Self-Consistent Field Theory and Its Applications by M. W. Matsen
Self-Consistent Field Theory and Its Applications by M. W. Matsen
Self-Consistent Field Theory and Its Applications by M. W. Matsen
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32 1 <strong>Self</strong>-consistent field theory <strong>and</strong> its applications<br />
where the tildes, ˜Dr α ≡Dr α P [r α ], denote that the functional integrals are weighted <strong>by</strong><br />
(<br />
P [r α ]=exp − 3 ∫ 1<br />
)<br />
2a 2 ds|r ′<br />
N<br />
α(s)| 2 (1.141)<br />
0<br />
so as to account for the internal entropy of each coarse-grained segment. The functional<br />
integrals over each r α (s) are, in principle, restricted to the volume of the system, V = AL.<br />
To ensure that the chains are properly grafted to the substrate, there are Dirac delta functions<br />
constraining the s =1end of each chain to z = ɛ; we will eventually take the limit ɛ → 0.<br />
Furthermore, there is a Dirac delta functional that constrains the overall concentration, ˆφ(r),<br />
to be uniform over all r ∈V.<br />
Although the expression for Z is inherently simple, its evaluation is far from trivial.<br />
Progress is made <strong>by</strong> first replacing the delta functional <strong>by</strong> the integral representation,<br />
∫ ( ∫<br />
)<br />
δ[1 − ˆφ]<br />
ρ0<br />
∝ DW exp dr W (r)[1 −<br />
N<br />
ˆφ(r)]<br />
(1.142)<br />
This expression is equivalent to Eq. (1.335) derived in the Appendix, but with k(x) substituted<br />
<strong>by</strong> −iρ 0 W (r)/N . The constants, ρ 0 <strong>and</strong> N, have no effect on the limits of integration, but<br />
the i implies that W (r) must be integrated in the complex plane along the imaginary axis.<br />
Inserting Eq. (1.142) into Eq. (1.140) <strong>and</strong> substituting ˆφ(r) <strong>by</strong> Eqs. (1.13) <strong>and</strong> (1.108)<br />
allows the integration over the chain trajectories to be performed. This results in the revised<br />
expression,<br />
∫<br />
Z ∝<br />
( ∫<br />
DW (Q[W ]) n ρ0<br />
exp<br />
N<br />
)<br />
dr W (r)<br />
(1.143)<br />
where<br />
∫<br />
Q[W ] ∝<br />
(<br />
˜Dr α exp −<br />
∫ 1<br />
0<br />
)<br />
ds W (r α (s)) δ(z α (1) − ɛ) (1.144)<br />
is the partition function of a single polymer in the external field, W (r). For convenience, the<br />
partition function of the system is reexpressed as<br />
∫ (<br />
Z ∝ DW exp − F [W ] )<br />
(1.145)<br />
k B T<br />
where<br />
( )<br />
F [W ] Q[W ]<br />
nk B T ≡−ln − 1 ∫<br />
AaN 1/2 V<br />
dr W (r) (1.146)<br />
There are a couple of subtle points that need to be mentioned. Firstly, the present calculation<br />
of Z allows the grafted ends to move freely in the z = ɛ plane, when actually they<br />
are grafted to particular spots on the substrate. As long as the chains are densely grafted, the<br />
only real consequence of treating the ends as a two-dimensional gas is that Q[W ] becomes<br />
proportional to A, the area available to each chain end. To correct for this, the Q[W ] in Eq.