For an asymmetric blend of chains with and segments and interaction parameter

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Soft condensed matter · Soft condensed matter

Soft Matter at the Thermal Scale, Ch Polymer solutions and blends

Problem

For an asymmetric blend of chains with

N_A

and

N_B

segments and interaction parameter

\chi

, the spinodal is the locus where the free-energy curvature vanishes,

\frac{1}{N_A\phi} + \frac{1}{N_B(1-\phi)} = 2\chi,

with

\phi

the volume fraction of species A. ((a)) Rearrange this condition into a quadratic equation for

\phi

a\phi^2+b\phi+c=0

, giving

a

b

and

c

in terms of

N_A

N_B

and

\chi

. ((b)) For a blend with

N_A = 500

N_B = 50

and

\chi = 0.030

, find both spinodal compositions, as volume fractions of A, to three significant figures.