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

1.6 Polymer Brushes 31
1.6.3 Diffusion Equation for a Parabolic Potential
In Section 1.2, we established that the path-integral definition of the partition function is equivalent
to the one based on the diffusion equation, just as Feynman’s path-integral formalism of
quantum mechanics coincides with Schrödinger’s differential equation. As in quantum mechanics,
the differential equation approach is generally the more practical method of calculating
q(z,z 0 ,s). Here, we demonstrate that it indeed gives identical results to those obtained in
the previous section, where the path-integral approach was used.
Normally the diffusion Eq. (1.24) has to be solved numerically, but there already exists a
known analytical solution for parabolic potentials (Merzbacher 1970). In fact, it was originally
derived for the wavefunction of a simple harmonic oscillator; once again, we benefit from the
analogy with quantum mechanics. Transforming the quantum mechanical wavefunction in
Merzbacher (1970), according to Eqs. (1.50)-(1.56) for the parabolic potential in Eq. (1.113),
gives
(
q(z,z 0 ,s)=
3
4sin(πs/2)
) 1/2 (
exp − 3π[(z2 + z0 2)cos(πs/2) − 2zz )
0]
4a 2 N sin(πs/2)
(1.138)
Although the derivation of this expression is complicated, it is relatively trivial to confirm that
it is in fact the proper solution to Eq. (1.24) for the initial condition, q(z,z 0 , 0) = aN 1/2 δ(z −
z 0 ).
Substituting z =0and s =1into Eq. (1.138), we find that q(0,z 0 , 1) = √ 3/4, consistent
with the path-integral calculation in Eq. (1.127). In terms of the partition function, the
distribution of the sN’th segment is given by
ρ(z; z 0 ; s) = q(0,z,1 − s)q(z,z 0,s)
(1.139)
q(0,z,1)
The derivation is essentially the same as that for Eq. (1.31). Inserting Eq. (1.138) and simplifying
with some basic trigonometric identities gives the identical expression to that in Eq.
(1.134), as required by the equivalence of the two approaches. Given this, the remaining results
of the preceding section follow. One slight difference is that we now obtain a specific
value for the unknown constant in Eq. (1.137), because the initial condition for q(z,z 0 ,s),
in effect, selects a proportionality constant for Eq. (1.127), but not one with any physical
significance. Nevertheless, this provides us with a standard reference by which to compare
calculations.
1.6.4 Self-Consistent Field Theory (SCFT) for a Brush
The parabolic potential used up to now is only approximate. The argument for it assumes that
the tension at the free chain-end is zero, but there is in fact a small entropic force that acts to
broaden the end-segment distribution, g(z 0 ) (Matsen 2002b). Here, the self-consistent field
theory (SCFT) is described for determining the field in a more rigorous manner. The theory
starts with the partition function for the entire system, which is written as
∫ ( )
∏
n
Z ∝ ˜Dr α δ(z α (1) − ɛ) δ[1 − ˆφ] (1.140)
α=1