Aalborg Universitet Topological model of alkali germanate glasses and exploration of the germanate anomaly

Aalborg Universitet

Topological model of alkali germanate glasses and exploration of the germanate anomaly

Welch, Rebecca S.; Wilkinson, Collin J.; Shih, Yueh-Ting; Bødker, Mikkel Sandfeld;

DeCeanne, Anthony V.; Smedskjær, Morten Mattrup; Huang, Liping; Affatigato, Mario; Feller, Steve A.; Mauro, John C.

Published in:

Journal of the American Ceramic Society

DOI (link to publication from Publisher):

10.1111/jace.17102

Creative Commons License CC BY-NC-ND 4.0

Publication date:

2020

Document Version

Accepted author manuscript, peer reviewed version Link to publication from Aalborg University

Citation for published version (APA):

Welch, R. S., Wilkinson, C. J., Shih, Y-T., Bødker, M. S., DeCeanne, A. V., Smedskjær, M. M., Huang, L., Affatigato, M., Feller, S. A., & Mauro, J. C. (2020). Topological model of alkali germanate glasses and exploration of the germanate anomaly. Journal of the American Ceramic Society, 103(8), 4224-4233.

https://doi.org/10.1111/jace.17102

DR MORTEN SMEDSKJAER (Orcid ID : 0000-0003-0476-2021) PROFESSOR LIPING HUANG (Orcid ID : 0000-0001-6121-5054) DR JOHN C. MAURO (Orcid ID : 0000-0002-4319-3530)

Article type : Article

Topological model of alkali germanate glasses and exploration of the germanate anomaly

Rebecca S. Welch¹, Collin J. Wilkinson², Yueh-Ting Shih³, Mikkel S. Bødker⁴, Anthony V.

DeCeanne²,Morten M. Smedskjaer⁴, Liping Huang³, Mario Affatigato¹, Steve A. Feller¹ and John C.

Mauro^2,*

1Department of Physics, Coe College, Cedar Rapids, IA, USA

2Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA, USA

3Department of Materials Science and Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA

4Department of Chemistry and Bioscience, Aalborg University, Aalborg, Denmark

*Corresponding author: jcm426@psu.edu

Keywords: Statistical mechanics; Topological constraint theory; Germanate glasses; Germanate anomaly; Modeling

Abstract

Accepted Article

Germanate glasses are of particular interest for their excellent optical properties as well as their abnormal structural changes that appear with the addition of modifiers, giving rise to the so-called germanate anomaly. This anomaly refers to the non-monotonic compositional scaling of properties exhibited by alkali germanate glasses and has been studied with various spectroscopy techniques.

However, it has been difficult to understand its atomic scale origin, especially since the germanium nucleus is not easily observed by NMR. In order to gain insights into the mechanisms of the germanate anomaly, we have constructed a structural model using statistical mechanics and topological constraint theory to provide an accurate prediction of alkali germanate glass properties.

The temperature onsets for the rigid bond constraints are deduced from in-situ Brillouin light scattering, and the number of constraints is shown to be accurately calculable using statistical methods. The alkali germanate model accurately captures the effect of the germanate anomaly on glass transition temperature, liquid fragility, and Young’s modulus. We also reveal that compositional variations in the glass transition temperature and Young’s modulus are governed by the O-Ge-O angular constraints, whereas the variations in fragility are governed by the Ge-O radial constraints.

Accepted Article

1. Introduction

Germanate glasses are known for their excellent optical properties,^1,2 making them suitable for application in optical systems^3-5 due to their low transmission losses⁶ and low phonon energies⁷ compared to many silicates.⁸ They also feature high solubility for rare earth elements,^9,10 which has facilitated interest in the development of germanate-based optical sensors,¹¹ solar cells,¹² lasers,^13,14 and waveguides.^9,15 Alkali germanates are particularly interesting for their unique composition dependence of properties such as density,^16,17 refractive index,¹⁶ elastic moduli,¹⁸ and glass transition temperature (Tg).¹⁹ In silicate glasses, the addition of alkali results in a monotonic scaling of many properties, corresponding to the formation of non-bridging oxygens (NBOs).²⁰ However, in germanate systems, the trends are largely non-monotonic upon addition of alkali oxides. For instance, the density of alkali silicates increases gradually with added modifier,^20,21 but for alkali germanates, the density passes through a maximum, and then decreases.¹⁹ This characteristic reversal in property scaling is known as the germanate anomaly.^16,22,23 However, the structural origin of this anomaly has been difficult to characterize, mainly because the ⁷³Ge isotope is not suitable for characterization using nuclear magnetic resonance (NMR) spectroscopy.²⁴ Recent implementation of increased field strength in NMR systems has allowed for exploration of ⁷³Ge crystalline compounds where it has been shown that Ge can be investigated, although with significant difficulty.^25,26 Specific coordination environments still remain inconclusive for germanate glasses; however, there is potential for insights in the future.²⁴ In general, other spectroscopy techniques are needed to determine such local environments including infrared (IR) spectroscopy,^16,27 extended X-ray absorption fine structure (EXAFS),^28,29 X-ray absorption near edge structure (XANES),³⁰ X-ray photoelectron spectroscopy (XPS),^28,31 X-ray and neutron diffraction,^32-34 and Raman spectroscopy.17,27,35-38 The main issue with the use of these techniques is the disagreement within the community regarding interpretation of the data. As such, it has been difficult to precisely determine germanium speciation and how it changes in the anomalous composition regime.

The qualitative conclusion is that there exists an increase in the Ge coordination number, and that this change plays a critical role in the scaling of all the glass properties.³⁹ That is, the formation of higher coordinated units (GeOn, designated as Nⁿ units) has been suggested to cause the densification of alkali germanates,^16,27,40 while the formation of NBOs causes the characteristic reversal in property

Accepted Article

trends.^23,28 However, the current debate is which high-coordinated unit forms, the N⁵ unit30,32,35,41,42 or N⁶ unit,19,23,27,28,33,34,38,40 and which contributes the most to the anomalous property scalings.

