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Article

Revisiting the Dependence of Poisson’s Ratio on Liquid Fragility and Atomic Packing Density in Oxide Glasses

by
Martin B. Østergaard
1,
Søren R. Hansen
1,
Kacper Januchta
1,
Theany To
1,
Sylwester J. Rzoska
2,
Michal Bockowski
2,
Mathieu Bauchy
3 and
Morten M. Smedskjaer
1,*
1
Department of Chemistry and Bioscience, Aalborg University, 9220 Aalborg East, Denmark
2
Institute of High-Pressure Physics, Polish Academy of Sciences, 01-142 Warsaw, Poland
3
Department of Civil and Environmental Engineering, University of California, Los Angeles, CA 90095, USA
*
Author to whom correspondence should be addressed.
Materials 2019, 12(15), 2439; https://doi.org/10.3390/ma12152439
Submission received: 3 July 2019 / Revised: 29 July 2019 / Accepted: 29 July 2019 / Published: 31 July 2019
Browse Figures
Versions Notes

Abstract

:
Poisson’s ratio (ν) defines a material’s propensity to laterally expand upon compression, or laterally shrink upon tension for non-auxetic materials. This fundamental metric has traditionally, in some fields, been assumed to be a material-independent constant, but it is clear that it varies with composition across glasses, ceramics, metals, and polymers. The intrinsically elastic metric has also been suggested to control a range of properties, even beyond the linear-elastic regime. Notably, metallic glasses show a striking brittle-to-ductile (BTD) transition for ν-values above ~0.32. The BTD transition has also been suggested to be valid for oxide glasses, but, unfortunately, direct prediction of Poisson’s ratio from chemical composition remains challenging. With the long-term goal to discover such high-ν oxide glasses, we here revisit whether previously proposed relationships between Poisson’s ratio and liquid fragility (m) and atomic packing density (Cg) hold for oxide glasses, since this would enable m and Cg to be used as surrogates for ν. To do so, we have performed an extensive literature review and synthesized new oxide glasses within the zinc borate and aluminoborate families that are found to exhibit high Poisson’s ratio values up to ~0.34. We are not able to unequivocally confirm the universality of the Novikov-Sokolov correlation between ν and m and that between ν and Cg for oxide glass-formers, nor for the organic, ionic, chalcogenide, halogenide, or metallic glasses. Despite significant scatter, we do, however, observe an overall increase in ν with increasing m and Cg, but it is clear that additional structural details besides m or Cg are needed to predict and understand the composition dependence of Poisson’s ratio. Finally, we also infer from literature data that, in addition to high ν, high Young’s modulus is also needed to obtain glasses with high fracture toughness.

