An OrthoBoXY-method for various alternative box geometries.

We have shown in a recent contribution [Busch and Paschek, J. Phys. Chem. B, 2023 127, 7983-7987] that for molecular dynamics (MD) simulations of isotropic fluids based on orthorhombic periodic boundary conditions with "magical" box length ratios of Lz/Lx = Lz/Ly = 2.7933596497, the computed self-diffusion coefficients Dx and Dy in x- and y-direction become system size independent. They thus represent the true self-diffusion coefficient D0 = (Dx + Dy)/2, while the shear viscosity can be determined from diffusion coefficients in x-, y-, and z-direction, using the expression η = kBT·8.1711245653/[3πLz(Dx + Dy - 2Dz)]. Here we present a more generalized version of this "OrthoBoXY"-approach, which can be applied to any orthorhombic MD box of any shape. In particular, we would like to test, how the efficiency is affected by using a shape more akin to the cubic form, albeit with different box length ratios Lx/Lz ≠ Ly/Lz and Lx < Ly < Lz. We use NVT and NpT simulations of systems of 1536 TIP4P/2005 water molecules as a benchmark and explore different box geometries to determine the influence of the box shape on the computed statistical uncertainties for D0 and η. Moreover, another "magical" set of box length ratios is discovered with Ly/Lz = 0.57804765578 and Lx/Lz = 0.33413909235, where the self-diffusion coefficient in x-direction becomes system size independent, such that D0 = Dx.

Computing the frequency-dependent NMR relaxation of 1H nuclei in liquid water.

We present a computational framework for reliably determining the frequency-dependent intermolecular and intramolecular nuclear magnetic resonance (NMR) dipole-dipole relaxation rates of spin 1/2 nuclei from Molecular Dynamics (MD) simulations. This approach avoids the alterations caused by the well-known finite-size effects of translational diffusion. Moreover, a procedure is derived to control and correct for effects caused by fixed distance-sampling cutoffs and periodic boundary conditions. By construction, this approach is capable of accurately predicting the correct low-frequency scaling behavior of the intermolecular NMR dipole-dipole relaxation rate and thus allows for the reliable calculation of the frequency-dependent relaxation rate over many orders of magnitude. Our approach is based on the utilization of the theory of Hwang and Freed for the intermolecular dipole-dipole correlation function and its corresponding spectral density [L.-P. Hwang and J. H. Freed, J. Chem. Phys. 63, 4017-4025 (1975)] and its combination with data from MD simulations. The deviations from the Hwang and Freed theory caused by periodic boundary conditions and sampling distance cutoffs are quantified by means of random walker Monte Carlo simulations. An expression based on the Hwang and Freed theory is also suggested for correcting those effects. As a proof of principle, our approach is demonstrated by computing the frequency-dependent intermolecular and intramolecular dipolar NMR relaxation rates of 1H nuclei in liquid water at 273 and 298 K based on the simulations of the TIP4P/2005 model. Our calculations are suggesting that the intermolecular contribution to the 1H NMR relaxation rate of the TIP4P/2005 model in the extreme narrowing limit has previously been substantially underestimated.

Role of Hydrogen Bond Defects for Cluster Formation and Distribution in Ionic Liquids by Means of Neutron Diffraction and Molecular Dynamics Simulations.

Defects fundamentally govern the properties of all real materials. Correlating molecular defects to macroscopic quantities remains a challenge, particularly in the liquid phase. Herein, we report the influence of hydrogen bonds (HB) acting as defects in mixtures of non-hydroxyl-functionalized ionic liquids (ILs) with an increasing concentration of hydroxyl-functionalized ILs. We observed two types of HB defects: The conventional HBs between cation and anion (c-a), and the elusive HBs between cations (c-c) despite the repulsive Coulomb forces. We use neutron diffraction with isotopic substitution in combination with molecular dynamics simulations for measuring the geometry, strength, and distribution of mobile OH defects in the IL mixtures. In principle, this procedure allows relating the number and stability of defects to macroscopic properties such as diffusion, viscosity, and conductivity, which are of utmost importance for the performance of electrolytes in batteries and other electrical devices.

The Influence of Deuterium Isotope Effects on Structural Rearrangements, Ensemble Equilibria, and Hydrogen Bonding in Protic Ionic Liquids.

