Tricriticality in AnBn Star Polymers Controlled by Arm Number

Symmetric A_nB_n star polymer melts offer a molecular route for tuning the character of a microphase order-disorder transition (ODT). Earlier second-order random phase approximation (RPA) work by Olvera de la Cruz and Sanchez established that the spinodal is located at (χN)_s ≃ 10.495 and does not depend on n [1,2]. A subsequent fourth-order calculation by Dobrynin and Erukhimovich showed that the quartic term changes sign for n > 5, indicating that the spinodal criterion alone cannot determine the mean-field ODT once higher-order stabilization becomes important [3]. In this work, we revisit this problem by deriving the lamellar free energy through sixth order in RPA and comparing the result with self-consistent field theory (SCFT). The quartic coefficient changes sign at n_tc ≃ 5.4475, and the sixth-order coefficient remains positive in the neighborhood of this point. Consequently, melts with n < n_tc enter the ordered state continuously at the spinodal, whereas those with n > n_tc exhibit a first-order ODT with (χN)_ODT < (χN)_s. Near this tricritical point, the separation between the ODT and the spinodal varies quadratically with n - n_tc, in quantitative agreement with the SCFT results. The origin of this behavior can be traced to correlations between distinct arms sharing the same junction; their relative weight increases with n and eventually drives the quartic coefficient negative. The noninteger value of n_tc may be viewed as an effective architecture realizable by blending neighboring stars, such as A₅B₅ and A₆B₆. These results identify a rare physical example of tricriticality associated with finite-wavenumber microphase separation and provide a controlled mean-field reference point for future studies of fluctuation effects.

References
[1] L. Leibler, Macromolecules 13, 1602 (1980).
[2] M. Olvera de la Cruz and I. C. Sanchez, Macromolecules 19, 2501 (1986).
[3] A. V. Dobrynin and I. Y. Erukhimovich, Macromolecules 26, 276 (1993).