Dynamics of Fracture in Silica and Soda-Silicate Glasses: From Bulk

Oct 21, 2015 - *E-mail: [email protected]. ... Glasses containing nanovoids and atomic defects (such as under- and overcoordinated silicon and...
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Dynamics of Fracture in Silica and Soda-Silicate Glasses: From Bulk Materials to Nanowires Alfonso Pedone,*,† Maria Cristina Menziani,† and Alastair N. Cormack‡ †

Department of Chemical and Geological Sciences, University of Modena and Reggio Emilia, Via G. Campi 103, 41125 Modena, Italy Kazuo Inamori School of Engineering, New York State College of Ceramics, Alfred University, Alfred, New York 14802, United States

‡

ABSTRACT: Classical molecular dynamics simulations are used to investigate the fracture mechanism, intrinsic strength, strain at failure and elastic modulus of silica and soda-silicate bulk glasses and nanowires. The latter have been generated by using a new casting approach described in this paper for the first time. The results show that large systems have to be used to reproduce the brittle fracture mechanism of silicate glasses; the appropriate dimensions of the simulation boxes depend on the glass composition. Whereas for silica glass an ideal brittle fracture is observed with models containing 30k atoms, for soda-silicate glasses models with more than 60k atoms should be used. Glasses containing nanovoids and atomic defects (such as under- and overcoordinated silicon and oxygen atoms) are less brittle than flaw-free bulk glasses. The main finding, shown here for the first time, is that the presence of atomic defects and/or modifier cations allows the material to rearrange its structure and absorb the stresses caused by mechanical deformation, the former by transforming from high energy point defects to more stable configurations and the latter by saturating NBOs formed during the gradual breaking of the Si−O bonds that starts soon after the strain at failure is reached. In general, silica nanowires are characterized by lower mechanical properties with respect to bulk models because of the slightly higher amount of atomic defects (3-fold Si, nonbridging oxygens, and small rings) on their surfaces compared to that found in bulk glasses. These defects are not present in soda-silicate nanowires whose surfaces are rich in sodium ions that compensate the negative charge of nonbridging oxygens.

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INTRODUCTION

For this reason, many previous experimental and theoretical efforts have been devoted to the investigation of the fracture process and strength of oxide glasses. Glasses are considered brittle materials since they fracture instantaneously as soon as the maximum stress that they can sustain is reached. No appreciable deformation is observed, and the fracture surfaces are generally flat and perpendicular to the applied tensile stress.3,4 The practical strength of glasses is several orders of magnitude smaller than the intrinsic theoretical stress (computed on the basis of the bonding energies) because it is controlled by the presence of microscopic cracks or flaws present on their surfaces. These act as stress concentrators giving way to the rapid crack propagation that characterizes the brittle fracture mechanism.5−7 Several researchers have shown that the strength of carefully prepared E-glass and silica glass fibers can reach extremely high values with very small standard deviations when tested in liquid nitrogen.8,9 More recently, silica nanowires with diameters ranging between 60 and 300 nm have been manufactured, and

Silicate glasses play a key role in several technological fields such as medicine, photonics, optics, electronics, telecommunications, clean energy, and waste management. However, despite the great number of properties and functionalities that can be easily conferred to these materials by changing composition and/or processing technologies, their poor mechanical properties have always been considered as the main limitations to application with high levels of tensile stress.1,2 The strength, toughness, and elastic properties of silicate glasses are a major impediment for further advancement in the production of flexible display, solar modules, next-generation touch-screen devices, large-scale and high-altitude architectural glazing, lightweight packaging, ultrastiff composites, highcapacity optical fibers for telecommunications, and versatile building blocks for future micro- and nanoscale photonic circuits and components. Therefore, a better understanding of the fracture mechanisms and intrinsic strength of silicate glasses is of crucial importance for the design of new ultrastrong materials for addressing new social challenges in energy, medicine, and communication systems. © 2015 American Chemical Society

Received: September 4, 2015 Revised: October 14, 2015 Published: October 21, 2015 25499

