Handbook of Solvents - George Wypych - ChemTech - Ventech!

194 Christian Wohlfarth
where:
Qm mass based degree of swelling
Qv volume based degree of swelling
m1 absorbed mass of the solvent
mN mass of the dry polymer network
ν1 absorbed volume of the solvent
νN volume of the dry polymer network
ρ1 density of the solvent
ρN density of the dry polymer network
Since Qv =1/ϕ2 , usually volume changes were measured. If the sample swells
isotropically, the volume change can be measured by the length change in one dimension.
Calculation of solvent activities from measurements of the swelling degree needs a statistical
thermodynamic model. According to Flory, 46 a closed thermodynamic cycle can be constructed
to calculate the Gibbs free energy of swelling from the differences between the
swelling process, the solution process of the linear macromolecules, the elastic deformation
and the crosslinking. The resulting equation can be understood in analogy to the
Flory-Huggins relation, Equation [4.4.13a] with r →∞, and reads:
2
13 /
( 1 ) c 1( 2
2)
Δμ / RT = ln − ϕ + ϕ + χϕ + ν V Aηϕ − Bϕ
[4.4.63]
1 2 2 2
where:
νc network density νc = ρN/Mc V1 molar volume of the pure liquid solvent 1 at temperature T
η memory term
A microstructure factor
B volume factor
Mc molar mass of a network chain between two network knots
ϕ 2 equilibrium swelling concentration = 1/Qv χ Flory-Huggins χ-function
A numerical calculation needs knowledge of the solvent activity of the corresponding
homopolymer solution at the same equilibrium concentration ϕ2 (here characterized by the
value of the Flory-Huggins χ-function) and the assumption of a deformation model that provides
values of the factors A and B. There is an extensive literature for statistical thermodynamic
models which provide, for example, Flory: 46 A = 1 and B = 0.5; Hermans: 214 A=1
andB=1;James and Guth 215 or Edwards and Freed: 216 A=0.5andB=0.Adetailed explanation
was given recently by Heinrich et al. 203
The swelling equilibrium depends on temperature and pressure. Both are related to the
corresponding dependencies of solvent activity via its corresponding derivative of the
chemical potential:
⎛ ∂T
⎞ 1 ∂μ 1 T ∂μ 1
⎜
⎟
∂ϕ ⎟
⎝ 1 ⎠ S ∂ϕ H ∂ϕ
P 1 1 P T 1 1
=
⎛ ⎞ ⎛ ⎞
⎜
⎟
= ⎜
Δ ⎝ ⎠ Δ ⎜
⎟
, ⎝ ⎠PT
,
[4.4.64]
where:
ϕ1 equilibrium swelling concentration of the solvent
ΔS1 differential entropy of dilution at equilibrium swelling
ΔH1 differential enthalpy of dilution at equilibrium swelling where ΔH1 =TΔS1 The first derivative in Equation [4.4.64] describes the slope of the swelling curve.
Since the derivative of the chemical potential is always positive for stable gels, the positive