We report strong isotope effects for the protic ionic liquid triethylammonium methanesulfonate [TEA][OMs] by means of deuterium solid-state NMR spectroscopy covering broad temperature ranges from 65 K to 313 K. Both isotopically labelled PILs differ in non-deuterated and fully deuterated ethyl groups of the triethyl ammonium cations. The N-D bond of both cations is used as sensitive probe for hydrogen bonding and structural ordering. The 2 H NMR line shape analysis provides the deuteron quadrupole coupling constants and the characteristics of a broad heterogeneous phase with simultaneously present static and mobile states indicating plastic crystal behavior. The temperatures where both states are equally populated differ by about 80 K for the two PILs, showing that deuteration of the ethyl groups in the trialkylammonium cations tremendously shifts the equilibrium towards the static state. In addition, it leads to a significant less cooperative transition, associated with a significantly reduced standard molar transition entropy.

Hydrogen bond redistribution effects in mixtures of protic ionic liquids sharing the same cation: non-ideal mixing with large negative mixing enthalpies.

We report a joint experimental and theoretical study characterising the hydrogen bond (HB) redistribution in mixtures of two different protic ionic liquids (PILs) sharing the same cation: triethylammonium-methanesulfonate ([TEA][OMs]) and triethylammonium-trifluoromethanesulfonate ([TEA][OTf]). The mixing behaviour deviates strongly from ideality, exhibiting large negative energies of mixing. In the PIL, the [TEA] cation acts as a HB donor, being able to donate a single HB. Both, the [OMs] and the [OTf] anions can act as HB acceptors, which can accept multiple HBs via their respective SO3-groups. We use a combination of molecular dynamics (MD) simulations, calorimetry, and 1H-NMR chemical shift measurements to determine the difference in HB strength between the two species to be about 13 kJ mol-1, favouring the [TEA]-[OMs] interaction. Based on our MD simulations we are able to formulate a lattice model, discriminating between HB and nonspecific intermolecular interactions. We demonstrate that, due to the ordered structure of the PILs, mostly the HB interactions contribute to the mixing energy. This allows to us to connect the equilibrium of HBs to each of the two anion species with the mixing energies by a simple relation, which is obeyed by both, MD-simulation as well as experimental calorimetry and 1H-NMR chemical shift data.

An exact a posteriori correction for hydrogen bond population correlation functions and other reversible geminate recombinations obtained from simulations with periodic boundary conditions. Liquid water as a test case.

The kinetics of breaking and re-formation of hydrogen bonds (HBs) in liquid water is a prototype of reversible geminate recombination. HB population correlation functions (HBPCFs) are a means to study the HB kinetics. The long-time limiting behavior of HBPCFs is controlled by translatoric diffusion and shows a t-3/2 time-dependence, which can be described by analytical expressions based on the HB acceptor density and the donor-acceptor inter-diffusion coefficient. If the trajectories are not properly "unwrapped," the presence of periodic boundary conditions (PBCs) can perturb this long-time limiting behavior. Keeping the trajectories "wrapped," however, allows for a more efficient calculation of HBPCFs. We discuss the consequences of PBCs in combination with "wrapped" trajectories following from the approximations according to Luzar-Chandler and according to Starr, each deviating in a different fashion from the true long-time limiting behavior, but enveloping the unperturbed function. A simple expression is given for estimating the maximum time up to which the computed HBPCFs reliably describe the long-time limiting behavior. In addition, an exact a posteriori correction for systems with PBCs for "wrapped" trajectories is derived, which can be easily computed and which is able to fully recover the true t-3/2 long-time behavior. For comparison, HBPCFs are computed from MD simulations of TIP4P/2005 model water for varying system sizes and temperatures of 273 and 298 K using this newly introduced correction. Implications for the computations of HB lifetimes and the effect of the system-size are discussed.

Counting cations involved in cationic clusters of hydroxy-functionalized ionic liquids by means of infrared and solid-state NMR spectroscopy.

