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

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n-mer in phase P, for large values of n, will be maximized if the volume

of the P phase is kept to a minimum compared to phase Q. Stated simply,

the longer chains are concentrated in one of the phases, while the

shorter chains are relatively unaffected by the phase separation and

distribute themselves evenly and in proportion to the volume of the

phases. As a practical matter, then, keeping the volume of the dilute

phase as large as possible allows the optimum partitioning of the largest

molecules into the smaller but more discriminating concentrated phase.

The relative amounts of various n-mers in a fractionated sample is

examined numerically in the following example.

Example 8.4

Evaluate A in Eq. (8.64), assuming the polymer with n = 200 is divided

equally between the two phases and taking the ratio of phase volumes R to

be 10-1, 10~2, and 10"3. Use these A values to evaluate the relative

amounts of n-mer in the two phases for polymers with n = 100, 400, 600,

and 800. Comment on the significance of the numerical results.

Solution

The ratio of Eq. (8.66) to Eq. (8.67) gives the ratio of the

concentrations of n-mers in phases P and Q: fn p/fn Q = ReAn. Taking this

ratio to be unity for n = 200 gives ReA(20°) = 1, which is readily solved

for A using the R values given. Once these A values are obtained, fn p/fn

q can be evaluated for the required n values. For the phase volume ratios

under consideration, the corresponding values of A are listed below; also

tabulated are the ratios fn P/fn Q for the various n's:

R = Vp/Vq: 10"1 io-2 10~3

An=200' 1.15 X 10~2 2.30 X 10"2 3.45 X 10"2

100 0.32 0.099 0.032

400 9.95 99.0 985

600 99.9 9850 9.77 X 10s

800 990 9.80 X 10s 9.69 X 10(r)

Note that phase P is more concentrated in polymer, although smaller in

volume than phase Q. It is apparent from these values that the

combination of large n's

540

Thermodynamics of Polymer Solutions

and large differences in the volumes of the separated phases gives rise

to the most efficient segregation of polymer between phases. Also note

that the requirement that 200-mers be evenly distributed between the

phases creates the situation in which 100-mers are present in higher

proportion in the more dilute phase.

•

Figure 8.5 illustrates the sort of separation this approach predicts.

Curve A in Fig. 8.5 shows the weight fraction of various n-mers plotted

as a function of n. Comparison with Fig. 6.7 shows that the distribution

is typical of those obtained in random polymerization. Curve Â shows the

distribution of molecular weights in the more dilute phase-the coacervate

extract-calculated for the volumes of the two phases in the proportion

100:1. The distribution in the concentrated phase is shown as curve Ñ; it

is given by the difference between curves A and B.

In practice, such a fractionation experiment could be carried out by

either lowering the temperature or adding a poor solvent. In either case

good temperature control during the experiment is important. Note that

the addition of a poor solvent converts the system to one containing

three components, so it is apparent that the two-component Flory-Huggins

model is at best only qualitatively descriptive of the situation. A more

accurate description would require a

n

Figure 8.5 Theoretical plots of weight fraction n-mers versus n for

unfractionated polymer (A), the dilute phase (B), and the concentrated

phase (C) (drawn with R = 10~2). (Adapted from Ref. 1.)

Polymer Fractionation

541

triangular phase diagram. The onset of precipitation is marked by the

appearance of turbidity. In keeping with the principle outlined above,

only a small volume of the precipitated phase is allowed to form. Then

the sample is allowed to stand undisturbed until the two phases can be

physically separated. This step can require quite a long wait.

This procedure is then repeated by decreasing the solvent goodness

even further by another decrease in temperature or addition of

precipitant. In this manner a set of fractions such as those shown in

Fig. 8.6 are obtained from the initial distribution. In Fig. 8.6 curve A

again represents the initial distribution. Eight fractions are obtained

by precipitating successive portions of the polymer of progressively

lower molecular weights as shown, until the dilute phase contains only

the lowest molecular weight fraction as a residue. The curves in Fig. 8.6

are calculated for a phase volume ratio of 1000:1.

Figure 8.6 shows that the individual fractions still contain a

considerable range of chain lengths, with this effect becoming less

pronounced in the later cuts. In addition, there is a definite overlap

among the fractions. Nevertheless, the approach results in a sharpening

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