1. Introduction

Poisson’s ratio (ν) is the negative ratio of the transverse strain to the longitudinal strain of a material under uniaxial stress in the elastic regime. It relates to the shear modulus (G) and bulk modulus (B), as
ν = 3 B − 2 G 6 B + 2 G
For isotropic materials in three dimensions [1], this limits ν to be within −1 and 0.5, as the values of G and B are always positive. Different material families and compositions exhibit pronounced diversity in their elastic properties and thus Poisson’s ratio. Materials with ν ~ 0.5 are highly incompressible and tend to deform through shape change, while materials with ν ~ 0 are highly compressible. So-called auxetic materials, with negative values of ν, swell under tension [2,3,4]. At ν ~ 0.2, a transition between two different types of stress patterns in frozen-in solid has been reported, namely shear and uniform deformation [5]. Various macroscopic properties have been linked to Poisson’s ratio [6], including some outside the elastic regime, such as densification [7], connectivity [8], and ductility [9].
Oxide glasses exhibit interesting properties such as transparency, high hardness, high chemical durability (in many cases), and low-cost of raw materials. The brittleness of oxide glasses has been a major hindrance for their use in various engineering and functional applications [10]. As the crack tip formation and growth mechanisms are not well understood, it is challenging to design ductile oxide glasses. Post-processing approaches such as chemical strengthening [11] are thus currently used to improve the mechanical performance. However, molecular dynamics (MD) simulations suggest that silicate glasses can exhibit some nanoscale ductility [12,13], and it is also possible for silica glass to feature ductility induced by electron-beam irradiation [14]. Interestingly, as shown for metals [15] and metallic glasses [9], high G/B ratio (and thus low Poisson’s ratio) favors brittleness. In other words, a correlation between fracture energy (Gfrac), i.e., energy required to create two new fracture surfaces, and Poisson’s ratio has been observed, which also manifests itself by a brittle-to-ductile (BTD) transition around νBTD = 0.32 not only for metallic glasses but various non-crystalline solids (Figure 1) [6,9,16].
The problem for oxide glasses is the fact that they mostly exhibit ν < 0.30, with only few oxide glasses reported with ν > 0.34 [24,25]. As such, the existence of a BTD transition for oxide glasses needs additional verification. However, recent MD simulations on permanently densified SiO2 glasses have confirmed the existence of a BTD transition, although the value of νBTD was found to depend on the average coordination number [26]. Moreover, a recent study has explained the empirical BTD transition based on microscopic dynamical properties [16], building on the observation that ductility is closely related to the secondary β-relaxation [27,28], while Poisson’s ratio is proposed to be related to the effective Debye-Waller factor. The study suggests that ductile materials can withstand deformation at higher rates because they exhibit faster β-relaxation [16].
In an attempt to overcome the brittleness of oxide glasses, it is thus of great interest to discover high-ν oxide glasses (ν > 0.32). Unfortunately, there are presently no composition-dependent models available for predicting ν, and thus, inefficient Edisonian trial-and-error composition design is currently utilized [29]. It is therefore of interest to find predictable surrogates for Poisson’s ratio. Most notably, liquid fragility (m) has been proposed to be positively correlated with the ratio of bulk and shear moduli (and thus Poisson’s ratio) for a broad range of glassy systems covering covalent and hydrogen-bonded, van der Waals and ionic glasses, i.e., a range of organic molecules, oxide, halogenide, and chalcogenide glasses [30,31]. Angell’s liquid fragility is defined as the slope of the base-10 logarithm of viscosity versus Tg-scaled inverse temperature curve at Tg, where Tg is the glass transition temperature (m = d log (η)/d log (Tg/T) at Tg) [32]. This is the fragility index used in this work, although we note that other definitions of fragility exist [33]. The proposed m-ν relation is of interest, since tools such as topological constraint theory [34,35] and coarse-graining (related to structural connectivity) [36,37] can be used to predict m. Since the original study by Novikov and Sokolov in 2004 [30], a similar m-ν dependence has been found for metallic glasses, although the change in fragility with modulus ratio varies for different systems [38,39,40]. It has been noted that the correlation is only observed within a narrow range of m due to the limited amount of data on bulk metallic glasses [40]. The proposed linear relationship between m and the bulk-to-shear modulus has been seriously questioned by Yannopoulos and Johari [41], who have argued that some data points were erroneously plotted, showing that no general correlation for neither organic, inorganic, nor metallic glasses exists when including more data. The lack of correlation between m and elastic properties has then been suggested to be due to the strong sensitivity of ν to temperatures above Tg, as strong melts (low m) exhibit Poisson’s ratio that is almost constant before and after the glass transition, while fragile melts (high m) show a significant change in Poisson’s ratio above Tg [8]. There have been studies supporting the m-ν correlation. For example, building on a proposed relation between Poisson’s ratio and packing density (see below) Duval et al. [42] argue that the relation between m and ν is due to the structural fluctuations being breathing-like (with change of volume) in strong liquids and shear-like (without change of volume) in fragile liquids. Greaves et al. [6] have also argued that the correlation between m and ν depend on the glass system, showing linear correlations for binary alkali silicates and metallic glasses, but with different slopes.
Besides liquid fragility, Poisson’s ratio has been suggested to be positively correlated with the atomic packing density (Cg) [6,8,43], which is defined as the ratio between the volume occupied by the ions and the corresponding effective volume of glass. Cg could potentially be a good surrogate for Poisson’s ratio, since the compactness of the sample affects the vibrational modes [42] and materials with a high Cg should exhibit relatively strong interatomic interactions [16]. Based on experimental data, an empirical relation between ν and Cg has been proposed (ν = 0.5–1/7.2Cg) [43], but it has been found to overestimate the Poisson’s ratio for borate and phosphate glasses and underestimate it for germanate and aluminate glasses [8]. A reason for an overestimated Poisson’s ratio of borate glasses might be the low average coordination number as explained for the prediction of Young’s modulus [44].
The purpose of this work is to revisit the validity and universality of the proposed m-ν and Cg-ν correlations. This is done to determine whether prediction of liquid fragility or atomic packing density can be used to guide the discovery of oxide glasses with high Poisson’s ratio (ν > 0.32), which are expected to be ductile following the relation in Figure 1. To do so, we perform an extensive literature review to obtain liquid fragility, density, and Poisson’s ratio data for various glass systems. Since experimental data on oxide glasses with ν > 0.30 are scarce, we also synthesize a total of 20 new oxide glasses, particularly aluminoborate and zinc borate glasses as these have been found to have relatively high Poisson’s ratio [45,46]. To further expand the dataset, we also determine the missing property (e.g., m if only Cg and ν are known) from previously synthesized glasses in our laboratory [18,46,47,48,49]. Moreover, we subject selected oxide glasses to high-temperature densification to induce a higher Cg value in bulk samples and then probe whether it correlates with an expected increase in ν. Finally, we also discuss the implications of the findings for designing tough oxide glasses.

