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Encyclopedia of Physical Science and Technology EN008C-602 July 25, 2001 20:31

866 Macromolecules, Structure

N x = Np (x −1) (1 − p). (2)
The number average and weight average degrees of poly-
merization are calculated from
¯
X n = N x x N x , (3)
¯
X w = W x x W x = (N x x)x N x x , (4)
where W x is the weight fraction of molecules of degree of
polymerization x. It is evident that
FIGURE 6 Melting curves for partially crystalline polymers as a ¯
X n = 1/(1 − p), (5)
function of molecular weight.
and it can be shown that
¯
II. MOLECULAR WEIGHT AND SIZE X w = (1 + p)/(1 − p). (6)
¯
¯
The polydispersity is characterized by X w / X n or (1 + p).
A. Molecular Weight Distributions Thus, as the reaction approaches complete conversion,
and Averages
¯
¯
(p → 1), X w /X n → 2. A monodisperse polymer on the
We have already seen that macromolecular materials, other hand, is one where all chains have identical molec-
¯
¯
whether prepared by chain or step methods, usually con- ular weights, and X w /X n = 1. Monodispersity is difficult
tain a broad distribution of molecular weights. Two im- to achieve in synthetic polymers, but occurs often for
portant averages specify the molecular weight distribution biomolecules.
of a polymer. These are the number average molecular On a number basis, there are more smaller chains than
¯
¯
weight M n and the weight average molecular weight M w . larger ones. However, these short chains comprise a small
We shall first illustrate the derivation of related quanti- fraction of the total weight. Figure 7 illustrates this point. It
ties, the number average and weight average degrees of shows the most probable distribution of polymer weights
¯
¯
polymerization X n and X w . Statistical derivations of these for an addition polymer of number average degree of poly-
¯
¯
¯
averages depend upon the propagation mechanism, so we merization X n of 100. Note that X n and X w are different
shall treat step polymerization and chain polymerization by a factor of 2.
separatelyinthefollowingsections.Monodispersityisdif-
ficult to achieve in step or chain growth polymerization,
2. Chain Polymerization
but a very low polydispersity (1.04) is possible with living
polymerization. Statistical derivations of number average and weight av-
erage molecular weights for chain polymerization follow
arguments similar to those outlined in Eqs. (1)–(6). In the
1. Step Polymerization
Let us consider a difunctional monomer, A—B. We shall
let p be the probability that one end (let us say A) has
reacted at time t. Therefore the probability of finding an
unreacted A group is (1 − p). Since we want to calculate
the number of molecules that are a particular number of
units (let us say x) in length, we need to determine the
probability of finding a chain which is x units long. Such
a chain would have (x − 1) A groups that had reacted,
and this probability would be p (x −1) . One A group would
be unreacted, and its probability would be (1 − p). So,
the total probability of finding a chain x units in length is
given by
p (x −1) (1 − p). (1)
FIGURE 7 Weight fraction of degree of polymerization of x for an
If there are N molecules, the fraction of them that are x addition polymer having a number average degree of polymeriza-
2
units in length, N x , is given by tion ¯ X n of 100; W x is calculated by W x = (x/ ¯ X ) exp(−x/ ¯ X n ).
n