DOI: 10.1021/acs.jpcc.5b08657 J. Phys. Chem. C 2015, 119, 25499−25507

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The Journal of Physical Chemistry C

a time step of 2 fs. Coulombic interactions were calculated by the Ewald summation method31 with a cutoff of 12 Å and an accuracy of 10−4. The short-range interaction cutoff was set to 5.5 Å. As for the glass nanowires, these were generated by using a casting approach. The parallelepiped glass structures representing the bulk glasses were inserted in carbon nanotubes with lengths equal to the z-side of the simulation box and diameters a few angstroms greater than the diagonal of the quadratic xy face of parallelepiped structures. An exponential repulsive potential (of the type Ae−r/ρ) between the carbon atoms and the glass constituents (O, Si, Na, and Ca ions) was applied, with parameters adjusted to confine the glass inside a cylinder, the diameter of which allows reproducing in the best possible way the densities of the experimental glasses investigated. In particular, for all the systems investigated the A and ρ parameters of each C-ion pair were set to 10 000 eV and 0.8 Å−1. Once the initial structures had been prepared, the systems were melted and cooled to 300 K by using the same protocol described above. In the final step of the protocol, the carbon nanotube was removed, and the glass nanowire structures were relaxed for 60 ps at 300 K by using the Nose−Hoover NPT ensemble.30 A schematic representation of the casting approach is reported in Figure 1.

strengths in excess of 10 GPa were measured for the thinner wires.10 This has led to the conclusion that measured pristine fiber strength values represent “intrinsic strengths” of flaw- and crack-free glasses.11 Despite intense investigations, the fracture mechanism of glasses is still controversial. An issue of current interest is the possibility of plastic deformation at crack tips in silicate glasses at room temperature. In fact, when the fracture process of soda-lime silicate glasses is analyzed at the nanometric scale with an atomic force microscope the formation of cavities 20 nm long and 5 nm deep was observed ahead of the crack tip, and cracks advanced via the coalescence of the cavities.12,13 Although prediction of the failure strength of glasses can be achieved by using the continuum theory of Fracture Mechanics, the study of the dynamics of crack propagation and of the mechanism of fracture at the atomic level requires the use of molecular dynamics simulations techniques.14 Since the first investigation by Soules and Busby15 in 1983, several MD simulations of the response of silica glass to an applied tensile load have been carried out.16−23 These studies have revealed that the mechanical properties (Young’s modulus, strength, failure strain, and fracture mechanism) depend on both experimental parameters (strain rate, the cooling rate of the melt) and computational details (force-field, system dimension, and the statistical ensemble).18,19,23 A striking similarity between these simulations is that the fracture process is initiated by the nucleation, growth, and coalescence of nanoscale voids pre-existing in the bulk. However, it is important to note that in several of the aforementioned investigations a pronounced plastic deformation was observed and ascribed to computational artifacts. In this work, MD simulations are employed to understand the effect of composition, system dimension, and presence of nanometric defects on the mechanical properties and the fracture mechanism of SiO2 and 20Na2O·80SiO2 bulk glasses and nanowires.

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COMPUTATIONAL DETAILS MD simulations have been carried out by using a modified version of the DLPOLY program.24 The PMMCS pairwise interatomic potentials based on the rigid ionic model developed by us has been employed.14,25 This force field reproduces with good accuracy the structure, transport, and mechanical properties of oxide glasses26−28 and allows one to handle systems containing hundreds of thousands of atoms in a reasonable time even with clusters made up of dozens of CPUs. The structural models of the SiO2 and 20Na2O·80SiO2 (NS20) bulk glasses containing about 30k and 60k atoms were obtained by using the melt-quench approach.14 The systems were heated at 5000 K, a temperature more than adequate to bring them to the liquid state in the framework of the adopted force field. The melt was then equilibrated for 100 ps and subsequently cooled continuously to 300 K with a nominal cooling rate of 5 K/ps. The temperature was decreased by 0.01 K every time step using the Evans thermostat,29 and the volume of the simulation boxes was kept fixed to those needed to reproduce the experimental densities at room temperature. Subsequently, another 100 ps of equilibration at constant energy and 60 ps at constant pressure (P = 0 kbar) using the Nose−Hoover algorithm30 was performed at 300 K. In all the simulations, the integration of the equation of motion was performed using the Verlet leapfrog algorithm with

Figure 1. Schematic representation of the protocol used to generate glass nanowires.

