Cone and Plate Rheometers

In a cone and plate rheometer, the polymer melt is situated between a flat and a conical plate. In most rheometers, the cone is rotating and the plate is stationary however, this is not absolutely necessary. The basic geometry is shown in Fig. 6.21. 

If the cone rotates with an angular velocity and the cone angle p is very small, the shear rate in the fluid is given by [57]  

Because of the conical geometry, the shear rate is uniform throughout the fluid. The torque necessary to rotate the cone is related to the shear stress by  

the viscosity can be directiy determined from measurement of torque and rotationai speed. If the fluid between the cone and piate is viscoeiastic, the piates wiii be pushed apart when the fluid is sheared. The force with which the piates are pushed apart F is reiated to the first normal stress difference Nj in the fluid  

The first normai stress difference is an accurate indicator of the viscoeiastic behavior of a fluid. Thus, with the cone and plate rheometer, one can accurateiy determine some viscoeiastic characteristics of a fluid. 

The solutions to these equations are summarized in Table S.4.1 and derived in the Sections 5.4.1-S.4.6. 

Most common instrument for normal stress measurements Simple working equations homogeneous deformation Nonlinear viscoelasticity G(t, y) 

Useful for low and high viscosity materials High viscosity limited by elastic edge failure 

Low viscosity limited by inertia corrections, secondary flow, and loss of sample at edges 

This section will present the derivations for a cone and plate rheometer. A schematic of the device is shown in Fig. A3.2. The main discussion is presented in Section 3.6.2. 

The velocity in vector format for the above cone and plate rheometer is  

We now assume that we have simple shear flow in the p direction and that the gradient is in the (-r0) direction. From continuity, dv / 0 = 0, which leads to  

Using the parameters in the above figure, the deformation rate tensor is defined as follows  

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

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MODELS BASED ON DECOUPLED FLOW EQUATIONS -SIMULATION OF THE FLOW INSIDE A CONE-AND-PLATE RHEOMETER... 

Petera, J. and Nassehi, V., 1995. Use of the finite element modelling technique for the improvement of viscometry results obtained by cone-and-plate rheometers. J. Non-Newtonian Fluid Mech. 58, 1-24. 

There are a number of techniques that are used to measure polymer viscosity. For extrusion processes, capillary rheometers and cone and plate rheometers are the most commonly used devices. Both devices allow the rheologist to simultaneously measure the shear rate and the shear stress so that the viscosity may he calculated. These instruments and the analysis of the data are presented in the next sections. Only the minimum necessary mathematical development will he presented. The mathematical derivations are provided in Appendix A3. A more complete development of all pertinent rheological measurement functions for these rheometers are found elsewhere [9]. 

There would be a minimum of 80 data sets needed to generate this data for one temperature. Because of the time involved, usually about 10 to 15 shear rate data points are generated at each temperature. The plot of the viscosity as a function of shear rate at 270°C is presented in Fig. 3.22. The viscosity below a shear rate of 5 1/s would be best taken using a cone and plate rheometer. The wall friction for the capillary rheometer between the piston and the rheometer cylinder wall would likely cause a force on the piston of the same order as the force due to the flow stress. 

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] ... 

Figure 3.26 Complex viscosity measured using a cone and plate rheometer. The data are for a GPPS resin with an MFR of 1.5 dg/min (5 kg, 200 °C) measured at 225 °C. The data are from Fig. 3.22...

G storage modulus as measured using a cone and plate rheometer G" loss modulus as measured using a cone and plate rheometer J(t) creep compliance... 

R radius of the capillary die flow path for a capillary rheometer or the radius of a cone and plate rheometer... 

Appendix A3 Rheological Calculations for a Capillary Rheometer and for a Cone and Plate Rheometer... 

The flow behaviour of polymeric electrophotographic toner systems containing carbon black varying in surface area and concentration were determined using a cone and plate rheometer [51]. As the concentration of carbon black was increased, the viscosity at low shear rates become unbounded below a critical shear stress. The magnitude of this yield stress depended primarily on the concentration and surface area of the carbon black flller and was independent of the polymer (polystyrene and polybutyl methacrylate) and temperature. It was postulated that at low shear rates the carbon black formed an independent network within the polymer which prevented flow. 

