42). 

Figure 7 DMTA scan for (a) normal, and (b) shear gradient processed polymer.

For the calculation of shear stress, the time-dependent impeller power, particle diameter dp and viscosity v according to v = K/9 with the representative shear gradient y = for the non Newtonian broth (see equation (17) [28]) were used. 

When studying systems with mixed fluid and solid directions, it is important to keep in mind that each solid direction should be allowed to breathe and fluid directions need to be scaled isotropically or constrained to a constant value. Allowing two fluid directions to fluctuate independently from one another allows the simulation cell to become flat like a pancake, which we certainly would like to avoid. As an example, consider Figure 15, in which a lamellar block copolymer phase is sheared. The convention would be to have the shear direction parallel to x and the shear gradient direction parallel to y. No reason exists for the simulation cell to distort such that Lxz = Lyz = 0 would not be satisfied on average, so one may fix the values of Lxz and Lyz from the beginning. As a result, one solid direction exists plus two fluid directions. We can also constrain Lxx to a constant value, because the shear direction will always be fluid and another fluid direction can fluctuate. This result means that we should allow the simulation cell to fluctuate independently in only the directions of... 

Figure 15 A lamellar block copolymer phase is reoriented through external shear. The initial phase has the direction of the lamellae parallel to the shear gradient direction. The most stable state would be to orient the director parallel to the shear and shear gradient direction. However, the reorientation process gets stuck before true equilibrium is reached. The stuck orientation is relatively stable, because the lamellae have to be broken up before they can further align with respect to the shear flow. Reprinted with permission from Ref. 56.

When the shear rate reaches a critical value, secondary flows occur. In the concentric cylinder, a stable secondary flow is set up with a rotational axis perpendicular to both the shear gradient direction and the vorticity axis, i.e. a rotation occurs around a streamline. Thus a series of rolling toroidal flow patterns occur in the annulus of the Couette. This of course enhances the energy dissipation and we see an increase in the stress over what we might expect. The critical value of the angular velocity of the moving cylinder, Qc, gives the Taylor number ... 

The Taylor vortices described above are an example of stable secondary flows. At high shear rates the secondary flows become chaotic and turbulent flow occurs. This happens when the inertial forces exceed the viscous forces in the liquid. The Reynolds number gives the value of this ratio and in general is written in terms of the linear liquid velocity, u, the dimension of the shear gradient direction (the gap in a Couette or the radius of a pipe), the liquid density and the viscosity. For a Couette we have ... 

Assuming an attractive potential only, given by equation 5.26, Smoluchowski showed that the frequency of collisions per unit volume between particles of radii ax and a2 in the presence of a laminar shear gradient y is given by ... 

Table 2). All the radii have a certain molar mass dependence. The magnitudes of these radii, however, can deviate strongly from each other. These differences result from the fact that they are physically differently defined. The radius of gyration, R, is solely geometrically defined the thermodynamically equivalent sphere radius, R-p is defined by the domains of interaction between two macromolecules, or in other words, on the excluded volume. The two hydrodynamic radii R and R result from the interaction of the macromolecule with the solvent (where the latter differs from R by the fact that in viscometry the particle is exposed to a shear gradient field).

Dynamit-Nobel AG, NethP, Appl 6411854 (1966) CA 65, 6992(1966) (Liquid esters of nitric acid and aromatic nitro compds can be gelatinized with polymers of unsatd acids or unsatd alcohols and their derivs. The advantages of these polymers include increased safety of manipulation and increased rate of gelatinizacioii. Thus a 60/40—NG/ NGc soin was mixed with 3 wt %-of finely powd polymethylacrylates and, after 1.5 hrs, had a viscosity of 4250 cp at 20° under shearing gradient of 15 sec )... 

A specially designed thin-film machine can be used to process very viscous, non-Newtonian materials. The apparatus can also be used to remove solvents from polymers and polycondensation processes having viscosities exceeding 10,000 poises. The Luwa thin-film machine has a small clearance between the heated wall and rotor blade. This clearance results in high shear gradients and considerably reduces apparent viscosity. The increased turbulence and improved surface renewal that ensue improve reaction velocities and aid the required forced product flow on the walls of the apparatus. 

The viscosity curve of a typical non-Newtonian product such as polystyrene is plotted in Figure 8 as a function of the shear gradient. Because of the small gap between rotor and heating wall, together with the relatively high rotor speed, the thin-film machine works within the limits of shear gradient of 1000 to 10,000 sec 1. 

Figure 8. Viscosity curves for polystyrene expressed as a function of shear gradient. Fields of applications for Luwa thin-fUm machines type M and type HS (Luwa Fumtruder)...

Figure 13 Course taken by shear gradient S, concentration X, viscosity n, and product temperature T along the length of the processing ana discharge zone of a high viscosity machine (Luwa Filmtruder)...

In this thin-film machine, the small clearance between heated wall and rotor blade, together with the high peripheral blade velocity, results in high shear gradients, whereby the apparent viscosity in the film is considerably reduced. The resulting increased turbulence and better surface renewal improve heat transfer, increase reaction velocities, and aid the required forced product flow on the wall. On the basis of test... 

Figure 3. Concentration dependence of viscosity for benzene solutions of some block copolymers and one polybutadiene (P 5) at 25°C. Viscometers with long capillaries were used and viscosities were extrapolated to zero shear gradient (34).

If velocity or shear gradients are present and are sufficiently large, the frequency of collisions depends on the volume fraction of solids and the mean velocity gradient. Assuming that sedimentation is slow compared to other collision mechanisms, the overall aggregation rate, -dN/dt, is ... 

In this case, expression (6.33) for an oscillatory shear gradient gives the dynamic modulus of the system... 

The energy required to melt the polymer components is largely transmitted via the screw shafts the heat flow through the barrel wall serves only to form a melt film on the wall [6]. This melt film is important to create an adhesion of the polymer on the wall, thus generating a shear gradient [11]. To achieve this, the temperature of the barrel wall must be higher than the softening point of the polymer. 

A method based on Mooney s diagram. This involves representing the variations in wall shear gradient defined by... 

In fact, the fiber contribution to the shear viscosity of a fiber suspension at steady state is modest, at most. The reason is that, without Brownian motion, the fibers quickly rotate in a shear flow until they come to the flow direction in this orientation they contribute little to the viscosity. Of course, the finite aspect ratio of a fiber causes it to occasionally flip through an angle of n in its Jeffery orbit, during which it dissipates energy and contributes more substantially to the viscosity. The contribution of these rotations to the shear viscosity is proportional to the ensemble- or time-averaged quantity (u u ), where is the component of fiber orientation in the flow direction and Uy is the component in the shear gradient direction. Figure 6-21 shows as a function of vL for rods of aspect... 

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