In hydroxy-functionalized ionic liquids, two types of hydrogen bonding coexist: the conventional H-bonds between cation and anion (c-a) and those between cation and cation (c-c), although the interaction between like-charged ions is supposed to be much weaker due to the repulsive Coulomb forces. Counting the cations involved in either (c-a) or (c-c) clusters is a challenge. For that purpose, we recently performed neutron diffraction (ND) measurements and molecular dynamics (MD) simulations at and above room temperature accompanied by NMR solid-state experiments in the glassy state of the ILs. In principle, these methods are suitable for determining the populations of (c-a) and (c-c) cluster species. For different reasons we could only address single temperatures and/or small temperature intervals above 300 K. The by far largest temperature range with reasonable efforts is accessible by simple infrared (IR) spectroscopy. However, counting (c-a) or (c-c) hydrogen bonds is a difficult task due to the different transition dipole moments resulting in varying intensities and broad vibrational bands. Here we present a method for deriving the number of cations involved in (c-a) ion pairs from IR spectra in the OH stretch region. This procedure provides access to the equilibria of (c-a) and (c-c) hydrogen bonds as a function of temperature allowing derivation of the transition enthalpy.

Isolating the role of hydrogen bonding in hydroxyl-functionalized ionic liquids by means of vaporization enthalpies, infrared spectroscopy and molecular dynamics simulations.

The enthalpy of vaporization is mainly the amount of the energy needed for transferring quantities from the liquid into the gas phase. It simply describes the energy required to overcome the interaction energy between quantities if those evaporate as monomers as is the case for molecular liquids. The situation for ionic liquids (ILs) is more complex. We do not know the delicate composition of different types of interaction, neither for the liquid nor for the gas phase. Additionally, we have to consider that ILs evaporate as ion pairs which carry substantial interaction energy of all kind into the vapor phase. In this study, we measured the vaporization enthalpies of well-selected hydroxyl-functionalized and non-hydroxyfunctionalized ILs. In particular, we focussed on the case of hydroxyl-functionalized ILs providing possible hydrogen bonding between cation and anion in the liquid as well as in the gas phase. With infrared spectroscopy, we showed that all the hydroxyl groups are involved in hydrogen bonding in the liquid state of the ILs. However, molecular dynamics simulations showed that the evaporating ion pairs also include this hydrogen bond. A detailed analysis of the potential energies for all IL constituents showed that the hydrogen bond hinders favourable interaction between the polarizable ring of the cations and the anions leading to higher vaporization enthalpies for the hydroxyl-functionalized ILs.

The Double-Faced Nature of Hydrogen Bonding in Hydroxy-Functionalized Ionic Liquids Shown by Neutron Diffraction and Molecular Dynamics Simulations.

We characterize the double-faced nature of hydrogen bonding in hydroxy-functionalized ionic liquids by means of neutron diffraction with isotopic substitution (NDIS), molecular dynamics (MD) simulations, and quantum chemical calculations. NDIS data are fit using the empirical potential structure refinement technique (EPSR) to elucidate the nearest neighbor H⋅⋅⋅O and O⋅⋅⋅O pair distribution functions for hydrogen bonds between ions of opposite charge and the same charge. Despite the presence of repulsive Coulomb forces, the cation-cation interaction is stronger than the cation-anion interaction. We compare the hydrogen-bond geometries of both "doubly charged hydrogen bonds" with those reported for molecular liquids, such as water and alcohols. In combination, the NDIS measurements and MD simulations reveal the subtle balance between the two types of hydrogen bonds: The small transition enthalpy suggests that the elusive like-charge attraction is almost competitive with conventional ion-pair formation.

Hydrogen Bonding Between Ions of Like Charge in Ionic Liquids Characterized by NMR Deuteron Quadrupole Coupling Constants-Comparison with Salt Bridges and Molecular Systems.

We present deuteron quadrupole coupling constants (DQCC) for hydroxyl-functionalized ionic liquids (ILs) in the crystalline or glassy states characterizing two types of hydrogen bonding: The regular Coulomb-enhanced hydrogen bonds between cation and anion (c-a), and the unusual hydrogen bonds between cation and cation (c-c), which are present despite repulsive Coulomb forces. We measure these sensitive probes of hydrogen bonding by means of solid-state NMR spectroscopy. The DQCCs of (c-a) ion pairs and (c-c) H-bonds are compared to those of salt bridges in supramolecular complexes and those present in molecular liquids. At low temperatures, the (c-c) species successfully compete with the (c-a) ion pairs and dominate the cluster populations. Equilibrium constants obtained from molecular-dynamics (MD) simulations show van't Hoff behavior with small transition enthalpies between the differently H-bonded species. We show that cationic-cluster formation prevents these ILs from crystallizing. With cooling, the (c-c) hydrogen bonds persist, resulting in supercooling and glass formation.

