The Cone-and-Plate Viscometer

The appealing feature of the cone-and-plate principle is that for small angles (4 less than ca. 0.052 rad) the rate of shear is uniform 

Therefore, if fi/cti is the true rate of shear and Eqn 4-30 gives the shear stress, a simple calculation suffices to obtain the viscosity, whether the flow is Newtonian or non-Newtonian. 

The most readily available cone-and-plate viscometer is the Per ranti-Shirley instrument, described in detail by Van Wazer zt al. 12], 

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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. 

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 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. 

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... 

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. 

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. 

R. M. Turian and R. B. Bird, Viscous Heating in the Cone-and-Plate Viscometer. II. Temperature Dependent Viscosity and Thermal Conductivity, Chem. Eng. Sci., 18,689 (1963). 

The cone-and-plate viscometer is an in vitro flow model used to investigate the effects of bulk fluid shear stress on suspended cells. Anticoagulated whole blood specimens (or isolated cell suspensions) are placed between the two platens (both of stainless steel) of the viscometer. Rotation of the upper conical platen causes a well-defined and uniform shearing stress to be applied to the entire fluid medium as described by Konstantopolous et al. (1998). The shear rate (y) in this system can be readily calculated from the cone angle and the speed of the cone using the formula i/ = where y is the shear rate in sec-1, mis the... 

The mechanical force most relevant to platelet-mediated thrombosis is shear stress. The normal time-averaged levels of venous and arterial shear stresses range between 1-5 dyn/cm2 and 6 10 dyn/cm2, respectively. However, fluid shear stress may reach levels well over 200 dyn/cm2 in small arteries and arterioles partially obstructed by atherosclerosis or vascular spasm. The cone-and-plate viscometer and parallel-plate flow chamber are two of the most common devices used to simulate fluid mechanical shearing stress conditions in blood vessels. 

Parallel Plate Viscometer, This instrument resembles the cone and plate viscometer, except that it has a flat horizontal rotating plate in place of the cone. The shear rate within the narrow gap of the two plates is not as uniform as for the cone and plate viscometer. The limiting shear rates for the parallel plate viscometer are similar to those of the cone and plate instrument. This type of a viscometer is suitable for rheological measurements of suspensions and emulsions. 

The cone-and-plate viscometer is one of the rotational methods of measuring the polymer viscosity. It consists of a fiat horizontal plate and a cone with an obtuse angle. The cone touches the plate at its tip and rotates at a constant speed. The melt is charged into the gap forming between the horizontal plate and the cone. The rotational velocity determines shear rate and the torque applied gives shear stress. Shear rate is constant across the gap, thus it eliminates the need for non-Newtonian behavior of the melt. In a plate-plate viscometer, the cone is replaced by a second flat plate. The Couette viscometer is comprised of two concentric cylinders where one can be rotated at a constant speed. 

The cone-and-plate viscometer is another type of rotational viscometer and is schematically shown in Fig. 10.4. For any time-independent fluid, the following equations apply if the... 

Oscillatory measurements using the cone-and-plate viscometer are sometimes carried out to demonstrate the elastic behavior of a viscoelastic fluid [10]. The fluid in the viscometer is subjected to an oscillatory strain imposed on the bottom surface while the response of the shearing stress is measured on the top surface. If the phase shift between the input strain and the output stress is 90°, the sample is purely viscous if it is 0°, the sample is completely elastic. A measured phase shift between 0° and 90° demonstrates that the fluid is viscoelastic. 

An analogous result is obtained if the plate rotates and the cone and plate viscometer is the Weissenberg Rheogoniometer. It consists of a plate that can be rotated at different speeds by means of a constant speed motor-cum-gear assembly (Figure 3.34). The speed or rotation is measured accurately by means of a transducer. 

The cone and plate viscometer gives reliable experimental data over an extensive range of shear rates (10 -10 sec ). Not only can it be used to measure viscosities in simple shear, but it can also be used to determine the dynamic properties of viscoelastic materials. The unit is also set up to measure the normal stresses exhibited by viscoelastics, i.e., those perpendicular to the plane of shear. 

The method and type of equipment used also vary with the type of adhesive. As an example, the viscosity of electrically conductive adhesives is measured according to ASTM D1824, Apparent Viscosity of Plastisols and Organosols at Low-Shear Rates by Brookfield Viscosity The viscosity of typical electrically conductive die-attach adhesives is measured with a Brookfield HBT viscometer with Spindle TB and Speed 5. For higher-viscosity conductive adhesives and for underfill adhesives, a Brookfield RVT or RVF viscometer is used with Spindles 6 or 7 at speeds of 4—10.4 rpm. Another Brookfield viscometer, the Cone-and-plate viscometer with a CP-51 spindle is used for low-to-intermediate viscosity adhesives. Finally, the Brookfield HAT and HBT instruments are used for the high-viscosity (1-2 million cP at 1 rpm) adhesives typically used in SMT applications. ... 

In the cone-and-plate viscometer, there are similar, though perhaps somewhat less severe, problems associated with the outer edge (129). 

The cone and plate viscometer measures the dynamic viscosity of modified bituminous binders. Although the test method has been developed for modified bituminous binders, it may also be suitable for other bituminous binders. 

For the range of values encountered with colloidal suspensions, a concentric cylinder rotating viscometer is often used to measure the viscosity. Less often used are the cone and plate viscometer and the parallel plate viscometer. For liquids and solutions, a capillary viscometer is often used. Information on several types of viscometers can be found in Ref. 55. 

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... 

Figure 10.22 The cone-and-plate viscometer, showing the cone on top, a rotating plate, and couple attached to the cone. The inset shows a form of tmncation employed in many instruments (52).

The cone and plate viscometer is a widely used instrument for shear flow rheological properties of polymer systems [21-32]. The principal features of this viscometer are shown schematically in Figure 3.1. The sample whose rheological properties are to be measured is trapped between the circular conical disc at the bottom and the circular horizontal plate at the top. The cone is connected to the drive motor which rotates the disc at various constant speeds while the plate is... 

The cone and plate viscometer can be used for oscillatory shear measurements as well. In this case, the sample is deformed by an oscillatory driver which may be mechanical or electromagnetic. The amplitude of the sinusoidal deformation is measured by a strain transducer. The force deforming the sample is measured by the small deformation of a relatively rigid spring or tension bar to which is attached a stress transducer. On account of the energy dissipated by the viscoelastic polymer system, a phase difference develops between the stress and the strain. The complex viscosity behavior is determined from the amplitudes of stress and strain and the phase angle between them. The results are usually interpreted in terms of the material functions, p, G, G" and others [33-40]. 

Oscillatory shear measurements can be done with the parallel-disc arrangement in a similar manner as in the case of the cone and plate viscometer and similarly the material functions, and others... 

Besides the coaxial device, the cone-and-plate viscometer is also used. In this device an inverted cone faces a solid plate and the apex of the cone just touches the plate. The measured liquid is in the free gap. The viscosity of the measured fluid is computed from the torque on the cylinder-driving shaft [4,11). 

The Mooney viscometer, used particularly in the rubber industry, is a variant of the cone-and-plate viscometer it restricts the sample to a disk-shaped cavity (ASTM D1646) [4]. 

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