Water in hybrid perovskites: Bulk MAPbI3 degradation via super-hydrous state

Here, first-principles density functional theory calculations are presented which reveal how water incorporation in hybrid halide perovskite [CH3NH3]PbI3 (MAPbI3) catalyzes the phase transition to the ([CH3NH3]PbI3.H2O edge-sharing) monohydrate (colorless) phase, eliminating its favorable photovoltaic properties. First, fundamental chemical and electrostatic interactions between water and each component of MAPbI3 are analyzed, demonstrating their dependence on water concentration. Second, the energetics of incorporated water is explored, leading to the discovery of spontaneous phase segregation into dry regions and regions with more than one water per formula unit—termed the “super-hydrous state.” Third, the properties of the super-hydrous state are analyzed, including the acceleration of octahedron breaking and rearrangement by the high water density. This reveals the phase transformation to be a bulk process, initiated at the super-hydrous regions. This paper concludes with a discussion of how this super-hydrous model explains disparate recent experimental observations concerning the water-induced transition from (black) perovskite to edge-sharing PbI2 (yellow) phase. © 2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5087290


I. INTRODUCTION
5][46][47][48] It consists of a perovskite (ABX 3 ) organic-inorganic hybrid lattice, where the methylammonium (MA) organic A cations sit in the cages formed by the inorganic (BX 3 ) lead iodide corner-sharing octahedral lattice.4][65][66][67][68][69] Although the presence of moisture has been found to have advantageous effects during the formation and crystallization of MAPbI 3 , leading to an improvement of the performance of solar cells, [70][71][72][73][74][75][76][77] prolonged exposure is known to cause its degradation back to PbI 2 .9][80][81][82] Just by placing the material in ambient conditions for 24 h, the PCE of the material drops by up to 80% due to its instability. 720][91][92] Experimental studies indicate that the degradation mechanism (from favorable black-phase ARTICLE scitation.org/journal/apmperovskite MAPbI 3 ) first goes through a transformation into a colorless edge-sharing monohydrate (MAPbI 3 ⋅H 2 O) phase. 93,947][108][109][110][111][112][113][114][115][116][117][118][119][120][121] Polarization dependence of water binding energy (BE) on MAPbI 3 (001) has been shown. 122Ab initio molecular dynamics (MD) calculations on different facets of MAPbI 3 (001) have also been performed. 100,123Force-field based MD has also been applied to study the surface interactions with liquid water and a layer-by-layer degradation. 124here have also been some theoretical studies on water in bulk MAPbI 3 .For example, Hall et al. studied the reversible water intercalation and related this to the water pressure dependence of the cell open-circuit voltage. 125They indicated a favorable interaction between H 2 O and a hydrogen of the ammonium group (in MA + ) for low H 2 O concentrations.This study involves water BEs up to 0.5 H 2 O (per MA) concentration.They observe that the BE per H 2 O becomes less exothermic for increasing water concentration (up to the highest coverage they study, i.e., 0.5).Grancini et al. made the same observation about the favorability of H 2 O-MA + interactions and consider this as the driving force for water incorporation, but they only studied 0.25 H 2 O (per MA) concentration. 101Jong et al. studied water intercalation in 1 H 2 O (per MA) concentration. 126

II. MOTIVATION
Despite the advances in understanding the instability of perovskite MAPbI 3 , there are still important unsolved questions.Specifically, we address four such questions: (i) What is the nature of bonding between water and MAPbI 3 , and what are the fundamental forces driving it?To answer this, we perform density functional theory (DFT)-based electronic structure analysis along with appropriately designed numerical experiments.(ii) How does the energetics of water bonding in MAPbI 3 help us to determine the arrangement of water in the solid?To answer this, we compute the concentration dependence of water bonding geometries and energetics, varying the (H 2 O:MA) ratio from 1 32 to 2. (iii) How can the cornersharing perovskite MAPbI 3 structure readily and in low (room) temperatures turn into an edge-sharing monohydrate phase with a dramatically different structure?To understand this, we analyze the high concentration limit of water intercalation and explain how a super-hydrous state acts as a transient bridge between cornerand edge-sharing structures.(iv) Is MAPbI 3 degradation a bulk or surface-driven mechanism?To answer this, we take few experimental puzzles and explain how a super-hydrous bridge and a bulk degradation mechanism can rationalize them.Some of these experiments include the following: (a) the isotropic and homogeneous reversible formation of the monohydrate phase independent of the depth in the film; 93 (b) the scaling behavior of moisture-induced grain degradation in polycrystalline perovskite MAPbI 3 ; 127 (c) the observation that grain-boundary passivation enhances the stability of the sample. 128

III. COMPUTATIONAL METHODS
We perform calculations using density functional theory (DFT) 129,130 with plane wave basis sets using the Quantum Espresso. 131Ultrasoft GBRV pseudopotentials 132 and Perdew, Burke, Ernzerhof (PBE) exchange correlations (XC) 133 have been used.Dispersion interactions are accounted for using the Grimme DFT-D2 method. 134,135][138][139] We use a kinetic energy cutoff for wave functions (Ecut) equal to 476 eV and the charge cutoff being 10 times larger.The sampling for k-space is equivalent to 3 × 3 × 2 for the tetragonal MAPbI 3

