Handbook of Solvents - George Wypych - ChemTech - Ventech!

362 Semyon Levitsky, Zinoviy Shulman
The differences between other relaxation times in the both spectra are more essential:
the distribution, predicted by the KRZ model, is much narrower than that predicted by the
Rouse theory.
The KSR and Rouse models were subjected to numerous experimental tests. A reasonably
good agreement between the theoretical predictions and experimental data was demonstrated
for a variety of dilute polymeric solutions. 14 Further advance in the
molecular-kinetic approach to description of relaxation processes in polymeric systems
have brought about more sophisticated models. 15,16 They improve the classical results by
taking into account additional factors and/or considering diverse frequency, temperature,
and concentration ranges, etc. For the aims of computer simulation of the polymeric liquid
dynamics in hydrodynamic problems, either simple approximations of the spectrum, F 1(λ),
or the model of subchains are usually used. Spriggs law 17 is the most used approximation
Z
λ = λ / k , z ≥
[7.2.29]
1k 11 2
The molecular theory predicts strong temperature dependence of the relaxation characteristics
of polymeric systems that is described by the time-temperature superposition
(TTS) principle. 18 This principle is based on numerous experimental data and states that
with the change in temperature the relaxation spectrum as a whole shifts in a self-similar
manner along t axis. Therefore, dynamic functions corresponding to different temperatures
are similar to each other in shape but are shifted along the frequency axis by the value a T; the
latter is named the temperature-shift factor. With ωa T for an argument it becomes possible
to plot temperature-invariant curves Re{G 1*(ω a T)} and Im{G 1*(ω a T)}. The temperature
dependence of a T is defined by the formula
a
T
( )
( p 0 s 0 )
( T0) T0 p() T − s()
T
( TT ) ( T) − ( T)
ρ η η
=
ρ η η
[7.2.30]
The dependence of viscosity on the temperature can be described by the activation
theory 19
−1 −1
[ E ( RT ) ( T0 T 1) ] 0 E ( RT0) ( T0 T 1)
[ ]
η = η exp / − , η = η exp / − [7.2.31]
p p0 p G 0
s s s G
where:
Ep,Es activation energies for the solution and solvent, respectively
The Es value is usually about 10 to 20 kJ/mol. For low-concentrated solutions of polymers
with moderate molecular masses, the difference between these two activation energies,
ΔE=Ep-Es, does not exceed usually 10 kJ/mol. 18,20 For low-concentrated solutions of
certain polymers in thermodynamically bad solvents negative ΔE values were reported. 20
The Newtonian viscosity of solution related to the polymer concentration can be evaluated,
for example, using Martin equation 18
where:
( ) []
η / η = 1 + c~ exp k c~ , c~ = c η
[7.2.32]
p s M
~ c reduced concentration of polymer in the solution
[η] intrinsic viscosity