Atomistic investigation into the mechanical behaviour of crystalline and amorphous TiO₂ nanotubes

Abstract

Titanium dioxide (TiO₂) nanotubes are appealing to research communities due to their excellent functional properties. However, there is still a lack of understanding of their mechanical properties. In this work, we conduct molecular dynamics (MD) simulations to investigate the mechanical behaviour of rutile and amorphous TiO₂ nanotubes. The results indicate that the rutile TiO₂ nanotube has a much higher Young's modulus (∼800 GPa) than the amorphous one (∼400 GPa). Under tensile loading, rutile nanotubes fail in the form of brittle fracture but significant ductility (up to 30%) has been observed in amorphous nanotubes. This is attributed to a unique ‘repairing’ mechanism via bond reconstruction at under-coordinated sites as well as bond conversion at over-coordinated sites. In addition, it is observed that the fracture strength of rutile nanotubes is strongly dependent on their free surfaces. These findings are considered to be useful for development of TiO₂ nanostructures with improved mechanical properties.

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1. Introduction

TiO₂ is a multifunctional semiconductor material with a large refractive index,¹ high energy conversion efficiency^2–5 and high photocatalytic activity.^6–9 Nanosized TiO₂ has also demonstrated unique physical and chemical properties. For example, owing to the quantum-size effect,¹⁰ the energy gap between the conduction and valence bands of TiO₂ nanoparticles exhibits a blue shift,¹¹ as compared to bulk TiO₂.¹² Different opto-electronic properties have also been observed for TiO₂ nanotubes.¹³ TiO₂ with distinctive nanostructures were synthesized by various methods.^14–16 Recently, TiO₂ nanotube arrays have demonstrated great potential for applications in sensors,¹⁷ dye-sensitized solar cells,^18,19 drug delivery and tissue engineering.^20–23 To take full advantage of the superior electrical and biomedical properties of TiO₂ nanotubes, it is critical to maintain the mechanical integrity in relevant structures and devices. A better understanding of the mechanical performance of TiO₂ nanotubes is essential. Recently, nanoindentation technique has been adapted to explore the mechanical behaviour of TiO₂ nanotube arrays.^24,25 Significant densification and fracture of nanotubes were observed in these experiments. Hirakata et al. proposed a stress-based fracture model to estimate the fracture strength of TiO₂ nanotube arrays.²⁶

For individual TiO₂ nanotubes, owing to nanoscale dimensions, it is still a challenge to experimentally evaluate their mechanical properties, and corresponding deformation and failure mechanisms. On the other hand, atomistic simulations provide an alternative approach. In this work, we conducted MD simulations to investigate the mechanical properties (i.e. Young's modulus, fracture strength and yield strength), and deformation and failure mechanisms of individual TiO₂ nanotubes. The effects of crystal structures and geometrical parameters on the mechanical properties of single TiO₂ nanotube were also discussed.

2. Simulation methods

TiO₂-nanotube models with different geometrical parameters (i.e. inner radius and wall thickness) and microstructures (i.e. rutile crystal and amorphous) were built by seamlessly wrapping TiO₂ thin films in a self-programmed code, as shown in Fig. 1. Such code implements the spatial transformation method to convert thin films into nanotubes. The axis of rutile nanotube is aligned with [001] direction. The geometrical parameters of these TiO₂-nanotube models are listed in Table 1. It is found that there are minor differences between the thicknesses and inner radii of thin films before wrapping and corresponding nanotubes. The nanotube lengths in all simulation models were chosen as 30 nm. Periodic boundary conditions were applied to all simulation models along the length direction.

Fig. 1 (a) Schematics of TiO₂ nanotube wrapped from TiO₂ thin film; (b) TiO₂ nanotube with rutile structure and (c) TiO₂ nanotube with amorphous structure. (c) also illustrates nanotube inner radius R_in and wall thickness h.

