Self-Consistent Field Theory and Its Applications by M. W. Matsen

1.2 Gaussian Chain in an External Field 11
molecular weight and bulk mass densities for various common polymer types, such as those
depicted in Fig. 1.2. Given the chemical composition and thus the molecular weight of a
monomer, it is then possible to calculate the statistical segment length, a, and the bulk segment
density, ρ 0 . However, the values obtained depend on the definition of the segment (i.e., the
value selected for m). Of course, none of the physical quantities can be affected by such a
choice, and indeed this is always the case. If, for example, the number of monomers in each
segment is doubled, then N decreases by a factor of 2 and a increases by √ 2, such that the
quantity, R 0 = aN 1/2 , remains unchanged.
The freedom to redefine segments is linked to the scaling behavior and thus the universality
of polymeric systems. However, it can be confusing because the size of N no longer has any
absolute meaning, or in other words, it is impossible to judge whether a polymer is long or
short based solely on its value of N. Fortunately, there is a special invariant definition,
( ) R
3 2
N≡ 0
= a 6 ρ 2 0
N/ρ N (1.12)
0
with a real physical significance that provides a true measure of the polymeric nature of the
molecule. It is based on the ratio of the typical volume spanned by a polymer, R 0 3 , to its
physical volume, N/ρ 0 . Thus, small values of N correspond to compact molecules and large
values imply open configurations with many intermolecular contacts. Be aware that some
authors assume, without mentioning, that N = N by innocuously setting the segment volume
to ρ −1
0 = a 3 .
1.2 Gaussian Chain in an External Field
, each small enough such that their local
environment is more or less homogeneous, but also large enough so that they obey Gaussian
statistics with a statistical segment length, a. The coarse-grained trajectory of the polymer
(e.g., the path in Fig. 1.4(b)) is then specified by a function, r α (s), where the parameter,
0 ≤ s ≤ 1, runs along the backbone of the polymer such that equal intervals of s have the
same molecular weight. (The subscript α will be used to label different molecules when we
eventually treat complete systems.) Having specified the trajectory, a dimensionless segment
concentration can be defined as
The preceding section examined polymers in an ideal homogeneous environment, where no
forces were exerted on the chains. In the more interesting situations depicted in Fig. 1.3, the
polymers do experience interactions that vary with respect to position, r. Their net effect can
generally be represented by a static field, w(r), determined self-consistently from the overall
segment distribution in the system. However, we ignore for now the source of the field and
start by developing the statistical mechanics for a single polymer acted upon by an arbitrary
external field. Although fields are generally real quantities, the derivations that follow hold
even when w(r) is complex; this fact will be needed in the derivation of self-consistent field
theory (SCFT).
We assume that the field varies slowly enough that the chain can be divided into N appropriately
sized coarse-grained segments of volume, ρ −1
0
ˆφ α (r) = N ∫ 1
ds δ(r − r α (s)) (1.13)
ρ 0
0