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

Macromolecules, Structure 869

According to this method, the entropy and enthalpy of
mixing are given by
S = −k(N 1 ln v 1 + N 2 ln v 2 ), (15)
H = χkTN 1 v 2 , (16)
where χ is the Flory–Huggins interaction parameter. Be-
cause G = H − T S, the free energy of mixing is
given by

G = kT (N 1 ln v 1 + N 2 ln v 2 + χ N 1 v 2 ). (17) FIGURE 10 (a) Dilute coils swollen in a good solvent, below the
overlap concentration. (b) Transition region c where the coils be-
∗
gin to overlap. (c) Semidilute region where coils overlap and dis-
This master free-energy relationship can be used to ob- 0.5
play ideal size proportional to χ .
tain experimentally measurable quantities. These quanti-
ties can be used to determine molecular weight, as we shall
C. Light Scattering
see in the rest of this section.
For example, the chemical potential of a nonelectrolyte Light scattering from polymer solutions can be used to
7
−1
4
solution can be expressed as measure molecular weights between 10 and 10 g mol .
This method is very useful, as it provides the weight av-
2 3
µ 1 = −RTV 1 A 1 C 1 + A 2 C + A 3 C + ··· , (18) erage molecular weight, the second virial coefficient, and
2
2
the radius of gyration.
and the osmotic pressure π is Polymer characterization by light scattering takes ad-
vantage of the fact that light is scattered whenever a beam
π = − µ 1 /V 1 , (19) of light impinges on matter. The electromagnetic wave in-
teracts with the induced oscillating dipole in the molecule,
where V 1 is the molecular volume of the solvent and x
thereby emitting light. We shall first develop the expres-
is the number of chain segments. Here, A 2 is the second
sions for light scattering from small particles. We then
virial coefficient. When combined, Eqs. (18) and (19) give
amend these expressions to account for the situation when
the wavelength of light and the size of the particle are
−1 2
π/C 2 = RTM + RTA 2 C 2 + RTA 3 C + ··· . (20)
2 2 comparable. This latter condition is generally true for
polymers.
We shall see later that light scattering measurements
For dilute solutions of small molecules, we can treat
provide experimental determination of the second virial
each molecule as an independent point scatter, and the
coefficient.
scattering intensity will be proportional to the number of
The Flory–Huggins theory holds for semidilute solu-
molecules. The Rayleigh’s ratio R θ , is the ratio between
tions, but does not address the consequence that very
the scattered intensity (observed at angle θ from incident)
dilute polymer solutions must be discontinuous. Other
and the incident beam. It is given by
theoretical treatments have been developed. The Flory–
4
4
2
2
Krigbaum theory treats the system in terms of excluded R θ = 8π να (1 + cos θ)/λ , (21)
volume effects. In this treatment, the θ temperature is where ν is the number of molecules, α is the molecular
the temperature at which the partial molar free energy polarizability, and λ is the wavelength of light. The molec-
from polymer–solvent interactions is zero, and the poly- ular polarizability α is related to the dielectric constant D
mer adopts its unperturbed dimensions. The second virial or to the refractive index n of the medium. In systems other
coefficient goes to zero at the θ point. than gases, destructive interference occurs and reduces the
A corresponding state theory has been set forth to cor- intensity of the scattered light.
rect the shortcomings of the above treatments. This the- The amount of this reduction in scattering intensity
ory can predict the volume changes that occur on mix- can be accounted for through fluctuation theory. In this
ing to 10–15% of the experimental value. More recently,
treatment the local fluctuation of the dielectric constant
deGennes has developed scaling concepts to describe the
is evaluated. Physically, the fluctuation in the dielectric
concentration region c (where polymer chains in solution
∗
constant arises from density or concentration fluctuations.
begin to overlap), between the dilute (c < c ) and semidi-
∗
According to Debye, the mean square concentration fluc-
lute (c > c ) regimes (see Fig. 10). Scaling concepts make
∗
tuation is given by
correct predictions about the dependence of the second kT
virial coefficient on the number of links in a given poly- 2 , (22)
( c 2 ) = 2 2
mer chain. d G dc 2