For a symmetric polymer blend with and

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Thermodynamic potentials and the formal structure of thermodynamics

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

Problem

For a symmetric polymer blend with NA=NB=N=600N_A = N_B = N = 600 and χ=0.0036\chi = 0.0036, someone argues: "The free energy of mixing per site at ϕ=0.5\phi = 0.5 is f(0.5)=1Nln⁡(0.5)+0.25χ=−2.6×10−4 kBTf(0.5) = \frac{1}{N}\ln(0.5) + 0.25\chi = -2.6\times10^{-4}\,k_BT which is negative, so mixing lowers the free energy below that of the two pure components. The ϕ=0.5\phi = 0.5 mixture is therefore stable." ((a)) Find the error in this argument. ((b)) Decide, correctly, whether the ϕ=0.5\phi = 0.5 mixture is stable, by computing the quantity that actually decides stability.

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