Second critical shear rate

Theoretically, a polymer solution viscosity at different shear rates could have three regimes, as shown schematically in Figure 6.6. At very low shear rates that are below the first critical shear rate, the polymer solution behaves like a Newtonian fluid. The viscosity is independent of shear rate. At intermediate shear rates that are above the first critical shear rate and below the second critical shear rate, the polymer solution behaves like a pseudoplastic fluid. Here, the viscosity decreases with shear rate. At high shear rates that are above the... 

The second critical shear rate is much higher (i.e., 100 times Chauveteau, 1981) than the first one. The first critical shear rate is equal to the inverse of the longest rotational relaxation time k in the solution. Dilatancy starts as soon as the product of Rouse relaxation time and the maximum stretch rate, e, is greater than 4 (Chauveteau, 1981). The Rouse relaxation time demarcates the onset of entanglement effects (Roland et al., 2004). Chauveteau reported that the ratio of shear rate y to the maximum stretch rate e at the contraction was about 2.5 by laser anemometry for similar polymer solutions and flow geometries. Therefore, the second stretch rate (elongation rate) corresponds to the product of shear rate and Rouse relaxation time equal to 10. 

One can see that in Run 1 a mixture of two HDPE was used, both of which had rather low power-law index values (0.40 and 0.42), and the second one had a significantly lower value of the critical shear rate (300 s ). What is worse, the regrind, used in the amount of 15% in the final mix, had an exceptionally low critical shear rate (50 s ) along with a rather low value of the power-law index (0.32). This means that the processing window for this combination was quite narrow, and a slight increase in a flow rate would result in melt fracture, board roughness, sharkskin, and other profile defects (Figs. 17.10-17.14, and 17.19-17.23). 

Super-extrusion is a technique whereby the shear stress in the die land is above the critical shear rate for melt fracture see Section 7.5.3.2. This is possible with polymers that exhibit a second stable region above the melt fracture region. Linear polymers such as HDPE, EEP, and PEA exhibit super-extrusion behavior. The melt fracture behavior can be determined on a capillary rheometer by running a polymer melt at different shear rates and observing the corresponding condition of the extru-date. A typical flow curve for a linear polymer is shown in Eig. 11.43. 

Solution Rheology. Solutions of polyacrylamides tend to behave as pseudoplastic fluids in viscometric flows. Dilute solutions are Newtonian (viscosity is independent of shear rate) at low shear rates and transition to pseudoplastic, shear thinning behavior above a critical value of the shear rate. This critical shear rate decreases with the polymer molecular weight, polymer concentration, and the thermodynamic quality of the solvent. A second Newtonian plateau at high shear rates is not readily seen, probably because of mechanical degradation of the chains... 

Bentley and Leal have measured droplet shapes and critical conditions for droplet breakup over a wide range of capillary numbers, viscosity ratios, and flow types. The flow type is conveniently controlled in an apparatus called a four-roll mill, in which a velocity field is generated by the rotation of four rollers in a container of liquid (see Fig. 1-15). By varying the rotation rate of one pair of rollers relative to that of a second pair, velocity fields ranging from planar extension to nearly simple shear can be produced near the stagnation point. 

To accelerate the diffusion rate and shorten the time for the formation of gas/polymer solutions, we must raise the temperature and shorten the diffusion distance. This is done by deforming the two-phase mixture of polymer and gas through shear distortion to decrease the diffusion path. This type of deformation occurs in an extruder under laminar-flow conditions. The bubbles are stretched by the shear field of the two-phase mixture and eventually break up to minimize the surface energy when a critical Weber number is reached (4). The disintegrated bubble size is calculated to be about 1 mm and the initial striation thickness after bubble disintegration is calculated to be about twice the bubble diameter (5). This striation thickness decreases with further shear, and the gas diffusion occurs faster as a result of the increase in the surface area and the decrease in striation thickness. The striation thickness in an extruder is estimated to decrease to about 100 /xm. At this thickness, the diffusion time is about 1 min in PET, from 10 to 20 s in polystyrene (PS), polyfvinyl chloride) (PVC), and high density polyethylene (HDPE), and in the range of a few seconds in low density polyethylene (LDPE). 

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