2. Experimental

2.1. Sample Preparation

Oxide glasses were prepared by the traditional melt-quenching technique using reagent grade chemicals (see Table 1). Three families of glasses were synthesized, namely Zn-borates, aluminoborates, and Ca-Zr-silicates. Zn-borates were prepared from H3BO3 (Hoenywell, North Caroline, USA) and ZnO (VWR, Leuven, Belgium), and doped with La2O3 (Sigma-Aldrich, Steinheim, Germany), Ta2O5 (Sigma-Aldrich, Steinheim, Germany), and/or GeO2 (Alfa Aesar, Massachusetts, USA). Aluminoborates were prepared from H3BO3 and Al2O3 (Sigma-Aldrich, Steinheim, Germany) with additions of BaCO3 (ChemPUR, Karlsruhe, Germany), MgCO3 (Acros Organics, New Jersey, USA), CaCO3 (Sigma-Aldrich, Steinheim, Germany), Li2CO3 (Merck, Darmstadt, Germany), Cs2CO3 (Sigma-Aldrich, Steinheim, Germany), Ga2O3 (Sigma-Aldrich, Steinheim, Germany), and/or Ta2O5 (Sigma-Aldrich, Steinheim, Germany). Ca-Zr-silicates were produced from SiO2 (Merck, Darmstadt, Germany), ZrO2 (Hoenywell, North Caroline, USA), and CaCO3. All glasses were post-annealed for 30 min at around Tg (Tg ± 5 °C) (Tg is determined by differential scanning calorimetry), prior to density and Poisson’s ratio characterization to ensure similar thermal history.
Some of the synthesized glasses (sample size approx. 13 × 13 × 2.5 mm3) were then subjected to isostatic compression at their respective ambient pressure Tg value in a nitrogen gas pressure chamber containing a multizone cylindrical furnace [50]. The applied pressure was 1 GPa and the compression time was 30 min, which is needed for obtaining a fully densified structure [51]. After the treatment the samples were first cooled to room temperature, then relaxed to ambient pressure at room temperature (but the glasses remain partially densified).

2.2. Characterization

We determine the values of m, Cg, and ν for both the newly-synthesized glasses and those missing from previous studies [18,46,47,48,49], as shown in Table 1. First, the densities (ρ) of the glasses were determined using the Archimedes principle with ethanol as the immersion medium. The measured density and chemical composition were used to calculate the molar volume (Vm) and in turn atomic packing factor (Cg) using Equations (2) and (3), respectively.
V m = 1 ρ ∑ i x i M i
C g = 1 V m ∑ i x i V i .
Here xi, Mi, and Vi are the mole fraction, molar mass, and ionic volume (or metallic radii), respectively, of each compound. Structural assumptions regarding valence and coordination number of each cation are described in detail in the Supporting Information, while the anionic oxygen, nitrogen, and fluorine radii were assumed to be 1.35, 1.46, and 1.285 Å, respectively, as reported by Shannon [53]. The packing density of metallic glasses was calculated by using the metallic radii of the pure metal [54], as suggested by Rouxel [8].
Given the small sample size for many of the studied glasses (due to their poor glass-forming ability), we did not determine the liquid fragility (m) using direct viscosity measurements. Instead we determined it using differential scanning calorimetry (DSC) on a STA 449F1 instrument (Netzsch, Selb, Germany). Small disc-shaped specimens of 40–60 mg and Ø ~ 4 mm were prepared for measurements. DSC upscans were performed in Pt crucibles and argon atmosphere (50 mL/h) at different heating rates subsequent to cooling the glasses from well above the glass transition at the same rate. The heating/cooling rates were 5, 10, 20, and 30 °C/min. The fragilities were corrected for a systematic error using Equation (4), as described in Zheng et al. [55].
m = 1.289 ( m DSC − m 0 ) + m 0
Here, m, mDSC, and m0 are the liquid fragility determined from viscosity, the liquid fragility determined from DSC, and the fragility of a perfectly strong glass that equals 14.97, respectively.
Samples were ground using SiC paper to obtain coplanar surfaces. The longitudinal and transverse wave velocities (VL and VT, respectively) were measured by an ultrasonic thickness gauge (38DL Plus; Olympus, Tokyo, Japan) using the pulse-echo method with 20 MHz delay line. The thickness of the samples were measured with a digital micrometer (Mitutoyo, Kawasaki, Japan) with a precision of 0.01 mm. Poisson’s ratio (ν) was calculated from VL and VT, following Equation (5). For literature studies, which did not report the value of ν, it was calculated either from wave velocities using Equation (5), or from the other elastic moduli based on the isotropic nature of the oxide glasses (see, e.g., Equation (1)).
ν = V L 2 − 2 V T 2 2 ( V L 2 − V T 2 )

3. Results and Discussion

3.1. Studied Compositions

The compositions of the oxide glasses synthesized and/or characterized in this study are given in Table 1, along with the values of Cg, Tg, m, and ν. The glasses are made of various network formers, covering silicates, borates, aluminoborates, and aluminoborosilicates with various network-modifying oxides. The glasses also exhibit a wide range of Poisson’s ratio values (approximately from 0.21 to 0.34), and some glasses thus exhibit ν > νBTD. However, it is outside the scope of the present study to determine the fracture toughness of these glass samples, which, in most cases, are too small in size to be tested via self-consistent fracture toughness methods [10]. The liquid fragility values range from 22 to 60, while the atomic packing density ranges from 0.48 to 0.61. The zinc borate glasses generally feature the highest values of Cg and ν.
We have identified literature data of Poisson’s ratio using the SciGlass database, in addition to searching traditional glass journals using keywords such as “Poisson’s ratio”, “mechanical properties”, and “elastic properties”. Furthermore, we added the keyword “fragility” to obtain liquid fragility data on similar glass systems. In order to obtain Cg data, we have identified studies reporting density, molar volume, or Cg values.

