Nature of proton dynamics in a polymer electrolyte membrane, nafion: a first-principles molecular dynamics study

Proton Transport in Perfluorinated Ionomer Simulated by Machine-Learned Interatomic Potential

Polymers are a class of materials that are highly challenging to deal with using first-principles methods. Here, we present an application of machine-learned interatomic potentials to predict structural and dynamical properties of dry and hydrated perfluorinated ionomers. An improved active-learning algorithm using a small number of descriptors allows to efficiently construct an accurate and transferable model for this multielemental amorphous polymer. Molecular dynamics simulations accelerated by the machine-learned potentials accurately reproduce the heterogeneous hydrophilic and hydrophobic domains formed in this material as well as proton and water diffusion coefficients under a variety of humidity conditions. Our results reveal pronounced contributions of Grotthuss chains consisting of two to three water molecules to the high proton mobility under strongly humidified conditions.

Controlling Hydronium Diffusivity in Model Proton Exchange Membranes

Fuel-cell-based proton exchange membranes (PEMs) show great potential as cost-effective and clean energy conversion devices. In our recent work, we found that for the low-hydrated model PEMs with a inhomogeneous water distribution and a sulfonate anionic functional end group (SO₃^-), the H₃O⁺ reacts with SO₃^- according to SO₃^- + H₃O⁺ ↔ SO₃H + H₂O, indicating that the anions in PEMs become active participants in the hydronium diffusion. In this work, we use fully atomistic ab initio molecular dynamics simulations to elucidate the optimal conditions that would promote the participation of SO₃^- in the hydronium diffusion mechanism by increasing the H₃O⁺/SO₃^- reactivity, thus increasing the hydronium diffusivity along the cell. The results presented in this work allow us to suggest a set of design rules for creating novel, highly conductive PEMs operating at high temperatures under a nonuniform water distribution using a linker/anion with a relatively high pK_a such as (CH₂)₂SO₃. We expect that the discovery of these key design principles will play an important role in the synthesis of high-performing materials for emerging PEM-based fuel cell technologies.

Molecular dynamics simulations of proton conducting mediums containing phosphoric acid

An anhydrous proton conducting electrolyte is a key material to realise medium-temperature fuel cells that can drastically simplify heat radiation systems in transportation applications. However, practical applications are limited by...

Evaluation of proton transport and solvation effect in hydrated Nafion membrane with degradation

In proton exchange membrane fuel cells (PEMFCs), free radicals easily attack ionomers, resulting in membrane degradation.

Longitudinal strand ordering leads to shear thinning in Nafion

Proton-exchange membrane fuel cells (PEMFC) offer a promising energy generation alternative for a wide range of technologies thanks to their ecological friendliness and unparalleled efficiency. At the heart of these electrochemical cells lies the membrane electrode assembly with its most important energy conversion components, the Proton Exchange Membrane. This component is created through the use of printing techniques and Nafion inks. The physicochemical properties of the ink, such as its viscosity under shear, are critical for the finished product. In this work we present non-equilibrium Molecular Dynamics simulations using a MARTINI based coarse-grained model for Nafion to understand the mechanism governing the shear viscosity of Nafion solutions. By simulating a Couette flow and calculating density maps of the Nafion chains in these simulations we shed light on the process that leads to the experimentally observed shear thinning effects of Nafion solutions under flow. We observe rod-shaped Nafion microstructures, 3 nm in size on average, when shear flow is absent or low. Higher shear rates instead break these structures and align Nafion strands along the direction of the flow, resulting in lower shear viscosities. Our work paves the way for a deeper understanding of the dynamic and mechanical properties of Nafion including studies of more complex CL and PEM inks.

A Molecular Dynamics Simulation Based Investigation of the Proton Conductivity of Anhydrous Pyrazole Doped Poly(Vinylphosphonic Acid) Composite System