Uniaxial tensile tests were conducted along the z-axis in both bulk glasses and nanowires by rescaling the atomic positions using the formula r = (1 + ε)r0 where r and r0 are displaced and previous positions and ε is the strain tensor. The effective strain rate was set to 1 × 109 s−1. During tension tests, the box sides perpendicular to the strained one were allowed to relax anisotropically using a modified version of the Nose−Hoover NPT algorithm in which the z-direction remains fixed at each strain. The stress tensor was computed at each step by using the formula 1 σαβ = − (∑ piα piβ /mi + ∑ riαfiβ ) V i i where piα, riα, and f iα are the α components of the momentum, positions, and forces acting on the ith atom. The true stress tensor was then transformed into a nominal (or engineering) stress tensor whose normal components are defined by σnom = σT·Si/S0, in which Si and S0 are the instantaneous and initial cross-sectional area perpendicular to the loading direction. After the stress−strain diagrams were 25500

DOI: 10.1021/acs.jpcc.5b08657 J. Phys. Chem. C 2015, 119, 25499−25507

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The Journal of Physical Chemistry C produced, the elastic properties were obtained by direct analysis of curves. All tests were performed at (constant) room temperature and zero external pressure.

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RESULTS AND DISCUSSIONS Structure of Bulk Glasses and Nanowires. The short and medium range structure of silica and soda-silicate glasses has been already investigated by means of MD simulations employing the PMMCS potentials in previous works.19,26,27 Therefore, in this section we will only compare the salient structural features of glass nanowires with that of bulk glasses with the same compositions. The longitudinal and sectional views of silica and NS20 glass nanowires are depicted in Figure 2. The nanowires are about

Figure 3. Number density ρi(R) (top panels) and fraction χi(R) (bottom panel) of different species as a function of their distance from the nanowire central z-axis. The larger statistical fluctuations seen toward the core reflect the decreasing number of atoms contained in volume elements for R → 0.

The atomic fraction and density in each shell were calculated as χi(R) = ni(R)/N(R) and ρi(R) = ni(R)/V(R) and where ni and N are the time-average number of atoms of species i and of all species found in the shell, respectively, while V is the volume of the shell. The number density trends reported in the middle panels of Figure 3 reveal significant departures from the bulk density of all species starting beyond 32−33 Å from the central axis. These deviations are used to define the surface region of the nanowires, as the external shell (6−7 Å thick) where the distribution is significantly different from the bulk. The bulk and surface regions are also highlighted by the relative atomic fractions reported in the bottom panels. These show that the surface of the silica nanowire is rich in oxygen atoms, whereas that of the NS20 glass nanowire is rich in Na atoms. The average coordination numbers CNi, the NBO and BO relative fractions, and the distribution of Qn silicate species going from the inner to the external (surface) regions of the nanowires are reported in Figure 4. From these plots, it is evident that the relative fraction of NBOs increases at the surface, and in the case of the silica nanowire, some three-

Figure 2. Longitudinal and sectional view of the (a) silica and (b) NS20 glass nanowires. Yellow, red, and blue spheres represent silicon, oxygen, and sodium ions, respectively.

10.4−10.6 nm long with a diameter of 7.5 nm. After constant pressure relaxations at 300 K the density of both the silica and the NS20 nanowires slightly decreases (2.13 and 2.20 g/cm3, respectively) with respect to the experimental values (2.20 and 2.39 g/cm3, respectively), in contrast to what happens for bulk glasses (2.26 and 2.44 g/cm3, respectively). The structural analysis reveals that the pair distribution functions, bond angle distributions, and the Qn distributions computed for the inner region of the nanowires are very similar to those found in bulk glasses and thus in good agreement with experimental data as already demonstrated in previous papers. In general, the different densities observed between the nanowires and bulk glasses are due to a small expansion of the nanowire surfaces with respect to the inner regions. In order to characterize the nanowires, we have analyzed several structural parameters, focusing, in particular, on how the structure changes in moving from the internal to the surface region. Figure 3 shows number density and fraction profiles of each species (O, Si, Na) going from the core to the surface of the nanowire. Given the approximately cylindrical shape of the nanowires, the profiles were calculated by dividing the volume in concentric cylindrical shells of thickness ΔR = 1 Å, located at distance (R ± ΔR) from the central axis.