Although, even the existence of these units has been debated since the first discovery of the germanate anomaly.^36,37,43 For instance, Raman spectroscopy analysis of sodium germanate glasses conducted by Henderson and Fleet¹⁷ suggested that there is no evidence for the formation of higher coordinated units. Instead the densification is caused by the network saturation of three-membered rings of GeO4 tetrahedra, while the density decrease is a result of the formation of new Q³ units, where Qⁿ refers to the number of bridging oxygens in a GeO4 tetrahedral unit. A study by Hoppe,⁴⁴ who extracted packing densities of potassium germanates from neutron and X-ray diffraction, argued that there exists a combination of five-coordinated trigonal bipyramidal units and six-coordinated octahedra and that both of these structural units cause the anomaly. However, it was argued that there exists great difficulty in differentiating concentrations of GeO5 versus GeO6 using neutron and X-ray diffraction.^45,46 Recent work conducted by Hannon et al.³⁹ tried to distinguish these units by applying the Wright et al. model⁴⁷ for borate glasses to alkali germanate families. This model was previously used to predict boron coordination, in which there also exists an anomaly.⁴⁸ It was concluded that GeO5 is more abundant than GeO6 when applied to cesium germanates and that GeO5 has the largest contribution to the anomalous variation in, for instance, Tg.³²

Another proposed model for alkali germanate families was introduced by Kiczenski et al.,¹⁹ who combined the existing alkali silicate model^20,21 with results obtained by Jain et al.⁴⁰ of rubidium germanates. From their work, the packing fractions of individual germanate structural units were determined. It was concluded that GeO6 had the highest packing fraction and that the packing density maximum was caused by void filling, which was also previously proposed by Weber,⁴⁹ as well as concurrent coordination changes. The anomaly in the Tg trend was observed at ~16 mol% alkali oxide, with the anomalous behavior believed to be caused by the formation of six-coordinated germanium.

The germanium models found in the literature, which have been developed based on experimental studies around the anomalous compositional regime, have resulted in conflicting conclusions.^50,51 In a review by Henderson,⁵⁰ it has become clear that several areas pertaining to the germanate anomaly still remain ambiguous and must be clarified, including a need for the

Accepted Article

explanation of what drives the structural reorganization of this glass system as well as why this system differs from that of alkali silicate glasses. An in-depth model which has a physical basis would provide a fundamental understanding of the structural changes that occur in the anomalous regime, but such development has been limited in the past with mainly experimental fitting techniques or referencing of GeO₂ crystalline structures, such as in the case of ab initio molecular simulations.⁵¹

More recently, there has been an introduction of a new structural model (not yet applied to germanate systems) which has shown significant potential in accurately capturing coordination environments based on physical parameters. This statistical mechanics model, initially derived by Goyal and Mauro,⁵² focuses on the noncentral hypergeometric distribution of modifiers in a glass network. By fitting Boltzmann weighting factors to each type of network modifier-former interaction, the model enables a statistical prediction of the composition dependence of network former speciation in mixed-modifier compositions. An extension of this model has been developed by Bødker et al.,⁵³ showing accurate prediction of structure speciation in alkali phosphate systems using calculated enthalpies from known fictive temperatures (Tf). When compared to experimental results, this model features excellent agreement in Qⁿ-speciation.

Topological constraint theory (TCT) has also been shown to be a valuable model in predicting various glass properties. It quantifies the network rigidity of glasses by calculating the atomic degrees of freedom relative to the number of constraints that exist between network-forming atoms. This framework is based on the original work of Phillips and Thorpe⁵⁴ who sought to simplify theories of glass formation by focusing on the constraints in the network, specifically, interactions pertaining to the flexibility of bond angles and bond lengths tied to compositional differences. The theory was later extended by Mauro et al.⁵⁵ who took the temperature dependence of the constraints into consideration.

Predictions based on this model have been found to be in excellent agreement with experimental results for Tg,⁵⁶ hardness,^57,58 Young’s modulus,^59,60 fragility,^61,62 heat capacity jump at the glass transition,⁶³ as well as other mechanical properties of glasses.⁶⁴

The structural model developed by Kiczenski et al.¹⁹ shows good agreement with experimental data and captures the anomalous behavior of alkali germanates. However, the model was fitted to densities from literature whereas the statistical mechanics model⁵³ is capable of providing a structural model which is based on thermodynamic parameters. Therefore, by applying these parameters we can

Accepted Article

obtain a new model for the alkali germanate system which has physical implications. The results can thus be used with topological constraint theory to predict properties such as Tg, fragility, and elastic modulus once the constraint onset temperatures have been determined with in-situ Brillouin light scattering. By combining the statistical mechanics model with topological constraint theory, we present an over-arching method which can be used to characterize structural and macroscopic properties of alkali germanates simultaneously. Thus, capturing the anomalous properties found in alkali germanate systems, as well as providing insights into what drives these structural changes.

2. Model

2.1. Structure of Alkali Germanate Glasses

The statistical mechanics model⁵³ is used to determine the speciation of glass structural units as a function of the modifier concentration. As the number of modifiers (i.e., alkali) increases, there is a direct impact on the number of network former species available (i.e., germanium) since each modifier can be thought of as “drawing” a network former from a pool of finite network former species in order to form the glass matrix. Each draw changes the number of remaining species and concurrently changes the probability of forming specific coordinated units depending on enthalpically and entropically favored configurations. Thus, the probability (p) of a species type (i) on a given draw (), i.e., addition of a modifier, depends on the degeneracy of the species (g) and the accumulated number of successful draws (n). It is also dependent on the previous number of attempts (j), the number of distinct species in the pool (), and the weighting factor (w). The probability is given by⁵²

,

𝑝_𝑖,𝑤= ^{(𝑔𝑖― 𝑛𝑖,𝜔 ―1)𝑤𝑖}

∑^Ω

𝑖= 1∑^{𝜔 ―1}

𝑗= 0(𝑔_𝑖― 𝑛_𝑖,𝑗)𝑤_𝑖

where the weighting factor is then given by the Boltzmann distribution and with each network former draw, the probability changes accordingly. This drawing will also occur at the relevant temperature at which the structure is frozen in, i.e., the fictive temperature (Tf),

Accepted Article

, (3) 𝑤_𝑖= exp

( ^―

)

where Hi is the enthalpy associated with forming an association with the drawn species of type i and k is the Boltzmann’s constant. It is assumed that the fictive temperature is equal to the glass transition temperature. In order to calculate the Hi model parameters based on the relevant structural evolution, a Python script was developed based on a basin-hopping algorithm which determines the global minimum in the error between the fit and the experimental values. Further details can be found in the extensive works of Bødker et al.^53,65,66 Due to the difficulty of using NMR spectroscopy to study germanate glasses, the model was not fit to experimental NMR data, but rather fitted to the proposed structural model by Kiczenski et al.¹⁹ shown in Figure 1. It is worth noting that this structural model is, in turn, based on the experimental work of Jain et al.⁴⁰ for the rubidium germanate system.