√
2 × √ 2 × 2 supercell containing 48 atoms.The Ecut and k-mesh have been chosen to achieve a convergence of better than 0.01 eV for calculating binding energies (BEs).This was checked by increasing Ecut up to 1360 eV and increasing the k-space sampling to the equivalent of 9 × 9 × 6 (for more details, see Sec. 1 of the supplementary material).The BE is defined such that a more negative number means more exothermic For calculating the structures with(out) water, we perform variable-cell relaxations (vc-relax), permitting the atomic coordinates and all components of the lattice vectors to relax.Many (in some concentrations ≈100) educated guess-structures (based on the knowledge of H 2 O bonding discussed in this work) have been used as initial guesses (starting points) to ensure identification of the global energy minimum.
We successfully achieve a convergence of better than 0.01 eV (for BEs) with respect to our convergence parameters; this is the precision we use in quoting calculated energetics.The theoretical model we propose is self-consistent up to this limit.Furthermore, we have identified the error that might be associated with the absolute values of our BEs (compared to experiments).We have pinpointed the possible errors generated by short and long-range corrections to our XC functional, by benchmarking our results against hybrid functionals and different schemes for inclusion of dispersion interactions.For more details, see Sec. 2 of the supplementary material.

A. Fundamentals of water bonding in MAPbI 3
Below, we delve into the nature of water bonding in MAPbI 3 .
To better understand the system by analyzing different contributions to the bonding, we also perform numerical gedanken experiments on molecular MA + (with negative jellium background) and PbI − 3 (with positive jellium background).

H 2 O-MA + bond
In much of the literature, the bond between water and organic MA + group in MAPbI 3 has been named a hydrogen bond. 140lthough here we are not concerned with the semantics, the bonding between MA + and water is fundamentally different from a canonical hydrogen bond, e.g., O-H⋯O bonds in a water network.The main Thus, the hole on the MA + makes a great contribution to the bond.
To better understand the spatial distribution of this hole and how this changes upon water bonding, we use the Bader charge analysis (BCA). 142,143As can be seen in Table I, the hole on MA + is mostly located on N-bound hydrogens (H N ).Thus, these hydrogens are more "proton-like."This is the reason H 2 O binds to one of these hydrogens.In Table II, after H 2 O binds to an H N (we denote this specific hydrogen by H * N ), (a) the H * N becomes even more proton-like, i.e., the hole redistributes itself to move more toward an H 2 O lone-pair, and (b) the associated N-H bond is elongated.This shows itself as the blue region (electron depopulation) localized on this hydrogen in Fig. 1(c).5][146] The common cause is creating a deeper electrostatic potential well (and a stronger associated electric field) that can more strongly polarize and bind the H 2 O lone-pair. 144In Table II, in addition to localization of the hole on H * N , MA + electrons also get more localized on the CH 3 group to decrease the electron-electron repulsion with the polarized lone-pair.There is also a small charge transfer from H 2 O to MA + which stems from some degree of covalency and orbital mixing.
The role of local electric fields and electrostatic potential wells (ESP wells) in enhancing water bonding via lone-pair polarization has been recently pointed out in the context of inorganic surface chemistry; 144 we extend such a picture to the bulk chemistry of hybrid organic-inorganic systems.The chemically active lone-pair orbitals of water are its HOMO (perpendicular to its dipole) and HOMO-1 (parallel to its dipole), which are shown in Figs.1(a) and 1(b).Figures 1(c) and 1(d) show the water lone-pair (specifically HOMO-1) polarization upon interacting with MA + .The lone-pair polarization puts more electron charge density closer to the protonlike H * N , close to which a deep electrostatic potential well (and associated strong electric field) exists [this charge movement is indicated by the yellow arrow along O-MA bond in Fig. 1(c)].Figures 1(c) and 1(d) reveal that the charge density differences are virtually identical using B3LYP and PBE-D2 XC functionals.It can also be seen that there is some electron charge movement toward the CH 3 group (as also mentioned above in the context of changes in Bader charges) to minimize the electron-electron repulsion between the polarized lone-pair and the MA.The latter charge redistribution is indicated by a yellow arrow along the N-C bond in Fig. 1(c).
In addition to the electrostatics, there is also a (less dominant) covalent component to the H 2 O-MA + bond.This is noticeable in the BCA presented above, showing itself as a small charge transfer from H 2 O to MA + .This covalency also shows up as a bifurcation of water HOMO-1.For more details, see Sec. 3 of the supplementary material.