Figure 4. Average coordination numbers CNi (top panels); NBO and BO relative fractions (middle panels); and distribution of Qn silicate species going from the inner to the external (surface) regions of the nanowires. 25501

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The Journal of Physical Chemistry C coordinated Si atoms are present. However, a quantitative analysis of such defects reveals that 1% of 3-Si, 0.21% of NBOs, and 0.67% of TBOs are present. These values are slightly higher than those computed for flaw-free bulk models in which the amounts of five-coordinated Si atoms, NBOs, and TBOs are 0.1%, 0.2%, and 0.2%, respectively. This feature is probably a consequence of the casting approach employed to produce the nanowires since more defects are found when the cutting approach is used.23 Regarding the Qn distributions, the percentage of Q4 species remains constant to the bulk value up to R = 32 Å and then drastically decreases to give place to the rise of Q3 and Q2 species. The latter are the dominant species in the outer surface of silica nanowires. As for the NS20 nanowire, the plots reveal that sodium concentration is quite constant in the inner region (below 32 Å) and increases at the surface of the nanowires. In this case, silicon is always 4-fold coordinated, whereas oxygen atoms are present as NBOs. The Qn distribution in the inner shells is similar to that observed in the bulk (57% of Q4, 37% of Q3, and 6% of Q2 species), whereas the concentration of Q3 and Q2 species increases at the nanowire surface at the expense of the of Q4 species. Moreover, in this case, a small amount of Q1 species is also observed at the glass surface. Another important feature of the nanowire surface is the reduced Na−O coordination. In particular, the number of oxygen nearest-neighbors (computed with a cutoff of 3.0 Å) is reduced from 4.6 in the bulk (a value in line with previous MD simulations25,26,32 employing the same cutoffs and lower than that obtained in other studies employing larger cutoffs33) to 3.5−3.0 in the surface. Therefore, the increased fraction of the sodium ions exposed at the surface is not fully balanced by the available oxygen atoms, so that the latter are unable to complete the ideal bipyramidal coordination of Na in sodasilicate glasses.33−35 Consequently, sodium ions exposed at the nanowire surfaces will have a more pronounced Lewis acidity with respect to those in the bulk glasses, and in the presence of moisture in the environment, the nanowire/environment interfaces will be dominated by Na−water interactions. Thus, NS20 glass nanowires are expected to be very reactive. Uniaxial Tension Tests in Bulk Glasses: Flaw-Free Silica Glass. In a previous study, we investigated the effect of the statistical ensemble (NVT and NPT), the strain rate, and the temperature on the stress−strain diagrams of a structural model of a flaw-free silica glass containing 12k atoms.19 Most of the previous uniaxial tension MD simulations employed the NVT ensemble in which the lateral axes were fixed while the tensile axis was elongated.18 We showed that in these simulations glasses are forced to break in a brittle manner because a triaxial tensile stress is imposed on the system. The better and more realistic approach to investigate the fracture mechanism of silicate glasses is that using the NPT method in which the lateral axes are free to relax while the tensile axis is elongated. In the stress−strain diagrams obtained by using the NPT approach, four distinct regions were identified, as reported in Figure 5(a). The first region coincides with the elastic regime, where stress increased linearly with strain. We observed that at low strain the deformation is taken up by the gradual opening of the Si−O−Si bond angle, whereas at high strain it is taken up by the stretching of Si−O bonds. In the yield region, the stress varies very slowly with strain, and the first Si−O bond breaks at

Figure 5. (a) Stress−strain diagram of silica glass model containing 12k atoms: (1) is the elastic region, (2) the yielding region, (3) the unstable region, and (4) the zero-stress region. (b) Comparison between the stress−strain diagrams of silica glass models containing 12k (black curve) and 30k (red curve) atoms. (c) Comparison of the stress−strain diagram of the 30k silica glass models containing a notch 1.0 (blue) and 2.0 (green) nm long.