Table 1 Geometrical parameters of TiO₂ nanotubes with rutile and amorphous structures

Rutile	Inner radius Rin (Å)	Thickness h (Å)	Amorphous	Inner radius Rin (Å)	Thickness h (Å)
No. 1	7.6756	8.56510	No. 1	7.27395	10.07025
No. 2	7.4366	13.6460	No. 2	7.19950	14.7030
No. 3	7.2829	18.5438	No. 3	6.88030	19.7068
No. 4	26.575	13.7300	No. 4	25.3300	14.7000
No. 5	37.475	13.1100	No. 5	36.6850	14.3800
No. 6	48.420	13.2900	No. 6	47.4650	14.1600

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The amorphous structure in the TiO₂ models was achieved via a melt-and-quench procedure, which is very popular for formation of amorphous solids.^27,28,32 Firstly, rutile TiO₂ thin films were melted from 300 K to 10 000 K under the Nosé–Hoover NVT ensemble.^29,30 After reaching to 10 000 K, the Nosé–Hoover NPT ensemble^29,30 with a constant pressure of 1 bar was performed to further equilibrate the system. Then, rapid quenching at a rate of 4.85 × 10¹³ K s⁻¹ was applied to the TiO₂ thin films under the Nosé–Hoover NVT ensemble until 300 K. The finally obtained simulation models, which are equilibrated under the Nosé–Hoover NPT ensemble at 300 K, indicate that atomistic structures of amorphous TiO₂ thin films are approximately independent of quench rates in the range of ∼10¹² to 10¹⁴ K s⁻¹. Hence, only one quench rate was considered here.

Matsui and Akaogi proposed a pairwise potential (MA potential), which is derived from typical Born–Huggins–Meyer (BHM) potential,³¹ for the simulations of TiO₂ materials.³³ Numerous computational studies have demonstrated that MA potential can accurately describe geometries and properties of TiO₂ crystal (i.e. rutile, anatase and brookite) as well as amorphous TiO₂ structures.^33–36 It has also been applied to simulations of both nanophase solid TiO₂ (ref. 37) and nanophase liquid TiO₂.³⁸ Comparing different potentials for TiO₂ simulation, Collins et al.³⁹ pointed out that MA potential is the most suitable one for TiO₂ simulation. The MA potential is also a type of ionic model, it includes Buckingham interaction (first and second terms in eqn (1)) and pairwise coulombic interaction (third term in eqn (1)), which can be expressed in the following formula:^33,37,38

where r_ij is the distance between ions i and j with their charges q_i and q_j, respectively; A_i, B_i and C_i are fitting parameters, and depend on Ti or O ion; and f is a standard force of 4.184 kJ Å⁻¹ mol⁻¹. Compared to typical BHM potential, an attractive term −D_iD_j/r_ij⁸ (D_i and D_j are fitting parameters) is excluded in MA potential. All the potential parameters are adapted from previous simulation studies,^33,37,38 and are listed in Table 2. Ti and O ions are assigned with constant partial charges of q_Ti = +2.196 and q_O = −1.098, respectively. In our MD simulations, the ratio of Ti and O atom numbers was exactly 1 : 2 to keep all models' charge neutral. In this work, the cut-off distances for both Buckingham and pairwise coulombic interactions were chosen as 8.0 Å.

Table 2 Potential parameters of MA potential

Ion	A (Å)	B (Å)	C (Å^3 kJ^1/2 mol^−1/2)
Ti	1.1823	0.077	22.5
O	1.6339	0.117	54.0

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All MD simulations were carried out using the large-scale atomic/molecular massively parallel simulator (LAMMPS)⁴⁰ with the MA potential, which has been used for simulating mechanical properties of nanomaterials in our previous works.^41–43 TiO₂-nanotube models were firstly relaxed to a minimum energy state using the conjugate gradient energy minimization method until the maximum atomic force was brought below 10⁻⁸ eV Å⁻¹. Tensile loadings were then carefully applied by stretching nanotube models along their longitudinal directions at a strain rate of 0.1% ps⁻¹ and with a time step of 0.5 fs. The Nosé–Hoover NVT ensemble was performed to maintain nanotube models at 300 K. Since the dimensions of all nanotube models are large enough to satisfy the thermodynamic limit, the virial theorem^44,45 can be used to evaluate the atomic stress and determine stress–strain (σ–ε) curves under tensile loadings. Specifically, the atomic stress can be expressed as:

where σ^α_ij is the virial stress of atom α; Ω^α is atomic volume of atom α; m^α is atomic mass of atom α; v^α_i and v^α_j are velocities of atom α along i and j directions; r_αβ^j is the distance between α and β atoms along i direction; f_αβ^j is the force applied by atom β on atom α along i direction. All atomistic structures and deformation of TiO₂ nanotubes were visualized by the open-source ATOMEYE package.⁴⁶

