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

20 1 Self-consistent field theory and its applications
where
c † i = 1 V
∫
dr f i (r)q † (r, 1) (1.69)
In practice, it is impossible to complete the infinite sums in Eqs. (1.66) and (1.68), but the
eigenvalues are appropriately ordered such that the terms become less and less important.
Therefore, the sums can be truncated at some point, i = M, where the number of terms
retained, M +1, is guided by the desired level of accuracy.
1.5.2 Ground-State Dominance
The previous expansion for q(r,s) in Eq. (1.66) becomes increasingly dominated by the first
term involving the ground-state eigenvalue, λ 0 , the further s is from the end of the chain. For
a high-molecular-weight polymer, the approximations,
and
q(r,s) ≈ c 0 exp(−λ 0 s)f 0 (r) (1.70)
q † (r,s) ≈ c † 0 exp(−λ 0(1 − s))f 0 (r) (1.71)
become accurate over the vast majority of the chain. In this case, it follows that the partition
function of the entire chain is
Q[w] ≈Vc 0 c † 0 exp(−λ 0) (1.72)
and that the segment concentration is
φ α (r) ≈
N
ρ 0 V f 0 2 (r) (1.73)
These simplifications permit us to derive an analytical Landau-Ginzburg free energy expression
for the entropic energy, f e , of the polymer as a function of its concentration profile,
φ α (r). Our derivation will assume that concentration gradient,
∇φ α (r) = 2N
ρ 0 V f 0(r)∇f 0 (r) (1.74)
is zero at the extremities (i.e., boundaries) of our system, V.
The first step of the derivation is to multiply Eq. (1.59) for i =0by f 0 (r) and integrate
over V, producing the result,
ρ 0
N
∫
dr w(r)φ α (r) = a2 N
6V
∫
dr f 0 (r)∇ 2 f 0 (r)+λ 0 (1.75)
Next, this expression for the field energy along with Eq. (1.72) for Q[w] are inserted into Eq.
(1.34) giving the expression,
∫
f e
k B T = − a2 N
dr f 0 (r)∇ 2 f 0 (r) − ln(c 0 c † 0
6V
) (1.76)