3.2. Poisson’s Ratio vs. Packing Density

Figure 2 shows the dependence of Poisson’s ratio on atomic packing density for various glass systems, covering both the present results and literature data. Due to the large number of points in the center of the plot, Figure 2 shows the density of Cg vs. ν data points for the various glass families, which can be identified in Figure S1 in the Supporting Information. Data for the following glass types are included: metallic [9,20,56,57,58,59], oxynitrides [8,20,60,61], pure oxides [8,20,62,63,64,65,66,67,68,69,70,71,72,73], alkali silicates [8,18,24,63,65,72,74,75], alkali borates [18,64,67,68,76,77,78], alkaline earth silicates [8,79], alkali-alkaline earth borates [80], alkali-alkaline earth silicates (including Pb, Ti, and Fe) [8,20,62,65,67,73,81,82,83,84,85], phosphosilicates [82], germanosilicates [86], aluminosilicates [8,18,60,61,69,70,87,88,89,90,91,92,93,94,95], zinc borates [20,45,76,96], lead borates [20,45,65,96,97,98], aluminoborates [18,46,87,99], germanates [87,100], aluminoborosilicates [18,20,101], borosilicates [18,20,62,67,71,101], borates containing bismuth or tellurium [97,102,103,104,105], halogenides (flourides and oxyflourides) [62,106,107,108,109], vanadates [110,111], tellurites [66,112,113,114,115,116,117,118,119,120,121,122,123,124], phosphates [20,66,68,106,125,126,127,128,129,130,131], rare earth aluminates [8], and oxycarbides [8,20]. In the Supporting Information, we discuss the assumptions (based on structural data) used to calculate Cg with information from Refs. [18,46,49,53,66,68,76,81,86,87,90,96,97,99,110,111,122,123,125,126,128,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148].
We do not observe a strong correlation between Cg and ν (Figure 2), although an overall positive correlation might be apparent, in agreement with the earlier work of Rouxel [8]. However, we note that the present glasses in Table 1 as well as those from literature show a broad range of ν values for the same Cg value, e.g., around Cg ~ 0.48 (Figure 2). The majority of the data cluster in the center of the diagram, showing an approximate sigmoidal-like trend with a transition of ν at Cg = 0.5. That is, within a limited range of Cg, ν increases from around 0.18 to 0.28 for a majority of the glasses, followed by a smaller increase towards ν = 0.40 for metallic glasses with Cg ~ 0.75. As seen in Figure S1 in the Supporting Information, the Makishima-Mackenzie model [43] does not describe the Cg vs. ν trend as well as the sigmoidal-like trend described by Rouxel [8]. Finally, we should note that the majority of studies included in Figure 2 are on silicate and borate glasses, which exhibit similar Cg values. This is the origin of the clustering of data around Cg = 0.5 in the plot. On the other hand, many of the data points for phosphate (ν = 0.25–0.3) and tellurite (ν = 0.2–0.25) glasses are different, with relatively high Cg (>0.5) and low Cg (<0.5) values, respectively.
In addition to composition variation, the properties of bulk glasses can also change permanently due to post-treatment, such as isostatic high-temperature densification [51,149], which always leads to an increase in Cg (note that the change in Cg is measured ex situ under ambient conditions, after the glass is fully decompressed). As shown in Figure 3, the pressure-induced increase in Cg does not systematically result in an increase in ν, casting doubt on the universality of the proposed Cg-ν correlation. Zinc borate, aluminoborate, and sodium borate glasses feature a decrease in ν, while SiO2 and aluminotitanophosphate glasses feature an increase in ν upon isostatic compression at 1 GPa around Tg. The soda-lime borate glasses show a complex behavior with a monotonic increase in ν and Cg with increasing pressure (0–0.57 GPa) for low total modifier content (15 mol%), while ν first increases for pressures up to 0.2 GPa but then decreases at higher pressures for glasses with higher total modifier content (25 and 35 mol%). The lack of decrease in ν with increasing Cg could be due to the interplay of changing coordination numbers, bond lengths, and bond angles. Densification usually causes two phenomena (i) an increase in Cg, e.g., due to decreased modifier-oxygen bond lengths [150], which typically leads to an increase in Poisson’s ratio, and (ii) an increase in the network connectivity, e.g., through increasing coordination number of network formers, which typically decreases the Poisson’s ratio [8]. These two competitive effects make it difficult to understand the effect of pressure (densification) on the Poisson’s ratio. Finally, we note that a general problem with the calculation of Cg is the various assumptions needed when insufficient structural data are available [8].