H 2 O-PbI −
3 bond Here, to identify and better understand the different binding motifs in the full MAPbI 3 system, we design another gedanken experiment: take the fully vc-relaxed MAPbI 3 supercell, remove the MA + cations, and substitute them with a uniform positive jellium.We then relax the remaining PbI − 3 cell, fixing the lattice parameters.The goal is to understand how the inorganic backbone can bind H 2 O, independent of specific organic groups.
Figure 2 depicts the structure of the bare PbI − 3 , the H 2 O binding geometry to the Pb 2+ cation, and the associated charge density difference plots.In our PbI − 3 model, as well as MAPbI 3 , the Pb 2+ cations are encaged at the center of I 6 octahedra.Such a symmetric  in Fig. 2(b).This water-induced distortion (symmetry-breaking) has the same principles as the adsorbate-induced lifting in water-surface bonds. 144Also similar to the case of surfaces, the H 2 O bonding geometry is such that the HOMO (perpendicular to H 2 O classical dipole) points toward Pb 2+ .The HOMO thus becomes polarized, creating a dipole perpendicular to the water geometric dipole.This is shown in Fig. 2(c) by the orange arrow pointing toward Pb 2+ .
H 2 O bonding to the inorganic framework is a triaxial phenomenon [see Fig. 2(c)], associated with three charge accumulation centers: one between O and Pb 2+ and two between H H 2 O and I − s [marked by I 1 and I 2 in Fig. 2(c)].A water lone-pair (mostly HOMO) polarizes toward the positive center on Pb, while some charge density on Pb gets re-distributed (to minimize electron-electron repulsion with the polarized lone-pair and create a more positive site toward which the lone-pair polarizes) and localized in the two I − -H + axes [between H 2 O hydrogens and I 1 and I 2 in Fig. 2(c)].This can be considered as a hydrogen bond between the H 2 O hydrogens and the lattice I − ions.Also for a similar reason, some charge density on Pb redistributes to the opposite side of the atom [this can be noticed as the red center at the top-left of the marked Pb in Fig. 2(c)].Both the shape of charge density re-distribution and the projected density of states (PDOS) [see Fig. 3(a)] teach us that this triaxial bonding is driven by electrostatics rather than covalency.In Fig. 3(a), the four H 2 O peaks below the Fermi energy (E F ) correspond to eight states occupied by electrons in molecular H 2 O.In Fig. 3(a), the H 2 O chemically active orbitals (HOMO and HOMO-1) remain virtually molecular-like (associated with small broadening and sharp peaks) upon the triaxial interaction with the inorganic backbone.This is the fingerprint of a non-covalent bond.For more information on triaxial bonding and short continuum of states above 5 eV, see Sec. 4 of the supplementary material.Here, the octahedron is not as distorted as the other Pb-bonding mode (or Pb bonding mode in PbI − 3 ); thus, less mechanical deformation cost is imposed upon the system.For more details, see Sec. 6 of the supplementary material.

Water bonding in edge-sharing MAPbI 3 ⋅H 2 O monohydrate
The reversible degradation of corner-sharing perovskite MAPbI 3 (black phase) to the edge-sharing MAPbI 3 ⋅H 2 O monohydrate (colorless phase) is a precursor to the irreversible degradation to edge-sharing stacked hexagonal PbI 2 (yellow phase) (see Fig. 5). 93,147,148The mechanism behind water binding in the colorless phase is not different from the cases discussed above.This is explained in Sec.7 of the supplementary material.Ranging from n = 0 to 1, each H 2 O binds to the H N group on MA + , and thus, it might not be obvious why there is a strong n-dependence on H 2 O BE.The answer to this is multi-fold.First, each H 2 O perturbs (distorts) the inorganic backbone, to maximize the attractive H-bonding to I − and to minimize the electrostatic repulsion between the O 2p lone-pair and I − .Thus, intercalated water in these materials (even if it is directly interacting with MA + in lower concentrations) is surrounded by a network of distortions in the inorganic framework.Thus, increasing the H 2 O concentration (from n = 0 to 1) leads to unfavorable interactions between these distortions and destabilizes the H 2 O binding.Second, the hole-localization on MA + is also n-dependent.As mentioned previously, the hole on MA + becomes concentrated on the H N interacting with H 2 O, to make it more proton-like.In the case of two water molecules interacting with the same MA + , an extra effect changing the BE will be their competition for this hole.In addition to the previous two effects which destabilize H 2 O binding, a third (stabilizing) effect in higher concentrations is the possibility of forming H-bonding networks between neighboring H 2 O molecules.The existence of these sometimes opposing effects makes it hard to a priori make a unique guess for the H 2 O binding geometry in MAPbI 3 .In Fig. 7(b), our calculations reveal at n = 1, which is four H 2 O per our simulation unit cell, the ground-state configuration is the one in which two of the H 2 O molecules form a H-bond with each other and through that make a connected chain of (H 2 O-MA-H 2 O-H 2 O-MA), leaving the fourth H 2 O out of the chain to bind to two neighboring MA + .In Fig. 6, at (n > 1) range, there is a wide "valley of stability," which we call a super-hydrous state.In this plot, we have only investigated concentrations as high as n = 2, but it is possible that a super-hydrous state continues to be (more) stable even with a larger number of H 2 O molecules per unit-cell.As can be seen in Fig. 6, water prefers to phase-segregate into regions of high and low concentrations.Thus, the way to the stable edge-sharing MAPbI 3 ⋅H 2 O monohydrate (colorless) phase (which as shown in Fig. 6 is a local thermodynamic sink for the hydrated system) is not a straight way that simply connects a stoichiometric water intercalated H 2 O-MAPbI 3 (perovskite) phase to the colorless phase, but rather a "longer" path which passes through the phase-segregated valley of stability, i.e., the super-hydrous nH 2 O-MAPbI 3 (n > 1) phase (see Fig. 6).It is only in this super-hydrous state that the structure has enough freedom (associated with lower transformation barriers) to readily re-arrange into an edge-sharing structure.This is due to the broken octahedra and under-coordinated (watercoordinated) Pb 2+ and I − ions (see Fig. 7), in addition to emergence of water H-bond networks.Relevant features that can be seen in, e.g., Fig. 7

