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.

Ho2C2 Cluster with Flexible Configurations inside a Large C2(61)-C92 Cage.

Carbide clusterfullerenes (CCFs) have been of great concern due to their potential applications in materials science, in which the internal carbide cluster plays vital roles in the stability and properties of CCF. However, there still remains a debate about what configuration is ideal for the internal carbide cluster. In this work, we isolated two isomers (I and II) of Ho2C94 and studied them by means of mass spectrometry, UV-vis-NIR spectroscopy, and cyclic/differential pulse voltammetry. A combined study of single-crystal X-ray diffraction (SC-XRD) and density functional theory (DFT) computation ascertains isomer-I as Ho2C2@C2(61)-C92, in which the Ho2C2 cluster displays variable configurations from planar zigzag to folded butterfly with very small distortion energy (∼10 kJ/mol). This study hence confirms that the internal carbide cluster is intrinsically flexible over a broad geometrical range in a relatively large fullerene cage, where the nanoscale compression effect is almost negligible.

Computer modeling of polymer stars in variable solvent conditions: a comparison of MD simulations, self-consistent field (SCF) modeling and novel hybrid Monte Carlo SCF approach.

Computer-aided modeling is a systematic approach to grasp the physics of macromolecules, but it remains essential to know when to trust the results and when not. For a polymer star, we consider three approaches: (i) Molecular Dynamics (MD) simulations and implementing a coarse-grained model, (ii) the self-consistent field approach based on a mean-field approximation and implementing the lattice model due to Scheutjens and Fleer (SF-SCF) and (iii) novel hybrid Monte Carlo self-consistent field (MC-SCF) method, which combines a coarse-grained model driven by a Monte Carlo method and a mean-field representation driven by SF-SCF. We compare the performance of these approaches under a wide range of solvent qualities. The MD approach is formally the most exact but suffers from reasonable convergence. The mean-field approach works similarly in all solvent qualities but is quantitatively least accurate. The MC-SCF hybrid allows us to combine the benefits of the simulation route and the effective performance of SCF. We consider the center-to-end distance Rce, the radius of gyration Rg2 of the star and the polymer density profiles φ(r) of polymer-segments in it. All three methods show a good qualitative agreement one to another. The MC-SCF method is in good agreement with the scaling predictions in the whole range of solvent quality values showing that it grasps the essential physics while remaining computationally in bounds.

Intramolecular micellization and nanopatterning in pH- and thermo-responsive molecular brushes.

Conformational transitions and nanoscale self-organization triggered in double pH- and thermo-responsive molecular brushes by varying environmental conditions are studied by means of analytical mean-field theory and numerical Scheutjens-Fleer self-consistent field modelling. Such molecular brushes are composed of multiple thermo-responsive side chains end-grafted onto the main chain (backbone) and are capable of acquiring ionic charges via reversible (de)protonation of the monomer units. Competition of long-range Coulomb repulsion with short-range solvophobic interactions leads to complex patterns in the intramolecular self-organization of molecular brushes. In particular, we observed formation of pearl necklace-like structures with multiple dense nanodomains formed by weakly ionized collapsed side chains and stabilized by a fraction protruding into the solution and strongly ionized ones. Such structures are thermodynamically stable in a certain parameter range and can be termed as intramolecular micelles. The stimuli-induced intramolecular nanopatterning occurs via a sequence of quasi-first order phase transitions corresponding to splitting/fusion of collapsed domains accompanied by jumps in the average degree of ionization and macromolecular dimensions. A re-entrant sequence of transitions is observed when the salt concentration is used as a control parameter. These theoretical predictions provide guidelines for design of smart unimolecular devices, for example multicompartment nanocarriers of active substances or nanosensors.

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.

Ho2O@D3(85)-C92: Highly Stretched Cluster Dictated by a Giant Cage and Unexplored Isomerization.

For endohedral metallofullerenes (EMFs), it has been well established that the cage shape and size should match those of the endohedral cluster. As a result, sufficient cluster-cage interaction can be achieved, which is essential for mutual stabilization. Nevertheless, how a small endohedral cluster nests in a giant fullerene has been less explored. Herein, we report a pair of large oxide-cluster fullerene (OCF) isomers, denoted as Ho2O@C92-I and -II. Crystallographic studies reveal that major isomer-I possesses a D3(85)-C92 cage with a highly stretched Ho2O cluster inside, which contributes to achieving regular metal-cage contacts. Density functional theory (DFT) computations also reveal the predominant abundance of the D3(85) isomer relative to the other two possible minor species including C1(67) and C2(64) isomers. Moreover, electrochemical (EC) studies verify that the isomers exhibit almost identical redox behaviors, indicating their similar cage structures. On the basis of the remarkable topological similarity of D3(85) and C1(67) isomers, isomer-II is likely to be Ho2O@C1(67)-C92, though it remains to be confirmed. Our studies thus provide new insights into the cage-cluster interplay and cage isomerization, both contributing to a better understanding of large EMFs.

