Probing the Soft and Nanoductile Mechanical Nature of Single and

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Probing the Soft and Nano-Ductile Mechanical Nature of Single and Polycrystalline Organic-Inorganic Hybrid Perovskites for Flexible Functional Devices Jingui Yu, Mingchao Wang, and Shangchao Lin ACS Nano, Just Accepted Manuscript • DOI: 10.1021/acsnano.6b05913 • Publication Date (Web): 02 Dec 2016 Downloaded from http://pubs.acs.org on December 7, 2016

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Probing the Soft and Nano-Ductile Mechanical Nature of Single and Polycrystalline Organic-Inorganic Hybrid Perovskites for Flexible Functional Devices Jingui Yu†, Mingchao Wang†, Shangchao Lin* Department of Mechanical Engineering, Materials Science and Engineering Program, FAMU-FSU College of Engineering, Florida State University, Tallahassee, Florida 32310, USA. *Corresponding author contact information: [email protected] †

These authors contribute equally to this work.

Abstract: Although organic-inorganic hybrid perovskites have been extensively investigated for promising applications in energy-related devices, their mechanical properties, which restrict their practical deployment as flexible and wearable devices, have been largely unexplored at the atomistic level. Towards this level of understanding, we predict the elastic constant matrix and various elastic properties of CH3NH3PbI3 (MAPbI3) using atomistic simulations. We find that single-crystalline MAPbI3 is much stiffer and exhibits higher ultimate tensile strength than polycrystalline samples, but the later exhibit unexpected, greatly enhanced nano-ductility and fracture toughness, resulting from the extensive amorphization during the yielding process. More interestingly, polycrystalline MAPbI3 exhibits inverse Hall-Petch grain-boundary strengthening effect, in which the yield stress is reduced when decreasing the grain size, due to amorphous grain boundaries. By monitoring the centro-symmetry parameter and local stress evolution we 1 ACS Paragon Plus Environment

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confirm the soft and nano-ductile nature of defective MAPbI3 with a crack. By conducting atomic stress decomposition, we attribute such fracture toughness primarily to the strong electrostatic interactions between the ionic components. The observed limited brittle fracture behavior is attributed to the transformation of partial edge dislocations to disordered atoms (nanovoid formation). A significant plastic deformation region is observed when nanovoids enlarging and coalescing with adjacent ones, which ultimately leads to crack propagations via ionic-chain breaking. After comparing with traditional inorganic energy-related materials, we find that hybrid perovskites are more compressible and can absorb more strain energy before fracture, which makes them well suited for wearable functional devices with high mechanical flexibility and robustness. Keywords: hybrid perovskite, polycrystalline thin films, fracture mechanics, crack propagation, microstructural evolution, molecular dynamics

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Very recently, organic–inorganic hybrid perovskites, such as the prototypical methyl ammonium lead iodide (CH3NH3PbI3 or MAPbI3), have emerged as promising energy-related materials, such as light absorbers in photovoltaic (PV) cells, emitters in light-emitting diodes (LEDs),1 or potential thermoelectric materials with high figure of merit.2, 3 They provide the possibility of integrating useful organic and inorganic characteristics within a single crystalline molecular-scale hybrid, enabling unique electronic,4 optical,5 thermoelectric,2, 7

3

and surface6,

properties. These ambipolar semiconductors have attracted increasing attention due to their

scalable, low-temperature and low-cost synthesis processes, small and direct electron bandgaps, high extinction coefficients, and high carrier mobility.8, 9 With the growing interest in using organic-inorganic hybrid perovskites for PV and LED applications, it is essential to advance the existing understanding of their degradation pathways and mechanical stabilities, in particular, fracture mechanics considering cracks or defects, beyond the ideal picture of elasticity and plasticity for pristine samples. It is well-known that as-synthesized crystalline materials always possess invisible cracks and defects at the microscopic level. The initiation and propagation of these cracks or defects dictate the material’s failure and lifetime, rather than the ultimate stress-strain predicted for pristine, crack-free materials. There has been rapid progresses in studies of the electronic band structures and exciton diffusion pathways of hybrid perovskites,9,

