Page 86 - Engineering Plastics Handbook

60 Introduction

The WLF equation for a does not work for polymers at temperatures
T
o o
100 C (232 F) above their T . For these polymers an Arrhenius formula
g
is used.
To calculate the time-temperature superposition shift factor for poly-
o o
mers at temperatures about 100 C (232 F) above their T , the following
g
Arrhenius formula is used.
∆ E
η= Aexp
RT
g
where η= viscosity
A = Arrhenius constant
∆E = activation energy for flow
R = gas law constant
g
T = temperature, K

From this expression, the time-temperature superposition shift factor
a becomes [2]
T
 ∆ E  1 1  
a = exp   −  
T
  R  T T ref  
g
Another route to calculate the shift factor is to equate the natural log
of a (ln a ) to free volume ƒ when ƒ is assumed to be linearly depend-
T
T
ent on temperature.
Loss Modulus

Viscoelastic behavior can be viewed as three fundamental modulus char-
acteristics: G*or E*= complex modulus, G′ or E′= storage or dynamic mod-
ulus, and G′′ or E′′ = loss or viscous modulus. The moduli are related by
the angle of phase lag δ in stress-to-strain phase lag. They are derived from
measurements of the complex modulus and phase angle δ relationships of
stress to strain, by dynamic mechanical analysis (DMA) using a
®
Rheometric Solids Analyzer, RSA, TM supplied by TA Instruments [19].
Further information on loss modulus, storage modulus, and DMAis found
in Chap. 2, “Products and Designs,” and Chap. 3, “Properties.”
To calculate the loss modulus as a function of time [2], use
T ρ
G ′′ = G( ′′ ) ref ref
T ref T test ρ
T
test test
where ′′ = loss modulus as a linear viscoelastic property at
G
T ref
reference temperature
G ′′ = loss modulus at test temperature
T ref
T ref = reference temperature, K