random and a representative, average chain length is unknown.
This problem can only be solved with better analysis of the experimentally realized crosslinking structure or crosslinkers which result in less random structures. On top of this uncertainty, it seems possible that a repeated, more controlled experiment for determiningQm(whereVgel=V/2 is strictly obeyed) could improve agreement with the simulations significantly. It seems, however, more reasonable to report the gel swellingQm as a function of the salt concentration in the supernatant solution.
These data are straightly available from simulations but, unfor-tunately, not from the experiments as they were performed in [116]. Despite all these possible reasons for deviations our gel models can reproduce significant trends in the experimental data.
Therefore, we conclude that the basic physical driving forces of gel swelling are incorporated in the models.
qualitatively the swelling behavior for differently crosslinked SN hydrogels. In general, the simulations overestimatedQm, which can be ascribed to several simplifications that were made and to the rather undefined structure of the hydrogels synthesized via free radical polymerization. Furthermore, MD simulations showed in agreement with the experiments that the water ab-sorbency of an IPN is much smaller at low salinities compared to its related SN, whereas the differences between both network types vanish at higher salt concentrations.
The work detailed in this chapter is in part submitted as:
J. Landsgesell, P. Hebbeker, O. Rud, R. Lunkad, P. Košovan, C. Holm. “Grand-Reaction Method for Simulations of Ionization Equilibria and Ion Partitioning in a Broad Range of pH and Ionic Strength” In:ChemRxiv(2019)
URL:https://doi.org/10.26434/chemrxiv.9741746.v2 Reacting systems in contact with a supernatant solution (i.e. a reservoir) are ubiquitous in chemical research, especially in colloid and polymer science. Such a setup is widely used in applications to separate or purify substances [124], like in medicine [125, 126, 127] or water purification [128,129,130].
Other applications are osmotic motors [12], sensors [131] or sim-plyweak polyelectrolyte gels(which we investigate in chapter 6). A schematic view of the systems we want to investigate is depicted in Figure 5.1.
The equilibrium state of the ionization reaction in such systems is determined by the pH and by the concentration of other ions in the system. The presence of charged polymers affects the partitioning of small ions between the system and the reservoir.
The change in ion concentrations affect the reaction equilibrium,
System Reservoir
H+ H+ Cl
-OH -Cl -HA
A-HA HA
Na+ Na+
Na+
H+ OH
-Cl -H+
Figure 5.1: A system exchanging small ions with a reservoir. The macro-molecules inside the system take part in chemical reactions.
Different colors represent different particle types. In total, the problem requires describing at least six particle types:
four for the ionic species in the reservoir (H+, OH−, Na+, Cl−) and two additional types for the different ionization states of the monomers in the system (HA, A−).
which in turn affects the ion partitioning in a non-trivial feedback loop.
This feedback loop cannot be captured using the constant pH method [48] or the reaction ensemble in its original formulation.
The constant pH method cannot be applied due to the fact that it would need to be used with the pH value in the system containing an unknown Donnan-partitioning contribution∆(see Equation (5.8)). The reaction ensemble on the other hand needs to be enhanced by a grand-canonical simulation protocol and it turns out that this is not straight forward. We, therefore, introduce the grand-reaction method for coarse-grained simulations of acid-base equilibria in a system coupled to a reservoir at a given pH and concentration of added salt.
To perform a coarse-grained simulation of a reactive phase in equilibrium with a multi-component solution phase, we need to
solve two strictly separated tasks:
1. Prescribing the exact composition of thereservoir
2. Incorporating chemical reactions in the simulation of the system
The first task has its own challenges, related to the definition of pH, which is why we devote Section 5.1to defining pH and describing the ionic strength in an aqueous solutions. In Section 5.2, we introduce an idealized Donnan model which allows to understand the main consequences of coupling a reactive system to a reservoir. In Section 5.3, we present how to faithfully represent a reservoir (defined by a pH value and reservoir salt concentration) with interacting particles. Using this reservoir, we can then impose chemical reactions in the system, which is done in Section 5.4. The approach we present there is aimed at avoiding sampling problems. Finally, the results for an example system of a weak polyelectrolyte solution, separated from the reservoir by a semipermeable membrane, are shown in Section 5.6. In the Chapter6we use the grand-reaction method, presented here, for describing weak polylelectrolyte gels.
5.1 pH and Ionic Strenght
pH is a measure for the chemical potential µH+ of H+ and is defined by IUPAC [22] using the relative activityaH+:
pH=−log10(aH+)=−log10(cH+γH+/c ), (5.1)
with the mean activity coefficient1γH+ =exp(βµexH+). The chemi-cal potential of H+and OH−ions are coupled in aqueous solution via the auto-dissociation of water:
Kw=aH+aOH− = cH+γH+cOH−γOH−
c 2 =10−14,
The variation of ionic strength with pH is depicted in Figure 5.2. The impact of pH on the ionic strength implies that even strong polyelectrolytes react to changes in pH due to a modified electrostatic screening. Therefore, a strong polyelectrolyte is more collapsed at extreme pH than around pH=7, where the ionic strength is minimal.
In contrast to simulations of strong polyelectrolytes, where the species of particles are not important, the information about the species of a particle has to be resolved in simulations in-volving chemical reactions: Chemical reactions are by definition sensitive to the chemical potential of specific particle species (see the defining equation for the equilibrium constant (2.12)).
An important part of the chemical potential of a species is the ideal part, mainly described by the concentration of the species.
Therefore, this information has to be represented in simulations of weak polyelectrolyte solutions.
1If the mean activity coefficient is used in the definition of pH, then we use that the activity of H+is defined byaH+ =exp(β(µH+−µH+)), where we useµH+=µH+−zH+e0ψ, i.e. the total (electro-)chemical potential with the (mean) electric potential contribution subtracted (compare [132,133]). The reason for this is, that the mean activity coefficientγ±=p
exp(β(µex+ +µex−)) is defined using a sum of the excess chemical potential of a positive and a negative ion and therefore the electric potential contribution to the excess chemical potential is canceled, compare Section 2.5.3
0 2 4 6 8 10 12 14 0
2 4 6 8 10
·10−2
pH IonicstrengthIres[mol/L]
Figure 5.2: Schematic representation of ionic strengthIresin a salt free solution:Iresvaries, when varying the pH of a solution by adding strong acid, to lower pH, or base, to increase pH.
Note 9
Note that it is always possible to distinguish whether a parti-cle is of a given species or not. This is not in contradiction to the principle of indistinguishable particles: the principle just states that particles belonging to the same species cannot be distinguished.
In coarse-grained simulations it is common to neglect the chem-ical nature of a species if it has interactions which are, approx-imately, identical to another species. For example, when we deal with coarse-grained simulations of strong polyelectrolytes,
it is natural to ignore the fact that there is a certain amount of H+, Na+, OH−and Cl−ions, because the strong polyelectrolyte interacts with all ions of same charge in the same way. Therefore, only the resulting ionic strength is of importance. This insight allows to simplify the simulations and coarse grain the chemical nature of the ion species away so that one only has to speak about positive and negative ions.