10

but very few experimental or theoretical

investigations have been reported on their mechanical properties. Similar to silicon-based11, 12 solar cells and inorganic LED devices (such as those using InGaN, ZnSe, GaP, and etc.), there are great concerns that applications of hybrid perovskites in energy-related devices may be limited by their brittleness and lack of mechanical flexibility compared to organic PV and OLED (organic LED) materials based on conductive polymers or small molecules. In particular, the 3 ACS Paragon Plus Environment

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stress state (residue or thermal stresses, cyclic bending, and surface scratching) and the crystallinity of the hybrid perovskite layer have strong impacts on the solar spectrum absorption performance, optical bandgap, and the resulting energy conversion efficiency.13 It also has large impacts on the structural phase transition,14 material degradation and mismatch at interfaces, and the resulting device lifetime, as studied in recent experiments on making flexible and fatigue resistant hybrid perovskite solar cells.15, 16 Experimental methods, while certainly of great value, have limitations in determining the long-term stability of hybrid perovskites, measuring their fracture mechanical properties, and decoupling the various atomistic interaction contributions to the above properties. Indeed, direct measurements of the elastic, plastic, and fracture responses of hybrid perovskites through tensile testing are challenging, since high sample quality (with large crystal sizes, or ideally, be single crystalline rather than polycrystalline) and large samples are required. For example, so far only nano-indentation tests have been carried out very recently to measure the hardness and deduce the Young’s modulus (10 ~ 20 GPa) of MAPbX3 (X = Br, I, Cl) at the room temperature.17, 18 On the other hand, theoretical and computational methods, from first-principle calculations to force field-based molecular dynamics (MD) simulations, can predict materials properties that are difficult to access experimentally. Recently, the elastic constants and elastic mechanical properties of pristine hybrid perovskites MABX3 (B = Sn, Pb; X = Br, I) have been determined using the density functional theory (DFT) for their cubic, tetragonal and orthorhombic phases.19 Despite the above DFT19 and experimental17,

18

studies on the elastic properties of hybrid

perovskites, fundamental understanding and quantification of their fracture mechanical properties and failure mechanism are still missing. Current experimental techniques have difficulty in obtaining high quality bulk perovskite samples to probe crack propagation, while 4 ACS Paragon Plus Environment

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DFT calculations cannot handle length scales larger than 10 nm and time scales longer than 10 ns, both are relevant for probing fracture mechanics. Therefore, large-scale MD simulations are well-suited for studying the fracture mechanics of hybrid perovskites to understand their failure mechanisms and predict the fracture toughness as measured by the critical strain energy release rate and the critical stress intensity factor for crack propagation. With the above in mind, we first computed the elastic constant matrix of pristine MAPbI3 using force field-based molecular mechanics (MM) calculations, for the cubic, tetragonal and orthorhombic phases. We used the first-principle derived classical model potential for hybrid perovskites (MYP), recently developed by Mattoni et al.20, 21 We utilized these elastic constants to determine the elastic properties of MAPbI3, and compared with our MD simulation results for pristine single and polycrystalline MAPbI3 considering the finite temperature effect, together with previous DFT and experimental results. Then we carefully explored the fracture mechanical behavior and microstructural evolution of defective single-crystalline MAPbI3 (with a crack) under tensile loading using MD simulations (see Methods for details). The elastic, plastic, and fracture mechanical properties of MAPbI3 were characterized and found to be very soft and nano-ductile, with major contributions from interatomic Coulombic interactions (determined through atomic stress decomposition, see Eq. 8 in Methods), an important feature for ionic crystals. We paid special attention to the underlying mechanism of the nano-ductility observed during the microstructural evolution. A local plastic deformation region was found at the nanoscale, while only localized partial edge dislocations were observed near the crack tip. The movement of these dislocations forms nanovoids, which enlarge and coalesce with adjacent nanovoids to achieve crack propagations. Finally, we compared the mechanical properties, in


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particular, stiffness, fracture toughness, and flexibility, of hybrid perovskites against existing energy-related inorganic PV, LED, and thermoelectric materials. RESULTS AND DISCUSSION We first validated the capability of the current MYP force field (see Methods for details) to reproduce the elastic properties of hybrid perovskites. The MYP potential has been shown to well capture the structural (temperature dependent lattice constants for different phases), dynamical (rotations of MA+ cations and distortions of the PbI6 octahedra), and some mechanical properties (bulk modulus and cohesive energy) of MAPbI3.20,

