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

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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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