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

1.7 Polymer Blends 55
1.7.6 Grand-Canonical Ensemble
Up to this point, the polymer blend has been treated in the canonical ensemble (Hong and
Noolandi 1981), where the numbers of A- and B-type polymers are fixed. However, the
statistical mechanics of multicomponent blends can be equally well performed in the grandcanonical
ensemble (Matsen 1995a), where n A and n B are permitted to fluctuate with chemical
potentials, μ A and μ B , controlling their respective averages. The grand-canonical partition
function, Z g , is related to the canonical one, Z, by the expression,
Z g =
∞∑
∞∑
n A=0 n B=0
z nA
A
znB B Z (1.245)
where z A =exp(μ A /k B T ) and z B =exp(μ B /k B T ). When the melt is treated as incompressible
such that n A + n B = n, one of the chemical potentials becomes redundant and the
definition can be simplified to
∞∑ ∞∑
Z g = z nA Z (1.246)
n A=0 n B=0
with z =exp(μ/k B T ). This is evaluated by inserting the expression for Z from Eq. (1.196)
and summing over n A and n B . The sum over n A is carried out using the Taylor series expansion,
∞∑ 1
( ρ0
)
n A ! N zQ nA
( ρ0
)
A[W A ] =exp
N zQ A[W A ]
(1.247)
n A=0
with an equivalent expansion for the sum over n B . With the summations completed, the
partition function reduces to
∫
(
Z g ∝ DΦ A DW A DW B exp − F )
g[Φ A ,W A ,W B ]
(1.248)
k B T
where
F g
nk B T = −z Q A[W A ]
− Q B[W B ]
+ 1 ∫
dr[χNΦ A (1 − Φ A ) −
V V V
W A Φ A − W B (1 − Φ A )] (1.249)
Just as before, F g [Φ A ,W A ,W B ] can be readily evaluated but the functional integration in Eq.
(1.248) cannot. The saddle-point approximation is therefore once again applied, producing
the identical self-consistent conditions,
φ A (r)+φ B (r) =1 (1.250)
w A (r) − w B (r) =χN(1 − 2φ A (r)) (1.251)
derived in the canonical ensemble. However, φ A (r) and φ B (r) are now defined by
φ A (r) ≡ − z V
φ B (r) ≡ − 1 V
DQ A [w A ]
Dw A (r)
DQ B [w B ]
Dw B (r)
= z
=
∫ 1
0
∫ 1
0
ds q A (r,s)q † A
(r,s) (1.252)
ds q B (r,s)q † B
(r,s) (1.253)