Engineering & Science
Mayo-Lewis Copolymer Equation & Azeotropic Feed Ratio
Calculate instantaneous copolymer chain composition F_1 = (r_1*f_1^2 + f_1*f_2) / (r_1*f_1^2 + 2*f_1*f_2 + r_2*f_2^2) and drift-free azeotropic monomer feed f_1,azeo = (1 - r_2) / (2 - r_1 - r_2).
Why batch copolymer reactors suffer composition drift—except right at the azeotropic feed!
When two vinyl monomers M_1 and M_2 undergo free-radical copolymerization, growing chain radicals M_1• and M_2• rarely add both monomers at equal speeds! Frank R. Mayo and Frederick M. Lewis (JACS, 1944) defined monomer reactivity ratios r_1 = k_11 / k_12 and r_2 = k_22 / k_21 and derived the instantaneous copolymer mole fraction: F_1 = (r_1 * f_1^2 + f_1 * (1 - f_1)) / (r_1 * f_1^2 + 2 * f_1 * (1 - f_1) + r_2 * (1 - f_1)^2)!
When both r_1 < 1 and r_2 < 1 (alternating tendency, like Styrene M_1 + Methyl Methacrylate M_2 at 60 °C: r_1 = 0.52, r_2 = 0.46), there exists a golden azeotropic monomer feed f_1,azeo = (1 - r_2) / (2 - r_1 - r_2) where F_1 = f_1 identically so zero composition drift occurs across conversion! At f_1 = 0.30 styrene feed, chains incorporate styrene faster (F_1 = 0.3710 > 0.30, depleting styrene!), whereas charging the exact azeotrope f_1 = 0.5294 locks F_1 = 0.5294!
| Styrene (M1, r1=0.52) + MMA (M2, r2=0.46) Feed | Instantaneous Chain F1 | Batch Drift Direction (F1 - f1) | Reactor Control Strategy |
|---|---|---|---|
| Styrene-Lean Feed (f1 = 0.3000) | 0.3710 | +0.0710 (Depletes Styrene) | Semi-Batch Styrene Makeup Needed |
| Azeotropic Feed (f1 = 0.5294) | 0.5294 (= f1!) | 0.0000 (Zero Drift!) | Uniform Optical-Grade SAN/SMMA |
| Styrene-Rich Feed (f1 = 0.8000) | 0.7311 | -0.0689 (Enriches Styrene) | Semi-Batch MMA Makeup Needed |
Mayo-Lewis Instantaneous Copolymer Mole Fraction F_1
Evaluates (r_1*f_1^2 + f_1*(1-f_1)) / (r_1*f_1^2 + 2*f_1*(1-f_1) + r_2*(1-f_1)^2) for monomer feed mole fraction 0 <= f_1 <= 1.
=IF(OR(B2 < 0, B2 > 1, B3 < 0, B4 < 0), "Invalid Reactivity Inputs", IF(OR(B2 = 0, B2 = 1), B2, ROUND((B3 * (B2 ^ 2) + B2 * (1 - B2)) / (B3 * (B2 ^ 2) + 2 * B2 * (1 - B2) + B4 * ((1 - B2) ^ 2)), 4)))Drift-Free Azeotropic Monomer Feed Fraction f_1,azeo = (1 - r_2) / (2 - r_1 - r_2)
Checks whether an internal azeotrope exists (either both r_1 < 1 and r_2 < 1, or both r_1 > 1 and r_2 > 1) and returns f_1,azeo.
=IF(OR(B3 < 0, B4 < 0), "Invalid Reactivity Inputs", IF(OR(AND(B3 < 1, B4 < 1), AND(B3 > 1, B4 > 1)), ROUND((1 - B4) / (2 - B3 - B4), 4), "No Internal Azeotrope"))Try the example
Use the original values or change them to test your understanding.