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Fig. 2-9 Number-fraction distribution curve for linear polymerization. Plot 1, p 0:9600; plot 2, p 0:9875; plot 3, p 0:9950. After Howard [1961] (by permission of Iliffe Books, London and Elsevier, Oxford).

(E27)



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number basis there are more monomer molecules than any polymer species regardless of the extent of reaction. Although the number of monomer molecules decreases as p increases, they are still the most plentiful species. The situation is quite different for the weight distribution of molecular weights. On a weight basis, the proportion of low-molecular-weight species is very small and decreases as p increases. The maxima in Fig. 2-10 occur at x 1= ln p , which is very near the number-average degree of polymerization given by Eq. 2-27. The experimental determination of a molecular weight distribution is frequently obtained in an integral form in which the combined or cumulative weight fraction Ix of all polymer molecules having degrees of polymerization up to and including x are plotted against x. For this purpose it is useful to express the Flory distribution function in terms of Ix . This is done by summing wx values from Eq. 2-89





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(E28)

Ix X wx 1 p 2 X xp x 1 2-90

Fig. 2-10 Weight fraction distribution plot for linear polymerization. Plot 1, p 0:9600; plot 2, p 0:9875; plot 3, p 0:9950. After Howard [1961] (by permission of Iliffe Books, London and Elsevier, Oxford).

where p is a positive real number, v is the speed, c is the speed of light, and 'Y = 1/ V1 - (v/c)2. Then, as lim-y_= hb) = c, we find that

Ix 1 px xpx 1 p 2-91

lim {

2 3 4

The limit of Ix at high values of x is 1 and is reached at progressively higher values of x for higher conversions. 2-7b Breadth of Molecular Weight Distribution

g c5(x - ct),

The number- and weight-average degree of polymerization X n and X w can be derived from the number and weight distribution functions, respectively. The number- and weight-average molecular weights have been de ned by Eqs. 1-15 and 1-17. Dividing Eq. 1-15 by the weight Mo of a structural unit yields the number-average degree of polymerization as

(F.29)

Xn xNx xN x Nx 2-92

where in this case,

X n xpx 1 1 p 2-93

(F.30)

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Xn 1 1 p 2-27

At first sight the integral in (F.30) seems to vanish, as a change of variables Xl = xp for the integral of the first term would make it identical to the integral of the second term. However, we must take extra care here because the separate integration of each term diverges. So the right procedure is first to consider

X w xwx 2-94

X w x2 px 1 1 p 2 2-95

(F.31) and take the limit of 1] - 00 only at the end. Using (F.31) in (F.30) with b = p for the first term in the integrand of (F.30) and b = 1 for the second, we obtain

2 3 4

Xw 1 p 1 p 2-96

[:dx{k-~}

Xw 1 p Xn 2-97

r I 1 +(1]/p)2 +1]/p ~ +1]1 V1 = 71':'~ n V1 + (1]/p)2 -1]/p V1 + rp -1]

The ratio X w =X n is synonymous with the ratio M w =M n discussed in Sec. 1-4. It is a measure of the polydispersity of a polymer sample. The value of X w =X n increases with the extent of reaction and approaches 2 in the limit of large extents of reaction. The ratio X w =X n is also referred to as the polydispersity index (PDI). The most probable distribution of Flory is generally well established, although its experimental veri cation has been somewhat limited. Direct evidence for the most probable

71-=

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