The Pennsylvania State University

The Graduate School

ADVENTURES IN HIGH DIMENSIONS: UNDERSTANDING

GLASS FOR THE 21ST CENTURY

A Dissertation in

Material Science and Engineering

by

Collin James Wilkinson

© 2021 Collin James Wilkinson

Submitted in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

May 2021

ii The dissertation of Collin Wilkinson was reviewed and approved by the following:

John Mauro Professor of Materials Science and Engineering Chair, Intercollege Graduate Degree Program Associate Head for Graduate Education, Materials Science and Engineering Dissertation Advisor Chair of Committee

Seong Kim Professor of Chemical Engineering Professor of Materials Science and Engineering

Ismaila Dabo Associate Professor of Materials Science and Engineering

Susan Sinnott Professor of Materials Science and Engineering Professor of Chemistry Head of the Department of Materials Science and Engineering

iii Abstract

Glass is infinitely variable. This complexity stands as a promising technology for the 21st century since the need for environmentally friendly materials has reached a critical point due to climate change. However, such a wide range of variability makes new glass compositions difficult to design. The difficulty is only exaggerated when considering that not only is there an infinite variability in the compositional space, but also an infinite variability thermal history of a glass and in the crystallinity of glass-cearmics. This means that even for a simple binary glass there are at least 3 dimensions that have to be optimized. To resolve this difficulty, it is shown that energy landscapes can capture all three sets of complexity (composition, thermal history, and crystallinity).

The explicit energy landscape optimization, however, has a large computational cost. To circumvent the cost of the energy landscape mapping, we present new research that allows for physical predictions of key properties. These methods are divided into two categories: compositional models and thermal history models. Both models for composition and thermal history are derived from energy landscapes. Software for each method is presented. As a conclusion, applications of the newly created models are discussed.

iv Table of Contents

List of Figures ...... vi

List of Tables ...... xi

Acknowledgments...... xii

Chapter 1 The Difficulty of Optimizing Glass ...... 1

1.1 Energy Landscapes ...... 6 1.2 Fictive Temperature ...... 8 1.3 Topological Constraint Theory ...... 15 1.4 Machine Learning ...... 19 1.5 Goals of this Dissertation ...... 20

Chapter 2 Software for Enabling the Study of Glass ...... 22

2.1 ExplorerPy ...... 22 2.2 RelaxPy ...... 28

Chapter 3 Understanding Nucleation in Liquids ...... 30

3.1 Crystallization Methods ...... 33 3.1.A Mapping and Classifying the Landscape ...... 35 3.1.B Kinetic Term for CNT ...... 38 3.1.C Degeneracy calculations ...... 41 3.1.D Free Energy Difference ...... 42 3.1.E Interfacial Energy ...... 44 3.2 Results & Discussion ...... 46 3.3 Conclusions ...... 51

Chapter 4 Expanding the Current State of Relaxation ...... 52

4.1 A thought experiment to expand our understanding of ergodic phenomenon: The Relativistic Glass Transition ...... 52 4.1.A Relativistic Liquid ...... 56 4.1.B Relativistic Observer ...... 60 4.2 Temperature and Compositional Dependence of the Stretching Exponent ...... 64 4.2.A Deriving a Model ...... 67 4.2.B Experimental Validation ...... 73 4.3 Conclusion ...... 81

Chapter 5 Glass Kinetics Without Fictive Temperature ...... 82

5.1 Background of the Adam Gibbs Relationship...... 82

v 5.2 Methods ...... 84 5.3 Results ...... 86 5.3.A Adam-Gibbs Validation ...... 86 5.3.B MYEGA Validation ...... 87 5.3.C Adam-Gibbs and Structural Relaxation ...... 89 5.3.D Landscape Features ...... 91 5.4 Topography-Property Relations ...... 94 5.5 Barrier Free Description of Thermodynamics ...... 99 5.7 Toy Landscapes for the Design of Glasses and Glass Ceramics ...... 102 5.7 Discussion ...... 107

Chapter 6 Enabling the Prediction of Glass Properties ...... 109

6.1 Controlling Surface Reactivity ...... 109 6.2 Elastic Modulus Prediction ...... 125 6.3 Ionic Conductivity ...... 135 6.4 Machine Learning Expansion ...... 145

Chapter 7 Designing Green Glasses for the 21st Century ...... 152

7.1 Glass Electrolytes ...... 152 7.2 Hydrogen Fuel Cell Glasses ...... 158

Chapter 8 Conclusions ...... 167

References ...... 169

vi List of Figures

Figure 1. The Volume-Temperature (VT) diagram: at the highest temperature there exists the equilibrium liquid which as it is quenched can either become a super cooled liquid or crystallize. Crystallization causes a discontinuity in the volume. The super- cooled liquid upon further quenching departs from equilibrium and transitions into the glassy state. Reproduced from Fundamentals of Inorganic Glass Science with permission from the author1...... 5

Figure 2. The schematic for the flow of the program. Beginning in the top right corner and running until the condition in the pink box is satisfied. Yellow diamonds represent checks and blue operations...... 25

Figure 3. The enthalpy landscape for SiO2. This was explored using the command above and then plotted using PyConnect. The plot is a disconnectivity graph where each terminating line represents an inherent structure and tracing where two lines meet describes the activation barrier. The potentials are taken from the BKS potential 94...... 27

Figure 4: (A) The energy landscape for a 256-atom barium disilicate glass-ceramic, the x- axis being an arbitrary phase space and the y-axis being the potential energy calculated from the Pedone et al. potentials36. The colors are indicative of the crystallinity, where the blue basin is the initial starting configuration. The landscape shows the lowering of energy associated with partially crystallizing the sample. (B) The energy landscape relationship between the cutoff for crystalline and super- cooled liquid states for 256 atoms is shown. This shows a clear drastic energy change occurring around the cutoff value of 1.0 Å...... 37

Figure 5. An example interfacial structure between the crystalline phase on the left and the last sequential SCL/glass phase on the right for a barium disilicate system. The gray atoms are barium, silicon is shown in red, and blue represents oxygen...... 43

Figure 6. (Top) The relaxation time and free energy difference (Middle) shown as a function of temperature for each system size. The experimental values for kinetics and thermodynamics come from ref.125 and from the heat capacity data taken from 126, respectively. It is clear to see that driving force shows good agreement across all systems however the kinetics terms only converge for the 256 and 512 atoms systems. (Bottom) The fit used to calculate the interfacial energy as a function of temperature...... 45

Figure 7. The nucleation curve for barium disilicate for converged systems as predicted using the model presented in this work. The data referenced can be found in refs. 113,128,129...... 47

Figure 8. The surface energy with respect to temperature for the work presented here...... 50

Figure 9. An Angell diagram created using the MYEGA expression 55. With a very strong glass (m~17), a highly fragile glass (m~100) and a pure borate glass (m~33) 17,134,135.

vii The infinite temperature limit is from the work of Zheng et al.54, and the glass transition temperature is from the Angell definition...... 54

Figure 10. The relativistic glass transition temperature for B2O3 glass...... 57

Figure 11. The equilibrium viscosity curves for a borate glass travelling at different fractions of light speed. All of the viscosities approaching the universal temperature limit for viscosity...... 59

Figure 12. The modulus needed to satisfy the condition for the glass transition...... 61

Figure 13 (Top) The predicted glass transition with the anomalous behavior occurring at v=0.44c. (Bottom) Viscosity plot showing a dramatic reduction of the viscosity as the observer approaches the speed of light...... 63

Figure 14. b predicted and from literature showing good agreement with a total root- mean-square error of 0.1. The fit for organic systems is given by ln  = − 6.6l n K and for inorganic systems by ln  = − 7.5l n K ...... 72

Figure 15. The equilibrium model proposed with the experimental points showing good agreement between the experimentally measured data points and the equilibrium derived model. RMSE was 0.02 for Corning© JadeTM (A) and less than 0.01 for 154,155 SG80 (B). (C) The model fit for B2O3 experimental data . The fragility and glass 17 transition temperature of the B2O3 are taken from the work of Mauro et al...... 75

Figure 16. The stretching exponent calculated as described in the text for a Gaussian distribution of barriers. This plot shows that the distribution of barriers has a large effect on the stretching exponent. A Tg cannot be described since there is no vibrational frequency included in the model, though the glass transition temperature should be the same for all distributions since the mean relaxation time is the same for all distributions at all temperatures. The deviation is given in ln eV units...... 78

Figure 17. (Top) The Prony series parameters as a function of the stretching exponent. Each color designates one term in the series. (Bottom) The output from RelaxPy v.2.0 showing the stretching exponent effects on the relaxation prediction of Corning© JadeTM glass. Each quadrant shows one property that is of interest for relaxation experiments. In particular, it is interesting to see the dynamics of the stretching exponent during a typical quench...... 80

Figure 18. The viscosity (left) and landscape (right) predictions for three common systems. The first system is newly calculated in this work while the latter two come from our previous works72,89. It is seen that the viscosity predicted from the AG model is very accurately able to reproduce the experimental viscosity curves from the MYEGA model. The last system is a potential energy landscape while the others are enthalpy landscapes...... 87

Figure 19. The configurational entropy comparisons between the three major viscosity models, which validates the main underlying assumption of the MYEGA model. The

viii VFT and AM are unable to capture the physics of configurational entropy, therefore ruling ...... 89

Figure 20. Estimated bulk viscosity of B2O3. The fit to the bulk viscosity used the configurational entropy from the enthalpy landscape with the barrier 0.0155 eV (compared to the shear barrier of 0.0149 eV) and the infinite limit allowed to vary (10-2.63 Pa·s for bulk viscosity). In Figure 18, the configurational entropy is confirmed for the shear viscosity, thus confirming the AG for both shear and bulk viscosities. Sidebottom data are from Ref. 154...... 90

Figure 21 (A) The histogram for the energy minima for each of the test systems showing a good fit with the log normal distribution. This distribution will then be a valid form to calculate the enthalpy distribution of the model presented in the next session. (B) The configurational entropy from the model showing the accuracy of

the scaling of the entropy predicted by the MYEGA model. The S value was fit for each system...... 93

Figure 22 (A) Comparison between the randomized method (histogram) and the deterministic method (vertical lines) showing good agreement between the maximum in the histogram and the value predicted by the deterministic technique, validating the approach. It is worth noting that the 100-basin distribution is a very wide distribution where the total number of basins is less than the number of points used in the calculation. This is done for a variable number of basins with the number of basins shown in the legend. (B) The dependence of fragility and the glass transition temperature vs. the distribution of states and the number of basins...... 99

Figure 23. The driving forces for different example glasses calculated using a combination of the MAP model, RelaxPy, and the toy landscape model. The parameters for each glass can be found in Table 4...... 102

Figure 24 The predicted outputs from the toy landscape method showing predictions of the enthalpy and entropy under a standard quench for barium disilicate. This prediction does not require fictive temperature or any such assumptions about the evolution of the non-equilibrium behavior...... 105

Figure 25. The prediction for nucleation and growth from the 5 parameters. The volume in nucleation is assumed to be on the order of one cubic angstrom while the a parameter in growth is assumed to be around one nm, both are in good agreement for estimates in literature. The values for the orange points are taken from these works128,167...... 107

Figure 26. Representative example of the initial non-hydrated sodium silicate used in the hydration models. Color scheme: Si atom (ivory), O atom (red), and Na atom (blue). The z-axis is elongated to allow space for an insert of water...... 114

Figure 27. Initial (a) and final (b) states of the water/glass interface. Note that only the top surface in contact with water is shown here...... 116

ix Figure 28. (Top) Schematic of water binding energy calculations. (Bottom) An example of the electronic-structure DFT calculation using semilocal exchange-correlation functionals finding the binding energy of a water ‘pixel’ to the surface...... 119

Figure 29. Example contour surface showing the average coordination per atom on the glass surface, for the first run at 300 ps...... 121

Figure 30. ReaxFF MD-derived water binding energies plotted versus the number of constraints for surface atoms at the local pixel. Results show a distinct maximum, in which there is a near hydrophilic-hydrophobic transition of the surface. The error bars represent the standard deviation. A second system with 1500 atoms was performed to show convergence of the ReaxFF MD results...... 123

Figure 31. The Young’s modulus prediction and experimentally determined values for 0.3Na2O·0.7(ySiO2·(1-y)P2O5) glasses. The root mean square error (RMSE) values of the model predictions are: 6.41 GPa for constraint density, 3.13 GPa for free energy density, and 7.74 GPa for angular constraint density...... 128

Figure 32 The Young’s modulus residuals for different prediction methods, sorted from minimum to maximum error. The free energy density model gives the most accurate results. The constraint density has a RMSE of 6.1 GPa, the angular density has a RMSE of 20 GPa, and the energy density has a RMSE of 5.9 GPa...... 129

Figure 33. (A) Temperature dependence of the Young’s modulus from theory and experiment for 10% Na2O 90% B2O3. Using the previously fitted onset temperatures, the only free parameters are then the vibrational frequency and the heating time in which their product was fitted to be 14000. Where each dip in the modulus corresponds to a constraint no longer being rigid as heated through each onset. The onsets were fitted from the compositional dependence and only the width of the transition was fit which may account for the discrepancy around the inflection. The data was fit using a least-squares method, and the resultant fit is shown as the calculated method. The fit has an R2 of 0.94. (B) The contribution from each constraint to the overall modulus...... 131

Figure 34. The Young’s modulus prediction (using the same fitting method described in Fig. 3) and experimentally determined values for (A) zGe·(100-z)Se with an R2 of 2 0.93 and (B) xLi2O·(100-x)B2O3 glasses with an R of 0.986...... 133

Figure 35. The structure for the initial minimum energy configuration showing the boron (blue) network with interconnecting oxygens (red) and the interstitial sodium ions (yellow)...... 139

Figure 36. The energy barrier between the two sites. Oxygen is blue, boron is red, and sodium is ivory. The barrier is overestimated compared to experimental data; this could be from several sources of error such as potential fitting, thermal history fluctuations, or sampling too few transitions. The line is drawn as a guide to the eye. ... 139

Figure 37. Snapshots of the NEB calculation. On the top is the total network as a function of reaction coordinates. The middle shows the local deformation around the ion of

x any atom that moves, in between inherent structure mandating a relaxation force. The color shows the degree of deformation...... 141

Figure 38. Different network formers and the prediction of the activation barrier from our model compared with activation barriers from literature. (A) Sodium silicate predictions and experimental values241, the error is calculated from the error in the fragility when fitting the data. (B) Lithium phosphate activation energy235 predicted with topological constraint theory and compared with the experimental values. (C) Predictions over two different systems of alkali borates232, sodium and lithium, with a reported R2 of 0.97...... 143

Figure 39 The dependence of infinite viscosity limit for other viscous parameters. (Top left) shows the distribution of the infinite temperature limit in the database after limits exerted on the system. (Top right) The distribution of the infinite viscosity limit vs the glass transition. (Bottom left) The relationship between fragility and infinite temperature limit of viscosity. (Bottom right) The infinite temperature limit of viscosity vs. the key metric predicted by SR...... 149

Figure 40 The prediction of the scaling of the activation barriers for a common sodium borosilicate system...... 157

Figure 41 The relationship between the glass transition and the proton conductivity. This is justified two ways one through the relationship of the entropy of diffusion and glass formation (the Adam-Gibbs model) and through the fact that water is known to depress the glass transition...... 162

Figure 42 (A) The glass transition prediction vs the experimental values showing a good correlation. (B) The confusion matrix of the random forest method used to determine the glass forming region. Over top the constraints at the glass transition provided by each oxide species is listed. Since the objective is to decrease Tg while staying in the glass forming region, we will attempt to minimize use of elements that increase the glass transition (nc > 1.7)...... 166

xi List of Tables

Table 1. The key properties considered for commercial application. The optical properties have been omitted since it is physical unrealistic to expect quantitative predictions of quantum-controlled phenomena from a classical description of glass structure...... 18

Table 2 Variable definitions...... 65

Table 3 Measured temperatures and their corresponding relaxation values for Corning© JadeTM glass and Sylvania Incorporated’s SG80...... 74

Table 4. A table of parameter values for the three example glasses used in Figure 23. The distribution of underlying inherent structure energies and the glass transition temperature (500 K) were kept the same while the total number of basins were allowed to vary...... 101

Table 5 Density of the simulated sodium silicate glasses after relaxation at 300K...... 112

Table 6. System configurations for sodium silicate glass-water reactions...... 117

Table 7 Fitted values from this analysis compared to those reported in the literature. The disparity between the constraints evaluated with molecular dynamics most likely come from the speed in which the samples are quenched...... 132

Table 8. Hyperparameters for different neural networks after hyper-optimizations...... 151

Table 9. A table with some ionic conductivity models and the parameters needed for them as well as the disadvantages for each. These are not the only models but are representative of those commonly used in literature...... 154

Table 10. The predicted compositions based on the optimization scheme proposed...... 157

Table 11. The compositions synthesized in this work. These compositions were predicted by minimizing the cost function described in Eq. (123). OP* is the variant that was melted after OP partially crystallized. B-OP appeared to have surface nucleation in some spots but was cut and removed before APS treatment...... 165

xii Acknowledgments

“To Love Another Person is to See the Face of God” -Victor Hugo, Les Miserables

I have been truly blessed to love and be loved by a plethora of mentors, friends, colleagues, and family across my academic journey. I have to first thank my advisor, Dr. John Mauro, who has been as kind, patient, and caring as any mentor I have ever met and in the process has made me a better scientist and a better person. His mentorship was built on top of those who first found and shaped me into a scientist, Dr. Ugur Akgun and Dr. Steve Feller, without whom my scientific career would not be possible. Along with these individuals I’ve had the pleasure to learn so much about glass and life from Dr. Madoka Ona, Dr. Firdevs Duru, Dr. Mario Affatigato, Dr. Doug Allan, Dr.

Ozgur Gulbiten, and Dr. Seong Kim.

Rebecca Welch, Arron Potter, and Anthony DeCeanne are my partners in all things.

Rebecca is my partner in research and in life. Her opinion, support, and patience has been indispensable to me. Arron is my oldest colleague and friend; he is the one that I share the highs of research and friendship with. Anthony has put up with me non-stop for about 5 years now. He is the most patient and kind friend you could ask for. Without these three individuals I could not have done anything presented in this work. I additionally have to thank Karan Doss, Aubrey Fry, Caio

Bragatto, Daniel Cassar, Brenna Gorin, Mikkel Bodker, Greg Palmer, Katie Kirchner, Kuo-Hao

Lee, and Yongjian Yang. I have the privilege to call these individuals both my friends and colleagues.

The last people I have to thank are my family. I want to thank my brothers (Sam & Quinn), my mom, and my dad. My mom has helped me explore the universe through a love of reading and her constant unwavering support. My dad was the first to show me the wonders of science with rockets as a kid; as a teenager he supported my fledgling interest as he drove hundreds of miles to every corner of this country, and he is the person for whom this dissertation is dedicated.

Chapter 1 The Difficulty of Optimizing Glass

Glass is a complex world-changing material. Though it has existed for thousands of years, the surface of its true potential is only just being scratched. To unlock the potential of glass for new applications and to enable the maximum benefit to society, we must be able to design new glasses with wide ranging properties faster and cheaper than ever before [1]–[5]. This is particularly important now that the need for ecologically responsible materials is increasing due to climate change. In order to facilitate such glass design, we must return to first principles and build a picture of glass from the ground up encompassing both compositional and thermal history dependencies into our understanding of glass properties[3], [6]. The present accepted description of glass is a non-crystalline, non-ergodic, non-equilibrium material that appears solid and is continuously relaxing towards the super-cooled equilibrium liquid state[1]. This definition gives insights into the nature of glass and includes points of particular interest in this dissertation.

The first point to consider is that glass is non-crystalline, meaning that it contains no long- range order. This gives glass one of its key advantages: being infinitely variable. Since a glass is not limited by having to reach stoichiometric crystalline structures, there is no limit to the number of possible glasses or glass structures. This gives glass its possibilites. Glass has been considered as a candidate for everything from hydrogen fuel cells[7] to batteries[8] and for applications ranging from commercial smart phone covers[9] to bioactive applications[10]. The infinite variability also explains the difficulty of designing glasses. Consider a system with three constituents, for example, SiO2, Na2O, and B2O3. This then means there are two independent compositional dimensions that must be fully explored to find an optimal composition for an application. If this space were discretized to 1 mol% spacing, that would give over 5000 unique

2 glass candidates that would need to be studied to reach the optimal design. A five-component glass would correspondingly have over 9 million unique glasses. More generally, for a glass with up to

C components there are C −1 dimensions over which to optimize. If instead one considers a crystal, there is not a continuously variable space but instead only discrete points.

The other ramification of glass being non-crystalline is that it must bypass the region of crystallization at a rate much faster than the rate of nucleation[11]. This is inherent in the definition of glass and is shown in Figure 1. Figure 1 is called the volume-temperature (VT) diagram and is key to understanding the nature of glass. The bypassing of the crystallization region leads to a super-cooled liquid, and during quenching the material departs from equilibrium and enters the glassy state. The region of departure is called the glass transition temperature range or simply the glass transition. To understand why this occurs, we must first understand the concept of ergodicity.

Ergodicity was defined by Boltzmann to mean that the time-average value of a property is equal to the ensemble-average value of that property[12], [13]. Now that ergodicity is defined, we can start to consider the implications of the non-ergodic nature of glass. Since it is non-ergodic, the ensemble average will not equal the time average on human timescales; however, the ergodic hypothesis says that in the limit of long time they will be equal[12]. In order for both statements to be true, glass must evolve to be ergodic and thus, is inherently unstable. To explore this concept further, we must understand the glass transition process and define a timescale. If there exists a liquid at some temperature, T , and is then perturbed to T+ dT , there will be some inherent time

associated with its relaxation towards equilibrium which is described by  struct , the structural relaxation time. There is a separate relaxation time when the system is mechanically perturbed

called the stress relaxation time,  stress . To then understand whether the resulting material is a glass

or a liquid, we need to compare  struct with the observation time. If the observation time is much longer than the relaxation time, then what is observed over the course of a experiment is the

3 equilibrium liquid properties. Conversely, if the observation time is much shorter than the relaxation time, then the observed properties will not be ergodic and as such, we will not sample the properties of the equilibrium liquid. We define this non-ergodic state as the glassy state.

However, since in the limit of long time the system must become ergodic once again then over time all glasses will return to an ergodic nature through a process called relaxation [14].

The last point considered in the definition of the glass is that it is non-equilibrium. Not only is glass non-equilibrium, it is also unstable and continuously progressing towards the super-cooled liquid state (e.g., relaxation). Relaxation is inherent in all glasses, and we can see this for a model system in the VT diagram (Figure 1). In this diagram, there are two glasses (of the same composition), quenched at different rates (both of which bypass crystallization). In the slow-cooled glass there is a lower volume (higher density) compared to the fast-cooled glass. This difference is due to the fact that the slow-cooled glass has been allowed to relax more fully compared to the other. This adds at least one additional dimension of optimization (such as quench rate from a high temperature to room temperature) in which glasses must be designed in.

This dissertation is dedicated to understanding the infinite variability of glass with respect to crystallization, composition, and thermal history with tools being implemented and designed so that the challenge of optimizing over these vast spaces can be done with higher efficiency, lower cost, and with less experimental work load. These tools are all inspired by the energy landscape description of glass. To design the optimal glass for any application, we must understand the relationship between the C −1 compositional dimensions, a minimum of one thermal history dimension, and the properties of a material. There are many ways to explore these relationships, each with its advantages and disadvantages. The methods to optimize over these thermal history and compositional dimensions include energy landscapes[15], fictive temperature[16], topological constraint theory[17], and machine learning[3]. Of these energy landscapes, are the most fundamental with each other technique being related back to the energy landscape in some form.

4 Though they are the most fundamental, they are also the most difficult, time consuming, and have the largest barrier to entry for use. To then build our understanding of each technique, we will start with energy landscapes then explore how through a series of assumptions we can arrive at the other methods.

5

Figure 1. The Volume-Temperature (VT) diagram: at the highest temperature there exists the equilibrium liquid which as it is quenched can either become a super cooled liquid or crystallize. Crystallization causes a discontinuity in the volume. The super-cooled liquid upon further quenching departs from equilibrium and transitions into the glassy state. Reproduced from Fundamentals of Inorganic Glasses with permission from the author[1].