21

Force field-based MM

simulations are conducted using the volume perturbation method in GULP22, 23 to calculate the first-order elastic constant matrix [cij] and compliance constant matrix [sij] of the orthorhombic, tetragonal and cubic phases of MAPbI3. The calculated cij values are summarized in Table S1 in the Supporting Information, and they are consistent with the DFT results,19 although small deviations were observed for c44. Overall, MM calculations predict slightly higher elastic constants than DFT. Based on the Voigt–Reuss–Hill (VRH) approximation,24 elastic constants cij can be utilized to evaluate various mechanical properties of MAPbI3, including Young’s modulus E, bulk modulus K, shear modulus G, and Poisson’s ratio ν (see Table 1). The VRH approach combines the upper (Voigt) and the lower (Reuss) bounds of the mechanical properties as the macroscopic effective mechanical properties.


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Table 1. Comparison between elastic properties of pristine (except for the last row for the defective MD model) MAPbI3 obtained using MM (from this work), MD (from this work), DFT19 calculations, and 17, 18

available experimental nano-indentation results

: bulk modulus (K), shear modulus (G), Young’s

modulus (E), and Poisson’s ratio (ν). The subscript “VRH” denotes the Voigt–Reuss–Hill approximation, which is applied for MM and DFT calculations only. For MD (predicted both E and ν) and experimental (measured E and used ν from DFT) results, K and G were estimated based on K  G

E and 31  2 

E , assuming that MAPbI3 is isotropic and homogeneous. 21   

MAPbI3 Phases (Methods)

KVRH (GPa)

GVRH (GPa)

EVRH (GPa)

νVRH

Orthorhombic (DFT)19

18.1

3.6

15.0

0.36

Orthorhombic (MM)a Tetragonal (DFT)

17.80

9.27

23.68

0.28

19

12.2

3.7

12.8

0.33

a

20.25

10.51

26.87

18

18

Tetragonal (MM)

Tetragonal (Experiment)

13.9

5.4

17

10.4 ~ 10.7 , 14.0 ~ 14.3

0.28 18

0.3319

Cubic (DFT)19

16.4

8.7

22.2

0.28

Cubic (MM)a

18.08

11.91

29.29

0.23

13.11

5.70

14.94

0.31

Polycrystalline Cubic (MD)a

2.51 ~ 4.87

2.15 ~ 3.21

5.08 ~ 7.89

0.16 ~ 0.23

Defective Single-crystalline Cubic (MD)a

12.32

5.36

14.04

0.31

Single-crystalline Cubic (MD)a

a

This work

The mechanical anisotropy is another important factor in the fracture mechanics of MAPbI3, since the formation and propagation of micro-cracks are often related to the elastic anisotropy. To better describe the anisotropic elastic behavior of each phase, we constructed the three-dimensional (3D) surface contour plots for E with respect to different crystallographic directions, as depicted in Figure 1(a)-(c) (see Methods for details). In principle, for an isotropic 7 ACS Paragon Plus Environment

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crystal, the 3D surface contour should be spherical. As shown in Figure 1(a)-(b), the orthorhombic and tetragonal phases show strong elastic anisotropy. The E value in the [010] direction for the orthorhombic phase and [001] direction for the tetragonal phase have the lowest values. This is consistent with the fact that both directions have the largest stacking spaces and thus lowest packing densities. While for the cubic phase (Figure 1(c)), higher symmetry contributes to its weaker elastic anisotropy: the values of E along x, y, and z axes are all ~25.5 GPa. The MM-predicted Young’s modulus E for MAPbI3 is in the range of 14 ~ 35 GPa for the different phases. These values are higher than those measured by recent nano-indentation tests17, 18

(see Table 1) due to the neglect of temperature.


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Figure 1. (a)-(c) Crystal structures of MAPbI3 under the orthorhombic, tetragonal and cubic phases, and the corresponding three-dimensional (3D) illustrations of the surface contours of the MM-predicted Young’s modulus E. The x, y, and z axes used here reflect the [100], [010], and [001] crystallographic directions. (d) Stress-strain curve of the pristine cubic single and polycrystalline MAPbI3 (consisting of 6, 10 and 14 grains) under tensile loading.  zz* is the effective tensile stress along the z axis considering the Poisson’s effect, and ε is the engineering strain along the z axis. The simulation setup and the chemical structure of MAPbI3 are shown in the top right corner.


