Handbook of Solvents - George Wypych - ChemTech - Ventech!

182 Christian Wohlfarth
where:
I excess
excess scattering intensity
T absolute temperature
Δ mixG Gibbs free energy of mixing.
ϕi volume fractions of component i (= i th polymer species in the molecular distribution)
P(θ) properly averaged particle scattering factor
θ scattering angle from the transmitted beam
The determinant in the denominator is to be calculated at constant temperature and
2
2
pressure. It reduces to the single second derivative ( ∂ Δmix G/ ∂ ϕ2) P,T = ( ∂μ1/ ∂ϕ2)
P,T for
the case of a strictly binary monodisperse polymer solution. The average particle scattering
factor is of primary importance in studies of the size and shape of the macromolecules, but it
is merely a constant for thermodynamic considerations.
Conventionally, the so-called Rayleigh factor (or ratio) is applied:
R
() θ
excess
I r
≡
IV
0 0
2
2 ( 1+ cos θ)
where:
R(θ) Rayleigh factor
I0 incident intensity of unpolarized light
r 2
square of the distance between sample and detector
V0 detected scattering volume
and, neglecting P(θ), Equations [4.4.39 and 4.4.40] can be transformed to:
where:
R
() θ
≡
RTKϕ V
2 1
( ∂μ / ∂ϕ )
1 2
PT ,
[4.4.40]
[4.4.41]
V1 partial molar volume of the solvent in the polymer solution at temperature T
ϕ 2 volume fraction of the monodisperse polymer
K optical constant
The optical constant for unpolarized light summarizes the optical parameters
of the experiment:
2
4
( / 2)
/ ( 0)
2 2
K ≡ 2π n dn dc N λ
o PT , Av
[4.4.42]
where:
no refractive index of the pure solvent
n refractive index of the solution
c2 mass by volume concentration c2 =m2/ν NAv Avogadro’s number
λ0 wavelength of light in vacuum
For dilute polymer solutions, the partial derivative in Equation [4.4.41] is a weak function
of composition and the scattering intensity increases roughly proportional to the volume
fraction of the polymer. While Equation [4.4.41] permits any light scattering data to be