6 1.1 Energy Landscapes

Energy landscapes were first developed by Goldstein and Stillinger, who proposed the concept as a way to understand the evolution of a material under complex conditions [18]–[20].

Since Stillinger’s seminal work [18], energy landscapes have become important in the study of protein folding [15], [21], [22], network glasses [6], [13], and other complex materials. The concept of energy landscapes has become commonplace and is readily evoked when describing the time evolution of a material [20], [22]–[26]. Despite energy landscapes being commonplace in some fields, the procedure for mapping a landscape remains difficult due to the associated computational challenges [15], [27]. To map an energy landscape, one must find a set of local minima and the first order saddle point connecting each pairwise combination of those minima. Each minimum is called an inherent structure and represents a structurally/chemically stable state of the material. A basin is the set of structures that converge to the same inherent structure upon minimization [18].

In glass science, the energy landscape is used to justify complex behaviours which are most commonly calculated using molecular dynamics simulations (MD) [13], [28], [29]. It is easy to understand why MD is used so frequently used when considering the ease of use of common MD packages. For example, LAMMPS [30] is a common, user-friendly MD program with a low barrier to entry, allowing individuals with little programming experience to run complex MD simulations.

Inspired by this accessibility, as well as the explicit Python bridge to LAMMPS, we have developed a software that can be used to explore energy landscapes with a variety of exploration techniques.

Similar softwares [15], [31]–[35] have been created to map landscapes; however, each program has limitations and may not be suitable for all purposes while ours is a general purpose software.

The calculation of the energy for landscapes is done using an empirical interatomic potential that can be fit using either experimental data and a series of MD simulations or using ab initio methods. One common potential form is the Lennard-Jones form (LJ),

7

126   VLJ =−4   r  r    (1) in which and  are fitting parameters for every pair-wise set of atoms in a system, V is the potential energy contributed by the two-atom interactions, and r is the distance between the two ions in question. Though the LJ form is simple and easy to parameterize it is unable to recreate complicated material behaviours for a real glass. As such, a more complicated potential is often used, such as a Morse potential. A commonly used silicate glass potential is the Pedone potential which consists of Morse potential with an additional columbic term and repulsive term [36] and is given by,

2 Z Z e 2 C VYa rr=+−−−−+ij 1 exp1  . (2) Pedone rr( 0 ) 12

Z is the charge of a given ion, e is the charge of an electron, and Y, C, a, and r0 are fitting parameters.

It is also worth noting that the Pedone et al. potentials are only accurate when considering the interactions between each individual ion species with oxygen. For instance, Si-O, Na-O, and O-O are all explicitly parameterized while Si-Na only interacts coulombically (the first term).

Interatomic potentials are discussed here only briefly, but there are extensive resources available for those readers who wish to learn more[37]–[39].

Once an energy landscape has been mapped, the thermodynamics and kinetics for a given system can be easily calculated [33], [40]. The thermodynamic driving force is given by the difference between the free energy of the current inherent structure and the free energy of the lowest state, while the kinetics is the rate in which the system crosses the barrier between the basins. This can be calculated explicitly for arbitrary timescales and temperatures with another software,

KineticPy [41]. The energy landscape provides insights into the detailed nature of material behaviour. Though energy landscapes are very powerful, they are also difficult to map. To, approximate the behavior of the landscape then, we use approximations such as fictive temperature,

8 topological constraint theory (TCT), or machine learning. Fictive temperature is a way to estimate the effects of the thermal history dimension[42] while TCT is used to estimate the compositional dependence of properties[43].

1.2 Fictive Temperature

Fictive temperature is an approximation used to find the change in the occupational probabilities on an energy landscape[16], [44]. Instead of considering every possible combination of occupational probabilities like the energy landscape, it instead approximates the occupational probability using a single temperature at which the landscape would be in equilibrium. It is essentially a measure of how close/far a glass is from equilibrium. This approximation allows for us to calculate the relaxation dynamics of a glass without needing any computationally heavy energy landscapes. However, this single parameter description oversimplifies relaxation[14], [16].

The functional form for glass relaxation dates back to 1854 when Kohlrausch fit the residual charge of a Leyden jar with[45]

 t gtg( )(0)exp=− (3) 

where t is time, τ is the relaxation time constant, β is the stretching exponent, and g(t) is the relaxation function of some property (e.g., volume, conductivity, viscosity, etc.); Eq. (3) is referred to as the stretched exponential relaxation function (SER). The stretching exponent is bounded from

0 < β ≤ 1, where the upper limit (β = 1) represents a simple exponential decay and fractional values of β represent stretched exponential decay.

For over a century the physical origin of the stretched exponential relaxation form was one of the greatest unsolved problems in physics until 1982 when Grassberger and Proccacia[46] were able to derive the general form based on a model of randomly distributed traps that annihilate

9 excitations and the diffusion of the excitations through a network. This model, however, provided no physical meaning for β, which was still treated as an empirical fitting parameter. In 1994,

Phillips[47], [48] was able to extend the diffusion trap model by showing that β=1 at high temperatures and at low temperatures, the stretching exponent can be derived based on the effective dimensionality of available relaxation pathways. He in turn expressed the stretching exponent,

df  = (4) df + 2 where d is the dimensionality of the network and f is the fraction of relaxation pathways available.

He then proposed a set of ‘magic’ numbers for common scenarios. Assuming a three-dimensional network with all pathways activated (d = 3,f = 1), a value of β= 3/5 is obtained. The second case is a three dimensional network with only half of the relaxation pathways activated (d =3,f = 1/2) yielding a value of β= 3/7. A third magic value β= 1/2 was found for a two dimensional technique with all pathways activated (d = 2,f = 1). β = 3/5 occurs for stress relaxation of glasses under a load, because both long- and short-range activation pathways are activated, whereas a value of β = 3/7 is obtained for structural relaxation of a glass without an applied stress. The value was confirmed by

Welch et al.[9] when the value was measured over a period of 1.5 years at more than 600◦C below the glass transition temperature of Corning Gorilla Glass. Though this model was able to reproduce the limiting values it was criticized widely in the community. Critics argued that the model was simply created such that it reproduced the stretching exponent values of certain experiments and ignored many other experimental results that appeared to disagree with the model. The model also fails to interpolate between the low temperature values Phillips predicted and the high temperature limit.

The work from Phillips and the stretched exponent gave a form for relaxation, however there remained no method to instantaneously state the distance from equilibrium of a glass. To account for this problem, an additional thermodynamic variable (or order parameter), called fictive

10 temperature (Tf) was proposed. Early research on relaxation in glasses dates back to the work of

Tool and Eichlin in 1932[49], and Tool in 1946[50] whose works originally proposed a temperature at which a glass system could be in equilibrium without any atomic rearrangement. Tool suggested that this fictive temperature was sufficient to understand the thermodynamics of a glassy system.

Originally the fictive temperature was treated as a single value that was some function of thermal history (Tf[T(t)]). The evolution of fictive temperature as proposed by Tool is then given by,

dTTtTTt()(())− ff= (5) dtTT  (,f )

In which  for stress relaxation is given by,

 (TtT(), f ())Tt( )  = (6) G

In Eq. (6)  is the shear viscosity and G is the shear modulus. This means that to predict the time evolution of a glass only three things are required: shear modulus, the viscosity as a function of temperature and fictive temperature, and the stretching exponent. Though fictive temperature qualitatively reproduced the results they were looking for, subsequent experiments have shown that the concept of a single fictive temperature is inadequate[51], [52].

A key experiment was performed by Ritland in 1956[51]. Ritland took several samples with different thermal histories but the same measured fictive temperature; thus, based on Tool’s equation, both should have had identical relaxation properties. Ritland, however, showed that the refractive index evolved differently between samples. To account for these differences, the use of multiple fictive temperatures was suggested[52]. This worked because the stretched exponential form could be approximated with a Prony series,

N exp−Kt  w exp − K t . (7) ( )  ii  i

11

In the Prony series N is the number of terms while wi and K i are fitting parameters. This means that each term could represent one fictive temperature and this collection of fictive temperatures would reproduce the overall experimentally observed form. Ritland’s experiment showed that materials held isothermally at their fictive temperatures with varying thermal histories can give different results for the relaxation of a given property. Thus, as a glass network relaxes the fictive temperature will also shift. Narayanaswamy[52] was able to apply Ritland’s concept in a highly successful engineering model with multiple fictive temperatures, but the physical meaning of multiple fictive temperatures remains elusive.

In general, there is reason to be skeptical of the concept of fictive temperature. Fictive temperature was not derived but instead was an empirical approximation that Tool needed to gain an early quantitative understanding. Recently, Mauro et al.[16] used energy landscapes to try to

understand the underlying validity of fictive temperature. T f implies that the occupational

probability ( pi ) is given by,

1 Hi pi =−exp . (8) QkT 

Q is the partition function, k is Boltzmann’s constant, and Hi is the enthalpy of the given state.

However, when the actual probability evolution predicted by fictive temperature was compared to the probability calculated through a systematic Monte Carlo approach the results were found to be drastically different (even when many fictive temperatures were tried). This, implied that fictive temperature is insufficient to describe the glassy state and must be replaced with a new method.

Despite the inadequacies of fictive temperature, state-of-the-art modeling techniques for glass relaxation still relies on the concept. In these techniques a set of fictive temperatures (usually

12) are used to calculate 12 parallel relaxing fictive temperatures[42]. This is because there has been no alternative method able to reach a prediction of the evolution of glass under different

12 thermal histories. The current state-of-the-art model is called the Mauro-Allan-Potuzak (MAP)[44] model of non-equilibrium viscosity and was derived from a more fundamental expression for viscosity called the Adam-Gibbs model (AG) which is given by[53],

B log(,1010TTf )log=+ . (9) TScf(,)TT

B is the barrier for a cooperative rearrangement, l og10  is the infinite temperature limit of

-2.93 viscosity and according to Zheng et al.[54] it is approximately 10 Pa s, and Sc is the configurational Adam-Gibbs entropy. This entropy has long since been a source of debate and will be addressed later in chapter 5.

In order to calculate the equilibrium part of the viscosity the MAP model relies on an earlier

Mauro, Yue, Ellison, Gupta, and Allan (MYEGA) equilibrium viscosity model[55]. This model was derived based on earlier work by Naumis[23], [56], Gupta and Mauro[57], and the AG model.

These models showed that the configurational entropy can be related to the degrees of freedom,

SfNkc = ln  . (10) f is the degrees of freedom per atom, N is the number of atoms, and  is the number of degenerate states per floppy mode. To then build in temperature dependence, they assumed that the degrees of freedom can be modeled using an Arrhenius form (with activation barrier H * ),

H * f( T )=− d exp . (11) kT

When Eqs. (9), (10), and (11) are combined into an equilibrium form the equilibrium viscosity is then given by,

CH* log10= l og 10  + exp  . (12) T kT 

Where C is equivalent to,

13 B C = . (13) dk N ln 

Through a change in variables this can be written in terms of three common variables that allow for all viscosity equations to be written with three variables specified by Angel[58]: the glass

transition temperature, the fragility, and l og10  . The glass transition temperature is defined as the isokom where the liquid has a viscosity of 1012 Pa s. The fragility is defined as[58],

d log  m  10 . (14) Tg d  T TTg =

The three most common viscosity models (MYEGA, Vogel-Fulcher-Tammann (VFT)[59], and

Avramov-Milchev (AM)[60]) are listed below in terms of the same parameters,

TTggm  log(12101010= log)exp11log +−−− , (15) TT1o2− l g10  

2 (12log− 10  ) log1010= log  + , (16) T m −+−112log   10   Tg  and

m/(12− log10 ) Tg log10101= log +−(12 log 0 ) . (17) T

To incorporate non-equilibrium effects with these equilibrium models the MAP model proposes,

log10= x log 10 MYEGA (T , m , T g )+− (1x )log 10  ne ( T f , T , m , T g ) . (18)

14 l og10 ne is the non-equilibrium viscosity whose form and compositional dependence can be found elsewhere[42], [44], [61]. x is the ergodicity parameter and is of crucial importance to understanding glass regardless of the model being used[12]. In the MAP model it is given by,

m 3.244 min,(TTf ) x = . (19) max, TT ( f )

Though the MAP model was derived from the best understanding of energy landscapes of glasses at the time it has a few flaws that prevent it from perfectly predicting the different relaxation behaviors of glass. A few of the short-comings are:

1. The method ignores crystallization. For a comprehensive model showing the response of

glass to temperature, the possibility of crystallization must be included. Currently, no

relaxation model includes this possibility.

2. It is built around the concept of fictive temperature. This could be replaced in a future

model but currently at present it carries with it all the failures of the fictive temperature

picture of glass dynamics, adding only the relaxation time coming from landscape-derived

values.

3. The MAP model does not consider temperature dependence of the stretching exponent.

This is typically circumvented by choosing a fixed stretching exponent.

4. The parameters needed for the non-equilibrium viscosity in the MAP model are

experimentally challenging to measure. To circumvent this Guo et al.[61] published

methods to estimate the values based on fragility and the glass transition, but the

approximations were only tested on a few samples with similar fragilities and chemistries.

5. The MAP model only predicts stress relaxation. Since the publication of the MAP model,

it has been shown that only stress relaxation time is related to the shear viscosity[62].

Structural relaxation must be incorporated in future models.

15 For a complete understanding of the thermal-history dependence of glass, all of these questions must be rigorously answered.

1.3 Topological Constraint Theory

TCT is one way to estimate the compositional dependence of glasses. It uses a set of assumptions that connects the underlying structure to glass properties through understanding of the energy landscape. These assumptions include the temperature dependence of the rigidity, the same assumptions used in the derivation of the MYEGA models, and that the effects of thermal history on structure are minimal. TCT was originally proposed by Gupta and Cooper (GCTCT) to try to understand the role of rigid polytopes on glasses properties[63]. Gupta and Cooper fundamentally understood that the ability for glasses to form was reliant on the transformation of a liquid with many degrees of freedom to a glass with a fixed degree of freedom. Since the structure of the glass is the same as the structure of the liquid at the glass transition and the degrees of freedom of the glass ( f ) changes during the transition to glass, it was possible to gain insights into the transformation process. This approach is fundamentally an approximation of the highly multidimensional landscape where only approximate structural information is used to understand the entropy of the energy landscape. Though the core idea was useful, it was “clunky” and difficult to make calculations with.

To improve upon the ideas of GCTCT, Phillips and Thorpe (PTTCT)[64] proposed a mathematically equivalent form where the degrees of freedom could be simply found by

considering the number of constraints ( nc ) around each network-forming atom in the glass. They showed that the degrees of freedom were related to the number of constraints by,

f=− d nc . (20)

16 The number of constraints could then be calculated by considering the two-body and three-body interactions (first and second terms) as a function of the mean coordination of the network forming species,

r nr=+− 23. (21) c 2

This led to a qualitative language to discuss glass with a few quantitative insights. The biggest quantitative insight comes in the form of the ‘Boolchand Intermediate Phase’[65]–[68]. If it has positive degrees of freedom, then a network is ‘floppy’ and retains high kinetic energy (as it flips back and forth between two inherent structures). If the system has negative degrees of freedom, then the network is stressed-rigid and there is additional energy stored in the stress associated with over-coordination. If the number of constraints is equal to the number of dimensions, then the resulting system will have zero degrees of freedom and there is no stored stressed energy or kinetic energy from flipping between basins. Boolchand went searching for this perfectly ‘isostatic’ glass and found a non-zero window of compositions that were close to isostatic but had properties drastically different compared to both stress rigid and floppy glasses. This is now called the

Boolchand Intermediate Phase.

Despite the success in the prediction of the intermediate phase TCT remained largely qualitative until the work by Naumis[23], [56] and Gupta & Mauro[57]. Naumis pointed out that the degrees of freedom are a ‘valley’ of continuous low energy so that the structure can freely move between basins of the landscape. Since these valleys are deformation pathways, Naumis pointed out that they will dominate the configurational entropy of the system. Building on this Mauro and

Gupta wrote that since the entropy is proportional (Eq. (10)), we could then write the glass transition temperature as a function of the degrees of freedom and the AG model,

B Tg = . (22) f(12 − lgo 10  ) Nk ln 

17 Since everything in Eq. (22) is approximately a constant between compositions except for f , we could then write the expressions in terms of reference values (subscript r) and accurately scale the glass transition of new compositions by knowing the changing structures:

Tdngrcr,,( − ) Tg = . (23) dn− c

The last insight, provided by Mauro and Gupta was that the number of constraints change as a function of temperature. They realized that each constraint has a temperature dependence which is controlled by the interaction energy of the constraints. This makes intuitive sense because as the thermal energy becomes greater than that of the barriers additional valleys in the energy landscape form. This also allowed them to expanded TCT to include quantitative predictions of fragility.

The understanding of the relationships between the configurational entropy, the number of constraints, the degrees of freedom and, the viscous properties lead to a revolution in understanding glass properties. Soon after the work by Mauro & Gupta multiple works came out predicting mechanical, chemical, and viscous properties of glass knowing only the underlying structure.

Though this framework is powerful, it is not a complete description without further parameterizations with each model requiring some fitting parameters or reference values. In addition, there are still key properties that are missing from the standard topological approach that are needed to design commercial glasses.

Ultimately, MGTCT needs be expanded to include the properties that are needed for commercial applications. A list of composition dependent properties needed for understanding the performance of commercial glasses are shown in Table 1. Properties taken from Ref. [3] show what properties must be controlled to manufacture technological glasses. All models take additional inputs and there are multiple ways to link outputs from some models to inputs of the other. Choice of model vary depending on the use case.

18 Table 1. The key properties considered for commercial application. The optical properties have been omitted since it is physical unrealistic to expect quantitative predictions of quantum- controlled phenomena from a classical description of glass structure. Properties TCT Machine Learning Other methods

Glass Transition [57], [69] [70] Energy Landscapes: [71]

Fragility [57], [69] This work. This work

Relaxation MAP model: [44], [61]

Crystallization This work: [72]

MD/MC: [73]–[75]

Melting Temperature This work. MD/This work:[72]

Chemical Durability [76], [77] [79], [80] Linear Methods: [81]

This work: [78]

Density [82] MD, Packing: [83]

CTE This work. MD

Hardness [5], [84], [85] MD

Young’s Modulus [86], This [82], This work. MD

work: [87]

Activation Barrier for Conductivity MD, This work: [88]

Batch Cost Economic Calculation.

Ion Exchange Properties Empirical Relationships: [2]

19 1.4 Machine Learning

All the methods discussed so far are physically informed techniques. The techniques presented are inherently powerful because their very nature is informed by reality. Unfortunately, this means that some knowledge about the system is required. These techniques are useless without the required parameterizations. However, a class of algorithms called ‘Machine Learning’ (ML) require no such parameterization. Instead, ML uses large quantities of data to determine trends linking any physically connected input to output allowing for accurate predictions of properties across wide compositional spaces. This is favorable with a large of volume data because the required inputs are minimal to achieve a prediction. The difficulty (and often expense) in this technique is gathering large quantities of high-quality data to enable this method.

The predominant ML method used in this work is neural networks (NN). Neural networks work by constructing a network of neurons. This method was designed in such a way to mimic a human brain so the network is elastic enough to be modified, improving results. Each neuron takes in a series of inputs received from the previous layer (or input data) multiplies by a weighting factor, sums the values, and then performs some ‘activation function’ on them. By changing the weights, it is then possible to recreate any function given sufficient data. This is a preferred technique because it packages into a simple, easy-to-use software. Neural Networks (once trained) can be used by anyone, stored in a small file, can recreate any function, is computationally cheap, and does not require as much data as some other ML techniques.

This is an interesting method to estimate the effects of the composition on the landscape and corresponding properties because we cannot understand the underlying mathematics. It is fundamentally a numerical process. Thus, even though we know that the properties of a glass are dominated by the topography of the landscape, how the NN figures out the relationship between the chemistry and properties is not understood. It is possible to then use the NN to give us the

20 physical parameters we need for other models building multiple methods together to achieve new insights.

1.5 Goals of this Dissertation

To enable the next generation of glass design, we must enable a new era of computationally methods being used to produce accurate predictions of the behavior of glass. This includes both the thermal history dependence of glass (in terms of relaxation and crystallization) and the compositional dependence especially those properties listed in Table 1. The goals of this thesis will be to address some of these outstanding questions in a systematic way that is practical for an external user to understand and implement. Predicting thermal history effects are covered in

Chapters 3-5 while compositional effects are predicted in Chapters 6-7. The breakdown of each following Chapter as it relates to these questions is listed below:

21 • Chapter 2: Software. Unfortunately, there is no standardized software for the

calculation of relaxation or energy landscapes. I present two new codes to

standardize these calculations so that the process is reproduceable and standardized.

• Chapter 3: Crystallization. This section will address new computational ways to

calculate crystallization focusing on predicting nucleation rates and how to improve

our nucleation calculations.

• Chapter 4: Expanding the Current State of Relaxation. This chapter will focus on

expanding the MAP model with a temperature dependent form of the stretching

exponent as well as understanding the role of ergodicity in glass relaxation.

• Chapter 5: Relaxation and Crystallization without Fictive Temperature. Chapter 5

will focus on discussing relaxation and crystallization without fictive temperature

by proposing a new relaxation model that is reliant on creating ‘Toy Landscapes’.

This approach removes assumptions about the evolution of the occupational

probabilities.

• Chapter 6: Enabling Prediction of Properties. This chapter will use TCT and ML to

address the compositional dependance of different key properties needed for

enabling the faster design of new glass compositions.

• Chapter 7: Designing ‘Green’ Glasses. Applying information learned throughout

this dissertation to create new glass compositions.

• Chapter 8: Conclusions.

22 Chapter 2

Software for Enabling the Study of Glass

To being our systematic investigation it is important to have a suite of tools that implement current state-of-the-art models. Though many tools exist for structure and topological constraint theory there were no codes that existed for the calculation of relaxation using the MAP model and there were no codes that were capable of exploring the landscapes that we are interested in. To correct this, we present two new novel codes:

ExplorerPy for mapping energy landscapes and RelaxPy for calculating MAP dynamics[42], [89].

2.1 ExplorerPy

Mapping an energy landscape allows for modeling a range of traditionally inaccessible processes, such as glass relaxation. The advantage of this software (over other codes to map energy landscapes) is that it only requires knowledge of LAMMPS to run. Also, our software contains a choice of three methods to map the landscape. The first of the three different methods implemented to explore energy landscapes in this work is eigenvector following [90]. Eigenvector following has been used in the past to generate landscapes that predict the glass transition and protein folding pathways [44], [91]. This implementation follows the work of Mauro et al. [90], [92] and uses an independent Lagrange multiplier for every eigenvector. Eigenvector following excels at finding small barriers quickly; however, it rarely finds unique structural changes in complex materials

(especially in systems with periodic boundary conditions). The second method that we have implemented, following in the steps of Niblett et al. [27], is a combination of molecular dynamics and nudged elastic band methods (NEB) [34], [35]. To find the transition points, MD is run at a

23 fixed temperature for a fixed amount of time, the structure is minimized, and then NEB is used to find the saddle point between the initial structure and the new inherent structure. Details of the nudged elastic band method can be found elsewhere [34]. Using these techniques, it is possible to map a sufficiently large landscape to make kinetic and thermodynamic measurements possible.

These two methods benefit from the fact that they work well when the volume is varied and together enable computationally inexpensive mapping of enthalpy landscapes [71], [92] (where the volume is no longer fixed and instead the pressure is specified). However, it is not recommended to generate a large enthalpy landscape from MD in this software due to the computational cost involved.

Although molecular dynamics is a powerful technique, there is a large computational cost to simulate a small amount of time.

The third method is a new technique created in an attempt to circumvent the computational cost of MD while still finding large structural changes. We have named this technique “eigen treiban” (ET). This technique relies on shoving along an eigenvector with a multiplier rather than stepping. After the system is shoved the system is minimized and if the minimized structure exhibits a displacement in atomic position is larger than a threshold a NEB calculation is performed to find the saddle point. By shoving in the direction of an eigenvector the goal is to reproduce large changes due to vibrational modes. The fixed multiplier (m0) is calculated by taking a target distance (d) and the largest displacement magnitude out of the atoms j (vij) in a given eigenvector direction i (ai),

d m0 = . (24) max[nij ]

To convert to the displacement magnitude from an eigenvector in three dimensions we use the expression,

2 2 2 vij = ai[3j] +ai[3j +1] + ai[3j + 2] . (25)

The dimension j has the length of the number of atoms in the systems.

