Order-Disorder Tricriticality in $\mathrm{A}_n \mathrm{B}_n$ Star Polymer Melts

Tricriticality usually requires tuning an additional thermodynamic parameter. Here we show that, in symmetric $\mathrm{A}_n\mathrm{B}_n$ star-polymer melts, the arm number $n$ itself plays this role and drives the order--disorder transition (ODT) from second order to first order. By developing a sixth-order free-energy expansion within the random phase approximation and comparing it with self-consistent field theory (SCFT) calculations, we analytically identify a tricritical arm number, $n_{\mathrm{tc}}\approx 5.4475$. For $n<n_{\mathrm{tc}}$, the lamellar ordering transition remains continuous and occurs at the spinodal point, $(\chi N)_{\mathrm{s}}\approx 10.495$. For $n>n_{\mathrm{tc}}$, the transition becomes first order, and $(\chi N)_{\mathrm{ODT}}$ shifts below $(\chi N)_{\mathrm{s}}$ with a quadratic dependence near the tricritical point. SCFT calculations confirm the predicted transition character and phase-boundary shift. The origin of this behavior is traced to inter-arm correlations generated by the common junction. We further show that the noninteger tricritical arm number can be effectively realized in binary mixtures of star polymers. This provides a rare analytically tractable example of architecture-induced tricriticality in a microphase-separating polymer system.