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Published May 24, 2022 | Supplemental Material
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Supernatant Phase in Polyelectrolyte Complex Coacervation: Cluster Formation, Binodal, and Nucleation


This work studies the structure and thermodynamics of the supernatant phase in polyelectrolyte complex coacervation, a relatively unexplored area. By combining the cluster theory in dilute solution with our recently developed mean-field theory for inhomogeneous polyelectrolyte solutions (Zhang, P.; Wang, Z.-G. Macromolecules 2021, 54, 10994), we systematically investigate the structure of finite-sized clusters formed by oppositely charged polyions in symmetric dilute solutions and how these clusters affect the binodal, spinodal, and nucleation for polyelectrolyte complex coacervation. We find that both the polyion concentration deep inside the cluster and the interfacial tension decrease with increasing the cluster size, reaching their respective bulk coexistence values with corrections inversely proportional to the cluster radius. The polyion concentration in the supernatant phase at coexistence is several orders of magnitude higher than that obtained under the uniform mixing approximation. For most relevant conditions away from the critical point, the supernatant phase consists predominantly of polyion pairs. By examining the nucleation barrier in supersaturated solutions, we can determine a pseudo-spinodal when the barrier is a few multiples of the thermal energy. The location of this pseudo-spinodal is similarly shifted to much higher concentrations than predicted under the uniform mixing approximation. Making the volume approximation for the clusters, we obtain simple analytical expressions for the cluster formation free energy, the modified binodal, and the pseudo-spinodal. In particular, we propose a simple approximate formula for estimating the concentration of the coexisting supernatant phase in terms of the chain length, interfacial tension, and the polyion concentration in the coacervate phase.

Additional Information

© 2022 American Chemical Society. Received: February 15, 2022; Revised: April 13, 2022; Published: May 12, 2022. The authors thank Prof. Isamu Kusaka for valuable discussion on the interfacial tension calculation for finite-sized droplets. P.Z. acknowledges the financial support provided by the National Natural Science Foundation of China (22073016 and 21803011) and the award of Shanghai Dongfang Scholar. Z.-G.W. acknowledges financial support from the Hong Kong Quantum AI Lab Ltd. The authors declare no competing financial interest.

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August 22, 2023
October 24, 2023