Links:

Temperature Dependence
of Polymer Viscosity

Polymers exhibit behavior during flow and deformation which depends on both temperature and time (frequency), that is, the viscoelastic response of a polymer will depend on the shear history. Only for Newtonian liquids, and in a first approximation for low molecular weight polymers, is the viscosity independent of the shear history. In this case, the temperature dependence of the viscosity may be decribed by an Arrhenius equation of the form

where T is the absolute temperature, R the universal gas constant, A a material constant, and E the activation energy. Knowledge of the two material-dependent parameters E and A allows for the prediction of the viscosity for any temperature. The two parameters are usually determined from a plot of log η against 1/T which yields a straight line:

For most polymer melts a straight line can only be drawn over a fairly small temperature range of about 50°C. In general, the lower the shear rate and the molecular weight the better the agreement.

Doolittle¹ provided another equation that is much more accurate for entangled polymer systems. He postulated that the viscosity is an exponential function of the reciprocal of the fractional free volume f:

or

where A and B are constants and f is the free volume fraction,

f = v_f / v = (v - v_hc) / v

v_hc is the inaccessible volume and v the measured (molar) volume. Similar to the (molar) polymer volume, it can be assumed that the fractional free volume f increases linearly with temperature. Choosing the free volume at the glass transition temperature Tg as the reference state, the free volume as a function of temperature may be written as

f = f_g +  α_f (T - T_g)

where f_g is the fractional free volume at Tg and α_f is the thermal expansion coefficient. It has been suggested that the two parameters have the univerals values

f_g ≈ 0.025
α_f ≈ 4.8 x 10^-4 K^-1

Williams, Landel and Ferry (1955) have suggested that the viscosity η at a temperature T may be related to the viscosity η_g at the glass transition temperature:²

or

where a_T is the so called WLF shift factor. Using the universal values of f_g and α_f  yields two new constants:

C₁ = B / f_g ≈ 17.44
C₂ = f_g / α_f = 51.6 K

The two constants are not universal at all. Only polymers with isoviscose behavior will obey the WLF equation. For all other polymers, the two parameters might be very different.

Many polymers have melt viscosities of about 10¹³ poise at their glass transition temperature. Lets assume a polymer has a glass transition temperature of about 373 K then the viscosity at 423 K is:

or η = 2.61 x 10⁴ Poise.

References & Notes

1. A.K Doolittle, Journal of Applied Physics, Vol. 22, 12, 1471 - 1475 (1951)

2. M.L Williams, R.F. Landel and J.D. Ferry, J. Am. Chem. Soc. 77, 3701-3707 (1955)