High water concentration: Facile octahedra breaking and structural changes
As we have seen previously, the Pb binding mode for H 2 O is a higher energy but still competitive mode for lower H 2 O concentrations (Fig. 4).For n = 0.25, the MA + and Pb 2+ water binding modes are −0.60 and −0.42 eV, respectively.Although the Pb bonding modes are higher energy at low coverage, as the water molecules keep distorting and expanding the MAPbI 3 cell, upon intercalation, there reaches a point (at n > 1) at which the Pb bonding modes are energetically favorable for the system.This causes the structures with broken octahedra to actually become the ground state of the system in the presence of enough water.The exposed Pb 2+ cations can generate strong electrostatic potential wells that can effectively polarize the H 2 O lone-pair and make strong bonds to it. 144To do this, it is necessary that the symmetry of the PbI 6 octahedron is broken, creating angles of approach along which the water can see a large effective positive charge.As mentioned previously, this is associated with a mechanical deformation energy; thus, Pb bonding modes are not the ground state for water binding at low coverages.By contrast, at higher H 2 O concentrations, the structure is automatically pushed in the direction of distortion and disconnectivity.Now, there are enough water molecules that, through forming strong bonds (lone-pair, hydrogen or triaxial bonds) to the MAPbI 3 ionic components and via hydrogen bonding networks with other H 2 O molecules, stabilize such distortions (e.g., under-coordinated Pb 2+ sites).
As can be seen in Fig. 8, at near-zero H 2 O concentration regime, the cell volume remains almost constant, or even decreases slightly by water intercalation, mostly due to dispersion interactions. 149As the water molecules (and their associated distortions in the inorganic-backbone) get closer (by increasing n), the lattice expands to accommodate these distortions and added volume of water molecules.Up to n = 1, the added cell volume by water addition is small.Changing n from 0.5 to 1 by the addition of two Å3 .The corner-sharing perovskite MAPbI 3 structure can be considered to be broken for n > 1.Here, water begins to interact with the Pb 2+ cations by breaking the PbI 6 octahedra.The large volume expansion per additional H 2 O, along with the broken octahedra and associated distortions, gives the superhydrous structure more freedom to re-arrange into the monohydrate edge-sharing phase.This is another indicator [in addition to the thermodynamic phase-segregation argument (Fig. 6)] that a low (room) temperature transition to the monohydrate phase is facilitated through a super-hydrous pathway.The fact that at n > 1, the structure is broken and addition of each water molecule introduces even more distortion and under-coordinated sites, indicates that the super-hydrous state might not be a stable phase (deep enough Gibbs free energy local minimum).Such a phase rather acts as a transient intermediate state between the perovskite and edge sharing structure.
The fact that something special happens at around (n > 1) is not only evident in volume-vs-concentration plot (Fig. 8) but also in bandgap-vs-concentration plot [see Fig. 9(a)].The blue curve depicts the changes in the bandgap of the water intercalated system.The orange curve corresponds to a gedanken experiment done to understand the role of water in changing the materials' bandgap: we remove the H 2 O molecules from the relaxed structure, freeze the rest of the structure, and find the electronic structure to calculate the bandgap.It can be seen that in the (0 ≤ n ≤ 1) range, addition of water generally increases the bandgap, 150 and both the hydrated (blue) and H 2 O-removed (orange) curves virtually overlap with each other.This gap change is due to the structural deformation and lattice expansion. 151,152Generally speaking, the addition of water reduces the connectivity of the MAPbI 3 host structure, weakening bonds, narrowing the bands, and increasing the bandgap. 152,153It can be seen in Fig. 9(a) that in the (1 < n ≤ 2) range, the orange and blue curves begin to deviate from each other.Here, the effect of water cannot be reduced to merely changing the geometric properties of the host MAPbI 3 structure, rather as in Figs.9(b) and 9(c), the water molecules have a "chemical" role in dictating the bandgap.It can be noted by comparing Figs.9(b) and 9(c) that at higher water concentrations (e.g., n = 2), the water orbitals (more significantly) constitute both the valence band minimum (VBM) and conduction band maximum (CBM), due to an increased covalency with I − and Pb 2+ of the inorganic backbone.This is caused as a result of broken octahedra and water directly making bonds to the inorganic backbone (i.e., the Pb 2+ ).