the intrinsic failure strain. In the third region, a dramatic drop in the stress with increasing strain is observed, as a consequence of the gradual rupture of the Si−O bonds, which can then rearrange to recover the structure. In this region, called the unstable region, nanovoids with radius of ∼5.0 Å are initially created far from one another, but they then start to coalesce to make up the critical void that grows and breaks the material. In the final region, the stress is zero, and the two fractured surfaces of the sample are formed. While the elastic region was found to be almost independent of the strain rate and the operating temperature employed, more pronounced changes were observed for the yielding and the unstable regions. In brief, an increase in both the intrinsic strength and strain to failure occurs with the strain rate. Moreover, when the latter is greater than 1010 s−1 the unstable region expands, and the stress−strain diagram resembles that of viscous material rather than a brittle one. This phenomenon was also observed when the operating temperature was increased. To understand the effect of the system size on the mechanical properties of MD-derived glass structures we have compared the stress−strain diagrams of silica glass containing 30k and 12k atoms computed at 300 K with the NPT approach and strain rate of 109 s−1 (Figure 5(b)). The figure shows that the size does not affect the elastic and yield regions. In fact, both systems present an intrinsic strength of about 10.9 GPa and strain at failure (strain corresponding to the maximum of the stress−strain diagram) of about 14.6%. These values, which are reported in Table 1, are in general good agreement with experimental values measured at room temperature for silica fibers. In fact, Smith and Michalske36 reported intrinsic strengths in the range 11−14 GPa, whereas Bogatyrjov et al.37 reported a strain to failure of 14.5% and intrinsic strengths of 10.2 GPa for a tin-coated silica fiber. The more striking difference between the two diagrams appears in the unstable region whose width is halved for the bigger system. This means that with the increase of the system 25502

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The Journal of Physical Chemistry C Table 1. Mechanical Properties (Strength, Strain at Failure, and Young’S Modulus) Extracted from the Stress−Strain Diagrams of the Systems Investigated glass SiO2 12k SiO2 30k SiO2 30k with Notch 1.0 nm long SiO2 30k with Notch 2.0 nm long SiO2 Nanowire SiO2 Exp. NS20 30k NS20 60k NS20 Nanowire NS20 Exp.41

strength (GPa)

strain at failure (%)

Young’s modulus

10.9 10.9 9.0

14.5 14.5 -

78.0 78.1 76.5

8.2

-

75.5

8.7 11−1436 5.1 5.1 4.6 11.0

13.0 14.537 17.0 17.0 17.3 19.0

73.2 72.039 50.5 50.5 37.2 57.5

size the fracture becomes more brittle. Figure 6 shows the formation and evolution of the crack for the bigger system. This Figure 7. (a) Sectional view of the silica glasses containing a spherical pore (cyan sphere) with diameter of 1.0 nm and a pore of 2.0 × 1.0 × 1.0 nm in size and represented by three overlapping cyan spheres. (b) Perspective view of the pore with size 2.0 × 1.0 × 1.0 nm. The blue sticks represent the SiO4 groups forming the wall of the pore.

these structures revealed that a small amount of under- and overcoordinated silicon atom (0.50%) NBOs (0.51%) and TBOs (0.40%) are present. These values are slightly higher than those computed for flaw-free models in which the amounts of under- and overcoordinated Si atoms, NBOs, and TBOs are 0.1%, 0.2%, and 0.2%, respectively. Figure 5(c) shows the stress−strain diagrams simulated for these two models at 300 K, whereas the mechanical properties (strength, strain at failure, and Young’s modulus) are reported in Table 1. The intrinsic strength decreases from 10.9 GPa for the “flaw-free” model to 9.0 and 8.2 GPa for the models with notches 1.0 and 2.0 nm long. Contrary to what one would expect, it seems that the fracture mechanism of such defective models is less brittle than that of flaw-free models since the unstable region slightly expands. This behavior can be ascribed to the size of the artificial nanovoids created which are more similar to those of the voids that form in “flaw-free” glass models than to the notches usually present in macroscopic samples. However, the insertion of the artificial nanovoids provides alike useful physical insights because it allows us to better investigate the factors that can give rise to nanoductility phenomena in silicate glasses.38 As already noted earlier, the artificial nanovoids bear with them atomic defects such as 3-fold and 5-fold coordinated Si atoms, NBOs, and BOs. A detailed visual analysis of the evolution of such defects during the uniaxial tensile tests has revealed that they play an important role in the rising of nanoductility. Among the aforementioned defects, TBOs, NBOs, and 5-coordinated Si atoms seem to be the most important ones since they are able to sustain plastic deformation by converting them from high-energy point defects to more stable structures. Figure 8 reports two mechanisms of structural relaxation occurring in the unstable region of the stress−strain diagram during uniaxial tensile tests of the defective models. In