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MAPbI3 exhibits three types of temperature-dependent crystal structures, that is, the orthorhombic, tetragonal and cubic phases. For many MAPbI3-based functional devices (such as solar cells and LED), their working temperatures are higher than the tetragonal-to-cubic phase transition temperature (327 K).25 It is well-known that MM calculations usually over-predict the elastic constants due to the neglect of temperature. Therefore, we carried out mechanical tensile test on pristine cubic single and polycrystalline MAPbI3 using MD simulations at 350 K (see Figure 1(d)) to directly determine the various elastic properties of MAPbI3, as listed in Table 1. The effective tensile stress  zz* (considering the Poisson’s effect for the fixed simulation box dimensions, see Eq. 9 in Methods) is computed during the tensile loading process and shown in Figure 1(d) as a function of the engineering strain ε. For pristine single-crystalline MAPbI3, the resulting ultimate tensile stress and yield strain are 0.449 GPa and 5.0%, respectively. From the linear elastic region of the stress-strain curve and separate MD simulations with varying normal pressure tensors under the NPT ensemble, we found that the Young’s modulus E = 14.94 GPa and the Poisson’s ratio ν = 0.31, respectively, which were then used to estimate the bulk and shear modulus K and G in Table 1 for the cubic phase (MD). The K/G ratio has been previously identified as a measure of ductility of hybrid perovskites,19 while lower ν values were connected with larger compressibility or deformability of solid materials. The high K/G ratios of 1.14 ~ 5.03 and low ν values of 0.16 ~ 0.36 obtained in experimental nano-indentations and various theoretical methods (DFT and our MM and MD simulations) suggest that hybrid perovskites could be ductile for extensive deformation, primarily attributed to the low shear modulus G.


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Compared to MM results, we found that the temperature effect considered in MD simulations has a great influence on the predicted elastic properties and improves the predicted elastic properties compared to experiments17, 18. With the above validations, we are confident about the capability of the MYP force field utilized here to capture the mechanical properties of MAPbI3. Since MAPbI3 are deployed as thin films composed of small grains in hybrid perovskite solar cells, here we also evaluate the mechanical properties of pristine polycrystalline MAPbI3. We studied three polycrystalline MAPbI3 models possessing different averaged grain sizes of 8.25 ~ 12.60 nm (or the inverse of the number of grains from 14 to 6, see Figure S1 in the Supporting Information). Based on the stress-strain curves in Figure 1(d), their Young’s moduli E and Poisson’s ratio ν are in the ranges of 5.08 ~ 7.89 GPa and 0.16 ~ 0.23, respectively. These values are lower than those of single-crystalline MAPbI3, demonstrating the mechanical deterioration induced by grain boundaries. Remarkably, there exist obvious yielding regions in the stress-strain curves. The corresponding yield stresses and yield strains for all three polycrystalline MAPbI3 are in the ranges of 0.133 ~ 0.181 GPa and 3.4% ~ 4.6%, respectively, which are much lower than the ultimate stress and strain of single-crystalline MAPbI3. It is also unveiled that the yield stress is reduced when decreasing the averaged grain size. Such trend in hybrid perovskite (see Figure S1) is in good agreement with the recently observed inverse Hall-Petch (grain boundary strengthening) effect in polycrystalline grapheme.26 Reductions of the yield stress here can be attributed to the amorphous structures at the grain boundaries. The amorphization process normally weakens the fracture strength of solids,27 and therefore, 11 ACS Paragon Plus Environment

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polycrystalline MAPbI3 with smaller grains, which correspondingly contain larger fractions of amorphous structures at the grain boundaries, will exhibit lower yield stress.

Figure 2. Microstructural evolution of pristine cubic-phase single and polycrystalline MAPbI3 (6, 10 and 14 grains) during the tensile loading process. Plots (a)-(c) are simulation snapshots colored based on the centro-symmetry parameter (CSP, defined by Eq. 11 in Methods). For single-crystal, the corresponding strain values are 5.0%, 15.4% and 35.0% for (a1), (b1) and (c1), respectively. For polycrystals, the corresponding strain values are 4.7%, 32.2% and 58.2% for (a2)-(a4), (b2)-(b4) and (c2)-(c4), respectively.

