Taking the minimum with respect to the coil sizeR, one calculates@F
@R¼ 0 (4.20)
and one can have
Rn^3 =ðdþ^2 Þ (4.21)This is the so-calledFlory-Fisher scaling law(De Gennes 1979 ). The critical
exponentn¼1 in (4.21) at the dimensionalityd¼1;n¼3/4 atd¼2;n¼3/5 at
d¼3; andn¼1/2 atd¼4. These critical exponents are consistent with that of
self-avoiding walks obtained above from the computer simulations. The scaling law
for the ideal chain model occurs only in 4D space of SAWs. In 3D space, the
renormalization group theory yields the critical exponent asn¼0.588 0.001,
which is in good consistency with the computer simulation results (Le Guillou and
Zinn-Justin 1977 ).
Why is Flory’s mean-field estimation so successful? De Gennes provided an
explanation in his book (De Gennes 1979 ). On the one hand, the strong correlation
along the chain was underestimated by the average internal concentration, because of
an inhomogeneous distribution of chain units inside the coil. On the other hand, the coil
elasticity was also underestimated by the Gaussian distributions, because of volume
exclusions among the chain units. Both underestimations cancel each other, which
leads to a reasonable scaling relationship for the coil size in athermal dilute solutions.
4.2.3 Single-Chain Conformation in Athermal
Concentrated Solutions
In concentrated polymer solutions, although the overall distribution of chain units is
almost homogeneous over the space, the distribution of those chain units in a given
chain is still localized. The single-chain conformation appears like a coil. If we
label the chain ends with the fluorescent chemical groups, their correlation lengths
can be measured experimentally. Figure4.5 demonstrates that there exists a
so-called ‘correlation hole’in the radial distribution of fluorescent ends of other
chains (De Gennes 1979 ). This correlation hole implies that the nearest neighbors
of each chain unit belong very likely to the same polymer chain.
The high degree of penetration among polymer chains actually leads to an
effective attraction between the chain units, and this attraction screens out the
expansion effect due to the volume exclusion. This statement was first quantita-
tively described by Edwards (Edwards 1966 ). Thescreening effectcan be attributed
to the anisotropic packing of local chains around each chain unit. To understand this
effect, let’s consider those solute molecules each of which is formed by two
4.2 Single-Chain Conformation in Polymer Solutions 51