Since pressure driven viscometers employ non-homogeneous flows, they can only measure steady shear functions such as viscosity, 77(7). However, they are widely used because they are relatively inexpensive to build and simple to operate. Despite their simplicity, long capillary viscometers give the most accurate viscosity data available. Another major advantage is that the capillary rheometer has no free surfaces in the test region, unlike other types of rheometers such as the cone and plate rheometers, which we will discuss in the next section. When the strain rate dependent viscosity of polymer melts is measured, capillary rheometers may provide the only satisfactory method of obtaining such data at shear rates... 

Measurement of the flow properties of non-Newtonian fluids is typically accomplished via rotational techniques. The rotational methods fall into two basic types, concentric cylinder and cone and plate rheometers. In a concentric cylinder rheometer, a bob is placed inside a cylinder so that the fluid to be studied may be placed into the gap between the cylinders. This arrangement helps approximate a uniform shear rate throughout a sample by shearing only a thin film of sample fluid between... 

In the cone and plate rheometer, a cone-shaped bob is placed against a flat plate so that the fluid to be studied may be placed into the gap between the lower face of the cone and the upper face of the plate. Again, in the Searle method, the cone is rotated while in the Couette method the plate turns. In each case, the torque on the cone is measured. Figure 6.5 shows a Searle-type cone and plate arrangement. For this arrangement the shear stress is given by ... 

Furthermore, when the cone-and-plate rheometer is outfitted with pressure taps at various radial positions, the experimentally obtained pressure distribution is found to be increasing with decreasing radial distance. This, as we will see later, enables us to compute the secondary normal stress difference, namely, x22 — T33, where direction 3 is the third neutral spatial direction. 

We note that with the cone-and-plate rheometers, fracture of the polymer melt is observed at shear rates exceeding 10 2 or 10 1 s-1. Fracture is initiated at the melt-air interface at the perimeter. This has been attributed to the fact that the elastic energy becomes greater than the energy required to fracture the polymer melt at those shear rates (22). Irrespective of the origin of the fracture, it limits the operation of the cone-and-plate instrument to below the previously mentioned shear rates. 

Figure E3.2b presents the primary normal stress difference data for LDPE, and Fig. E3.2c presents the primary and secondary normal stress-difference data for a 2.5% polyacrylamide solution, again using a cone-and-plate rheometer.

Fig. E3.2c Values for -(in - t22), (t22 - T33) and the ratio -(in - t22)/(t22 - t33) for 2.5% acrylamide solution measured with a cone-and-plate rheometer. [Reprinted with permission from E. B. Christiansen and W. R. Leppard, Trans. Soc. Rheol., 18, 65 (1974).]...

This section will be devoted to the Newtonian viscosity i]0, that is to situations where the shear rate is proportional to the shear stress. This is the case under steady-state conditions at low shear rates. Although rj0 may be directly measured at low shear rates in a cone and plate rheometer, it is in general not measured directly but found by extrapolation of viscosity values, as measured in a capillary rheometer, as a function of shear rate ... 

FIG. 15.13 Non-Newtonian shear viscosity r/(q) at 170 °C vs. shear rate, q, for the polystyrene mentioned in Fig. 15.12, measured in a cone and plate rheometer (O) and in a capillary rheometer ( and ) and the dynamic and complex viscosities, rj (w) (dotted line), rj (w) (dashed line) and i (< ) (full line), respectively, as functions of angular frequency, as calculated from Fig. 15.12. From Gortemaker (1976) and Gortemaker et al. (1976). Courtesy Springer Verlag. 

FIG. 15.46 Viscosity, 77, and first normal stress difference, Nh of Vectra 900 at 310 °C as functions of shear rate, according to Langelaan and Gotsis (1996). The first normal stress coefficient, Yi, is estimated from N, by the present author. ( ) Capillary rheometer ( ) and ( ) cone and plate rheometer ( ) complex viscosity rj (A) non-steady state values of the cone and plate rheometer. Courtesy Society of Rheology. 

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