The influence of like-charge attraction on the structure and dynamics of ionic liquids: NMR chemical shifts, quadrupole coupling constants, rotational correlation times and failure of Stokes-Einstein-Debye.

Ion pairing is one of the most fundamental atomic interactions in chemistry and biology. In contrast, pairing between like-charged ions remains an elusive concept. So far, this phenomenon was observed only for large-scaled structures, assemblies, stabilizing frameworks, or in aqueous solution wherein like-charge attraction is supported by mediating water molecules. Recently, we reported the formation of cationic clusters in pure ionic liquids (ILs) which all include hydroxyl groups (OH) for possible hydrogen bonding. In such structures like-charge repulsion is overcome by cooperative hydrogen bonds. The vibrational bands in the OH-stretch region of the infrared spectra can be clearly assigned to H-bonded ion pairs (c-a) or to H-bonded cationic clusters (c-c). The equilibrium between both types of ionic clusters can be controlled by using the same cation but differently strong interacting anions. In the present work, we study the influence of the cationic cluster formation on structural and dynamical NMR properties of ionic liquids, where we know that they form cationic clusters to different extent. First, we measure proton chemical shifts, δ1H, and determine deuteron quadrupole coupling constants, χD, from a calculated relation between both NMR properties. Reliable χD values for the liquid phase are a prerequisite for calculating reorientational correlation times, τOH, from measured deuteron relaxation times, T1. It is shown that the correlation times are significantly influenced by the amount of cationic clusters present in the IL. The Stokes-Einstein-Debye (SED) relation is valid for the ILs wherein H-bonded ion pairs (c-a) are the dominant species. With increasing cationic cluster (c-c) formation of e.g. cyclic tetramers, SED breaks down because of the structural heterogeneities.

Revisiting imidazolium based ionic liquids: Effect of the conformation bias of the [NTf2] anion studied by molecular dynamics simulations.

We study ionic liquids composed of 1-alkyl-3-methylimidazolium cations and bis(trifluoromethyl-sulfonyl)imide anions ([C n MIm][NTf2]) with varying chain-length n = 2, 4, 6, 8 by using molecular dynamics simulations. We show that a reparametrization of the dihedral potentials as well as charges of the [NTf2] anion leads to an improvement of the force field model introduced by Köddermann, Paschek, and Ludwig [ChemPhysChem 8, 2464 (2007)] (KPL-force field). A crucial advantage of the new parameter set is that the minimum energy conformations of the anion (trans and gauche), as deduced from ab initio calculations and Raman experiments, are now both well represented by our model. In addition, the results for [C n MIm][NTf2] show that this modification leads to an even better agreement between experiment and molecular dynamics simulation as demonstrated for densities, diffusion coefficients, vaporization enthalpies, reorientational correlation times, and viscosities. Even though we focused on a better representation of the anion conformation, also the alkyl chain-length dependence of the cation behaves closer to the experiment. We strongly encourage to use the new NGOLP (Neumann, Golub, Odebrecht, Ludwig, Paschek) force field for the [NTf2] anion instead of the earlier KPL parameter set for computer simulations aiming to describe the thermodynamics, dynamics, and also structure of imidazolium-based ionic liquids.

Rotational and translational dynamics and their relation to hydrogen bond lifetimes in an ionic liquid by means of NMR relaxation time experiments and molecular dynamics simulation.

