Linear viscoelastic behavior of bidisperse polystyrene blends: experiments and slip-link predictions

Abstract

The linear viscoelastic behavior of three well-entangled linear monodisperse polystyrene melts and their blends is investigated. The monodisperse melts are blended in a 1:1 weight ratio to obtain three polystyrene bidisperse blends for which the linear viscoelastic behavior is also measured. Special attention is paid to controlling sample size and solvent content, and checking for consistency in the high-frequency regime. We also attempt to estimate uncertainty quantitatively. The experimental results agree well with the discrete slip-link model, a robust mesoscopic theory that has been successful in predicting the rheology of flexible entangled polymer liquids and gels. Using recently developed analytic expressions for the relaxation modulus, predictions of the monodisperse samples are made. The parameters for the model are obtained from the low-frequency crossover of one experiment. Using this parameter set without adjustment, predictions over the fully accessible experimental frequency range are obtained for the monodisperse samples and their blends with very good agreement.

Appendix A: Error propagation

The well-known expressions for experimentally determining G^′ and G^″ in parallel disk flow are given by

$$\begin{array}{@{}rcl@{}} G^{\prime}=\frac{2MH}{\pi R^{4}\theta}\cos[\delta], \end{array} $$

(A1)

$$\begin{array}{@{}rcl@{}} G^{\prime\prime}=\frac{2MH}{\pi R^{4}\theta}\sin[\delta], \end{array} $$

(A2)

where R is the sample radius, H is the gap, M is the torque, 𝜃 is the deflection angle, and δ is the phase angle. It is clear from Eqs. A1 and A2, the moduli have a strong dependence on R, which means that small uncertainties can have a large impact on measured values. The unknown shape of the sample free surface leads to an estimated uncertainty in R of ± 0.5 mm (Kashyap 2011). In order to propogate these errors through to the measured values of G^′ and G^″, we use the following error propagation expressions:

$$\begin{array}{@{}rcl@{}} {\Delta}_{G^{\prime}}&=&\sqrt{\left( \frac{\partial G^{\prime}}{\partial M}{\Delta}_{M}\right)^{2}+ \left( \frac{\partial G^{\prime}}{\partial H}{\Delta}_{H}\right)^{2}+ \left( \frac{\partial G^{\prime}}{\partial R}{\Delta}_{R}\right)^{2}}\\ & &\overline{+\left( \frac{\partial G^{\prime}}{\partial\theta}{\Delta}_{\theta}\right)^{2}+ \left( \frac{\partial G^{\prime}}{\partial\delta}{\Delta}_{\delta}\right)^{2}} \end{array} $$

(A3)

$$\begin{array}{@{}rcl@{}} {\Delta}_{G^{\prime\prime}}&=&\sqrt{\left( \frac{\partial G^{\prime\prime}}{\partial M}{\Delta}_{M}\right)^{2}+ \left( \frac{\partial G^{\prime\prime}}{\partial H}{\Delta}_{H}\right)^{2}+ \left( \frac{\partial G^{\prime\prime}}{\partial R}{\Delta}_{R}\right)^{2}}\\ & &\overline{+\left( \frac{\partial G^{\prime\prime}}{\partial\theta}{\Delta}_{\theta}\right)^{2}+ \left( \frac{\partial G^{\prime\prime}}{\partial\delta}{\Delta}_{\delta}\right)^{2}} \end{array} $$

(A4)

After taking the corresponding partial derivatives using Eqs. A1 and A2 and substituting back into Eqs. A3 and A4, we obtain the following error in G^′ and G^″,

$$\begin{array}{@{}rcl@{}} {\Delta}_{G^{\prime}}&=&\frac{2MH}{\pi R^{4}\theta}\cos[\delta]\sqrt{\left( \frac{{\Delta}_{M}}{M} \right)^{2}+ \left( \frac{{\Delta}_{H}}{H}\right)^{2}+ \left( \frac{4{\Delta}_{R}}{R}\right)^{2}}\\ & &\overline{+\left( \frac{{\Delta}_{\theta}}{\theta}\right)^{2}+ \left( {\Delta}_{\delta}\frac{\sin[\delta]}{\cos[\delta]}\right)^{2}}, \end{array} $$

(A5)

$$\begin{array}{@{}rcl@{}} {\Delta}_{G^{\prime\prime}}&=&\frac{2MH}{\pi R^{4}\theta}\sin[\delta]\sqrt{\left( \frac{{\Delta}_{M}}{M} \right)^{2}+ \left( \frac{{\Delta}_{H}}{H}\right)^{2}+ \left( \frac{4{\Delta}_{R}}{R}\right)^{2}}\\ & &\overline{+\left( \frac{{\Delta}_{\theta}}{\theta}\right)^{2}+ \left( {\Delta}_{\delta}\frac{\cos[\delta]}{\sin[\delta]}\right)^{2}} . \end{array} $$

(A6)

Since most (if not all) commercial rheometers used to measure G^∗ only report values for the torque (M) and rotation angle (𝜃) amplitudes and the phase angle (δ), determining uncertainties for these quantities is a challenge. In Table 4, we give estimates for these along with those for the sample geometry (H, R). In Fig. 8, we present the uncertainty of measured G^′ and G^″ for PS105 due to uncertainty in the radius, gap, torque, phase, and deflection angles. It can be observed that the slip-link prediction with self-consistent parameters obtained from PS206 is within this error.

Table 4 Uncertainties in the sample radius and instrument torque, gap, deflection, and phase angles for linear oscillatory measurements (Kashyap 2011)

Fig. 8

Error propagation of the sample radius, gap, torque, phase, and deflection angles to the moduli