Temperature Dependences of the Viscoelastic Response of Polymer Systems

Abstract

The most successful temperature dependence for the viscous flow [1,2], viscoelastic response [1], dielectric dispersion [3-5], nuclear magnetic resonance response [6-8] and dynamic light scattering [9-10] of polymers and supercooled liquids with various chemical structures is the Williams, Landel, and Ferry (WLF) equation [11,12] log J 0 s (T)h(T) J 0 s (T 0)h(T 0) % log t(T) t(T 0) ¼ log a T ¼ À C 1 (T À T 0) C 2 þ T À T 0 , (26:1) where J 0 s is the steady state recoverable compliance; h is the shear viscosity; t is a retardation or relaxation time; a T is the timescale shift factor; T 0 is the chosen reference temperature ; and C 1 and C 2 are characterizing constants. J 0 s (T) is a very weak function of the temperature. In fact in the temperature range where T=T g varies from 1.2 to 2.0, J 0 s has been found to be independent of temperature [13]. Therefore its variation is often ignored. It will be ignored in this chapter. Some authors identify magnitude variations with temperature which are reported as b T ¼ J 0 s (T)=J 0 s (T 0). Williams, Landel, and Ferry [12] reported that such an expression is valid for polymers over the temperature range T g < T < T g þ 100. When T g is chosen as the reference temperature, i.e., when the response curves measured at different temperatures are shifted primarily along the time or frequency scales to superimpose upon the response curve measured at T g , it was initially noted that the constants C g 1 and C g 2 assume values close to 17.448 and 51.68, respectively, for 17 polymers [12]. Individual treatment of the data on a wide range of polymers indicates that C g 1 may take values between 158 and 268 and C g 2 between 208 and 1308. The fit extends to temperatures below T g if the polymer is at its equilibrium density. The WLF expression has been shown [12] to be related to the Vogel-Fulcher-Tammann-Hesse equation [14-16], log t i ¼ log A þ (C=2:303) T À T 1 , (26:2) for h or t where A, C and T 1 are empirical constants. It follows that log a T ¼ C=2:303 T À T 0 À C=2:303 T 0 À T 1 : (26:3) This is identical to the WLF expression provided the Vogel parameters and the WLF parameters are related as C ¼ 2:303C 1 C 2 (26:4) and T 0 À T 1 ¼ C 2 : (26:5) 26.2 RELATION OF WLF EQUATION TO FREE VOLUME The WLF equation for a T has been rationalized in terms of Doolittle's free volume theory [17]. According to this theory 455