The Cone and Plate

A flat plate forms the lower element and the upper element is conical in shape with a cone angle close to n rad. Advantages of such an arrangement are that only small sample volumes are needed, the mass of the cone can be kept low to minimise the moment of inertia, and they are easy to 

What we measure experimentally at a given shear rate is the moment of the force on the cone, which gives us the twist in the spring or load cell. That is, the sum of the forces on each element dr wide multiplied the distance from the centre r  

If D is the angular displacement of the spring with a spring constant of Cs, i.e. the torque required to give unit angular displacement, then the stress is 

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Using the described algorithm the flow domain inside the cone-and-plate viscometer is simulated. Tn Figure 5.17 the predicted velocity field in the (r, z) plane (secondary flow regime) established inside a bi-conical rheometer for a non-Newtonian fluid is shown. 

Normal Stress (Weissenberg Effect). Many viscoelastic fluids flow in a direction normal (perpendicular) to the direction of shear stress in steady-state shear (21,90). Examples of the effect include flour dough climbing up a beater, polymer solutions climbing up the inner cylinder in a concentric cylinder viscometer, and paints forcing apart the cone and plate of a cone—plate viscometer. The normal stress effect has been put to practical use in certain screwless extmders designed in a cone—plate or plate—plate configuration, where the polymer enters at the periphery and exits at the axis. 

Assuming that the melt viscosity is a power law function of the rate of shear, calculate the percentage difference in the shear stresses given by the two methods of measurement at the rate of shear obtained in the cone and plate experiment. 

Now the cone and plate gives true shear rate whereas the ram extruder uses apparent shear rate. The Non-Newtonian correction factor is... 

Therefore the true shear rate on the cone and plate is equivalent to a shear rate of 0.69(1.18) = 0.817 on the ram extruder... 

Mechanical rheometry requires a measurement of both stress and strain (or strain rate) and is thus usually performed in a simple rotating geometry configuration. Typical examples are the cone-and-plate and cylindrical Couette devices [1,14]. In stress-controlled rheometric measurements one applies a known stress and measures the deformational response of the material. In strain-controlled rheometry one applies a deformation flow and measures the stress. Stress-controlled rheometry requires the use of specialized torque transducers in conjunction with low friction air-bearing drive in which the control of torque and the measurement of strain is integrated. By contrast, strain-controlled rheometry is generally performed with a motor drive to rotate one surface of the cell and a separate torque transducer to measure the resultant torque on the other surface. 

Another approximation to planar Couette conditions can be found in the cone-and-plate cell, shown in Figure 2.8.3. The angular speed of rotation of the cone is taken to be Q (in radians per second) while the angle of the cone is a (in radians) and is generally small, say 4-8°. A point in the fluid is defined by spherical polar (r, 0, ()>), cylindrical polar (q, z, cj)) or Cartesian (%, y, z) coordinates, where Q = y = rsin0 and z = rcos0. 

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. 

Assays. Nitrogen assays to determine 1-amidoethylene unit content were done by Kjeldahl method. Limiting viscosity numbers were determined from 4 or more viscosity measurements made on a Cannon-Fenske capillary viscometer at 30°C. Data was extrapolated to 0 g/dL polymer concentration using the Huggins equation(44) for nonionic polymers and the Fuoss equation(45) for polyelectrolytes. Equipment. Viscosities were measured using Cannon-Fenske capillary viscometers and a Brookfield LV Microvis, cone and plate viscometer with a CP-40, 0.8° cone. Capillary viscometers received 10 mL of a sample for testing while the cone and plate viscometer received 0.50 mL. 

There are two main types of viscometer rotary instruments and tubular, often capillary, viscometers. When dealing with non-Newtonian fluids, it is desirable to use a viscometer that subjects the whole of the sample to the same shear rate and two such devices, the cone and plate viscometer and the narrow gap coaxial cylinders viscometer, will be considered first. With other instruments, which impose a non-uniform shear rate, the proper analysis of the measurements is more complicated. 

The Mooney arrangement of a bob with a conical base is an attractive design as it is relatively easy to fill and uses the base area to enhance the measurement sensitivity. However the cone angle must be such that the shear rates in both the cone and plate and concentric cylinder sections are the same. This means that the gap between the cylinders must be very slightly larger than the gap at the edge of the cone and plate if a constant shear rate is required. Unfortunately the DIN standard bob is poor in this respect. 

The rheometer most often used to measure viscosity at low shear rates is the cone and plate viscometer. A schematic of a cone and plate rheometer is found in Fig. 3.24. The device is constructed with a moving cone on the top surface and a stationary plate for the lower surface. The polymer sample is positioned between the surfaces. Two types of experiments can be performed the cone can be rotated at a constant angular velocity, or it can be rotated in a sinusoidal function. The motion of the cone creates a stress on the polymer between the cone and the plate. The stress transferred to the plate provides a torque that is measured using a sensor. The torque is used to determine the stress. The constant angle of the cone to the plate provides an experimental regime such that the shear rate is a constant at all radii in the device. That is, the shear rate is independent of the radial position on the cone, and thus the shear stress is also independent of the position on the cone. 