We now discuss the surprising nano-ductility of polycrystalline MAPbI3. In Figure 1(d), the yielding regions of the three polycrystalline MAPbI3 are more extensive than that of the single-crystalline MAPbI3. To explore the underlying yielding mechanism, the simulated 12 ACS Paragon Plus Environment

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deformation configurations of both single and polycrystalline MAPbI3 at different tensile strains are shown in Figure 2. For single-crystalline MAPbI3, it deforms homogeneously during tensile loading (Figure 2(a1)). When reaching the ultimate tensile stress (~ 0.449 GPa), a phase transformation (amorphization) from crystalline to amorphous begins, but is mitigated and suppressed by the formation of nanovoids (Figure 2(b1)). The adjacent nanovoids gradually merge (Figure 2(c1)), which is also accompanied by significant stress drops. For polycrystalline MAPbI3, Figures 2(a2)-(c2), (a3)-(c3) and (a4)-(c4) show that there are extensive and continuous phase transformations in the yielding region. The crystalline structure is gradually absorbed by the amorphous grain boundaries and becomes highly amorphous without forming any nanovoids. Similar phase transformations were also confirmed to be a dominant yielding mechanism in polycrystalline aragonites (carbonate minerals).28 Here the large fraction of amorphous structures in polycrystalline MAPbI3 is believed to greatly facilitate such amorphization process and hence, results in the large yielding region. Interestingly, the stress enhancement in the yielding region for the 10 and 14-grain cases (compared to the 6-grain case) also reveals the strain hardening behavior of polycrystalline MAPbI3 with smaller grain sizes. This may be due to the grain rotation and grain-boundary sliding29, 30 phenomena observed here (see Movies S1, S2 and S3 in the Supporting Information).


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Figure 3. Local von Mises stress distribution of pristine single and polycrystalline MAPbI3 (with 6, 10 and 14 grains) during tensile loading. Plots (a)-(c) are colored based on the values of local von Mises stresses. For single-crystal, the corresponding strain values are 5.0%, 15.4% and 35.0% for (a1), (b1) and (c1), respectively. For polycrystals, the corresponding strain values are 4.7%, 32.2% and 58.2% for (a2)-(a4), (b2)-(b4) and (c2)-(c4), respectively.

The von Mises stress distribution is relatively uniform in single-crystalline MAPbI3, as shown in Figure 3(a1). Local stress concentrations lead to lots of nanovoids (see Figures 2(b1) and 3(b1)), which then emerge into a crack, releasing some stress (see Figure 3(c1)). There are some stress concentrations at the grain boundaries, as shown in Figures 3(a2)-(a4), for polycrystalline MAPbI3. Smaller grain-size models possess more grain boundaries and show lots of stress concentrations at the grain boundaries, which lead to lower yield stresses and the inverse Hall-Petch effect. As the transformation from crystalline to amorphous structure 14 ACS Paragon Plus Environment

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proceeds, the stress concentrations will move from the grain boundaries to other regions, as shown in Figures 3(b2)-(b4). After yielding, the system becomes unstressed and the local von Mises stresses reduces to small values, as shown in Figures 3(c2)-(c4). The single crystalline MAPbI3 is then selected in MD simulations to investigate its microstructural evolution during fracture and the resulting fracture toughness. We constructed a defective model for the single-crystalline MAPbI3 with a rectangular crack at the center and applied the tensile test (see Figure 4(a)). The central crack is terminated with the more stable charge-neutral (001) surfaces (i.e., [MAI]0 or [PbI2]0). The effective tensile stress  zz* is computed during the tensile loading process and shown in Figure 4(a) as a function of the engineering strain ε. From the linear elastic region of the stress-strain curve, we estimated that the Young’s modulus E is ~ 14.04 GPa (see Table 1, lower than that of the pristine case due to the presence of crack), in perfect agreement with the experimental nano-indentation results in the range of 10.4 ~ 14.3 GPa.17, 18 We also estimated the corresponding bulk and shear modulus, K and G, for the defective model in Table 1, in very good agreement with the experimental results. The predicted ultimate tensile stress and yield strain are 0.319 GPa and 4.4%, respectively, determined from the linear elastic region. The sharp drop of  zz* after the linear elastic region reflects the brittle fracture nature, but it is then followed by a very slow decay of stress (stringing effect) until complete failure (  zz* approaches zero) at a large strain of 44.7%. Interestingly, MAPbI3 does not behave like traditional ceramics or inorganic oxide-based perovskites (such as SrTiO331) that are classical brittle materials. A significant plastic deformation region is observed 15 ACS Paragon Plus Environment

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with a constant stress level (half of the ultimate tensile stress), which occurs at a strain range of 8.1~ 24.4%. These observations imply strong electrostatic attractions between MA+ cations and octahedral PbI6 cages (discussed later).

