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Macromolecules, Structure 867
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present case we are dealing with free radical reactions methods for measuring M n and M w . However, before we
and need to make several assumptions. First, the concen- can discuss experimental techniques for measuring molec-
tration of the monomer is considered to remain constant ular weights of polymers, we must digress briefly into the
throughout the polymerization. Second, we assume that solution properties of macromolecules.
the rate of initiation is constant throughout the polymer-
ization. Finally, we assume that termination occurs either B. Solution Properties of Macromolecules
bydisproportionationorbychaintransfer(shortlyweshall
Solutions of macromolecules differ greatly from solutions
examine the effects of relaxing this third assumption).
of small molecules. In dilute solutions of small molecules,
A given radical can either propagate or terminate. We
the molecules are effectively separated from each other by
shall assign p to the probability that the radical will prop-
solvent. In contrast, dilute solutions of macromolecules
agate. We can then obtain the probability of finding an
have high local concentrations of monomer units because
x-mer, which requires x − 1 propagation steps [probabil-
ity = p (x −1) ] and one termination [probability = (1 − p)]. the chain segments are covalently bonded together. Fur-
thermore, solutions of small molecules contain materials
As before, the probability of finding a chain x units long
that are generally monodisperse in molecular weight. So-
is then given by
lutions of polymers, on the other hand, contain macro-
p (x −1) (1 − p), (7) molecules that have a dispersion in molecular weight.
In a general sense, liquids made up of small molecules
and the number of molecules N x is
are Newtonian in behavior. That is, they obey Newton’s
N x = Np (x −1) (1 − p), (8) equation defining the viscosity of a fluid as the coefficient
of proportionality between the shear stress and the velocity
where N is the total number of molecules. The number
gradient. Liquids containing polymers, on the other hand,
average and weight average degrees of polymerization are
display non-Newtonian behavior. Some examples of non-
obtained as before, and are given by
Newtonian behavior are shown schematically in Fig. 8.
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X n = 1/(1 − p), (9) These examples are described more fully in the figure
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X w = (1 + p)/(1 − p). (10) legend.
When propagation is highly favored over termination, 1. Polymer Chain End-to-End
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p → 1 and X w / X n → 2. Distance and Radius of Gyration
So far we have considered the probability of finding
In a “good” solvent, where polymer–solvent interactions
an x-mer when termination occurs by disproportionation
are favorable, a polymer chain adopts a somewhat ex-
or chain transfer. These termination mechanisms are uni-
tended shape. In a “poor” solvent, the polymer tends to
molecular in polymer—that is, only one polymer chain is
coil back on itself because polymer–polymer interactions
involved in these reactions. Termination could also occur
are more favorable than polymer–solvent interactions. We
by chain combination, in which case the termination step
shall see later that these interactions are temperature-
requires two chains and is thus bimolecular in polymer.
dependent and are exactly balanced at the θ temperature.
A chain combination to produce an x-mer requires x − 2
propagation steps [probability = p (x −2) ] and two termina- In an elementary approach, we can treat a polymer chain
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2
tion steps [probability = (1 − p) ]. In this case, X n and as being freely jointed. In such a model there are no restric-
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X w are given by tions on the bond angles between monomer units. Statisti-
cal random walk calculations show that a polymer with x
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X n = 2/(1 − p), (11) monomer units each of length I at θ conditions has a root
2 1/2 given by
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X w = (2 + p)/(1 − p), (12) mean square (rms) end-to-end distance ¯ r
2 1/2 = lx 1/2 . (13)
¯ r
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and thus X w / X n → 1.5 as p → 1.
This freely jointed chain model is clearly unreasonable for
a real polymer, as bond hybridizations place restrictions
3. Relationship between Degree of
on bond angles. We shall see later (Section IV) that this
Polymerization and Molecular Weight
model can be improved by taking these restrictions into
Now that we have derived expressions for the number av- account. The actual end-to-end distance can be determined
2 1/2
erage and weight average degrees of polymerization, we according to Eq. (68). The radius of gyration ¯ s is the
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can convert these averages into M n and M w by multiplying rms distance of the units of a chain from its center of
the former quantities by the monomer molecular weight. gravity. The radius of gyration and the actual end-to-end
In the following sections we shall describe experimental distance are related by