Handbook of Solvents - George Wypych - ChemTech - Ventech!

208 Christian Wohlfarth
dynamic behavior of polymer systems. The main goal was first to overcome the restrictions
of Flory’s equation of state to the liquid state, to improve the calculation of the compressibility
behavior with increasing pressure and to enable calculations of fluid phase equilibria
at any densities and pressures from the dilute gas phase to the compressed liquid including
molecules differing considerably in size, shape, or strength of intermolecular potential energy.
More recently, when more sophisticated methods of statistical mechanics were developed,
deeper insights into liquid structure and compressibility behavior of model polymer
chains in comparison to Monte Carlo modelling results could be obtained applying thermodynamic
perturbation theory. Quite a lot of different equations of state have been developed
up to now following this procedure; however, only a limited number was applied to real systems.
Therefore, only some summary and a phenomenological presentation of some equations
of state which have been applied to real polymer fluids should be given here, following
their historical order.
The perturbed-hard-chain (PHC) theory developed by Prausnitz and coworkers in the
late 1970s 320-322 was the first successful application of thermodynamic perturbation theory
to polymer systems. Since Wertheim’s perturbation theory of polymerization 323 was formulated
about 10 years later, PHC theory combines results from hard-sphere equations of simple
liquids with the concept of density-dependent external degrees of freedom in the
Prigogine-Flory-Patterson model for taking into account the chain character of real polymeric
fluids. For the hard-sphere reference equation the result derived by Carnahan and
Starling 324 was applied, as this expression is a good approximation for low-molecular
hard-sphere fluids. For the attractive perturbation term, a modified Alder’s 325 fourth-order
perturbation result for square-well fluids was chosen. Its constants were refitted to the thermodynamic
equilibrium data of pure methane. The final equation of state reads:
PV
RT
2
4y −2y
mA
= 1+
c + c
3 ∑∑
~ ~
( y −1)
n m
V T
nm
m n
[4.4.94]
where:
05 .
y packing fraction with y = V/(V0τ) and τ = ( π / 6) 2 = 0. 7405 (please note that in a
number of original papers in the literature the definition of y within this kind of
equations is made by the reciprocal value, i.e., τV0/V) c degree of freedom parameter, related to one chain-molecule (not to one segment)
V0 hard-sphere volume for closest packing
Anm empirical coefficients from the attractive perturbation term
The reduced volume is again defined by V ~
~
= V/V0 and the reduced temperature by
T= T/T* . The coefficients Anm are given in the original papers by Beret 320,321 and are considered
to be universal constants that do not depend on the chemical nature of any special
substance. The remaining three characteristic parameters, c, T* and V0 , have to be adjusted
to experimental PVT-data of the polymers or to vapor-liquid equilibrium data of the pure
solvents. Instead of fitting the c-parameter, one can also introduce a parameter P* by the relation
P* = cRT*/V0. In comparison with Flory’s free-volume equation of state, PHC-equation
of state is additionally applicable to gas and vapor phases. It fulfills the ideal gas limit,
and it describes the PVT-behavior at higher pressures better and without the need of temperature
and/or pressure-dependent characteristic parameters, such as with Flory’s model.
Values for characteristic parameters of polymers and solvents can be found in the original
literature. A review for the PHC-model was given by Donohue and Vimalchand, 326 where a