To study macroscopic systems with coarse grained simulations one typically simulates a micro- scopic part of this macroscopic system. By reducing the size of the simulated system one introduces finite size effects. In this work we study the finite-size effects in the reaction ensemble, which is used to simulate reactive system. We calculate the finite-size effects in a non-interacting systems by explicitly calculating the partition function. This approach provides high precision data at low computational costs. For a grand canonical insertion/deletion of a pair of particles our results reproduces previously published results, validating our approach. Further, we show that a sim- ple isomerization reaction is not affected by finite size effects. For a decomposition reaction we show that previous estimates were overestimating the finite-size effects, and one can simulate much smaller systems while avoiding the finite-size effects. For previously studied acid-base equilibria the finite-size effects are only relevant at extreme conditions. The tool we provide allows to a priori estimate the finite-size effects and find the limits of the applicability of the reaction ensemble.

We developed a new 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, that we term the Grand-reaction method. More generally, it can be used for simulations of any reactive system coupled to a reservoir of a known composition. Conceptually, it can be regarded as an extension of the reaction ensemble, combining explicit simulations of reactions within the system and Grand-canonical exchange of particles with the reservoir. To demonstrate its strength, we applied our method to a solution of weak polyelectrolytes in equilibrium with a reservoir. Our results show that the ionization and swelling of a weak polyelectrolyte are affected by the Donnan effect due to the partitioning of ions and by the polyelectrolyte effect due to electrostatic repulsion along the chain. Both effects lead to a similar shift in ionization and swelling as a function of pH, albeit for different physical reasons. By comparison with published results, we showed that neglecting one or the other effect may lead to erroneous predictions or misinterpretations of results. In contrast, the Grand-reaction method accounts for both effects on the results and allows us to quantify them. Finally, we outline possible extensions and generalizations of the method and provide a set of guidelines for its safe application by a broad community of users.

The swelling of polyelectrolyte hydrogels has been often explained using simple models derived from the Flory-Rehner model. While these models qualitatively predict the experimentally observed trends, they also introduce strong approximations and neglect some important contributions. Consequently, they sometimes incorrectly ascribe the observed trends to contributions which are of minor importance under the given conditions. In this work, we investigate the swelling properties of weak (pH-responsive) polyelectrolyte gels at various pH and salt concentrations, using a hierarchy of models, gradually introducing various approximations. For the first time, we introduce a three-dimensional particle-based model which accounts for the topology of the hydrogel network, for electrostatic interactions between gel segments and small ions and for acid-base equilibrium coupled to the Donnan partitioning of small ions. This model is the most accurate one, therefore, we use it as a reference when assessing the effect of various approximations. As the first approximation, we introduce the affine deformation, which allows us to replace the network of many chains by a single chain, while retaining the particle-based representation. In the next step, we use the mean-field approximation to replace particles by density fields, combining the Poisson-Boltzmann equation with elastic stretching of the chain. Finally, we introduce an ideal gel model by neglecting the electrostatics while retaining all other features of the previous model. Comparing predictions from all four models allows us to understand which contributions dominate at high or low pH or salt concentrations. We observe that the field-based models overestimate the ionization degree of the gel because they underestimate the electrostatic interactions. Nevertheless, a cancellation of effects on the electrostatic interactions and Donnan partitioning causes that both particle-based and field-based models consistently predict the swelling of the gels as a function of pH and salt concentration. Thus, we can conclude that any of the employed models can rationalize the known experimental trends in gel swelling, however, only the particle-based models fully account for the true effects causing these trends. The full understanding of differences between various models is important when interpreting experimental results in the framework of existing theories and for ascribing the observed trends to particular contributions, such as the Donnan partitioning of ions, osmotic pressure or electrostatic interactions.

Weak ampholytes are ubiquitous in nature and commonly found in artificial pH-responsive systems. However, our limited understanding of their charge regulation and the lack of predictive capabilities hinder the bottom-up design of such systems. Here, we used a coarse-grained model of a flexible polymer with weakly ionisable monomer units to quantitatively analyse the ionisation behaviour of two oligopeptides. Our model predicts differences in the charge states between oligopeptides and monomeric amino acids, showing that conformational flexibility and electrostatic interactions between weak acid and base side chains play a key role in the charge regulation. By comparing our simulations with experimental results from potentiometric titration, capillary zone electrophoresis and NMR, we demonstrated that our model reliably predicts the charge state of various peptide sequences. Ultimately, our model is the first step towards understanding the charge regualtion in flexible disordered proteins, and towards using predictive bottom-up design of responsive ampholytes to tailor their properties as a function of charge and pH.

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.

