Power-law molecular-weight distributions dictate universal behaviors in highly polydisperse polymer solutions

Abstract: Polydispersity is a universal feature of synthetic polymers and biological molecules in the cytoplasm. However, its quantitative impact on collective behavior remains poorly understood because conventional metrics, such as the polydispersity index, fail to capture broad, non-Gaussian size distributions. Here, we develop an experimental platform in which polyethylene glycol (PEG) solutions are engineered to follow tunable power-law molecular-weight distributions spanning an extensive range, from $M = 1$ kg/mol to $10^{4}$ kg/mol. By systematically varying the $M$ distribution exponent $a$, we identify a robust regime ($1 < a \lesssim 2.5$) in which the viscosity scaling exponent in the entangled regime, the overlap concentration $c^{\ast}$, and the entanglement concentration ${c_{\mathrm{e}}}$ all exhibit pronounced maxima that exceed monodisperse limits. This amplification minimizes as the upper cutoff $M_{\max}$ is reduced, with the system approaching monodisperse behavior. The enhanced rheology arises from a competition between long-chain-dominated entanglement and short-chain-mediated void filling, demonstrating that the whole shape of the molecular-weight distribution plays a decisive role. Consequently, these collective behaviors cannot be reproduced by simply tuning the average molecular weight. Together, our results establish the power-law exponent $a$ as a quantitative control parameter that links polymer entanglement, soft packing, and molecular crowding in highly polydisperse systems.