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.

Ho2O@C84: Crystallographic Evidence Showing Linear Metallic Oxide Cluster Encapsulated in IPR Fullerene Cage of D2d(51591)-C84.

Fullerene C84 is the third-most-abundant species after C60 and C70. In the past decade, a variety of C84-based clusterfullerenes have been well-studied experimentally, which otherwise do not include oxide clusterfullerenes (OCFs). In this work, we report a comprehensive inspection of Ho2O@C84, including its mass, spectroscopic, crystallographic, electrochemical (EC), and density functional theory (DFT) studies. Importantly, crystallographic data reveal an IPR cage of D2d(51591)-C84 with a linear endohedral Ho-O-Ho cluster, indicating that the compression effect of the C84 cage is less pronounced compared to that of a smaller cage. The experimentally observed structure is confirmed by DFT computations, which also verify its superior stability. Further studies suggest that Ho2O@C84 has reduced EC and HOMO-LUMO gaps compared to those of empty species, again demonstrating the effect of cluster encapsulation.

Ho2O@C74: Ho2O Cluster Expands within a Small Non-IPR Fullerene Cage of C2(13333)-C74.

Steering the cluster configuration inside a fullerene cage has been one of most interesting topics in the field of fullerenes, since the physical property of a cluster fullerene may be modified accordingly. It has been well-recognized that the cluster configuration can be tuned via altering the cage size. Typically, the carbide cluster and the oxide cluster are experimentally seen to be curled up within a small fullerene cage whereas they are expanded in a large cage. In this work, a new oxide cluster fullerene Ho2O@ C2(13333)-C74 is prepared and isolated. The single-crystal X-ray diffraction (XRD) study reveals that the Ho2O cluster, however, expands within the small non-IPR cage of C2(13333)-C74 with a Ho-O-Ho angle of >170°, indicating that cluster configuration is highly related to the cage shape and cage structure as well. The DFT computation demonstrates that the cluster-to-cage electron-transfer obviously enhances the aromaticity of the motif containing the fused-pentagon pair and hence stabilizes the non-IPR cage of C2(13333)-C74. In addition, the electrochemical and magnetic properties of Ho2O@ C2(13333)-C74 are studied to further investigate the effect of endohedral Ho2O cluster.

Isolation and Structural Characterization of Er@ C2 v(9)-C82 and Er@ C s(6)-C82: Regioselective Dimerization of a Pristine Endohedral Metallofullerene Induced by Cage Symmetry.

Two Er@C82 isomers have been isolated and unambiguously characterized as Er@ C2 v(9)-C82 and Er@ C s(6)-C82, respectively, by single-crystal X-ray diffraction. Er@ C s(6)-C82 is identified as a dimeric structure in the crystalline state, but dimerization does not occur for Er@ C2 v(9)-C82 under identical crystallization conditions, indicating a cage-symmetry-induced dimerization process. Density functional theory calculations reveal that the major unpaired spin resides on a special C atom of Er@ C s(6)-C82, which leads to regioselective dimerization. Calculations also found that the dimeric structure of Er@ C s(6)-C82·Ni(OEP) is much more stable than the two monomers, suggesting a thermodynamically favorable dimerization process. Vis-near-IR spectrometric and electrochemical results demonstrate that the electronic structure of Er@C82 isomers is Er3+@C823-, instead of the theoretically proposed Er2+@C822-.

Adamantylidene Addition to M3 N@Ih -C80 (M=Sc, Lu) and Sc3 N@D5h -C80 : Synthesis and Crystallographic Characterization of the [5,6]-Open and [6,6]-Open Adducts.

