Reply to the 'Comment on "Simulations of ionization equilibria in weak polyelectrolyte solutions and gels"' by J. Landsgesell, L. Nová, O. Rud, F. Uhlík, D. Sean, P. Hebbeker, C. Holm and P. Kosovan, Soft Matter, 2019, 15, 1155-1185.

Levin and Bakhshandeh suggested in their comment that (1), we stated in our recent review that pH-pKA is a universal parameter for titrating systems, that (2), we omitted to mention in our review the broken symmetry of the constant pH algorithm, and that (3), a constant pH simulation must include a grand-canonical exchange of ions with the reservoir. As a reply to (1), we point out that Levin and Bakhshandeh misquoted and hence invalidated our original statement. We therefore explain in detail under which circumstances pH-pKA can be a universal parameter, and also demonstrate why their numerical example is not in contradiction to our statement. Moreover, the fact that pH-pKA is not a universal parameter for titrating systems is well known in the pertinent literature. Regarding (2), we admit that the symmetry-breaking feature of the constant pH algorithm has escaped our attention at the time of writing the review. We added some clarifying remarks to this behavior. Concerning (3), we point out that the grand-canonical coupling and the resultant Donnan potential are not features of single-phase systems, but are essential for two-phase systems, as was shown in a recent paper by some of us, see J. Landsgesell et al., Macromolecules, 2020, 53, 3007-3020.

Multivalent counterions accumulate in star-like polyelectrolytes and collapse the polymer in spite of increasing its ionization.

We used computer simulations to explore the dissociative and conformational behaviour of branched weak polyelectrolytes with multivalent counterions. We compared simulated titration curves and chain sizes in the presence of added salt of various valencies, keeping the total charge of salt constant. We showed that multivalent counterions enhance ionization of the weak polyelectrolytes, in spite of collapsing of the chains. We provided evidence that such an effect is absent in systems with only monovalent counterions at the same ionic strength, and thus cannot be attributed to electrostatic screening. We attributed it to strong ion-ion correlations that we quantified by comparing potentials of mean force with the mean electrostatic potentials. Finally, we used the partition coefficient to quantify the ability of star-like polyelectrolytes to capture multivalent ions, that is important for water-treatment applications. Our work provides fundamental understanding of the mechanism of polyelectrolyte collapse and ionization response upon addition of multivalent ions.

Simulations of ionization equilibria in weak polyelectrolyte solutions and gels.

This article recapitulates the state of the art regarding simulations of ionization equilibria of weak polyelectrolyte solutions and gels. We start out by reviewing the essential thermodynamics of ionization and show how the weak polyelectrolyte ionization differs from the ionization of simple weak acids and bases. Next, we describe simulation methods for ionization reactions, focusing on two methods: the constant-pH ensemble and the reaction ensemble. After discussing the advantages and limitations of both methods, we review the existing simulation literature. We discuss coarse-grained simulations of weak polyelectrolytes with respect to ionization equilibria, conformational properties, and the effects of salt, both in good and poor solvent conditions. This is followed by a discussion of branched star-like weak polyelectrolytes and weak polyelectrolyte gels. At the end we touch upon the interactions of weak polyelectrolytes with other polymers, surfaces, nanoparticles and proteins. Although proteins are an important class of weak polyelectrolytes, we explicitly exclude simulations of protein ionization equilibria, unless they involve protein-polyelectrolyte interactions. Finally, we try to identify gaps and open problems in the existing simulation literature, and propose challenges for future development.

Local pH and effective pKA of weak polyelectrolytes - insights from computer simulations.

In this work we study the titration behavior of weak polyelectrolytes by computer simulations. We analyze the local pH near the chains at various conditions and provide molecular-level insight which complements the recent experimental determination of this quantity. Next, we analyze the non-ideal titration behaviour of weak polyelectrolytes in solution, calculate the effective ionization constant and compare the simulation results with theoretical predictions. In contrast with the universal behaviour with respect to chain length, we find non-universality and deviations from theory with respect to polymer concentration and permittivity of the solvent. The latter we explain in terms of counterion condensation and ion correlation effects, which lead to reversal of the non-ideal titration behaviour at very low permittivities. We discuss the impact of these findings on the interpretation of experimental results.