Water shuttling to wet regions through bulk MAPbI 3
The energetics of water binding vs concentration in Fig. 6 shows that water intercalation and segregation into high and low concentration regions is energetically favored for the hydrated MAPbI 3 system.But how favorable is the associated dynamics?We calculate the water diffusion barrier (in low concentration n ≈ 0.0625 or 1 16 ) along different possible pathways across the perovskite material.The corresponding minimum-energy paths (MEPs) for water diffusion in the ab plane (a and b represent short axes of the tetragonal phase) and along the c axis (the longer direction) are found and depicted in Figs. 10 and 11.The diffusion barrier for the latter is found to be 0.39 eV, while for the former, it is 0.44 eV.These small values, along with the small barriers reported in the literature for water entrance from the vacuum into the material, 116,122 mean that MAPbI 3 is isotropically permeable to water molecules.Such fast diffusion of water in low concentrations is also supported by experimental observations and has been described by Müller et al. as fast and inconspicuous. 154n important point to note in the MEPs along both long-and short-axis diffusion pathways is the critical role of the electrostatic bond between the proton-like H N group of MA + and the water lonepair (HOMO).It can be seen in Fig. 10 that the first MA + rotates such that its H N group always points at the H 2 O HOMO.After reaching the TS2, the water HOMO is locked to the H N group of the second MA + and both rotate to bring the H 2 O into its final state.The same effect can be seen for the long-axis diffusion (see Fig. 11).Here, the H N group of the first MA + group is locked into the H 2 O HOMO, and both MA + and H 2 O rotate until water reaches its local energy minimum at the bottom of its initial cell (image #10 of Fig. 11).Then, the second MA + (at the bottom cell) begins to rotate and bring its H N group closer to the H 2 O.The H N group then (electrostatically) locks into the water HOMO, and the MA + and H 2 O rotate with each other to lead H 2 O to its final state.

D. Connection to experiments
Our proposed degradation mechanism via bulk water shuttling and a phase-segregated super-hydrous state can be readily applied to understand some important experimental observations in the literature.Leguy et al. 93 reported that the formation of the monohydrate edge-sharing MAPbI 3 (colorless) phase is independent of the depth in the film, which is isotropic and homogeneous.Wang et al. reported the scaling behavior of moisture-induced grain degradation in polycrystalline MAPbI 3 and mention that grains degrade along the in-plane direction (from grain boundaries toward their center) rather than the out-of-plane direction (from surface down), and that the necessary duration for films to degrade showed a linear relationship with the grain size. 127 an effective mechanism to enhance the stability of the sample via grain-boundary passivation with hydrophobic agents. 128As shown in Fig. 6, although the presence of a low concentration (n ≲ 0.1) of water molecules is energetically favorable for the system, but as the energy vs concentration has a positive slope for 0 ≤ n ≤ 1, each H 2 O makes the local environment more hydrophobic.On the contrary, if there is a nucleus of a larger concentration of H 2 O molecules (n ≈ 2), the addition of extra water molecules does not affect the energetics of the rest due to the zero slope of energy vs concentration in this limit.Consequently, such a nucleus grows, which can then transform the local environment toward the lower-energy colorless phase (see Fig. 6).We propose that these initial high concentration nuclei are located in grain boundaries due to a large concentration of structural and stoichiometric defects and the related free volume. 155Next, since the H 2 O molecules can easily shuttle through the material, 154 they find these high-concentration nuclei ("wet" regions) get adsorbed into them, leaving the "center" of the grains as low concentration ("dry") regions behind.This is the reason why the degradation propagates "in-plane" from the grain boundaries toward the center. 127,156We propose that the homogeneity reported by Leguy et al. 93 and the thickness independent degradation rate mentioned by Wang et al. are the result of two operating principles: (i) The details of the grain boundaries are irrelevant in the degradation process, and they only provide the initial nuclei for water phase-segregation.(ii) The supply of water molecules that drives the degradation forward are not provided via grain boundaries, but by water molecules readily diffusing through the bulk of the material.Leguy et al. attributed the homogeneous and isotropic progression of degradation to rapid transport of water molecules along the grain boundaries, 93 while we propose that the water shuttling is done by the bulk of the MAPbI 3 rather than the grain boundaries.Here, the grain boundaries are only responsible for providing initial high-concentration water nuclei for phase segregation.As the degradation proceeds and the monohydrate phase propagates in-plane toward the grain center, 127 what used to be a grain boundary is now a monohydrate-perovskite interface.It is not guaranteed that such an interface independent of local details of the sample keeps universal water permittivity characteristics.On the other hand, a bulk-driven water transport mechanism guarantees such universality.
It should be emphasized that the transformation of an overstoichiometric super-hydrous state [nH 2 O-MAPbI 3 , (n > 1)] to a stoichiometric edge-sharing MAPbI 3 ⋅H 2 O monohydrate locally leads to an excess of water molecules.These can then join the other over-stoichiometric regions and help catalyze the transformation of the rest of the sample into the edge-sharing phase.
We note that the super-hydrous state is a transient rather than a stable state, which acts as a bridge between the perovskite and edge-sharing structures.Thus, isolating and directly characterizing it as a stable phase might not be experimentally viable.
Nevertheless, this state has some unique characteristics that can be searched for as potentially experimentally testable fingerprints.This includes (a) the over-stoichiometric high concentration of H 2 O molecules, (b) the aforementioned Pb 2+ -H 2 O bonds, and (c) the volume expansion (≈13%) relative to the perovskite phase and (≈7%) relative to the edge-sharing monohydrate.In a real sample, depending on the local constitution and topography, the formation of the super-hydrous state might lead to a stress field in the interface with the rest of the material (either perovskite or edgesharing).An effective encapsulation scheme or interfacing with a hard material, especially in the limit of smaller perovskite nanoparticles, can in fact penalize the super-hydrous state, since transformation to a higher volume state at higher pressure incurs a greater Gibbs free energy penalty.Studying the effect of external pressure on the super-hydrous state can be the subject of future work.
The super-hydrous bulk-degradation model can help shed light upon the experimental evidence regarding the role of heterostructure/heterojunctioning 157,158 and the dimensionality engineering [159][160][161][162][163][164][165] on the stability of the halide-perovskite-based systems.Quan et al. 166 have studied the stability of a class of mixed-cation Ruddlesden-Popper (RP) 167,168 quasi-2D perovskite films [(PEA) 2 (MA)n− 1 PbnI 3n+1 , 1 < n < ∞] and have found an improved stability for smaller values of n.In addition to (a) the larger concentration (per Pb) of hydrophobic CH groups on the long-chain cations 168 and (b) the thermodynamic arguments originally presented by Quan et al. based on DFT-based formation enthalpies, 166 one can rationalize the kinetics behind this stability via our bulk degradation model.Decreasing n effectively reduces the dimensionality of the system, suppresses the bulk-driven degradation mechanism, and improves the stability.A central idea in our model is the role of initial high concentration nuclei necessary for creation and expansion of the super-hydrous regions.Based on the notion of electrostatics-driven 144 water bonding mechanism developed in this work, such initial nuclei tend to form in regions possessing high density of under-coordinated Pb 2+ cations which can create effective 3D bonding motifs for stabilization of initial H 2 O networks (e.g., nano-droplets).The dimensionality engineering schemes (e.g., quasi-2D structures with smaller n) can help eliminate such regions.Additionally, in such RP quasi-2D structures, even if the density of defects in a layer is high enough to create a super-hydrous region and degradation, they do not propagate to other layers separated by hydrophobic long organic cations.This should be contrasted with the discussed case of continuous 3D hybrid perovskites.Heterojunctioning can also be used for defects' suppression of perovskite's interfaces.Additionally, as mentioned before, interfacing (via strong contacts with harder materials) can help the stability (especially of smaller halide perovskite particles) by restraining the lattice constants and penalizing the higher volume super-hydrous state.Heterostructuring can also physically seal off the perovskite sample from the inbound water molecules, similar in philosophy to the encapsulation schemes. 169,170Jana and Kim have recently shown such protective interfaces do not necessarily need to be made with foreign materials or molecules but by transformation of the peripheral layers of the halide perovskites [to Pb(OH) 2 in their work]. 171][174][175] Finally, it is worthy to note that Walsh and co-workers have studied defect (specifically vacancies) chemistry in bulk hydrated perovskite MAPbI 3 with stoichiometric H 2 O:MA ratio [nH 2 O-MAPbI 3 (n = 1)]. 126,176They studied the effect of water on vacancy formation energetics and enhancing vacancy-mediated ion migration.This is indeed an important theoretical venue in understanding the bulk water chemistry, yet our approach in this work was different in philosophy.We propose that the degradation through a superhydrous state can proceed without the need of pre-existing vacancies and can take place in a perfect bulk MAPbI 3 single crystal (given the initial high water concentration nuclei).In this mechanism, aggregation of water molecules naturally distorts the structure and catalyzes the phase transformation.Nonetheless, our super-hydrous bulk-degradation model suggests that at grain boundaries the defect chemistry of halide perovskites becomes particularly important to help understand the (in)stability of these materials.Such a bulkdriven mechanism should also be contrasted with the recent surfacedriven mechanism proposed by Fan et al. for the thermal degradation of MAPbI 3 at higher temperatures and in the absence of water. 40,177