With the recognition of the multiple advantages of proton transport membranes that can operate under anhydrous conditions and offer promising opportunities as fuel cells working at high temperatures, a number of such membranes have been developed, but the proton transport mechanism of these materials has not been fully understood. In this work, a theoretical investigation based on molecular dynamics simulations is carried out on a system that is very similar to a real anhydrous proton transport membrane. The location and type of hydrogen bonds have been precisely identified by intermolecular pair correlation functions. Furthermore, analysis of the proton coordination numbers shows that more protons are located in the neighborhood of the oxygen atoms of poly(vinyl phosphonate anion) than in the neighborhood of the nitrogen atoms of pyrazole. The proton conductivity, 1.06 × 10^-3 Scm^-1, is obtained by the self-diffusion coefficient of the protons at 423 K, which is reasonably close to the experimentally measured value, 2 × 10^-4 Scm^-1. In addition, the analysis of the proton trajectories provides us with the proton transfer mechanism in an anhydrous membrane: (a) proton hopping between the oxygen atoms of poly(vinyl phosphonate anion) and (b) proton hopping between two pyrazole molecules. Therefore, the network of the hydrogen bond is the pathway to transport protons via the processes of hydrogen bond forming and breaking.

Effect of Electrolyte on the Proton Transport through Graphane in the Electrochemical Cell: A First-Principles Study

The proton transport behaviors through graphane in the electrochemical environment are not only determined by the film but also correlated with the properties of the electrolytes. Here, the effect of electrolytes is studied for this transport process. The step of proton transfer from electrolyte to graphane is the rate-determining step of the whole transport process in most of the studied cases and is indeed influenced much by the electrolytes, while the following steps are affected little. Its energy barrier increases significantly with the number of water molecules but only fluctuates with the number of mimicked Nafion molecules until the bulk case. This barrier could be further affected by the hydration number of Nafion and be reduced by increasing local proton concentrations. The dynamical effect of the environment and the nuclear quantum effect are found to further reduce the energy barrier of the transport process but by a relatively small amount.

Effect of Elevated Temperatures on the States of Water and Their Correlation with the Proton Conductivity of Nafion

For the first time, we report the effects of elevated temperatures, from 80 to 100 °C, on the changes in the states of water and ion-water channels and their correlation with the proton conductivity of Nafion NR212, which was investigated using a Fourier transform infrared spectroscopy study. Experimentally, three types of water aggregates, protonated water (H+(H2O) n ), nonprotonated hydrogen (H)-bonded water (H2O···H2O), and non-H-bonded water, were found in Nafion, and the existence of those three types of water was confirmed through ab initio molecular dynamics simulation. We found that the proton conductivity of Nafion increased for up to 80 °C, but from 80 to 100 °C, the conductivity did not increase; rather, all of those elevated temperatures showed identical conductivity values. The proton conductivities at lower relative humidities (RHs) (up to 50%) remained nearly identical for all elevated temperatures (80, 90, and 100 °C); however, from 60% RH (over λ = 4), the conductivity remarkably jumped for all elevated temperatures. The results indicated that the amount of randomly arranged water gradually increased and created more H-bonded water networks in Nafion at above 60% RH. From the deconvolution of the O-H bending band, it was found that the volume fraction f i (i=each deconvoluted band) of H-bonded water for elevated temperatures (>80-100 °C) increased remarkably higher than for 60 °C.

Theoretical analyses on water cluster structures in polymer electrolyte membrane by using dissipative particle dynamics simulations with fragment molecular orbital based effective parameters

The mesoscopic structures of polymer electrolyte membrane (PEM) affect the performances of fuel cells. Nafion® with the Teflon® backbone has been the most widely used of all PEMs, but sulfonated poly-ether ether-ketone (SPEEK) having an aromatic backbone has drawn interest as an alternative to Nafion. In the present study, a series of dissipative particle dynamics (DPD) simulations were performed to compare Nafion and SPEEK. These PEM polymers were modeled by connected particles corresponding to the hydrophobic backbone and the hydrophilic moiety of sulfonic acid group. The water particle interacting with Nafion particles was prepared as well. The crucial interaction parameters among DPD particles were evaluated by a series of calculations based on the fragment molecular orbital (FMO) method in a non-empirical way (Okuwaki et al., J. Phys. Chem. B, 2018, 122, 338–347). Through the DPD simulations, the water and hydrophilic particles aggregated, forming cluster networks surrounded by the hydrophobic phase. The structural features of formed water clusters were investigated in detail. Furthermore, the differences in percolation behaviors between Nafion and SPEEK revealed much better connectivity among water clusters by Nafion. The present FMO-DPD simulation results were in good agreement with available experimental data.

The mesoscopic structures of polymer electrolyte membrane (PEM) affect the performances of fuel cells.

