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.

Can oppositely charged polyelectrolyte stars form a gel? A simulational study.

We present a Langevin molecular dynamics study of an equimolar mixture of monodispersed oppositely charged di-block four-armed polyelectrolyte stars. We used an implicit solvent coarse-grained representation of the polyelectrolyte stars, and varied the length of the terminal charged blocks that reside on each arm. By varying the polymer concentration, we computed PV diagrams and determined the free-swelling equilibrium concentration with respect to a pure water reservoir as a function of the charged block length. We investigated various structural properties of the resulting equilibrium structures, like the number of ionic bonds, dangling arms, isolated stars, and cluster sizes. The ionic bonds featured a broad distribution of the number of arms involved and also displayed a distribution of net charges peaked around the neutral ionic bond. Our main result is that for charged block length equal to 4 and 5 ionized beads the resulting macro-aggregate spans the box and forms a network phase. Furthermore, we investigated the restructuring dynamics of ionic bonds; the results suggested both short bond lifetimes and a high frequency of ballistic association/dissociation events. Bonds result strong enough to yield a stable gel phase, but they are still weak enough to allow network restructuring under thermal fluctuations.

Erratum: Modeling Gel Swelling Equilibrium in the Mean Field: From Explicit to Poisson-Boltzmann Models [Phys. Rev. Lett. 122, 208002 (2019)].

This corrects the article DOI: 10.1103/PhysRevLett.122.208002.

Influence of weak groups on polyelectrolyte mobilities.

The ionization of dissociable groups in weak polyelectrolytes does not occur in a homogenous fashion. Monomer connectivity imposes constraints on the localization of the dissociated (charged) monomers that affect the local electric potential. As a result, the mean bare charge along a weak polyelectrolyte can vary depending on the proximity to topological features (e.g. presence of crosslinks or dangling ends). Using reaction-ensemble Monte-Carlo simulations we calculate the dissociation inhomogeneities for a few selected PE configurations, linear, rod-like, flexible four-arm star, and a star with stiff arms. An ensemble preaverage is used to obtain the annealed bare charge profile for these different polymer configurations. Using molecular dynamics simulations within a Lattice-Boltzman fluid, we investigate how the electrophoretic mobility is affected by the bare charge inhomogeneities arising from the annealed weak polyelectrolytes. Surprisingly, the mobility obtained for the situations corresponding to the predicted charge profile for annealed weak polyelectrolytes are not significantly different than the mobility obtained when all the monomers have an identical charge (under the constraint that the total polyelectrolyte bare charge is the same). This is also true for the stiff rod-like variants where conformational changes induced from the localization of the monomer charges are negligible. In salty solutions, we find that counterions are affected by the electric potential modulations induced by the topological features. Since the counterions crowd in regions where the electric potential caused by the dissociated monomers is highest, they wash-out the bare charge inhomogeneities and contribute to a more uniform effective backbone charge.

Modeling Gel Swelling Equilibrium in the Mean Field: From Explicit to Poisson-Boltzmann Models.

We develop a double mean-field theory for charged macrogels immersed in electrolyte solutions in the spirit of the cell model approach. We first demonstrate that the equilibrium sampling of a single explicit coarse-grained charged polymer in a cell yields accurate predictions of the swelling equilibrium if the geometry is suitably chosen and all pressure contributions have been incorporated accurately. We then replace the explicit flexible chain by a suitably modeled penetrable charged rod that allows us to compute all pressure terms within the Poisson-Boltzmann approximation. This model, albeit computationally cheap, yields excellent predictions of swelling equilibria under varying chain length, polymer charge fraction, and external reservoir salt concentrations when compared to coarse-grained molecular dynamics simulations of charged macrogels. We present an extension of the model to the experimentally relevant cases of pH-sensitive gels.

Poly(sodium acrylate) hydrogels: synthesis of various network architectures, local molecular dynamics, salt partitioning, desalination and simulation.

Various poly(sodium acrylate) hydrogels with different architectures, such as single networks, interpenetrating double networks and surface crosslinked hydrogels, are synthesized with a systematic change in their degree of crosslinking. The influence of these 3D structures on the absorbency of aqueous NaCl solutions is investigated. The local polymer mobility in water is probed in the form of transverse (T2) 1H-relaxation at a low field, which allowed confirming the structural aspects of the studied network topologies. Salt partitioning between the gel and the surrounding solution phase in NaCl solutions with an initial salt concentration of c0 = 0.017-0.60 mol L-1 (≙1-35 g L-1) is investigated. The data are compared with an idealized mean-field Donnan model, which fit the experimental findings only under the assumption of a drastically reduced effective charge density of feff ≈ 25 mol% independent of the hydrogel used. The unequal salt distribution allows desalination of salt water by applying an external pressure to a swollen hydrogel to recover its water which has a lower salinity. The specific energy needed to desalinate 1 m3 was estimated to be 6-18 kW h m-3. This value decreases with a lower degree of swelling independent of the network topology. Besides the experiments, simulations based on a Poisson-Boltzmann mean-field model and MD simulations are performed to determine the degree of swelling and salt partitioning as a function of c0 for different hydrogels. Both simulations describe qualitatively the experimental data, where deviations can be ascribed to model simplifications and the imperfect structure of the hydrogels synthesized via free radical polymerization.

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.

Computer Simulations of Static and Dynamical Properties of Weak Polyelectrolyte Nanogels in Salty Solutions.

We investigate the chemical equilibria of weak polyelectrolyte nanogels with reaction ensemble Monte Carlo simulations. With this method, the chemical identity of the nanogel monomers can change between neutral or charged following the acid-base equilibrium reaction HA ⇌ A- + H⁺. We investigate the effect of changing the chemical equilibria by modifying the dissociation constant K a . These simulations allow for the extraction of static properties like swelling equilibria and the way in which charge-both monomer and ionic-is distributed inside the nanogel. Our findings reveal that, depending on the value of K a , added salt can either increase or decrease the gel size. Using the calculated mean-charge configurations of the nanogel from the reaction ensemble simulation as a quenched input to coupled lattice-Boltzmann molecular dynamics simulations, we investigate dynamical nanogel properties such as the electrophoretic mobility µ and the diffusion coefficient D.

Wang-Landau Reaction Ensemble Method: Simulation of Weak Polyelectrolytes and General Acid-Base Reactions.

We present a novel method for the study of weak polyelectrolytes and general acid-base reactions in molecular dynamics and Monte Carlo simulations. The approach combines the advantages of the reaction ensemble and the Wang-Landau sampling method. Deprotonation and protonation reactions are simulated explicitly with the help of the reaction ensemble method, while the accurate sampling of the corresponding phase space is achieved by the Wang-Landau approach. The combination of both techniques provides a sufficient statistical accuracy such that meaningful estimates for the density of states and the partition sum can be obtained. With regard to these estimates, several thermodynamic observables like the heat capacity or reaction free energies can be calculated. We demonstrate that the computation times for the calculation of titration curves with a high statistical accuracy can be significantly decreased when compared to the original reaction ensemble method. The applicability of our approach is validated by the study of weak polyelectrolytes and their thermodynamic properties.