Page 515 - Carrahers_Polymer_Chemistry,_Eighth_Edition

478 Carraher’s Polymer Chemistry

E , for this jump, which is dependent of the size and shape of the diffusing molecule and the size of
D
the holes in the membrane, that is, the free volume. For permeation, the activation energy is on the
order of 20–40 kJ/mol. The values for E are somewhat higher.
D
The rate of diffusion, D, and the rate of permeability, P, increase exponentially as shown by the
Arrhenius equation for diffusion Equation 13.20.
D = D e −E /RT (13.20)
D
o
where D is the rate of diffusion at some base temperature, R is the ideal gas constant, and T is the
o
Kelvin temperature.
Diffusion can be described by Fick’s law Equation 13.21,
 dc 
F =− D   dx   (13.21)

where F is the weight of diffusate crossing a unit area per unit time and is proportional to the con-
centration gradient, dc/dx. The proportionality constant D is directly related to the pressure differ-
ential across the membrane and inversely related to the membrane thickness.
When the diffusion coeffi cient D is dependent on concentration, the diffusion process is said to
be Fickian. In such cases, D is inversely related to solubility, S, and to permeability, P, as follows:
P
D
S (13.22)

It has been suggested that linear alkanes diffuse through the holes of membranes by alignment
with the segments of the organic polymer chains. Such alignments are more difficult for branched

alkanes so that they diffuse more slowly.
The diffusion coeffi cient D is inversely related to the cross-link density of vulcanized rubbers.
When D is extrapolated to zero concentration of the diffusing small molecules, it is related to the
distance between the cross-links. Thus, as the cross-link density increases D becomes smaller, as

expected. Further, the diffusion coefficient is less for crystalline polymers in comparison to the
same polymer except in the amorphous state. In fact, this can be roughly stated as follows:
D = D (1 − x) (13.23)
a
c
where D is the diffusion constant for the crystalline material, D is the diffusion constant for the
c
a
amorphous material, and x is the extent of crystallinity.
The permeability coeffi cient P is related to the diffusion coeffi cient D and the solubility coeffi -
cient S as shown by Henry’s law:
P = DS (13.24)
Thus, the permeability values are high when the solubility parameter of the diffusion molecules
are similar to that of the polymer fi lm.
While some polymers exhibit non-Fickian diffusion below the T , many of these become Fickian
g
as the temperature is raised to above the T .
g
13.16 SUMMARY

1. The American Society for Testing and Materials (ASTM) and comparable organizations
throughout the world have established meaningful standards for the testing of polymers.
2. Spectroscopic techniques that are useful for small molecules are equally as important with
macromolecules. These techniques give both structural data and data related to the morphol-
ogy of the polymers.

9/14/2010 3:42:19 PM
K10478.indb 478 9/14/2010 3:42:19 PM
K10478.indb 478