The influence of spacer chains on the intramolecular complexation in star-shaped heteroarm (miktoarm) polymers is investigated. To overcome the mutual attraction of different polymeric components present in a miktoarm star with different homopolymeric arms, spacer chains of different length are attached to the core of the star at three different positions. In most of the investigated cases, this leads to diblock copolymer arms within the miktoarm star. Hereby, the inner spacer separates the outer blocks from their attractively interacting homopolymeric arms. The effect on the intramolecular complexation and the structure of the star polymer is obtained by Monte Carlo simulations of a simple bead-spring model. Then, long spacers can completely prevent the complexation. Both, local shielding by the spacer chains and the increased distance between the complex-forming polymers due to the spacer chains inhibit the complex formation. For a range of spacer positions and lengths, an equilibrium between a system forming a complex and a complex free system is found. The spacer chains can be used as a tool to tune the intramolecular complexation.

Complex-forming copolymers (polymers whose monomeric units of different type attract each other) are often discussed to be a pathway to achieve self-assembled structures, which differ from the classical micellization scheme. Here, coarse-grained Monte Carlo simulations are performed to investigate the aggregation behavior of miktoarm star polymers/diblock copolymers, which are able to form complexes between the constitutional blocks. These results are compared with the corresponding polymers without complexation, but allowing segregation of one of the blocks (selective solvent scenario). By analyzing the radial distribution function and the effective polymer-polymer interaction the occurrence of aggregates is probed. While the segregating copolymers form larger aggregates, the complex-forming copolymers are stable against aggregation up to concentrations in the order of the overlap concentration. These results are in line with former experimental findings.

Segregation is a well-known principle for micellization, as solvophobic components try to minimize interactions with other entities (such as solvent) by self-assembly. An opposite principle is based on complexation (or coacervation), leading to the coassembly/association of different components. Most cases in the literature rely on only one of these modes, though the classical micellization scheme (such as spherical micelles, wormlike micelles, and vesicles) can be enriched by a subtle balance of segregation and complexation. Because of their counteraction, micellar constructs with unprecedented structure and behavior could be obtained. In this feature, systems are highlighted, which are between both mechanisms, and we study concentration, architecture, and confinement effects. Systems with inter- and intramolecular interactions are presented, and the effects of polymer topology and monomer sequence on the resulting structures are discussed. It is shown that complexation can lead to altered micellization behavior as the complex of one hydrophobic and one hydrophilic component can have a very low surface tension toward the solvent. Then, the more soluble component is enriched at the surface of the complex and acts as a microsurfactant. Although segregation dominates for amphiphilic copolymers in solution, the effect of the complexation can be enhanced by branching (change of architecture). Another possibility to enhance the complexation is by confining copolymers in a (pseudo-) 2D environment (like the one available at liquid-liquid interfaces). These observations show how new structural features can be achieved by tuning the subtle balance between segregation and complexation/solubilization.

An adaptive algorithm optimizing single-particle translational displacement parameters in Metropolis Monte Carlo simulations is presented. The optimization is based on maximizing the mean square displacement of a trial move. It is shown that a large mean square displacement is strongly correlated with a high precision of average potential energy. The method is here demonstrated on model systems representing a Lennard-Jones fluid and a dilute polymer solution at poor solvent conditions. Our adaptive algorithm removes the need to provide values of displacement parameters in simulations, and it is easily extendable to optimize parameters of other types of trial moves.

The intramolecular complex formation occurring in miktoarm star polymers and block copolymers is investigated experimentally and by means of Monte Carlo simulations. The driving force of the complex formation is an attractive interaction between the different blocks. In the simulations, a simple bead-spring model with an attractive Lennard-Jones potential is used to mimic the complexation. The effect of topology (arm number) and composition (arm length) is simulated. Experimentally, star-shaped polymeric systems and diblock copolymers of poly(propylene oxide) (PPO) and poly(dimethylaminoethyl methacrylate) (PDMAEMA) of varying degrees of polymerization are investigated. By quantifying the complex formation with help of fluorescence spectroscopy of a polarity-sensitive probe, we show that a complex is formed preferably at high degrees of polymerization of PDMAEMA and a high number of PDMAEMA arms. These results are in accordance with the simulations: we show that the actual amount of PPO in the complex correlates with the fluorescence dye uptake.

