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# Polymer Chemistry. The Basic Concepts - Himenz P.C.

Himenz P.C. Polymer Chemistry. The Basic Concepts - Copyright, 1984. - 736 p.
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for calculating such
(à)

(b)
Figure 8.3 Volume fraction polymer in equilibrium phases for chains of
different length, (a) Theoretical curves drawn for the indicated value of
n, with the interaction parameter as the ordinate. Note that x increases
downward. (Redrawn from Ref. 6.) (b) Experimental curves for the
molecular weights indicated, with temperature as the ordinate. [Reprinted
with permission from A. R. Shultz and P. J. Flory, J. Am. Chem. Soc.
74:4760 (1952), copyright 1952 by the American Chemical Society.]
534 Thermodynamics of Polymer Solutions
Phase Separation
535
a phase diagram. The procedure is somewhat involved, so we dispense with
the details and merely show the results of such calculations. Figure 8.3a
shows the fraction of polymer in the equilibrium phases plotted against x
f°r polymers with the indicated values of n. Increasing values of x are
toward the bottom of Fig. 8.3a, so the curves have the same shape as
would be obtained if temperature were the ordinate. Several features of
these theoretical phase diagrams are noteworthy:
1. The miscibility gap becomes progressively more lopsided as n
increases. This means that ô2 c occurs at lower concentrations and that
the tie line coordinates-particularly for the more dilute phase-are lower
for large n.
2. For the case of n -+ °°, the limiting values of ô2 c and xc are
shown to be 0 and 0.5, as required by Eqs. (8.59) and (8.60),
respectively.
3. Increasing positive values of x-moving downward in Fig. 8.3-
correspond to more endothermic values of AHm. Interpreting the latter in
terms of Eq. (8.49) means that systems for which 52 - is large might show
a miscibility gap for a given n, while complete miscibility is obtained
for the same polymer in a solvent for which 52 - is smaller. Decreasing
the solvent "goodness" by the addition of a less suitable solvent may
induce phase separation, at least for those molecules of large n.
4. If the poorer solvent is added incrementally to a system which is
poly-disperse with respect to molecular weight, the phase separation
affects molecules of larger n, while shorter chains are more uniformly
distributed. These ideas constitute the basis for one method of polymer
fractionation. We shall develop this topic in more detail in the next
section.
The curves shown in Fig. 8.3a are theoretical phase diagrams based
on the Flory-Huggins model. Comparing theoretical predictions with
experimental data thus provides our first opportunity to test the model.
Figure 8.3b shows phase diagrams for polyisobutylene samples with the
molecular weights indicated in diisobutyl ketone. Temperature variation
is used to change the solvent "goodness." The broken lines in Fig. 8.3b
are theoretical. We observe that the theory is qualitatively accurate,
but that there is considerable discrepancy in quantitative detail. In
particular, the experimental curves are considerably broader than
predicted by theory. Generally speaking, the theory is more successful in
accounting for xc than for ô2 c. It should be noted, however, that
critical phenomena are extremely sensitive to small variations in a
model-remember that at a critical point it only takes an infinitesimal
variation to push the system into different regions of phase behavior-so
Fig. 8.3b is not the best way to test the Flory-Huggins theory.
A far more satisfactory test of the Flory-Huggins theory is based on
the chemical potential. According to Eqs. (8.13) and (8.20),
536
Thermodynamics of Polymer Solutions
Hi - fiie = RT In
Pi
Pi
(8.61)
in which the vapor pressures are measurable quantities. According to the
Flory-Huggins theory }ii - }i° is given by Eq. (8.53). Combining these
two results gives
RT In
Pi
Pi
= RT
1Ï0! + (1 - ~ ) 02 +X022
(8.62)
which suggests that a plot of In (Pi/Pi°) - In 0t - (1 - 1/n) 02 versus
Ô22 should give a straight line of slope x- Figure 8.4 is such a plot for
several different polymer-solvent systems. In view of the complexity of
the phenomena
Ô22
Figure 8.4 Experimental test of Flory-Huggins theory by Eq. (8.62) for
the systems indicated. (From Ref. 3, used with permission.)
Polymer Fractionation
537
involved, the ability of the theoretical function to describe
experimental results is remarkably good. It is interesting to note that
the system for which the poorest agreement is observed-polystyrene in
chloroform-is one in which it is easy to rationalize an acid-base-type of
interaction between the hydrogen in CHCI3 and the electrons in the phenyl
groups. A chemical type interaction such as this is expected to show more
complicated concentration effects than the Flory-Huggins model predicts.
In the discussion of osmotic pressure in Sec. 8.8 we shall see that the
latter experiment can also be used to measure and thereby test the Flory-
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