which can be obtained from Flory-Krigbaum theory on dilute polymer solutions
(see Sect.8.3.3), wherenis the specific volume of polymersr^1 ,v 1 is the molar
volume of solvent, andF(x)is a chain-length-related function. WhenT¼y,
A 2 ¼0. This implies the cancellation of two-body interactions between attraction
and repulsion. We can also call this state asunperturbed state, when the polymer
coils behave like that in an ideal solution.
According to (4.44) in the Flory-Krigbaum theory for dilute solutions, when the
solvent is the polymer (x!1), the molar volume of the solventv 1 !1too, then
A 2 !0. This implies that the bulk phase is the theta solvent of polymers. With this
approach, Flory had already recognized the unperturbed chain conformation in
concentrated polymer solutions (Flory 1953 ).
Extrapolating Flory-Huggins theory to the dilute limit (beyond the assumption of
the theory) also provides
Dme 1 ¼kTðw1
2
Þf^22 (4.45)wherewis the Flory-Huggins interaction parameter (see Sect.8.2.4). At the theta
point,w¼1/2. This is exactly coincident with the critical point of phase separation
wC¼1/2 for the infinitely long polymers (see Sect.9.1). This coincidence implies
that at the theta point, the correlation length of concentration fluctuations diverges.
The strong correlation is normally conducted by the strong chemical bonds along the
chain, which will be truncated at two chain ends. Therefore, many physical properties
of polymers exhibit a scaling relationship with respect to the chain length. This
probably is the reason why de Gennes’ great effort to introduce the scaling analysis
into polymer physics becomes so successful. He first made analogue of the spin
correlation of a ferromagnet with zero-component limit of magnetization, to the
scaling exponents of self-avoiding random walks with zero approaching of the
inversed chain lengths (De Gennes 1972 ). The zero approaching ends with the infinite
chain length for the theta point, where the correlation length along the chain diverges,
and all the chain units lose their internal freedom in strong thermal fluctuations along
the chain. The strong correlation exhibits a structural self-similarity along the chain,
which legitimates wide applications of the renormalization group theory (Freed
1987 ) and the self-consistent-field theory (Edwards 1965 ; Helfand 1975 ) in polymer
systems. In summary, the critical point for collapse transition of a single chain reflects
the thermodynamic nature of the theta point, corresponding to the Boyle point, at
which the real gas behaves like an ideal gas. The physical states of amorphous
polymers are not so far away from their theta states; therefore, polymer properties
often exhibit a scaling relationship with respect to the chain length.
Near the theta point, the osmotic pressure of the dilute solution is closely related
to the chain length of polymers, as demonstrated in (4.43). In semi-dilute solutions,
however, the osmotic pressure is related to the degree of interpenetration of
polymer coils, no longer related to the chain length. Using the blob model, we have
xC^3 =^4 (4.46)58 4 Scaling Analysis of Real-Chain Conformations