Theoretical Investigation of the H2O2-Induced Degradation Mechanism of Hydrated Nafion Membrane via Ether-Linkage Dissociation

A H₂O₂-induced degradation mechanism is presented for the hydrated Nafion membrane proceeding through the dissociation of the ether linkages of the side chains. Although the durability of proton-exchange membrane fuel cells clearly depends on the degradation rate of the membrane, typically Nafion, the degradation mechanism still has not been resolved. It has often been assumed that the principal mode of degradation involves OH^• radicals; in contrast, we show here that a H₂O₂-induced degradation mechanism is more likely. On the basis of state-of-the-art theoretical calculations and detailed comparison with experimental results, we present such a mechanism for the hydrated Nafion membrane, proceeding through the dissociation of the ether linkage of the side chains, with a relatively low activation energy. In this mechanism, (H₂O)_λHO₃S-CF₂-CF₂-O-O-H (λ is the hydration number) is obtained as a key degradation fragment. Possible subsequent decomposition-reaction mechanisms are also elucidated for this fragment. The calculated vibrational spectra for the intermediates and products proposed in these mechanisms were found to be consistent with the experimental IR spectra. Further consideration of this H₂O₂-mediated degradation mechanism could greatly facilitate the search for ways to combat membrane degradation.

Nitrogen-Mediated Graphene Oxide Enables Highly Efficient Proton Transfer

Two-dimensional (2D) graphene and graphene oxide (GO) offer great potential as a new type of cost-efficient proton-exchange membranes (PEM) for electrochemical devices. However, fundamental issues of proton transfer mechanism via 2D membranes are unclear and the transfer barrier for perfect graphene are too high for practical application. Using ab initio molecular dynamic simulations, we screened the proton transfer barrier for different un-doped and nitrogen doped GO membranes, and clarified the corresponding transfer mechanisms. More significantly, we further identify that N-mediated GO can be built into a highly efficient PEM with a proton transfer rate of seven orders of magnitude higher than an un-doped case via. a proton relay mechanism between a ketone-like oxygen and a pyridine-like nitrogen across the vacancy site. The N-doped 2D GO is also impermeable to small molecules, and hence a highly efficient PEM for practical applications.

Transport in Proton Exchange Membranes for Fuel Cell Applications-A Systematic Non-Equilibrium Approach

We hypothesize that the properties of proton-exchange membranes for fuel cell applications cannot be described unambiguously unless interface effects are taken into account. In order to prove this, we first develop a thermodynamically consistent description of the transport properties in the membranes, both for a homogeneous membrane and for a homogeneous membrane with two surface layers in contact with the electrodes or holder material. For each subsystem, homogeneous membrane, and the two surface layers, we limit ourselves to four parameters as the system as a whole is considered to be isothermal. We subsequently analyze the experimental results on some standard membranes that have appeared in the literature and analyze these using the two different descriptions. This analysis yields relatively well-defined values for the homogeneous membrane parameters and estimates for those of the surface layers and hence supports our hypothesis. As demonstrated, the method used here allows for a critical evaluation of the literature values. Moreover, it allows optimization of stacked transport systems such as proton-exchange membrane fuel cell units where interfacial layers, such as that between the catalyst and membrane, are taken into account systematically.

Molecular dynamics simulation of Keggin HPA doped Nafion® 117 as a polymer electrolyte membrane

Nafion®/heteropoly acid (HPA) composite membranes and the impact of the anionic charge of HPA on water and hydronium dynamics were investigated using molecular dynamics simulation.

Proton transport mechanism of imidazole, triazole and phosphoric acid mixtures from ab initio molecular dynamics simulations

We performed first principles molecular dynamics simulations to elucidate the mechanism and role of 1,2,3-triazole in proton transport while it is mixed with phosphoric acid (PA) and a phosphoric acid imidazole mixture. PA doped imidazole based polymer acts as an efficient polyelectrolyte membrane for fuel cells. The conductivity of this membrane increases when triazole is added to the system. For the first time we performed ab initio molecular dynamics simulations of complex mixtures of PA, imidazole and triazole. We have quantitatively estimated the structural diffusion and vehicular motion of protons. We found that upon the addition of triazole in PA and the PA imidazole mixture, the structural diffusion of protons increases significantly. The mechanism of proton transport is different when triazole is added to the mixture. We have also identified two different paths for structural diffusion (constructive and non-constructive) that contribute to long and short range proton transport.

