Handbook of Solvents - George Wypych - ChemTech - Ventech!

152 Christian Wohlfarth
derivative of the Gibbs free energy of mixing with respect to the amount of substance of the
solvent:
0 ∂ Δ
Δμ1
= μ1 − μ1
=
∂ 1
⎛ n ⎞ mixG
⎜
⎟
⎝ n ⎠
T, p, nj≠1 [4.4.11]
where:
ni amount of substance (moles) of component i
n total amount of substance (moles) of the mixture: n = Σni ΔmixG molar Gibbs free energy of mixing.
For a truly binary polymer solutions, the classical Flory-Huggins theory leads to: 46,47
or
where:
ΔmixG / RT = x lnϕ + x lnϕ
+ gx ϕ [4.4.12a]
1 1 2 2 1 2
( )
ΔmixG / RTV = ( x / V )ln ϕ + x / V lnϕ
+ BRTϕ
ϕ [4.4.12b]
1 1 1 2 2 2 1 2
xi mole fraction of component i
ϕi volume fraction of component i
g integral polymer-solvent interaction function that refers to the interaction of a solvent
molecule with a polymer segment, the size of which is defined by the molar volume of
the solvent V1 B interaction energy-density parameter that does not depend on the definition of a
segment but is related to g and the molar volume of a segment Vseg by B = RTg/Vseg V molar volume of the mixture, i.e. the binary polymer solution
Vi molar volume of component i
The first two terms of Equation [4.4.12] are named combinatorial part of
Δ mixG, the third one is then a residual Gibbs free energy of mixing. Applying Equation
[4.4.11] to [4.4.12], one obtains:
or
or
where:
2
Δμ 1 ( 1 ϕ 2) 1 ϕ 2 χϕ 2
1 ⎛ ⎞
/ RT = ln − + ⎜ − ⎟ + [4.4.13a]
⎝ r ⎠
⎡
⎛ ⎞ ⎤
χ = ⎢ μ − ( 1−ϕ ) −⎜1− ⎟ϕ
⎥ ϕ
⎣
⎝ ⎠ ⎦
1
/ RT ln /
r
2
Δ 1 2 2 2
⎡
⎛ ⎞ ⎤
χ = ⎢lna
−ln ( 1−ϕ ) −⎜1− ⎟ϕ
⎥/
ϕ
⎣
⎝ r ⎠ ⎦
1
1 2 2
2
2
[4.4.13b]
[4.4.13c]
r ratio of molar volumes V 2/V 1, equal to the number of segments if V seg =V 1