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Macromolecules, Structure 891

and the NMR chemical shift and other spectroscopic pa-
rameters. In this section we shall describe briefly some of
the methods for experimentally observing polymer chain
conformations in solution. Combined with such experi-
mental measurements, the results from RIS calculations
provide valuable insight into the structure and conforma-
tion of macromolecules.

FIGURE 34 Representative conformations of polymers: (a) 3 1
helix (gt)(gt)(gt) viewed down the helix axis; and (b) syndiotactic 1. The Characteristic Ratio, C ∞
(gg)(tt) sequences found in polypropylene.
Recall from our previous discussion (Section II.F) that
the intrinsic viscosity [η] is obtained by extrapolating the
specific viscosity, η sp to zero concentration:
define the energies of various conformers. The matrices

include the effects of both first- and second-order steric η − η 0
[η] = lim c. (69)
interactions. Using the RIS model and matrix operations, C 2 →0 η 0
any conformation-dependent property of a polymer chain
In this expression, η and η 0 are the polymer solution and
can be calculated, although such a calculation requires
solvent viscosities, respectively, and c is the polymer con-
some estimate of the bond rotational states and their asso-
centration in g/dL, or grams per deciliter.
ciated energies.
Flory has shown that the intrinsic viscosity may also be
A detailed account of RIS model and associated calcu-
expressed as
lations is beyond the scope of this article. However, we
2 3/2
summarize several of the conclusions concerning the re- [η] = φ r /M, (70)
lationship between structure and conformation of vinyl
where φ is a universal constant equal to approximately
polymers that can be drawn from a combination of RIS 21
2.5 × 10 and M is the polymer molecular weight.
calculationsandexperimentalresults.Isotacticchainstend 2
The mean squared end-to-end distance is r (see
to assume alternating gauche and trans rotational states.
Section II.B.1), which in a non-θ solvent may be some-
These ··· (gt)(gt)(gt) ··· sequences favor the formation 2
what greater than the unperturbed dimensions ¯ r . In real
of a 3 1 helix. [A 3 1 helix makes one turn for every three solvents the degree of expansion of the molecular coil is
designated as α, such that
monomer units. When viewed along the helix axis the 3 1
helix exhibits threefold symmetry, as shown in Fig. 34(a).] 2 2 2
r = α ¯ r . (71)
As we have seen, such helical sequences will have equal
probability of being either left- or right-handed. In so- For a θ solvent, where the expanding effect of excluded
lution these conformations will reverse chirality rapidly volume is exactly balanced by polymer–polymer interac-
and at random, as the conformation states have very short tions, α = 1.
lifetimes. The characteristic ratio C ∞ is the ratio between the un-
Syndiotactic chains, in contrast to the isotactic se- perturbed and the random walk dimensions and is given by
quences discussed above, generally favor tt conforma- 2 2
C ∞ = r /xl (72)
tions. There is a substantial occurrence of helical confor-
mations of the type ··· (gg)(tt) (gg)(tt) ··· . Figure 34(b) where x is the number of chain segments and l is the
shows a view of this helix along the helix axis. This is length of a single monomer unit, as before.
a special case found for syndiotactic polypropylene. We For all real polymers, C ∞ is greater than unity and is
shall return to this structure later (Section IV.C.3). In so- a measure of departure from a freely jointed or freely
lution the conformations responsible for such a helical rotating model.
2
sequence also undergo rapid interconversions. Substituting into Eq. (72) for r , we can obtain a re-
lationship between C ∞ and [η]:
1 [η] θ M 2/3
B. Experimental Observation of Chain C ∞ = (73)
xl 2 φ
Conformation in Solution
If we let m be the monomer molecular weight,
Examples of the conformation-dependent properties that 2/3
can be calculated from the RIS model include the dipole m [η] θ M
C ∞ = (74)
moment, the molar Kerr constant, the characteristic ratio, Ml 2 φ