Page 96 - Carrahers_Polymer_Chemistry,_Eighth_Edition

Molecular Weight of Polymers 59

the theta temperature. The theta (θ) temperature is the lowest temperature at which a polymer of

infinite molecular weight is completely miscible with a specific solvent. The coil expands above the

theta temperature and contracts at lower temperatures.
Physical properties of polymers, including solubility, are related to the strength of covalent bonds,
the stiffness of the segments in the polymer backbone, the amount of crystallinity/amorphous, and
the intermolecular forces between the polymer chains. The strength of the intermolecular forces
is directly related to the CED, which is the molar energy of vaporization per unit volume. Since
intermolecular attractions of solvent and solute must be overcome when a solute (here the polymer)
dissolves, CED values may be used to predict solubility.
When a polymer dissolves, the first step is often a slow-swelling process called solvation in

which the polymer molecules swell by a factor δ, which is related to CED. Linear and branched
polymers dissolve in a second step, but network polymers remain in a swollen condition. In the dis-
solving process, external polymers are initially “dissolved” exposing additional polymer chains to
the solvent, and so on eventually resulting in the entire polymer mass being dissolved. Thus, poly-
mer solubility often takes considerably longer than the solubility of smaller molecules.
As early as 1926, Hildebrand showed a relationship between solubility and the internal pressure
of the solvent and in 1931, Scatchard incorporated the CED concept into Hildebrand’s equation.
This led to the concept of a solubility parameter, δ, which is the square root of CED. Thus, as shown
below, the solubility parameter, δ, for nonpolar solvents is equal to the square root of the heat of
vaporization per unit volume.

 ∆ E  1/ 2
δ =   = (CED) 1/ 2 or δ 2 = CED (3.2)
 V 

According to Hildebrand, the heat of mixing a solute and a solvent is proportional to the square
of the difference in solubility parameters, as shown below, where ϕ is the partial volume of each
component, namely, solvent γ and solute φ . Since, typically, the entropy term favors solution and
2
1
the enthalpy term acts counter to the solution, the objective is to match solvent and solute so that
the difference between their δ values is small, resulting in a small enthalpy acting against solubility
occurring.
∆H = φ φ (δ − δ ) 2 (3.3)
2
m
2
1
1
The solubility parameter concept predicts the heat of mixing liquids and amorphous polymers.
It has been experimentally found that generally any nonpolar amorphous polymer will dissolve in a
liquid or mixture of liquids having a solubility parameter that generally does not differ by more than
1/2
½
± 1.8 (cal/cc) . The Hildebrand with units of (cal/cc) is preferred over these complex units giving
for the previous expression ±1.8 H.
The solubility parameter concept is based on obtaining a negative Gibbs’ free energy. Thus,
as the term ∆H approaches zero, ∆G will have the negative value required for solution to occur
m
because the entropy term favors solution occurring. As noted before, the entropy (S) increases in the
solution process hence the emphasis is on achieving low values of ∆H .
m
For nonpolar solvents, which were called regular solvents by Hildebrand, the solubility param-
eter is equal to the square root of the difference between the enthalpy of evaporation (H ) and the
v
product of the ideal gas constant (R) and the Kelvin (or Absolute) temperature (T) divided by the
molar volume (V), as shown in the following equation:
E 
δ =    ∆ V   1/ 2 =    ∆ H − RT    1/ 2 (3.4)
v
V

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