E. Connection to other hybrid-perovskites
The chemical bonding analysis and the degradation model hereby presented were centered on MAPbI 3 as representative of the hybrid-perovskites class.9][180][181][182][183][184][185] A critical feature of the superhydrous state is water binding with the inorganic cation (IC), which is originally located in a symmetric cage of the halide anions.Water binding to the inorganic backbone requires the mechanical distortion of the octahedron and breaking of this symmetry.The strength of the chemical bond between the IC and the halide has a determining role in dictating the energetics of this mechanical deformation, thus the relative favorability and energetics of the super-hydrous state.The relative sizes of the IC and the halide also dictate their bond length, which is a factor that dictates the local electrostatics that determines the ESP well, lone-pair polarization, and the strength of the H 2 O-IC bond. 144A possible direction for future research will be to study the periodic trends in the energetics of the super-hydrous state (as a measure of the material's stability) vs the combination of different IC and halides.It might also be possible that some simple descriptors can be found to map the combinations-space to the aforementioned energetics.In principle, the effect of changing the organic cation's type and the associated trends can also be understood based on differences in (a) direct electrostatic interaction with water, (b) stabilizing interaction with the (bare and distorted) inorganic octahedra, (c) interaction with the intercalated H 2 O bound to the distorted octahedra [see the discussions in Fig. 4(e) and Sec.6 of the supplementary material), and (d) changing the water diffusion barrier through the bulk material (as shown above, interaction between water and the organic cation plays a central role in the water diffusion process).

