Page 121 - Carrahers_Polymer_Chemistry,_Eighth_Edition

84 Carraher’s Polymer Chemistry

For linear polymers at their theta temperature, that is, the temperature where the chain attains
unperturbed dimensions, the Flory equation resembles the Mark–Houwink equation, where α is
equal to 1.0 as shown below.

3
[η] = KM α = K′ M 1/2 (3.33)
1/2
The intrinsic viscosity of a solution, like the melt viscosity, is temperature dependent and
decreases as temperature increases.

[η] = Ae E/RT (3.34)

While the description of viscosity is complex, the relative viscosity is directly related to the

flow-through times using the same viscometer as shown in Equation 3.29, where t and t are the
o

flow times for the polymer solution and solvent, respectively, and the density of the solution (ρ) and
solvent (ρ ) are related as in Equation 3.35.
o
η t ρ
η = ρ t = η r (3.35)
0 0 0
Since the densities of the dilute solution and solvent are almost the same, they are normally can-
celled giving Equation 3.36.
η t
η = t = η r (3.36)
0 0

Thus, the relative viscosity is simply a ratio of flow times for the polymer solution and solvent.
Reduced viscosity is related to the LVN by a virial equation 3.37
η sp = η η 2 η 3 2 +⋅⋅⋅
k
c
k

1
c [] + [] + [] c (3.37)
For most solutions, Equation 3.37 reduces to the Huggins viscosity relationship, Equation 3.38,
η sp 2
η
η
k
c = [] + [] c (3.38)
1
which allows [η] to be determined from the intercept of the plot of η /c versus c and is the basis for
sp
the top plot given in Figure 3.17.
Another relationship often used in determining [η] is called the inherent viscosity equation and
is given in Equation 3.39.
η
η
η
2

ln r = [] k 2 [] c (3.39)
c
Again, a plot of In η /c versus c gives a straight line with the intercept [η] (or LVN) after extrap-
r
olation to zero polymer concentration. This is the basis of the lower plot in Figure 3.17. While k and
1
k are mathematically such that
2
k + k = 0.5 (3.40)
1 2
many systems appear not to follow this relationship.

9/14/2010 3:37:11 PM
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K10478.indb 84 9/14/2010 3:37:11 PM