24 This software uses the system and potentials created from a LAMMPS input scripts, and as such has a wide range of access to different potentials and possible systems. The only input that has to be provided is the file which would be used for molecular dynamics in

LAMMPS. If the script has been used for molecular dynamics, simply by eliminating any thermostats or barostats as well as any “run” commands it is possible to use it as an input script to

ExplorerPy (it is also recommended to eliminate any calls to “Thermo” because it may interfere with the extraction of variables from the underlying LAMMPS framework). If the LAMMPS input script successfully sets up a simulation all other variables and parameters can be controlled from command line options as described in the “Illustrative Example” section. The initial structure will be taken from the input script as well.

ExplorerPy is a software that has been written in Python with modular functions and ease of use in mind. It can be run from a Linux terminal with all major options being set via the command line. The software then generates a list of inherent structure energies stored in the min.dat file and stores the transition point energy for each pairwise inherent structure in the tsd.dat file. All of the transition points and minimum are stored in the same directory from which it is run as .xyz files.

The software architecture is shown in Figure 2. Details of the stepping procedure for eigenvector following can be found elsewhere[93]. The variables “MD frequency” and “ET frequency” are set as command line options at the time of the launch of the software.

25

Figure 2. The schematic for the flow of the program. Beginning in the top right corner and running until the condition in the pink box is satisfied. Yellow diamonds represent checks and blue operations.

The software is capable of mapping an energy landscape for an arbitrary system, i.e., for any system that can be simulated in LAMMPS. This allows for accurate prediction of the evolution of complex thermodynamics and kinetics. The software requires no programming and is controllable via command line options. ExplorerPy prints all of the basins, transition states, inherent volumes, as well as information about the curvature of the system so that the vibrational frequency is calculable. This is all of the information required to calculate a complex range of phenomena over a specified long timescale. ExplorerPy is only set up to work within the LAMMPS metal units.

To run the software there are only a few parameters that have to be set for each mode. The force threshold (-force), curvature threshold (-curve), and step size (-stepsize) when running the command line option have to be set to allow for a stable exploration of a given landscape.

These seem to be potential-specific and for a given potential appear to be fairly stable. It is best to start large for the force threshold and small for the eigen threshold and step size then modify as

26 necessary. If eigenvector following is not being used there is no need to adjust these values. Beyond this there are several optional command line options (only the file (-file) command is required):

▪ -file [file input]: this is a standard lammps input, if the input sets up a lammps run it will work to set this up. There should not be any ‘thermo’ calls in the lammps script.

▪ -press [specified pressure in atmospheres]: if this is specified the system will always be minimized to this pressure, while if not specified the volume remains fixed.

▪ -num [number of basins to explore]

▪ -time [max time to run in hours] if this time is reached before the number of basins are found, the simulation will stop

▪ -thresh [minimum energy for barrier to keep]

▪ -center: if this is given as a flag the system will be set to the anchor location each step.

▪ -anchor [atom selection for the position to be held constant]

▪ -md [frequency of molecular dynamics search] : If a custom MD LAMMPS file is not provided it defaults to starting all of the atoms at 3000 K and running for 50 fs.

▪ -shove [frequency of eigen treiban]

▪ -custom [file used for MD exploration]

▪ -help [List all commands possible in software with these descriptions]

An example run would then be done the following way:

[mpirun -np 5 python3.7 Explorer.py -file in.SiO2 -num 1000 -force 1.0 -curve 1e-5 -stepsize 0.12 -press 0.0 -md 11 -time 48]

This would start a run with five parallel processors searching for 1000 basins with an MD step being ran every 11 searches with a maximum time of 48 hours. The results of this command are shown in Figure 3. Below we have attached an example of in.SiO2 the output and additional scripts related to the example has been uploaded to the associated GitHub.

27

Figure 3. The enthalpy landscape for SiO2. This was explored using the command above and then plotted using PyConnect. The plot is a disconnectivity graph where each terminating line represents an inherent structure and tracing where two lines meet describes the activation barrier. The potentials are taken from the BKS potential [94].

 https://github.com/lsmeeton/pyconnect

28 2.2 RelaxPy

RelaxPy script is written using Python syntax and relies on the numpy, scipy, and matplotlib libraries. It solves the lack of a unified relaxation code in our community and is based on the MAP model[44], [61] interpretation of multiple fictive temperatures and the shear modulus/viscosity form of the Maxwell equation[62], [95]. It is designed to be run within a Linux terminal with command line options or inside a Python interpreter. It can be used by calling the name of the script, followed by the input file, the desired output file name, and finally an optional tag for all fictive components to be displayed as well as printed to a specified output file (formatted as comma separated values).

The RelaxPy package consists of an algorithm that iterates over time in order to determine the values of viscosity (η), relaxation time (τ ), and current fictive temperature (Tf ) given the user- supplied thermal history and material property values. The purpose of this code is to enable calculation of the relaxation behavior’s change over time by using the MAP model using only experimentally-attainable parameters. Tool’s equation is solved for each term in a Prony series approximation of relaxation finding the change in each Tfi component for a given time (t) to accurately reproduce the stretched exponential form. The program then iterates over the entire thermal history of the sample using a user-specified time step, dt, allowing for calculation of the overall fictive temperature over time. The values for the Prony series fit (wi and Ki) are taken from the database originally generated by Mauro and Mauro[96] and expnded to include any value of the stretching exponent. The values were fit by Mauro and Mauro with a hybrid fitting method based on the number of terms in the desired Prony series and the magic value from Phillips. The thermal history is also defined in the original input file using linear interpolation. The initial fictive temperature components are assumed to be at equilibrium (Tfi = T(0)).

29 To solve for the non-equilibrium viscosity used the MAP expression is used with the approximate scaling of values proposed by Guo et al.[61]. The software is capable of performing the calculations needed for a variety of thermal histories and processing methods desired by both industry and academia, with a focus on the fictive temperature. This software is then fundamentally limited by the issues associated with the MAP model and the compositional approximations made. For instance, this software will need to be further modified to include structural relaxation when such a model is proposed[62].

An example input code is listed below:

# Denotes Comments Tg : 794 . 0 m: 36.8 C: 149.4 dt : 1.0 Beta : 3/7 # Start Temp, End Temp, Time [s] 1000, 10, 1300 10, 100, 1000 100, 30, 500 # Assumes Linear Interpolation

This example input states that over the space of 1300 seconds an initial glass sample is cooled from

1000 ◦C to 10 ◦C. Then it is heated to 100◦C over 1000 seconds and then finally down to room temperature in the space of 500 seconds. All temperatures are given in Celsius.

This software aims to provide the first open-source software for modeling of glass relaxation behavior using the MAP model of viscosity by creating a package that can easily and quickly be used to approximate the fictive temperature components of a glass as well as the composite fictive temperature. RelaxPy allows for researchers to calculate the evolution of macroscopic properties of glass during a relaxation process (assuming a fixed stretching exponent) and will facilitate the calculation based on a user-supplied thermal history. RelaxPy can be seamlessly adopted by most glass research groups and removes the guessing associated with determining a fictive temperature, allowing for a fully quantitative, comparable discussion of Tf simultaneously creating a tool that can easily be expanded for the entire community as new physics is discovered.

30 Chapter 3

Understanding Nucleation in Liquids

Crystallization cannot be ignored in any material system at temperatures below its liquidus temperature, although extreme resistance to crystallization is reported in some organic and inorganic substances[97]. Crystallization (and by extension nucleation) is critical in systems ranging from ice to thin films[98]–[100]. Glasses are non-crystalline substances; therefore, crystallization kinetics will determine the necessary thermal history to avoid crystallization and obtain a glassy state. Despite the absolute importance of crystallization to the study and design of glasses there is no current method to predict the nucleation of crystals in this event. One of the main challenges in the production of glass-ceramics is finding the optimal thermal history for the desired crystallite distributions. There are two phenomenological steps in the process of phase transformation from a supercooled liquid to a crystal: a nucleation step and a growth step. The growth step is described well by Wilson-Frenkel´s theory,[1] but the underlying physics of nucleation remains elusive. A stable nucleus is the precursor to a crystal, comprising a nano-sized periodic assemblage of atoms. A clear understanding of the relationship between liquid kinetics and crystal nucleation has not been established despite decades of research.

The study of nucleation is particularly critical for glass-ceramics. Since their discovery by

S. Donald Stookey, glass-ceramics have become prevalent in our society through products such as stovetops and Corningware[73], [101], [102]. Glass-ceramics are a composite material comprising at least one crystalline phase embedded in a parent glassy phase, benefitting from the properties of both the glassy and crystalline phases[1], [103]. One of the reasons glass-ceramics are ubiquitous is their relatively easy production; glass-ceramics can be formed through standard glass forming procedures, using additional heat treatments. Additionally, there is a wide range of accessible

31 properties through tuning both the chemical composition of the parent glass and the thermal history, and hence the microstructure, of the heat-treated glass-ceramics[101], [103]–[106].

Glass-ceramic production consists of first synthesizing a glass and then generating a microstructure through the two-stage ceramming process[73], [103], [107]. The nucleation step remains poorly understood, with classical nucleation theory (CNT) giving widely varying results depending on how the parameters are determined[1], [73], [108]. The inability to consistently get accurate predictions for a nucleation curve means that choosing an appropriate nucleation temperature must be done experimentally, which can prove to be one of the most challenging and time-consuming aspects of properly designing a glass-ceramic. The difficulty is due to the magnitudes of the crystal nucleation rates not being predictable with CNT or any other current model.

This difficulty in designing a material is apparent when considering the high dimensional phase space of parameters that can adjusted. We haver presesnted some tools developed to help optimize over this ‘glassy’ space, such as RelaxPy[42], KineticPy[41], topological constraint theory[109], or various machine learning models[4], [70], [110], [111]. However, a glass-ceramic system has at least four additional phase dimensions (two for the nucleation/growth temperatures and two for nucleation/growth times) that need to be optimized (depending on the parameterization of the experiment, it is also possible to add another four dimensions relating the heating and cooling between each step). Therefore, a glass-ceramic system has a minimum of d+5 dimensions over which the system must be optimized, and yet those four additional dimensions are the least understood. Designing methods to easily optimize over this large space with feasible computational methods is of utmost importance to reaching a new generation of custom materials.

Experimental studies of nucleation have used techniques ranging from differential scanning calorimetry (DSC) to electron microscopy to try to capture predictive capability in the

‘ceramic’ phase-space (the four additional dimensions). These studies have successfully captured

32 the shape of the nucleation curve but fail to provide the information needed to calculate the temperature-dependence of the nucleation rate. The shape of the curve can be determined through

DSC and with great effort an approximation of the rate can be made however, this technique is too laborusome to be efficient[112]–[114]. Since the parameters have been inaccessible to traditional computational approaches and experimental determination, all previous studies of nucleation have required at least one variable to be fit. If a nucleation curve could be predicted only using a computational method or with an efficient simple DSC method, the rate at which new glass- ceramics products are created would increase. In addition, the cost of producing glass-ceramics would likely decrease and more complex microstructures, and therefore more unique properties, could be explored.

Crystal nucleation has been historically difficult to simulate and measure experimentally, because the phenomena happen on spatial scales of nanometers but over many orders of magnitude of time scales. CNT is the most widely applied model despite its supposed failure. To test CNT’s assumptions an interest in computational work has grown. Recent approaches using grand canonical Monte Carlo have successfully predicted the short time dynamics of nuclei formation[73]. This approach involves use of a continuous solvation model for the liquid and has enabled greater computational insights; however, it remains computationally expensive[74].

Molecular dynamics (MD) fails to capture nucleation in glass-forming systems due to the need for longer spatial and temporal scales to capture nucleation effects accurately. Some attempts have been made using Monte Carlo, MD, or empirical approaches to replace classical nucleation theory

(CNT) with some alternate method where the parameters can be directly measured[73], [115]–

[117]. All nucleation models and Monte Carlo models attempt to describe the rate that a supercooled liquid system transitions to a crystalline structure as a function of temperature.

Explicitly mapping the energy landscape would fundamentally explain the underlying physics of nucleation. An energy landscape of a system is given by

33

UUxyzxyzxyz= (,,,,,,...,,,),111222 N N N (26) where U is the potential energy, and x, y, and z are the locations of each of the N atoms in a periodic box. The locations of these atoms give rise to an energy as parameterized by an interatomic potential. An energy landscape can be constructed by mapping the relationship of the local minima and the first order saddle points between pairs of minima22-27. Energy landscapes have become crucial to studies of biomolecules, glasses, and catalysts[21], [26], [44] because landscapes show atomic configurational changes and calculate the associated kinetics and thermodynamics. In this study, we will explore the energy landscape of a barium disilicate system capturing the glassy and crystalline basins enabling a test of CNT by calculating all needed parameters (interfacial free energy, kinetic barrier, and free energy difference of the supercooled liquid and the equilibrium state).

3.1 Crystallization Methods

CNT expresses the steady-state nucleation rate (I) as the product of a kinetic and a thermodynamic factor,

−W * IZD= Te ()exp,  (27) kT where Ze is the Zeldovich factor, which incorporates the probability of a critical sized nucleus that will form a new phase independent of other nuclei, D(T) is the kinetic factor often given by an

Arrhenius expression over sufficiently small temperature ranges, W* is the work associated with generating a thermodynamically stable nucleus that will continue to grow rather than dissipate (i.e., the critical work) if temperature and pressure are kept constant, k is Boltzmann’s constant, and T is absolute temperature. The Zeldovich factor is typically treated as a constant with a value ~0.1[118].

34 Here, the kinetic term is expressed as D(T) since, in this work, the kinetics are explicitly calculated at every temperature, and it is not treated as a simple Arrhenius expression. In CNT, the work W is given by the work to generate a spherical isotropic nucleus[1] , which consists of a volume term and a surface energy term. It is given by

4 WrGTrT=+32()4(), (28) 3 where G is the Gibbs free energy difference between the crystalline state and the corresponding liquid, r is the radius, and  is the interfacial energy between the crystalline and liquid phases. Eq.

(28) neglects the strain energy term that arises from the formation of a solid nucleus in the supercooled liquid. It is assumed that the liquid relaxes fast enough so that strain energy is not significant for nucleation in liquids[11].

Determining the critical radius (i.e., taking the derivative of Eq. (28) with respect to the radius and setting the expression equal to 0) allows one to solve for the critical work, and the expression becomes

16 ()T 3 W * = . (29) 3(GT())2

Three temperature dependent functions remain: D, G , and  . Though there are existing methods in literature to approximate each of these quantities, the approximations often include large errors and assumptions about the behavior of the system[119]. Often, experimentally the kinetic function is approximated with the Stokes-Einstein relationship, the G term is calculated from heat capacity curves (assuming that the macroscopic energy is similar to the nanoscopic energy), and

 is fitted[119]. In the calculation presented here only configurational energies are considered while vibrational contributions are ignored. Ignoring the vibrational component and the reliance on empirical potentials are the most likely sources of error in this work.

35 The energy landscape for this investigation was generated using ExplorerPy[89] starting with the orthorhombic barium disilicate crystalline structure[89]. Barium disilicate was chosen since it is a commonly studied glass-ceramic with multiple reliable experimental datasets being found in literature[75], [113], [120]. The rest of this section is divided into four parts, starting with a discussion of mapping the landscape and the following three sections are a discussion of calculating D, G , and  , respectively.

3.1.A Mapping and Classifying the Landscape

The ExplorerPy energy landscape exploration employed the Eigen-Trieban technique in the canonical ensemble with systems consisting of 128, 256, and 512 atoms[89]. The number of atoms was chosen to achieve convergence of the results while also balancing the large computational cost associated with energy landscape exploration. A smaller system was also calculated with 64 atoms; however, this systems was too small to accurately discern energies enough to make the nucleation calculation. The Eigen-Trieban technique is a newly developed method to explore energy landscapes using a method of shoving along the eigenvectors and minimizing the potential energy, which is followed by using nudged elastic band calculations to find the associated transition points[89]. This technique allows for larger transitions to be found while retaining the ability to calculate the vibrational frequencies. The Pedone et al. potentials[36] were used and the volume of the cell was fixed, due to simulation instability when pressure was fixed. The Pedone et al. potentials were chosen since they have reproduced key parameters for both the crystalline and glassy state in previous Monte Carlo studies as well as in the training set of the potential[73], [74], [121]. Though the Eigen-Trieban method may work well for predicting crystallization and finding large barriers, it may be insufficient to describe glass relaxation or

36 viscosity, since it rarely finds the small structural transitions that dominate the Adam-Gibbs relationship. Note that these exceptions are inconsequential for predictions of nucleation found in the work presented here33.

The barium disilicate system was explored for 2500 basins for 128 and 256 atoms and 4000 for 512 atoms since more basins were needed to converge the liquid state distribution of energies.

This number of basins was shown to be sufficient since multiple paths from the initial crystalline starting point to SCL states were obtained. The distribution of energies of the SCL were found to converge and the resulting energy landscape is seen in Figure 4.

A crystal parameter was then defined from the root mean square displacement (RMSD) for each atom from the nearest lattice site of the same species (  ). This average value is used to characterize whether a basin is a crystal or SCL. The RMSD approach is not the only method to assign a crystallinity value[122]; however, it is one of the simplest and, as such, was chosen for this application. To then categorize each basin as either a crystalline basin or SCL basin, a plot of the minimum energy of the SCL states versus the cutoff of RMSD was produced. This plot for 256 atoms shows a clear transition in the energies around a cutoff value of 1. Å, which is, therefore, taken as the cutoff radius of which basins are considered crystals. The same analysis was repeated for each system size. This analysis is shown in Figure 4. Any basin with a RMSD less than the cutoff value is considered a crystalline state. This cutoff was found to be generally insensitive to the results with any values ranging between 0.9 Å and 1.1 Å changing the resulting prediction of nucleation by less than one order of magnitude but shifting the peak temperature by up to 40 K.

This cutoff also reproduced the energy difference between the SCL and crystal in molecular dynamics simulations of the materials.

37 [

Figure 4: (A) The energy landscape for a 256-atom barium disilicate glass-ceramic, the x-axis being an arbitrary phase space and the y-axis being the potential energy calculated from the Pedone et al. potentials[36]. The colors are indicative of the crystallinity, where the blue basin is the initial starting configuration. The landscape shows the lowering of energy associated with partially crystallizing the sample. (B) The energy landscape relationship between the cutoff for crystalline and super-cooled liquid states for 256 atoms is shown. This shows a clear drastic energy change occurring around the cutoff value of 1.0 Å.

38

3.1.B Kinetic Term for CNT

The transition rate was calculated by the inverse of the average relaxation time from the

SCL to the crystalline states. The relaxation time ( ) is defined by the relaxation function ( ),

t , (30) =−0 e x p 

where t is time and 0 is the scaling of the relaxation function. The reason we are able to calculate the kinetics of nucleation but not viscosity is because the nucleation kinetics are dominated by the transitions between two states that we specifically searched for when exploring the energy landscapes, while the viscosity is defined by the entropy dominating small transitions described by the AG.

The AG relationship shows that the viscosity is dominated by the configurational entropy of the system. Experimentally the kinetic term is often calculated with the Stokes-Einstein relationship,

kT D()T = , (31) 6a where a is the radius of the diffusing species and  is the viscosity of the liquid. Interestingly, this approximation gives reasonable results when investigating experimental data, since the viscosity is a measurement of the entropy of small cooperative liquid rearrangements. Conversely, the kinetics that govern nucleation are large structural rearrangements from a supercooled liquid to a crystal. It is nonetheless possible that these cooperative rearrangements and the large diffusion processes for nucleation are on the same order of magnitude which is why the Stokes-Einstein relationship gives

39 a reasonable approximation in experimental parameterizations. When the Stokes-Einstein relationship is used, there is an implicit assumption that the large nucleating rearrangements are proportional to the small barriers that govern viscosity[53].

A mean relaxation time can be found by averaging over a system of master equations,

dp i (32) =−().gKpgKpijijjiji dt ji

In this expression, g is the degeneracy of a microstate, pi is the probability of occupying the microstate, and Kij is a transition rate between inherent structure i and inherent structure j. The equilibrium probability of occupying a microstate can be given by a Boltzmann probability (where

Z is the partition function and U is the potential energy of the inherent structure),

1 U i . (33) pgii=−exp ZkT 

By substituting the value of p j in Eq. (32) for a two-state model (1 =+ppij==12) we write,

dpi =(gi K ji(1 − p i) − g j K ij p i ) . (34) dt

Isolating the probabilities and splitting the differential, we have

dp i = dt , (35) gijiijiji Kpg(1 Kp−−) which when solved via integration takes the form,

 ln()gi Kg jii Kg−+ jij K iji p t = . (36) −+()gi Kg jij K ij

Solving for the occupational probability yields,

g K−exp  − g K + g K t i ji( j ij i ji ) pi = . (37) gi K ji+ g j K ij

This shows an analytical solution for the relaxation time in a two-state model,  ,

40 1  = . (38) ggKijijijK +

The two-state model approximation is made to enable the calculation of the mean relaxation time between the liquid basins to the crystalline basins. The transition rates between the basins are given

by the vibrational frequency ( ) and the barrier between the states ( Uij ),

 −Uij  Kij = exp. (39) kT

The probability of transitioning from a given liquid basin i to a crystalline basin j is given by

1 −Uij pij = exp, (40) YkT  where Y is the partition function with respect to all transitions from the current SCL basin to all crystalline basins as only these transitions are considered to calculate the relaxation time,

Uij Y =−exp . (41) j kT

The total expected relaxation time of the system making the transition is then given by the probability of occupying a basin, the probability of transitioning from one basin to another specific basin, and the associated relaxation time,

1  = pp . (42) iij  ii j gi Kg jij+ K j i

The first summation is over the liquid states and the second summation is performed just over crystalline states to capture the mean relaxation time from a supercooled liquid to a crystal. We

14 fixed the vibrational frequency to a value of 1.7 × 10 Hz (i.e., about one jump attempt per 5 femtoseconds), which was calculated using the average of the eigenvalues of each basin ( ) and the average mass of the atoms (m) using,

41

1   = . (43) 2 m

3.1.C Degeneracy calculations

To obtain an appropriate value for the degeneracy, an understanding the phase-space volume of the basin is needed, since the degeneracy is linked to the volume of the basin[40]. By taking the displacement between the basin coordinates of each atom and the transition point we can define a radius of the basin, R. Using R and a spherical approximation we can calculate the phase- space volume degeneracies,

 3/2N gR= 3N . (44) iiN gcry+1 2

In which ( x) is Euler’s gamma function. The degeneracy is then normalized by a constant such that the liquid-liquid state transitions had a value of 100 s at the glass transition (980 K) to be in agreement with Angell[58]. The normalization step allows for a quantitative prediction of the liquidus temperature to confirm the accuracy of our model. The liquidus value found is approximately 1200 K, 1715 K, and 1950 K for 128, 256, and 512 atoms respectively. The experimentally determined liquidus temperatures range from 1680 K to 1710 K[113], [115], [120].

It is worth noting that changing the exponent drastically changes the thermodynamics while having minor effects on the kinetics and the normalization value drastically changes the kinetics while keeping the thermodynamics nearly constant. We chose the denominator to satisfy the liquid relaxation constraint. It is also feasible to choose the exponent to satisfy a constraint on the liquidus temperature. The normalization value is an important to consider because for larger system a change in 0.01 can result in a change in the kinetics by ~10 orders of magnitude.

42 3.1.D Free Energy Difference

Due to limitations in the exploration of the energy landscape requiring a fixed volume, we will approximate the difference in the Gibbs free energy by calculating the difference of the

Helmholtz free energy. This approximation is reasonable due to the similar densities between the crystalline and glassy phases under ambient pressure[73], [74]. The first step is determining the mean energy difference of all the basins, and the second step was repeating the process for only the liquid basins. The mean energy was calculated with the following expression,[123]

(45) GkTppUkTgp=+− i lnln.iii ( i ) i

In Eq. (45), i is the set of relevant basins for the conditions we are interested in. The difference between the mean free energy of the liquid basins and the entire landscape basin energies is the

G parameter. This free energy difference is typically experimentally calculated by integrating heat capacities; however, numerous approximations exist. These approximations are often linear approximations normalized to the liquidus temperature. In this work we find that the driving force appears to be predominantly linear, which is in good agreement with the approximations that are often used for the driving force[124].