3. Results and discussions

3.1. Structural characteristics of crystalline and amorphous TiO₂ nanotubes

The structural characteristics of rutile and amorphous TiO₂ nanotubes obtained at 300 K have been compared with bulk rutile TiO₂ at the same temperature. Here, the bulk rutile TiO₂ model was built with the dimensions of 10 nm × 10 nm × 10 nm and applied with periodic boundary conditions along all directions. As observed in radial distribution functions (RDFs) (Fig. 2), bulk rutile has two obvious sharp peaks within the bond cut-off radius of R_Ti–O = 3.0 Å.³⁴ These two peaks correspond to two different Ti–O bond lengths (1.9486 Å and 1.9802 Å) in rutile TiO₂, in agreement with the experiment.⁴⁷ In Fig. 2(a), the RDFs of rutile nanotubes have more small sharp peaks compared with that of bulk rutile TiO₂, within the cut-off radius of R_Ti–O = 3.0 Å. Here the peak sharpness can be attributed to the crystalline arrangement of rutile nanotubes. And more sharp peaks result from a large number of under-coordinated Ti and O atoms on the existing free surfaces of nanotubes, which have different Ti–O bond lengths from those inside nanotubes. The RDFs of rutile nanotubes are also insensitive to the nanotube wall thickness and inner radius, indicating little variation of bond length distribution.

Fig. 2 Radial distribution functions (RDFs) g(r_Ti–O) of Ti–O pairs in (a) rutile bulk (ru-bulk) and rutile nanotubes (ru-no. 1–6), and (b) amorphous bulk (a-bulk) and amorphous nanotubes (a-ru-no. 1–6).

In contrast, smooth RDF peaks are observed in amorphous TiO₂ (bulk and nanotubes), as shown in Fig. 2(b). Generally, it is difficult to estimate the accurate atoms distributions in amorphous material. Petkov et al.⁴⁸ investigated the atomic structure of amorphous TiO₂ by electron and X-ray diffraction. They found that the first smooth peak in RDFs of amorphous TiO₂ centered at around 1.96 Å. Our simulation also indicates the first smooth peak in RDFs located around 1.96 Å, confirming the validation and accuracy of current models. Smooth peaks observed in RDFs are common for amorphous materials.^49–53 The smooth peaks for amorphous nanotubes can be well explained by the disordered reconstruction of both inner and surface Ti–O bonds during amorphization, which have wide-range bond length distribution. In addition, Fig. 2(b) also indicates the RDFs of amorphous nanotubes are independent of nanotube sizes.

We also investigated the coordination number distributions of both Ti and O atoms in rutile and amorphous TiO₂ nanotubes, k_Ti–O (Ti as the central atom) and k_O–Ti (O as the central atom). As shown in Fig. 3, only coordination number k_O–Ti = 3 and k_Ti–O = 6 exist in rutile bulk TiO₂, which is in accordance with Ti₃O₆ unit cell of rutile. In rutile nanotubes, the fractions of k_O–Ti = 3 and k_Ti–O = 6 decrease and the fractions of under-coordination numbers such as k_O–Ti = 1, 2 and k_Ti–O = 3 become non-zero. This is due to that the structure of rutile nanotube gets truncated by the free surfaces, causing many under-coordinated atoms on the free surfaces. To achieve energy minimization, these under-coordinated Ti or O atoms tend to interact with nearby O or Ti atoms, causing distortion and reconstruction of surface microstructure. In amorphous TiO₂ nanotube, the fractions of k_O–Ti = 3 and k_Ti–O = 6 decrease more as compared to rutile nanotube. This is because that a number of under-coordinated and over-coordinated atoms are introduced to the amorphous TiO₂ nanotubes, resulting from the amorphization process. The corresponding non-zero coordination numbers (e.g. k_O–Ti = 2, 4 and k_Ti–O = 5, 7) are shown in Fig. 3. The mean coordination numbers k̄_Ti–O = 5.89 and k̄_Ti–O = 2.88 are in good agreement with those in previous results.^48,52

Fig. 3 Coordination number distribution in rutile bulk (ru-bulk) and nanotubes (ru-2, 4, 6), and amorphous TiO₂ nanotubes (a-2, 4, 6) with (a) Ti atom as the center atom (Ti–O) and (b) O atom as the center atom (O–Ti) at 300 K.

