Editing Gel (section)

=== Modified Donnan equilibrium of polyelectrolyte gels ===
Consider a [[hydrogel]] made of [[polyelectrolyte]]s decorated with [[weak acid]] groups that can ionize according to the reaction

: <math>\text{HA} \rightleftharpoons \text{A}^- + \text{H}^+</math>

is immersed in a salt solution of physiological concentration. The degree of [[ionization]] of the [[polyelectrolyte]]s is then controlled by the <math>\text{pH}</math> and due to the charged nature of <math>\text{H}^+</math> and <math>\text{A}^-</math>, [[electrostatic interactions]] with other ions in the systems. This is effectively a reacting system governed by [[Acid dissociation constant|acid-base equilibrium]] modulated by electrostatic effects, and is relevant in [[drug delivery]], sea water [[desalination]] and [[Dialysis (chemistry)|dialysis]] technologies. Due to the elastic nature of the gel, the dispersion of <math>\text{A}^-</math> in the system is constrained and hence, there will be a partitioning of salts ions and <math>\text{H}^+</math> inside and outside the gel, which is intimately coupled to the [[polyelectrolyte]] degree of ionization. This ion partitioning inside and outside the gel is analogous to the partitioning of ions across a semipemerable membrane in classical [[Gibbs–Donnan effect|Donnan]] theory, but a membrane is not needed here because the gel volume constraint imposed by network elasticity effectively acts its role, in preventing the macroions to pass through the fictitious membrane while allowing ions to pass.<ref name=":2">{{Cite journal |last1=Landsgesell |first1=Jonas |last2=Hebbeker |first2=Pascal |last3=Rud |first3=Oleg |last4=Lunkad |first4=Raju |last5=Košovan |first5=Peter |last6=Holm |first6=Christian |date=2020-04-28 |title=Grand-Reaction Method for Simulations of Ionization Equilibria Coupled to Ion Partitioning |url=https://pubs.acs.org/doi/10.1021/acs.macromol.0c00260 |journal=Macromolecules |language=en |volume=53 |issue=8 |pages=3007–3020 |doi=10.1021/acs.macromol.0c00260 |bibcode=2020MaMol..53.3007L |issn=0024-9297}}</ref>

The coupling between the ion partitioning and polyelectrolyte ionization degree is only partially by the classical [[Gibbs–Donnan effect|Donnan]] theory. As a starting point we can neglect the electrostatic interactions among ions. Then at equilibrium, some of the weak acid sites in the gel would dissociate to form <math>\text{A}^-</math>that electrostatically attracts positive charged <math>\text{H}^+</math> and salt cations leading to a relatively high concentration of <math>\text{H}^+</math> and salt cations inside the gel. But because the concentration of <math>\text{H}^+</math> is locally higher, it suppresses the further ionization of the acid sites. This phenomenon is the prediction of the classical Donnan theory.<ref>{{Cite book |url=https://www.worldcat.org/oclc/48383405 |title=Electrostatic effects in soft matter and biophysics |date=2001 |publisher=Kluwer Academic Publishers |others=Christian, Ph. D. Holm, Patrick Kékicheff, Rudolf Podgornik, North Atlantic Treaty Organization. Scientific Affairs Division, NATO Advanced Research Workshop on Electrostatic Effects in Soft Matter and Biophysics |isbn=1-4020-0196-7 |location=Dordrecht [Netherlands] |oclc=48383405}}</ref> However, with electrostatic interactions, there are further complications to the picture. Consider the case of two adjacent, initially uncharged acid sites <math>\text{HA}</math> are both dissociated to form <math>\text{A}^-</math>. Since the two sites are both negatively charged, there will be a charge-charge repulsion along the backbone of the polymer than tends to stretch the chain. This energy cost is high both elastically and electrostatically and hence suppress ionization. Even though this ionization suppression is qualitatively similar to that of Donnan prediction, it is absent without electrostatic consideration and present irrespective of ion partitioning. The combination of both effects as well as gel elasticity determines the volume of the gel at equilibrium.<ref name=":2" /> Due to the complexity of the coupled acid-base equilibrium, electrostatics and network elasticity, only recently has such system been correctly recreated in [[Computer simulations of fluids|computer simulations]].<ref name=":2" /><ref>{{Cite journal |last1=Blanco |first1=Pablo M. |last2=Madurga |first2=Sergio |last3=Mas |first3=Francesc |last4=Garcés |first4=Josep L. |date=2019-11-12 |title=Effect of Charge Regulation and Conformational Equilibria in the Stretching Properties of Weak Polyelectrolytes |url=https://pubs.acs.org/doi/10.1021/acs.macromol.9b01160 |journal=Macromolecules |language=en |volume=52 |issue=21 |pages=8017–8031 |doi=10.1021/acs.macromol.9b01160 |bibcode=2019MaMol..52.8017B |hdl=2445/156380 |s2cid=208747045 |issn=0024-9297|hdl-access=free }}</ref>