3.3. Poisson’s Ratio vs. Liquid Fragility

Next, we revisit the correlation between liquid fragility and Poisson’s ratio. Considering first the oxide glasses from Table 1 with ν ≥ 0.28, we find no apparent correlation between ν and m (Figure 4). Hence, the data for these oxide glass-formers with relatively high ν values support the criticism of the Novikov and Sokolov correlation [8,41]. Figure 5 further tests the m-ν correlation by including literature data on various oxide glass formers (Figure 5a) and all types of glass families (Figure 5b). The oxide glass-formers include pure oxides (SiO2, B2O3, GeO2) [30,151], borates [18,64,67,76,78,151,152,153], silicates [18,89,152,154], aluminoborates [18], aluminoborosilicates [18,101], borosilicates [18,101], and tellurites [114,121,155]. For these systems, we highlight two observations. First, multiple liquid fragility data are obtained for pure oxides though having same (or very similar) Poisson’s ratio, e.g., three data points are shown in Figure 5a for SiO2 (ν = 0.145 [31]). Second, for some glass systems the values of m and ν are obtained from different studies when only one of the properties is reported. In those cases, we have compared the density, molar volume, or atomic packing density values to ensure the similarity of the materials. The additional glasses in Figure 5b include metallic [38,39,40,156], ZIF-62 [157], organic [30,31], ionic [30,151], halogenide [30], and chalcogenide glasses [30].
There appears to be a weak positive correlation between liquid fragility and Poisson’s ratio, but it is significantly scattered and not universal as previously reported for organic, inorganic, and metallic glasses [41]. Here, we have also included oxide glasses, but these do not strengthen the possible correlation between m and ν. We note that Greaves et al. [6] have shown a positive m-ν correlation with varying slope for each glass system. The oxide glass systems used in that study were binary silicates, but as seen in Figure 5a, there is no strong correlation when considering a wider range of oxide glass families. It thus appears that the correlation can only be found within very narrow compositional variations, as those in binary sodium or potassium silicates. When considering all the glass families (Figure 5b), it is also evident that no universal correlation is observed, since a very wide range of ν values (around 0.15 to 0.40) is seen for a relatively narrow range of m values (around 25 to 35).

3.4. Implications for Design of Tough Oxide Glasses

In order to design mechanically tough oxide glasses, it is important to control the factors that influence the fracture toughness (KIc). Therefore, besides fracture energy (Gfrac) and Poisson’s ratio (Figure 1), we analyze the effect of Young’s modulus (E) on KIc. Under plane strain, we have [158],
K I c = G f r a c   E ( 1 − ν 2 )
To understand what criteria need to be fulfilled to design high-KIc oxide glasses, we have calculated Gfrac for various systems based on the measured literature values of KIc, E, and ν. Glasses from literature include metallic [9,20,159,160,161,162,163], silicates [18,19,20,21,82,164,165,166], borates [20,166,167], phosphates [126,166], tellurites [166], chalcogenides [20,166,168,169], flourides and oxyflourides [164,166], oxynitrides [20], and oxycarbides [20]. As discussed, there is a pronounced effect of ν on Gfrac when ν exceeds ~0.32 (Figure 1). In contrast, there is no correlation between E and ν (see Figure S2 in the Supporting Information), whereas there is an expected correlation between KIc and Gfrac (see Figure S3 in the Supporting Information). Even though only ν affects Gfrac, KIc is also increasing with E (Figure 6). This highlights the importance of tailoring future glass composition with a combination of high Gfrac and E. In turn, this confirms the importance of producing high-ν oxide glasses (due to the ν-Gfrac relation) in order to improve the fracture toughness. We note that the correlation between KIc and E for metallic glasses is vague (Figure 6), which might be due to the difficulty in measuring KIc of metallic glasses [170,171], while ν, in contrast, is easy to measure. In summary, there are three ways to increase the fracture energy: (i) increase the ductility (or by surrogate Poisson’s ratio), (ii) increase the ultimate strain at constant E, and/or (iii) increase E at constant ultimate strain. All of these increase the area under the stress-strain curve, but note that (ii) and (iii) assume that fracture remains fully brittle.

4. Conclusions

We have tested the validity of previously proposed relationships between Poisson’s ratio on the one hand and liquid fragility and atomic packing density on the other hand. This was done by performing an extensive literature review and by preparing new oxide glasses, especially within the zinc borate and aluminoborate families that exhibit Poisson’s ratio (ν) above 0.30, up to 0.34. This is relevant for oxide glasses, since these are believed to undergo a brittle-to-ductile transition for ν~0.32. Although two overall increasing trends in Poisson’s ratio with both liquid fragility and atomic packing density are observed, it is also clear that no universal relationships are observed when considering the wide range of compositions herein, including oxide, metallic, halogenide, chalcogenide, ionic, and organic glass families. This work suggests that additional structural details besides, e.g., packing density, are needed to predict the Poisson’s ratio of oxide glasses.