Ab initio molecular dynamics simulations of aqueous triflic acid confined in carbon nanotubes

Ab initio molecular dynamics simulations were performed to investigate the effects of nanoscale confinement on the structural and dynamical properties of aqueous triflic acid (CF3SO3H). Single-walled carbon nanotubes (CNTs) with diameters ranging from ∼11 to 14 Å were used as confinement vessels, and the inner surface of the CNT were either left bare or fluorinated to probe the influence of the confined environment on structural and dynamical properties of the water and triflic acidic. The systems were simulated at hydration levels of n = 1-3 H2O/CF3SO3H. Proton dissociation expectedly increased with increasing hydration. Along with the level of hydration, hydrogen bond connectivity between the triflic acid molecules, both directly and via a single water molecule, played a role on proton dissociation. Direct hydrogen bonding between the CF3SO3H molecules, most commonly found in the larger bare CNT, also promoted interactions between water molecules allowing for greater separation of the dissociated protons from the CF3SO3(-) as the hydration level was increased. However, this also resulted in a decrease in the overall proportion of dissociated protons. The confinement dimensions altered both the hydrogen bond network and the distribution of water molecules where the H2O in the fluorinated CNTs tended to form small clusters with less proton dissociation at n = 1 and 2 but the highest at n = 3. In the absence of nearby hydrogen bond accepting sites from H2O or triflic acid SO3H groups, the water molecules formed weak hydrogen bonds with the fluorine atoms. In the bare CNT systems, these involved the CF3 groups of triflic acid and were more frequently observed when direct hydrogen bonding between CF3SO3H hindered potential hydrogen bonding sites. In the fluorinated tubes, interactions with the covalently bound fluorine atoms of the CNT wall dominated which appear to stabilize the hydrogen bond network. Increasing the hydration level increased the frequency of the OH···F (CNT) hydrogen bonding which was highly pronounced in the smaller fluorinated CNT indicating an influence on the confinement dimensions on these interactions.

An adaptive finite-element method for large-scale ab initio molecular dynamics simulations

A snapshot of ab initio molecular dynamics simulations for a polymer electrolyte membrane at low hydration.

Persistent Subdiffusive Proton Transport in Perfluorosulfonic Acid Membranes

Proton transport (PT) in solutions of small amphiphiles in water has previously been shown to be subdiffusive for long times. The present study analyzes simulations of hydrated perfluorosulfonic acid (PFSA) membranes in order to determine whether PT is also subdiffusive in these important amphiphilic systems. We show that PT is indeed subdiffusive for several hundred picoseconds for all hydration levels examined, and the subdiffusive behavior is highly dependent on water concentration. We also investigate the caging of the excess proton using a recently developed technique and show that the excess proton exhibits caging effects up to at least 1 ns in PFSA systems. In order to fully characterize the long-time behavior of PT in PFSAs, these results demonstrate that multiple nanosecond trajectories are needed, well beyond the current capabilities of ab initio molecular dynamics.

Ab initio molecular dynamics simulation of proton hopping in a model polymer membrane

We report the results of ab initio molecular dynamics simulations of a model Nafion polymer membrane initially equilibrated using classical molecular dynamics simulations. We studied three hydration levels (λ) of 3, 9, and 15 H2O/SO3(-) corresponding to dry, hydrated, and saturated fuel cell membrane, respectively. The barrier for proton transfer from the SO3(-)-H3O(+) contact ion pair to a solvent-separated ion pair decreased from 2.3 kcal/mol for λ = 3 to 0.8 kcal/mol for λ = 15. The barrier for proton transfer between two water molecules was in the range from 0.7 to 0.8 kcal/mol for the λ values studied. The number of proton shuttling events between a pair of water molecules is an order of magnitude more than the number of proton hops across three distinct water molecules. The proton diffusion coefficient at λ = 15 is about 0.9 × 10(-5) cm(2)/s, which is in good agreement with experiment and our previous quantum hopping molecular dynamics simulations.

