Handbook of Solvents - George Wypych - ChemTech - Ventech!

4.4 Measurement of solvent activity 153
χ Flory-Huggins interaction function of the solvent
The segment number r is, in general, different from the degree of polymerization or
from the number of repeating units of a polymer chain but proportional to it. One should
note that Equations [4.4.12 and 4.4.13] can be used on any segmentation basis, i.e., also
* *
with r = V2 / V1
on a hard-core volume segmented basis and segment fractions instead of
volume fractions, or with r=M2/M1 on the basis of mass fractions. It is very important to
keep in mind that the numerical values of the interaction functions g or χ depend on the chosen
basis and are different for each different segmentation!
From the rules of phenomenological thermodynamics, one obtains the interrelations
between both parameters at constant pressure and temperature:
χ ϕ ( )
∂g
∂g
= g + 1 = g − 1 −ϕ2
[4.4.14a]
∂ϕ
∂ϕ
1
1
g = ∫ χdϕ ϕ
1
ϕ
0
1
1
2
[4.4.14b]
A recent discussion of the g-function was made by Masegosa et al. 50 Unfortunately, gand
χ-functions were not always treated in a thermodynamically clear manner in the literature.
Sometimes they were considered to be equal, and this is only true in the rare case of
composition independence. Sometimes, and this is more dangerous, neglect or misuse of
the underlying segmentation basis is formed. Thus, numerical data from literature has to be
handled with care (using the correct data from the reviews 5-8,11 is therefore recommended).
A useful form for their composition dependencies was deduced from lattice theory by
Koningsveld and Kleintjens: 51
where:
β
g = α +
and
( 1−γϕ
)
χ = α +
β( 1−γ)
( 1−γϕ
)
2 2
α acts as constant within a certain temperature range
β describes a temperature function like β = β 0 + β1
/T
γ is also a constant within a certain temperature range.
Quite often, simple power series are applied only:
n
∑
χ = χ ϕ and g =
i=
0
i
i
2
n
∑
i=
0
i
i+
1
⎛ χ ⎞⎛1−ϕ
⎞ 2
⎜ ⎟⎜
⎟
⎝i+
1⎠
⎜
⎝ 1−ϕ
⎟
2 ⎠
2
[4.4.15]
[4.4.16]
where:
χi empirical fitting parameters to isothermal-isobaric data
Both interaction functions are also functions of temperature and pressure. An empirical
form for these dependencies can be formulated according to the rules of
phenomenological thermodynamics:
( ) ( )
g = β + β / T + β + β / T Por χ = a + b / T + c + d / T P [4.4.17]
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