Measurement of pressure coefficient of melt viscosity: drag flow versus capillary flow

Abstract

The pressure coefficient of viscosity of poly(α-methylstyrene-co-acrylonitrile) was measured using a high-pressure sliding plate rheometer (HPSPR) and two types of capillary rheometer: a piston-driven device with a throttle at the exit [piston capillary rheometer with throttle (PCRWT)] operated at a fixed flow rate, and a counter-pressure nitrogen capillary rheometer (CPNCR) operated at a fixed pressure drop. In the HPSPR, the pressure, shear rate, density, and viscosity are all uniform throughout the sample, while the analysis of capillary data is complicated by the axial pressure gradient and the radial shear rate gradient. The polymer was found to be piezorheologically simple, and the HPSPR data indicated that the pressure coefficient of viscosity β ≡ dln(a _P)/dP decreased slightly with increasing pressure at high pressure. While β from PCRWT data from different laboratories and instruments agreed fairly well, the β values were on average about 2/3 of that from the HPSPR. The CPNCR yields β about 18% lower than that of the HPSPR.

Appendix

Equation 26 should be concave-up in a semi-log plot, while Eq. 22 gives a linear prediction. However, both predictions by Eqs. 22 and 26 look straight in Fig. 15, and one can thus ask two questions: (1) why two different equations give similar straight predictions in a semi-log plot and (2) why β _throttle and β, obtained by fitting to each equation giving the same prediction, are different.

• (1) Both ${\mathrm{d}\ln \left( {\Delta P} \right)} \mathord{\left/ {\vphantom {{\mathrm{d}\ln \left( {\Delta P} \right)} {\mathrm{d}P_\mathrm{e} }}} \right. \kern-\nulldelimiterspace} {\mathrm{d}P_\mathrm{e} }$ and ${\mathrm{d}^2\ln \left( {\Delta P} \right)} \mathord{\left/ {\vphantom {{\mathrm{d}^2\ln \left( {\Delta P} \right)} {\mathrm{d}P_\mathrm{e}^2 }}} \right. \kern-\nulldelimiterspace} {\mathrm{d}P_\mathrm{e}^2 }$ of Eq. 26 are positive, so that the solid line predicted by Eq. 26 in a semi-log plot is mathematically concave-up and rising, having certain curvature, but this looks straight in Fig. 15, to the eye. At high P _e (above 100 MPa), the prediction by Eq. 26 with the same parameters shows an upswing, so that the prediction of Eq. 26 will not seem straight any longer.

• (2) Since $\Delta P_0 \beta _{\mathrm{sr}} \exp \left( {\beta _{\mathrm{sr}} P_\mathrm{e} } \right)<0.2$ in Eq. 26, we can use the Maclaurin series expansion of the natural logarithm in Eq. 26, and Eq. 26 becomes Eq. 35:

  $$\label{Equ35} \begin{array}{lll} \Delta P&=&\frac{1}{\beta _{\mathrm{sr}} }\sum\limits_{j=1}^\infty {\frac{\left[ {\Delta P_0 \beta _{\mathrm{sr}} \exp \left( {\beta _{\mathrm{sr}} P_\mathrm{e} } \right)} \right]^j}{j}} \\ &=&\Delta P_0 \exp \left( {\beta _{\mathrm{sr}} P_\mathrm{e} } \right)+\beta _{\mathrm{sr}} \frac{\left[ {\Delta P_0 \exp \left( {\beta _{\mathrm{sr}} P_\mathrm{e} } \right)} \right]^2}{2} \\ &&+{\kern2pt}\beta _{\mathrm{sr}}^2 \frac{\left[ {\Delta P_0 \exp \left( {\beta _{\mathrm{sr}} P_\mathrm{e} } \right)} \right]^3}{3}+\cdot \cdot \cdot \end{array} $$

  (35)

  The first term of Eq. 35 is the same as the right-hand side of Eq. 22:

  $$ \label{Equ36} \Delta P=\Delta P_0 \exp \left( {\beta _{\mathrm{throttle}} P_\mathrm{e} } \right) $$

  (22)

  If β _sr = β _throttle, the sum of the terms after the first term in Eq. 35 must be zero or negligible because Eqs. 22 and 35 show the almost same prediction for ΔP. However, this summation is well larger than zero, and thus β _sr < β _throttle to give the same prediction. It should be noted that the straight prediction of Eq. 26 in Fig. 15 is not because Eq. 26 can approximate to the first term of Eq. 35 but because the prediction of the whole Eq. 26 can look straight to the human scale even though it is not mathematically.