Supplementary Materials

The following are available online at https://www.mdpi.com/1996-1944/12/15/2439/s1, Calculation of atomic packing density (Cg) and the structural assumptions made. Figure S1: Dependence of Poisson’s ratio (ν) on atomic packing density (Cg) for various glass systems, including those from Table 1. References for literature data are given in the main text. Cg is calculated according to Equation (3), building on the structural assumptions described in the Supporting Information. The errors associated with ν and Cg are smaller than the size of the symbols (0.01 and 0.002, respectively). The empirical Makishima-Mackenzie model (MM-model, solid line) [43] is also represented (black line). Figure S2. Dependence of Young’s modulus (E) on Poisson’s ratio (ν) for various glass systems. References for all data are given in the main text. Errors on E and ν are estimated to be smaller than 2 GPa and 0.01, respectively. Figure S3. Dependence of measured fracture toughness (KIc) on the calculated fracture energy (Gfrac) for various glass systems (references are given in the main text). Note that the axes are logarithmic and that KIc (together with E and ν) are used in the calculation of Gfrac based on Equation (6) from the main manuscript. Errors in KIc and Gfrac are estimated to be smaller than 0.05 and 15%, respectively.

Author Contributions

M.M.S. conceived the study. M.B.Ø., S.R.H., K.J., S.J.R. and M.Bockowski performed the experiments. M.B.Ø., K.J., T.T. and M.M.S. analyzed the data. M.M.S. and M.B. supervised the research. M.B.Ø. and M.M.S. wrote the manuscript with revisions from K.J., M.Bauchy and T.T. All authors participated in discussing the data.

Funding

This work was supported by the Independent Research Fund Denmark (grant no. 8105-00002). M.Bauchy acknowledges funding from the National Science Foundation under Grants No. 1562066, 1762292, and 1826420.

Acknowledgments

The authors thank D. Möncke (Alfred University) for providing access to SciGlass.

Conflicts of Interest

The authors declare no conflict of interest.