In order to utilize the statistical mechanics model,⁵³ several possible structural transitions must be considered. The more typical transition in the Q units when a modifier atom (M) is added is represented as:

(4) Q^𝑛+ M→Q^{𝑛 ―1}

However, to account for the germanate anomaly, two additional transitions must be considered:

(5) Q⁴+2M→N⁶

and

. (6)

N⁶+ M→Q³+2M

The enthalpy change of each of the three structural transitions are then fit to minimize the error between the present statistical mechanical model and the model defined by Kiczenski et al.¹⁹ In order to correctly capture the anomaly, the mechanism outlined in Eq. 4, where the addition of two alkali converts a four coordinated unit to a six coordinated one, was replaced with Eq. 5 at 18 mol%; this will be justified shortly. The parameters are shown in Table 1 and the fitted model is shown in Figure 2 with an overlaid comparison of the Kiczenksi et al. model in Figure 3. The fictive temperatures used to obtain the relative enthalpy values are shown later in Figure 5(a). When the enthalpies were fitted with the statistical mechanics model, there was a failure to accurately reproduce the anomaly.

Therefore, there must be some limiting factor that disables all of the Q⁴ units being converted entirely

Accepted Article

to N⁶ units. As such, a cutoff was implemented at 18 mol%, which was fitted to the structural model proposed by Kiczenksi et al.¹⁹

2.2. Topology of Alkali Germanate Glasses

Once the structural parameters have been established, it is possible to calculate a variety of properties using topological constraint theory. Three relevant constraint types are considered with TCT:  constraints which are linear constraints between germanium and oxygen atoms (Ge-O),  constraints which represent the angular O-Ge-O bonds, and  constraints which represent the Ge-O- Ge bond angles. Since the mean coordination numbers have here been determined from the statistical mechanics model, it is possible to calculate the number of each constraint which allows for the prediction of various industrially significant properties such as Tg, fragility, and elastic modulus. The number of constraints per atom is calculated in the following equations, as outlined by Wilkinson et al.:⁶⁷

𝑛_𝛼= 2

[

(

)]

, (6) 𝑛_𝛽=

[

(

)]

. 𝑛_𝛾=

[

(

)]

In order to predict the glass transition temperature, the work by Gupta and Mauro⁶⁸ has shown that Tg

is proportional to the configurational entropy, Sc,

, (7)

𝑇_𝑔(𝑥)

𝑇_𝑔(𝑥_𝑟)

=

𝑐[𝑇_𝑔(𝑥),𝑥]

for composition, x, where the subscript r denotes a reference value. The work of Naumis^69,70 has shown that the configurational entropy is also proportional to the degrees of freedoms, f, such that

.

𝑇_𝑔(𝑥)

𝑇_𝑔(𝑥_𝑟)

=

𝑔(𝑥),𝑥]

=

𝑔(𝑥),𝑥]

Here, the degrees of freedom are related to the number of constraints (n) and the dimensionality of the network (d). Eq. (9) enables prediction of the glass transition temperature when 𝑛[𝑇_𝑔(𝑥_𝑟),𝑥_𝑟] is the number of constraints that are rigid at Tg. After determining the number of constraints, the temperature dependence of the constraints must also be calculated using a probabilistic model⁵⁵ where

Accepted Article

. (9) 𝑞_𝑐(𝑇) = [1―exp (―^∆𝐹

𝑐∗

𝑘𝑇)]^𝑣𝑡

Here, qc is the probability of constraint c being rigid, ∆𝐹_𝑐^∗ is the activation barrier for breaking the constraint, and t is the frequency times time which corresponds to the number of escape attempts. To calculate the activation barrier, the onset temperature (Tc) previously fitted from BLS is used in

. (10)

∆𝐹_𝑐^∗ =―𝑘𝑇_𝑐ln

(

)

This temperature dependence also enables prediction of liquid fragility (m), which is defined as

. (11)

𝑚(𝑥) =^{∂log10𝜂(𝑇,𝑥)}

∂

[

] |

𝑇=𝑇𝑔(𝑥)

This has been shown to be equivalent to the topological expression

(12) 𝑚(𝑥) =𝑚₀

(

^|

)

where m0 is the lower limit of fragility and found to be 14.97.⁷¹ In addition to enabling the topological prediction of fragility, constraint theory can also be used to predict the temperature and compositional dependence of Young’s modulus.^59,60 The free energy density (∆𝐹_𝑐) of the constraints can be written as

(13)

∆𝐹_𝑐=^{𝜌(𝑥)𝑁}_𝑀(𝑥)^𝐴∑

𝑙=𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡𝑠∆𝐹_𝑙𝑛_𝑙(𝑥)𝑞_𝑙(𝑇)

in which  is the density, M is the molar mass, NA is Avogadro’s number, ql(T) is the onset function for each respective constraint as outlined in Eq. 10, and c is the type of constraint. The free energy density can be converted to the Young’s modulus (E) with

. (14)

𝐸=_𝑑∆𝐹^𝑑𝐸_𝑐[∆𝐹_𝑐(𝑥)― Δ𝐸^′]

Here, ∆𝐸^′ is the free energy density intercept fitted to experimental Young’s modulus values.

This provides a temperature and compositional dependence of Young’s modulus as a function of the free energy associated with the magnitude of each constraint.