V. CONCLUSIONS
We discussed the fundamental underlying physics and chemistry behind water bonding in MAPbI 3 by extending concepts from the recent theory of lone-pair-surface bonds. 144The low concentration bonding mode for water is a relatively strong electrostatic bond between the polarized water lone-pair (mostly HOMO) and the proton-like ammonium H of MA + .The important electrostatic role of the extra hole on MA + in the H 2 O binding energy (BE) was discussed.This hole redistributes itself closer to H 2 O to enhance its bonding.The possibility of Pb-bound modes for water adsorption was also discussed.Here, the PbI 6 octahedron must be broken to create the appropriate electrostatics for H 2 O lone-pair bonding.Such a bonding mode is not the ground state for lower water concentrations, but as water concentration increases (n = H 2 O:MA > 1), H 2 O-Pb bonds readily form and structures with broken PbI 6 octahedra occur as ground states.The role of H 2 O molecules in increasing the bandgap of water-intercalated phases was discussed.In under-stoichiometric concentrations (n = H 2 O:MA < 1), bandgap change is led by mechanical effects leading to a smaller degree of connectivity in the system, while in the over-stoichiometric regime (n = H 2 O:MA > 1), the water has a chemical effect in widening the bandgap of the system, mostly through the formation of covalent bonds with under-coordinated Pb 2+ sites.
By calculating the concentration dependence of water BE in MAPbI 3 , we showed that the stoichiometric water intercalated configuration is only a local energy maximum (in H 2 O BE vs concentration space), i.e., the system would rather phase-segregate into low-and high-concentration regions.The high-water-concentration (wet) regions, which we call a super-hydrous state, are structures with expanded volume and broken PbI 6 octahedra, in which the corner-sharing MAPbI 3 perovskite structure is ready to re-arrange itself into the edge-sharing MAPbI 3 ⋅H 2 O monohydrate phase.This transient intermediate state gives the organic and inorganic components of MAPbI 3 enough freedom to re-arrange into the colorless edge-sharing phase.This colorless phase is a local thermodynamic sink, even more energetically favorable than water intercalation in very low concentrations.Thus, the system (water intercalated perovskite MAPbI 3 ) has a thermodynamic tendency to form such segregated hydration states and then degrade into the colorless monohydrate.
We find small and virtually isotropic diffusion barriers (≈0.4 eV) through bulk perovskite MAPbI 3 .Thus, one should perceive MAPbI 3 as a water-permeable structure that is readily filled with a small concentration of water molecules.These water molecules distort the inorganic-backbone (although not directly bonding with it at small concentrations) and through these distortions repel other H 2 O molecules.Thus, water in low concentration renders the sample more hydrophobic.Only once the water concentration is significantly increased (n = H 2 O:MA > 1), the water molecules and their associated distortions become cooperative and hydrophilic, leading to a local super-hydrous state, further disrupting and expanding the inorganic backbone upon joining additional itinerant H 2 O molecules.
Our model of super-hydrous mediated bulk degradation can be used to rationalize some important experimental observations including the macroscopically homogeneous and isotropic degradation of perovskite MAPbI 3 into the colorless monohydrate in multigrain samples, and the in-plane direction of degradation growth in a single MAPbI 3 grain. 93,127,128In such a model, the grain boundaries, which possess a large concentration of defects, are natural nuclei at which the initial high water concentration for phase segregation can occur and from which the degradation propagates.5][216][217] The role of surfaces on the (opto)electronic properties of hybrid perovskites has been previously investigated. 218Based on our results, surfaces do not have a direct influence on the degradation mechanism.However, passivation of the whole surface area of the sample can lead to water insulation of the sample, thus indirectly helping the stability.Similar to the grain-boundaries surfaces can also host a large number of defects.Nevertheless, 3D bonding motifs can occur much easier in the defected grain boundaries as opposed to the defected surfaces, and the existence of such favorable binding matrices means that the degradation nucleates in grain boundaries (rather than surfaces) and propagates in-plane toward the center of the grains (upon joining other phase-segregated H 2 O molecules) as experimentally observed. 127

SUPPLEMENTARY MATERIAL
See supplementary material for more details on the computational methods, convergence tests, and further electronic structure analyses.

