Locking the Phase Angle
Accurate phase angle measurements are extremely important for dynamic measurements on a rheometer. Without them the viscoelastic parameters become meaningless. When evaluating a combined motor and transducer, or single head rheometer, it is critical to understand what affects the measurement of the phase angle and if the rheometer reports accurate phase angle. Users of rheometers generally understand that the sample phase angle can vary between 0° and 90°. What is not generally understood is the concept of the total, or raw phase in a single head rheometer. The raw phase consists of the sample phase plus the phase due to the inertia of the motor and geometry. The range of the total, or raw phase is between 0° and 180° as shown schematically in Figure 1. The x-axis represents strain, the applied deformation, and the arrows represent the resultant stress, or torque, vectors. The stress response for a purely elastic material will be in phase with the strain yielding a phase angle of 0°. For a purely viscous material, the stress response will be in phase with the strain rate producing a phase angle of 90°. The inertia torque contribution from motor and geometry can be 180° in phase angle. The instrument measures the raw or total phase angle. The sample phase angle needed to calculate viscoelastic properties is determined by subtracting the inertial contribution from the total phase angle.
As the raw phase signal is directly measured by the rheometer, it is an easily accessible signal that should be reported. The benefit of viewing the raw phase is that it allows one to assess the amount of phase correction being applied to the measurement. But this is not the case with most rheometer manufacturers. Many vendors practice bad science by “locking” or “clamping” the phase angle. They impose artificial limits of 0° and 90° on the reported phase angle. It is simple to demonstrate why locking the phase is bad science. Consider Figure 2, which shows results from a frequency sweep on a Cannon Viscosity standard S60 (viscosity ~ 100 mPa.s @ 25°C).
The Figure shows the complex viscosity, phase angle (sample), and raw phase (motor torque) for the S60 as a function of frequency. At low frequencies, the phase and raw phase are very close or equivalent in magnitude, as inertia is negligible. As the frequency increases, and inertia becomes more significant, the raw phase quickly increases, eventually reaching a maximum of 180°. The reported sample phase, which is corrected for inertia contributions, is 90° up until a frequency of ~ 40 rad/s. Above this frequency, the measurement becomes so dominated by the inertial contribution that it becomes difficult to separate out the true sample phase using a correction, and error is observed in the corrected phase. The increase in the magnitude of the complex viscosity clearly demonstrates that the measurements are being dominated by inertia (See High Inertia Rheometer Design for more details). Recognize that the phase angle is not used in the calculation of complex viscosity.
To determine if a rheology manufacturer is locking the phase angle, have them repeat the test in Figure 2. If the viscosity deviates from Newtonian behavior and the phase angle is exactly 90° (shown as orange dots in Figure 2), then the phase is locked. Although the phase angle is not used in the calculation of complex viscosity, the same inertial issues cause both the increase in viscosity and the inaccuracies in the sample phase. If the viscosity is correct, the inertia is not dominating the measurement and sample phase angle can be corrected and reported accurately. Ask the rheometer manufacture how the phase angle can be exactly 90° if the viscosity is wrong? Locking the phase angle like this may make data look smoother because the range of delta is bound, but doing this makes the data meaningless.
TA Instruments takes a more responsible approach. We don’t hide what really happens. First, our rheometers are designed to minimize inertia (as much as five times lower than competitive designs). We proudly publish the inertia specifications of all of our AR-SERIES Rheometers. Second, we do not use the “bad science technique” of locking the phase angle. TA Instruments’ software reports both raw phase angle and the sample phase angle to help our users determine the effect of inertia.