Handbook of Solvents - George Wypych - ChemTech - Ventech!

4.2 Effect of system variables on solubility 125
where:
Δμ 1
2
lna1 = = ln(
1−
ϕ2) + ϕ2 + χ1ϕ 2
[4.2.2]
RT
χ =zΔ ε kT
*
/ [4.2.3]
1 12
* * * *
Δε12 = 05 .( ε11 + ε 22 ) − ε12,a1-solvent
activity, ε11, ε22
- energy of 1-1 and 2-2 contacts
formation in pure components, ε12- energy of 1-2 contacts formation in the mixture, μ 1- chemical potential of solvent.
The critical value of χ1 sufficient for solubility of polymer having large molecular
mass is 0.5. Good solvents have a low χ1 value. χ1 is a popular practical solubility criterion
and comprehensive compilations of these values have been published. 6-9
Temperature is another factor. It defines the difference between polymer and solvent.
Solvent is more affected than polymer. This distinction in free volumes is stipulated by different
sizes of molecules of polymer and solvent. The solution of polymer in chemically
identical solvent should have unequal free volumes of components. It causes important
thermodynamic consequences. The most principal among them is the deterioration of compatibility
between polymer and solvent at high temperatures leading to phase separation.
The theory of regular solutions operates with solutions of spherical molecules. For the
long-chain polymer molecules composed of segments, the number of modes of arrangement
in a solution lattice differs from a solution of spherical molecules, and hence it follows
the reduction in deviations from ideal entropy of mixing. It is clear that the polymer-solvent
interactions differ qualitatively because of the presence of segments.
Some novel statistical theories of solutions of polymers use the χ1 parameter, too.
They predict the dependence of the χ1 parameter on temperature and pressure. According to
the Prigogine theory of deformable quasi-lattice, a mixture of a polymer with solvents of
different chain length is described by the equation: 10
( ) ( )
Rχ1 = A r / T + BT / r
[4.2.4]
A A
where:
A, B constants
rA number of chain segments in homological series of solvents.
These constants can be calculated from heats of mixing, values of parameter χ1, and
from swelling ratios. The Prigogine theory was further developed by Patterson, who proposed
the following expression: 11
χ1 = ν τ
2
⎛ ⎞
⎜ ⎟ +
⎝ ⎠
⎛ U CP
⎞
⎜ ⎟
RT ⎝ R ⎠
1 2 1 2
[4.2.5]
where:
U1 configuration energy (-U1 - enthalpy energy)
CP1 solvent thermal capacity
ντ , molecular parameters
The first term of the equation characterizes distinctions in the fields of force of both
sizes of segments of polymer and solvent. At high temperatures, in mixtures of chemically
similar components, its value is close to zero. The second term describes the structural con-