Additions of adamantylidene (Ad) to M3 N@Ih -C80 (M=Sc, Lu) and Sc3 N@D5h -C80 have been accomplished by photochemical reactions with 2-adamantyl-2,3'-[3H]-diazirine (1). In M3 N@Ih -C80 , the addition led to rupture of the [6,6]- or [5,6]-bonds of the Ih -C80 cage, forming the [6,6]-open fulleroid as the major isomer and the [5,6]-open fulleroid as the minor isomer. In Sc3 N@D5h -C80 , the addition also proceeded regioselectively to yield three major isomeric Ad mono-adducts, despite the fact that there are nine types of C-C bonds in the D5h -C80 cage. The molecular structures of the seven Ad mono-adducts, including the positions of the encaged trimetallic nitride clusters, have been unambiguously determined through single-crystal XRD analyses. Furthermore, results have shown that stepwise addition of Ad to Lu3 N@Ih -C80 affords several Ad bis-adducts, two of which have been isolated and characterized. The X-ray structure of one bis-adduct clearly revealed that the second Ad addition took place at a [6,6]-bond close to an endohedral metal atom. Theoretical calculations have also been performed to rationalize the regioselectivity.

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.

Eu@C72: Computed Comparable Populations of Two Non-IPR Isomers.

Relative concentrations of six isomeric Eu@C 72 -one based on the IPR C 72 cage (i.e., obeying the isolated-pentagon rule, IPR), two cages with a pentagon-pentagon junction (symmetries C 2 and C 2 v ), a cage with one heptagon, a cage with two heptagons, and a cage with two pentagon-pentagon fusions-are DFT computed using the Gibbs energy in a broad temperature interval. It is shown that the two non-IPR isomers with one pentagon-pentagon junction prevail at any relevant temperature and exhibit comparable populations. The IPR-satisfying structure is disfavored by both energy and entropy.

The Unanticipated Dimerization of Ce@C2v (9)-C82 upon Co-crystallization with Ni(octaethylporphyrin) and Comparison with Monomeric M@C2v (9)-C82 (M = La, Sc, and Y).

We report that Ce@C2v (9)-C82 forms a centrosymmetric dimer when co-crystallized with Ni(OEP) (OEP = octaethylporphyrin dianion). The crystal structure of {Ce@C2v (9)-C82 }2 ⋅2[Ni(OEP)]⋅4 C6 H6 shows that a new C-C bond with a bond length of 1.605(5) Šconnects the two cages. The high spin density of the singly occupied molecular orbital (SOMO) on the cage and the pyramidalization of the cage are factors that favor dimerization. In contrast, the treatment of Ni(OEP) with M@C2v (9)-C82 (M = La, Sc, and Y) results in crystallization of monomeric endohedral fullerenes. A systematic comparison of the X-ray structures of M@C2v (9)-C82 (M = Sc, Y, La, Ce, Gd, Yb, and Sm) reveals that the major metal site in each case is located at an off-center position adjacent to a hexagonal ring along the C2  axis of the C2v (9)-C82 cage. DFT calculations at the M06-2X level revealed that the positions of the metal centers in these metallofullerenes M@C2v (9)-C82 (M = Sc, Y, and Ce), as determined by single-crystal X-ray structure studies, correspond to an energy minimum for each compound.

Computed Relative Populations of D2 (22)-C84 Endohedrals with Encapsulated Monomeric and Dimeric Water.

Water monomer and dimer encapsulations into D2 (22)-C84 fullerene are evaluated. The encapsulation energy is computed at the M06-2X/6-31++G** level, and it is found that the monomer and dimer storage in C84 yields an energy gain of 10.7 and 17.4 kcal mol(-1) , respectively. Encapsulation equilibrium constants are computed by using partition functions based on the M06-2X/6-31G** and M06-2X/6-31++G** molecular data. Under high-temperature/high-pressure conditions, similar to that for the encapsulation of rare gases in fullerenes, the computed (H2 O)2 @C84 -to-H2 O@C84 ratio is close to 1:2.

Mean squared displacement from fluorescence correlation spectroscopy.

Under certain conditions, the mean squared displacement (MSD) can be retrieved from fluorescence correlation spectroscopy (FCS) measurements. However, in the general case this procedure is not valid, and the apparent MSD obtained from FCS data may substantially differ from the true one. In this work we discuss under which conditions this procedure can be applied. Furthermore, we use computer simulations to obtain the MSD and the apparent MSD for the diffusion of a single polymer chain under various approximations. Based on the simulation results we discuss the reliability of the apparent MSD obtained from FCS, showing that it systematically deviates from the true MSD. We also propose a general procedure to verify the reliability of the apparent MSD by measurements at various focal spot sizes.

Charge-controlled nano-structuring in partially collapsed star-shaped macromolecules.