3.2. Mechanical properties of crystalline and amorphous TiO₂ nanotubes

To understand the mechanical performance of rutile and amorphous TiO₂ nanotubes, σ–ε curves under tensile loadings are analyzed in Fig. 4. From these σ–ε curves, we can see approximately linear behavior when the strain is in the range of 0–3%, where the slopes denote the Young's moduli of TiO₂ nanotubes. For rutile nanotubes (Fig. 4(a)), their Young's moduli are insensitive to inner radius and thickness (about 800 GPa). The modulus of rutile TiO₂ nanowires estimated by MD simulation is about ∼688 GPa.⁵⁴ Normally, the significant free-surface effect due to large surface-to-volume ratio can lead to much higher Young's moduli of nanostructured materials than those of bulk counterparts.^55,56 Hence rutile TiO₂ nanotubes possess larger surface-to-volume ratios due to the existence of inner wall surface, thereby having a higher Young's moduli than those of rutile TiO₂ nanowires. For all amorphous TiO₂ nanotubes (Fig. 4(b)), their Young's moduli (approximate 400 GPa) is independent of inner radius and thickness, only half value of the rutile nanotubes. The decrease of Young's modulus is associated with the structural disorder introduced during the amorphization process, which has also been observed in other materials.^57,58 Under the tensile loading up to the critical (yield or fracture) strain, the stress distribution in amorphous nanotubes is more non-uniform than that in rutile ones. This means amorphous nanotubes contain more Ti–O bonds which cannot efficiently bear external loading, and thus possesses lower Young's moduli than rutile ones.

Fig. 4 Stress–strain (σ–ε) curves of (a) rutile and (b) amorphous TiO₂ nanotubes at 300 K.

Considering the structure evolution during loading, rutile nanotubes generate reversible elastic deformation within a small strain range and no structural failure can be observed, as shown in Fig. 5(a). Then, a drop of stress σ occurs, triggered by crack initiation in nanotube walls, as shown in Fig. 5(b). Obviously, the fracture strength of rutile nanotubes, namely the maximum stresses in σ–ε curves (Fig. 4(a)), is determined by the elastic limit before the crack initiation. The crack propagates across the cross-section area of the nanotubes, as shown in Fig. 5(c). After reaching fracture strength, the nanotube starts to lose the load capacity until the complete failure, Fig. 5(d). As shown in Table 3, the fracture strength of rutile nanotubes is in the range of σ_f = 28.55–62.58 GPa, which are greater than those of rutile nanowires (∼10 GPa).⁵⁴ Remarkably, except for rutile nanotube no. 2, increase of the average inner and outer radii (see Table 3) results in the enhancement of fracture strength. This trend unveils the possible strong free-surface effect on fracture strengths of rutile nanotubes. The atoms near the surface of the crystalline nanotube reside in a local environment different from the interior, which can result in the non-zero surface stress.⁵⁹ With increasing the average radius, such surface stress is reduced, leading to increased fracture strength.⁶⁰ Similar free-surface effect also appears in metal nanowires, where non-zero tensile surface stress can enhance their tensile yield stress.^61,62

Fig. 5 Snap shots of tensile deformation of rutile TiO₂ nanotube no. 2 from (a) initially elastic deformation, (b) crack formation, (c) crack propagation, to (d) total failure under the tensile strain range of 2–15%.

Table 3 Fracture strength (or yield stress) of TiO₂ nanotubes

(GPa)	No. 1	No. 2	No. 3	No. 4	No. 5	No. 6
Rutile	40.95	28.55	36.63	45.34	54.37	62.58
Amorphous	18.60	18.91	18.18	18.65	19.28	20.15

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As shown in Fig. 4(b), all amorphous TiO₂ nanotubes deform elastically within the strain range of 0–10%. Different from σ–ε curves of rutile nanotubes, however, significant ductility (about 30%) is observed in their σ–ε curves after the yielding. The corresponding yield stress σ_y and ε_y are around 19 GPa and 10%, respectively. Similar to the Young's modulus, the yield stresses of amorphous nanotubes are lower than fracture strengths of rutile ones. During yielding no obvious cracks can be observed in Fig. 6(b). As shown in Fig. 6(c), the nanotube starts to experience necking at about 30% strain. This is followed by final failure via shear deformation, corresponding to a failure strain of about 45%, Fig. 6(d). As compared to rutile type nanotubes, these amorphous nanotubes demonstrate significant ductility. Note that the fracture strength and yield stress obtained from MD simulations are generally higher than those from experiments.⁶³ This is attributed to the large number of structural defects in TiO₂ nanotubes formed during synthesis process.

