The cone-and-plate geometry

In the cone-and-plate geometry, the test sample is contained between an upper rotating cone and a stationary flat plate (see Figiue 2.5, upper). In the example shown, the cone is 40 mm in diameter, with a cone angle of V 59 relative to the plate, and a truncation of 51 pm. 

The small gap size dictates the practical constraints for the geometry a gap-to-maximum particle (or aggregate) size ratio of 100 is desirable to ensure the adequate measurement of bulk material properties. This geometry is, therefore, limited to systems containing small particles or aggregates, and the strain sensitivity is fixed. Normal stress differences may be determined from pressure and thrust measurements on the plate. 

The form factors for the cone-and-plate geometry are as follows  

Many experimentalists employ a sea of liquid around the cone (often referred to as a drowned edge ), partly in an attempt to satisfy the requirement that the velocity field be maintained to the edge of the geometry. 

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In the Couette cell the shear stress varies signficantly with radial position across the gap as r2. Should a more uniform stress environment be required then the cone-and-plate geometry may be used [17]. An example apparatus is shown in Figure 2.8.7. 

On the other hand, the principles of some additional rheological measurements needed for comparison, will briefly be reviewed in this section. The cone-and-plate geometry has already been mentioned above. With such an arrangement the apex of a rather flat cone rests on a flat... 

As has already been pointed out in Sections 1.3 and 1.5, the slit-geometry is interesting for two reasons. First, it enables the measurement of flow birefringence in the 1—3 plane. Second, it furnishes the possibility to investigate polymer melts at high shear rates, where the cone-and-plate geometry fails. In the present section it remains to give a short description of the apparatus. 

Using the cone and plate geometry, stress growth experiments have also been performed using different temperatures and different shear rates. Correct tangential (X+(t,Y) Tj+(t,Y)) and normal stress (Ni(t,Y) Vi(t,Y)) data were... 

When applied to geometries with moving boundaries, such as the cone-and-plate geometry, the Helfrich argument suggests that the flow should be concentrated in thin zones of width proportional to mesh size, and hence there should be apparent slip. 

Figure 4 Images (a) and shear rate profiles (h) from a cetylpyridinium chloride sodium salicylate solution of wormlike micelles at different imposed average shear rates in the cone-and-plate geometry. The dark and bright hands show regions of fast moving approaching and receding regions of fluid (from ref 57)...

Matsumoto et al. [1986] reported that in the cone-and-plate geometry, the storage, G , and loss, G , shear moduli of uniform, non-rigid spheres decrease monotonically with test time (or number of shearing cycles). G and G were observed to decrease by 4 decades, but steady state shearing for 15 seconds returned them to the initial values. Since the phenomenon depended on the rigidity as well as on the uniformity of shape and size, development of a structure during the dynamic test must be postulated. 

Viscosity measurements were made at 50 C on a Rheometrics Mechanical Spectrometer Model RMS-7200 in steady shear mode with the cone and plate geometry. 

In the capillary and Couette (as well as rotating parallel-plate) geometries, a gradient of shear exists in the radial direction. The cone and plate geometry has the advantage that constant shear strain and shear rate are applied at all radial distances. When the cone angle m is very small, tu- < 0.1 rad, analysis of the equations of motion indicates that the shear stress on the plate is... 

The large gap sizes available can be used to overcome the limitations encormtered using the cone-and-plate geometry, such as its sensitivity to... 

Whorlow [1992] notes that, of the many methods which have been proposed for the measurement of various combinations of the first and second normal stress differences, N and N2 respectively, few can give reliable estimates of N2- Combined pressure gradient and total force measurements in the cone-and-plate geometry, or combined cone-and-plate and plate-plate force measurements, appear to give reliable values [Walters, 1975] and satisfactory results may also be obtained from techniques based on the measurement of the elevation of the surface of a liquid as it fiows down an inclined open duct [Kuo and Tanner, 1974],... 

Fluid inertia effects have been found to be very small for the cone-and-plate geometries typically supplied with these instruments. While inertial corrections are foimd to be imimportant for the parallel plate geometries, for shearing gaps of the order of 2 mm or less (except possibly for very thin fluids), they must be taken into accoimt in the concentric cylinder geometry (especially for high-density, mobile fluids). Evaluation methods are available for p, in the case of cylindrical and plane Couette flow, taking into account fluid inertia [Aschoff and Schummer, 1993]. 

Rotational rheometer n. An instrument for measuring the viscosity of molten polymers (any many other fluid types) in which the sample is held at a controlled temperature between a stator and a rotor. From the torque on either element and the relative rotational speed, the viscosity can be inferred. The most satisfactory type for melts is the cone-and-plate geometry, in which the vertex of the cone almost touches the plate and the specimen is situated between the two elements. This provides a uniform shear rate throughout the specimen. It may be operated in steady rotation or in an oscillatory mode. 

This empirical rule seems to hold up quite well for most polymers. Using this rule, it is possible to determine viscosity data up to 500 s with a cone and plate rheometer by applying an oscillatory motion to the cone. This would be impossible if a steady rotational motion was applied to the cone. In steady shear measurements on a cone and plate rheometer, the maximum shear rate that can be measured is around 1 s, which is much too low for applications to extrusion problems. The same is true for measurements in the parallel plate test geometry. Thus, the dynamic measurement extends the shear rate measurement range considerably, while still being able to take advantage of the cone and plate geometry. 

The most important device is the cone-and-plate viscometer see Figure 10.22 (52). The advantage of the cone-and-plate geometry is that the shear rate is very nearly the same everywhere in the fluid, provided the gap angle, 9q, is small. The shear rate in the fluid is given by... 

While the cone and plate geometry is the preferred arrangement to obtain the steady viscometric functions, it is limited to low shear rates — usually, to those less than 10 s . At higher shear rates encountered in processing ( 10-10 s ), it is customary to resort to capillary rheometry to measure the shear viscosity. Unfortunately, the normal stress differences cannot be obtained from this test. To get N at high shear rates one can, however, employ a slit device based on the so-called hole pressure effect [21]. 

The cone and plate geometry shown in Figure 2.3.1 is a common one for measuring viscosity. 

A sketch of the cone and plate geometry is shown in Figure 5.4.1. Spherical coordinates are the proper ones for the problem note also Figure 5.1.2. If we assume ... 

The shear heating of Newtonian (Thrian and Bird, 1963) and power law fluids (TVirian, 1965) has been studied in the cone and plate geometry. For the case that both cone and plate are isothennal at To, Tbrian s power law analysis predicts the maximum temperature rise (see also Bird et al., 1987, p. 226 note hBr = Na)... 

The cone and plate geometry has the advantage that the first normal stress coefficient, /, (see Table 11.1), can be easily calculated by measuring the thrust on the cone, f, according to... 

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