3. Experimental Methods

In order to obtain the onset temperatures needed for the temperature-dependent constraint model, a binary glass composition of 0.2K2O·0.8GeO2 was studied using in-situ Brillouin light scattering (BLS) spectrometry with a light source of 532 nm Verdi V2 DPSS green laser. The sample was

Accepted Article

synthesized by mixing reagent grade K2CO3 and GeO2 powders in an alumina crucible before being melted in a bottom-loading box furnace. The sample was heated from 693 to 1773 K at 5 K/min., followed by a 1-hour dwell at 1773 K. The melt was poured into a mold before being annealed overnight (approximately 15 hours) at 658 K. The density of the annealed glass was measured by the Archimedes’ method using ethanol as the liquid medium. For the BLS measurement, the annealed glass was cut and optically polished to 100–200 μm in thickness with parallel top and bottom surfaces using 600 grit silicon carbide sandpaper and cerium oxide slurry. To monitor the changes of elastic moduli as a function of temperature, BLS measurement was taken through the top fused quartz window of a Linkam TS1500 heating stage. The polished glass sample was heated from room temperature to temperature beyond the Tg at a heating rate of 50 K/min. Once the temperature inside the heating stage had stabilized for 5 min., BLS spectrum was collected in the emulated platelet geometry (EPG). Details of the experimental set-up and the light scattering geometry can be found elsewhere.^72,73 From the measured longitudinal (VL) and transverse sound (VT) velocities in BLS, Young’s modulus (E) can be calculated from the longitudinal modulus (Mx) and the shear modulus (S) by using the following equations:

, (15)

𝑀_𝑥=𝜌𝑉²_𝐿

, (16)

𝑆=𝜌𝑉²_𝑇

. (17)

𝐸=𝑆^{3𝑀 ― 4𝑆}_{𝑀 ― 𝑆}

In this study, the density ( ) was assumed to be constant within the measured temperature range. This 𝜌 is due to the volumetric thermal expansion being a low value (on the order of 10^-6/K)⁷² which has a relatively small influence on the calculated modulus (less than 1%).

4. Results

It has been shown previously that Brillouin scattering allows for fast and convenient quantification of constraint onset temperatures.⁵⁹ The average density of the studied potassium germanate glass needed for Brillouin measurements was found to be 3.49834 g/cm³ with a standard deviation of 0.00242 for five runs. The fitted model and experimental data for the temperature dependence of Young’s modulus is shown in Figure 4. The fitted constraint onset temperatures are T_

Accepted Article

= 508 K, T = 1636 K, and T = 320 K; though there appears to be a transition around 775 K, this is extremely close to the glass transition and with a lower Tit is not possible to reproduce the other properties accurately. The number of escape attempts, vt, was fitted to be 15000. Once these values are established, it is possible to calculate the glass transition temperature and liquid fragility.

Experimental rubidium germanate Tg values, obtained from Kiczenski et al.,¹⁹ were used to compare with the model calculations (Figure 5(a)). Literature values for potassium germanate fragilities were taken from the work by Shelby⁷⁴ and are overlaid in Figure 5(b). Figure 5(c) shows the compositional dependence of the modulus with the two empirical parameters (∆𝐹_′𝑐 and 𝑑𝐸/𝑑∆𝐹_𝑐) fit to literature data of potassium germanates.⁷⁵ The density values for these calculations were taken by linearly extrapolating the densities also reported by Kiczenski et al.¹⁹ Figure 6 shows the contribution of , , and  constraints associated with the influence of N⁶ units as a function of composition and temperature.

5. Discussion

The model proposed by Kiczenski et al.¹⁹ predicts that the N⁶ unit is the predominant high- coordinated unit in the anomalous region for binary sodium, lithium, potassium, and rubidium germanates. This configuration is believed to be the most efficient packing arrangement and is able to accurately account for the observable increases in density.¹⁹ Here, the statistical mechanics model is considered in order to provide a more detailed coordination environment which is physically meaningful and based on enthalpic values. The results enable use of topological constraint theory to predict properties such as Young’s modulus, fragility, and Tg. From the statistical mechanics prediction of speciation, we found a consistent presence of the N⁶ unit even at concentrations above 18 mol% alkali oxide (Fig. 2). This contrasts the structural model by Kiczenski et al. where the N⁶ reaches a maximum around 20 mol% before decreasing in concentration. Since our model shows that this unit continues to exist at the same fraction even past the anomalous region, we suggest that the N⁶ unit is the most energetically favorable. After reaching 40 mol% alkali oxide, N⁶ declines rapidly at the same composition where the Q² unit emerges, thus making the Q² more energetically preferred at higher alkali percentages.

Accepted Article

Due to the well documented coordination change of boron in borate glasses, in which there is a maximum in the fraction of tetrahedral boron in modified borate glasses, it is unsurprising that the abnormalities in the germanate systems have often been compared.^32,39 The in-depth exploration and popularity of the borate anomaly is attributed to the NMR active behavior of ¹⁰B^48,76,77 and ¹¹B.^76,78 Several structural models aimed at successfully predicting the borate anomaly have been proposed including the model developed by Wright et al.⁴⁷ and the random pair model developed by Gupta.⁷⁹ With the former model, an alternating network of three and four coordinated borate structural units were hypothesized and the premise of this model has been applied to the germanate structural model by Hannon et al.³⁹ In contrast, the random pair model also accurately predicts speciation in borate and borosilicate glasses,^63,80 although the experimental evidence for the presence of B^IV-O-B^IV linkages has been questioned.^81,82 The model captures the boron coordination change by proposing that tetrahedral borate units are bridged together and incorporated into the network as pairs. However, no such correlation can be deduced with the random pair model if applied to germanate systems.

Since the model by Hannon et al.³⁹ provides a detailed description of the coordination environment for the alkali germanate glass system, it is then possible to input this information into topological constraint theory to predict the same properties shown previously. Thus, allowing a direct comparison of the proposed structural model here, which only considers the influence of N⁶ units, to that of Hannon et al.’s, which only assumes the N⁵ unit. With the two models in Figure 7, we can notice that both provide good agreement in the initial upward slopes for the glass transition temperature and elastic modulus. However, discrepancies between the two models became apparent in the tail ends as well as in the fragility graph, leading to an overprediction of the experimental values with the Hannon et al. model. While the influence of the N⁶ unit is debated in literature,30,32,41,42 the agreement between experimental results with our proposed model after direct comparison with that of Hannon et al., suggests that six coordinated germanium does in fact have a significant impact in the germanate anomaly. Therefore, the presence of five-coordinated germanium units plays a lesser role in the overall determination of the properties studied herein.