43

Figure 5. An example interfacial structure between the crystalline phase on the left and the last sequential SCL/glass phase on the right for a barium disilicate system. The gray atoms are barium, silicon is shown in red, and blue represents oxygen.

44

3.1.E Interfacial Energy

To calculate the interfacial energy of the interface between the crystalline phase and the

SCL, the initial basin (pure crystalline) and the last basin (glass basin) were annealed at 1700 K for

1 ns and then cooled to a temperature of 800 K over a period of 10 ns. The same process was repeated for a composite where the initial stable crystal and the glass were placed next to each otherThe structure was multiplied 10 times to create a wall of a large interface to reduce energetic fluctuations. A small section of the composite structure is shown in Figure 5. The difference between the addition of the energy of the crystalline phase and glassy phase and the composite gives the energy of the interface, which was normalized to the size of the surface area. Multiple arrangements of the composite were tested, but anisotropy was found to be minimal with the fluctuations of the energy being larger than the differences from different tested directions. The final expression used for the interfacial free energy (in eV/Angstrom2) is

 = 5.047+100.03−6T 040 .

45

Figure 6. (Top) The relaxation time and free energy difference (Middle) shown as a function of temperature for each system size. The experimental values for kinetics and thermodynamics come from ref.[125] and from the heat capacity data taken from [126], respectively. It is clear to see that driving force shows good agreement across all systems however the kinetics terms only converge

46 for the 256 and 512 atoms systems. (Bottom) The fit used to calculate the interfacial energy as a function of temperature.

3.2 Results & Discussion

The results for the free energy difference, the kinetic parameter, and the surface energy are shown in Figure 6. These parameters, as a function of temperature, are used to calculate the nucleation rates in the expression

0.116  3 I =−exp 2 . (46)  ()TG3() kT

The 0.1 being an approximate value of the Zeldovich factor. Using the parameters shown in Figure

6 and solving Eq. (46), the nucleation rates are shown in Figure 7 along with experimental data for validation. At first glance, we notice that there is about 7 order of magnitude differences between data and prediction. Considering that theses potentials were trained on glass and were not intended for this purpose we consider this as excellent agreement. This method also shows agreement below the peak, but the accuracy falls off more quickly below the peak temperature. It is likely that the data below the peak is severely underestimated due to a failure to reach steady state nucleation rates. Reliable steady state nucleation rates are not available for this system and as such a direct comparison below the peak temperature is not directly available. It is worth mentioning that the failure of CNT of predicting the order of magnitude of nucleation rates is well reported and can amount to dozens of orders of magnitude when a constant value of  is used[75], [113], [115],

[119], [127]; thus, the proposed method has an advantage when comparing with the current classical nucleation theory toolset. Moreover, the prediction of Tmax (~950 K) is also very close to the experimental value (985 K).

47

Figure 7. The nucleation curve for barium disilicate for converged systems as predicted using the model presented in this work. The data referenced can be found in refs. [113], [128], [129].

48

To further compare CNT and the model presented here, we decided to compare the results of the fitted and this work’s calculated surface energies, as seen in Figure 8. The results show good agreement, all achieving the same order of magnitude, implying that experimental methods can achieve a close approximation to the values required for CNT. The prediction of nucleation is extremely sensitive to the value of the surface energy with small changes leading to orders of magnitude change. It stands to reason then that further work that needs to be approached through computation and experiments is developing a rigorous model to predict the interfacial free energy.

The other assumption that is used experimentally is the Stokes-Einstein relationship. To compare the landscape results of the transition rate from the SCL to crystalline states and the

Stokes-Einstein relationship, the liquid state relaxation was calculated. The small barriers that govern the Adam-Gibbs relationship were not found with this exploration method; hence, quantitative comparisons are not possible (and relaxation calculations should not be made); however, the relaxation times can be qualitatively compared. The liquid state relaxation rate is proportional to the viscosity and is given by the expression[44],

pi  l =  . (47) i  Kij j

To understand the validity of the Stokes-Einstein relationship we can write a series of theoretically true proportionalities,

TT DT()  . (48) l

The values of the experimental kinetics fail to align with the kinetics calculated in this work. This is expected, due to the fact that the viscosity/liquid relaxation is dominated by the set of small transitions in the landscape, while the nucleation transition is defined by the transition of one region to another, with a substantially higher activation barrier. This comparison may also account for

49 some part of the experimental overestimation below the peak, with the kinetic term calculated dropping off at a faster rate than that calculated from the experimental viscosity. Figure 6 also shows evidence that it may be inappropriate to use the Stokes-Einstein relationship for nucleation, and as such we may need to re-visit the experimental models that often rely on the assumption of the Stokes-Einstein relationship. This observation casts some doubt in the legitimacy of describing the kinetic factor as a diffusivity, which is standard for CNT. For more information on this CNT failure, please see Refs. [75], [113], [115], [119], [125], [127].

Nucleation is a complicated concept to investigate and model. Unlike the crystalline phase, where the atomic positions are clearly defined, the details of the atomic coordinates and bond configurations of the liquid sites must be described in terms of statistical distributions. Due to this added complication, mean-field distributions are typically used to describe non-crystalline sites[14], [16], [108], [130]–[133]. The incorporation of fluctuations could more accurately predict each liquid site’s transition to the crystalline phase. Fluctuations in the degeneracy could be one reason that the nucleation rate is over predicted when the temperature is less than the peak nucleation temperature. A given G varies with the environment and depends on the specific liquid state, not the mean value. Therefore, the rate of nucleation from all supercooled liquid states to the crystal state is not constant, which is why fluctuations need to be considered and why energy landscapes are such a powerful technique. This is especially highlighted when considering that different degeneracies yield drastically different heat capacity. Energy landscapes are extremely useful because they consider a large distribution in free energies and show all structural fluctuations possible. Fluctuations in cluster size and degeneracy influence Adam-Gibbs configurational entropy, which directly impact the transition rate between the crystal and liquid state.

50

0.014

0.012 ) 2 0.01

0.008

0.006

0.004 Surface Energy (eV/Å Energy Surface This Work 0.002 Xia et al. (2017) Ref. 23 Fokin et al. (2016) Ref. 41 0 800 850 900 950 1000 1050 1100 Temperature [K]

Figure 8. The surface energy with respect to temperature for the work presented here.

51

3.3 Conclusions

Though this is not the first time CNT has been examined computationally in an attempt to understand if the theory is valid[74], [116], [117], it is the first parameterization of each variable independently in an attempt to calculate the resulting nucleation rate. Not only are all the variables calculated when used in the expression for CNT, the liquidus temperature is reproduced, and the nucleation rate is reasonably predicted. This is, as far as the authors know, also the first time a real energy landscape for a complicated glass-ceramic system is being reported. This method leveraging energy landscapes alongside CNT enables for a reasonably fast calculation of the nucleation curve of a glass forming liquid. In future work, this model could be used for either developing commercial glass-ceramics or for predicting the critical rate needed to maintain a glass. This model can also provide additional insights into the physics of nucleation and the validity of CNT, without the constraints of fitting parameters. This method fundamentally has the power to connect the 3N dimensions of the energy landscape to the compositional dimensions and thus allowing for a unique powerful method of predicting nucleation parameters as a result simplifying the crystalline dimensions in which a glass/glass-ceramic must be optimized over.

Chapter 4

Expanding the Current State of Relaxation

Before we consider a new approach to relaxation (optimizing our understanding of the 3N dimensions) there are first missing pieces in the current state specifically the temperature and compositional dependence of the stretching exponent. However, even before expanding the current state of the art relaxation methods we also need to understand ergodicity. In this section we will start with a simple though experiment to understand the temporal effects of relaxation and then in the second section combine it with fictive temperature to create a new model for the stretched exponent.

4.1 A thought experiment to expand our understanding of ergodic phenomenon: The

Relativistic Glass Transition

The prophet Deborah sang in the book of Judges, “The mountains flowed before the Lord”

[6], [13]. However, the question the prophet failed to ask was what happens if the mountains are moving at relativistic speeds. The focus of this section is to highlight the temporal effects on ergodicity in the context of special relativity and the reference frame of the observer. In order to understand relativistic effects on materials, one must consider a well-characterized material that actively relaxes but appears solid on our timescale, viz., pure B2O3 glass [134], [135]. Glass is a particularly good candidate to study the effects of time dilation on a material since it undergoes a kinetic transition known as the glass transition (Tg), which has multiple definitions that will be considered.

53 The first definition is in terms of the aptly named “Deborah Number,” which was proposed by Reiner [136] to offer a view of fluidity and equilibrium mechanics. It was later expanded to explain the origin of the glass transition and the transition from an ergodic liquid system to a non- ergodic solid-like glassy state[12], which is the fundamental definition of the glass transition [11],

[12], [137]. It is important to note here that the breakdown of ergodicity is relative to the observation timescale and is reflected in the Deborah number (D), which is defined as

t D = (49)  where t is the (external) observation time and  is the relaxation time of the material, which can be expressed through the Maxwell relation [42]. By definition, the Deborah Number equals 1 at the glass transition temperature, Tg [13], [137].

Another definition of the glass transition temperature is due to Angell, who defined Tg as the temperature where the viscosity of the supercooled liquid is 1012 Pa s [58], [138]. The Angell diagram is an important plot relevant to viscosity and the glass transition, where the abscissa is Tg/T and the ordinate axis is log10( ). In the Angell diagram, the value of the viscosity at Tg and in the

limit of infinite temperature are fixed, and the difference in the temperature scaling of the Tg/T- normalized viscosity is the slope of the log10( ) curve at Tg/T = 1, which Angell defined as the fragility (m) of the supercooled liquid. Using this definition, one can establish the consistency of the Angell plot at relativistic speeds. An example of the Angell plot is shown in Figure 9.

54

Figure 9. An Angell diagram created using the MYEGA expression [55]. With a very strong glass (m~17), a highly fragile glass (m~100) and a pure borate glass (m~33) [17], [134], [135]. The infinite temperature limit is from the work of Zheng et al.[54], and the glass transition temperature is from the Angell definition.

55 Using the two previous definitions it is expressed,

Gt D ==1 (50) Tg  which can be re-written as,

(T ) g = t (51) GT( g )

In order to solve this equation, one must know the shear modulus at the glass transition temperature, recent improvements in topological constraint theory have led to the ability to predict properties with varying temperature and composition. Using a recent topological constraint model for elastic moduli proposed by Wilkinson et al. [139], one can predict the temperature dependence of the modulus,

n q Tx N ΔΔΔFnq TFnq++ TFnq TxN  dGdG c c c CA( ) ( ) (   ( )    ( ) ( )) ( ) A G ==  d FMd FM cc

(52).

In Eq. (5), nc is the number of topological constraints (c), qc is the constraint onset function as described by Mauro et al. [69], Fc is the free energy of the constraint,  is the density of the

푑퐺 system, M is the molar mass, and is a scaling factor. This expression was evaluated as a 푑훥퐹푐 function of temperature with the total modulus coming from Kodama et al. [135] and is equal to 8

GPa at room temperature. The B2O3 structure consists entirely of three-coordinated boron [140],

[141].

Using the modulus predicted in Eq. (52) numerical observation time is expressed,

56

1012 Pa s t ==141 s (53) 7.0910 Pa9

Using this description of the static (v/c=0 where v is the speed of the glass and c is the speed of light) glass transition it is possible now to describe the relativistic glass transition for which there are considered two separate cases. It is worth noting that this is not the first case study of relativistic viscosity, a fairly large basis of literature dealing with ideal fluids near large gravitational bodies [142]. There is also some work also relating the relaxation time in a gravitational field to the nonequilibrium entropy relaxation time [143], [144]. This work emphasizes the theoretical effects of special relativity on real liquids, their glass transitions, and the properties there dependent on.

4.1.A Relativistic Liquid

The first case is when a sample of a glass-forming system moves past a stationary observer at a speed approaching light giving the new observation time as,

t t = 0 (54) 

where t0 = 141 s (as previously calculated) and g is the Lorentz factor,

1  = (55) v2 1− c2

The Maxwell relation can be modified to account for relativistic effects by combining with Eq. (54)

,

(Tg ) = GT( g ) . (56) t0

57 This gives a condition that is temperature dependent to predict the new glass transition as a function of v, however, in order to accurately describe the relativistic behavior density must be considered.

Density will change by a factor of  2 , because both the mass and the volume will be affected and as such,

(Ttgg)(= GT) 0 . (57)

The prediction then for the glass transition as a function of the speed of light is shown in Figure

10.

Figure 10. The relativistic glass transition temperature for B2O3 glass.

58 Using the Angell plot one can calculate the viscosity using the adjusted glass transition temperature, which is used to understand the dynamics of the equilibrium liquid with the Mauro-Yue-Ellison-

Gupta-Allan (MYEGA) expression [55], [145]. The MYEGA equation needs 2 other parameters besides Tg to generate the viscosity curve: fragility (m), which is the slope at Tg in the Angell plot, and the infinite temperature viscosity (). The fragility will not change because one assumption in this model is the consistency of the Angell plot at all relativistic velocities, even relativistic. The infinite temperature viscosity which also will not change because this is the minimum possible viscosity for a liquid. The new viscosity curves for various speeds are shown in Figure 11.

59

Figure 11. The equilibrium viscosity curves for a borate glass travelling at different fractions of light speed. All of the viscosities approaching the universal temperature limit for viscosity.

60

By using the Deborah Number, it has been shown that at relativistic speeds will change the viscosity in order to maintain the Deborah number and in this case, it is shown that when the Lord would move past the mountains at relativistic speeds the mountains would not actually flow but instead hold still (even more so than they currently do). From this, it is inferred that any liquid moving past the earth at relativistic speeds will appear to be more solid and eventually appear as a glass.

4.1.B Relativistic Observer

In this case, we consider an observer moving past a sample of glass at relativistic speeds giving,

tt= 0 (58) which when the relativistic shear modulus  2 factor is included can be written as

 (Tg ) = GT( g ) (59) t0

Solving in the same manner as before it is shown which shear modulus is needed to satisfy the condition in Figure 12.

61

Figure 12. The modulus needed to satisfy the condition for the glass transition.

62 In Figure 12, the shear modulus at the glass transition temperature is shown; however the constraint theory mechanism being implemented to calculate the shear modulus has a built in upper limit when calculating the Tg past the point in which all the constraints are intact. Due to this limitation, there is an upper limit to our prediction. Nonetheless the calculated Tg over the available range are shown in Figure 13, as well as the related viscosities. It is seen clearly that at higher velocities the glass transition shifts towards zero, more dramatically at the higher temperatures.

This leads to the interesting result that if we take the limit of an observer moving past the earth at close to the speed of light, all glass would appear to be a liquid.

63

Figure 13 (Top) The predicted glass transition with the anomalous behavior occurring at v=0.44c. (Bottom) Viscosity plot showing a dramatic reduction of the viscosity as the observer approaches the speed of light.

64 Applying the mechanisms proposed by Einstein to the concept of the Deborah number, it has been shown that the viscosities will appear to change dramatically as a glass-forming system approaches relativistic speeds. This allows us to build an intuitive understanding of the differences between glass and liquids. Specifically, any material can appear either more or less fluid based on the velocity with the velocity with which they travel relative to the observer. This serves to highlight the importance of the role of the observer on the glass transition, and why even small changes in

DSC scan rates or sampling frequency can lead to drastically different measured glass transitions.

4.2 Temperature and Compositional Dependence of the Stretching Exponent

The mathematical form for glass relaxation ( ) was originally proposed by Kohlrausch in

1854 based on the decay of charge in a Leyden jar[9], [14], [29], [42], [45], [96], [146],

 (tt) =−exp( ( / ) ) , (60) where t is time, t is the relaxation time of the system, and b is the dimensionless stretching exponent.

Eq.(1) was originally proposed empirically and is known as the stretched exponential relaxation

(SER) function[42], [96]. The relaxation time, t, depends on the composition, temperature, thermal history, pressure, and pressure history of the glass, as well as the property being measured[62],

[147]. For example, the stress relaxation time of a glass can be written as[62]

 (TTf ,,, PP f )  s = (61) G( TTf ,,, PP f ) where h is the shear viscosity and G is the shear modulus of the glass-forming system[42].

(Variables are defined in Table 2.) In addition to using Eq.(61) to describe stress relaxation time, structural relaxation time has recently been hypothesized follow Eq.(61) with bulk viscosity and

65 the difference of bulk modulus between infinite and 0 frequency replacing shear viscosity and shear modulus[62]. Both h and G vary with the temperature (T) and thermal history of the glass (as quantified via the fictive temperature, Tf) as well as the pressure (P) and its corresponding history

(fictive pressure, Pf)[16].

To connect the stretching exponent to its physical origins, we can begin with a result from

Richert and Richert[148] that relates  to an underlying structural relaxation time distribution, and then develop equations that determine  from known quantities. Their expression,

22 2 1−  ln = 2 , (62) 6 

2 relates  to the variance of the logarithm ( ln ) of the structural relaxation time . For reference, all variables are defined in Table 2. The physical origins of the relaxation time  are related to the configurational entropy (Sc) as shown by the Adam-Gibbs equation[53], [55], [149],

B ln=+ln  , (63) TSc

where  is the infinite temperature relaxation time, T is the absolute temperature, and B represents the energy barrier for relaxation.

Table 2 Variable definitions.

Variable Definition t; ts Structural relaxation time; stress relaxation time b Kohlrausch exponent (i.e., the stretching exponent)

T; Tf; Tf,i Absolute temperature; fictive temp.; partial fictive temp.

P; Pf Pressure; fictive pressure

66 kB Boltzmann’s constant wi, ki, i=1 to N Prony series parameters

B Adam-Gibbs relaxation barrier

Sc Adam-Gibbs configurational entropy

B Distribution of activation barriers

2 Variance

Na Number of atoms f Topological degrees of freedom per atom

 Degenerate configurations per floppy mode

H Enthalpy barrier for Relaxation d Number of dimensions f* Fraction of activated relaxation pathways

S Adam-Gibbs entropy in the infinite-temperature limit m Kinetic fragility index m0 The limit of a strong liquid

6 Grouped unknowns B   NkaBln 

A Proportionality between fragility and distribution of barriers x A given composition

 S  5.640A1/2 

 Intercept of the linear model of .

67

4.2.A Deriving a Model

Combining Eqs.(62) and (63), Gupta and Mauro[150] proposed that the variance of the energy

2 barriers for relaxation ( B ) could be rewritten as

22 2 2 1−  Bc= TS ()T  2 . (64) 6 

Solving for the stretching exponent,

TS ()T  = c . (65) 62 + TS (T ) 2 Bc( )

The work of Naumis et al.[23], [56] has shown that the configurational entropy is proportional to the topological degrees of freedom in the network, a result that was used in the derivation of temperature-dependent constraint theory[57] and the MYEGA (Mauro-Yue-Ellison-Gupta-Allan) equation for the relaxation of supercooled liquids[55]. The MYEGA equation was derived by expressing the configurational entropy as

STxfTxNc (,)(,)ln= k a B . (66)

In Eq. (66), f is the topological degrees of freedom per atom, Na is the number of atoms, kB is

Boltzmann’s constant, x is a given composition and  is the number of degenerate configurations per floppy mode. The temperature dependence of the topological degrees of freedom was approximated using a simple two-state model,

* −Hx() f( x , T )== df 3exp . (67) kTB

Here, H(x) is the enthalpy barrier for relaxation, d is the dimensionality of the system, and f* is the

68 fraction of activated relaxation pathways. Combining Eqs. (65) and (67) as well as condensing the unknowns into the term defined by

6  = B (68)  NkaBln  we get

df *  = . (69) 2  * 2 + (df ) T

This can be compared to the prediction made by Phillips[48] for the stretching exponent at temperatures below the glass transition,

df *  = . (70) 2 + df *

Comparing the two expressions of Eq.(69) and Eq.(70), they would agree if

 =+44df * . (71) T

We will show later (Figure 12) an example where extrapolating our model prediction for  to room temperature (with T in Eq.(71) replaced by fictive temperature Tf) gives a result close to the Phillips value of Eq.(70). When the temperature T in Eq.(69) is high (much larger than ) then Eq.(69) gives us →1. Thus, the model can interpolate between the low-temperature (Phillips) value and the high-temperature value of 1. In our current model, we do not have access to a value for  in Eq.(68) so this model cannot assess whether the Phillips room temperature value for  is universal, i.e. whether Eq.(71) is always satisfied at room temperature.

Zheng et al.[151] showed that the Mauro-Allan-Potuzak (MAP) model for the relaxation time of the nonequilibrium glassy state[44] implies that the configurational entropy can be written as a function of thermal history (Tf), the fragility index (m), the fragility limit of a strong liquid

69

(m0), the glass transition temperature (Tg), and the limit of infinite temperature configurational

entropy ( S ) as,

T S g m Scf(T )= exp − − 1 . (72) ln10 Tf m0

The qualitative relationship between the distribution of activation barriers and fragility was proposed by Stillinger[18], [19], who suggested that the energy landscape of a strong liquid (low fragility) has a small distribution of activation barriers and that a higher fragility is associated with a broader distribution of activation barriers, i.e., a higher variance of the activation barriers. This leads to a phenomenological relationship that is here assumed to be valid based on Stillinger’s work on energy landscapes[18], [19],

2 m  B =−A1 , (73) m0 where A is some constant of proportionality. This expression was chosen because in the limit of a strong liquid there is an infinitely sharp distribution of activation barriers. Combining Eqs.(65),

(72), and (73), we obtain the temperature dependence of the stretching exponent for a liquid:

Tg m TSf  exp−− 1 Tmf 0  ().Tf = (74) 2 mm 2 2 2 2Tg   6( ln10) ATS− 1  + f  exp −  − 1  m00  Tf  m 

Fictive temperature appears in this new expression because we are deliberately expressing this function as an equilibrium model. Rewriting Eq. (74) in terms of one unknown () and grouping constants,

70

Tg m Tf exp1−− Tmf 0  (Tf ) = . (75) m 22 2Tg m  −+1  Tf exp1−− m00 Tf  m 

Eq. (75) is an expression for the stretching exponent as a function of thermal history, glass transition, and fragility index with only one unknown. In Eq. (75),

 S =  . (76) 6( ln10) A1/2

The only unknown for the compositional dependence of the stretching exponent is the value of D

. The composition-dependent part of Eq.(76) is since we approximate A to be independent of S¥ composition. The fragility dependence of was proposed by Guo et al. [61], S¥

é m(x) - m(xref )ù S (x) = S (x )expê ú , (77) ¥ ¥ ref m ëê 0 ûú where x is composition and xref is a reference composition in the same glass family. Seeking the simplest possible expression to approximate the unknown D, we take the natural logarithm of

Eq.(76) and of Eq.(77) and combine them to get

m ln  =+ ln  , (78) m0 with the additional definition

  Sxm ( refref x ) ( ) lnl= n   − . (79)  1/2  m  6( ln10) A  0

The result is that ln D varies linearly with fragility m and the intercept ln D is independent of composition (depends on one reference composition).

In Figure 9 we plot the data of Böhmer et al.[152] vs. the predicted exponent. The work of

71 Böhmer et al. is the summary of the literature data relating fragility index to the stretching exponent at the glass transition, which we use to fit values to Eq. (78). The dataset included in their work covers chalcogenide, oxide, and organic glasses. The fitting (consisting of least-squares minimization of the difference between the predicted stretching exponent and that which was reported) was done twice, once for organic and once for inorganic systems. During the fit it was assumed that Tg=Tf. Some assumption about thermal history was necessary since the individual thermal histories or Tf values are not known for this whole collection; since we are trying to track overall trends in  values vs. composition, this reasonable simplifying assumption is consistent with our program. Using Eqs.(75) and (78), the temperature and compositional dependence of the stretching exponent can be described with only one free parameter.

72

Figure 14. b predicted and from literature showing good agreement with a total root-mean-square error of 0.1. The fit for organic systems is given by ln = − 6.6ln K and for inorganic systems by ln  = −7.5ln K .

