Universal Scaling of Phase Diagrams of Polymer Solutions

A combination of microfluidic and small-angle laser light scattering enables us to map phase diagram of two polymers with different chain lengths in three solvents in an unprecedented speed. Each precisely measured phase diagram leads to a pair of critical temperature (Tc) and volume fraction of polymer (ϕc), in which ϕc is scaled to the chain length (N) as ϕc ∼ N–0.37±0.01. Accounting for a huge size difference of polymer chain and solvent molecules, an adjustable volume ratio (Rc) of solvent to polymer is introduced to generate a dimensionless reduced volume fraction ψ {= ϕ/[ϕ + Rc(1 – ϕ)]}, so that each skewed phase diagram is shifted and symmetrized with a symmetrical axis ψc = 0.325 ± 0.002. After normalized by a solvent and polymer (not chain length) dependent constant ψ0, all of the measured 17 phase diagrams collapsed into one master curve, |ψ – ψc|/ψ0 = ε0.325±0.007N0.152±0.004, where ε = |T – Tc|/Tc, a reduced temperature. As N → ∞, |ψ – ψc| ∼ |ϕ – ϕc|/ϕc nearby the critical point so that our results lead to |ϕ – ϕc| ∼ ε0.325±0.007N–0.22±0.01, where the scaling over the chain length is close to −2/9 predicted by Muthukumar. We have successfully placed the last jigsaw piece, i.e., the chain length dependence, in the phase diagrams of polymer solutions.