Fig. 6 Snap shots of tensile deformation of amorphous TiO₂ nanotube no. 2 under the tensile strain of (a) 2%, (b) 12%, (c) 30% and (d) 50%.

3.3. Deformation mechanisms of TiO₂ nanotubes

3.3.1 Rutile TiO₂ nanotubes. Fig. 7 shows the atomic structure evolution of a rutile nanotube under tensile loading. During initial elastic deformation, Ti–O bonds are stretched and rotated without bond breaking. TiO₆ octahedrons, which is the basic building block of TiO₂ are regularly arranged. When the stress increases to the fracture strength (point A), some defects (nanovoids) are initiated via bond breakings, as observed in the atomic structure from inset A in Fig. 7(b). The arrangement of TiO₆ octahedrons becomes disordered. Nanovoids increase with loading, as shown in the inset B and inset C in Fig. 7(b). Then, continuous bond breaking leads to the final failure, shown in inset D in Fig. 7(b). Similar fracture behaviour has been observed in crystalline brittle materials such as graphene and carbon nanotube. With increasing the tensile loading, the C–C bond breaking triggers crack propagation and fracture.^63,64 The structural defects inside may act as the sites of stress concentration and initiate the bond breaking.

Fig. 7 (a) Stress–strain (σ–ε) curve of rutile nanotube no. 6 and (b) bond breaking under tensile loading (insets A–D correspond to points A–D in the σ–ε curve).

3.3.2 Amorphous TiO₂ nanotube. Fig. 8 shows the microstructural evolution of amorphous TiO₂ nanotubes under tensile loading. Different from rutile nanotubes, TiO₆ octahedrons are randomly arranged within the nanotubes owing to the amorphous nature. When subjected to tensile loading, the Ti–O bonds experience stretching and rotation, and start to break (inset A in Fig. 8(b)), accompanied by stress drop from the peak in Fig. 8(a). It is interesting to note the significant plastic behaviour in the stress–strain curve. In contrast to the rutile TiO₂ nanotube, as shown in the insets B and C in Fig. 8(c), nanovoids are initiated at different locations which are considered to be pre-existing defects (e.g. under- and over-coordinated sites) and newly-formed defects (bond-broken sites). The coalescence of nanovoids (inset D and E in Fig. 8(c)) results in the final failure. Therefore, the under- and over-coordinated Ti sites play an important role in the deformation and failure mechanism of amorphous nanotubes.

Fig. 8 (a) Stress–strain (σ–ε) curve of amorphous TiO₂ nanotube no. 6, and (b) evolution of atomic structure under tensile loading (insets A–E correspond to points A–E in the σ–ε curve).

Note that some newly-formed nanovoids disappear during the tensile loading. For example, those nanovoids shown in inset A in Fig. 8(c) disappear but other nanovoids emerge at different locations, inset B Fig. 8(c). In other words, these nanovoids can be repaired during deformation. To understand this repairing mechanism, the evolution of Ti–O bonds with loading is examined, as illustrated in Fig. 9. In a localized area (point G, highlighted by the blue circle), the T–O bond shown in Fig. 9(a) is broken under tensile loading, Fig. 9(b). This is also accompanied by the formation of new bonds, shown in green in Fig. 9(c). The nanovoid at point G can be repaired via the formation of new bonds, as shown in Fig. 9(d).

Fig. 9 (a) Localized Ti–O bonds (highlighted by the blue circle), (b) bond breaking under loading, (c) new bonds formation (highlighted in green), and (d) repaired bonds.