73

4.2.B Experimental Validation

Experimental density measurements were made using Corning© JadeTM glass as described elsewhere[146]. Changes in the density are normalized to obtain the relaxation functions, which are fit with Eq. (60). The resulting values of  are shown in Table 3. A comparison in Figure 15 is shown with data obtained on a soda-lime silicate, SG80, where the stretching exponent was obtained using the measurement of the released enthalpy as a function of isothermal annealing time during relaxation below the glass transition. The measurement of the released enthalpy[153] relies on the change in the excess heat capacity in the glass transition range as a function of annealing time. Normalized released enthalpy relaxation functions for each temperature were fit with Eq.(60) to obtain the stretched exponent values for SG80. In Figure 15, parts A and B are samples measured below Tg while in part C the samples measured are at Tg and above. The values were then fit with

Eq.(75) where the only free parameter was D , with an equilibrium assumption (Tf = T).

74 Table 3 Measured temperatures and their corresponding relaxation values for Corning© JadeTM glass and Sylvania Incorporated’s SG80.

-1 Glass Temperature (K) Tg (K) Tg/T m D (K ) Beta t (s)

Corning© JadeTM 1031 1074 1.042 32 0.002 0.504 7249

Corning© JadeTM 1008 1074 1.065 32 0.002 0.447 32458

Corning© JadeTM 973 1074 1.103 32 0.002 0.395 419214

SG80 800 800 1 36 0.0057 0.632 9.2

SG80 783 800 1.021 36 0.0057 0.593 18.2

75

( (

A) B)

(

C)

Figure 15. The equilibrium model proposed with the experimental points showing good agreement between the experimentally measured data points and the equilibrium derived model. RMSE was TM 0.02 for Corning© Jade (A) and less than 0.01 for SG80 (B). (C) The model fit for B2O3 experimental data[154], [155]. The fragility and glass transition temperature of the B2O3 are taken from the work of Mauro et al.[17].

76

A combination of the MAP model for non-equilibrium shear viscosity[44], the model presented here primarily in Eq.(75), and the model presented by Wilkinson et al.[139] for the temperature dependence of elastic modulus, allows for fully quantitative modeling of stress relaxation behavior. The missing model required to understand structural relaxation is the bulk viscosity curve[62]. All previous relaxation (structural or stress) models[42], [61] have relied on approximations that use a constant exponent  and on a constant (temperature-independent) modulus value, whereas here, every parameter of Eq.(60) may be modeled as a function of temperature. Furthermore, in combination with the relaxation models described by Guo et al.[61], in which multiple fictive temperatures are described using a Prony series and a temperature- dependent modulus, one can construct a relaxation curve accounting for the temperature dependence and thermal history dependence of all relevant parameters:

 (Tf ) N  G() T t G() T ki ( Ttf ) expexp−− w T  . (80)  i ( f ) (TT ,) ) i=1 (TT, ff

Here, wi and ki are fitting parameters that are completely determined by the value of , and each term in the Prony series is denoted with the subscript i. Usually, 8 or 12 terms are included in the

Prony series[96]. Each term in the Prony series is assigned a partial fictive temperature (Tfi) whose relaxation is described by the simple exponential in the Prony series. Equilibrium conditions are assumed at the start of the simulation, which allows for a known set of starting probabilities within the energy landscape interpretation of relaxation. The same model also includes the fragility index and glass transition dependence of the non-equilibrium shear viscosity. This method is implemented in RelaxPy[42], as discussed in the next section. This serves as an approximation for the evolution of the of the non-equilibrium state; however the temperature dependence of the bulk viscosity and a replacement for fictive temperature need to be quantified to improve the understanding of the underlying physics[1], [16], [62].

77 Separately, we can explore the relationship between this model given by Eq.(75) and the underlying energy landscape. In order to better understand the stretching exponent , consider that there exists a full set of parallel relaxation modes within an energy landscape. The relaxation modes are then weighted by the occupational probability corresponding to a particular mode. This gives a series of transition rates with some probability prefactor and an associated relaxation time (scaled from the mean relaxation time), which gives rise to a Prony series form of Eq.(80). Thus, the distribution of relaxation times (or barriers) determines the evolution of the stretching exponent while the average barrier determines the mean relaxation time. If this Prony series description, in turn, describes the stretching exponent, we arrive at a physical description and origin of the stretching exponent. As the temperature approaches infinity, even though there is a distribution of activation barriers, the distribution of relaxation times approaches a Dirac delta function, and the stretching exponent approaches one (a simple exponential decay). As the temperature decreases, the distribution of relaxation times broadens due to the wide distribution of barriers; however, due to the broken ergodic nature of glass, the value of the stretching exponent will be controlled by the instantaneous occupational probability.

The activation barriers may be modeled, e.g., using a numerically random set of Gaussian distribution of barriers[156]. The probability of selecting an individual transition is then calculated using a Boltzmann weighting function; the transition also has an associated relaxation time with it.

The Prony series is then recreated with the probability multiplied with the simple exponential relaxation form. The sum of all of these terms is then fit with the stretched exponential form. Figure

11 shows the stretching exponent calculated for the mean activation energy of 1.0 eV and three choices of standard deviation,  = eee−−−2.521.5,and eV as shown in the legend, where Eq.(60) has been fit to the Prony series calculated from an equilibrium Boltzmann sampling of a Gaussian distribution of barriers. The temperature dependence for the stretching exponent matches the form

78 derived earlier in this work. This stretching exponent shape has been shown previously though not in a closed-form solution[156]. This secondary method not only validates the generalized form of

Eq.(75) but also highlights the validity of using the configurational entropy and barrier distributions as the underlying metric controlling the temperature dependence of the stretching exponent.

Figure 16. The stretching exponent calculated as described in the text for a Gaussian distribution of barriers. This plot shows that the distribution of barriers has a large effect on the stretching exponent. A Tg cannot be described since there is no vibrational frequency included in the model, though the glass transition temperature should be the same for all distributions since the mean relaxation time is the same for all distributions at all temperatures. The deviation is given in ln eV units.

79 To create software to model this complex behavior, the authors’ existing software RelaxPy[42] has been modified into a new version, RelaxPy v2.0. Instead of fixed values for the Prony series, the values are chosen dynamically to match the stretching exponent as predicted by Eq.(75). The database of values wi() and ki() was created with a fixed number of terms N in Eq.(80) (we chose

N=12). This database is available in the same Github repository as the software. The parameters were fit by starting with the values obtained for =3/7 reported by Mauro and Mauro[96], to make the new version of the model smoothly reproduce a prior optimal fit, then  was stepped by 0.01

(when less than 0.95) or by 0.001 (when greater than 0.95). Using the previously optimized values of wi and ki for starting values for each new , the wi and ki values were then varied to minimize the root-mean-square error between the stretching exponent form and the Prony series form, i.e. to satisfy Eq.(80) with least error. This list of compiled values makes up the contents of a database that the software accesses. The values of wi and ki used for a given  are calculated by finding the closest value in the database to the given value of . This allows for an efficient implementation of relaxation modeling. Figure 12 shows the fitted parameters to the Prony series and an example

RelaxPy output for Corning© JadeTM glass undergoing quenching at a rate of 10 K/min. Note that the room temperature stretching exponent value is 0.46, a close match the Phillips’s “magic” value of 3/7 (0.43)[157]. This prediction for Corning© JadeTM is a non-equilibrium prediction in contrast to the equilibrium high-temperature prediction shown in

80

Figure 17. (Top) The Prony series parameters as a function of the stretching exponent. Each color designates one term in the series. (Bottom) The output from RelaxPy v.2.0 showing the stretching exponent effects on the relaxation prediction of Corning© JadeTM glass. Each quadrant shows one property that is of interest for relaxation experiments. In particular, it is interesting to see the dynamics of the stretching exponent during a typical quench.

81 4.3 Conclusion

In this section we have sought to expand the current understanding of relaxation and improve the models that currently exists. We have proposed a model that accounts for the effects of special relativity as well as a model was derived through an understanding of the distribution of barriers for relaxation and the temperature dependence of the Adam-Gibbs entropy. The model outlined herein describes the temperature dependence of the stretching exponent,  , in glass relaxation.

The work done on the relativistic effects on relaxation illustrates the crucial role of the observer on all relaxation models and offers an extreme in which to test common relaxation models. The model for the stretching exponent not only considers the extremes at high and low temperature (when compared to the glass transition) as in the Phillips model, but also for any intermediate temperature as a function of the fictive temperature. This model does not have any explicit temperature dependence since it was assumed that an equilibrium model works well to describe the instantaneous distribution of relaxation times. Given the physical argument and the success of this model when tested by experiments and by another model, it is at least reasonable to formulate the temperature-dependence of  in terms of its fictive-temperature-dependence as we have done here.

Including both T and Tf is possible to consider but lies outside our current scope. Using previously derived compositional dependence for the MAP model, a fragility index dependence of the stretching exponent was defined and tested. The model was confirmed using multiple experimental datasets. In addition, a theoretical comparison to a distribution of landscape activation barriers was found to reproduce the same trends as the model.

Chapter 5 Glass Kinetics Without Fictive Temperature

Fictive temperature, as discussed through this text, is unable to capture key physical phenomena[16], [42], [51]. To circumvent this limitation, we rely on two different frameworks.

The Adam-Gibbs relationship and generalized features of the energy landscape. These two features allow us to create a method that is completely generalizable and does not involve fictive temperature. This method creates fake landscapes that generally capture the trends of glass based on experimental measurements. This new method we have called ‘Toy Landscapes’.

5.1 Background of the Adam Gibbs Relationship

Experimental evidence in support of the AG model comes from the work of Richet[158], in which they compared the predicted configurational entropy from viscosity curves with that obtained from DSC. The configurational entropy was calculated from DSC data using

T CCp− p, vib S=+ S() T dT . (81) cc0  T T0

In Eq. (81), Cp is the isobaric heat capacity, Cp,vib is the vibrational heat capacity, and T0 is the initial temperature from which the configurational entropy is integrated. The difference between the total heat capacity and the vibrational heat capacity is the heat capacity due to structural rearrangements.

Richet found good agreement between the measured viscosity curve and calorimetric configurational entropy at temperatures above the glass transition, confirming validity of the AG relationship. When applying Eq. equation reference goes here, all temperatures must remain above the glass transition, i.e., in the liquid or supercooled liquid state. The calculation is invalid upon

83 cooling through the glass transition due to the breakdown of ergodicity[12], [13], [159] and the irreversibility of the glass transition process. Application of Eq. (6) to temperatures below the glass transition leads to an incorrect calculation of excess entropy in the glassy state[160].

In this study, we expand on these previous works to understand the relationship between viscosity and the underlying energy landscape and provide new insights into the thermodynamics of liquids through a simplified “toy landscape” model constructed from experimentally measured viscosity parameters. The energy landscape framework is especially helpful for elucidating the thermodynamics and kinetics of supercooled liquid and glassy systems. Energy landscapes describe the evolution of all atomic transitions for kinetic processes in a system. To perform such calculations, information about the inherent structures and transition points in a landscape must be known. In this work, we solve the inverse problem of deducing realistic landscape parameters using the AG relationship and experimentally measured data. This information is used to construct a “toy landscape” model to describe glass relaxation processes. To achieve an accurate relaxation model, the following steps are taken to ensure the validity of the results:

A. Confirm the AG model for shear viscosity, using energy/enthalpy landscape calculations. B. Validate the assumptions made by the MYEGA model. C. Confirm the AG model for bulk viscosity. D. Explore the fundamental relations between viscous properties and topography of the landscape. E. Use the knowledge gained to propose a new thermodynamic model to calculate the driving force for glass relaxation and the scaling of the liquid’s free energy.

This chapter is organized into sections devoted to each of the topics above, preceded by a methods section. All of the sections are presented to validate a new model of glass relaxation that is sufficiently complex to achieve a calculation of the thermodynamics and kinetics of the liquid.

This new model will be based on understanding the landscape from experimental properties and we have called this approach the “toy landscape” model.

84 5.2 Methods

Potential energy landscapes (fixed volume) and enthalpy landscapes (fixed pressure) are powerful modeling techniques where atomic configurations are mapped in a 3N-dimensional phase space (or 3N+1 dimensions for enthalpy landscapes)[89]. The energy landscape approach is based on mapping the continuous 3N dimensional space to a discrete set of energy minima (called

“inherent structures”) and first-order saddle points (“transition points”). To perform this mapping, a systematic search for inherent structures is performed while also mapping the lowest-energy transition points between each pairwise combination of adjoining minima. The landscape itself is partitioned into basins, which represent the set of all configurations that minimize to a common inherent structure[161]. This combination of the basin and transition point information results in a topographic map of the 3N phase space (either a partial or whole mapping) that can be used to gain insights into the physics of any atomistic system such as proteins[91], liquids[18], [19], glasses[1],

[93], and nucleating crystals. At equilibrium, the probability distribution for occupying the various basins in the landscape is given by[13],

Ei gi exp − kT pi = , (82) E j  g j exp− j kT where Ei is the potential energy (or enthalpy) of the inherent structure, gi is the degeneracy of the basin, i is the basin index, and pi is the probability of occupying a given basin i. The denominator is a normalization factor given by the summation over all basins. Mauro et al.[44] previously used this formalism to study the non-equilibrium viscosity, calculate the evolution of configurational heat capacity, and elucidate the long-time relaxation kinetics of selenium glass[44], [93], [160]. For equilibrium systems, the configurational entropy is given by the Gibbs formula,

85

Skppci=−  ln i . (83) i

In order to validate the AG model, the landscapes of several common systems were explored using ExplorerPy[89]. ExplorerPy is a software program built specifically to map landscapes using the LAMMPS molecular dynamics package[30]. The three systems we explored are B2O3[162] (150 atoms and 2000 basins), SiO2[162] (150 atoms and 1000 basins), and BaO-

2SiO2[121] (128 atoms and 1500 basins). All systems used periodic boundary conditions. The initial structures for B2O3 and SiO2 were constructed by placing atoms corresponding to the chemistries of the systems on a random grid (with 1 Å spacing) corresponding to a density of 2.0 g/cm3 and equilibrated at 2500 K, then quenching at a rate of 1 K/ps to room temperature using the

Wang et al.[162] potentials in the NPT (constant pressure, number of atoms, and temperature) ensemble. The energy landscape of the BaO-2SiO2 system was taken from our previous work[72].

The SiO2 (constant pressure exploration) system was explored using ET with a push distance of 0.3

Å and a pressure of 1.0 atm. B2O3 was explored using isobaric MD at a temperature of 1500 K for

500-time steps of 1 fs, also at a pressure of 1 atm. The minimization (after MD or ET) was handled by the LAMMPS[30] steepest descent algorithm until the certainty in energy was smaller than 10-

4 eV. Both used a nudged elastic band spring constant of 1.0 eV/Å2. The energy distribution for all systems was fully sampled, which was confirmed by randomly checking if half the basin distributions matched the full distribution. It was also found that multiple structures in each exploration were discovered more than once, and a slightly smaller B2O3 system (130 atoms) was also explored and showed identical results to the 150-atom distribution. The insights gained from the interaction between the experimentally accessible viscous properties and the landscapes presented here will be generalized to create a simplified “toy landscape” model to capture the thermodynamics and kinetics of the glass-forming system.

86 5.3 Results

5.3.A Adam-Gibbs Validation

To predict viscosity using the Adam-Gibbs expression, there are only three required parameters: the configurational entropy (as a function of temperature), the barrier for cooperative rearrangements, and the infinite-temperature limit of viscosity. Using a landscape, the equilibrium configurational entropy can be obtained using Eq. (83). The infinite-temperature limit of shear viscosity is given by the work of Zheng et al.[54] where a systematic study of MYEGA fits to viscosity data revealed a common value of 10-2.93 Pa·s. This infinite temperature value of viscosity is used for all the experimental and computational viscosity curves shown in this work. The final parameter is the mean barrier for a cooperative rearrangement. In Figure 18, the barrier for cooperative rearrangements was chosen to accurately produce a viscosity of 1012 Pa·s at the experimentally measured value of the glass transition temperature. From these parameter values, the predicted Adam-Gibbs viscosity curve is plotted. The universal infinite limit, fragility, and glass transition values from references were inserted into the MYEGA equation to plot the

‘Experimental’ curves for B2O3, SiO2, and BaO·2SiO2 respectively[58], [69], [120]. When examining Figure 18, it is clear that the AG model is able to fundamentally recapture the viscosity.

87

Figure 18. The viscosity (left) and landscape (right) predictions for three common systems. The first system is newly calculated in this work while the latter two come from our previous works[72], [89]. It is seen that the viscosity predicted from the AG model is very accurately able to reproduce the experimental viscosity curves from the MYEGA model. The last system is a potential energy landscape while the others are enthalpy landscapes.

5.3.B MYEGA Validation

Now that we have confirmed that the configurational entropy of the landscape works well to describe the viscosity via the AG relationship, a definitive temperature-dependent form is needed in order to make more general predictions. The key approximation made in deriving the MYEGA model is that the configurational entropy follows an Arrhenius form. By testing this Arrhenius

88 form, we can evaluate this assumption in the derivation of the MYEGA model while simultaneously gaining insights into the thermodynamics of the liquid and supercooled liquid states. A strong liquid

( m  15 ) has a configurational entropy that is nearly constant as a function of temperature while a fragile liquid ( m 15 ) is a liquid in which the configurational entropy is highly temperature dependent. Since we are testing the MYEGA equation, it is also worthwhile to consider alternate viscosity expressions. Assuming validity of the Adam-Gibbs equation, the configurational entropy deduced from the viscosity experiments is given by

B Sc = , (84) T (logl10 (T) − og10  ) where the viscosity function is given by a viscosity model such as Vogel-Fulcher-Tammann[59]

(VFT) equation, the Avramov-Milchev[60], [163] (AM) equation, or the MYEGA expression.

The resulting entropy predicted by each viscosity model (using experimental data for fragility and the glass transition) is compared to the calculated configurational entropy from the

landscape for B2O3 in (Tg = 518 K and m = 33 [69] were used for the experimental values). The figure clearly shows that the MYEGA equation, which assumes that configurational entropy scales in an Arrhenius fashion, reproduces the calculated configurational entropy from the landscape the most accurately out of the three considered viscosity models. This also means that the configurational entropy form of the AM and VFT models are not physical and cannot be used to recreate physical results. Since the temperature dependence of the entropy has a fixed mathematical form and is dominated by the distribution of basins, we now understand that the distribution of basins must also have a fixed form, and that form must control experimentally accessible properties.

This is the key insight that will allow for the development of the toy landscape model.

89

Figure 19. The configurational entropy comparisons between the three major viscosity models, which validates the main underlying assumption of the MYEGA model. The VFT and AM are unable to capture the physics of configurational entropy, therefore ruling

5.3.C Adam-Gibbs and Structural Relaxation

Although the Adam-Gibbs model has been validated for shear viscosity and has provided

the functional basis of the MYEGA model, let us now consider whether the temperature

dependence of the configurational entropy can also predict the bulk viscosity (also known as the

volume viscosity). Recent work has highlighted the importance of bulk viscosity, viz., that the

structural relaxation time is proportional to the bulk viscosity[62]. Hence, the temperature

dependence of the bulk viscosity provides key information needed to predict structural relaxation

times.[62] Following Scherer[164], the relationship between the structural relaxation time ( structural

) and the bulk viscosity ( B ) is given by,

B=−(KK 0 ) structural , (85)

90 where K is the infinite frequency bulk modulus and K0 is the zero-frequency bulk modulus. In order to calculate bulk viscosity for B2O3 the zero-frequency and infinite-frequency modulus is taken from these works respectively [165], [166]. The experimental structural relaxation times were taken from the work of Sidebottom et al.[154]. In Figure 20, the bulk viscosity is shown as well as the Adam-Gibbs comparison.

Figure 20. Estimated bulk viscosity of B2O3. The fit to the bulk viscosity used the configurational entropy from the enthalpy landscape with the barrier 0.0155 eV (compared to the shear barrier of 0.0149 eV) and the infinite limit allowed to vary (10-2.63 Pa·s for bulk viscosity). In Figure 18, the configurational entropy is confirmed for the shear viscosity, thus confirming the AG for both shear and bulk viscosities. Sidebottom data are from Ref. [154].

In Figure 20, the fitted values of B and log10   can be used to gain insights into the difference between structural and stress relaxations since the configurational entropy is universal between the two. The value of B/k (the barrier for cooperative re-arrangements in natural units) for shear viscosity was 173 K, while for bulk viscosity it was 180 K. This difference of 7 K in the barrier leads to an order of magnitude difference in the fitted viscosity curves. This is a small

91 difference and explains the variance in the values at Tg. The other difference is in the infinite temperature value of viscosity, which is -2.93 log(Pa·s) for shear viscosity (from Zheng et al.) and

-2.63 log(Pa·s) for bulk viscosity. This implies that the MYEGA raw form is accurate to predict the structural relaxation time as well as the stress relaxation (though the values of the pre- exponential factor and the composite constant will change between the two forms). However, the form in terms of fragility and the glass transition will need to be modified to accurately describe the structural relaxation time. The form assumes the viscosity of interest has a value of 1012 Pa·s at the glass transition temperature for shear viscosity, which does not have to be the value for the bulk viscosity curves at the glass transition. Further research is needed to understand the appropriate bulk viscosity at the glass transition and the infinite temperature of bulk viscosity.

5.3.D Landscape Features

As shown here, the configurational entropy is governed by a Boltzmann sampling of a probabilistic distribution of states, n(E). This distribution of the inherent structures is empirically found to follow a log-normal distribution, as shown in Figure 21. While we are not claiming a deeper meaning to the underlying origin of the log-normal distribution, we may adopt this form of the probability density function since it accurately describes the underlying distribution of microstates. This is the key insight needed to develop a new model because this means that the fragility and glass transition are fundamentally related to the parameters of the distribution of basins on the landscape (number of basins and standard distribution of the enthalpy).

The configurational entropy in the MYEGA model is given by,

 Sm Tg  Sc =exp − − 1 , (86) ln10T  12− log10 

92 where S is the infinite temperature limit of the configurational entropy. In Figure 21 we also show the computed value of Eq. (83) as a function of temperature compared to that of Eq. (86), which shows excellent agreement between the configurational entropy and the MYEGA model.

This test also supports Stillinger’s[18], [19] view of fragility being correlated to the ‘roughness’ of the landscape topography. According to Stillinger’s view, if the distribution of inherent structure energies is narrow, then the configurational entropy cannot have a large temperature dependence, leading to a low fragility. Alternatively, if the distribution is broad, then there must be a temperature dependence to the configurational entropy, which necessarily results in a higher fragility.

93

(A)

(B) (

B)

Figure 21 (A) The histogram for the energy minima for each of the test systems showing a good fit with the log normal distribution. This distribution will then be a valid form to calculate the enthalpy distribution of the model presented in the next session. (B) The configurational entropy from the model showing the accuracy of the scaling of the entropy predicted by the MYEGA model. The

S value was fit for each system.

94 5.4 Topography-Property Relations

Now that we have confirmed the validity of the Adam-Gibbs and MYEGA equations from the energy landscape approach, we can draw some key conclusions regarding the thermodynamic properties of liquids. For example, heat capacity has been related to fragility[112] and is of importance for glass manufacturing since it is related to the cost of heating and forming a glass.

Isobaric heat capacity is defined as

H Cp = , (87) T P where H is enthalpy, T is temperature, and P is pressure. The mean configurational enthalpy calculated from an enthalpy landscape is given by

Hpconficonf H = i , . (88)

The heat capacity has contributions from both vibrational and configurational modes in liquids, while in glasses the configurational mode is mostly lost. For a reversible process, the configurational heat capacity can be written as a function of the configurational entropy,

Sc CTp, conf = . (89) T P

By combining Eq. (86) and Eq. (89) we can write the expression for configurational heat capacity of a liquid as

* TTgg mmSH  *  CSHp, conf =−−−  =− 1 exp1exp. (90) TTTT12−− log121010 log 

From fluctuation theory, the heat capacity can be expressed in terms of the variance of the enthalpy

2 ( H ),

95 1 C =  2 . (91) pconfH, kT 2

It is important to note that the variance of the enthalpy is not equivalent to the variance of the log- normal distribution. The variance of the enthalpy includes a Boltzmann distribution over the log- normal distributed basins. This gives a direct relationship between the enthalpy fluctuations and the viscosity parameters. If we further adopt the log-normal distribution as the effective distribution for all liquid basins of all enthalpy landscapes, then the distribution is only a function of the

common viscosity parameters ( S , m , Tg , and l og10  ). The viscosity parameters will only change the variance of the distribution since the mean of the distribution is inconsequential (the zero-point enthalpy can be adjusted to an arbitrary value without changing the calculation of the configurational entropy). The parameters of the log-normal distribution can then be fitted to reproduce the configurational entropy’s temperature dependence, thus creating a ‘toy’ enthalpy landscape for us to play with for the purpose of thermodynamic and kinetic calculations. Generating the landscape is a numerical calculation as the number of basins and standard deviation must be chosen such to reproduce key phenomena.