Figure 6. Formation and evolution of crack for the silica glass containing 30k atoms. Percent strain is reported.

evolution is very similar to that previously observed with the NVT approach for smaller systems (see Figure 7 of ref 19). In this case the sequential breaking of the Si−O bonds occurs and results in the rapid formation of nanovoids, of radius around 5.0 Å, that coalesce more or less instantaneously to break the material in 5 ps, as soon as the first Si−O bond is broken. Uniaxial Tension Tests in Bulk Glasses: Defective Silica Glass Models. In flaw-free glass structures the voids must first form, grow, and coalesce before the material fractures. We could wonder what would happen if nanovoids with dimensions similar to the critical ones are already present in the simulation box. To investigate this issue we have generated two structural models, one containing a spherical void with diameter of 1.0 nm and another containing a nanovoid with sizes 2.0 × 1.0 × 1.0 nm. The first void has been created by inserting in the center of the simulation box a fictitious atom with radius of 0.5 nm (a Morse function with parameters D = 1 × 10−6 eV, a = 3.0000 Å−1, and re = 5.0 Å was applied between the center of the fictitious atom and the glass constituents) and melting the system as described in the computational details. The second void has been created by placing in the center of the simulation box three fictitious atoms with radius of 0.5 nm each. The three fictitious atoms were displaced by 0.5 nm along the x axis and kept fixed during the glass generation. In any case, the interaction between the fictitious atoms and the silica matrix was only repulsive so that no Si and O atoms could enter into the nanovoids. A sectional view of the defective structural models created in this way is shown in Figure 7a. Figure 7b displays a perspective zoomed view of the cavity with size 2.0 × 1.0 × 1.0 nm. The analysis of 25503

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Figure 9. Stress−strain diagrams of the silica glass model containing 30k atoms and of the NS20 glass models containing 30k and 60k atoms at 300 K.

(NS20 glass) in the computation and from 0.145 to 0.19 in the experiment, respectively.41 On the contrary, the results reported in Table 1 reveal that the intrinsic strength of the NS20 glass is almost half of that of the silica glass (5.1 vs 10.9 GPa) in disagreement with Lower et al.41 who estimated an intrinsic strength for NS20 glass fibers of 11 GPa, very similar to that measured for silica glass. However, the failure strengths of the sodium silicate glasses reported by Lower et al.,41 calculated from the failure strains (determined through two-point bending measurements) and the respective “zero strain” elastic modulus, are considerably greater than those reported elsewhere for similar glasses42 and in fact, for several compositions, exceed the strength calculated for the silica fibers. As also stated by Lower et al., the strengths of Nasilicate glass fibers reported in their paper are overestimates because they did not account for “strain softening”, a decrease in the elastic modulus noted in earlier tensile studies of alkalicontaining glass fibers43 and in our calculations. The addition of Na2O to the glass leads to less polymerized tetrahedral network structures containing percolating channels of (mobile) Na ions, and it is plausible to expect a reduction of the strength with respect to silica glass. Figure 10 shows the distribution of sodium ions in the bulk and on the surface of the NS20 glass before and after fracture. As expected, our simulations reveal that the glass breaks in concomitance with the Na-rich zones. Moreover, the NS20 glass models show a fracture mechanism less brittle than silica (sometimes called nanoductility38) since a wider “unstable region” is observed for the 30k model. This is due to the higher mobility of Na ions that are able to follow the deformation and recover the structure by saturating the new NBOs formed during the fracture of the glass. However, as observed for silica glass, the employment of bigger systems reduces the “ductility” of the model. Yuan et al.23 explained the system size effect by resorting to the ratio between the fracture surface energy (FSE) and the elastic strain energy (ESE) of the system considered. They observed that in order to allow the system to undergo a brittle fracture the ESE should overwhelm the FSE; i.e., the ESE/FSE ratio should exceed 1.0. Otherwise, the system does not have enough elastic strain energy to promote the brittle fracture when deformed beyond the elastic limit. Other possible factors are related to the minor impact of atomic defects in big systems and to their lower configurational energy with respect to small systems.44 As already observed for the silica glass, the size of the model does not affect the elastic region of the diagram and thus the value of the Young’s modulus. By linear fitting of the stress− strain curve in the strain range 0−5% we have obtained a value