The under-coordinated atoms in the amorphous TiO₂ nanotube may contribute to such repairing mechanism. Randomly organized Ti–O bonds in the amorphous nanotube are stretched, rotated or broken under loading. On the other hand, the under-coordinated O atoms or surface dangling O atoms are inclined to react with nearby under-coordinated Ti atoms, leading to the formation of new bonds. The breaking or forming of bonds will change the local structure and atom distribution, as well as the local stress and strain distribution. For example, if a bond gets broken, this will release the tensile stress to the section close to it. If this happens near a nanovoid, it will cause the shrinkage of the nanovoid. This can benefit the formation of new Ti–O bonds across the nanovoid and the repairing of it. The competition between the breaking and formation of new bonds may delay the formation of nanovoids with critical sizes required for brittle fracture, contributing to the ductility observed in the σ–ε curves. With the increase of tensile loading, more nanovoids are formed but less are repaired in amorphous nanotubes. In such case, the coalescence of neighboring nanovoids starts, and the stress begins to drop gradually (Fig. 8(b)). When the tensile loading further rising, the coalescence of a series of nanovoids occurs (inset D in Fig. 8(c)), which leads to the total failure of amorphous nanotubes, shown in inset E in Fig. 8(c).

The over-coordinated (such as 7-fold) Ti sites also attribute to the ductility observed in the amorphous nanotubes. As shown in Fig. 10, the over-coordinated sites can be converted into the regular-coordinated (6-fold) Ti sites during tensile deformation, which can create new bonds, increase strain energy dissipation and ductility beyond the elastic limit.⁶⁵ Since a small fraction of over-coordinated Ti sites exist in the amorphous structure, the breaking of excessive Ti–O bonds at those 7-fold coordination sites enables the plastic deformation through such strain energy dissipation process. Similar phenomena have been observed at over-coordinated Si sites in amorphous silica glass.^65–67 Therefore, although the disordered structure of amorphous TiO₂ nanotube lowers the Young's modulus, its under- and over-coordinated atomic structure contribute to the ‘repairing’ and ‘conversion’ mechanism, which promotes new bond formation and breaking, and associated strain energy dissipation and ductility under loading.

Fig. 10 (a) Atomic configuration of the amorphous TiO₂ nanotube under tensile loading, (b) over-coordinated (7-fold) site, and (c) regular-coordinated (6-fold) site. Solid blue lines represent the existing Ti–O bonds, and dashed blue line represents the broken Ti–O bond.

4. Conclusions

In present work, we have conducted MD simulations to investigate the mechanical behaviour of both rutile and amorphous TiO₂ nanotubes, with a focus on the effects of geometric parameters and crystal structure. The results demonstrate that rutile nanotubes have higher Young's modulus and fracture strength than those of amorphous TiO₂ nanotubes. This can be attributed to the regular crystalline structure in the rutile TiO₂ nanotubes. However, amorphous TiO₂ nanotubes surprisingly exhibit greater ductility via a ‘repairing’ mechanism. In other words, the external loading can effectively promote bond reconstructions at under-coordinated Ti sites and strain energy dissipation at over-coordinated Ti sites, increasing the overall ductility. Furthermore, we find that fracture strengths of rutile TiO₂ nanotubes strongly depend on their free surfaces. The decrease of induced surface stress can render the enhancement of their fracture strength.

Acknowledgements

This research was supported by the Australian Research Council Discovery Project (DP150101717). Y. N. Xu acknowledges the financial support from the China Scholarship Council and a Top-Up scholarship from Queensland University of Technology.