To quantify the dependence of these parameters on the degeneracy and temperature, it is helpful to develop a descriptive formal model. The cumulative probability distribution of a normal distribution (with zero mean) is given by,

1 log)10 (H p = 1+ erf , (92) 2    2  where  is the standard deviation of the log-normal distribution, erf is the error function, H is the enthalpy, and p is the probability. To then calculate the equal probability spacing for a log-normal distribution, we solve for H for the set of probabilities, p , which range from  /2 to 1−  /2

96

, with a step-size of  . This then gives enthalpies of basins equally spaced by the cumulative probability distribution, and as a result, each basin has equal degeneracy,

−1 H =10. 21 efr 2p− (93)

This set of basins carries with it an Arrhenius activation barrier for the configurational entropy that is only a function of  and the degeneracy of each basin (if the  is chosen such that the energy distributions converge). This creates a model that fully describes the thermodynamics of the liquid, and though it currently lacks any kinetic component, it offers the possibility to be a useful tool when used in conjunction with Eqs. (82), (83), and (88). This new tool is denoted the “toy landscape model” (TLM). It is called the toy landscape model because we are generating a landscape that is not the real landscape but is enough to reproduce key experimental properties without needing expensive computational calculations. It is a ‘toy’ for us to play with without needing to worry about upfront computational cost.

To confirm the validity of TLM and the way it is constructed, we can calculate the activation barrier from this deterministic method and for a random set of inherent structures with the same number of basins and distributions. For this purpose, we used temperatures ranging from

1000 to 2000 K for both the deterministic and random models with a step size of 0.001 for TLM.

Excellent agreement between the random and deterministic methods is shown in Figure 22A. This analysis allows us to understand that the thermodynamics of the system are fundamentally linked to the viscous flow behavior. Though this technique was created to target viscosity and relaxation, it is generally applicable to all configurational thermodynamics of liquids and glasses. It is worth noting that all the thermodynamics will be relative to a 0 eV energy state defined at equilibrium at

0 K, and as such the enthalpy differences at some fixed temperature must be known to compare over compositional spaces.

97 In Figure 22B we see the same method being used to understand the relationship between the viscous parameters and the parameters of the landscape. This is the key to the numerical calculation of the ‘toy landscape’ model. The toy landscape is a numerical calculation where we use experimentally accessible values to generate a landscape whose thermodynamics should reproduce key thermodynamic phenomena. In Figure 22B we show that the fragility and glass transition are systematically changed as the number of basins and standard deviation of the enthalpy is changed. When this idea is combined with the basin calculation in Eq. (93) (giving equal degeneracies for all basins) then the complete set of enthalpy basins is easily attainable, giving access to the fundamental thermodynamics of the system.

98

( (A) A)

(B)

99 Figure 22 (A) Comparison between the randomized method (histogram) and the deterministic method (vertical lines) showing good agreement between the maximum in the histogram and the value predicted by the deterministic technique, validating the approach. It is worth noting that the 100-basin distribution is a very wide distribution where the total number of basins is less than the number of points used in the calculation. This is done for a variable number of basins with the number of basins shown in the legend. (B) The dependence of fragility and the glass transition temperature vs. the distribution of states and the number of basins.

5.5 Barrier Free Description of Thermodynamics

Throughout this work we have systematically explored the Adam-Gibbs model of viscosity. An important insight that we have gained is that the configurational entropy is free of any barriers. This is of particular interest because the Arrhenius slope of the activation barrier is linked to the glass transition and fragility. The glass transition temperature is known to be related to the set of activation barriers as it corresponds to a breakdown of ergodicity. What is found here is that the fragility is the “conversion factor” to account for the slope of the configurational entropy

(which, as previously identified, is related to the distribution of basins). This means that the barriers themselves do not matter until the material is close to the breakdown of ergodicity, and thus the equilibrium liquid thermodynamics and kinetics can be described without explicit information about the saddle points in the landscape.

Understanding the role of the topography in the properties of glasses and liquids lead to the most important insights derived from the AG model. Since the configurational entropy is the same for both stress and structural relaxation, simple viscous parameters allow us to understand fundamental thermodynamic quantities of the system without needing to map computationally expensive energy/enthalpy landscapes. The process of converting experimentally accessible properties to a usable landscape for thermodynamic calculations is what we are calling the toy landscape model. To show the utility of toy landscapes, we can calculate the driving force for glass relaxation. This information can give insights into the physics of relaxation by dynamically

100 calculating the Gibbs free energy of the equilibrium liquid at any temperature which defines a driving force for glass relaxation. To do this calculation of the driving force for glass relaxation, we consider the ergodicity parameter ( x ) defined by the Mauro-Allan-Potuzak (MAP) model of glass relaxation as the ratio between the glass entropy and the equilibrium liquid entropy:

SxSglass = liquid . (94)

To get information about the enthalpy, we consider an instantaneous Boltzmann distribution on the

landscape given by an equilibrium distribution at some elevated temperature, T f ,

1 H HH g =−exp i . (95) glassi i  kT Hi f  gi exp − kTf

This formalism in conjunction with TLM gives us the ability to explore the temperature and compositional dependence of the driving force for relaxation. The enthalpy of the liquid is calculated from an equilibrium distribution on the landscape. The driving force (  ) is defined as,

 =−+−HHTSxTT1 ( glassfl( qui ) liquid ( ) ) i d ( ) . (96)

For our purposes this gives a nice approximation for the driving force, improving on previous

approximations given by (TTf − )[16], [42]. TLM does not include a kinetic feature to calculate the occupational probability in each basin, however when coupled to a kinetic model such as the

MAP model[44], a complete description of the underlying thermodynamics of glass relaxation can be considered. For this calculation of the driving force the kinetics are calculated through the MAP model, but this is not necessarily a requirement. In Figure 23, the compositional and temperature dependence of the driving force is shown for three glasses with identical glass transitions and varying distributions of basins (as listed in Table 1). The calculations are performed using the

RelaxPy script and MAP model with the stretching exponent fixed at the Phillips value of 3/7,

101 explanations of which are available elsewhere[42]. The systems were quenched from 700 K to 300

K at a rate of 10 K/min. Figure 23 clearly shows how simply changing the total number of basins creates a larger driving force for glass relaxation. This simple change of the number of basins drastically increases the entropic contribution and due to the loss of ergodicity explains the increase of the driving force of relaxation. The kinetics could be calculated through any relaxation model or directly from the landscape if some barrier approximations are made.

Table 4. A table of parameter values for the three example glasses used in Figure 23. The distribution of underlying inherent structure energies and the glass transition temperature (500 K) were kept the same while the total number of basins were allowed to vary.

Sample Total Basins [-] Distribution of States [log eV] m Fragility [-] [K] Tg −1 14.93

1 100 0.2 837 40 2 500 0.2 537 31 3 1000 0.2 466 28

102

Figure 23. The driving forces for different example glasses calculated using a combination of the MAP model, RelaxPy, and the toy landscape model. The parameters for each glass can be found in Table 4.

5.7 Toy Landscapes for the Design of Glasses and Glass Ceramics

The concept of a toy landscape can go even further, building a method to predict the dynamics of a glass system based on simple inputs. This is possible because in the previous section we have related the distribution of basins on an energy landscape to the experimental viscosity which allows for thermodynamic insights. Building on this, we can assume that the barriers ( H * ) for transitioning between states is equal to that of the barrier mean for the viscous relaxation,

* H= mkTg ln10 . (97)

This means that the only key factor needed to then predict the dynamics of a landscape using the metabasin approach is the vibrational frequency. However, the relaxation time at the glass transition

103 is known to be a constant of 100s and as such we can choose a vibrational frequency that reproduces this fixed point. This gives an entire description of relaxation without any need for fictive temperature. An example of the results predicted through this method are shown in Figure 24. This prediction only required knowledge of the glass transition temperature and the fragility and from this a pure prediction of relaxation is made.

104

105

Figure 24 The predicted outputs from the toy landscape method showing predictions of the enthalpy and entropy under a standard quench for barium disilicate. This prediction does not require fictive temperature or any such assumptions about the evolution of the non-equilibrium behavior.

This method is also not limited to predicting the relaxation of glasses but can also capture the crystallization. It is known from the earlier work on nucleation presented in Chapter 3, that a simple two state model was able to capture the underlying physics of the system. Thus, if we know the free energy of the crystal and barrier to crystallization, we can find predict key crystallization

106 phenomena. By fitting the heat flow peak magnitude and location this gives enough information to infer the barrier and the enthalpy of the crystal. The degeneracy (and thus the entropy) of the crystal can then be calculated through knowledge of the liquidus temperature,

HcrySCLl−GT( ) cry = exp. (98) kTl

This then gives everything for the model presented in Chapter 3, from only 1 calorimetry experiment, the glass transition, the fragility, and the liquidus temperature. To predict nucleation a series of estimates were done running the toy landscape at different temperatures until the temperature with the fastest crystallization rate was found. This was then assumed to be the maximum rate of nucleation. The value of the interfacial energy was then chosen in such a way so that the predicted curve reproduces the peak at the same temperature. This gives all the parameters needed for a prediction using CNT. It is also possible to make a prediction of the growth rate of crystals using this expression,

kT  G U = 1exp−− . (99) 2  6akT 

The results for these estimates are shown in Figure 25. The reason why growth is more accurate compared to nucleation is that the kinetic factor of growth is governed by viscosity while nucleation is governed by a kinetic factor that includes an assumed vibrational frequency.

These methods have a plethora of assumptions built in such as that the vibrational frequency of basins undergoing relaxation is equal to the vibrational frequency when undergoing crystallization, however as more information is gained about a particular system (such as more crystallization peak as a function of temperature) the information could be built into the system to improve crystallization predictions. If the peak size is known as a function of scan rate/temperature even the enthalpy as a function of temperature could be known. This provides a whole system for the real estimates of crystallization can be improved.

107

14 0.00 Predicted Rate Rodrigues et al. 12 2.00

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1

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nuclei

log 4 14.00 2 Growth Prediction 00 1000 1100 1200 1300 16.00 Temperature elvin Cassar et al. 0 1 .00 00 1100 1300 1 00 00.00 1100.00 1300.00 1 00.00 1 00.00 Temperature Temperature

Figure 25. The prediction for nucleation and growth from the 5 parameters. The volume in nucleation is assumed to be on the order of one cubic angstrom while the a parameter in growth is assumed to be around one nm, both are in good agreement for estimates in literature. The values for the orange points are taken from these works[128], [167].

5.7 Discussion

The impact of the Adam-Gibbs model on glass science is hard to overstate, and it has become foundational to ideas that will enable the design of next generation glasses. The Adam-

Gibbs equation still has more to offer the glass community in terms of fundamental understanding and practical applications. In this work we have shown that the Adam-Gibbs relationship is consistent with both shear and bulk viscosities. Also, we have validated the main assumption of the

MYEGA model, giving insights into the scaling of configurational entropy from the underlying enthalpy landscape. Using this knowledge of the relationship between the Adam-Gibbs model with the underlying enthalpy landscapes, we have developed a simplified “toy landscape” approach for modeling glass relaxation. The toy landscape is constructed numerically by understanding the distribution of inherent structures and transition points in the landscape and then relating these distributions to experimentally measured parameters of the MYEGA viscosity model. This

108 simplified landscape model can then be used to calculate the underlying thermodynamics and kinetics of the system and represents a practical application of energy landscapes to understand glass relaxation, nucleation, and other relevant thermodynamic properties.

This also closes our discussion on understanding the thermal history effects on glass. What we have shown is that the dynamics of a glass system can be understood through energy landscapes and fundamentally the same insights can be recreated through ‘Toy Landscapes’. This approach allows fundamentally for the complex 3N dimensional landscape to be summarized and abbreviated into a smaller landscape that can easily be implemented for any glass. However, this new parameterization relies on knowing some key properties that are only due to the location in the compositional phase space and for a complete picture of the system to be created we must also create a sufficient approximation for the compositional phase space shown in the next 2 chapters.

Chapter 6 Enabling the Prediction of Glass Properties

So far models have been created that describe the very nature of the thermal history effects on glasses. However, as an input to these models we have to know about the compositional dependence of key parameters, namely the glass transition and the fragility. In this chapter we extend compositional models to other key properties needed for the design of glasses. Here, these key properties that have not been readily predictable are approached using both machine learning and

TCT models. The TCT models are focused on surface reactivity, Young’s modulus, and ionic conductivity. The ML addresses CTE, fragility, melting point, as well as Young’s modulus.

6.1 Controlling Surface Reactivity

Glass surfaces, especially their interactions with water[168]–[171], are of the utmost importance to nuclear waste glass, cover glass, and many other modern applications[106]. It is generally accepted that the outermost surface of oxide glasses readily reacts with water molecules, where water molecules chemisorb on an as-formed glass surface and populate it with surface hydroxyls. Recent models attempt to quantitatively explain the change in properties observed on hydroxylated glass surfaces[77], [168], [172], [173], and most agree that the surface readily becomes hydrated because it is more stable to have a bonded hydroxyl group than to have dangling bonds. In glass/water interactions hydrolysis and diffusion lead to two forms of adsorbed water: chemisorbed and physisorbed[171], [174]–[176]. Both affect the network differently and are controlled by different processes: physisorption by hydrogen bonding[175] and chemisorption by reaction with the surface to form bonded hydroxyl groups,

110

≡ Si − O − Si ≡ +H2O → ≡ Si − OH + OH − Si ≡.

Although binding energy studies have been performed using molecular dynamics simulations[177], a specific study of the surface reactivity, in tandem, has not been conducted. Recent advancements in reactive force field modeling allow the direct observation of both diffusion and surface reactivity,[38], [178]–[180] and several studies have found surface reactivity constants consistent with those reported experimentally[173], [178], [181].

Recently, TCT has been used to predict various bulk properties of glassy systems such as

Vickers hardness and fragility[17], [182], [183]. Based on the TCT description of the glass network, glassy materials have been postulated to have a so-called ‘intermediate phase’ in which the atomic structure of a material will self-organize so as to be isostatically constrained (n=3), as explored extensively in various works[65]–[67], [133], [184], [185]. In this case, it has been shown that such materials exhibit anomalous behavior in certain properties, such as an high hardness and lower free energy[43], [67], [77].

TCT has also been used to understand the impact of water on bulk glass structure and properties. Potter et al.[168] found that, by accounting for the impact of chemisorbed (dissociated) water on glass network connectivity (viz., breaking Si-O-Si bonds) and the impact of physisorbed

(molecular) water on strain of the Si-O-Si bond angle, the glass transition temperature (Tg) could be predicted. This model could be further expanded to include other properties (such as modulus[139] or dissolution kinetics[77]) predicted by TCT, such as the work done by Liu et al.[186] on calcium-silicate-hydrate gel. An extensive study relating topological constraints[6],

[17], [57] and surface energy was performed by Yu et al.[172], specifically focusing on the transition from hydrophilic to hydrophobic behavior on silica surfaces. Their work used reactive molecular dynamics to model the change in surface energy of a silicate glass, and then correlate it with the number of surface constraints present, with the work focusing primarily on the global average of the surface.

111 To investigate the effects of glass network topology on the surface reactivity, we model the hydration of a silicate glass was modelled using molecular dynamics simulations according to the following procedures. Initially, bulk sodium silicate glasses were simulated (150 atoms with a molar composition of 70SiO2·30Na2O) using the Teter potential[187]. For a bulk glass, this pair potential is known to accurately simulate sodium silicate glasses and has been extensively studied to investigate various properties, including structural features, transport of sodium ions, and vibrational density of states[188]. The size of the system is initially set to achieve the experimentally measured density of 2.466 g/cm3 [189]. A total of 35 Si-atoms, 85 O-atoms, and 30

Na-atoms were randomly inserted within a periodic cubic box of a = 12.686 Å, and the initial configuration was energy-minimized to avoid any overlaps prior to the glass formation. The glass was held for 0.5 ns with a constant number of atoms, volume, and energy (a microcanonical, or

NVE ensemble). Since a thermal fluctuation of ~400 K (from 2000 K to 2400 K) was observed during the microcanonical run, the glass was allowed to evolve further at 2400 K for 0.5 ns with a constant number of atoms, volume, and temperature (a canonical, or NVT ensemble). Within the same ensemble, the melted system was then cooled with a constant cooling rate of 0.5 K/ps, and after the temperature had reached 300 K, it was equilibrated for 1 ns. Finally, a constant pressure of 1 atm was applied to the system (NPT ensemble) for another 1 ns to ensure no significant fluctuations of the density during the equilibration process. Three different sodium silicate glasses of the same nominal composition were constructed via the aforementioned procedure using different initial atomic positions. These repeated simulations compensate for the limitation of small sample size by allowing us to capture the statistical behavior of glass surfaces during the glass- water reactions. The final densities of the sodium silicate are presented in Table 5. These final structures were used as starting configurations for the glass-water reactions. MD simulations with the Teter potential creating three glass networks were carried out using the Large-scale

Atomic/Molecular Massively Parallel Simulator (LAMMPS) package[30].

112 Table 5 Density of the simulated sodium silicate glasses after relaxation at 300K.

Simulation box dimension a (cubic) [Å] * Density [g/cm3] **

Run1 12.979 2.303

Run2 12.630 2.499

Run3 12.737 2.437

* Initial dimension: a=12.686 Å

3 ** Experimental density (70SiO2·30Na2O mol%): 2.466 g/cm

113 After the bulk sodium silicate glasses were obtained, a reactive potential was employed to model the glass surface and subsequent glass-water interface, as the system of interest for this study required characterization of the reactive processes during the simulation of surface phenomena. In this study, all reactive MD simulations are performed with the Na/Si/O/H parameterization using the ReaxFF reactive force field framework15. The ReaxFF parameters are trained using a first- principles data set that describes water interaction at the sodium silicate glass-water interface.

Further details of the ReaxFF methodology and its potential forms can be found in earlier publications by van Duin et al.[38], [179], [181], [190].

114

Figure 26. Representative example of the initial non-hydrated sodium silicate used in the hydration models. Color scheme: Si atom (ivory), O atom (red), and Na atom (blue). The z-axis is elongated to allow space for an insert of water.

115

As shown in Figure 26, a free surface was first created by expanding the c parameters of each of the equilibrated bulk glasses. This process results in a vacuum region above and below the two surfaces of the glass slab along the z-direction. These exposed surfaces were relaxed at 300 K for

100 ps with the ReaxFF reactive force field. Following the relaxation of both top and bottom surfaces, the vacuum region was filled with water molecules. The number of water molecules that are inserted in the vacuum region was controlled to have a density of ~0.99 g/cm3. In addition, all water molecules were kept at least 2 Å from the uppermost and lowermost atoms of the glass surface to prevent any initial close contact with the surface. Glass-water reaction simulations were carried out at 300 K with the configurations shown in Table 6 for 500 ps in the NVT ensemble.

From these runs, trajectories at every 100 ps were obtained for the surface reactivity analyses.

Figure 27 shows the initial and final positions of glass surface reaction with water, at 0 and 500 ps, respectively.

116

a) b)

Figure 27. Initial (a) and final (b) states of the water/glass interface. Note that only the top surface in contact with water is shown here.

117

Table 6. System configurations for sodium silicate glass-water reactions.

Simulation cell [Å3] Number of water molecules

Run1 12.979 × 12.979 × 38.94 146

Run2 12.630 × 12.630 × 37.89 134

Run3 12.737 × 12.737 × 38.21 138

118

The hydrated surfaces of sodium silicate glasses from each run were used to investigate the adsorption behavior of a water molecule. Since the time evolution of surface reactivity is of key interest, the binding energy of a water molecule to the hydrated surface was calculated at 0, 100,

200, 300, 400 ps. In order to evaluate the local heterogeneity of reactivity imposed by varied glass surface structures, each surface was divided into a 10 by 10 grid and the binding energy at each site was mapped across the grid. The water molecule position in the z-direction from top and bottom surface was maintained to be 2.0 Å from the outermost atom during the energy calculation. The binding energies (Eb) were calculated as below, where the energy of a surface with adsorbed water

(Esw) was taken with respect to the surface energy (Es) and the energy of molecular water (Ew).

Ebsw=−−EEEsw (100)

A negative binding energy would indicate that water adsorption to the surface site at the corresponding grid is thermodynamically favorable. The binding energy map may then be used to locate where water binding would be most stable based on the ReaxFF calculations. From three independent glass-water reaction boxes, a total of six hydrated surfaces are obtained (a top and bottom surface from each run), increasing the statistical reliability of this analysis in the ensemble, despite the small overall system size of any one box. The binding energy mapping process is shown in Figure 28.

119

surface

10×10 grid ~2.0 Å

Figure 28. (Top) Schematic of water binding energy calculations. (Bottom) An example of the electronic-structure DFT calculation using semilocal exchange-correlation functionals finding the binding energy of a water ‘pixel’ to the surface.

120 Similar calculations are carried out at an electronic structure level using the DFT framework as implemented within the Vienna Ab-initio Simulation Package (VASP) software. Total energies are computed from the static geometries obtained from the reacted ReaxFF structures using the

Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional with projector augmented wave

(PAW) pseudopotentials to describe the ion cores. A kinetic energy cutoff of 500 eV was used for the expansion of the plane waves. Due to the large non-crystalline structure, the Brillouin zone is only sampled at the Γ-point with 0.1 eV of Methfessel-Paxton smearing to help electronic convergence. Due to the computational cost associated with these DFT simulations, the results were primarily used to elucidate the electronic interactions between the water molecule and the glass surface.

To count the topological constraints in the structure generated from the ReaxFF MD simulations, an algorithm was developed to systematically exclude all non-bridging oxygens, alkali, and hydrogens from the structure. The number of surface rigid constraints (in the pixel) around each network-forming atom was then calculated; an example contour plot generated using this procedure is shown in Figure 29. The surfaces were then converted to a 10×10 pixel grid, and the average coordination of network-forming atoms in each pixel calculated. The depth of each pixel taken was to be 4 Å below the surface.

121

Figure 29. Example contour surface showing the average coordination per atom on the glass surface, for the first run at 300 ps.

122

Figure 30 shows the results from the MD binding energy studies, where a clear transition in the region around 3.0 constraints/atom is shown. This region is isostatic because the number of constraints is equal to the number of translational degrees of freedom, and the width highlighted in the figure is evidence for the existence of an intermediate phase. In order to confirm that the binding energies had indeed converged, a larger system of 1500 atoms was simulated. Good quantitative agreement was found between the 150- and 1500-atom systems. In this work, the isostatic behavior occurs over a wide range in which certain silicon network sites dramatically increase the binding sites (binding energy approaching 0). The state or structure with nc outside of the intermediate range will readily interact with water. Only those in the isostatic region are likely to remain unaffected after the interaction with water; thus, they are the rate limiting species in dissolution of silica glass.

123

⟨푑퐻−푂퐺푙푎푠푠 ⟩ = 3.45 Å

⟨푑퐻−푂퐺푙푎푠푠 ⟩ = 2.48 Å

Figure 30. ReaxFF MD-derived water binding energies plotted versus the number of constraints for surface atoms at the local pixel. Results show a distinct maximum, in which there is a near hydrophilic-hydrophobic transition of the surface. The error bars represent the standard deviation. A second system with 1500 atoms was performed to show convergence of the ReaxFF MD results.

