Polymer Physics

as thecohesive energy density(CED), with the unit of cal/cm^3 or J/cm^3. Further-
more, Hildebrand defined thesolubility parameter(Hildebrand 1936 )as

dð

DE

V

Þ^1 =^2 (4.10)

Therefore, (4.9) can be simplified as

DUmix

Vmix

¼f 1 f 2 ðd 1 d 2 Þ^2 (4.11)

Only whenDUmix/Vmixis smaller thanTDSmix/Vmixcan a polymer be potentially
dissolved in the solvent. For a given polymer, its solvent can be selected according
to the following empirical rules:

1. If the polymer and the solvent are both polar, their polarities should be close to
   each other;


2. If the polymer and the solvent are both nonpolar, their solubility parametersd
   should be close to each other.
   In short, the rules above can be summarized into one sentence, “Like likes like”.
   The solubility parameters of common solvents and polymers can be found from
   the conventional handbooks for physical chemistry or polymers. It is also possible
   to estimate the solubility parameter of polymers with the method of molar attractive
   constants according to

d 2 ¼r

SFi

M

(4.12)

whereris the polymer density,Mis the molar mass of the repeating units, and the
attractive constant for various chemical groupsFican be found in the Hoy’s
Table (Hoy 1985 ). In practice, a convenient way to dissolve a solid polymer sample
is to use a mixture of different solvents. The effective solubility parameter of the
mixed solvents can be adjusted by changing their compositions according to

d 1 dAfAþdBfB (4.13)

For more general solution systems containing polar molecules, Hansen proposed
the concept of three-dimensional solubility parameter, as given by

d^2 ¼d^2 dþd^2 pþd^2 h (4.14)

which includes the contributions of dispersion forcesdd, dipole-dipole interactionsdp,
and hydrogen bondsdh(Hansen 1967 ). Recently, polymer dissolution in the solvents
has been reviewed by Miller-Chou and Koenig (Miller-Chou and Koenig 2003 ).

48 4 Scaling Analysis of Real-Chain Conformations