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

1.6 Polymer Brushes 33
(1.146) is divided by A, which is necessary for F [W ] to be a proper extensive quantity that
scales with the size of the system. Alternatively, we could imagine that the chain ends are
free to move, but then the molecules would become indistinguishable and Z would have to
be divided by a factor of n! to avoid the Gibbs paradox (Reif 1965). Either way, we arrive
at the same expression for F [W ] apart from additive constants. That bring us to the second
point that additive constants to F [W ] are of no consequence, since they can be absorbed into
the proportionality constant omitted in Eq. (1.145). This fact has been used to insert an extra
factor of aN 1/2 into Eq. (1.146) to make the argument of the logarithm dimensionless.
Even though W (r) takes on imaginary values, Q[W ] is solved by the exact same methods
derived in Section 1.2, and thus F [W ] is readily computed. However, the functional integral
in Eq. (1.145) cannot be performed exactly. To proceed further, we use the fact that F [W ]
is an analytic functional, which means that the path of integration can be deformed without
altering the integral. A standard trick to solve such integrals, the method of steepest descent
(Carrier et al. 1966), is to deform the path from the imaginary axis to one that passes through
a point where
DF [W ]
DW (r)
=0 (1.147)
in a direction such that the imaginary part of F [W ] remains constant. At such a point, the real
part of F [W ] has a saddle shape, and consequently it is denoted as a saddle point . Changing
the path of integration in this way concentrates the non-zero contribution of the integral to the
neighborhood of the saddle point, allowing for a convenient expansion of which the first term
gives
(
Z ≈ exp − F [w] )
(1.148)
k B T
where w(r) is the solution to Eq. (1.147). Thus, F [w] becomes the SCFT approximation to
the free energy of the system. It follows from the functional differentiation in Eq. (1.147),
that w(r) must be chosen such that
where
φ(r) =1 (1.149)
φ(r) ≡−V D ln(Q[w])
Dw(r)
(1.150)
is the segment concentration of n polymers in the external field, w(r); see Eq.
〈
(1.32).
〉
The quantity, φ(r), can also be identified as the SCFT approximation for ˆφ(r) , but the
explanation for this is often glossed over. It is justified by starting with the formal definition,
〈 〉
ˆφ(r) ≡ 1 ∫ ( )
∏
n ˜Dr α δ(z α (1) − ɛ) ˆφ(r)δ[1 −
Z
ˆφ] (1.151)
α=1