Handbook of Solvents - George Wypych - ChemTech - Ventech!

4.2 Effect of system variables on solubility 131
Phase diagrams are characterized by critical temperatures, spinodals and binodals. A
binodal is a curve connecting equilibrium structures of a stratified system. A spinodal is a
curve defining boundary of metastables condition (Fig.4.2.3).
Binodals are evaluated experimentally by light scattering at cloud point, 29 by volume
changes of coexisting phases, 30 or by the electron probe R-spectral analysis. 31
It is possible to calculate phase behavior, considering that binodals correspond to a
condition:
where:
′ =
( Δμ ) ( Δμ
)
i i
″
i a component of a solution
′ ″
, changes of a chemical potential in phases of a stratified system
( Δμ i) ( Δμ
i)
The equation of the spinodal corresponds to the condition
( ΔG) ∂( Δμ
)
2
∂
∂ϕ
2
i
i
[4.2.11]
= = 0
[4.2.12]
∂ϕ
i
where:
ΔG the Gibbs free mixing energy
ϕi volume fraction of a component of a solvent.
At a critical point, binodal and spinodal coincide
3 ( ΔG) ∂ ( ΔG)
2
∂
∂ϕ
= [4.2.13]
3
∂ϕ
2
i i
In the elementary case of a two-component system, the Flory-Huggins theory gives
the following solution: 3
⎡ ⎛ r ⎞
⎤
i
Δμ i = RT ϕ i + ⎜ − ⎟
2
⎢ln
1 ( 1− ϕ i) + riχ1ϕi
⎥
⎜
⎝ r ⎟
⎣⎢
j ⎠
⎦⎥
[4.2.14]
where:
ri, rj numbers of segments of corresponding component.
The last equation can be solved if one takes into account the equality of chemical potentials
of a component in two co-existing phases of a stratified system.
⎛
ln ′ + ⎜
x ⎞
⎛
i
2
− ⎟ ′ + ( ′ ) = ln ′′ + ⎜
x
ϕi 1 ϕ j xiχij
ϕi ϕi
1−
⎜ ⎟
⎝ x
⎜
j ⎠
⎝ x
REFERENCES
i
j
⎞
⎟ϕ′′
+ ′′
j xiχij
ϕi
⎟
⎠
( )
1 P.J. Flory, J. Chem. Phys., 9, 660, (1941).
2 P.J. Flory, J. Chem. Phys., 10, 51 (1942).
3 P.J. Flory, Principles of polymer chemistry, Cornell University Press, Ithaca, 1953.
4 M.L. Huggins, J. Chem. Phys., 9, 440 (1941).
2
[4.2.15]