Characterization of Relaxation Time Through Parallel Superposition

Introduction

The relaxation time or the distribution function of the relaxation times of polymers has been the subject of research for decades and the associated constitutive equations are well known [1]. However, these relaxation times are typically measured in the linear range. This note will investigate the behavior of relaxation times outside the linear range using a TA Instruments™ Discovery™ Rheometer. To do so, the parallel superposition technique combining the typical frequency sweeps experiments with steady state flow will be used.

Experimental

The HR-30 rheometer used in this work is a stress-controlled rheometer with magnetic and air bearing control. The geometry used was plate-plate with a 40 mm diameter. A parallel superposition experiment is performed in essentially the same way as a standard oscillation experiment.

In its standard mode, the oscillation technique is confined to samples which are effectively at rest. Therefore, any rheological information gained refers only to this unperturbed condition, but the viscoelastic properties of the sheared system are often of more interest. In the technique of parallel superposition, the motions of steady shear and oscillation are applied to the sample simultaneously. The oscillatory motion may be either across the shear field (orthogonal superposition) or in the direction of the shear (parallel superposition).

At low amplitudes the shear stress can be expressed as:

The complex parallel superposition modulus G* can be decomposed into in-phase and out of phase components, G′ and G″ respectively. If the amplitude of oscillation is sufficiently small, then one motion will not interfere with the response of the other, and the oscillation can be regarded as a linear perturbation of the steady shearing motion.

Using the parallel superposition technique, data, the influence of the shear rate/shear stress can be observed, both in the first Newtonian plateau and the pseudoplastic part of the flow curve. The influence of shear stress and shear rate on the distribution of the relaxation times can also be observed.

To demonstrate this, steady state stresses of 0, 0.79, 3.97, 7.97, 39.7, 79.7, 179.5, 239.2, 319, 478.5, 638, and 975 Pa were applied to polydimethysiloxane (PDMS) samples. The frequency sweep range was 0.01 Hz – 100 Hz.

The flow curve in Figure 2 shows a Newtonian plateau until approximately 300 Pa where the pseudoplastic range starts.

Table 1 shows the values of the crossover frequency and the crossover modulus for parallel superposition experiments. It is observed that the mean relaxation time is almost independent of the steady stress at the stress levels used in these experiments. Nevertheless, the modulus crossover decreases even at stresses inside the Newtonian plateau. The inverse of the crossover point gives information about what the dominant relaxation time is.

Table 1. Stress, Frequency and modulus crossover of each frequency sweep