In literature, comparisons between silicates and germanates have also been long discussed since they share a large number of similarities, particularly, that both silica and germanate glasses exist in the four-coordinated state when no modifiers are present, and that both exist in a higher coordination

Accepted Article

state if exposed to high pressure.^83-86 Furthermore, molecular dynamics studies by Micoulaut⁸⁷ have found even more similarities, especially pertaining to their respective liquid states. At the local structure, there are no observable differences between the two systems and differences only occurred at the long-range order while at low temperatures. Since both silicon and germanium share the same coordination when no modifiers are present as well as the same valence number, the difference between these two elements is the electron configuration, most notably the d orbital interactions. In vitreous silica, the conversion from four to a higher coordination state while under pressure has been discussed to be caused by the compression of oxygen atoms surrounding the silicon, which induces interactions between the oxygen lone pair orbitals with the empty 3d orbital. As such, the utilization of the empty 3d orbital causes the coordination number increase and leads to a more stable local environment.⁸⁸ Since germanium naturally has a 3d orbital while silicon does not, it is possible that the presence of such an orbital makes the change from four to six coordination easier than what is observed in silica.

Despite the structural similarities between silicates and germanates, results from topological constraint theory begin to elucidate the qualitative differences between the two systems. When constraint theory is applied to silicate systems, it has been shown that linear  constraints dictate most properties, while  constraints play a lesser role.⁸⁹ However, when applied to the germanate system, the  constraint is much weaker, having an onset temperature of only 330 K, while in silica it is

~1700 K.⁸⁹ Thus, the  constraints play only a minor role throughout each compositional change as seen in Figures 5(a) and 5(c). From Figure 5(c), we can note the unusually high fraction of  constraints, which has been rarely observed in previous topological constraint theory applications,⁵⁵ as well as little influence of  and  constraints contributing to the elastic modulus. Since the angular  constraint refers to the O-Ge-O bond within the germanate structure while the  constraint refers to the Ge-O bond, then the bond interactions from the  constraints are the strongest and most rigid.

This greatly contrasts previous results of topological constraint theory in other systems where  constraints show the highest contribution to properties such as Tg, Young’s modulus, as well as hardness due to its high onset temperature.^57,59,62

6. Conclusions

Accepted Article

A statistical mechanics model is derived for alkali germanate glasses, successfully providing a structural view of these glasses based on thermodynamic considerations. The resulting coordination environments from the model were implemented into topological constraint theory and used to predict industrially relevant properties such as Tg, fragility, and Young’s modulus. The strong agreement between predicted values and experimental values suggests that six coordinated germanium units have a significant impact on such properties. In addition, the use of topological constraint theory has given us new insights on the influence of constraints found within the glass system, mainly that the  constraint dictates Tg and elastic modulus. The combination of the statistical mechanics model with topological constraint theory has enabled a convenient method for industrial and academic use, as well as establishing a more complete picture of the binary alkali germanate glass family.

Acknowledgments

MMS acknowledges funding from the Independent Research Fund Denmark (grant no. 7017-00019).

RSW, MA, and SAF acknowledge funding from the United States National Science Foundation (grant no. DMR-1746230).

Tables

Table 1. Enthalpy parameters calculated using Eq. 2, from the statistical mechanics model⁵³ for the considered structural transitions. Explicit details of the calculations are extensive and can be found in previous works by Bødker et al.^53,65,66 The first transition outlined, Q⁴ N→ ⁶, is the baseline transition

which was used for normalization.

Transition Relative enthalpy (kJ/mol)

Q⁴ N→ ⁶ 0

N⁶ + M Q→ ³ + 2M 80.4

Q³ Q→ ² 132.8

Accepted Article

Figures Captions

Figure 1. The structural model proposed by Kiczenski et al.,¹⁹ showing that the increase in N⁶ units occurs only until about 20% modifier oxide (x), after which it continuously decreases.

Figure 2. Statistical mechanical model,⁵³ showing the compositional changes of each Q-species with added alkali.

Figure 3. Comparison between the model by Kiczenski et al.¹⁹ and the proposed model using statistical mechanics.⁵³

Figure 4. Young’s modulus as a function of temperature of the tested 0.2K2O·0.8GeO2 glass.

Experimental error bars are smaller than the points. The Tg was predicted with topological constraint theory as outlined in the methods. Good agreement is observed when the temperature is not close to Tg. At temperatures above the Tg, the poor agreement is due to the change from an elastic to viscous response.

Figure 5. Composition dependence of experimental and predicted glass properties. The contribution of each constraint is determined by taking the predicted property value multiplied by the number of that constraint and divided by the total number of active constraints. (a) Predicted glass transition overlaid with glass transition temperatures of rubidium germanate glasses collected by Kiczenski et al.¹⁹ Error bars for experimental Tg values are the same size as the points (±7°C). (b) The liquid fragility of potassium germanates calculated by fitting the onset temperatures to Brillouin scattering results are shown and then overlaid with experimental fragilities.⁷⁴ Experimental error for fragilities is 5%. (c) Elastic modulus prediction at room temperature with ∆F^′_c = 0.08 GPa and dE/d∆F_c = 15

Accepted Article

GPa/eV. The experimental data of potassium germanate glasses was taken by Osaka et al.⁷⁵ Experimental error values were not reported.

Figure 6. The number of rigid , , and  constraints as calculated from topological constraint theory vs. composition with the glass transition region labeled. The dashed line is located at 18 mol% of alkali oxide and corresponds to the property changes associated with the germanate anomaly.

Figure 7. The two different models showing the compositional dependence of the previously shown properties. The model by Hannon et al.³⁹ (dashed) assumes an alternating network of GeO4 and GeO5

units while the proposed model (solid line) is derived from statistical mechanics with the initial assumption that GeO6 units are formed, and not GeO5. (a) Shows the glass transition temperature predictions, (b) the fragility, (c) the elastic modulus.