Modeling the Phase Transition in Hydrophobic Weak Polyelectrolyte Gels under Compression.

One of the emerging water desalination techniques relies on the compression of a polyelectrolyte gel. The pressures needed reach tens of bars, which are too high for many applications, damage the gel and prevent its reuse. Here, we study the process by means of coarse-grained simulations of hydrophobic weak polyelectrolyte gels and show that the necessary pressures can be lowered to only a few bars. We show that the dependence of applied pressure on the gel density contains a plateau indicating a phase separation. The phase separation was also confirmed by an analytical mean-field theory. The results of our study show that changes in the pH or salinity can induce the phase transition in the gel. We also found that ionization of the gel enhances its ion capacity, whereas increasing the gel hydrophobicity lowers the pressure required for gel compression. Therefore, combining both strategies enables the optimization of polyelectrolyte gel compression for water desalination purposes.

Water Desalination Using Polyelectrolyte Hydrogel: Gibbs Ensemble Modeling.

Polyelectrolyte hydrogels can absorb a large amount of water across an osmotic membrane as a result of their swelling pressure. On the other hand, the insoluble cross-linked hydrogel network enables dewatering under the influence of external (thermal and/or mechanical) stimuli. Moreover, from a thermodynamic perspective, a polyelectrolyte hydrogel is already an osmotic membrane. These properties designate hydrogels as excellent candidates for use in desalination, at the same time avoiding the use of expensive membranes. In this article, we present our recent theoretical study of polyelectrolyte hydrogel usage for water desalination. Employing a coarse-grained model and the Gibbs ensemble, we modeled the thermodynamic equilibrium between the coexisting gel phase and the supernate aqueous salt solution phase. We performed a sequence of step-by-step hydrogel swellings and compressions in open and closed systems, i.e., in equilibrium with a large and with a comparably small reservoir of aqueous solution. The swelling in an open system removes ions from the large reservoir, whereas the compression in a closed system decreases the salt concentration in the small reservoir. We modeled this stepwise process of continuous decrease of water salinity from seawater up to freshwater concentrations and estimated the energy cost of the process to be comparable to that of reverse osmosis.

Interactions of star-like polyelectrolyte micelles with hydrophobic counterions.

Hydrophobicity of a counterion has a profound effect on the interaction with polyelectrolytes similar to that of multivalency. Specifically, understanding this interaction in weak polyelectrolyte micelles might assist in developing nanocarriers for pH-controlled encapsulation and release. We used star-like weak polyelectrolyte micelles of polystyrene-block-poly(2-vinyl pyridine) (PS-P2VP) with fixed aggregation number as a model polyelectrolyte, and cobalt bis(1,2-dicarbollide) (COSAN) as a model hydrophobic anion. We used NMR to assess the mobility of the polymer segments in the presence of varying amounts of COSAN, and at varying protonation degrees of the polyelectrolyte. Same experiments with indifferent electrolyte (NaCl) were used as a control. Furthermore, we used coarse-grained simulations to obtain a detailed picture of the effect of hydrophobic counterions on the conformation of the micelles. A small amount of hydrophobic counterions causes morphological changes within the micelles, whereas a bigger amount causes precipitation. This was confirmed both in simulations and in experiments. Furthermore, adsorption of the counterions induces ionization of the collapsed segments of the polyelectrolyte. Although the COSAN/P2VP system is rather specific, the generic model used in the coarse-grained simulations shows that the observed behavior is a consequence of synergy of hydrophobic and electrostatic attraction between polyelectrolytes and hydrophobic counterions. Our study provides general insights into the molecular mechanisms of these interactions.