Figure 4. (a) Stress-strain curve during the tensile fracture and crack propagation of defective single-crystalline MAPbI3. The defective model of MAPbI3 (cubic phase structure) with a crack is shown

in the plot. The stress-strain curve is colored to denote the various deformation and fracture stages during the tensile test. (b) Decomposed contributions from various interatomic interactions to the total stress-strain curve. The total effective stress was decomposed into contributions from short- and long-range Coulombic, vdW, bond, angle, and dihedral interactions. The non-bonded pairwise contribution to the stress is the sum of the vdW and Coulombic (short-range) contributions.

The colored mapping of the local von Mises stress (defined by Eq.10 in Methods) of MAPbI3 during the tensile test is shown in Figure 5(a)-(c). The crack does not propagate under a strain of 4.4% as shown in Figure 5(a), but some stress concentration is found near the crack tip with local disordering of the crystal structure. The crack first gets blunt when the strain is 8.1%, then a number of nanovoids (small cavitations) are formed during the initial stage of plastic 16 ACS Paragon Plus Environment

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deformation in Figure 5(b). However, the crack blunting and generation of nanovoids generally reflect the plastic deformation feature which is rarely observed in brittle ceramic materials.32 The number of nanovoids gradually increases when increasing the strain, which was also observed in defective amorphous/glassy ceramic and polymeric materials under tensile loading.33, 34 When the strain reaches 12.7%, the nanovoids in the vicinity of the crack tip coalesce quickly during the last stage of plastic deformation. The crack gradually propagates when the strain is increased up to 24.4%. Complete fracture finally occurs at the strain of 44.7% in Figure 5(c), and the system becomes unstressed and the local von Mises stress reduces to a small value, which also indicates that the strain energy is gradually released in the process of crack propagation.


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Figure 5. Crack propagation at different stages: plots (a)-(c) are colored based on the local von Mises stress; plots (d)-(f) are colored based on the CSP. The strains are 4.4%, 11.9% and 44.7% in (a), (b) and (c), respectively.

To decouple the effects from various interatomic interactions to the total stress, Figure 4(b) illustrates contributions, including van der Waals (vdW, modeled by the Buckingham and Lennard-Jones potentials here), short- and long-range (via Ewald summation) Coulombic, and bonded (i.e., covalent bonding, angular bending, and dihedral rotation within MA+) interactions, to the total effective stress. We found that the contributions from both long-range Coulombic and bonded interactions have negligible impact on the effective stress. Meanwhile, the contribution from non-bonded pairwise interactions (sum of the vdW and short-range Coulombic interactions) plays a leading role. The contributions from the short-range Coulombic interactions to the effective stress are positive, as shown in the inset in Figure 4(b), suggesting that net electrostatic attractions hold the ionic components together and fight against the external tensile stress. On the other hand, vdW interactions generate net repulsive (negative) stress and follow the direction of the external tensile stress. The magnitude of the short-range Coulombic contribution is larger than that of the vdW contribution, suggesting that the former is the dominant factor in determining the mechanical properties of MAPbI3. By probing the microstructural evolution at the atomistic level, we captured and examined the various brittle and ductile responses, including the competing processes of brittle fracture (nanovoids formation) and amorphization35 (PbI6 octahedra disordering) in the 18 ACS Paragon Plus Environment

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single-crystalline MAPbI3. The colored CSP mapping illustrates the degrees of local defects and partial dislocations during the tensile test, confirming the nano-ductility (featured by nanovoid formation and evolution) of

