Handbook of Solvents - George Wypych - ChemTech - Ventech!

4.4 Measurement of solvent activity 211
by Chen and Kreglewski, 356 and a hard-sphere pair-correlation function for the chain term as
following the arguments of Wertheim. The Huang-Radosz-form of the SAFT-equation of
state without an association term reads: 355
PV
RT
2
4y −2y
5y −2
mDnm
⎡ u
= 1+
r + ( 1−r)
+ r
3 m
( y − ) ( y − )( y − )
∑∑
1
1 2 1
~
n m ⎣kT V
⎢
⎤
⎦
⎥
n
[4.4.97]
where:
Dnm empirical coefficients from the attractive perturbation term
k Boltzmann’s constant
r chain segment number
In comparison to the PHC equation of state, the new term between the Carnahan-Starling
term and the double sum accounts for chain-bonding. The terms are proportional to the
segment number,r,ofthechain molecule. However, the hard-sphere volume, V0, is now a
slight function of temperature which is calculated according to the result of Chen and
Kreglewski: 356
[ 1 012 ( 3 0 ) ]
00
V = V − . exp − u / kT
[4.4.98]
0
3
where:
V 00
hard-sphere volume at T=0K
u0 well-depth of a square-well potential u/k = u0/k (1 + 10K/T) with 10K being an
average for all chain molecules.
The ratio u0/k or u/k is analogous to the characteristic parameter T* in the equations
above. There are two additional volume and energy parameters if association is taken into
account. In its essence, the SAFT equation of state needs three pure component parameters
which have to be fitted to equilibrium data: V 00 ,u0/k and r. Fitting of the segment number
looks somewhat curious to a polymer scientist, but it is simply a model parameter, like the
c-parameter in the equations above, which is also proportional to r. One may note additionally
that fitting to specific volume PVT-data leads to a characteristic ratio r/M (which is a
specific r-value), as in the equations above, with a specific c-parameter. Several modifications
and approximations within the SAFT-framework have been developed in the literature.
Banaszak et al. 357-359 or Shukla and Chapman 360 extended the concept to copolymers.
Adidharma and Radosz 361 provides an engineering form for such a copolymer SAFT approach.
SAFT has successfully applied to correlate thermodynamic properties and phase
behavior of pure liquid polymers and polymer solutions, including gas solubility and supercritical
solutions by Radosz and coworkers 355,357-359,361-368 Sadowski et al. 369 applied SAFT to
calculate solvent activities of polycarbonate solutions in various solvents and found that it
may be necessary to refit the pure-component characteristic data of the polymer to some
VLE-data of one binary polymer solution to calculate correct solvent activities, because
otherwise demixing was calculated. Groß and Sadowski 370 developed a “Perturbed-Chain
SAFT” equation of state to improve for the chain behavior within the reference term to get
better calculation results for the PVT- and VLE-behavior of polymer systems. McHugh and
coworkers applied SAFT extensively to calculate the phase behavior of polymers in supercritical
fluids, a comprehensive summary is given in the review by Kirby and McHugh. 371
They also state that characteristic SAFT parameters for polymers from PVT-data lead to