We report a concerted theoretical and experimental effort to determine the reorientational dynamics as well as hydrogen bond lifetimes for the doubly ionic hydrogen bond +OH⋯O- in the ionic liquid (2-hydroxyethyl)trimethylammonium bis(trifluoromethylsulfonyl)imide [Ch][NTf2] by using a combination of NMR relaxation time experiments, density functional theory (DFT) calculations, and molecular dynamics (MD) simulations. Due to fast proton exchange, the determination of rotational correlation times is challenging. For molecular liquids, 17O-enhanced proton relaxation time experiments have been used to determine the rotational correlation times for the OH vectors in water or alcohols. As an alternative to those expensive isotopic substitution experiments, we employed a recently introduced approach which is providing access to the rotational dynamics from a single NMR deuteron quadrupolar relaxation time experiment. Here, the deuteron quadrupole coupling constants (DQCCs) are obtained from a relation between the DQCC and the δ 1H proton chemical shifts determined from a set of DFT calculated clusters in combination with experimentally determined proton chemical shifts. The NMR-obtained rotational correlation times were compared to those obtained from MD simulations and then related to viscosities for testing the applicability of popular hydrodynamic models. In addition, hydrogen bond lifetimes were derived, using hydrogen bond population correlation functions computed from MD simulations. Here, two different time domains were observed: The short-time contributions to the hydrogen lifetimes and the reorientational correlation times have roughly the same size and are located in the picosecond range, whereas the long-time contributions decay with relaxation times in the nanosecond regime and are related to rather slow diffusion processes. The computed average hydrogen bond lifetime is dominated by the long-time process, highlighting the importance and longevity of hydrogen-bonded ion pairs in these ionic liquids.

The Relation between Vaporization Enthalpies and Viscosities: Eyring's Theory Applied to Selected Ionic Liquids.

Key properties for the use of ionic liquids as electrolytes in batteries are low viscosities, low vapor pressure and high vaporization enthalpies. Whereas the measurement of transport properties is well established, the determination of vaporization enthalpies of these extremely low volatile compounds is still a challenge. At a first glance both properties seem to describe different thermophysical phenomena. However, eighty years ago Eyring suggested a theory which related viscosities and vaporization enthalpies to each other. The model is based on Eyring's theory of absolute reaction rates. Recent attempts to apply Eyring's theory to ionic liquids failed. The motivation of our study is to show that Eyring's theory works, if the assumptions specific for ionic liquids are fulfilled. For that purpose we measured the viscosities of three well selected protic ionic liquids (PILs) at different temperatures. The temperature dependences of viscosities were approximated by the Vogel-Fulcher-Tamann (VFT) relation and extrapolated to the high-temperature regime up to 600 K. Then the VFT-data could be fitted to the Eyring-model. The values of vaporization enthalpies for the three selected PILs predicted by the Eyring model have been very close to the experimental values measured by well-established techniques. We conclude that the Eyring theory can be successfully applied to the chosen set of PILs, if the assumption that ionic pairs of the viscous flow in the liquid and the ionic pairs in the gas phase are similar is fulfilled. It was also noticed that proper transfer of energies can be only derived if the viscosities and the vaporization energies are known for temperatures close to the liquid-gas transition temperature. The idea to correlate easy measurable viscosities of ionic liquids with their vaporization enthalpies opens a new way for a reliable assessment of these thermodynamic properties for a broad range of ionic liquids.

A simple guiding principle for the temperature dependence of the solubility of light gases in imidazolium-based ionic liquids derived from molecular simulations.

We have determined the temperature dependence of the solvation behavior of a large collection of important light gases in imidazolium-based ionic liquids with the help of extensive molecular dynamics simulations. The motivation of our study is to unravel common features of the temperature dependent solvation under well controlled conditions, and to provide a guidance for cases, where experimental data from different sources disagree significantly. The solubility of molecular hydrogen, oxygen, nitrogen, methane, krypton, argon, neon and carbon dioxide in the imidazolium based ionic liquids of type 1-n-alkyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide ([Cnmim][NTf2]) with varying alkyl side chain lengths n = 2, 4, 6, 8 is computed for a temperature range between 300 K and 500 K at 1 bar. By applying Widom's particle insertion technique and Bennet's overlapping distribution method, we are able to determine the temperature dependent solvation free energies of those selected light gases in simulated imidazolium based ionic liquids with high statistical accuracy. Our simulations demonstrate that the magnitude of the solvation free energy of a gas molecule at a chosen reference temperature and that of its temperature-derivatives are intimately related to one another. We conclude that this "universal" behavior is rooted in a solvation entropy-enthalpy compensation effect, which seems to be a defining feature of the solvation of small molecules in ionic liquids. The observations lead to simple analytical relations, determining the temperature dependence of the solubility data based on the absolute solubility at a certain reference temperature. By comparing our results with available experimental data from many sources, we can show that our approach is particularly helpful for providing reliable estimates for the solvation behavior of very light gases, such as hydrogen, where conflicting experimental data exist.