Accepted Article

References

1. Anan’ev AV, Bogdanov VN, Champagnon B, Ferrari M, Karapetyan GO, Maksimov LV, Smerdin SN, Solovyev VA. Origin of Rayleigh scattering and anomaly of elastic properties in vitreous and molten GeO2. J. Non-Cryst. Solids. 2008; 354(26): 3049-58.

2. Sakaguchi S, Todoroki S, Rigout N. Optical properties in ternary germanate glasses. J. Non- Cryst. Solids. 1996; 196: 58-62.

3. Baccaro S, Cecilia A, Chen G, Du Y, Montecchi M, Wang H, Wang S. Effects of irradiation on transmittance of cerium doped germanate glasses in the ultraviolet and visible regions.

Nucl. Instr. and Meth. in Phys. Res. B. 2002; 191(1-4): 352-355.

4. Kochanowicz M, Dorosz D, Zmojda J, Miluski P, Dorosz J. Effect of temperature on upconversion luminescence in Yb3+/Tb3+co-doped germanate glass. Acta Physica Polonica A. 2013; 124(3): 471-473.

5. Luo Y, Qu C, Bhadu A, Mauro JC. Synthesis and characterization of K2O-ZnO-GeO2-SiO2

optical glasses. J. Non-Cryst. Solids. 2019; 503-504: 308-312.

6. Sakaguchi S, Todoroki S. Optical properties of GeO2 glass and optical fibers. Appl. Opt. 1997;

36(27): 6809-6814.

7. Xu R, Tian Y, Hu L, Zhang J. Broadband 2 μm emission and energy-transfer properties of thulium-doped oxyfluoride germanate glass fiber. Applied Physics B. 2011; 104(4): 839-844.

8. Brawer SA, White WB. Raman spectroscopic investigation of the structure of silicate glasses.

I. The binary alkali silicates. J. Chem. Phys. 1975; 63: 2421.

9. Li HY, Shen LF, Pun EYB, Lin H. Dy³⁺-doped germanate glasses for waveguide-typed irradiation light sources. J. Alloys and Compounds. 2015; 646: 586-591.

10. Pan Z, Morgan SH, Loper A, King V, Long BH, Collins WE. Infrared to visible upconversion in Er³⁺ doped lead germanate glass: effects of Er³⁺ ion concentration. J. Appl. Phys. 1995; 77:

4688.

11. Pisarski WA, Pisarska J, Lisiecki R, Ryba-Romanowski W. Er³⁺/Yb³⁺ co-doped germanate glasses for up-conversion luminescence temperature sensors. Sensors and Actuators A:

Physical. 2016; 252: 54-58.

Accepted Article

12. Zhang Q, Zhu B, Zhuang Y, Chen G, Liu X, Zhang G, Qiu J, Chen D. Quantum cutting in Tm³⁺/Yb³⁺-codoped lanthanum aluminum germanate glasses. J. Amer. Ceram. Soc. 2010;

93(3): 654-657.

13. Zhang Y, Lu C, Feng Y, Sun L, Yaru N, Xu Z. Effects of GeO2 on the thermal stability and optical properties of Er³⁺/Yb³⁺ -codoped oxyfluoride tellurite glasses. Materials Chemistry and Physics. 2011; 126(3): 786-790.

14. Jayasimhadri M, Cho E, Jang K, Lee HS, Kim SI. Spectroscopic properties and Judd-Ofelt analysis of Sm³⁺ doped lead-germanate-tellurite glasses. J. Phys. D. Appl. Phys. 2008; 41(17):

175101.

15. Zhu CL, Pun EYB, Wang ZQ, Lin H. Upconversion photon quantification of holmium and erbium ions in waveguide-adaptive germanate glasses. Applied Physics B. 2017; 123: 64.

16. Murthy MK, Ip J. Some physical properties of alkali germanate glasses. Nature. 1964; 201:

285-286.

17. Henderson GS, Fleet ME. The structure of glasses along the Na2O·GeO2 join. J. Non-Cryst.

Solids. 1991; 134(3): 259-269.

18. Shaw RR. Sonic velocities, elastic properties, and microhardness of sodium germanate glasses. J. Amer. Ceram. Soc. 1971; 54(3): 170-171.

19. Kiczenski TJ, Ma C, Hammarsten E, Wilkerson D, Affatigato M, Feller S. A study of selected physical properties of alkali germanate glasses over wide ranges of composition. J. Non-Cryst.

Solids. 2000; 272(1): 57-66.

20. Varshneya AK, Mauro JC. Fundamentals of Inorganic Glasses. 3^rd edn. Elsevier: Amsterdam, Netherlands; 2019.

21. Tischendorf B, Ma C, Hammersten E, Venhuizen P, Affatigato M, Feller S. The density of alkali silicate glasses over wide compositional ranges. J. Non-Cryst. Solids. 1998; 239(1-3):

197-202.

22. Amos RT, Henderson GS. The effect of alkali cation mass and radii on the density of alkali germanate and alkali germano-phosphate glasses. J. Non-Cryst. Solids. 2003; 331(1-3): 108-

Accepted Article

23. Ivanov AO, Evstrop’ev KS. On the structure of simple germanate glass. Dokl. Akad. Nauk.

SSSR. 1962; 145(4): 797-800.

24. Michaelis VK, Kroker S. 73Ge solid-state NMR of germaium oxide materials: experimental and theoretical studies. J. Phys. Chem. C. 2010; 114(49): 21736-21744.

25. Takeuchi Y, Nishikawa M, Yamamoto H. High-resolution solid-state 73Ge NMR spectra of hexacoordinated germanium compounds. Magnetic Resonance in Chemistry. 2004; 42(11):

907-909.

26. Michaelis VK, Aguiar PM, Terskikh VV, Kroeker S. Germanium-73 NMR of amorphous and crystalline GeO2. Chem. Commun. 2009; 31: 4600-4662.

27. Kamitsos EI, Yiannopoulos YD, Karakassides MA, Chryssikos GD, Jain H. Raman and infrared structural investigation of xRb2O·(1-x)GeO2 glasses. J. Phys. Chem. 1996; 100(28):

11755-11765.

28. Huang WC, Jain H, Marcus MA. Structural study of Rb and (Rb, Ag) germanate glasses by EXAFS and XPS. J. Non-Cryst. Solids. 1994; 180(1): 40-50.

