Engineering & Science
Flory-Huggins Polymer Solution Theory: Mixing Free Energy & Critical Chi (chi_c)
Calculate dimensionless Flory-Huggins lattice free energy of mixing Delta_Gm / (n_site * R * T) and the critical spinodal phase-separation parameter chi_c = 0.5 * (1 + 1 / SQRT(N))^2.
Understand why long polymer chains phase-separate so easily!
Mixing two small-molecule liquids enjoys huge combinatorial entropy (1 - phi)*LN(1 - phi) + phi*LN(phi). When Paul Flory and Maurice Huggins placed N-segment polymer chains of volume fraction phi_2 onto a solvent lattice (phi_1 = 1 - phi_2), connecting N monomers into a single macromolecule slashed the polymer's translational entropy by a factor of N! Per lattice site in units of R*T, dimensionless Flory-Huggins mixing free energy is: Delta_Gm / (N_sites * R * T) = (1 - phi_2) * LN(1 - phi_2) + (phi_2 / N) * LN(phi_2) + chi * phi_2 * (1 - phi_2)!
Here chi is the Flory-Huggins polymer-solvent interaction parameter: chi < 0.50 denotes a good solvent, chi = 0.50 is the Flory theta point for infinite chains, and liquid-liquid demixing (spinodal phase separation) sets in once chi exceeds the critical threshold chi_c = 0.5 * (1 + 1 / SQRT(N))^2! For a 100-segment polymer (N = 100), chi_c = 0.5 * (1.10)^2 = 0.6050—and as N -> infinity, chi_c -> 0.50!
| Solvent Regime (phi_2 = 0.20, N = 100) | Interaction Parameter chi | Delta_Gm / RT per Lattice Site | Miscibility Status (chi_c = 0.6050) |
|---|---|---|---|
| Good Solvent | chi = 0.45 (< chi_c) | -0.10973 | Completely Miscible Single Phase |
| Critical Demixing Point | chi_c = 0.5*(1 + 1/SQRT(100))^2 | chi_c = 0.6050 | Upper Critical Solution Threshold |
| Poor Solvent (Above chi_c) | chi = 0.85 (> chi_c) | -0.04573 (Concave Spinodal Dome!) | Phase Separates into Two Phases |
Dimensionless Flory-Huggins Mixing Free Energy per Lattice Site (Delta_Gm / RT)
Evaluates (1 - phi_2)*LN(1 - phi_2) + (phi_2 / N)*LN(phi_2) + chi * phi_2 * (1 - phi_2) for 0 < phi_2 < 1 and N >= 1.
=IF(OR(B2 <= 0, B2 >= 1, B3 < 1), "Invalid Lattice Inputs", ROUND(((1 - B2) * LN(1 - B2)) + ((B2 / B3) * LN(B2)) + (B4 * B2 * (1 - B2)), 5))Critical Interaction Parameter chi_c = 0.5 * (1 + 1 / SQRT(N))^2
Computes the exact critical chi above which a polymer of chain length N exhibits a miscibility gap.
=IF(B3 < 1, "Invalid Lattice Inputs", ROUND(0.5 * ((1 + (1 / SQRT(B3))) ^ 2), 4))Try the example
Use the original values or change them to test your understanding.