More than ever polymer science focuses on complex molecular structures and supramolecular assemblies. Microgels are responsive polymer materials and structures, which can be manipulated in e.g. charge or size by external parameters like pH or temperature variation. The investigated microgels are soft particulate polymer networks that can be dispersed in an aqueos medium. They reveal unique features providing new opportunities to develop smart bio‐inspired materials. In contrast to rigid colloidal particles, which lack the possibility to adapt their size and shape to enviromental requirements, microgels have switchable properties of form and function that make them very useful in a wide range of e.g. biological sciences and medical applications. They combine properties of dissolved macromolecules with those of colloidal particles. The direct visualization of the internal structure of materials is very important to analyse the spatial distribution of different compartments and, thus, to design novel materials with tailored properties. Microgels can be prepared with various morphologies and functions in different compartments. Careful analysis of the correlation between architecture and function requires powerful methods to visualize inner structure and compartmentalization in the nanometer range. Here, the direct visualization of different compartments within microgels using a combination of in situ and cryo transmission electron microscopy methods is shown. In particular, the challenge of determing the radial distribution of appropriately labeled compartments within single microgels and particles from 2D projections is adressed. Microgels with core‐shell architecture were obtained by precipitation polymerization. First a particle was synthesized and purified before a shell was synthesized on top by the seed and feed method. Core and shell have oppositely charged copolymers to create a two compartment amphoteric microgel system, that is alternately stained with gold and magnetite nanoparticles. [1], [2] For in situ liquid cell experiments, a thin layer of liquid was embedded between two hermetically sealed, electron transparent Si 3 N 4 ‐windows. The used holder is an in situ ‐liquid cell holder manufactured by Hummingbird Company and the microscope is a Zeiss Libra 200FE with an acceleration voltage of 200 kV. The resolution is mainly limited by the thickness of the liquid. Figure 1 shows a comparison between cryo TEM and in situ STEM. Due to the liquid layer thickness the resolution is limited in (b). Also the Brown emotion leads to a defocused and smudged image. Figure 2 shows the comparison of the radial distribution of nanoparticles according to the two images above calculated by a MATLAB routine. For the cryo TEM image in (b) the relative particle density as a function of the relative particle radius is plotted. (c) shows a 3D reconstruction of the model.

Compartmentalization in soft matter is important for segregating and coordinating chemical reactions, sequestering (re)active components, and integrating multifunctionality. Advances depend crucially on quantitative 3D visualization in situ with high spatiotemporal resolution. Here, we show the direct visualization of different compartments within adaptive microgels using a combination of in situ electron and super-resolved fluorescence microscopy. We unravel new levels of structural details and address the challenge of reconstructing 3D information from 2D projections for nonuniform soft matter as opposed to monodisperse proteins. Moreover, we visualize the thermally induced shrinkage of responsive core-shell microgels live in water. This strategy opens doors for systematic in situ studies of soft matter systems and their application as smart materials.

Binary diblock copolymers and corresponding ternary miktoarm stars are studied at oil-water interfaces. All polymers contain oil-soluble poly(propylene oxide) PPO, water-soluble poly(dimethylaminoethyl methacrylate) PDMAEMA and/or poly(ethylene oxide) PEO. The features of their Langmuir compression isotherms are well related to the ones of the corresponding homopolymers. Within the Langmuir-trough, PEO-b-PPO acts as the most effective amphiphile compared to the other PPO-containing copolymers. In contrast, the compression isotherms show a complexation of PPO and PDMAEMA for PPO-b-PDMAEMA and the star, reducing their overall amphiphilicity. Such complex formation between the blocks of PPO-b-PDMAEMA is prevented in bulk water but facilitated at the interface. The weakly-interacting blocks of PPO-b-PDMAEMA form a complex due to their enhanced proximity in such confined environments. Scanning force microscopy and Monte Carlo simulations with varying confinement support our results, which are regarded as compliant with the mathematical random walk theorem by Pólya. Finally, the results are expected to be of relevance for e.g. emulsion formulation and macromolecular engineering.

Star polymers with arms of two different homopolymers with a weak mutual attraction have been investigated using a coarse grained polymer model and Monte Carlo (MC) simulations. The effect of the polymer architecture on the complexation behavior has been studied for various numbers of arms (up to 8) starting from a diblock copolymer and compared to diblock copolymers as well as alternating linear copolymers. It was found that the star architecture promotes an internal complexation, which is indicated by a reduction of the end-to-end distance of the polymer chains. The local concentration of the weakly attractive partners is important to promote their interaction and to harvest the attraction. The results were compared to experimental data.

The site exchange of an anion moiety (O, N, F, Ne, P, S, Cl, and Ar) with an oxygen vacancy in fluorite-structured CeO2 was studied by means of density functional theory (DFT) calculations. The obtained activation energies of migration vary between 0.2 and 0.9 eV, and increase with the formal valence of the migrating ion; the size of the migrating ion appears to play a minor role. An analysis of ion displacements suggests that repulsive Coulombic interactions between the migrating anion and oxygen ions as the next-nearest neighbors in the saddle-point configuration provide the dominant contribution to the activation energy of migration. As well as emphasizing the ease with which anion moieties are mobile in AO2 fluorite materials, these results suggest a new paradigm for understanding fast oxygen-ion conducting materials.