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

1.5 Mathematical Techniques and Approximations 23
1.5.4 Random-Phase Approximation
Here, a Landau-Ginzburg free energy expression is derived for the entropic free energy, f e ,ofa
polymer molecule in a weakly-varying field, w(r). As before in Section 1.5.2, this calculation
will assume that the two ends of the chain are free, by imposing the initial condition in Eq.
(1.87). We will also use the invarience discussed at the end of Section 1.2 to fix the spatial
average of the field to ¯w =0. Given that w(r) only deviates slightly from zero, it is safe to
assume that the ensemble-averaged segment concentration, φ α (r), never strays far from its
spatial average,
¯φ α ≡ 1 V
∫
dr φ α (r) =
N
ρ 0 V
(1.90)
The first step is to find an approximate solution for q(k ≠0,s). Since the integral in Eq.
(1.85) involves w(k 1 ), which is small, the multiplicative factor of q(k − k 1 ,s) need not be
particularly accurate. Approximating it by q(k − k 1 , 0) from Eq. (1.87) gives
∂
q(k,s)
∂s
≈
∫
1
−xq(k,s) −
(2π) 3 dk 1 w(k 1 )q(k − k 1 , 0) (1.91)
= −xq(k,s) − w(k) (1.92)
where x ≡ k 2 a 2 N/6. Given that q(k, 0) = 0 for k ≠0, the solution becomes
where
q(k,s) ≈−w(k)h(x, s) (1.93)
h(x, s) ≡ 1 − exp(−sx)
x
(1.94)
The second step is to solve for q(k =0,s), in which case Eq. (1.85) reduces to
∂
∂s q(k,s)=− 1 ∫
(2π) 3 dk 1 w(k 1 )q(−k 1 ,s) (1.95)
Since we have set ¯w =0, it follows that w(k 1 )=0for k 1 =0, which in turn implies that the
approximation from Eq. (1.93) can be inserted inside the integral to give
∂
∂s q(k,s) ≈ 1 ∫
(2π) 3 dk 1 w(k 1 )w(−k 1 )h(x 1 ,s) (1.96)
The solution of this is
where
q(k,s) ≈ q(k, 0)
g(x, s) ≡
[
∫
1
1+
2(2π) 3 V
]
dk 1 w(−k 1 )w(k 1 )g(x 1 ,s)
(1.97)
2[exp(−sx)+sx − 1]
x 2 (1.98)