Characterization of Doubly Ionic Hydrogen Bonds in Protic Ionic Liquids by NMR Deuteron Quadrupole Coupling Constants: Differences to H-bonds in Amides, Peptides, and Proteins.

We present the first deuteron quadrupole coupling constants (DQCCs) for selected protic ionic liquids (PILs) measured by solid-state NMR spectroscopy. The experimental data are supported by dispersion-corrected density functional theory (DFT-D3) calculations and molecular dynamics (MD) simulations. The DQCCs of the N-D bond in the triethylammonium cations are the lowest reported for deuterons in PILs, indicating strong hydrogen bonds between ions. The NMR coupling parameters are compared to those in amides, peptides, and proteins. The DQCCs show characteristic behavior with increasing interaction strength of the counterion and variation of the H-bond motifs. We report the similar presence of the quadrupolar splitting pattern and the narrow liquid line in the NMR spectra over large temperature ranges, indicating the heterogeneous nature of PILs.

Deuteron quadrupole coupling constants and reorientational correlation times in protic ionic liquids.

We describe a method for the accurate determination of deuteron quadrupole coupling constants χD for N-D bonds in triethylammonium-based protic ionic liquids (PILs). This approach was first introduced by Wendt and Farrar for O-D bonds in molecular liquids, and is based on the linear relationship between the deuteron quadrupole coupling constants χD, and the proton chemical shifts δ(1)H, as obtained from DFT calculated properties in differently sized clusters of the compounds. Thus the measurement of δ(1)H provides an accurate estimate for χD, which can then be used for deriving reorientational correlation-times τND, by means of NMR deuteron quadrupole relaxation time measurements. The method is applied to pure PILs including differently strong interacting anions. The obtained χD values vary between 152 and 204 kHz, depending on the cation-anion interaction strength, intensified by H-bonding. We find that considering dispersion corrections in the DFT-calculations leads to only slightly decreasing χD values. The determined reorientational correlation times indicate that the extreme narrowing condition is fulfilled for these PILs. The τc values along with the measured viscosities provide an estimate for the volume/size of the clusters present in solution. In addition, the correlation times τc, and the H-bonded aggregates were also characterized by molecular dynamics (MD) simulations.

Non-ideal mixing behaviour of hydrogen bonding in mixtures of protic ionic liquids.

Ionic liquids (ILs) attract interest in science and technology as a result of their unique properties. Binary and ternary mixtures of ILs significantly increase the number of possible cation/anion combinations, resulting in targeted physical and chemical properties. In this work, we study the mixing behaviour of two protic ILs: triethyl ammonium methylsulfonate [Et3 NH][CH3 SO3 ] and triethylammonium triflate [Et3 NH][CF3 SO3 ]. We find a characteristic deviation from ideal mixing by means of low-frequency infrared spectroscopy. By using molecular dynamics simulations, we explain this behaviour as being the result of different strengths of anion/cation hydrogen bonding. This non-ideality of non-random H-bond mixing is also reflected in macroscopic properties such as the viscosity. Mixing suitable ILs may, thus, result in new ILs with targeted physical properties.

Hydrogen bonding in a mixture of protic ionic liquids: a molecular dynamics simulation study.

We report results of molecular dynamics (MD) simulations characterising the hydrogen bonding in mixtures of two different protic ionic liquids sharing the same cation: triethylammonium-methylsulfonate (TEAMS) and triethylammonium-triflate (TEATF). The triethylammonium-cation acts as a hydrogen-bond donor, being able to donate a single hydrogen-bond. Both, the methylsulfonate- and the triflate-anions can act as hydrogen-bond acceptors, which can accept multiple hydrogen bonds via their respective SO3-groups. In addition, replacing a methyl-group in the methylsulfonate by a trifluoromethyl-group in the triflate significantly weakens the strength of a hydrogen bond from an adjacent triethylammonium cation to the oxygen-site in the SO3-group of the anion. Our MD simulations show that these subtle differences in hydrogen bond strength significantly affect the formation of differently-sized hydrogen-bonded aggregates in these mixtures as a function of the mixture-composition. Moreover, the reported hydrogen-bonded cluster sizes can be predicted and explained by a simple combinatorial lattice model, based on the approximate coordination number of the ions, and using statistical weights that mostly account for the fact that each anion can only accept three hydrogen bonds.