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Materials 12 02439 g001 550
Figure 1. Dependence of fracture energy (Gfrac) on Poisson’s ratio (ν) for a range of materials, showing a brittle-to-ductile transition in the range of ν from 0.30 to 0.33. The figure is reproduced with the data from Lewandowski et al. [9] and Tian et al. [17]. We also extend it with new Gfrac data for silicate glasses [10,18,19,20,21], borate, chalcogenide, and metallic glasses [10,20], and graphene [22,23] obtained by single-edge pre-crack beam (SEPB), chevron notch (CN), single edge notch beam (SENB), indentation fracture (IF), or tensile testing methods. The error of ν and Gfrac is estimated to be 0.01 and 15%, respectively. The dashed line is a guide for the eye.
Figure 1. Dependence of fracture energy (Gfrac) on Poisson’s ratio (ν) for a range of materials, showing a brittle-to-ductile transition in the range of ν from 0.30 to 0.33. The figure is reproduced with the data from Lewandowski et al. [9] and Tian et al. [17]. We also extend it with new Gfrac data for silicate glasses [10,18,19,20,21], borate, chalcogenide, and metallic glasses [10,20], and graphene [22,23] obtained by single-edge pre-crack beam (SEPB), chevron notch (CN), single edge notch beam (SENB), indentation fracture (IF), or tensile testing methods. The error of ν and Gfrac is estimated to be 0.01 and 15%, respectively. The dashed line is a guide for the eye.
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Materials 12 02439 g002 550
Figure 2. Dependence of Poisson’s ratio (ν) on atomic packing density (Cg) for various glass systems, including those from Table 1. The scale represents the multiplicity of data points. References for literature data are given in the text. Cg is calculated according to Equation (3), building on the structural assumptions described in the Supporting Information. The errors associated with ν and Cg are 0.01 and 0.002, respectively. R2 value for a sigmoidal fit to the data is 0.162.
Figure 2. Dependence of Poisson’s ratio (ν) on atomic packing density (Cg) for various glass systems, including those from Table 1. The scale represents the multiplicity of data points. References for literature data are given in the text. Cg is calculated according to Equation (3), building on the structural assumptions described in the Supporting Information. The errors associated with ν and Cg are 0.01 and 0.002, respectively. R2 value for a sigmoidal fit to the data is 0.162.
Materials 12 02439 g002
Materials 12 02439 g003 550
Figure 3. Effect of high-temperature densification, as quantified by the increase in atomic packing factor (Cg), on the Poisson’s ratio (ν) of selected glasses: zinc borates (this study), aluminoborates (this study and Ref. [46]), soda-lime borates (Ref. [80]), sodium borate (Ref. [68]), SiO2 (Ref. [68]), and aluminotitanophosphates (Ref. [68]). The errors associated with ν and Cg are 0.01 and 0.002, respectively.
Figure 3. Effect of high-temperature densification, as quantified by the increase in atomic packing factor (Cg), on the Poisson’s ratio (ν) of selected glasses: zinc borates (this study), aluminoborates (this study and Ref. [46]), soda-lime borates (Ref. [80]), sodium borate (Ref. [68]), SiO2 (Ref. [68]), and aluminotitanophosphates (Ref. [68]). The errors associated with ν and Cg are 0.01 and 0.002, respectively.
Materials 12 02439 g003
Materials 12 02439 g004 550
Figure 4. Liquid fragility (m) for selected oxide glass-forming systems from this study (Table 1) plotted as a function of Poisson’s ratio (ν). No apparent correlation between m and ν is observed. The errors associated with m and ν are 1 and 0.01, respectively. R2 value for a linear fit to the data is 0.034.
Figure 4. Liquid fragility (m) for selected oxide glass-forming systems from this study (Table 1) plotted as a function of Poisson’s ratio (ν). No apparent correlation between m and ν is observed. The errors associated with m and ν are 1 and 0.01, respectively. R2 value for a linear fit to the data is 0.034.
Materials 12 02439 g004
Materials 12 02439 g005 550
Figure 5. Liquid fragility (m) as function of Poisson’s ratio (ν) for (a) oxide glass-formers (including both present compositions from Table 1 and literature data) and (b) various glass-formers from literature. References for literature data are given in the text. The dashed lines are guides for the eye, showing the trends for the majority of the data. The errors associated with m and ν are 1 and 0.01, respectively. R2 values for linear fits to the data are 0.078 and 0.178 for (a) and (b), respectively.
Figure 5. Liquid fragility (m) as function of Poisson’s ratio (ν) for (a) oxide glass-formers (including both present compositions from Table 1 and literature data) and (b) various glass-formers from literature. References for literature data are given in the text. The dashed lines are guides for the eye, showing the trends for the majority of the data. The errors associated with m and ν are 1 and 0.01, respectively. R2 values for linear fits to the data are 0.078 and 0.178 for (a) and (b), respectively.
Materials 12 02439 g005
Materials 12 02439 g006 550
Figure 6. Dependence of measured fracture toughness (KIc) on the measured Young’s modulus (E) for various glass systems (references are given in the text). Note that the axes are logarithmic. Errors in KIc and E are estimated to be smaller than 0.05 MPa m½ and 2 GPa, respectively.
Figure 6. Dependence of measured fracture toughness (KIc) on the measured Young’s modulus (E) for various glass systems (references are given in the text). Note that the axes are logarithmic. Errors in KIc and E are estimated to be smaller than 0.05 MPa m½ and 2 GPa, respectively.
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Table
Table 1. Atomic packing factor (Cg), glass transition temperature (Tg), liquid fragility (m), and Poisson’s ratio (ν) of various oxide glasses, either synthesized for this work or taken from previous studies [18,46,47,48,49,52]. The errors in Cg, Tg, m, and ν are estimated to be within ±0.002, 2 °C, 1, and 0.01, respectively.
Table 1. Atomic packing factor (Cg), glass transition temperature (Tg), liquid fragility (m), and Poisson’s ratio (ν) of various oxide glasses, either synthesized for this work or taken from previous studies [18,46,47,48,49,52]. The errors in Cg, Tg, m, and ν are estimated to be within ±0.002, 2 °C, 1, and 0.01, respectively.
Composition (mol%)Cg
(-)		Tg
(°C)		m
(-)		ν
(-)
Ca-Zr-Silicates45CaO-5ZrO2-50SiO20.523789.253.30.280
50CaO-5ZrO2-45SiO20.524806.252.00.288
Zn-Borates55ZnO-45B2O3 (a)0.566 (a)556.557.20.300 (a)
2La2O3-53ZnO-45B2O3 (a)0.565 (a)557.456.50.311 (a)
5La2O3-50ZnO-45B2O3 (a)0.572 (a)565.360.00.316 (a)
10La2O3-45ZnO-45B2O3 (a)0.580 (a)552.454.80.318 (a)
5La2O3-10GeO2-50ZnO-35B2O30.554576.341.40.311
2Ta2O5-53ZnO-45B2O30.581559.642.70.316
5Ta2O5-50ZnO-45B2O30.577563.748.30.315
2Ta2O5-55ZnO-43B2O30.583547.649.90.336
5Ta2O5-55ZnO-40B2O30.550563.748.30.317
10Sb2O3-55ZnO-35B2O30.498502.439.90.278
2La2O3-55ZnO-43B2O30.583533.038.90.302
5La2O3-55ZnO-40B2O30.551557.146.40.309
10La2O3-55ZnO-35B2O30.539542.242.20.310
2La2O3-2Ta2O5-53ZnO-43B2O30.617538.750.20.316
5La2O3-2Ta2O5-50ZnO-43B2O30.580539.544.60.325
5La2O3-5Ta2O5-50ZnO-40B2O30.569547.243.10.312
Aluminoborates25MgO-20Al2O3-55B2O3 (b)0.565 (b)636 (b)56.50.283
25CaO-20Al2O3-55B2O3 (b)0.551 (b)615 (b)54.80.220
25SrO-20Al2O3-55B2O3 (b)0.537 (b)590 (b)60.00.266
25BaO-20Al2O3-55B2O3 (b)0.545 (b)554 (b)57.20.290
18.75Li2O-6.25BaO-20Al2O3-55B2O30.531484.439.60.284
20Li2O-5MgO-20Al2O3-55B2O3(c)0.553 (c)482 (c)40.60.247
25Cs2O-20Al2O3-55B2O3 (d)0.479 (d)416 (d)48.80.319 (d)
25Cs2O-5Ga2O3-15Al2O3-55B2O30.480421.228.00.324
25Cs2O-10Ga2O3-10Al2O3-55B2O30.474418.725.30.320
25Cs2O-2Ta2O3-18Al2O3-55B2O30.474432.128.70.318
23Cs2O-2Ta2O3-20Al2O3-55B2O30.470433.622.20.316
21Cs2O-4Ta2O3-20Al2O3-55B2O30.475449.727.70.300
Aluminoborosilicates25Na2O-75SiO2 (e)0.49 (e)475 (e)33.30.25 (e)
25Na2O-12.5B2O3-62.5SiO2 (e)0.52 (e)539 (e)43.80.22 (e)
25Na2O-25B2O3-50SiO2 (e)0.55 (e)544 (e)48.70.22 (e)
25Na2O-37.5B2O3-37.5SiO2 (e)0.56 (e)525 (e)48.80.24 (e)
25Na2O-50B2O3-25SiO2 (e)0.56 (e)511 (e)50.60.25 (e)
25Na2O-62.5B2O3-12.5SiO2 (e)0.56 (e)495 (e)50.20.25 (e)
25Na2O-75B2O3 (e)0.56 (e)473 (e)51.40.27 (e)
25Na2O-12.5Al2O3-62.5SiO2 (e)0.49 (e)567 (e)32.80.23 (e)
25Na2O-12.5Al2O3-12.5B2O3-50SiO2 (e)0.51 (e)545 (e)50.40.24 (e)
25Na2O-12.5Al2O3-25B2O3-37.5SiO2 (e)0.52 (e)514 (e)43.30.25 (e)
25Na2O-12.5Al2O3-37.5B2O3-25SiO2 (e)0.52 (e)493 (e)46.40.26 (e)
25Na2O-12.5Al2O3-50B2O3-12.5SiO2 (e)0.52 (e)480 (e)48.90.27 (e)
25Na2O-12.5Al2O3-62.5B2O3 (e)0.52 (e)465 (e)39.00.29 (e)
25Na2O-25Al2O3-50SiO2 f)0.49 (e)792 (e)38.50.21 (e)
25Na2O-25Al2O3-12.5B2O3-37.5SiO2 (e)0.49 (e)611 (e)30.00.25 (e)
25Na2O-25Al2O3-25B2O3-25SiO2 (e)0.49 (e)511 (e)28.40.26 (e)
25Na2O-25Al2O3-37.5B2O3-12.5SiO2 (e)0.50 (e)468 (e)30.70.28 (e)
25Na2O-25Al2O3-50B2O3 (e)0.50 (e)459 (e)32.40.29 (e)
25Na2O-30Al2O3-45B2O3 (e)0.50 (e)528 (e)31,90.27 (e)
25Na2O-30Al2O3-32.5B2O3-12.5SiO2 (e)0.50 (e)469 (e)27.90.29 (e)
(a) Glasses and/or data are from Ref. [52]; (b) Glasses and/or data are from Ref. [48]; (c) Glasses and/or data are from Ref. [46]; (d) Glasses and/or data are from Ref. [18]; (e) Glass is from Ref. [49].