Using the cone and plate rheometer the angle Q is forced in a sinusoidal manner, leading to linear strain being introduced in the polymer. The shear strain, y, is a sinusoidal function of time t with a shear rate amplitude of % as follows ... 

Here t is the resulting shear stress, 6 is the phase shift often represented as tan(d), and (O is the frequency. The term 6 is often referred to as the loss angle. The in-phase elastic portion of the stress is To(cosd)sin(wt), and the out-of-phase viscous portion of the stress is To(sind)cos(complex modulus and viscosity, which can be used to extend the range of the data using the cone and plate rheometer [6] ... 

The strain for the cone and plate rheometer is as follows Now it follows that ... 

A number of techniques have been developed to measure melt viscosity. Some of these are listed in Table 3.8. Rotational viscometers are of varied structures. The Couette cup-and-bob viscometer consists of a stationary inner cylinder, bob, and an outer cylinder, cup, which is rotated. Shear stress is measured in terms of the required torque needed to achieve a fixed rotation rate for a specific radius differential between the radius of the bob and the cup. The Brookfield viscometer is a bob-and-cup viscometer. The Mooney viscometer, often used in the rubber industry, measures the torque needed to revolve a rotor at a specified rate. In the cone-and-plate assemblies the melt is sheared between a flat plate and a broad cone whose apex contacts the plate containing the melt. 

Various methods are used to examine the viscosity characteristics of metallized gels. Two types that have received extensive application are the cone and plate viscometer and the capillary viscometer. Both instruments can measure rheological characteristics at high shear rates, and the former is useful for low shear rate measurements as well. 

A cone and plate rotational type viscometer is used to obtain rheological data in the low-to-medium shear rate range. It gives a constant rate of shear across a gap, and therefore, equations for this instrument are simple when the angle is small (less than 3°). For this reason the cone and plate viscometer has become a standard tool... 

Figure 3 shows a curve of the 13 vol. % 4 graphite slurry in water as determined with a capillary viscometer. This is the same material examined on the cone and plate unit. The apparent viscosity is 1.6 poise at a shear rate of 103 sec. 1 and decreases to a value of 0.39 poise at a shear rate of 104 sec."1, the viscosity data being corrected for the true wall shear rate. The flow curves obtained from both instruments agree quite closely. 

We conclude this section with a few remarks about the cone-and-plate type of viscometer, sketched schematically in Figure 4.4. In this viscometer, the fluid is placed between a stationary plate and a cone that touches the plate at its apex. This apparatus also possesses cylindrical symmetry, but this time in order to indicate a location within the fluid we must specify not only r, the distance from the axis of rotation, but also the location within the gap between the cone and the plate, as measured by 0, the angle from the vertical (see Fig. 4.4). Mathematical analysis of this apparatus leads to the result... 

As is the case with all differential equations, the boundary conditions of the problem are an important consideration since they determine the fit of the solution. Many problems are set up to have a high level of symmetry and thereby simplify their boundary descriptions. This is the situation in the viscometers that we discussed above and that could be described by cylindrical symmetry. Note that the cone-and-plate viscometer —in which the angle from the axis of rotation had to be considered —is a case for which we skipped the analysis and went straight for the final result, a complicated result at that. Because it is often solved for problems with symmetrical geometry, the equation of motion is frequently encountered in cylindrical and spherical coordinates, which complicates its appearance but simplifies its solution. We base the following discussion on rectangular coordinates, which may not be particularly convenient for problems of interest but are easily visualized. 

For polymer melts another type of apparatus has been designed in order to measure flow birefringence in the same plane (17). This apparatus is of the cone-and-plate type. In this apparatus the light beam is directed in a radial direction. The principles of other arrangements, which were designed for the measurement of flow birefringence in a plane perpendicular to the plane of flow, will receive special attention in Section 1.5. 

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... 

There are some interesting points to be noted. First, it seems that also for polymer melts the normal stress differences (fin — fi22) and (fin—fi33) are practically equal. (Similar results have been obtained for melts of several polyethylenes.) Second, for the investigated polystyrene a practically quadratic dependence of nn — n33 on the shear stress is found up to the point of the inset of an extrusion defect. It is noteworthy that Fig. 1.9 shows no quadratic dependence of Pjd vs Ds, as would be expected for a second order fluid. Third, the measurements in the cone-and-plate apparatus have to be stopped at a shear stress at least one... 

Apparatuses for the Investigation of Polymer Melts 6.3.1. The Cone-and-Plate Apparatus... 

The cone-and-plate apparatus for the measurement of the flow birefringence of polymer melts has been described in detail very recently (77). Nothing particular has been changed on this apparatus meanwhile. For the present purpose, only the essential parts of this apparatus will be described. These parts are shown in Fig. 6.6. This figure shows a cross-... 

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. 

Yet another geometry is the cone and plate viscometer. This generally operates without a positive hydrostatic pressure and, although often used for plastics melts, is not suitable for rubbers because of excess slipping. 

This section describes common steps designed to measure the viscosity of non-Newtonian materials using rotational rheometers. The rheometer fixture that holds the sample is referred to as a geometry. The geometries of shear are the cone and plate, parallel plate, or concentric cylinders (Figure HI. 1.1). The viscosity may be measured as a function of shear stress or shear rate depending upon the type of rheometer used. 

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