Figure 8. Mechanism of structural relaxation occurring in the unstable region of the stress−strain diagram during the deformation of defective glass models. (a) Conversion of a TBO and a NBO to two BOs. (b) Conversion of a 5-fold coordinated Si atom and a BO to a 4-fold coordinated Si and a NBO, respectively.

particular, the upper panel shows the transformation of a TBO and a NBO connected to Si1 to two BOs. During the sample deformation the TBO first converts to a BO thanks to the breaking of the Si3−TBO bond, and then the 3-coordinated Si3 atom captures the NBO connected to Si1 leading to a relaxed structure. Instead, the bottom panel shows the transformation of a 5fold coordinated Si atom and a BO to a 4-fold coordinated Si and a NBO, respectively. It is worth noting that during the advancement of the deformation we observed that the Si atom colored in green oscillates between the two configurations reported until it finally converts to a stable 4-fold Si atom. Therefore, it seems that the structure can absorb the stresses caused by sample deformation until these types of atomic defects are present because their transformation (by bond breaking) leads to more stable structures. Another important mechanical property that can be extracted from the stress−strain diagrams is the Young’s modulus that gives an idea of the stiffness of the material. The zero-strain Young’s modulus (E0) was computed by fitting with a thirdorder polynomial the stress−strain diagram in the range 0−12% of strain. The Young’s modulus decreases from 78.1 GPa for the flaw-free model to 76.5 and 75.5 GPa for the models with notches of length 1.0 and 2.0 nm, respectively. All these values are in good agreement with the experimental value of 72 GPa,39 whereas the small softening of the structures observed with the insertion of the artificial nanovoids is in agreement with the results of experiments and computations on porous silica glasses.40 Uniaxial Tension Tests in Bulk Glasses: Soda-Silicate Glass. To investigate the effect of the composition on the mechanical properties of silicate glasses uniaxial tensile tests were performed also on two structural models of the NS20 glass. Figure 9 shows the comparison of the stress−strain diagrams of the silica glass model containing 30k atoms with those of structural models of the NS20 glass containing 30k and 60k atoms (the latter model is obtained by replicating the 30k cell along the z-dimension). A nice agreement between computed and experimental41 results is observed for the strain at failure. In fact, it increases from 0.15 (silica glass) to 0.17 25504

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Figure 11. Stress−strain curves of uniaxial tension for the bulk and nanowires of silica (top) and the NS20 (bottom) glasses.

reported by Brambilla et al. is probably due to the different methods used to prepare the fibers. In fact, in their experiments, the nanowire samples were manufactured by repeatedly fire-polishing the fiber surfaces. This process decreases the amount and the dimension of surface flaws and strengthens the ultimate strength of nanowires. Therefore, when the diameter gets smaller, the surface layer will play a more important role in determination of the nanowires’ mechanical properties. Instead, our nanowire models contain more surface flaws than bulk glasses and thus show poorer mechanical properties. However, in perfect agreement with Brambilla’s observation both the stress−strain diagram and the reduced necking shown in Figure 12 reveal that the fracture mechanism in silica nanowires is brittle.

Figure 10. Structural models of the (left) bulk at zero strain and (right) surfaces after fracture of the NS20 glass. Blue, yellow, and red spheres represent sodium, silicon, and oxygen ions, respectively.