124 The intermediate phase result is intuitive due to the implicit stability that comes with having an isostatic phase, which is both energetically favorable and stress-free. The larger negative binding energies in the over-constrained regions arise from large differences between the free energies of the chemically bonded state (chemisorbed) and unbound state (water in near proximity to the surface, but not bonded) in these regions; in this region, the chemical reaction with the water molecule allows the glass surface to reduce the number of incompatible constraints, which in turn alleviates local stress in the network. In other words, a local region undergoes an alleviation of stress when Q4 units convert to Q3 units. The larger negative binding energy values in the under- constrained regions can trace their origin to the sites available for bonding with the water molecule.

Their high number of degrees of freedom allows for the facile reaction of water, i.e., the only energy cost is for the oxygen to dissociate from the network. The isostatic network within the intermediate phase is able to preserve its structural integrity because the energy barrier to deform the network is high, and there is no localized stress to create an additional driving force for reaction with water.

Indeed, the isostatic network is the most energetically favorable arrangement of a non-crystalline structure. This intermediate phase is governed by the fluctuations of the topology that arise with the minimization of stresses. These stresses change the local topology (within the range of the fluctuations) so that they become isostatic[68]. This leads to a flat region where there is a constant binding energy at the surface. Though it appears to be an intermediate phase it is important to note that this may instead be simply a manifestation of an isostatic threshold (a peak in binding energies rather than a plateau).

The average number of surface constraints has been shown in previous work to be largely controlled by the annealing time[172]. The surface constraints may then be used to predict surface reactivity. However, it is worth noting that the local number of constraints is not constant in time[17]. Furthermore, Potter et al.[168] showed that molecular water in the network can radically shift the free energy of the γ constraint. Therefore, the number of surface constraints is dynamic

125 during the influx of water. Shifting constraint energy will also alter the fragility, which in turn alters the diffusion activation enthalpy. In the future, this may provide a path for developing more durable glasses – currently it shows that a homogeneous intermediate phase (isostatic) surface should be targeted to achieve maximum chemical durability.

6.2 Elastic Modulus Prediction

Tailoring the elastic properties of glass is important in the design of new advanced compositions to control the stiffness and damage resistance of a variety of glass products[2], [191]–[193]. Elastic moduli are defined based on the ratio of the stress exerted upon a material to the resulting strain.

While various elastic moduli can be defined (e.g., Young’s modulus, shear modulus, bulk modulus, and Poisson’s ratio), for an isotropic material such as glass, only two of these quantities are mutually independent[1], [191], [194].

Previous attempts to model elastic moduli have focused on either computationally costly molecular dynamics simulations or empirical fitting methods. The work by Makishima and

Mackenzie[195] offered a possible solution for predicting elastic moduli based on the dissociation energy per unit volume of the glass; however, topological changes in the glass network are ignored, and the model cannot account for the temperature dependence of modulus. With MD simulations, elastic moduli can be obtained by applying a stress and measuring the resulting strain on the simulation cell, assuming that accurate interatomic potentials are available[2]. Machine learning has also been applied to model elastic moduli using experimentally measured composition-property databases, with a high predictive ability being achieved[2], [196]. Previous analytical modeling techniques related to the topology of the glass-forming network have shown good qualitative agreement with compositional trends in modulus, but have a lack of quantitative accuracy[64],

[197]–[199].

126 It is common in the glass community to assume that Young’s modulus may scale linearly with hardness. While this is clearly an oversimplification, let us begin with previously derived models for glass hardness as a potential starting point, since glass hardness has received considerable attention in the context of topological constraint theory [85], [192], [200]. Several models have been proposed to explain the origin of glass hardness, all of which have a linear form

dH Hnxn= v  ()' −  (101) v dn where Hv is the Vickers hardness, x is chemical composition, n’ is the critical level of constraints needed for the substance to provide mechanical resistance to the indenter in three dimensions, and n is a measure of the constraint rigidity. Recent work on glass hardness[192] has shown that n can most accurately be defined in terms of either the density of rigid constraints (or, alternatively, the density of rigid angular constraints), given by

nx()(xN) n()x = cA (102) Mx()

where nc is either the number of constraints per atom or number of angular constraints per atom, ρ is the density of the composition, M is the molar mass, and NA is Avogadro’s number. Since hardness and elastic modulus are often considered to be correlated, one might surmise that the elastic modulus might also be some function of angular constraint density or total constraint density. However, such an approach is not able to give quantitatively accurate predictions of modulus.

Here we propose an improved model of Young’s modulus based on the free energy density of the topological constraints,

(xN) A FFci=  iin()() x q T (103) Mx( ) i= constrasint

where qn(T) is the onset function for each associated constraint to give temperature dependence to

127 the predictions and is determined by the free energy ∆Fn. The calculated values of Δ퐹c are then used to calculate Young’s modulus (E) by

dE E =−Fcc()x F '. (104) dFc

In this new model, the Young’s modulus is controlled by the free energy parameters related to the strength of each constraint. Each parameter in this model captures a key physical phenomenon with each parameter corresponding to a physical property. Moreover, we demonstrate that a common set of parameters can simultaneously capture both the temperature and compositional dependence of modulus, as well as predict other properties such as the glass transition temperature and fragility.

To demonstrate the validity of this temperature-dependent constraint model, Young’s modulus data are collected from literature for lithium borate[195], sodium borate[195], and germanium selenide glass systems[201]. Additionally, the sodium phosphosilicate[202] system was experimentally determined following the procedure described by Zheng et al.[183] using room temperature resonant ultrasound spectroscopy on prisms of dimensions 10 mm × 8 mm × 6 mm.

Analytical topological constraint models have already been published for each of these systems[17], [57], [202].

The proposed models for glass elasticity (the models previously proposed for hardness based on constraint density and angular constraint density, as well as the new model for modulus based on the free energy density of constraints) were optimized in each case, leaving the intercept

푑퐸 푑퐸 and slope ( , ) as adjustable parameters. The constraint onset temperatures were also optimized 푑푛 푑푈푐 for the free energy density model. Figure 31 and Figure 32 show the results for each model in the phosphosilicate and sodium borate systems, respectively.

128

Figure 31. The Young’s modulus prediction and experimentally determined values for 0.3Na2O·0.7(ySiO2·(1-y)P2O5) glasses. The root mean square error (RMSE) values of the model predictions are: 6.41 GPa for constraint density, 3.13 GPa for free energy density, and 7.74 GPa for angular constraint density.

129

Figure 32 The Young’s modulus residuals for different prediction methods, sorted from minimum to maximum error. The free energy density model gives the most accurate results. The constraint density has a RMSE of 6.1 GPa, the angular density has a RMSE of 20 GPa, and the energy density has a RMSE of 5.9 GPa.

130 We have also evaluated the predictive ability of the model in terms of the temperature dependence of the Young’s modulus. sing the 10% Na2O·90% B2O3 data reported by Jaccani and

Huang[191], the model and experimental predictions of the temperature dependence of the Young’s modulus are plotted in Figure 33. Here, the constraint onset temperatures are determined from the room temperature modulus data. The number of escape attempts was optimized, since this controls the width of each transition and is thermal history dependent.

131

Figure 33. (A) Temperature dependence of the Young’s modulus from theory and experiment for 10% Na2O 90% B2O3. Using the previously fitted onset temperatures, the only free parameters are then the vibrational frequency and the heating time in which their product was fitted to be 14000. Where each dip in the modulus corresponds to a constraint no longer being rigid as heated through each onset. The onsets were fitted from the compositional dependence and only the width of the transition was fit which may account for the discrepancy around the inflection. The data was fit using a least-squares method, and the resultant fit is shown as the calculated method. The fit has an R2 of 0.94. (B) The contribution from each constraint to the overall modulus.

It is also possible to validate the model by comparing the free parameters fitted to the Young’s modulus data vs. that of previously reported data, as shown in Table 7. Results for two additional systems are plotted in Figure 34 to show the general validity of the free energy density model.

132 Table 7 Fitted values from this analysis compared to those reported in the literature. The disparity between the constraints evaluated with molecular dynamics most likely come from the speed in which the samples are quenched.

Value Fitted Value (K) Literature (K) Method, Citation Silicate  Onset 2212 1986 MD (Potter et al.[168])

Silicate  Onset 818 1600 MD (Potter et al.[168])

Silicate  Onset 450-500 810 MD (Potter et al.[168])

Borate  Onset 921 Not Reported Fitting Parameter (Mauro, Gupta, and Loucks[69]) Borate  Onset 715 740-760 Fitting Parameter (Mauro, Gupta, and Loucks[69])

Borate  Onset 393 328 Fitting Parameter (Mauro, Gupta, and Loucks[69])

133

(

A)

(

B)

Figure 34. The Young’s modulus prediction (using the same fitting method described in Fig. 3) and 2 experimentally determined values for (A) zGe·(100-z)Se with an R of 0.93 and (B) xLi2O·(100- 2 x)B2O3 glasses with an R of 0.986.

134 In the previous work of Zheng et al.[192] concerning hardness of glass, the authors write,

“It should be noted that each bond constraint corresponds to a certain energy since different kind of bond has different bonding energy… and thus the constraint density also represents an energy per unit volume. In other words, hardness is correlated to the energy per unit volume. Our findings for the borosilicate and phosphosilicate systems are further evidence in support of this argument, since both the total constraint density and angular constraint density approaches give better prediction of glass hardness compared to models based on number of atomic constraints” which makes it clear that the prediction of the hardness/modulus should be closely related to the onset temperatures (i.e., free energies) of the associated constraints. The model that was previously proposed could be extended to modulus if all constraints had equal amounts of potential energy, but due to the drastic change in strengths between the constraints, the prediction fails.

Bauchy et al.[85] showed also that hardness in the calcium-silicate-hydrate system is controlled by the angular constraints, which led to the development of angular constraint density as the governing control for hardness. When the analysis of the density of angular constraints is considered through the free energy view, it becomes apparent why this method works effectively for some systems. Models for predicting the hardness and elastic modulus of glass attempt to explicitly connect the rigid bond energy to the macroscopic properties of the system. The assumption for the hardness models is that the energies for breaking each type of constraint are all approximately equal, and hence only the number of the constraints matters. The argument can then be extended for elasticity: since elastic modulus is a bulk material property, with the approximation of all constraints being equal strength, the number of constraints per volume should be related. To correct for this approximation, when a weighted sum of free energies is used, the estimation becomes significantly more accurate. Using the density of the glass, a precise free energy per volume of the rigid constraints can be calculated. The knowledge that the energy of the bonds is tied to the elastic modulus has been widely known but had not been quantified nor placed within the context of topological constraint theory[203].

135 6.3 Ionic Conductivity

Ionic conductivity, 휎, is related to the number of charge carriers, n, and the mobility of the carriers, 휇, as they diffuse through a network by[1]

= Ze n , (105) where Ze is the charge of the conducting species. Since the charge-carrying ion is constant within a given glass family, the ionic conductivity depends only on the mobility and the concentration of the charge carriers present. However, it is not typically feasible to measure 휇 and n independently of each other, and thus it is unclear which variable scales in an Arrhenius fashion, thereby controlling the ionic conductivity of a glass. In the weak electrolyte model by Ravaine and

Souquet[204]–[207], the mobility of the charge carriers is considered constant while the concentration scales in an Arrhenius function. Conversely, the strong electrolyte model assumes a constant number of charge carriers, while the mobility follows an Arrhenius function[208], [209].

Recent MD simulations from Welch et al.[210] support the weak electrolyte interpretation for alkali silicate glasses, concluding that ionic conductivity is dictated by the concentration of charge carriers. Previous reports in the literature also supporting this view have proposed several hypotheses to explain this conclusion;[208], [209], [211]–[213] however, further exploration is needed.

The simplest form of the temperature dependence of free ions is given by

E a (106) nn=−0 exp . kT

Here, n0 is the total possible number of mobile ions, and Ea is the activation barrier for an ion to diffuse. It is often assumed that the activation enthalpy is related to the energy needed to deform the network temporarily to allow for ion motion[193], [209], [210], [212]; however, it is our hypothesis that there is a coupling between the ion hopping event and the relaxation of the local

136 network, allows for a permanent relaxation or deformation along ion channels. This means that there is some cooperative relaxation of the network along the diffusion path in conjunction with an ionic hopping event. These cooperative relaxations are discussed explicitly in the Adam-Gibbs formalism[53] for describing the relationship between viscosity and configurational entropy, Sc:

B =  exp  (107) TSc

where B is an activation barrier,  is the viscosity, and  is the viscosity of the liquid in the limit of infinite temperature. In the high-temperature (low-viscosity) liquid state, there is a well-defined relationship between the viscosity and diffusion (Stokes-Einstein relation). However, this relationship breaks down in the low-temperature (high-viscosity) glassy phase.

Herein, we present a new model for predictions for the compositional dependence of ionic conductivity. We can write the activation free energy barrier, Ea,c, for an ionic hopping event as,

ETSa, cccc=− . (108)

Here, is an enthalpic barrier for the ion to hop to a neighboring site, TC is the configurational

temperature (a value that describes the distribution of energies of the ions), and Sc is the entropy of the activation barriers. These quantities are associated with configurational changes in the glass, indicated by the subscript c. In the weak electrolyte model, all charge carriers have the same mobility and, hence, the entropic effects are dominant. This indicates a small value of , an assumption that will be validated by results later in this work, leading to a simplified approximation for the activation energy at temperatures below the glass transition but suffiently high temperatures such that

ETSa, c c c . (109)

In the case of a nonequilibrium glass at low temperature, the system becomes trapped in a localized

137 region of the energy landscape known as a “metabasin,” with slow inter-metabasin transitions[156],

[161]. The MAP model of the nonequilibrium viscosity of glass relies on this partial confinement within metabasins, where variation of the activation barrier for relaxation (H*) is related to the fragility of the system by[44]

dm 1 * = . (110) dH kTg ln10

Here, m is the fragility index (as proposed by Angell[58]) and Tg is the glass transition temperature.

Splitting the differential in Eq. (110) and integrating, we can write

* H k T= g ml n1 0 . (111)

This relationship was also proposed by Moynihan et al.[214] for the activation enthalpy of relaxation of a glass[214], [215]. Setting the Arrhenius form of the barrier described in Eq. (111) equal to the Adam-Gibbs equation (Eq. (107)), we write

HB*  expexp.=  (112) kTTS c

Solving Eq. (112) for the configurational entropy and inserting the result into Eq. (109) we obtain the simple relationship:

TBc Ea  2 . (113) kTmg ln10

Dividing the above expression with that of a reference state r and assuming that B and is Tc are constant with respect to compositional variation, we have

E Tm a = g, rr . (114) Ea, rg T m

This new equation is the first to relate the activation free energy for ionic hopping in a glass to the glass transition temperature and fragility of its corresponding supercooled liquid. It also

138 predicts a coupling between ionic diffusion and the cooperative rearrangements described by the

Adam-Gibbs relation. This possibility was first proposed by Ngai and Martin[216], who showed a correlation between the product of the Kohlrausch stretching exponent (  ) and the activation barrier for relaxation with the activation barrier for ionic conductivity. This implies that the diffusion in glasses is not governed by an elastic component, but instead controlled by the surrounding cooperative rearrangements, i.e., the  -relaxation of the glass[14], in good qualitative agreement with previous results shown by Potuzak et al.[217] Further evidence comes from the polymer community, where the decoupling of relaxation and conductivity is reported, with work showing that the decoupling of viscosity and diffusion varies with the fragility and the glass transition temperature[218]. This is a general relationship, which we demonstrate later to be valid for binary alkali borate, alkali silicates, and alkali phosphate glasses as well as validating the atomistic diffusion method using computational techniques.

In order to test this proposed relationship between relaxation and diffusion through simulation, a glass with the composition 10 Na2O-90 B2O3 (mol %) was synthesized in molecular dynamics with 1150 atoms using the potentials of Wang et al.[162] The system was quenched at a rate of 1 K/ps from randomly placed atoms at 2500 K in the NPT ensemble to room temperature, as shown in Figure 35.[210] To find the transition point energy of the diffusing ion, a nudged elastic band (NEB) calculation was performed. An alkali ion was chosen and its closest alkali neighbor was moved to another location so that the NEB calculation could be performed between those two sites. All calculations were carried out using the LAMMPS software package. Eleven reaction coordinates were used in the NEB calculation, and the results are shown in Figure 36.

139

Figure 35. The structure for the initial minimum energy configuration showing the boron (blue) network with interconnecting oxygens (red) and the interstitial sodium ions (yellow).

Figure 36. The energy barrier between the two sites. Oxygen is blue, boron is red, and sodium is ivory. The barrier is overestimated compared to experimental data; this could be from several sources of error such as potential fitting, thermal history fluctuations, or sampling too few transitions. The line is drawn as a guide to the eye.

140 Although the transition point energy is key for determining the activation barrier for diffusion, here we are also interested in the dynamics of the atoms around the mobile ion. Since we have hypothesized here that the ion diffusion is dependent on the ability for the local region to relax and deform instead of elastically straining, we expect a local deformation (relaxation) along the ion path. Figure 37 shows that along the ion diffusion tunnel there is indeed a permanent displacement of the atoms.

141

Figure 37. Snapshots of the NEB calculation. On the top is the total network as a function of reaction coordinates. The middle shows the local deformation around the ion of any atom that moves, in between inherent structure mandating a relaxation force. The color shows the degree of deformation.

142 In the final configuration of Figure 37, the permanent deformation required to lower the energy of the system is clearly seen. If it were purely an elastic response to dilate a pathway, the displacement between two inherent structures should be confined to changes with only the mobile ion; however, this is not observed. Instead, along the pathway of the ion movement, the network forming atoms also deform and change local positions to minimize the energy, implying a net relaxation around the ion “pathways.” To apply this and show the validity of Eq. (114), the sodium borate, lithium borate, lithium phosphate, and sodium silicate systems were considered, with the model results compared to experimental values in Figure 38. To confirm this model literature values for activation barriers have been used. The experimental activation barriers for the borates are taken from the work of Martin[213], while the fragilities and the glass transition temperatures are from

Nemilov[219]. The viscosity curve of the silicate system was fit with the Mauro-Yue-Ellison-

Gupta-Allan (MYEGA) equation of viscosity[55] using the data from Poole[220] and Knoche et al.[221] with the activation barriers from Martinsen[222]. Lithium phosphate activation data was taken from Ngai and Martin[216] with the glass transition and fragility taken from Hermansen et al. and predicted values from topological constraint theory[223].

143

Figure 38. Different network formers and the prediction of the activation barrier from our model compared with activation barriers from literature. (A) Sodium silicate predictions and experimental values[222], the error is calculated from the error in the fragility when fitting the data. (B) Lithium phosphate activation energy[216] predicted with topological constraint theory and compared with the experimental values. (C) Predictions over two different systems of alkali borates[213], sodium and lithium, with a reported R2 of 0.97.

144 It is worth noting that this model is predicting a non-equilibrium kinetic property using equilibrium viscosity parameters, a connection that has been demonstrated in the past for viscosity and relaxation models[61]. In this work, we are applying the same principles to the diffusion of ions in a network. The diffusion and viscosity, in the case of equilibrium, are known to be intimately linked using the Stokes-Einstein expression (which breaks down for highly cooperative rearrangements near and below the glass transition temperature[224]). Here, we are no longer connecting the two properties explicitly, but instead connecting the activation barrier for diffusive hopping to the two key parameters governing the viscous flow of the supercooled liquid state.

The failure of the Stokes-Einstein expression near the glass transition is a well-known problem and has been studied extensively for purposes of predicting crystallization rates[167]. This occurs because there is a breakdown of the ergodicity of a system around the glass transition. In the MAP model,[44] the viscosity is controlled by the ergodic parameter, x, and is given by

p min(,)TTf x = . (115) max(,)TTf

Here, the exponent p is related to the sharpness of the ergodic to non-ergodic transition. Using the ergodic parameter, the overall viscosity ( ) is expressed as a linear combination of equilibrium

viscosity (eq ) and non-equilibrium viscosity (ne ) such that

log10(,TTf )=x log 10  eq ( T , T f ) + (1 − x )log 10  ne ( T , T f ). (116)

By combining the Stokes-Einstein expression and the model proposed in this work, we can define the diffusion of an ion through the network as

 kbaTE −  log10DTTD ( ,f )l= x og 10+−( 1x) log 10 0 exp  . (117) 6)a(, T Tf kT

Here, a is an empirical constant related to the size of the diffusing species, k is Boltzmann’s

145 constant, Ea is the activation barrier calculated with Eq. (114), and D0 is the limit of diffusion as

the temperature approaches infinity. D0 should be chosen such that at TT= f the diffusion is continuous. This can be justified because the preexponential factor is related to the entropy of the system, and the fictive temperature is the temperature at which the configurational entropy of the transition barriers becomes nearly constant. Only one parameter is left unknown, a, since the viscosity and fictive temperature can be calculated with an arbitrary thermal history using a relaxation modeling tool such as RelaxPy[42]. Implicit in RelaxPy are the assumptions pertaining to the viscosity made by Guo et al.[61]. A similar model was first proposed by Cassar et al.[167] for modeling diffusion during crystallization, although their model used an empirical hyperbolic tangent function to approximate the ergodic factor. In contrast, the current model can be considered explicitly as a function of thermal history and composition.

6.4 Machine Learning Expansion

Though the TCT models presented here for different properties are immensely useful, they do have limitations; the greatest of which is that intimate knowledge of the glassy structure is required to parameterize these models. This is due to the fact that the landscape is fundamentally informed by the underlying structure of the network and as such to predict the dynamics of the landscape information on the structure must be incorporated in some form. However, there is an alternate method in which the physical origin is not considered in the development of the model: machine learning. Machine learning comes in many forms and has become crucial to the study of materials in the modern era. There is a plethora of machine learning techniques however, in this paper we are focused on four key methods: linear fitting (LF) methods, random forest (RF), symbolic regression (SR), and neural networks (NN)[3], [80], [225], [226]. These methods are

146 explored extensively in literature and the exact underlying mathematics is beyond the scope of this work. In this work these methods have been used to create models for: number of constraints (LF), glass stability (RF), fragility (NN), melting point (NN), the infinite limit of viscosity (SR), Young’s modulus (NN), and coefficient of thermal expansion (NN).

To enable a linear fitting method for the number of constraints a simple, steepest descent algorithm was used to find the number of constraints due to each component. There is no direct experimental way to measure the number of constraints so instead a technique leveraging some other property must be used to approximate the number of constraints. In this case, hardness is related to the number of constraints through,

HAnvc=− 2.5 . (118)

In which A is an empirical scaling parameter that is dependent on load, indenter geometry, and

glass family. In this linear approach to determining constraints nc can be given by,

components (119) nxnckk=  c, k=0

Combining this expression with a hardness database grouped by composition (in this case Sciglass) allows for an approximate number of constraints of the system but due to A being a variable there must be a fixed constraint in the system and every set of data is given a unqiue value of A. The fixed constraint in this system is that silicon atoms are four coordinated. The value of A is chosen such that the error between the prediction and the hardness is minimized. The total process is as follows:

Choose a set of nck,

1. Calculate number of constraints for all compositions 2. Iterate over every dataset in Sciglass with > 1 datapoints and scale by a factor of A 3. Calculate total error for steepest descent algorithm 4. Descent error slope 5. Repeat until error has been minimized

147

This then leads to an estimate for the number of constraints for every atomic species which can then be used to estimate important properties such as chemical durability, hardness, and qualitatively can predict glass stability.

Two other techniques that are used are RF and SR. When implemented a grid search for the hyperparameters were used with a testing set used to determine the efficiency of each set of hyperparameters. Additionally, each model was trained repeatedly and the lowest error was taken.

The RF method was used to predict glass stability while undergoing a novel processing technique called alkali-proton substitution (APS). The SR method was used to understand the compositional dependence of the infinite temperature viscosity limit used in 3-parameter viscosity models. To do this the Sciglass database was taken (as implemented in GlassPy[227]), and sets of data that included more than eight datapoints were fit to the MYEGA equation. If the RMSE error of the fit was less than 0.01 Pa s, the fragility was greater than 17, and the value of the log infinite limit was

−2.935 it was accepted into the final database. Each of these criteria were added due to the well- known fact that this database is full of error and imprecise measurements. The symbolic regression was then performed through the grid search and the equation that performed best and reappeared is given as,

T log  =+ABg (120) 10  m

With A and B being parameters that varied depending on the hyper-parameters, the infinite limit threshold, and the accuracy threshold used when creating the database. We are currently in the process of refining values for A and B is ongoing.

