NORMAL STRESSES
4.4 Methods of measuring N, and
4.4.1 Cone-and-plate
It is probably true to say that the cone-and-plate geometry is the most popular for determining the normal stress differences. The basic geometry is shown sche- matically in Fig. 4.11 (see also Chapter 2). The test liquid is contained between a
Fluid
\
Fig. 4.11 Basic geometry for cone-and-plate flow.
rotating cone and a flat stationary plate. (Alternatively the plate is designed to rotate with the cone stationary, with a small advantage as regards alignment).
With respect to suitably chosen spherical polar coordinates, the physical compo- nents of the velocity vector at any point in the liquid are assumed to be (see Fig.
4.11)
sine
with the boundary conditions that the angular velocity is zero at the plate surface and at the surface of the cone,
where is the gap angle. It can be shown that the flow represented by eqn (4.6) is equivalent to a steady simple-shear flow with shear rate = sine and that, when the stress equations of motion are taken into account, we obtain for the shear stress a (see, for example, Walters 1975, Chapter 4):
and, for the normal stress differences,
where p is the density and is a constant to be determined from the boundary conditions. Equations (4.8) and (4.9) are in general incompatible (in the sense that a solution to (4.8) will not be a solution to (4.9) and vice versa), unless we make the following assumptions:
(i) inertial effects are negligible, which means setting p = in (4.9);
(ii) the angle between the cone and the plate is small enough to allow us to set
= 1 in which in practical terms means restricting the gap angle to be no greater than
With assumptions (i) and (ii), we have
there is a constant shear rate throughout the sample and it is independent of the form of the viscometric functions. Equations (4.8) and (4.9) are now compatible.
It is easy to show that the torque acting on the stationary plate of radius a is given by (cf. Chapter 2)
4.41 of measuring and 67
and that if is the pressure on the plate at a radius r, in excess of that due to atmospheric pressure, then
there is a dependence of on r and the slope of the r ) curve can be used to yield
+
Further, if the pressure is integrated over the plate, we obtain the total normal force F on the plate and it can then be shown that (Walters 1975, Chapter 4)This force acts in the direction of the axis of rotation and pushes the cone and plate apart. It is essentially the same force that produces the Weissenberg rod-climbing effect.
The above analysis tells us that the measurement of the rotational speed will give the shear rate and that measurement of the torque on the stationary plate will the shear stress. As regards the normal stress differences, there are two alternatives.
First, the force F gives N,; secondly the radial distribution of pressure gives
+
Hence, in principle, the two normal stress differences can be obtained if these two alternatives are both used.There is a basic conflict in the normal force measurement, since the force F tends to separate the cone from the plate. The consequence of such a separation, if it were allowed, is to upset the condition of uniform shear rate throughout the sample and to reduce the mean value of shear rate. The ideal measuring system should be to axial forces. For systems which are not rigid, a servo-mechanism is used to maintain the cone-plate gap.
Various potential sources of error have to be borne in mind when performing experiments in the cone-and-plate geometry. The more important are enumerated below and we refer the reader to the texts of Walters (1975) and (1980) for further details.
I Inertial effect
The origin and nature of the effect of inertia has already been mentioned. It gives rise to the so-called "negative normal stress effect", whereby the plates are pulled together and the measured value of the force F is smaller than the true value. The reduction in the force F is given by (Walters 1975)
This formula is used to correct experimental values: it can be seen that it is sensitive to the rotational speed and very sensitive to the plate radius.
11 Hole-pressure error
A major source of error can arise when the pressure-distribution method is used is known as the hole-pressure error (Broadbent et al. 1968). Any method of measuring pressure which relies on the use of a hole in the bounding surface gives a low result with elastic liquids owing to the stretching of the flow lines as they pass over the hole. The reduction is directly related to and is in fact used as a means of measuring N,. This method is described later in this chapter, where a more detailed description is given. The error is avoided by the use of stiff, flush-mounted pressure transducers.
111 Edge effects
"Shear fracture" places an upper limit on the usable shear rate range for highly elastic materials like polymer melts. It is observed as a sharp drop in all stress components, and at the same time a change in shape of the free surface can be seen, as well as a rolling motion in the excess liquid around the rim of the plates. A horizontal free surface forms in the test sample at the rim and grows towards the centre, hence reducing the sheared area. The limiting shear rate can be quite low, depending on the liquid and the cone dimensions. Expressed as a critical normal stress the limit is given by
where c is a constant of the liquid. For a given liquid, shear fracture is minimized if the cone radius and gap angle are small.
The name "shear fracture" was given to the effect by (1965) owing to its similarity to 'melt fracture', which limits the occurrence of steady flow of polymer melts in tubes. Tordella (1956) made the first systematic study of melt fracture and noted that when the effect is severe the stream of melt breaks up with an accompanying tearing noise.
Another edge effect, also pointed out by (1972) is ascribable to changes in contact angle and/or surface tension of the test liquid brought about by shear.
The effect is of potential importance when the test liquid possesses only small normal stresses.
I V Miscellaneous precautions
The alignment of the cone axis to be coincident with the rotational axis, the setting of the cone tip in the surface of the plate, and the minimizing of, or correction for, viscous heating are other important matters to be taken into account in accurate work.
4.4.2 Torsional
Torsional flow is shown schematically in Fig. 4.12 (see also Chapter 2). Clearly, commercial instruments which are designed to work in a cone-and-plate mode can be easily adapted to the parallel-plate geometry and vice versa.
of measuring N, and
Stotionorv
I I
plote
Fig. 4.12 Basic geometry for torsional flow.
In this case, with respect to suitably defined cylindrical polar coordinates, the velocity distribution can be taken to be
= = = (4.16)
subject to the boundary conditions at the two plates
where h is the gap between the plates.
Taking into account the fact that eqn. (4.16) is equivalent to a steady simple-shear flow, the stress equations of motion are satisfied, with the shear rate given by
= (4.18)
provided
Equation (4.18) implies that the shear rate is independent of the viscometric functions: it depends on radial distance r , but is constant across the gap for fixed r .
time, we see from eqn. (4.19) that we have to neglect inertia for compatibility and there is no essential restriction on the gap h , except of that this must not be too large that edge effects in a practical rheometer become important. The edge effects mentioned in connection with the cone-and-plate instrument apply here.
After some routine mathematics, it is possible to show that the viscosity function can be determined from measurements of the torque through the equation (cf.
Chapter 2):
where is the shear rate at the rim ( r = a). It can also be shown that
where F is again the total normal force on the plates. We see that total-force data yield the combination - at the shear rate at the rim. Clearly, total force measurements taken in the cone-and-plate and parallel-plate geometries can be combined to yield and separately. However, since is small and two separate experiments have to be performed, significant scatter in the final data can be anticipated unless there is very refined experimentation.
Of interest is the fact that, in principle, relatively high shear rates can be attained with small gaps h . This has been utilized in the so-called "torsional balance rheometer" of Binding and Walters (1976) to obtain normal stress data at shear rates in excess of In this form of the instrument, a predetermined external normal force is applied to the upper plate and the separation h is allowed to vary until this force balances the normal force generated by the liquid. Gap h is measured and eqns. (4.20) and (4.21) applied.