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The Flory–Huggins theory describes the phase separation of solutions containing polymers. Although it finds widespread application from polymer physics to materials science to biology, the concentrations that coexist in separate phases at equilibrium have not been determined analytically, and numerical techniques are required that restrict the theory’s ease of application. In this work, we derive an implicit analytical solution to the Flory–Huggins theory of one polymer in a solvent by applying a procedure that we call the implicit substitution method. While the solutions are implicit and in the form of composite variables, they can be mapped explicitly to a phase diagram in composition space. We apply the same formalism to multicomponent polymeric systems, where we find analytical solutions for polydisperse mixtures of polymers of one type. Finally, while complete analytical solutions are not possible for arbitrary mixtures, we propose computationally efficient strategies to map out coexistence curves for systems with many components of different polymer types. 

Water drops floating at an interface can be self-assembled and controlled using electrostatic effects. Uncharged drops can be moved around as desired, while charged drops exhibit spontaneous directed motion. 

Ionic liquids (ILs) are an extremely exciting class of electrolytes for energy storage applications because of their unique combination of properties. Upon dissolving alkali metal salts, such as Li or...