Hydrophobic polyelectrolytes exhibit intra-molecular nano-scale self-organization instead of macroscopic phase separation because of the interplay between short-range hydrophobic attraction and long-range electrostatic repulsion. We aim to unravel how the morphology of the intra-molecular nanostructures can be controlled through the topology of the macromolecule on one hand and by adjustable ionization on the other hand. Specifically, we focus on hydrophobic star-branched polyelectrolytes, composed of either strong or weak acidic monomers. While both collapse in a globule when uncharged, and expand to full stretching of arms at high ionization, they exhibit quite different intermediate scenarios. For the strong ones, we observe the formation of bundles of arms as the main structural motif, and for the weak ones the intramolecular micelle-like structure is found at the same overall charge of the macromolecule. Here intramolecular disproportionation leaves some arms in a collapsed virtually neutral core, while others are substantially ionized and stretched in the corona.

Isomeric Sc2O@C78 Related by a Single-Step Stone-Wales Transformation: Key Links in an Unprecedented Fullerene Formation Pathway.

It has been proposed that the fullerene formation mechanism involves either a top-down or bottom-up pathway. Despite different starting points, both mechanisms approve that particular fullerenes or metallofullerenes are formed through a consecutive stepwise process involving Stone-Wales transformations (SWTs) and C2 losses or additions. However, the formation pathway has seldomly been defined at the atomic level due to the missing-link fullerenes. Herein, we present the isolation and crystallographic characterization of two isomeric clusterfullerenes Sc2O@C2v(3)-C78 and Sc2O@D3h(5)-C78, which are closely related via a single-step Stone-Wales (SW) transformation. More importantly, these novel Sc2O@C78 isomers represent the key links in a well-defined formation pathway for the majority of solvent-extractable clusterfullerenes Sc2O@C2n (n = 38-41), providing molecular structural evidence for the less confirmed fullerene formation mechanism. Furthermore, DFT calculations reveal a SWT with a notably low activation barrier for these Sc2O@C78 isomers, which may rationalize the established fullerene formation pathway. Additional characterizations demonstrate that these Sc2O@C78 isomers feature different energy bandgaps and electrochemical behaviors, indicating the impact of SW defects on the energetic and electrochemical characteristics of metallofullerenes.

Isolation and Crystallographic Characterization of the Labile Isomer of Y@C82 Cocrystallized with Ni(OEP): Unprecedented Dimerization of Pristine Metallofullerenes.

Although the major isomers of M@C82 (namely M@C2v (9)-C82 , where M is a trivalent rare-earth metal) have been intensively investigated, the lability of the minor isomers has meant that they have been little studied. Herein, the first isolation and crystallographic characterization of the minor Y@C82 isomer, unambiguously assigned as Y@Cs (6)-C82 by cocrystallization with Ni(octaethylporphyrin), is reported. Unexpectedly, a regioselective dimerization is observed in the crystalline state of Y@Cs (6)-C82 . In sharp contrast, no dimerization occurs for the major isomer Y@C2v (9)-C82 under the same conditions, indicating a cage-symmetry-induced dimerization process. Further experimental and theoretical results disclose that the regioselective dimer formation is a consequence of the localization of high spin density on a special cage-carbon atom of Y@Cs (6)-C82 which is caused by the steady displacement of the Y atom inside the Cs (6)-C82 cage.

Hiding and recovering electrons in a dimetallic endohedral fullerene: air-stable products from radical additions.

Fullerenyl radicals can be generated by addition of a free radical to a fullerene surface, by nucleophilic addition followed by one-electron oxidation, or by thermal dissociation of singly bonded fullerene dimers. However, fullerenyl radicals are usually very reactive and generally cannot be isolated. On the contrary, we have found that the reactions of the dimetallic endofullerenes, La2@Ih-C80 and La2@D5h-C80, with 3-chloro-5,6-diphenyltriazine resulted in mono-addition of the triazinyl radical to the fullerene cages to yield isolable fullerenyl radicals. The unusual stability of these fullerenyl radicals arises from the confinement of the unpaired electron to an internal, metal-metal bonding orbital. Accordingly, the fullerene cage protects the radical center from other reactive species. Furthermore, we demonstrate that the fullerenyl radical adduct of La2@Ih-C80 reacts with toluene to afford additional benzylation. Interestingly, the benzylated derivative is diamagnetic in solution, while it forms a paramagnetic dimer when crystallized.