The critical product for microphase separation of a symmetric, monodisperse diblock copolymer is (χN)_c = 10.495 — commonly rounded to 10.5 — as derived from Leibler's mean-field theory (1980) using the random-phase approximation (RPA). 1The Journal of Chemical Physics

This value is obtained by finding the spinodal condition at which the inverse structure factor S^-1(q) first diverges at a finite wavevector q* (corresponding to a lamellar periodicity on the order of the chain's radius of gyration), rather than at q = 0 as in ordinary macrophase separation. For a perfectly symmetric composition (f = 0.5), this spinodal is simultaneously a critical point in mean-field theory. 2Macromolecules3Macromolecules

Key context and corrections beyond mean-field:

• The value 10.495 is strictly a mean-field prediction. Fluctuation corrections (Fredrickson–Helfand theory) shift the actual order-disorder transition (ODT) to higher χN, with the correction scaling as O(N^-1/3). Specifically, the fluctuation-corrected ODT satisfies $(\chi N)_c \approx 10.495 + 41.022\, N^{-1/3}$ evaluated at q*. 1The Journal of Chemical Physics

• For finite chain lengths, this fluctuation correction is significant, and the ODT becomes weakly first-order rather than the second-order (continuous) transition predicted by mean-field theory. 3Macromolecules4The Journal of Chemical Physics

• The three segregation regimes are anchored by this value: the weak segregation limit (WSL) is χN ≈ 10–11, the intermediate regime connects to the strong segregation limit (SSL) at approximately χN ≈ 50. 5Macromolecules6Langmuir

• For cyclic (ring) diblock copolymers, the ODT shifts to significantly higher values, with (χ_effN)_ODT predicted in the range 40.8–43.0, compared to ~25.6 for linear chains of the same N in simulation — illustrating that the classic 10.495 applies only to linear architectures. 4The Journal of Chemical Physics