Gels 2022, 8, 96. https://doi.org/10.3390/gels8020096 www.mdpi.com/journal/gels Article Evidence of Many-Body Interactions in the Virial Coefficients of Polyelectrolyte Gels Ferenc Horkay 1, * and Jack F. Douglas 2, * 1 Section on Quantitative Imaging and Tissue Sciences, Eunice Kennedy Shriver National Institute of Child Health and Human Development, National Institutes of Health, Bethesda, MD 20892, USA 2 Material Measurement Laboratory, Materials Science and Engineering Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA * Correspondence: horkayf@mail.nih.gov (F.H.); jack.douglas@nist.gov (J.F.D.) Abstract: Simulation studies of aqueous polymer solutions, and heuristic arguments by De Gennes for aqueous polyethylene oxide polymer solutions, have suggested that many-body interactions can give rise to the ‘anomalous’ situation in which the second osmotic virial coefficient is positive, while the third virial coefficient is negative. This phenomenon was later confirmed in analytic calculations of the phase behavior and the osmotic pressure of complex fluids exhibiting coopera- tive self-assembly into extended dynamic polymeric structures by Dudowicz et al. In the present study, we experimentally confirm the occurrence of this osmotic virial sign inversion phenomenon for several highly charged model polyelectrolyte gels (poly(acrylic acid), poly(styrene sulfonate), DNA, hyaluronic acid), where the virial coefficients are deduced from osmotic pressure measure- ments. Our observations qualitatively accord with experimental and simulation studies indicating that polyelectrolyte materials exhibit supramolecular assembly in solution, another symptomatic property of fluids exhibiting many-body interactions. We also find that the inversion in the varia- tion of the second (A2) and third (A2) virial coefficients upon approach to phase separation does not occur in uncharged poly(vinyl acetate) gels. Finally, we briefly discuss the estimation of the os- motic compressibility of swollen polyelectrolyte gels from neutron scattering measurements as an alternative to direct, time-consuming and meticulous osmotic pressure measurements. We con- clude by summarizing some general trends and suggesting future research directions of natural and synthetic polyelectrolyte hydrogels. Keywords: polyelectrolyte gel; phase separation; salt; multivalent ions; osmotic swelling pressure; osmotic compressibility; small angle neutron scattering; osmotic virial coefficient; swell- ing-deswelling transition 1. Introduction The phase behavior of materials ranging from gases to polymer solutions and gels, can often be understood semi-quantitatively in terms of the first few virial coefficients quantifying intermolecular interaction strength in the limit of low concentration [1–5]. The second (A2) and third (A3) virial coefficients, or their equivalent χi -interaction pa- rameters in a polymer solution context [6,7], have correspondingly been adopted as measures of intermolecular interactions having fundamental significance for materials classification, design, and characterization [8]. It is often implicit in this type of descrip- tion of phase behavior that the interactions between the molecules can be described by a single pairwise decomposable interactions such as well-known Lennard-Jones or square-well potentials. However, it is not clear if patterns of phase behavior based on this type of ‘simple fluid’ apply to ‘complex liquids’, such as ionic and polyelectrolyte solu- tions, polyelectrolyte gels, etc. In these systems, many-body interactions associated with Citation: Horkay, F.; Douglas, J.F. Evidence of Many-Body Interactions in the Virial Coefficients of Polyelectrolyte Gels. Gels 2022, 8, 96. https://doi.org/10.3390/gels8020096 Academic Editor: Annarosa Gugliuzza Received: 30 December 2021 Accepted: 2 February 2022 Published: 4 February 2022 Publisher’s Note: MDPI stays neu- tral with regard to jurisdictional claims in published maps and insti- tutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/license s/by/4.0/).