the single-crystalline MAPbI3 within a glassy region33 around the

crack surface, as shown in Figure 5(d)-(f). Large structural distortion and stress concentration exist near the crack tip with high CSP values in Figure 5(d), indicating that its structure is rearranged and suggesting that there might be some dislocations near the crack tip in response to strain. The region with high CSP values becomes larger when increasing the strain in Figure 5(e), accompanied by the enhanced disordering of the single-crystalline MAPbI3, and finally, forms a plastic deformation zone (glassy state of MAPbI3). Interestingly, the von Mises stress distribution remains almost unchanged for a large range of strain values without crack propagation. In Figure 5(f), the path of crack propagation is located within the region of high CSP values (see Movie S4 in the Supporting Information), indicating that the crack propagates through local defects. The existence of localized partial edge dislocations at the crack tip within MAPbI3 under small tensile strains is shown in Figure S2. This suggests that MAPbI3 has a certain degree of shear-induced ductility at the nanoscale. It is known that the polar character of perovskite oxides and their associated charge effects remarkably affect the formation of dislocations in perovskite oxides, such as SrTiO3.36 Similar charge effect in MAPbI3 may also play a notable role in their formation of dislocations, although dislocations in MAPbI3 are significantly more difficult to capture than those in metals. Moreover, atoms in MAPbI3 are quickly rearranged under a small 19 ACS Paragon Plus Environment

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strain of 4.4% and easily transit from ordered to disordered states via the slipping of dislocations. The formation of nanovoids near those disordered atoms leads to the observed brittle-fracture feature in MAPbI3. Although dislocation is a common phenomenon observed during plastic deformation, we found quite few dislocations in MAPbI3, while the stress-strain curve suggests a wide plastic deformation region. This indicates that the dislocation observed here is not the primary reason for the plastic deformation in MAPbI3. The underlying reason lies in the existence of strong Coulombic interactions and nanovoid expansion during plastic deformation. In order to testify this hypothesis, some simulation snapshots (Figure S3) are shown at various stages of plastic deformation. We found that the ductile behavior of MAPbI3 is mainly attributed to the enlarging of individual nanovoids and the coalescence of adjacent nanovoids. Figure S3 shows the process of nanovoid expansion and coalescence at different strain values. Ionic components in MAPbI3 are rearranged to form the nanovoids after the initial brittle fracture process. The two red ovals in Figure S3(a) highlight two representative nanovoids adjacent to each other at a strain of 18.4%. The two adjacent nanovoids are gradually enlarged (growing) near the crack tip when the strain is increased. Then, only ionic chains are present to connect the two sides of the material (necking) and separate the two enlarged nanovoids when the strain reaches 22.8% in Figure S3(b). The adjacent nanovoids finally merge under higher strain values, with the ionic chains completely broken (see Movie S5 in the Supporting Information). The complete breakage of the ionic chains near the crack tip will initiate crack propagation at a strain of 24.4% in Figure S3(c). From a 20 ACS Paragon Plus Environment

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lattice structure point of view, the plastic deformation in MAPbI3 is associated with the distortion and splitting of the building-block PbI6 octahedra, as shown in Figure S3(d)-(e). The splitting of PbI6 octahedra to PbI4 tetrahedra contributes to the irreversible crack propagation during tensile loading in Figure S3(f)-(g) (for more details, see Figure S4 in the Supporting Information). It has been reported that the initial failure in a perfect crystal is induced by atomistic disorders,37, 38 which promotes the formation of nanovoids, similar to those observed in Figure S3(a). The populating, growing, and coalescing of nanovoids with similar sizes observed here finally lead to ionic chain breakage in MAPbI3, similar to the fracture behavior observed in MD simulations for skutterudite CoSb339. To quantify nanovoid migration or population, the lengths of the simulated region populated with nanovoids as a function of strain are shown in Figure S5. These results indicate that the length of the region populated with nanovoids increases linearly with strain during the brittle fracture process, which is also directly correlated with the linear decrease in the effective stress in Figure 4(a) (see Movie S6 in the Supporting Information). Based on the method developed by Brochard et al.40 and Bauchy et al.,33,

34

we

determined the fracture mechanical properties of MAPbI3 at the nanoscale, without any specific assumptions about the mechanical behavior of the material during fracture. Common methods, such as the numerical and analytical J-integral41, 42 and fictitious crack model of Hillerborg,43 have been used to determine the fracture energy release rate for either brittle or ductile materials under the assumption that the material is homogeneous and continuous in the macroscale.34 Therefore, these methods are not applicable for studying MAPbI3 because it is 21 ACS Paragon Plus Environment