FIG. 1 .FIG. 2 .
FIG. 1.(a) and (b) depict the H 2 O chemically active orbitals (HOMO and HOMO-1).Cyan and magenta show different signs of the wavefunction.(c) and (d) show the electron charge density difference upon water bonding to MA + derived using (c) PBE-D2 and (d) B3LYP-D2 XC functionals.Red (blue) shows the regions populated (depopulated) by electrons.The yellow arrows in (c) show the direction of local charge density movement.Color code is O(red), H(white), C(black), and N(green).
3. H 2 O-MAPbI 3 bondsBased on this understanding of H 2 O-MA + and H 2 O-PbI − 3 bonds, we now analyze water bonding to MAPbI 3 .So far, there are two distinct water binding sites available for H 2 O: (a) interacting with an H N on MA + and (b) distorting an octahedron and interacting with the exposed Pb 2+ .Based on H 2 O BEs reported above, the bonding to MA + seems more favorable.This is also confirmed by our direct MAPbI 3 calculations and the literature.
system.We examine these, first for the case of relatively low 0.25 H 2 O:MA concentration.First, we consider the more energetically favorable case of H 2 O binding to H N on MA + [Figs.4(a) and 4(b)].One difference with the simpler case of bonding to a molecular MA + is that here water is oriented such that mostly the HOMO lone-pair interacts with the H * N .The HOMO is polarized, putting more charge density closer to the electrostatic potential well near the positively charged H * N .Another difference is the existence of H-bonds between H 2 O and nearby I − anions.This can be seen in Figs.4(a) and 4(b) as the red region between H (of H 2 O) and I 1 , which are 2.52 Å apart.Smaller isovalues in the charge density difference plots reveal another red region between the other H and (I 2 ) 2.73 Å apart: a weaker H-bond.Hence, this can also be considered as a triaxial water bonding environment.It should be noted that although the water does not appear to disrupt the perovskite MAPbI 3 inorganic network, one can quantify some distortions to the lattice structure (for more details, see Sec. 5 of the supplementary material).We now consider the Pb-bonding mode for H 2 O.There are two such modes as local energy minima [see Figs.4(c) and 4(e)].The less stable bonding mode [Fig.4(c)] is similar to the H 2 O bonding mode to PbI − 3 .In Fig. 4(d), one can notice a triaxial bonding between the H 2 O and the inorganic backbone.Water has disrupted the octahedron and exposed the Pb 2+ , stabilizing its polarized lone-pair in the local electrostatic potential well.It also forms H-bonds with I 1 and I 2 sitting 2.45 and 2.52 Å apart from its hydrogens, respectively.The second (more favorable) Pb 2+ bonding mode [Figs.4(e) and 4(f)] is not favored in the simple case of water in PbI − 3 .
3 and phase-segregationIn Fig.6, we plot the H 2 O BE (per molecule), which is the energy gain by addition of water to the material's structure, as a function of H 2 O concentration in perovskite MAPbI 3 .The general trend is as follows: in a very low concentration, the energy gain by H 2 O intercalation (H 2 O BE) into perovskite MAPbI 3 is ≈−0.68 eV per molecule.This limit is a local minimum in the space of BE vs H 2 O concentration (n, defined as H 2 O:MA ratio).As we increase H 2 O concentration, the BE per molecule decreases in magnitude (less exothermic) and we reach a maximum at around n = 1.Increasing n further leads to stabilization of H 2 O intercalation and we reach an almost flat valley at around n = 2, which we call a super-hydrous state [see Fig.7(d)].In Fig.6, we also depict the energy gain of turning the water-intercalated perovskite structure into a monohydrate edge-sharing phase (−0.28 eV per H 2 O) at n = 1.The energy values in Fig.6should be compared to the energy associated with the standard entropy of gas phase water (1 bar, 300 K) ∼−0.59 eV.The BEs are great enough that the water intercalation in MAPbI 3 becomes relevant, although such adsorption deprives the molecule of its gas phase translational entropy.
(d), are (a) I − anions connected to only one Pb 2+ , (b) Pb 2+ cations bonding to two H 2 O, and (c) connected Pb-H 2 O-H 2 O-MA-H 2 O-Pb strings.In other words, the catalytic role of water for readily driving the phase-transition (around room temperature) through smaller barriers emerges in such an over-hydrous phase.

FIG. 9 .
FIG. 9.The change in the MAPbI 3 bandgap upon water intercalation as a function of water concentration (H 2 O:MA) is shown in (a).The blue dots are the calculations done on the whole cell (H 2 O + MA), while the orange dots are gedanken experiments done by removing the water from the structure, freezing the rest, and solving for the electronic structure.In (b) and (c), the PDOS for two water concentrations (i.e., 1 and 2 H 2 O:MA) is shown.

FIG. 10 .
FIG. 10.Minimum-energy pathway for H 2 O diffusion at low concentrations in ab plane.A few important images are shown, including the first TS image #7 and second TS image #13.As a guide to the eye, the diffusing H 2 O is shown by the orange arrow.Color code is O (red), H (white), C (black), N (green), I (brown), and Pb (dark gray).

FIG. 11 .
FIG. 11.Minimum-energy pathway for H 2 O diffusion at low concentrations along the c axis.A few important images are shown, including the first TS image #6, the second TS image #15, and the third TS image #24.Color code is O (red), H (white), C (black), N (green), I (brown), and Pb (dark gray).

TABLE I .
Total charges (nuclear + electronic) computed using BCA on MA 0 and MA + (electron charge is negative).The average charge on N-bound H is denoted as HN and C-bound is denoted as HC .

TABLE II .
Total charges (nuclear + electronic) computed using BCA and bond lengths for bare MA + before binding to H 2 O and MA + after H 2 O bonding.
125,126Although studying the H 2 O bonding to MA + and PbI − 3 separately teaches us important lessons on water-MAPbI 3 bonds, there are significant complexities that arise only when considering the whole MAPbI 3

FIG. 6. The
H 2 O BE (per molecule) as a function of H 2 O concentration (H 2 O:MA) in MAPbI 3 perovskite are shown by black filled circles.Each H 2 O at n = 1 gains −0.28 eV stability if the structure changes to the edge-sharing phase: this is shown by the black star.The minimum-energy path to the edgesharing monohydrate passes through the super-hydrous state.