This method allows for an easy prediction of the infinite limit so if the glass transition and fragility are predicted through some other means (such as ML or TCT) a better prediction of the third parameter is known without just accepting -2.93 as the mean. Plots of the infinite limit vs the

148 fragility, glass transition, and Tg/m showing the clear dependance of the infinite limit on the term

Tg/m are shown in Figure 39.

149

Figure 39 The dependence of infinite viscosity limit for other viscous parameters. (Top left) shows the distribution of the infinite temperature limit in the database after limits exerted on the system. (Top right) The distribution of the infinite viscosity limit vs the glass transition. (Bottom left) The relationship between fragility and infinite temperature limit of viscosity. (Bottom right) The infinite temperature limit of viscosity vs. the key metric predicted by SR.

150 NN are perhaps the most widely discussed ML tool applied in literature. This makes sense given they are universal function approximators and are quite easy to train given sufficient data.

The most difficult part of creating a NN is adjusting the hyperparameters so that good accuracy can be found without causing overfitting. To avoid overfitting the data is divided into a testing set then a training set (just as for RF and SR) but instead of doing a grid search for the hyperparameters we used gradient boosted regression trees to find the optimal set of parameters. This consists of creating an initial tree then evaluating it based on a small set of data and finding the minimum point, then running a new set with perturbations around that point to create another tree. This is repeated until the value converges to a small error. In this work the value being determined is the lowest error of 100 sequentially trained neural nets with a set of hyperparameters. This is particularly useful because there is no concern for the type of parameters (string, int, floats) that determine the value at the end, so one can do a mixed parameter optimization that includes every conceivable variable. The values being optimized are given by,

• Learning rate [10-5, 10-1] (float on logarithmic scale) • Decay rate for Adam Optimization [10-6, 10-2] (float on logarithmic scale) • Hidden layer activation function ‘relu’, ‘selu’, ‘linear’, ‘tanh’, ‘sigmoid’ (string) • Output layer activation function ‘relu’, ‘selu’, ‘linear’, ‘tanh’, ‘sigmoid’ (string) • The number of nodes in each layer [8,1024] (int) • Batch size [1,256] (int) • Patience [5,250] (int)

In Table 8 we have listed the hyper-parameters that were used for the fragility, melting point,

Young’s modulus, and co-efficient of thermal expansion (CTE).

151 Table 8. Hyperparameters for different neural networks after hyper-optimizations.

Model Fragility Melting Young’s CTE Temperature Modulus

Learning Rate 0.035 0.018 0.027 0.0091 Decay Rate 0.00293 0.00095 0.00161 0.002 Hidden Layer RELU RELU RELU RELU Output Layer Linear Linear SELU RELU Number of Hidden 2 1 1 1 Layers Number of Nodes 235 506 514 470 Batch Size 4 118 211 125 Patience 25 250 23 22 RMSE 5 [-] 80.5 [K] 6.8 [GPa] 9.6 x 10-7 [1/ K]

152 Chapter 7

Designing Green Glasses for the 21st Century

Thus far the work presented in this dissertation has enabled a new set of design tools for everything from crystallization, glass relaxation, and optimizing over the ideal compositional space we can use them for designing new glasses. Building on these models and insights new applications of glass are enabled specifically three glasses: an ion conducting glass for batteries, a proton conducting glass for hydrogen fuel cells, and a commodity glass for consumers. All three are currently undergoing testing but experimental results have been reported back as of yet. Overall, both the hydrogen and ion conducting glasses have been designed thorough a combination of ML and underlying physical models and as such hare reported herein.

7.1 Glass Electrolytes

An emerging high-interest application of glass is solid state batteries. Glasses offer a possible solution to the growing energy crisis, however to realize this new paradigm new research methods are needed. The main barrier for any new batteries to be commercialized is a high conductivity at room temperature (> 10-3 S cm), high stability, ease of processing, and ultimately an alternative must be cheaper than current liquid state Li-ion batteries; glass is an attractive candidate due to the innate stability, a simple processing technique, the infinite variability, and the relatively low cost of production. In addition, the structure of a glass is a liquid-like structure, which may encourage ion migration like that seen in liquids. Despite all of these advantages only a few compositions have realized the requirements but further research is needed to find optimal compositions.

153 To optimize the composition for activation barriers we must understand the relationship between conductivity and structure. Structural effects can be propagated in two different methods based on the Arrhrenius expression for conductivity,

E a . (121) = 0 exp − kT

The two methods are through  0 (the infinite temperature conductivity) and the activation barrier

( Ea ). The pre-exponential factor is approximately a constant for simliar compositions while the activation barrier varies dramatically over small compositional spaces making it the larger of the two concerns. Many models have been presented to predict the compositional dependence of the activation barrier: the Anderson-Stuart (AS) model, the Christensen-Martin-Anderson-Stuart

(CMAS) model, the weak electrolyte (WE) model, Kohlrausch exponent model (KEM) from Ngai et al.[216] and there is the model proposed in this dissertation which we will call the Wilkinson viscous cooperative conductivity model (WVCC)[8], [88], [204], [207]–[209].

To optimize, machine learning (ML) is a powerful technique that is implemented. The best method to train the ML is to use the direct relationship between the compositions and the activation barrier, but this is not currently feasible due to a lack of central depository or database for this information. To make the prediction of the activation barrier we need use some model that is reliant on commonly accesible parameters. This rules out all models for this type of analysis except since

WVCC since the other models require some fitting parameters not accessible. Each models required parameters, fitted values (please note that the models typically include approximations for these values), and caveats are shown in Table 9.

154 Table 9. A table with some ionic conductivity models and the parameters needed for them as well as the disadvantages for each. These are not the only models but are representative of those commonly used in literature.

Model Required Parameters Fit Parameters Caveats AS Shear Modulus ‘Madelung’ Constant The Madelung Charge of Anion Covalency Parameter Constant is not smooth Charge of Carrier Doorway Radius1 as a function of Radii of Anion composition and the Radii of Carrier fitting parameters scale the activation energy non-proportionally.

CMAS Shear Modulus ‘Madelung’ Constant The same issues as AS Charge of Anion Doorway Radius1 but in addition the Charge of Carrier Jump Distance of Ion1 dielectric permittivity Radii of Anion lacks a database as Radii of Carrier well. Dielectric Permittivity

WE Charge Concentration Equilibrium Coefficient1 The equilibrium coefficient drastically changes the prediction non-linearly.

KEM Stretching Exponent Proportionality There is not enough Coefficient data available to know the compositional dependence of the stretching exponent.

WVCC Glass Transition Proportionality Due to the Fragility Index Coefficient proportionality constant the activation barrier can only be known in a local composition range.

1 There are ways to approximate this value based on additional data.

155 The WVCC model predicts that the activation barrier for ionic conductivity is given by,

A Ea = , (122) mTg in which A is a proportionality constant. This means that to predict the glass behavior we don’t need to know things about the glass but instead merely about the liquid state where the viscosity is readily available. Leveraging this technique, if the viscosity information as a function of compositions is available, then the local glass with the lowest activation barrier can be found by

minimizing 1 Tmg . To access this information multiple options are available such as topological constraint theory but, in this work, we will leverage ML since it is easily applied to a large compositional space.

To get the viscosity we will use neural networks (NN). A well trained NN for the glass

transition is readily available and as such will be used for our prediction of Tg . In the previous section there was presented a trained fragility NN which enables a complete optimization of the local composition based on WVCC. To generate a database of fragility values that the NN was trained on SciGlass viscosity values were fit with MYEGA model. The data had to fit following criteria:

1. The total number of data points for fitting had to be greater than 5 from the same literature source 2. The root-mean-square-error of the fit had to be less than 0.01 Pa s 3. The infinite temperature value of viscosity had to be within 3 orders of magnitude of the accepted logarithmic value (-2.93 log(Pa s)).

An additional neural network recently created by Cassar[228] was also used for an independent comparison. Cassar’s NN was based on a hybrid physical ML work and may offer better insights when extrapolating far from experimental datapoints. In this study to prove this concept we will only focus on P2O5, B2O3, Al2O3, and Li2O.

156 This gives an entire method to predict glass battery candidates. This technique will find all of the local minimum but due to the fact that there is a proportionality constant it is unknown which local minimum is the absolute minimum. This means that the composition with the lowest activation barrier in the space is identified but another technique is needed to investigate which one of the identified compositions is best. The method follows:

1. Randomize an initial glass candidate with the content of being normalized 2. Run the glass candidate through the NN, to find values of fragility and the glass transition 3. Perturb the composition to find the gradient 4. Step down the gradient 5. If the new gradient is 0, this is a local minimum and a possible glass candidate; return to 1. Otherwise, return to step 3.

This method cannot find global optima because there is no guarantee that the proportionality constant will stay constant across wide compositional spaces. To narrow it down an additional technique to compare the predicted values is needed (such as another activation barrier model, molecular dynamics simulations, a regression algorithm, or simply melt the proposed compositions). To further pair down in this work, a k-means algorithm was used along with the assumption that the proportionality constant was slowly changing over compositional space. This allowed us to take the best glass in each grouping predicted by k-means. The glasses we have predicted (using both the separate predictor NN and the method by Cassar et al.[228]) are listed

below in Table 10 while an example prediction for 1000 mTg (which should scale with the

activation barrier) in the Na2O – B2O3 – SiO2 system is shown in Figure 40. The glasses are currently undergoing melting at Coe College and Iowa State. It is worth noting that composition D is a well- known composition that is known to exhibit criteria close to the criteria for conductivity. If this method shows that these compositions are better than other compositions in the family found in literature then it can be expanded to include a wider range of component and hopefully finding a universal candidate for glass batteries.

157 Table 10. The predicted compositions based on the optimization scheme proposed.

Compositions Li2O Al2O3 P2O5 B2O3

A- This work NN 55 31 0 14

B- This work NN 44 1 5 50

C- Cassar 60 26 2 13

D- Cassar 50 0 0 50

Figure 40 The prediction of the scaling of the activation barriers for a common sodium borosilicate system.

7.2 Hydrogen Fuel Cell Glasses

An emerging, high-interest application of glass that is proton-conducting intermediate temperature fuel cells[7], [229]. A fuel cell, in short, is a device that turns chemical fuel into electricity without the use of combustion. It is often described as a chemical battery, due to the similarity of a fuel construction with batteries. One of the crucial components to a working intermediate range fuel cell is an electrolyte with a high proton migration. The electrolyte must be stable over a wide range of temperatures (up to 673 K) and a proton conductivity greater than 10-2

S cm-1 at the operating temperature[7]. Previously phosphate glasses near the metaphosphate compositions were used since they carried the appropriate number of residual protons for conductivity studies. However, this restricted the phase space to only n-2 dimensions drastically reducing the degrees of freedom of the problem and artificially restricting possible solutions. In

2013[230], to access the full phase space, a new method was developed where a sodium phosphate glass could have all the sodium ions replaced with protons. This allows high concentrations of protons to be achieved with a variety of starting compositions. A previous report[230] has summarized successful samples, these data are used as a training set for this study, with additional data on compositions that have failed taken from internal theses.

To choose the candidate material for each model, a cost function is defined. The cost function is an analytical function whose inputs are the fraction of each oxide component and the output is a value that rates the composition. The cost function consists as of many properties that are of interest.

In the application of oxide electrolytes for an intermediate range fuel cell, the two main properties to worry about are the ionic conductivity at high temperatures (we will choose 473 K due to this being the low end in which these materials are considered) and the relative stability of each phase.

159 This is especially important since the material will undergo the APS (alkali-proton substitution), in which the sodium will be replaced with protons in the bulk of the material. The cost function used in this work is given by,

()logx =− TK=473 −1000S . (123)

In which S is either a 0 or 1 (representing a crystal or a glass, respectively) and  is the ionic conductivity. The function is arbitrary and could be expanded to include as many terms are as needed in the goal of this glass. The stability is heavily weighted because it is more important than the conductivity and any sample that is not stable should not be considered. In order to predict each of these samples a hybrid physical/empirical approach is used. For predicting the ionic conductivity of a proton-swapped glass we began with an observed relationship between the glass transition temperature and the conductivity at 473 K. This empirical relationship is shown in Figure

41. The relationship, although empirical, can provide some insights into the physics of proton transferred glasses (no universality is claimed in this work). The activation barrier for these glasses has widely remained unchanged as noted in a previous work, due to the fact that the activation barrier is simply to disassociate the proton with the non-bridging oxygen (NBO) and once the proton is free interacts rarely with the network until it is rebounded to said network. Thus, the pre- factor can be understood as being related to the degeneracy of proton conduction pathways. When the network has lower configurational entropy there are fewer pathways for the proton to travel; in this case it is known that the configurational entropy is then the dominating effect for the glass transition[53].

The relationship between the glass transition temperature and the configurational entropy can be expanded through the Stokes-Einstein relationship evaluated at the glass transition,

kT  =nZeD = nZe g 7.3  10−22 nT . (124) T= Tg 6a 1012 g

160 In which a is approximately the size of a proton and n is the number of charge carriers (which is approximately a constant according to the weak electrolyte theory). This determines the intercept of the glassy form proton conductivity and then the activation barrier (slope) is found to be approximately the same for all proton conducting glasses meaning that the intercept is the dominant effect on the behavior of the glass and the only variable controlling the intercept is the glass transition.

Leveraging the configurational entropy’s relationship with the glass transition along with topological constraint theory we can write an expression for the glass transition, and as such gain predictive power for proton conductivity. To predict the glass transition temperature (Tg) we start with the Adam Gibbs model of viscosity[53]. Topological constraint theory then states that the degrees of freedom of the network is proportional to the configurational entropy[23], [56], [57].

Using the liquid state definition of the glass transition ( = 1012 Pa s) and the well-known infinite temperature of viscosity we can then write,

B 1 T = . (125) g 14.93lnkf 

Where f is the degrees of freedom. Letting the constants be equal to A and rewriting the expression in terms of constraints we arrive at a predictive formula for the glass transition[109],

A Tg = . (126) 3−  mnx c, x x

Where nc is the number of constraints provided by each component at the glass transition associated with each component, x, and their molar fraction, mx. The value of the constraints was then linearly parameterized to the glass transition data from literature[7], [230], [231]. This is not the most explicit approach to counting constraints; however, it is the most convenient when considering large phase spaces being optimized over. This glass transition temperature is then converted to conductivity using the empirical translation shown in Figure 41.

161 The other term in our cost function is that of stability. The question of what forms glass is a notorious question that goes back to the first serious days of research into materials. It has been shown that no one metric is a good predictor and no metric is universal. However, one must be used to rule out bad compositions from the start. To do this a Random Forest method is implemented.

Though this may fail in edge cases it will at least help identify the right area to explore. A comprehensive review of random forest methods and machine learning methods for prediction of glass properties can be found elsewhere[3], [4], [225]. The amount of information previously obtained about what forms glass in the compositional family we will be working on is not enough to use random forest methods on just the fraction of each phase. In order to circumvent this issue, we chose four physical parameters that would be used as features for the predictions.

• Total mol% oxide modifier: This was chosen because it is well known that modifiers break up the network and most glass forming theories are related to the network percentages. We considered the oxide versions of H, Na, Ba, Sr, and Ca as the modifiers. • Total mol% network former: Similarly, it is well known the network influences glass forming. We considered B, Ge, and P as the network forming cations. • Mean cation charge: Though there has been previous work on this topic it is rarely considered as a ‘normal’ predictor of glass forming capability. It is included here because these glasses are mainly invert and as such a higher field strength will increase the cohesive nature of the network. • Entropy of mixing: This is an important parameter to consider with so many components it may be stabilized by the entropy of mixing. The entropy of mixing (Sm) was calculated using the Gibbs entropy.

The Gibbs entropy is given by,

Smx=− k mln mx . (127) x

The random forest model consisted of 100 trees with a maximum depth of 3. The results of both the random forest model, the topological predictions, the optimization, and the resultant glass are shown in the results section. The optimization was limited to the range of each component that has been explored experimentally as well as an additional constraint of at least 5 mol% rare earth oxide and at least 5 mol% of boron oxide or germania, this is justified elsewhere[231].

162

Figure 41 The relationship between the glass transition and the proton conductivity. This is justified two ways one through the relationship of the entropy of diffusion and glass formation (the Adam- Gibbs model) and through the fact that water is known to depress the glass transition.

163 To explain the results, we will first start with the models listed to confirm their validity.

The first model is the prediction of the glass transition temperature. Figure 42 shows the accuracy of the glass transition prediction as well as the relative contribution of each component to the glass transition. In this figure any component that has a number of constraints greater then PO5/2 will increase the glass transition and as such decrease the conductivity. Thus, it is preferable to construct a glass with only components whose constraints at Tg are less than PO5/2, as long as the glass remains stable. Interestingly whether the data includes only the glass transitions of protonated glass or all glasses determines the role of La. Barium and sodium both decrease the glass transition playing the role of the more traditional modifier, however they do provide some rigidity to the network. Protons have a net negative effect, which we believe to be a result of the APS process where they are introduced as the sodium is leaving so this negative effect corresponds to a breaking of the rigidity that exists. It is also interesting to note that adding a boron phase to the sample increases the glass transition, though not as much as some of smaller alkaline earth samples, assumingly due to their increased field strength. The worst network former appears to be GeO2.

To calibrate the accuracy of the random forest model, a confusion matrix is shown. It shows that the RF always predict that a glass will be a glass; however there is some error when it forms a crystal with 23% of the time being misidentified. Though random forest methods are empirical we can derive some physical meaning by looking at the relative importance of each feature. From the random forest study of stability, we are able to rank the relative importance of each feature used.

The mean cation charge, mixing entropy, and percent network formers are all of considerable importance while the percent modifiers is less so.

Once these models were developed and checked the reliability of the cost function is confirmed. After which, the minimization of the cost function was performed and one glass was chosen as optimal. Boron was found to be preferential in the glass but was limited since the compositional space was bound by previous samples (an additional sample is undergoing

164 characterization where the boron content was slightly increased). The optimized sample when synthesized had some visible nucleation so the optimization while redone with limiting the component maxima to just the limits of those where compositions in which APS was successfully performed. This resulted in the OP* composition. The three samples currently are undergoing experimental characterization are listed in Table 11.

165 Table 11. The compositions synthesized in this work. These compositions were predicted by minimizing the cost function described in Eq. (123). OP* is the variant that was melted after OP partially crystallized. B-OP appeared to have surface nucleation in some spots but was cut and removed before APS treatment.

Name PO5/2 NaO1/2 BaO LaO3/2 GeO2 BO3/2

OP 44 36 10 5 3 2

OP* 49 36 5 5 3 2

B-VOP 44 36 10 5 1 4

166

Figure 42 (A) The glass transition prediction vs the experimental values showing a good correlation. (B) The confusion matrix of the random forest method used to determine the glass forming region. Over top the constraints at the glass transition provided by each oxide species is listed. Since the objective is to decrease Tg while staying in the glass forming region, we will attempt to minimize use of elements that increase the glass transition temperature (nc > 1.7).

167 Chapter 8

Conclusions

The rapid rate of new information in the current scientific climate and the infinite variability of glass both stand as both unique challenges and opportunities. In this dissertation, we have divided the influence of energy landscapes on the effects of properties into two independent spaces: the compositional and the thermal history dimensions. By building models that succinctly and accurately describe the dynamics of different hyper-coordinate (composition, thermal history, crystallinity) changes, the feasibility of designing new glasses for the challenges of the 21st century is obtainable.

Before designing new models, the current state-of-the-art models need to be implemented and understood. To reach this end, two softwares, based on previous work, were created: RelaxPy and ExplorerPy. RelaxPy is an implementation of the MAP model and though the MAP model is powerful when predicting the dynamics of the glass, RelaxPy ultimately showed that fictive temperature is insufficient at capturing the underlying physics and is intensive to parameterize.

ExplorerPy was created to standardize the approach to mapping energy landscapes. Energy landscapes served as the key method of understanding the deeper dynamics of glasses, glass- ceramics, and liquids.

The thermal history dependence of glass with respect to both relaxation and crystallization has been incorporated into the existence of new models called “toy landscapes”. The toy landscapes have built upon the physics of previous models such as the MAP model; however, due to the lower parametrization cost and the built-in increased physicality, toy landscapes pose as a tool to increase our understanding and speed-up the rate of new glass discoveries. This tool can deal with the complexity of the higher dimensional spaces due to crystallization and relaxation without assumptions concerning fictive temperature.

168 The compositional dimensions are the remaining dimensions that must be optimized when designing a glass or glass-ceramic. Building on previous work, models have been developed that enable predictions of Young’s modulus, surface reactivity, and ionic conductivity. However, there are additional properties that are needed and for those, we have proposed novel machine learning approaches. This dissertation has not enabled the design of glass for every application. However, it lays the groundwork and approaches to design glasses for society’s growing needs. Presented as well, are the methods used to design glasses that could satisfy the requirements for solid state glass electrolytes and hydrogen fuel cells. All of these techniques together promise to be a powerful new framework to build the glasses of the future.

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VITA

Collin Wilkinson was born in Mt. Carroll, IL. He attended Coe College as an undergraduate and studied glass resulting in a physics bachelor’s degree in 201 . He worked with Dr. Ugur Akgun and Dr. Steve Feller on modeling different key properties of glass properties/responses to external stimuli. Collin joined the group of Dr. John Mauro at Pennsylvania State University in 2018.

List of publications written by first author (or co-first author) Collin while at Penn State:

1. CJ Wilkinson, JC Mauro. Explorer.py: Mapping the Energy Landscapes of Complex Materials. SoftwareX; Submitted. 2. CJ Wilkinson, DR Cassar, AV DeCeanne, KA Kirchner, ME McKenzie, ED Zanotto, and JC Mauro. Energy Landscape Modeling of Crystal Nucleation. NPJ Computational Materialis; Submitted. 3. CJ Wilkinson, K Doss, O Gulbiten, DC Allan, JC Mauro. Fragility and Temperature Dependence of Stretched Exponential Relaxation in Glass-Forming Systems. JACERS; Submitted. 4. CJ Wilkinson, JC Mauro. Comment on “The Fragility of Alkali Silicate Glass Melts: Part of a niversal Topological Patter” by DL Sidebottom. JNCS (2020); ( 2 ) 11 5. CJ Wilkinson, K Doss, DR Cassar, RS Welch, CB Bragatto, JC Mauro. Predicting Ionic Diffusion in Glass From its Relaxation Behavior. Journal of Physical Chemistry B (2020); 124 (6) 1099-1103 6. Z Ding, CJ Wilkinson, J Zheng, Y Lin, H Liu, J Shen, SH Kim, Y Yue, J Rent, CJ Mauro, Q Zheng. Topological Understanding of the Mixed Alkaline Earth Effect in Glass. JNCS (2020); (526) 119696 7. CJ Wilkinson, AR Potter, RS Welch, CB Bragatto, Q Zheng, M Bauchy, M Affatigato, SA Feller, JC Mauro. Topological Origins of Mixed Alkali Effect in Glass. Journal of Physical Chemistry B (2019); 123 (34) 7482. 8. CJ Wilkinson, K Doss, SH Hahn, Nathan Keilbart, AR Potter, NJ Smith, I Dabo, ACT Van Duin, SH Kim, JC Mauro. Topological Control of Water Reactivity on Glass Surfaces: Evidence of a Chemically Stable Intermediate Phase. JPCL (2019); 10 (14) 3955. 9. CJ Wilkinson, K Doss, G Palmer, JC Mauro. The Relativistic Glass Transition: A Through Experiment. JNCSX (2019); 100018. 10. CJ Wilkinson, Q Zheng, L Huang, JC Mauro. Topological Constraint Model for the Elasticity of Glass-Forming Systems. JNCSX (2019); 100019. 11. AR Potter, CJ Wilkinson, SH Kim, JC Mauro. Effect of Water on Toplogical Constraints in Silica Glass. Scripta Materialia (2019); (160) 48-52. 12. CJ Wilkinson, E Pakhomenko, MR Jesuit, AV DeCeanne, B Hauke, M Packard, SA Feller, JC Mauro. Topological Constraint Model of Alkali Tellurite Glasses. JNCS (2018); (502) 172. 13. CJ Wilkinson, YZ Mauro, JC Mauro. RelaxPy: Python code for Modeling of Glass Relaxation Behavior. SoftwareX (2018); (7) 255.