Molecular dynamics simulations of water sorption in a perfluorosulfonic acid membrane

Atomistic molecular dynamics simulations are reported over a wide range of water contents and temperatures to obtain a better understanding of the structural and transport aspects of water sorption in Nafion, a perfluorosulfonic acid membrane, under equilibrium conditions. For the short Nafion chains studied, good agreement is found between the water sorption isotherms from simulations and experiments at intermediate hydration (2 ≲ λ ≲ 7, where λ is the number of water molecules per sulfonate group), suggesting that, in that range, the isotherm is insensitive to effects of polymer chain relaxation. If polymer chain relaxation were important for water sorption at these conditions, then the water uptake of experimental membranes, which contain very long chains, might be far from equilibrium, making it difficult to obtain agreement with equilibrated, short-chain simulations. At λ ≲ 7, strong water-sulfonate interactions, rather than chain relaxation, may control water sorption, despite the fact that chain relaxation time increases dramatically with decreasing hydration. Evidence for strong water-sulfonate interactions is found in the observation that sulfonate groups share water molecules in their first coordination shells at λ ≲ 7. Strong water-sulfonate interactions are also observed to influence transport properties like water diffusivity, and are as important for understanding these transport properties as larger-scale phenomena like morphology and percolation transitions. Finally, at low humidity (λ ≈ 1-2), rod-like hydrophilic clusters are observed, as well as a mechanism of water diffusion that differs qualitatively from that of water at high hydration (λ ≳ 7) and in the bulk, pure-component phase.

Proton conduction in exchange membranes across multiple length scales

Concerns over global climate change associated with fossil-fuel consumption continue to drive the development of electrochemical alternatives for energy technology. Proton exchange fuel cells are a particularly promising technology for stationary power generation, mobile electronics, and hybrid engines in automobiles. For these devices to work efficiently, direct electrical contacts between the anode and cathode must be avoided; hence, the separator material must be electronically insulating but highly proton conductive. As a result, researchers have examined a variety of polymer electrolyte materials for use as membranes in these systems. In the optimization of the membrane, researchers are seeking high proton conductivity, low electronic conduction, and mechanical stability with the inclusion of water in the polymer matrix. A considerable number of potential polymer backbone and side chain combinations have been synthesized to meet these requirements, and computational studies can assist in the challenge of designing the next generation of technologically relevant membranes. Such studies can also be integrated in a feedback loop with experiment to improve fuel cell performance. However, to accurately simulate the currently favored class of membranes, perfluorosulfonic acid containing moieties, several difficulties must be addressed including a proper treatment of the proton-hopping mechanism through the membrane and the formation of nanophase-separated water networks. We discuss our recent efforts to address these difficulties using methods that push the limits of computer simulation and expand on previous theoretical developments. We describe recent advances in the multistate empirical valence bond (MS-EVB) method that can probe proton diffusion at the nanometer-length scale and accurately model the so-called Grotthuss shuttling mechanism for proton diffusion in water. Using both classical molecular dynamics and coarse-grained descriptions that replace atomistic representations with collective coordinates, we investigated the proton conductivity of polymer membrane structure as a function of hydration level. Nanometer-sized water channels form torturous pathways that are traversed by the charges during fuel cell operation. Using a combination of coarse-grained membrane structure and novel multiscale methods, we demonstrate emerging approaches to treat proton motion at the mesoscale in these complex materials.

A review of molecular-level mechanism of membrane degradation in the polymer electrolyte fuel cell

Chemical degradation of perfluorosulfonic acid (PFSA) membrane is one of the most serious problems for stable and long-term operations of the polymer electrolyte fuel cell (PEFC). The chemical degradation is caused by the chemical reaction between the PFSA membrane and chemical species such as free radicals. Although chemical degradation of the PFSA membrane has been studied by various experimental techniques, the mechanism of chemical degradation relies much on speculations from ex-situ observations. Recent activities applying theoretical methods such as density functional theory, in situ experimental observation, and mechanistic study by using simplified model compound systems have led to gradual clarification of the atomistic details of the chemical degradation mechanism. In this review paper, we summarize recent reports on the chemical degradation mechanism of the PFSA membrane from an atomistic point of view.