of 50.5 GPa that compares well with that measured by Lower et al. on glass fibers (57.5 GPa).41 In a previous work,26 we computed the Young’s modulus of the NS20 glass using the same force field in combination with a static method based on the calculation of the second derivative of the total energy with respect to the cell strains. We found a value of 58.5 GPa which compares very well to the experimental data. The better agreement of the static approach is because this was the method exploited to fit the interatomic potentials. However, it must be highlighted that in the static approach the temperature and dynamical effects are not taken into consideration as done in the uniaxial tensile test method for which MD simulations were employed. Uniaxial Tension Tests in Glass Nanowires. After having understood the effect of the system dimensions, the presence of nanovoids, and the composition on the mechanical properties of bulk glasses we have studied the behavior of the silica and NS20 glass nanowires under tensile loading. Indeed, very recently Brambilla et al.10 showed that the mechanical properties of glasses in the nanoscale are extremely different from those found in macroscopic samples. They measured the ultimate strength of glass silica nanowires manufactured by a top-down fabrication technique, finding a value in excess of 10 GPa. The ultimate strength was found to increase for decreasing nanowire diameters, and scanning electron micrographs of the broken fragments showed a fragile rupture. Figure 11 reports the stress−strain diagrams of silica and NS20 glass nanowires in comparison with the one computed for bulk glasses. The mechanical properties extracted from these stress−strain diagrams are reported in Table 1. Our simulations show that silica nanowires present lower intrinsic strength (8.7 GPa) with respect to that of flaw-free bulk glasses (10.9 GPa), a reduced strain at failure (13% vs 15% for the bulk), and a slightly lower zero-strain Young’s modulus (73.2 vs 78.1 GPa). The apparent contradiction of our data with those

Figure 12. Snapshots of the evolution of atomic configuration during deformation of silica (top) and NS20 (bottom) glass nanowires.

The comparison between the stress−strain diagrams of the NS20 glass nanowire with that of the bulk reported in the bottom panel of Figure 11 shows that the two systems have similar intrinsic strengths (4.6 and 5.1 GPa for nanowire and bulk glasses, respectively) and strain at failure (17.3%). This is a consequence of the lower number of atomic defects present on the surface of the NS20 nanowire with respect to those present in the silica nanowire. In fact, as described in the previous section, the surface of the NS20 nanowire is rich in NBOs, 25505

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The Journal of Physical Chemistry C which are charge compensated by sodium ions, as in the bulk. Therefore, whereas in silica glass the formation of NBOs and TBOs implies the creation of under- and overcoordinated silicon atoms and small rings which are defects that reduce the intrinsic strength of the glass, in soda-silicate glasses silicon remains 4-fold coordinated, and the intrinsic strength does not decrease significantly. Regarding the fracture mechanism, the rupture of the NS20 nanowire is more brittle than that observed for the bulk (a narrower unstable region is observed), but a more pronounced necking of the sample is observed compared to silica (see Figure 12).

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ACKNOWLEDGMENTS

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REFERENCES

A.P. thanks the University of Modena and Reggio-Emilia for supporting this work by granting the internal project (Fondo di Ateneo per la Ricerca, FAR2014) entitled “Role of modular phyllosilicates for the capture and storage of CO2: an experimental and computational investigation”.

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CONCLUSIONS Molecular dynamics simulations have been used to perform uniaxial tensile in silico experiments on structural models of the bulk and nanowires of silica and soda-silicate glasses. The effect of the system size, composition, and presence of nanovoids on the simulated stress−strain diagrams of bulk glasses has been investigated. The system size does not affect the Young’s modulus, the intrinsic strength, and strain to failure of the investigated systems, but it has a huge impact on the fracture mechanism. An artificial ductility arises for small systems but disappears by increasing the system dimensions. Brittleness also decreases when nanometric voids (carrying atomic defects such as under- and overcoordinated silicon and oxygen ions) are created in the structure of flaw-free silica glass or when modifier cations are present. The former are able to absorb local stresses by transforming from high energy point defects to more stable configurations, whereas the latter, being more mobile than oxygen and silicon ions, are able to recover the structure by saturating NBOs formed during the gradual breaking of the Si−O bonds that starts soon after the strain at failure is reached. A new casting method to generate flaw-free glass nanowires was proposed. This is based on the melting and quenching of the glass into carbon nanotubes of proper diameters. The surface of amorphous silica nanowires is rich in NBOs and undercoordinated silicon atoms, whereas that of the NS20 glass is rich in sodium ions that charge-balance NBOs belonging to Q2 silicon species. With respect to flaw-free bulk models, simulated nanowires have slightly poorer mechanical properties because of the atomic defects present on their surfaces, whereas soda-silicate nanowires have strengths similar to that of bulk glasses. The agreement with experimental data further confirms that molecular dynamics simulation techniques are an essential tool for glass manufactures since they allow the interpretation and prediction of mechanical properties of amorphous materials with good accuracy.

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. All authors contributed equally. Notes

The authors declare no competing financial interest. 25506

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Article

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