Handbook of Solvents - George Wypych - ChemTech - Ventech!

4.4 Measurement of solvent activity 185
A2,A3, ... second, third, ... osmotic virial coefficients
M2 molar mass of the polymer
According to the scattering theory of polydisperse polymers (please see, for example,
the book written by Strobl), 147 the molar mass of the polymer, M2, is equal to the mass average,
Mw, of polydisperse polymers. The exact application of Equation [4.4.47] is to be made
again at the scattering angle θ = 0. The constant K is now given by:
( Av )
4 2 2 4
K ≡ e Δ z / m N c
[4.4.48]
where:
e electron charge
m electron mass
Δz excess number of electrons
c speed of light in vacuum
In principle, there is agreement between values of second virial coefficients from light
scattering or X-ray scattering. Okano et al. 167,168 applied SAXS to semidilute solutions of
polystyrene in cyclohexane in the poor solvent regime and obtained virial coefficients in
good agreement with liquid-liquid data from a coexistence curve. Takada et al. 169 provided a
more recent example for poly(vinyl methyl ether) in cyclohexane, Horkay et al. 170 for
poly(vinyl acetate) in toluene and poly(dimethyl siloxane) in octane. In comparison to data
from osmotic pressure and neutron scattering, they observed good agreement.
4.4.3.2.4 Neutron scattering
Neutron scattering is an important method for investigating conformation and dynamics of
polymer molecules, Higgins, 171 or polymer mixtures, Hammouda. 172 A recent presentation
of various techniques can be found in a new book edited by Pethrick and Dawkins. 26 Thermodynamics
of polymer solutions is not the first task in neutron scattering experiments. The
general set-up of the neutron scattering experiment is equivalent to the one used for light
scattering, but applying a neutron source, and elastic neutron scattering at small angles
(SANS) can be applied like light scattering or X-ray scattering to obtain second virial coefficients
in dilute solutions. Similarly to the scattering of photons, it is the difference in scattering
power between solvent molecules and polymer segments which determines the
absolute scattering intensity. Formally, the virial equation has the same form as Equations
[4.4.43 and 47], again neglecting P(θ):
and
where:
( ) / ( ) ( / )
2
Kc Σ θ = 1 M + 2Ac+ 3Ac+�
[4.4.49]
2 2 2 2 3 2
( 2 1 1 2 )
K = b −b
ρν , / N
[4.4.50]
spez Av
c2 mass by volume concentration c2 =m2/ν A2,A3, ... second, third, ... osmotic virial coefficients
M2 molar mass of the polymer
Σ( θ) differential scattering cross section per volume unit
K contrast factor for neutron scattering
b1,b2 densities of solvent and polymer scattering length
density of the solvent
ρ 1