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

1.6 Polymer Brushes 39
The coefficients of the average polymer concentration can be expressed as
φ i =
V
Q[w]
∑
I jk Γ ijk (1.180)
jk
where the components of the matrix, I, are defined as
∫ 1
I jk ≡ ds q j (s)q † k
(s) (1.181)
= ∑ 0
[ ]
exp(dm ) − exp(d n )
U jm U kn¯q m (0)¯q n † (1)
d
m,n
m − d n
(1.182)
Note that in the instances where d m = d n , the factor in the square brackets reduces to
exp(d m ).
Given the formula to evaluate the segment concentration, the coefficients of the field, w i ,
can now be adjusted to satisfy the incompressibility condition, φ(z) =1, which is expressed
in terms of coefficients as
φ i = I i (1.183)
In practice, the functional expansion has to be truncated at some value i = M, where M is
chosen based on an acceptable error tolerance. In principle, the first (M +1)components, w i ,
would be adjusted to satisfy the first (M +1)conditions for φ i . However, Eqs. (1.183) are not
completely independent, due to the fact that SCFT is not affected by an additive constant to
w(z); see the discussion at the end of Section 1.2. To rectify this problem, the first condition,
φ 0 = I 0 , can be replaced by
∑
I i w i =0 (1.184)
i
which sets the spatial average of w(z) to zero. Now the set of equations are independent,
and they can be solved by standard numerical techniques. There are various techniques that
can be used (Thompson et al. 2004); we generally use the Broyden algorithm, which is a
quasi-Newton-Raphson method. This requires a reasonable initial guess for w(z), but that can
be provided by the SST estimate in Eq. (1.113). Figure 1.7 shows the field, w(z), obtained
from SCFT (solid curves) for several different brush thicknesses compared against the SST
result (dashed curve) from Eq. (1.113). As expected, the SST prediction becomes increasingly
accurate as L →∞.
We have glossed over the fact that β was introduced in order to prevent a delta function
from developing in w(z) at z =0. The signature of a delta function is that the field coefficients,
w i , fail to converge to zero in the limit of large i. To prevent this, we could include the
extra condition, w M =0, in our Broyden iterations and use it to determine β. The estimate,
β ≈ 3L/a 2 N, by Likhtman and Semenov (2000) provides a suitable initial guess. In reality,
this initial guess turns out to be very accurate, and since the spectral method can cope with
a small delta function, we simply use the initial guess provided by Likhtman and Semenov
(2000).