29. Greaves GN, Fontaine A, Lagarde P, Raoux D, Gurman SJ. Local structure of silicate glasses.

Nature. 1981; 293: 611-616.

30. Wang HM, Henderson GS. Investigation of coordination number in silicate and germanate glasses using O K-edge X-ray absorption spectroscopy. Chemical Geology. 2004; 213(1-3):

17-30.

31. Martino DD, Santos LF, Almeida RM, Montemor MF. X-ray photoelectron spectroscopy of alkali germanate glasses. Surf. Interface Analysis. 2002; 34(1): 324-327.

32. Hannon AC. Ge-O coordination in cesium germanate glasses. J. Phys. Chem. B. 2007;

111(13): 3342-334.

33. Ueno M, Misawa M, Suzuki K. On the change in coordination of Ge atoms in Na2O-GeO2 glasses. Physica B+C. 1983; 120(1-3): 347-351.

34. Kamiya K, Yoko T, Itoh Y, Sakka S. X-ray diffraction study of Na2O-GeO2 melts. J. Non.

Cryst. Solids. 1986; 79(3): 285-294.

35. Martino DD, Santos LF, Marques AC, Almeida RM. Vibrational spectra and structure of alkali germanate glasses. J. Non-Cryst. Solids. 2001; 293-295: 394-401.

Accepted Article

36. Furukawa T, White WB. Raman spectroscopic investigation of the structure and crystallization of binary alkali germanate glasses. J. Mater. Science. 1980; 15(7): 1648-1662.

37. Yiannopoulos YD, Varsamis CPE, Kamitsos EI. Density of alkali germanate glasses related to structure. J. Non-Cryst. Solids. 2001; 293-295: 244-249.

38. Yang Z, Xu S, Hu L, Jiang Z. Density of Na₂O-(3-x)SiO₂-xGeO₂ glasses related to structure.

Materials Research Bulletin. 2004; 39(2): 217-222.

39. Hannon AC, Martino DD, Santos LF, Almeida RM. A model for the Ge-O coordination in germanate glasses. J. Non-Cryst. Solids. 2007; 353(18-21): 1688-1694.

40. Jain H, Kamitsos EI, Yiannopoulous YD, Chryssikos GD, Huang WC, Küchler R, Kanert, O.

A comprehensive view of the local structure around Rb in rubidium germanate glasses. J.

Non-Cryst. Solids. 1996; 203: 320-328.

41. Watanabe K, Sakai T. Molecular dynamics simulation of sodium germanate glasses and the germanate anomaly. Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B. 2017; 58(1): 15-20.

42. Karthikeyan A, Almeida RM. Structural anomaly in sodium germanate glasses by molecular dynamics simulation. J. Non-Cryst. Solids. 2001; 281(1-3): 152-161.

43. Henderson GS, Wang HM. Germanium coordination and the germanate anomaly. Eur. J.

Mineral. 2002; 14(4): 733-744.

44. Hoppe U. Behavior of the packing densities of alkali germanate glasses. J. Non-Cryst. Solids.

1999; 248(1): 11-18.

45. Hoppe U, Kranold R, Weber H-J, Neuefeind J, Hannon AC. The structure of potassium germanate glasses – a combined X-ray and neutron scattering study. J. Non-Cryst. Solids.

2000; 278(1-3): 99-114.

46. Hoppe U, Kranold R, Weber H-J, Hannon AC. The change of the Ge-O coordination number in potassium germanate glasses probed by neutron diffraction with high real-space resolution.

J. Non-Cryst. Solids. 1999; 248(1): 1-10.

47. Wright AC, Vedishcheva NM, Shakhmatkin BA. A crystallographic guide to the structure of borate glasses. MRS Proceedings. 1996; 455: 381.

48. Feller SA, Dell WJ, Bray PJ. ¹⁰B NMR studies of lithium borate glasses. J. Non-Cryst. Solids.

1982; 51(1): 21-30.

Accepted Article

49. Weber H-J. Bond volumes in crystals and glasses and a study of the germanate anomaly. J.

Non-Cryst. Solids. 1999; 243(2-3): 220-232.

50. Henderson GS. The germanate anomaly: what do we know? J. Non-Cryst. Solids. 2007; 353:

1695-1704.

51. Micoulaut M, Cormier L, Henderson GS. The structure of amorphous, crystalline and liquid GeO2. J. Phys.: Condens. Matter. 2006; 18(45): R753.

52. Goyal S, Mauro JC. Statistical mechanical model of bonding in mixed modifier glasses. J.

Amer. Ceram. Soc. 2018; 101(5): 1906-1915.

53. Bødker MS, Mauro JC, Goyal S, Youngman RE, Smedskjaer MM. Predicting Q-speciation in binary phosphate glasses using statistical mechanics. J. Phys. Chem. B. 2018; 122(3): 7609- 7615.

54. Phillips JC, Thorpe MF. Constraint theory, vector percolation and glass formation. Solid State Commun. 1985; 53(8): 699-702.

55. Mauro JC, Gupta PK, Loucks RJ. Composition dependence of glass transition temperature and fragility. II. A topological model of alkali borate liquids. J. Chem. Phys. 2009; 130: 234503.

56. Smedskjaer MM, Mauro JC, Sen S, Yue Y. Quantitative design of glassy materials using temperature-dependent constraint theory. Chem. Mater. 2010; 22(18): 5358-5365.

57. Smedskjaer MM, Mauro JC, Yue Y. Prediction of glass hardness using temperature-dependent constraint theory. Physical Review Letters. 2010; 105(11): 115503(1-4).

58. Smedskjaer MM. Topological model for boroaluminosilicate glass hardness. Front. Mater.

2014; 1(23):1-6.

59. Wilkinson CJ, Zheng Q, Huang L, Mauro JC. Topological constraint model for the elasticity of glass-forming systems. J. Non. Cryst. Solids: X. 2019; 2: 100019.

60. Yang K, Yang B, Xu X, Hoover C, Smedskjaer MM, Bauchy, M. Prediction of the Young’s modulus of silicate glasses by topological constraint theory. J. Non-Cryst. Solids. 2019; 514:

15-19.