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mechanically anisotropic and possesses unique fracture properties at the nanoscale. The method we used is based on the energy theory of fracture mechanics and uses thermodynamic integration (over the stress-strain curve during crack propagation) to determine the fracture toughness.44 This method has been successfully used to predict the fracture toughness of materials possessing significant plasticity due to nanovoids, such as silicate glasses33 and amorphous calcium silicate hydrate.34 During tensile loading, the internal stresses of a system increase from zero, together with the associated mechanical (strain) energy P. When the crack starts to propagate, the mechanical energy stored is released to create two new surfaces. Base on the energy theory of fracture mechanics, the Helmholtz free energy F can take the form of the mechanical energy P under the NVT ensemble. Therefore, the energy released per unit area (the xy plane in Figure 4) of the propagated crack defines the critical strain energy release rate Gc, an important measure of fracture toughness based on the Griffith’s theory of fracture:40, 45

Gc 

Lx L y A



Lmax z

L0z

 zz dLz

(1)

where ∆A∞ is the total area of the crack created after complete fracture (under a strain of 44.7%  here),  zz is the effective stress along the z direction, Lx (fixed), Ly (fixed) and Lz (varies from

L0z to Lmax z as the strain is increased from 0 to 44.7%) are the sizes of the simulation box in the x, y and z direction, respectively. Another important measure of fracture toughness is KIc, the critical stress intensity factor for the Mode I fracture that is commonly used in engineering applications. KIc denotes the ratio 22 ACS Paragon Plus Environment

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between the critical stress at the crack tip right before crack propagation and the external loading stress far away from the crack tip. KIc can be estimated from Gc based on the Irwin formula for the plane strain loading condition considered here and under the assumption of mechanical isotropy:46 1/ 2

GE  K Ic   c 2   1  

(2)

where E = 14.04 GPa is the Young’s modulus predicted by MD for the defective MAPbI3 model, ν = 0.31 is the Poisson’s ratio for the cubic-phase MAPbI3 predicted by MD and given in Table 1. The critical strain energy release rate Gc is calculated based on Eq. 1 by integrating the entire stress-strain curve in Figure 4(a), which leads to Gc = 0.94 J/m2, and subsequently based on Eq. 2, KIc = 0.12 MPa·m1/2. The fracture toughness observed in MAPbI3 is quite low compared to traditional inorganic perovskites such as LaCoO347 (0.73 ~ 0.98 MPa·m1/2) and BaTiO348, 49 (0.68 ~ 1.0 MPa·m1/2). Combined with its low elastic modulus, MAPbI3 is both soft and easier to fracture (from a stress point of view), but exhibits plasticity and ductility (flexible from a strain point of view) at the nanoscale which is absent in most inorganic perovskites. To quantify the degree of ductility of MAPbI3, we decomposed Gc into:33

Gc  Ge  G d

(3)

where Ge is the reversible, elastic contribution to the fracture energy release rate in the linear elastic stage, Gd is the irreversible, dissipated energy release rate during plastic deformation and crack propagation. The value of Gd for a perfectly brittle material should be equal to zero. We 23 ACS Paragon Plus Environment

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used the brittleness parameter, B= Ge/Gc, to quantify ductility, which is larger for more brittle materials. The computed value of B for MAPbI3 is 0.20, suggesting that it is even more ductile than amorphous glasses, such as sodium silicate (B = 0.73) and calcium aluminosilicate (B = 0.38).33 To further understand the unexpected ductility of MAPbI3, we evaluated the total surface energy of typical charge-neutral [MAI]0 and [PbI2]0 free surfaces along the (001) plane, which gives γ = γMAI + γPbI2 = 0.19 J/m2 (see Methods). The total surface energy, γ, should be very close to Ge = 0.188 J/m2 (determined from the linear elastic deformation stage as in Eq. 3) in order to initiate crack propagation, which is indeed the case here, which again validates our predicted fracture properties of MAPbI3. However, such brittle fracture through the creation of free surfaces doesn’t proceed for long (for 4.4 ~ 8.1% strains) before the plastic deformation begins. Here the much smaller value of Ge compared to that of Gc (i.e., very small B value) indicates more energy dissipation during the plastic deformation and fracture stage than that during the elastic stage, which leads to the ultimate large ductility. Notably, the underlying mechanism behind such trend (Ge Dislocations in Strontium Titanate. Acta Mater. 2012, 60, 329-338. 39 ACS Paragon Plus Environment

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Graphical Abstract


Probing the Soft and Nanoductile Mechanical Nature of Single and

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