References

1. J. H. Braun, A. Baidins and R. E. Marganski, Prog. Org. Coat., 1992, 20, 105–138.
2. U. Bach, D. Lupo, P. Comte, J. E. Moser, F. Weissortel, J. Salbeck, H. Spreitzer and M. Gratzel, Nature, 1998, 395, 583–585.
3. L. Etgar, P. Gao, Z. Xue, Q. Peng, A. K. Chandiran, B. Liu, M. K. Nazeeruddin and M. Grätzel, J. Am. Chem. Soc., 2012, 134, 17396–17399.
4. E. J. W. Crossland, N. Noel, V. Sivaram, T. Leijtens, J. A. Alexander-Webber and H. J. Snaith, Nature, 2013, 495, 215–219.
5. J. T.-W. Wang, J. M. Ball, E. M. Barea, A. Abate, J. A. Alexander-Webber, J. Huang, M. Saliba, I. Mora-Sero, J. Bisquert, H. J. Snaith and R. J. Nicholas, Nano Lett., 2014, 14, 724–730.
6. J. Wang, D. N. Tafen, J. P. Lewis, Z. Hong, A. Manivannan, M. Zhi, M. Li and N. Wu, J. Am. Chem. Soc., 2009, 131, 12290–12297.
7. S. N. Habisreutinger, L. Schmidt-Mende and J. K. Stolarczyk, Angew. Chem., Int. Ed., 2013, 52, 7372–7408.
8. J. Yu, J. Low, W. Xiao, P. Zhou and M. Jaroniec, J. Am. Chem. Soc., 2014, 136, 8839–8842.
9. B. Qiu, M. Xing and J. Zhang, J. Am. Chem. Soc., 2014, 136, 5852–5855.
10. A. J. Nozik and R. Memming, J. Phys. Chem., 1996, 100, 13061–13078.
11. N. Satoh, T. Nakashima, K. Kamikura and K. Yamamoto, Nat. Nanotechnol., 2008, 3, 106–111.
12. N. Sakai, Y. Ebina, K. Takada and T. Sasaki, J. Am. Chem. Soc., 2004, 126, 5851–5858.
13. J. Yan, S. Feng, H. Lu, J. Wang, J. Zheng, J. Zhao, L. Li and Z. Zhu, Mater. Sci. Eng., B, 2010, 172, 114–120.
14. M. Liu, H. Wang, C. Yan, G. Will and J. Bell, Appl. Phys. Lett., 2011, 98, 133113.
15. D. Regonini, C. Bowen, A. Jaroenworaluck and R. Stevens, Mater. Sci. Eng., R, 2013, 74, 377–406.
16. M. Liu, H. Wang, C. Yan and J. Bell, Mater. Focus, 2012, 1, 136–141.
17. J. Tang, B. Kong, Y. Wang, M. Xu, Y. Wang, H. Wu and G. Zheng, Nano Lett., 2013, 13, 5350–5354.
18. W. Guo, X. Xue, S. Wang, C. Lin and Z. L. Wang, Nano Lett., 2012, 12, 2520–2523.
19. M. Ye, D. Zheng, M. Lv, C. Chen, C. Lin and Z. Lin, Adv. Mater., 2013, 25, 3039–3044.
20. J. Lu, H. Li, D. Cui, Y. Zhang and S. Liu, Anal. Chem., 2014, 86, 8003–8009.
21. Y.-Y. Song, F. Schmidt-Stein, S. Bauer and P. Schmuki, J. Am. Chem. Soc., 2009, 131, 4230–4232.
22. J. Park, S. Bauer, A. Pittrof, M. S. Killian, P. Schmuki and K. von der Mark, Small, 2012, 8, 98–107.
23. K. Gulati, S. Ramakrishnan, M. S. Aw, G. J. Atkins, D. M. Findlay and D. Losic, Acta Biomater., 2012, 8, 449–456.
24. Y. N. Xu, M. N. Liu, M. C. Wang, A. Oloyede, J. M. Bell and C. Yan, J. Appl. Phys., 2015, 118, 145301.
25. G. A. Crawford, N. Chawla, K. Das, S. Bose and A. Bandyopadhyay, Acta Biomater., 2007, 3, 359–367.
26. H. Hirakata, K. Ito, A. Yonezu, H. Tsuchiya, S. Fujimoto and K. Minoshima, Acta Mater., 2010, 58, 4956–4967.
27. F.-L. Arthur, B. Etienne, A. Tristan, M. Samy, L. David and T. Konstantinos, J. Phys.: Condens. Matter, 2014, 26, 055011.
28. Y. Qi, T. Çağın, Y. Kimura and W. A. Goddard, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 3527–3533.
29. S. Nosé, J. Chem. Phys., 1984, 81, 511–519.
30. W. G. Hoover, Phys. Rev. A, 1985, 31, 1695–1697.
31. M. P. Allen and D. J. Tildesley, Computer simulation of liquids, Oxford university press, 1989.