61. Mauro JC, Ellison AJ, Allan DC, Smedskjaer MM. Topological model for the viscosity of multicomponent glass-forming liquids. International Journal of Applied Glass Science. 2013;

4(4): 408-413.

Accepted Article

62. Lee K, Zheng Q, Ren J, Wilkinson CJ, Yang Y, Doss K, Mauro JC. Topological model for Bi-

2O3-NaPO3 glasses. I. Prediction of glass transition temperature and fragility. J. Non-Cryst.

Solids. 2019; 521: 119534.

63. Smedskjaer MM, Mauro JC, Youngman RE, Hogue CL, Potuzak M, Yue Y. Topological principles of borosilicate glass chemistry. The Journal of Physical Chemistry B. 2011;

115(44): 12930-12946.

64. Bauchy M, Wang B, Wang M, Yu Y, Qomi MJA, Smedskjaer MM, Bichara C, Ulm F, Pellenq R. Fracture toughness anomalies: viewpoint of topological constraint theory. J. Acta Materialia. 2016; 121: 234-239.

65. Bødker MS, Sørensen SS, Mauro JC, Smedskjaer MM. Predicting composition-structure relations in alkali borosilicate glasses using statistical mechanics. Front. Mater. 2019; 6: 175.

66. Bødker M, Mauro JC, Youngman RE, Smedskjaer MM. Statistical mechanical modeling of borate glass structure and topology: prediction of superstructural units and glass transition temperature. J. Phys. Chem. B. 2019; 123(5): 1206-1213.

67. Wilkinson CJ, Pakhomenko E, Jesuit MR, DeCeanne A, Hauke B, Packard M, Feller S, Mauro JC. Topological Constraint Model of Alkali Tellurite Glasses. J. Non. Cryst. Solids. 2018;

502: 172–175.

68. Gupta PK, Mauro JC. Composition dependence of glass transition temperature and fragility. I.

A topological model incorporating temperature-dependent constraints. J. Chem. Phys. 2009;

130: 094503.

69. Naumis GG. Energy Landscape and Rigidity. Phys. Rev. E. 2005; 71: 026114.

70. Naumis GG. Glass Transition Phenomenology and Flexibility: An Approach Using the Energy Landscape Formalism. J. Non. Cryst. Solids. 2006; 352(42-49): 4865–4870.

71. Zheng Q, Mauro JC, Yue Y. Reconciling Calorimetric and Kinetic Fragilities of Glass- Forming Liquids. J. Non. Cryst. Solids. 2017; 456: 95–100.

72. Guerette M, Huang L. A simple and convenient set-up for high-temperature Brillouin light scattering. J. Phys. D: Appl. Phys. 2012; 45(27): 275302(1-7).

73. Jaccani SP, Huang L. Understanding sodium borate glasses and melts from their elastic response to temperature. International Journal of Applied Glass Science. 2016; 7(4): 452-463.

Accepted Article

74. Shelby JE. Viscosity and thermal expansion of alkali germanate glasses. J. Amer. Ceram. Soc.

1974; 57(10): 436-439.

75. Osaka A, Takahashi K, Ariyoshi K. The elastic constant and molar volume of sodium and potassium germanate glasses and the germanate anomaly. J. Non-Cryst. Solids. 1985; 70(2):

243-252.

76. Bray PJ. Nuclear Magnetic Resonance studies of glass structure. J. Non-Cryst. Solids. 1985;

73(1-3): 19-45.

77. Bray PJ, Feller SA, Jellison Jr GE, Yun YH. B¹⁰ NMR studies of the structure of borate glasses. J. Non-Cryst. Solids. 1980; 38-39(Part 1): 93-98.

78. Zhong J, Bray PJ. Change in boron coordination in alkali borate glasses, and mixed alkali effects, as elucidated by NMR. J. Non-Cryst. Solids. 1989; 111(1): 67-76.

79. Gupta PK. The random-pair model of four-coordinated borons in alkali-borate glasses. In Proceedings of the International Congress on Glass; New, Delhi, 1986.

80. Varshneya AK, Zanotto ED, Mauro JC. Perspectives on the scientific career and impact of Prabhat K. Gupta. J. Non-Cryst. Solids: X. 2019; 1: 100011.

81. Yu Y, Stevensson B, Edén M. Direct experimental evidence for abundant BO4-BO4 motifs in borosilicate glasses from double-quantum ¹¹B NMR spectroscopy. J. Phys. Chem. Lett. 2018;

9(21): 6372-6376.

82. Du L, Stebbins J. Nature of silicon-boron mixing in sodium borosilicate glasses: a high- resolution ¹¹B and ¹⁷O NMR study. J. Phys. Chem. B. 2003; 107(37): 10063-10076.

83. Williams Q, Jeanloz R. Spectroscopic evidence for pressure-induced coordination changes in silicate glasses and melts. Science. 1988; 239(4842): 902-905.

84. Tse JS, Klug DD, Page YL. High-pressure densification of amorphous silica. Phys. Rev. B.

1992; 46(10): 5933.

85. Itié JP, Polian A, Calas G, Petiau J, Fontaine A, Tolentino H. Pressure-induced coordination changes in crystalline and vitreous GeO2. Phys. Rev. Lett. 1989; 63(4): 398.

86. Micoulaut M. Structure of densified amorphous germanium dioxide. J. Physics: Condensed Matter. 2004; 16(1): L121.

Accepted Article

87. Micoulaut M. A comparative numerical analysis of liquid silica and germania. Chemical Geology. 2004; 213(1-3): 197-205.

88. Wu M, Liang Y, Jiang J, Tse JS. Structure and properties of dense silica glass. Scientific Reports. 2012; 2: 398.

89.Potter AR, Wilkinson CJ, Kim SH, Mauro JC. Effect of water on topological constraints in silica glass. Scripta Materialia. 2019; 160: 48-52.

Accepted Article

jace_17102_f1.png

Accepted Article

jace_17102_f2.png

Accepted Article

jace_17102_f3.png

Accepted Article

jace_17102_f4.png

Accepted Article

This article is protected by copyright. All rights reserved

Accepted Article

jace_17102_f6.png

Accepted Article

This article is protected by copyright. All rights reserved

Accepted Article