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Østergaard, M.B.; Hansen, S.R.; Januchta, K.; To, T.; Rzoska, S.J.; Bockowski, M.; Bauchy, M.; Smedskjaer, M.M. Revisiting the Dependence of Poisson’s Ratio on Liquid Fragility and Atomic Packing Density in Oxide Glasses. Materials 2019, 12, 2439. https://doi.org/10.3390/ma12152439

AMA Style

Østergaard MB, Hansen SR, Januchta K, To T, Rzoska SJ, Bockowski M, Bauchy M, Smedskjaer MM. Revisiting the Dependence of Poisson’s Ratio on Liquid Fragility and Atomic Packing Density in Oxide Glasses. Materials. 2019; 12(15):2439. https://doi.org/10.3390/ma12152439

Chicago/Turabian Style

Østergaard, Martin B., Søren R. Hansen, Kacper Januchta, Theany To, Sylwester J. Rzoska, Michal Bockowski, Mathieu Bauchy, and Morten M. Smedskjaer. 2019. "Revisiting the Dependence of Poisson’s Ratio on Liquid Fragility and Atomic Packing Density in Oxide Glasses" Materials 12, no. 15: 2439. https://doi.org/10.3390/ma12152439

APA Style

Østergaard, M. B., Hansen, S. R., Januchta, K., To, T., Rzoska, S. J., Bockowski, M., Bauchy, M., & Smedskjaer, M. M. (2019). Revisiting the Dependence of Poisson’s Ratio on Liquid Fragility and Atomic Packing Density in Oxide Glasses. Materials, 12(15), 2439. https://doi.org/10.3390/ma12152439

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