32. M. D. Kluge, J. R. Ray and A. Rahman, Phys. Rev. B: Condens. Matter Mater. Phys., 1987, 36, 4234–4237.
33. M. Matsui and M. Akaogi, Mol. Simul., 1991, 6, 239–244.
34. H. Vo Van, Nanotechnology, 2008, 19, 105706.
35. K. h. Thomas, T. Marcus, E. Henrik, L. Marc, R. Detlev and F. Thomas, J. Phys. D: Appl. Phys., 2013, 46, 325302.
36. C. F. Valeria, F. A. N. Christian, M. B. n. Oviedo, P. B. Franco, Y. O. Fabiana and G. S. n. Cristin, J. Phys.: Condens. Matter, 2013, 25, 115304.
37. V. N. Koparde and P. T. Cummings, J. Phys. Chem. B, 2005, 109, 24280–24287.
38. V. Hoang, H. Zung and N. H. B. Trong, Eur. Phys. J. D, 2007, 44, 515–524.
39. D. Collins and W. Smith, Council for the Central Laboratory of Research Councils, Daresbury Research Report DL-TR-96–001, 1996.
40. S. Plimpton, J. Comput. Phys., 1995, 117, 1–19.
41. M. C. Wang, C. Yan, L. Ma, N. Hu and M. W. Chen, Comput. Mater. Sci., 2012, 54, 236–239.
42. M. C. Wang and C. Yan, Sci. Adv. Mater., 2014, 6, 1501–1505.
43. M. C. Wang, Z. B. Lai, D. Galpaya, C. Yan, N. Hu and L. M. Zhou, J. Appl. Phys., 2014, 115, 123520.
44. M. Zhou, Proc. R. Soc. London, Ser. A, 2003, 459, 2347–2392.
45. D. H. Tsai, J. Chem. Phys., 1979, 70, 1375–1382.
46. L. Ju, Modell. Simul. Mater. Sci. Eng., 2003, 11, 173–177.
47. D. T. Cromer and K. Herrington, J. Am. Chem. Soc., 1955, 77, 4708–4709.
48. V. Petkov, G. Holzhüter, U. Tröge, T. Gerber and B. Himmel, J. Non-Cryst. Solids, 1998, 231, 17–30.
49. M. J. Demkowicz and A. S. Argon, Phys. Rev. B: Condens. Matter Mater. Phys., 2005, 72, 245206.
50. A. Kerrache, N. Mousseau and L. J. Lewis, Phys. Rev. B: Condens. Matter Mater. Phys., 2011, 84, 014110.
51. A. J. Kulkarni, M. Zhou and F. J. Ke, Nanotechnology, 2005, 16, 2749.
52. V. Van Hoang, Phys. Status Solidi B, 2007, 244, 1280–1287.
53. M. M. J. Treacy and K. B. Borisenko, Science, 2012, 335, 950–953.
54. L. Dai, C. Sow, C. Lim, W. Cheong and V. Tan, Nano Lett., 2009, 9, 576–582.
55. J. Diao, K. Gall and M. L. Dunn, J. Mech. Phys. Solids, 2004, 52, 1935–1962.
56. C. Q. Chen, Y. Shi, Y. S. Zhang, J. Zhu and Y. J. Yan, Phys. Rev. Lett., 2006, 96, 075505.
57. K. Xue and L.-S. Niu, J. Appl. Phys., 2009, 106, 083505.
58. K. Xue and L.-S. Niu, J. Appl. Phys., 2010, 107, 083517.
59. G.-F. Wang and X.-Q. Feng, Appl. Phys. Lett., 2009, 94, 141913.
60. X.-Y. Sun, Y. Xu, G.-F. Wang, Y. Gu and X.-Q. Feng, Phys. Lett. A, 2015, 379, 1893–1897.
61. K. Gall, J. Diao and M. L. Dunn, Nano Lett., 2004, 4, 2431–2436.
62. Z. Yang, Z. Lu and Y.-P. Zhao, Comput. Mater. Sci., 2009, 46, 142–150.
63. M. C. Wang, C. Yan, D. Galpaya, Z. B. Lai, L. Ma, N. Hu, Q. Yuan, R. X. Bai and L. M. Zhou, J. Nano Res., 2013, 23, 43–49.
64. T. Dumitrica, M. Hua and B. I. Yakobson, Proc. Natl. Acad. Sci. U. S. A., 2006, 103, 6105–6109.
65. F. Yuan and L. Huang, Sci. Rep., 2014, 4, 5035.
66. Y.-C. Chen, Z. Lu, K.-I. Nomura, W. Wang, R. K. Kalia, A. Nakano and P. Vashishta, Phys. Rev. Lett., 2007, 99, 155506.
67. K.-I. Nomura, Y.-C. Chen, R. K. Kalia, A. Nakano and P. Vashishta, Appl. Phys. Lett., 2011, 99, 111906.

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