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

1.7 Polymer Blends 45
at the end of Section 1.2, one can show that F [φ A ,w A ,w B ] is unaffected by an added constant
to either w A (r) or w B (r). These degrees of freedom can be removed by fixing their spatial
averages to ¯w A =¯w B =0, and adding an appropriate constant to Eq. (1.205) such that
w A (r) − w B (r) =2χN(φ − φ A (r)) (1.206)
With this revised field equation, the SCFT approximation of the free energy, F [φ A ,w A ,w B ],
given by Eq. (1.199) can be reexpressed as
F
nk B T = F ( )
( )
h
nk B T − φ ln QA [w A ]
QB [w B ]
− (1 − φ)ln
−
V
V
∫
χN
dr(φ A (r) − φ)(φ B (r) − 1+φ) (1.207)
V
where
F h
nk B T
= φ[ln φ − 1] + (1 − φ)[ln(1 − φ) − 1] + χNφ(1 − φ) (1.208)
1.7.2 Homogeneous Phases and Macrophase Separation
The simplest solution to the self-consistent field equations is that for uniform fields, w A (r) =
w B (r) =0, corresponding to a homogeneous mixture. In this case, the partition functions
reduce to Q A [w A ]=Q B [w B ]=V, and the segment concentrations simplify to φ A (r) =φ
and φ B (r) =1− φ. Inserting these results into Eq. (1.207), we find that F = F h . Indeed, Eq.
(1.208) is the well-known Flory-Huggins expression for the free energy of a homogeneous
blend of composition, φ. Figure 1.10(a) shows the free energy curve for χN =3. Notice that
there is a region, between the two inflection points denoted by diamond symbols, where the
free energy has negative curvature. This property implies that the system is unstable towards
macrophase separation, where the melt splits into A- and B-rich phases occupying separate
regions of the volume, V.
To determine whether or not the blend will macrophase separate, we need an expression
for the combined energy free, F s = F (1)
h
+ F (2)
h
, of two distinct phases of compositions,
φ (1) and φ (2) , respectively. If the blend is to macrophase separate, it has to obey some basic
conservation rules. First, the number of molecules in the two phases, n (1) and n (2) , must
satisfy
n (1) + n (2) = n (1.209)
and second, the number of A-type polymers in each phase must comply with
n (1) φ (1) + n (2) φ (2) = nφ (1.210)
These conservation laws imply that the fraction of molecules in phase (1) will be
n (1)
n = φ(2) − φ
φ (2) − φ (1) (1.211)