Proton transport in functionalised additives for PEM fuel cells: contributions from atomistic simulations

The conventional polymer electrolyte membrane (PEM) materials for fuel cell applications strongly rely on temperature and pressure conditions for optimal performance. In order to expand the range of operating conditions of these conventional PEM materials, mesoporous functionalised SiO(2) additives are developed. It has been demonstrated that these additives themselves achieve proton conductivities approaching those of conventional materials. However, the proton conduction mechanisms and especially factors influencing charge carrier mobility under different hydration conditions are not well known and difficult to separate from concentration effects in experiments. This tutorial review highlights contributions of atomistic computer simulations to the basic understanding and eventual design of these materials. Some basic introduction to the theoretical and computational framework is provided to introduce the reader to the field, the techniques are in principle applicable to a wide range of other situations as well. Simulation results are directly compared to experimental data as far as possible.

A comparative ab initio study of the primary hydration and proton dissociation of various imide and sulfonic acid ionomers

We compare the role of neighboring group substitutions on proton dissociation of hydrated acidic moieties suitable for proton exchange membranes through electronic structure calculations. Three pairs of ionomers containing similar electron withdrawing groups within the pair were chosen for the study: two fully fluorinated sulfonyl imides (CF(3)SO(2)NHSO(2)CF(3) and CF(3)CF(2)SO(2)NHSO(2)CF(3)), two partially fluorinated sulfonyl imides (CH(3)SO(2)NHSO(2)CF(3) and C(6)H(5)SO(2)NHSO(2)CF(2)CF(3)), and two aromatic sulfonic acid based materials (CH(3)C(6)H(4)SO(3)H and CH(3)OC(6)H(3)OCH(3)C(6)H(4)SO(3)H). Fully optimized counterpoise (CP) corrected geometries were obtained for each ionomer fragment with the inclusion of water molecules at the B3LYP/6-311G** level of density functional theory. Spontaneous proton dissociation was observed upon addition of three water molecules in each system, and the transition to a solvent-separated ion pair occurred when four water molecules were introduced. No considerable quantitative or qualitative differences in proton dissociation, hydrogen bond networks formed, or water binding energies were found between systems containing similar electron withdrawing groups. Each of the sulfonyl imide ionomers exhibited qualitatively similar results regarding proton dissociation and separation. The fully fluorinated sulfonyl imides, however, showed a greater propensity to exist in dissociated and ion-pair separated states at low degrees of hydration than the partially fluorinated sulfonyl imides. This effect is due to the additional electron withdrawing groups providing charge stabilization as the dissociated proton migrates away from the imide anion.

Molecular structure and transport dynamics in perfluoro sulfonyl imide membranes

We report a detailed and comprehensive analysis from classical molecular dynamics simulations of the nanostructure of a model of hydrated perfluoro sulfonyl imide (PFSI) membrane, a polymeric system of interest as a proton conductor in polymer electrolyte membrane fuel cells. We also report on the transport dynamics of water and hydronium ions, and water network percolation in this system. We find that the water network percolation threshold for PFSI, i.e. the threshold at which a consistent spanning water network starts to develop in the membrane, is found to occur between hydration levels (λ) 6 and 7. The higher acidity of the sulfonyl imide acid group of PFSI compared to the sulfonic acid group in Nafion, as computationally characterized in our earlier ab initio study (Idupulapati et al 2010 J. Phys. Chem. A 114 6904-12), results in a larger fraction of 'free' hydronium ions at low hydration levels in PFSI compared to Nafion. However, the calculated diffusion coefficients of the H(3)O(+) ions and H(2)O molecules as a function the hydration level are observed to be almost the same as that of Nafion, indicating similar conductivity and consistent with experimental data.

Atomistic simulation of water percolation and proton hopping in Nafion fuel cell membrane

We have performed a detailed analysis of water clustering and percolation in hydrated Nafion configurations generated by classical molecular dynamics simulations. Our results show that at low hydration levels H(2)O molecules are isolated and a continuous hydrogen-bonded network forms as the hydration level is increased. Our quantitative analysis has established a hydration level (λ) between 5 and 6 H(2)O/SO(3)(-) as the percolation threshold of Nafion. We have also examined the effect of such a network on proton transport by studying the structural diffusion of protons using the quantum hopping molecular dynamics method. The mean residence time of the proton on a water molecule decreases by 2 orders of magnitude when the λ value is increased from 5 to 15. The proton diffusion coefficient in Nafion at a λ value of 15 is about 1.1 × 10(-5) cm(2)/s in agreement with experiment. The results provide quantitative atomic-level evidence of water network percolation in Nafion and its effect on proton conductivity.