SinceB 12 ,B 11 andB 22 are all negative values for nonpolar molecules, according
to (4.7),B>0 implies that the absolute energy change for forming monomer-
monomer and solvent-solvent pairs is larger than the absolute energy change for
forming their corresponding mixing pairs, which is apparently unfavorable for the
formation of homogeneous mixtures. Therefore, a positiveBoften implies phase
separation in the polymer solutions. If we use the volume fraction of nonpolar
polymersf 2 as the variable, the thermodynamic boundary conditions for phase
separation are calledphase diagrams. Such a solvent (B>0) is calledpoor solvent.
In cases ofB¼0 andB<0, the solvent is calledathermal solventandgood
solvent, respectively. Another demarcation between the poor and good solvents,
known as the theta solvent (Fig.4.4), is also helpful for a theoretical analysis of
nonpolar polymer solutions. Furthermore, the traditional demarcation at athermal
solvent is useful for polar polymers, especially for the separation between hydro-
philic and hydrophobic polymers in aqueous solutions.
4.3 Single-Chain Conformation in Polyelectrolyte Solutions
In dilute solutions, polymer chains are far apart, and hence each polymer chain
behaves like an isolated single coil. The surrounding molecules of athermal solvent
can compensate the attractive interactions among the chain units, leaving only the
volume-exclusion interactions between chain units for the single coil. In another
word, the conformation of a single chain in athermal dilute solutions is similar to
the trajectory ofa self-avoiding random walk(SAW).
In an ideal model of single chains, the chain conformation can be treated in
analogy to a trajectory of random walks in a lattice space. For random walks of
certain steps, the total amount of possible paths isqn, whereqis the coordination
number of the lattice, andnthe number of steps along the walk path. For a real
single chain containing only the volume exclusion, the proper analogy becomes a
self-avoiding walk. The mathematical treatment to SAW turns out to be a big
Fig. 4.4 Illustration of the phase diagrams for phase separation in polymer solutions. The upside
is poor solvent, and the downside is good solvent. There are two kinds of demarcations: one is
athermal solvent withB¼0, and the other is theta solvent
4.2 Single-Chain Conformation in Polymer Solutions 49